Integrand size = 36, antiderivative size = 93 \[ \int \frac {A+B \sin (e+f x)}{(a+a \sin (e+f x))^2 (c-c \sin (e+f x))^3} \, dx=\frac {(A+B) \sec ^3(e+f x)}{5 a^2 f \left (c^3-c^3 \sin (e+f x)\right )}+\frac {(4 A-B) \tan (e+f x)}{5 a^2 c^3 f}+\frac {(4 A-B) \tan ^3(e+f x)}{15 a^2 c^3 f} \]
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Time = 0.15 (sec) , antiderivative size = 93, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.083, Rules used = {3046, 2938, 3852} \[ \int \frac {A+B \sin (e+f x)}{(a+a \sin (e+f x))^2 (c-c \sin (e+f x))^3} \, dx=\frac {(4 A-B) \tan ^3(e+f x)}{15 a^2 c^3 f}+\frac {(4 A-B) \tan (e+f x)}{5 a^2 c^3 f}+\frac {(A+B) \sec ^3(e+f x)}{5 a^2 f \left (c^3-c^3 \sin (e+f x)\right )} \]
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Rule 2938
Rule 3046
Rule 3852
Rubi steps \begin{align*} \text {integral}& = \frac {\int \frac {\sec ^4(e+f x) (A+B \sin (e+f x))}{c-c \sin (e+f x)} \, dx}{a^2 c^2} \\ & = \frac {(A+B) \sec ^3(e+f x)}{5 a^2 f \left (c^3-c^3 \sin (e+f x)\right )}+\frac {(4 A-B) \int \sec ^4(e+f x) \, dx}{5 a^2 c^3} \\ & = \frac {(A+B) \sec ^3(e+f x)}{5 a^2 f \left (c^3-c^3 \sin (e+f x)\right )}-\frac {(4 A-B) \text {Subst}\left (\int \left (1+x^2\right ) \, dx,x,-\tan (e+f x)\right )}{5 a^2 c^3 f} \\ & = \frac {(A+B) \sec ^3(e+f x)}{5 a^2 f \left (c^3-c^3 \sin (e+f x)\right )}+\frac {(4 A-B) \tan (e+f x)}{5 a^2 c^3 f}+\frac {(4 A-B) \tan ^3(e+f x)}{15 a^2 c^3 f} \\ \end{align*}
Leaf count is larger than twice the leaf count of optimal. \(237\) vs. \(2(93)=186\).
Time = 2.00 (sec) , antiderivative size = 237, normalized size of antiderivative = 2.55 \[ \int \frac {A+B \sin (e+f x)}{(a+a \sin (e+f x))^2 (c-c \sin (e+f x))^3} \, dx=\frac {\left (\cos \left (\frac {1}{2} (e+f x)\right )-\sin \left (\frac {1}{2} (e+f x)\right )\right ) \left (\cos \left (\frac {1}{2} (e+f x)\right )+\sin \left (\frac {1}{2} (e+f x)\right )\right ) (-240 B+54 (A+B) \cos (e+f x)-32 (4 A-B) \cos (2 (e+f x))+18 A \cos (3 (e+f x))+18 B \cos (3 (e+f x))-64 A \cos (4 (e+f x))+16 B \cos (4 (e+f x))-384 A \sin (e+f x)+96 B \sin (e+f x)-18 A \sin (2 (e+f x))-18 B \sin (2 (e+f x))-128 A \sin (3 (e+f x))+32 B \sin (3 (e+f x))-9 A \sin (4 (e+f x))-9 B \sin (4 (e+f x)))}{960 a^2 c^3 f (-1+\sin (e+f x))^3 (1+\sin (e+f x))^2} \]
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Result contains complex when optimal does not.
Time = 1.24 (sec) , antiderivative size = 136, normalized size of antiderivative = 1.46
method | result | size |
risch | \(-\frac {4 i \left (24 i A \,{\mathrm e}^{3 i \left (f x +e \right )}-6 i B \,{\mathrm e}^{3 i \left (f x +e \right )}+15 B \,{\mathrm e}^{4 i \left (f x +e \right )}+8 i A \,{\mathrm e}^{i \left (f x +e \right )}+8 A \,{\mathrm e}^{2 i \left (f x +e \right )}-2 i B \,{\mathrm e}^{i \left (f x +e \right )}-2 B \,{\mathrm e}^{2 i \left (f x +e \right )}+4 A -B \right )}{15 \left ({\mathrm e}^{i \left (f x +e \right )}+i\right )^{3} \left ({\mathrm e}^{i \left (f x +e \right )}-i\right )^{5} f \,c^{3} a^{2}}\) | \(136\) |
parallelrisch | \(\frac {-30 A \left (\tan ^{7}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )+\left (30 A -30 B \right ) \left (\tan ^{6}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )+\left (10 A +20 B \right ) \left (\tan ^{5}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )+\left (-50 A -10 B \right ) \left (\tan ^{4}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )+\left (-26 A -16 B \right ) \left (\tan ^{3}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )+\left (42 A -18 B \right ) \left (\tan ^{2}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )+\left (-18 A +12 B \right ) \tan \left (\frac {f x}{2}+\frac {e}{2}\right )-6 A -6 B}{15 f \,c^{3} a^{2} \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )+1\right )^{3} \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )-1\right )^{5}}\) | \(171\) |
derivativedivides | \(\frac {-\frac {-\frac {A}{4}+\frac {B}{4}}{\left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )+1\right )^{2}}-\frac {2 \left (\frac {A}{4}-\frac {B}{4}\right )}{3 \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )+1\right )^{3}}-\frac {2 \left (\frac {5 A}{16}-\frac {3 B}{16}\right )}{\tan \left (\frac {f x}{2}+\frac {e}{2}\right )+1}-\frac {2 \left (A +B \right )}{5 \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )-1\right )^{5}}-\frac {2 A +2 B}{2 \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )-1\right )^{4}}-\frac {\frac {3 A}{2}+B}{\left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )-1\right )^{2}}-\frac {2 \left (\frac {5 A}{2}+2 B \right )}{3 \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )-1\right )^{3}}-\frac {2 \left (\frac {11 A}{16}+\frac {3 B}{16}\right )}{\tan \left (\frac {f x}{2}+\frac {e}{2}\right )-1}}{a^{2} c^{3} f}\) | \(183\) |
default | \(\frac {-\frac {-\frac {A}{4}+\frac {B}{4}}{\left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )+1\right )^{2}}-\frac {2 \left (\frac {A}{4}-\frac {B}{4}\right )}{3 \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )+1\right )^{3}}-\frac {2 \left (\frac {5 A}{16}-\frac {3 B}{16}\right )}{\tan \left (\frac {f x}{2}+\frac {e}{2}\right )+1}-\frac {2 \left (A +B \right )}{5 \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )-1\right )^{5}}-\frac {2 A +2 B}{2 \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )-1\right )^{4}}-\frac {\frac {3 A}{2}+B}{\left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )-1\right )^{2}}-\frac {2 \left (\frac {5 A}{2}+2 B \right )}{3 \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )-1\right )^{3}}-\frac {2 \left (\frac {11 A}{16}+\frac {3 B}{16}\right )}{\tan \left (\frac {f x}{2}+\frac {e}{2}\right )-1}}{a^{2} c^{3} f}\) | \(183\) |
norman | \(\frac {\frac {6 A -4 B}{10 a f c}-\frac {4 \left (4 A -B \right ) \left (\tan ^{5}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{15 a f c}-\frac {A \left (\tan ^{10}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{a f c}+\frac {\left (14 A -16 B \right ) \left (\tan ^{2}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{10 a f c}+\frac {\left (6 A -4 B \right ) \left (\tan ^{8}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{2 a f c}+\frac {\left (2 A -8 B \right ) \left (\tan ^{6}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{3 a f c}-\frac {2 \left (8 A -2 B \right ) \left (\tan ^{7}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{3 a f c}-\frac {\left (16 A -4 B \right ) \tan \left (\frac {f x}{2}+\frac {e}{2}\right )}{5 a f c}+\frac {2 \left (8 A -2 B \right ) \left (\tan ^{3}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{15 a f c}-\frac {\left (38 A +28 B \right ) \left (\tan ^{4}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{15 a f c}}{a \left (1+\tan ^{2}\left (\frac {f x}{2}+\frac {e}{2}\right )\right ) \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )+1\right )^{3} c^{2} \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )-1\right )^{5}}\) | \(321\) |
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Time = 0.25 (sec) , antiderivative size = 116, normalized size of antiderivative = 1.25 \[ \int \frac {A+B \sin (e+f x)}{(a+a \sin (e+f x))^2 (c-c \sin (e+f x))^3} \, dx=-\frac {2 \, {\left (4 \, A - B\right )} \cos \left (f x + e\right )^{4} - {\left (4 \, A - B\right )} \cos \left (f x + e\right )^{2} + {\left (2 \, {\left (4 \, A - B\right )} \cos \left (f x + e\right )^{2} + 4 \, A - B\right )} \sin \left (f x + e\right ) - A + 4 \, B}{15 \, {\left (a^{2} c^{3} f \cos \left (f x + e\right )^{3} \sin \left (f x + e\right ) - a^{2} c^{3} f \cos \left (f x + e\right )^{3}\right )}} \]
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Leaf count of result is larger than twice the leaf count of optimal. 2674 vs. \(2 (82) = 164\).
Time = 9.03 (sec) , antiderivative size = 2674, normalized size of antiderivative = 28.75 \[ \int \frac {A+B \sin (e+f x)}{(a+a \sin (e+f x))^2 (c-c \sin (e+f x))^3} \, dx=\text {Too large to display} \]
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Leaf count of result is larger than twice the leaf count of optimal. 651 vs. \(2 (88) = 176\).
Time = 0.24 (sec) , antiderivative size = 651, normalized size of antiderivative = 7.00 \[ \int \frac {A+B \sin (e+f x)}{(a+a \sin (e+f x))^2 (c-c \sin (e+f x))^3} \, dx=\frac {2 \, {\left (\frac {A {\left (\frac {9 \, \sin \left (f x + e\right )}{\cos \left (f x + e\right ) + 1} - \frac {21 \, \sin \left (f x + e\right )^{2}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{2}} + \frac {13 \, \sin \left (f x + e\right )^{3}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{3}} + \frac {25 \, \sin \left (f x + e\right )^{4}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{4}} - \frac {5 \, \sin \left (f x + e\right )^{5}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{5}} - \frac {15 \, \sin \left (f x + e\right )^{6}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{6}} + \frac {15 \, \sin \left (f x + e\right )^{7}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{7}} + 3\right )}}{a^{2} c^{3} - \frac {2 \, a^{2} c^{3} \sin \left (f x + e\right )}{\cos \left (f x + e\right ) + 1} - \frac {2 \, a^{2} c^{3} \sin \left (f x + e\right )^{2}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{2}} + \frac {6 \, a^{2} c^{3} \sin \left (f x + e\right )^{3}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{3}} - \frac {6 \, a^{2} c^{3} \sin \left (f x + e\right )^{5}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{5}} + \frac {2 \, a^{2} c^{3} \sin \left (f x + e\right )^{6}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{6}} + \frac {2 \, a^{2} c^{3} \sin \left (f x + e\right )^{7}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{7}} - \frac {a^{2} c^{3} \sin \left (f x + e\right )^{8}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{8}}} - \frac {B {\left (\frac {6 \, \sin \left (f x + e\right )}{\cos \left (f x + e\right ) + 1} - \frac {9 \, \sin \left (f x + e\right )^{2}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{2}} - \frac {8 \, \sin \left (f x + e\right )^{3}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{3}} - \frac {5 \, \sin \left (f x + e\right )^{4}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{4}} + \frac {10 \, \sin \left (f x + e\right )^{5}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{5}} - \frac {15 \, \sin \left (f x + e\right )^{6}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{6}} - 3\right )}}{a^{2} c^{3} - \frac {2 \, a^{2} c^{3} \sin \left (f x + e\right )}{\cos \left (f x + e\right ) + 1} - \frac {2 \, a^{2} c^{3} \sin \left (f x + e\right )^{2}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{2}} + \frac {6 \, a^{2} c^{3} \sin \left (f x + e\right )^{3}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{3}} - \frac {6 \, a^{2} c^{3} \sin \left (f x + e\right )^{5}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{5}} + \frac {2 \, a^{2} c^{3} \sin \left (f x + e\right )^{6}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{6}} + \frac {2 \, a^{2} c^{3} \sin \left (f x + e\right )^{7}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{7}} - \frac {a^{2} c^{3} \sin \left (f x + e\right )^{8}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{8}}}\right )}}{15 \, f} \]
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Leaf count of result is larger than twice the leaf count of optimal. 221 vs. \(2 (88) = 176\).
Time = 0.34 (sec) , antiderivative size = 221, normalized size of antiderivative = 2.38 \[ \int \frac {A+B \sin (e+f x)}{(a+a \sin (e+f x))^2 (c-c \sin (e+f x))^3} \, dx=-\frac {\frac {5 \, {\left (15 \, A \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{2} - 9 \, B \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{2} + 24 \, A \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right ) - 12 \, B \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right ) + 13 \, A - 7 \, B\right )}}{a^{2} c^{3} {\left (\tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right ) + 1\right )}^{3}} + \frac {165 \, A \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{4} + 45 \, B \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{4} - 480 \, A \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{3} - 60 \, B \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{3} + 650 \, A \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{2} + 70 \, B \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{2} - 400 \, A \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right ) - 20 \, B \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right ) + 113 \, A + 13 \, B}{a^{2} c^{3} {\left (\tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right ) - 1\right )}^{5}}}{120 \, f} \]
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Time = 12.54 (sec) , antiderivative size = 183, normalized size of antiderivative = 1.97 \[ \int \frac {A+B \sin (e+f x)}{(a+a \sin (e+f x))^2 (c-c \sin (e+f x))^3} \, dx=\frac {\left (\frac {8\,A}{15}-\frac {2\,B}{15}-\frac {16\,A\,\sin \left (e+f\,x\right )}{15}+\frac {4\,B\,\sin \left (e+f\,x\right )}{15}\right )\,{\cos \left (e+f\,x\right )}^2+\frac {2\,A}{15}-\frac {8\,B}{15}-\frac {8\,A\,\sin \left (e+f\,x\right )}{15}+\frac {2\,B\,\sin \left (e+f\,x\right )}{15}}{a^2\,c^3\,f\,\left (2\,{\cos \left (e+f\,x\right )}^3\,\sin \left (e+f\,x\right )-2\,{\cos \left (e+f\,x\right )}^3\right )}-\frac {\frac {2\,A}{5}+\frac {2\,B}{5}-\frac {2\,A\,\sin \left (e+f\,x\right )}{5}-\frac {2\,B\,\sin \left (e+f\,x\right )}{5}}{a^2\,c^3\,f\,\left (2\,\sin \left (e+f\,x\right )-2\right )}-\frac {\cos \left (e+f\,x\right )\,\left (\frac {16\,A}{15}-\frac {4\,B}{15}\right )}{a^2\,c^3\,f\,\left (2\,\sin \left (e+f\,x\right )-2\right )} \]
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