Optimal. Leaf size=127 \[ -\frac {(A-B) (c-d) \cos (e+f x)}{5 f (a+a \sin (e+f x))^3}-\frac {(2 A c+3 B c+3 A d-8 B d) \cos (e+f x)}{15 a f (a+a \sin (e+f x))^2}-\frac {(2 A c+3 B c+3 A d+7 B d) \cos (e+f x)}{15 f \left (a^3+a^3 \sin (e+f x)\right )} \]
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Rubi [A]
time = 0.15, antiderivative size = 127, normalized size of antiderivative = 1.00, number of steps
used = 4, number of rules used = 4, integrand size = 33, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.121, Rules used = {3047, 3098,
2829, 2727} \begin {gather*} -\frac {(2 A c+3 A d+3 B c+7 B d) \cos (e+f x)}{15 f \left (a^3 \sin (e+f x)+a^3\right )}-\frac {(2 A c+3 A d+3 B c-8 B d) \cos (e+f x)}{15 a f (a \sin (e+f x)+a)^2}-\frac {(A-B) (c-d) \cos (e+f x)}{5 f (a \sin (e+f x)+a)^3} \end {gather*}
Antiderivative was successfully verified.
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Rule 2727
Rule 2829
Rule 3047
Rule 3098
Rubi steps
\begin {align*} \int \frac {(A+B \sin (e+f x)) (c+d \sin (e+f x))}{(a+a \sin (e+f x))^3} \, dx &=\int \frac {A c+(B c+A d) \sin (e+f x)+B d \sin ^2(e+f x)}{(a+a \sin (e+f x))^3} \, dx\\ &=-\frac {(A-B) (c-d) \cos (e+f x)}{5 f (a+a \sin (e+f x))^3}-\frac {\int \frac {-a (2 A c+3 B c+3 A d-3 B d)-5 a B d \sin (e+f x)}{(a+a \sin (e+f x))^2} \, dx}{5 a^2}\\ &=-\frac {(A-B) (c-d) \cos (e+f x)}{5 f (a+a \sin (e+f x))^3}-\frac {(2 A c+3 B c+3 A d-8 B d) \cos (e+f x)}{15 a f (a+a \sin (e+f x))^2}+\frac {(2 A c+3 B c+3 A d+7 B d) \int \frac {1}{a+a \sin (e+f x)} \, dx}{15 a^2}\\ &=-\frac {(A-B) (c-d) \cos (e+f x)}{5 f (a+a \sin (e+f x))^3}-\frac {(2 A c+3 B c+3 A d-8 B d) \cos (e+f x)}{15 a f (a+a \sin (e+f x))^2}-\frac {(2 A c+3 B c+3 A d+7 B d) \cos (e+f x)}{15 f \left (a^3+a^3 \sin (e+f x)\right )}\\ \end {align*}
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Mathematica [A]
time = 0.49, size = 176, normalized size = 1.39 \begin {gather*} \frac {\left (\cos \left (\frac {1}{2} (e+f x)\right )+\sin \left (\frac {1}{2} (e+f x)\right )\right ) \left (15 (A d+B (c+2 d)) \cos \left (\frac {1}{2} (e+f x)\right )-5 (2 A c+3 B c+3 A d+4 B d) \cos \left (\frac {3}{2} (e+f x)\right )-2 (-3 (3 A c+2 B c+2 A d+8 B d)+(2 A c+3 B c+3 A d-8 B d) \cos (e+f x)+(2 A c+3 B c+3 A d+7 B d) \cos (2 (e+f x))) \sin \left (\frac {1}{2} (e+f x)\right )\right )}{30 a^3 f (1+\sin (e+f x))^3} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.36, size = 151, normalized size = 1.19
method | result | size |
derivativedivides | \(\frac {-\frac {-8 A c +8 A d +8 B c -8 B d}{2 \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )+1\right )^{4}}-\frac {2 \left (4 A c -4 A d -4 B c +4 B d \right )}{5 \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )+1\right )^{5}}-\frac {2 A c}{\tan \left (\frac {f x}{2}+\frac {e}{2}\right )+1}-\frac {-4 A c +2 A d +2 B c}{\left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )+1\right )^{2}}-\frac {2 \left (8 A c -6 A d -6 B c +4 B d \right )}{3 \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )+1\right )^{3}}}{f \,a^{3}}\) | \(151\) |
default | \(\frac {-\frac {-8 A c +8 A d +8 B c -8 B d}{2 \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )+1\right )^{4}}-\frac {2 \left (4 A c -4 A d -4 B c +4 B d \right )}{5 \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )+1\right )^{5}}-\frac {2 A c}{\tan \left (\frac {f x}{2}+\frac {e}{2}\right )+1}-\frac {-4 A c +2 A d +2 B c}{\left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )+1\right )^{2}}-\frac {2 \left (8 A c -6 A d -6 B c +4 B d \right )}{3 \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )+1\right )^{3}}}{f \,a^{3}}\) | \(151\) |
risch | \(-\frac {2 \left (2 A c +15 B d \,{\mathrm e}^{4 i \left (f x +e \right )}-10 i A c \,{\mathrm e}^{i \left (f x +e \right )}+15 i A d \,{\mathrm e}^{3 i \left (f x +e \right )}-20 i B d \,{\mathrm e}^{i \left (f x +e \right )}-15 i B c \,{\mathrm e}^{i \left (f x +e \right )}+30 i B d \,{\mathrm e}^{3 i \left (f x +e \right )}-15 i d \,{\mathrm e}^{i \left (f x +e \right )} A +15 i B c \,{\mathrm e}^{3 i \left (f x +e \right )}-20 A c \,{\mathrm e}^{2 i \left (f x +e \right )}-15 A d \,{\mathrm e}^{2 i \left (f x +e \right )}-15 B c \,{\mathrm e}^{2 i \left (f x +e \right )}-40 B d \,{\mathrm e}^{2 i \left (f x +e \right )}+3 A d +7 B d +3 B c \right )}{15 f \,a^{3} \left ({\mathrm e}^{i \left (f x +e \right )}+i\right )^{5}}\) | \(203\) |
norman | \(\frac {-\frac {14 A c +6 A d +6 B c +4 B d}{15 f a}-\frac {2 A c \left (\tan ^{8}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{f a}-\frac {2 \left (2 A c +A d +B c \right ) \left (\tan ^{7}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{f a}-\frac {2 \left (34 A c +11 A d +11 B c +14 B d \right ) \left (\tan ^{4}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{5 f a}-\frac {2 \left (16 A c +9 A d +9 B c +2 B d \right ) \left (\tan ^{5}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{3 f a}-\frac {2 \left (14 A c +3 A d +3 B c +4 B d \right ) \left (\tan ^{6}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{3 f a}-\frac {2 \left (14 A c +9 A d +9 B c +4 B d \right ) \left (\tan ^{3}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{3 f a}-\frac {2 \left (18 A c +7 A d +7 B c +8 B d \right ) \left (\tan ^{2}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{5 f a}-\frac {\left (8 A c +6 A d +6 B c +4 B d \right ) \tan \left (\frac {f x}{2}+\frac {e}{2}\right )}{3 f a}}{\left (1+\tan ^{2}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )^{2} a^{2} \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )+1\right )^{5}}\) | \(324\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [B] Leaf count of result is larger than twice the leaf count of optimal. 797 vs.
\(2 (127) = 254\).
time = 0.31, size = 797, normalized size = 6.28 \begin {gather*} -\frac {2 \, {\left (\frac {A c {\left (\frac {20 \, \sin \left (f x + e\right )}{\cos \left (f x + e\right ) + 1} + \frac {40 \, \sin \left (f x + e\right )^{2}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{2}} + \frac {30 \, \sin \left (f x + e\right )^{3}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{3}} + \frac {15 \, \sin \left (f x + e\right )^{4}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{4}} + 7\right )}}{a^{3} + \frac {5 \, a^{3} \sin \left (f x + e\right )}{\cos \left (f x + e\right ) + 1} + \frac {10 \, a^{3} \sin \left (f x + e\right )^{2}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{2}} + \frac {10 \, a^{3} \sin \left (f x + e\right )^{3}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{3}} + \frac {5 \, a^{3} \sin \left (f x + e\right )^{4}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{4}} + \frac {a^{3} \sin \left (f x + e\right )^{5}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{5}}} + \frac {2 \, B d {\left (\frac {5 \, \sin \left (f x + e\right )}{\cos \left (f x + e\right ) + 1} + \frac {10 \, \sin \left (f x + e\right )^{2}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{2}} + 1\right )}}{a^{3} + \frac {5 \, a^{3} \sin \left (f x + e\right )}{\cos \left (f x + e\right ) + 1} + \frac {10 \, a^{3} \sin \left (f x + e\right )^{2}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{2}} + \frac {10 \, a^{3} \sin \left (f x + e\right )^{3}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{3}} + \frac {5 \, a^{3} \sin \left (f x + e\right )^{4}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{4}} + \frac {a^{3} \sin \left (f x + e\right )^{5}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{5}}} + \frac {3 \, B c {\left (\frac {5 \, \sin \left (f x + e\right )}{\cos \left (f x + e\right ) + 1} + \frac {5 \, \sin \left (f x + e\right )^{2}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{2}} + \frac {5 \, \sin \left (f x + e\right )^{3}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{3}} + 1\right )}}{a^{3} + \frac {5 \, a^{3} \sin \left (f x + e\right )}{\cos \left (f x + e\right ) + 1} + \frac {10 \, a^{3} \sin \left (f x + e\right )^{2}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{2}} + \frac {10 \, a^{3} \sin \left (f x + e\right )^{3}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{3}} + \frac {5 \, a^{3} \sin \left (f x + e\right )^{4}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{4}} + \frac {a^{3} \sin \left (f x + e\right )^{5}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{5}}} + \frac {3 \, A d {\left (\frac {5 \, \sin \left (f x + e\right )}{\cos \left (f x + e\right ) + 1} + \frac {5 \, \sin \left (f x + e\right )^{2}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{2}} + \frac {5 \, \sin \left (f x + e\right )^{3}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{3}} + 1\right )}}{a^{3} + \frac {5 \, a^{3} \sin \left (f x + e\right )}{\cos \left (f x + e\right ) + 1} + \frac {10 \, a^{3} \sin \left (f x + e\right )^{2}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{2}} + \frac {10 \, a^{3} \sin \left (f x + e\right )^{3}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{3}} + \frac {5 \, a^{3} \sin \left (f x + e\right )^{4}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{4}} + \frac {a^{3} \sin \left (f x + e\right )^{5}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{5}}}\right )}}{15 \, f} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 283 vs.
\(2 (127) = 254\).
time = 0.79, size = 283, normalized size = 2.23 \begin {gather*} -\frac {{\left ({\left (2 \, A + 3 \, B\right )} c + {\left (3 \, A + 7 \, B\right )} d\right )} \cos \left (f x + e\right )^{3} - {\left (2 \, {\left (2 \, A + 3 \, B\right )} c + {\left (6 \, A - B\right )} d\right )} \cos \left (f x + e\right )^{2} - 3 \, {\left (A - B\right )} c + 3 \, {\left (A - B\right )} d - 3 \, {\left ({\left (3 \, A + 2 \, B\right )} c + {\left (2 \, A + 3 \, B\right )} d\right )} \cos \left (f x + e\right ) - {\left ({\left ({\left (2 \, A + 3 \, B\right )} c + {\left (3 \, A + 7 \, B\right )} d\right )} \cos \left (f x + e\right )^{2} - 3 \, {\left (A - B\right )} c + 3 \, {\left (A - B\right )} d + 3 \, {\left ({\left (2 \, A + 3 \, B\right )} c + {\left (3 \, A + 2 \, B\right )} d\right )} \cos \left (f x + e\right )\right )} \sin \left (f x + e\right )}{15 \, {\left (a^{3} f \cos \left (f x + e\right )^{3} + 3 \, a^{3} f \cos \left (f x + e\right )^{2} - 2 \, a^{3} f \cos \left (f x + e\right ) - 4 \, a^{3} f + {\left (a^{3} f \cos \left (f x + e\right )^{2} - 2 \, a^{3} f \cos \left (f x + e\right ) - 4 \, a^{3} f\right )} \sin \left (f x + e\right )\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [B] Leaf count of result is larger than twice the leaf count of optimal. 1819 vs.
\(2 (121) = 242\).
time = 5.37, size = 1819, normalized size = 14.32 \begin {gather*} \text {Too large to display} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 0.51, size = 223, normalized size = 1.76 \begin {gather*} -\frac {2 \, {\left (15 \, A c \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{4} + 30 \, A c \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{3} + 15 \, B c \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{3} + 15 \, A d \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{3} + 40 \, A c \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{2} + 15 \, B c \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{2} + 15 \, A d \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{2} + 20 \, B d \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{2} + 20 \, A c \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right ) + 15 \, B c \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right ) + 15 \, A d \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right ) + 10 \, B d \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right ) + 7 \, A c + 3 \, B c + 3 \, A d + 2 \, B d\right )}}{15 \, a^{3} f {\left (\tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right ) + 1\right )}^{5}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 14.26, size = 245, normalized size = 1.93 \begin {gather*} \frac {2\,\cos \left (\frac {e}{2}+\frac {f\,x}{2}\right )\,\left (\frac {53\,A\,c}{4}+3\,A\,d+3\,B\,c+\frac {13\,B\,d}{4}-4\,A\,c\,\cos \left (e+f\,x\right )+\frac {3\,A\,d\,\cos \left (e+f\,x\right )}{2}+\frac {3\,B\,c\,\cos \left (e+f\,x\right )}{2}+B\,d\,\cos \left (e+f\,x\right )+\frac {25\,A\,c\,\sin \left (e+f\,x\right )}{2}+\frac {15\,A\,d\,\sin \left (e+f\,x\right )}{2}+\frac {15\,B\,c\,\sin \left (e+f\,x\right )}{2}+\frac {5\,B\,d\,\sin \left (e+f\,x\right )}{2}-\frac {9\,A\,c\,\cos \left (2\,e+2\,f\,x\right )}{4}-\frac {3\,A\,d\,\cos \left (2\,e+2\,f\,x\right )}{2}-\frac {3\,B\,c\,\cos \left (2\,e+2\,f\,x\right )}{2}-\frac {9\,B\,d\,\cos \left (2\,e+2\,f\,x\right )}{4}-\frac {5\,A\,c\,\sin \left (2\,e+2\,f\,x\right )}{4}+\frac {5\,B\,d\,\sin \left (2\,e+2\,f\,x\right )}{4}\right )}{15\,a^3\,f\,\left (\frac {5\,\sqrt {2}\,\cos \left (\frac {3\,e}{2}+\frac {\pi }{4}+\frac {3\,f\,x}{2}\right )}{4}-\frac {5\,\sqrt {2}\,\cos \left (\frac {e}{2}-\frac {\pi }{4}+\frac {f\,x}{2}\right )}{2}+\frac {\sqrt {2}\,\cos \left (\frac {5\,e}{2}-\frac {\pi }{4}+\frac {5\,f\,x}{2}\right )}{4}\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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