diff --git a/c-ext/pythoncapi_compat.h b/c-ext/pythoncapi_compat.h
index 7f028529..450d7ed6 100644
--- a/c-ext/pythoncapi_compat.h
+++ b/c-ext/pythoncapi_compat.h
@@ -19,58 +19,87 @@ extern "C" {
 #endif
 
 #include <Python.h>
-#include "frameobject.h" // PyFrameObject, PyFrame_GetBack()
+#include "frameobject.h"          // PyFrameObject, PyFrame_GetBack()
+
+
+// Compatibility with Visual Studio 2013 and older which don't support
+// the inline keyword in C (only in C++): use __inline instead.
+#if (defined(_MSC_VER) && _MSC_VER < 1900 \
+     && !defined(__cplusplus) && !defined(inline))
+#  define inline __inline
+#  define PYTHONCAPI_COMPAT_MSC_INLINE
+   // These two macros are undefined at the end of this file
+#endif
+
 
 // Cast argument to PyObject* type.
 #ifndef _PyObject_CAST
-#define _PyObject_CAST(op) ((PyObject *)(op))
+#  define _PyObject_CAST(op) ((PyObject*)(op))
+#endif
+#ifndef _PyObject_CAST_CONST
+#  define _PyObject_CAST_CONST(op) ((const PyObject*)(op))
 #endif
 
+
 // bpo-42262 added Py_NewRef() to Python 3.10.0a3
-#if PY_VERSION_HEX < 0x030a00A3 && !defined(Py_NewRef)
-static inline PyObject *_Py_NewRef(PyObject *obj) {
+#if PY_VERSION_HEX < 0x030A00A3 && !defined(Py_NewRef)
+static inline PyObject* _Py_NewRef(PyObject *obj)
+{
     Py_INCREF(obj);
     return obj;
 }
 #define Py_NewRef(obj) _Py_NewRef(_PyObject_CAST(obj))
 #endif
 
+
 // bpo-42262 added Py_XNewRef() to Python 3.10.0a3
-#if PY_VERSION_HEX < 0x030a00A3 && !defined(Py_XNewRef)
-static inline PyObject *_Py_XNewRef(PyObject *obj) {
+#if PY_VERSION_HEX < 0x030A00A3 && !defined(Py_XNewRef)
+static inline PyObject* _Py_XNewRef(PyObject *obj)
+{
     Py_XINCREF(obj);
     return obj;
 }
 #define Py_XNewRef(obj) _Py_XNewRef(_PyObject_CAST(obj))
 #endif
 
+
 // bpo-39573 added Py_SET_REFCNT() to Python 3.9.0a4
 #if PY_VERSION_HEX < 0x030900A4 && !defined(Py_SET_REFCNT)
-static inline void _Py_SET_REFCNT(PyObject *ob, Py_ssize_t refcnt) {
+static inline void _Py_SET_REFCNT(PyObject *ob, Py_ssize_t refcnt)
+{
     ob->ob_refcnt = refcnt;
 }
-#define Py_SET_REFCNT(ob, refcnt) _Py_SET_REFCNT((PyObject *)(ob), refcnt)
+#define Py_SET_REFCNT(ob, refcnt) _Py_SET_REFCNT(_PyObject_CAST(ob), refcnt)
 #endif
 
+
 // bpo-39573 added Py_SET_TYPE() to Python 3.9.0a4
 #if PY_VERSION_HEX < 0x030900A4 && !defined(Py_SET_TYPE)
-static inline void _Py_SET_TYPE(PyObject *ob, PyTypeObject *type) {
+static inline void
+_Py_SET_TYPE(PyObject *ob, PyTypeObject *type)
+{
     ob->ob_type = type;
 }
-#define Py_SET_TYPE(ob, type) _Py_SET_TYPE((PyObject *)(ob), type)
+#define Py_SET_TYPE(ob, type) _Py_SET_TYPE(_PyObject_CAST(ob), type)
 #endif
 
+
 // bpo-39573 added Py_SET_SIZE() to Python 3.9.0a4
 #if PY_VERSION_HEX < 0x030900A4 && !defined(Py_SET_SIZE)
-static inline void _Py_SET_SIZE(PyVarObject *ob, Py_ssize_t size) {
+static inline void
+_Py_SET_SIZE(PyVarObject *ob, Py_ssize_t size)
+{
     ob->ob_size = size;
 }
-#define Py_SET_SIZE(ob, size) _Py_SET_SIZE((PyVarObject *)(ob), size)
+#define Py_SET_SIZE(ob, size) _Py_SET_SIZE((PyVarObject*)(ob), size)
 #endif
 
+
 // bpo-40421 added PyFrame_GetCode() to Python 3.9.0b1
 #if PY_VERSION_HEX < 0x030900B1
-static inline PyCodeObject *PyFrame_GetCode(PyFrameObject *frame) {
+static inline PyCodeObject*
+PyFrame_GetCode(PyFrameObject *frame)
+{
     PyCodeObject *code;
     assert(frame != NULL);
     code = frame->f_code;
@@ -80,15 +109,20 @@ static inline PyCodeObject *PyFrame_GetCode(PyFrameObject *frame) {
 }
 #endif
 
-static inline PyCodeObject *_PyFrame_GetCodeBorrow(PyFrameObject *frame) {
+static inline PyCodeObject*
+_PyFrame_GetCodeBorrow(PyFrameObject *frame)
+{
     PyCodeObject *code = PyFrame_GetCode(frame);
     Py_DECREF(code);
-    return code; // borrowed reference
+    return code;  // borrowed reference
 }
 
+
 // bpo-40421 added PyFrame_GetCode() to Python 3.9.0b1
 #if PY_VERSION_HEX < 0x030900B1
-static inline PyFrameObject *PyFrame_GetBack(PyFrameObject *frame) {
+static inline PyFrameObject*
+PyFrame_GetBack(PyFrameObject *frame)
+{
     PyFrameObject *back;
     assert(frame != NULL);
     back = frame->f_back;
@@ -97,24 +131,31 @@ static inline PyFrameObject *PyFrame_GetBack(PyFrameObject *frame) {
 }
 #endif
 
-static inline PyFrameObject *_PyFrame_GetBackBorrow(PyFrameObject *frame) {
+static inline PyFrameObject*
+_PyFrame_GetBackBorrow(PyFrameObject *frame)
+{
     PyFrameObject *back = PyFrame_GetBack(frame);
     Py_XDECREF(back);
-    return back; // borrowed reference
+    return back;  // borrowed reference
 }
 
+
 // bpo-39947 added PyThreadState_GetInterpreter() to Python 3.9.0a5
 #if PY_VERSION_HEX < 0x030900A5
 static inline PyInterpreterState *
-PyThreadState_GetInterpreter(PyThreadState *tstate) {
+PyThreadState_GetInterpreter(PyThreadState *tstate)
+{
     assert(tstate != NULL);
     return tstate->interp;
 }
 #endif
 
+
 // bpo-40429 added PyThreadState_GetFrame() to Python 3.9.0b1
 #if PY_VERSION_HEX < 0x030900B1
-static inline PyFrameObject *PyThreadState_GetFrame(PyThreadState *tstate) {
+static inline PyFrameObject*
+PyThreadState_GetFrame(PyThreadState *tstate)
+{
     PyFrameObject *frame;
     assert(tstate != NULL);
     frame = tstate->frame;
@@ -123,16 +164,20 @@ static inline PyFrameObject *PyThreadState_GetFrame(PyThreadState *tstate) {
 }
 #endif
 
-static inline PyFrameObject *
-_PyThreadState_GetFrameBorrow(PyThreadState *tstate) {
+static inline PyFrameObject*
+_PyThreadState_GetFrameBorrow(PyThreadState *tstate)
+{
     PyFrameObject *frame = PyThreadState_GetFrame(tstate);
     Py_XDECREF(frame);
-    return frame; // borrowed reference
+    return frame;  // borrowed reference
 }
 
+
 // bpo-39947 added PyInterpreterState_Get() to Python 3.9.0a5
 #if PY_VERSION_HEX < 0x030900A5
-static inline PyInterpreterState *PyInterpreterState_Get(void) {
+static inline PyInterpreterState *
+PyInterpreterState_Get(void)
+{
     PyThreadState *tstate;
     PyInterpreterState *interp;
 
@@ -148,32 +193,59 @@ static inline PyInterpreterState *PyInterpreterState_Get(void) {
 }
 #endif
 
+
 // bpo-39947 added PyInterpreterState_Get() to Python 3.9.0a6
 #if 0x030700A1 <= PY_VERSION_HEX && PY_VERSION_HEX < 0x030900A6
-static inline uint64_t PyThreadState_GetID(PyThreadState *tstate) {
+static inline uint64_t
+PyThreadState_GetID(PyThreadState *tstate)
+{
     assert(tstate != NULL);
     return tstate->id;
 }
 #endif
 
+
 // bpo-37194 added PyObject_CallNoArgs() to Python 3.9.0a1
 #if PY_VERSION_HEX < 0x030900A1
-static inline PyObject *PyObject_CallNoArgs(PyObject *func) {
+static inline PyObject*
+PyObject_CallNoArgs(PyObject *func)
+{
     return PyObject_CallFunctionObjArgs(func, NULL);
 }
 #endif
 
+
 // bpo-39245 made PyObject_CallOneArg() public (previously called
 // _PyObject_CallOneArg) in Python 3.9.0a4
 #if PY_VERSION_HEX < 0x030900A4
-static inline PyObject *PyObject_CallOneArg(PyObject *func, PyObject *arg) {
+static inline PyObject*
+PyObject_CallOneArg(PyObject *func, PyObject *arg)
+{
     return PyObject_CallFunctionObjArgs(func, arg, NULL);
 }
 #endif
 
+
+// bpo-1635741 added PyModule_AddObjectRef() to Python 3.10.0a3
+#if PY_VERSION_HEX < 0x030A00A3
+static inline int
+PyModule_AddObjectRef(PyObject *module, const char *name, PyObject *value)
+{
+    Py_XINCREF(value);
+    int res = PyModule_AddObject(module, name, value);
+    if (res < 0) {
+        Py_XDECREF(value);
+    }
+    return res;
+}
+#endif
+
+
 // bpo-40024 added PyModule_AddType() to Python 3.9.0a5
 #if PY_VERSION_HEX < 0x030900A5
-static inline int PyModule_AddType(PyObject *module, PyTypeObject *type) {
+static inline int
+PyModule_AddType(PyObject *module, PyTypeObject *type)
+{
     const char *name, *dot;
 
     if (PyType_Ready(type) < 0) {
@@ -188,20 +260,17 @@ static inline int PyModule_AddType(PyObject *module, PyTypeObject *type) {
         name = dot + 1;
     }
 
-    Py_INCREF(type);
-    if (PyModule_AddObject(module, name, (PyObject *)type) < 0) {
-        Py_DECREF(type);
-        return -1;
-    }
-
-    return 0;
+    return PyModule_AddObjectRef(module, name, (PyObject *)type);
 }
 #endif
 
+
 // bpo-40241 added PyObject_GC_IsTracked() to Python 3.9.0a6.
 // bpo-4688 added _PyObject_GC_IS_TRACKED() to Python 2.7.0a2.
 #if PY_VERSION_HEX < 0x030900A6
-static inline int PyObject_GC_IsTracked(PyObject *obj) {
+static inline int
+PyObject_GC_IsTracked(PyObject* obj)
+{
     return (PyObject_IS_GC(obj) && _PyObject_GC_IS_TRACKED(obj));
 }
 #endif
@@ -209,20 +278,30 @@ static inline int PyObject_GC_IsTracked(PyObject *obj) {
 // bpo-40241 added PyObject_GC_IsFinalized() to Python 3.9.0a6.
 // bpo-18112 added _PyGCHead_FINALIZED() to Python 3.4.0 final.
 #if PY_VERSION_HEX < 0x030900A6 && PY_VERSION_HEX >= 0x030400F0
-static inline int PyObject_GC_IsFinalized(PyObject *obj) {
+static inline int
+PyObject_GC_IsFinalized(PyObject *obj)
+{
     return (PyObject_IS_GC(obj) && _PyGCHead_FINALIZED((PyGC_Head *)(obj)-1));
 }
 #endif
 
+
 // bpo-39573 added Py_IS_TYPE() to Python 3.9.0a4
 #if PY_VERSION_HEX < 0x030900A4 && !defined(Py_IS_TYPE)
-static inline int _Py_IS_TYPE(const PyObject *ob, const PyTypeObject *type) {
+static inline int
+_Py_IS_TYPE(const PyObject *ob, const PyTypeObject *type) {
     return ob->ob_type == type;
 }
-#define Py_IS_TYPE(ob, type) _Py_IS_TYPE((const PyObject *)(ob), type)
+#define Py_IS_TYPE(ob, type) _Py_IS_TYPE(_PyObject_CAST_CONST(ob), type)
+#endif
+
+
+#ifdef PYTHONCAPI_COMPAT_MSC_INLINE
+#  undef inline
+#  undef PYTHONCAPI_COMPAT_MSC_INLINE
 #endif
 
 #ifdef __cplusplus
 }
 #endif
-#endif // PYTHONCAPI_COMPAT
+#endif  // PYTHONCAPI_COMPAT