Integrand size = 33, antiderivative size = 85 \[ \int \frac {(A+B \sin (e+f x)) (c+d \sin (e+f x))}{(a+a \sin (e+f x))^2} \, dx=\frac {B d x}{a^2}-\frac {(A c+2 B c+2 A d-5 B d) \cos (e+f x)}{3 a^2 f (1+\sin (e+f x))}-\frac {(A-B) (c-d) \cos (e+f x)}{3 f (a+a \sin (e+f x))^2} \]
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Time = 0.15 (sec) , antiderivative size = 85, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.121, Rules used = {3047, 3098, 2814, 2727} \[ \int \frac {(A+B \sin (e+f x)) (c+d \sin (e+f x))}{(a+a \sin (e+f x))^2} \, dx=-\frac {(A c+2 A d+2 B c-5 B d) \cos (e+f x)}{3 a^2 f (\sin (e+f x)+1)}+\frac {B d x}{a^2}-\frac {(A-B) (c-d) \cos (e+f x)}{3 f (a \sin (e+f x)+a)^2} \]
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Rule 2727
Rule 2814
Rule 3047
Rule 3098
Rubi steps \begin{align*} \text {integral}& = \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))^2} \, dx \\ & = -\frac {(A-B) (c-d) \cos (e+f x)}{3 f (a+a \sin (e+f x))^2}-\frac {\int \frac {-a (2 B (c-d)+A (c+2 d))-3 a B d \sin (e+f x)}{a+a \sin (e+f x)} \, dx}{3 a^2} \\ & = \frac {B d x}{a^2}-\frac {(A-B) (c-d) \cos (e+f x)}{3 f (a+a \sin (e+f x))^2}+\frac {(A c+2 B c+2 A d-5 B d) \int \frac {1}{a+a \sin (e+f x)} \, dx}{3 a} \\ & = \frac {B d x}{a^2}-\frac {(A-B) (c-d) \cos (e+f x)}{3 f (a+a \sin (e+f x))^2}-\frac {(A c+2 B c+2 A d-5 B d) \cos (e+f x)}{3 f \left (a^2+a^2 \sin (e+f x)\right )} \\ \end{align*}
Leaf count is larger than twice the leaf count of optimal. \(180\) vs. \(2(85)=170\).
Time = 1.40 (sec) , antiderivative size = 180, normalized size of antiderivative = 2.12 \[ \int \frac {(A+B \sin (e+f x)) (c+d \sin (e+f x))}{(a+a \sin (e+f x))^2} \, dx=\frac {\left (\cos \left (\frac {1}{2} (e+f x)\right )+\sin \left (\frac {1}{2} (e+f x)\right )\right ) \left (2 (A-B) (c-d) \sin \left (\frac {1}{2} (e+f x)\right )-(A-B) (c-d) \left (\cos \left (\frac {1}{2} (e+f x)\right )+\sin \left (\frac {1}{2} (e+f x)\right )\right )+2 (A c+2 B c+2 A d-5 B d) \sin \left (\frac {1}{2} (e+f x)\right ) \left (\cos \left (\frac {1}{2} (e+f x)\right )+\sin \left (\frac {1}{2} (e+f x)\right )\right )^2+3 B d (e+f x) \left (\cos \left (\frac {1}{2} (e+f x)\right )+\sin \left (\frac {1}{2} (e+f x)\right )\right )^3\right )}{3 a^2 f (1+\sin (e+f x))^2} \]
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Time = 0.58 (sec) , antiderivative size = 110, normalized size of antiderivative = 1.29
method | result | size |
derivativedivides | \(\frac {-\frac {2 \left (A c -d B \right )}{\tan \left (\frac {f x}{2}+\frac {e}{2}\right )+1}-\frac {-2 A c +2 d A +2 B c -2 d B}{\left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )+1\right )^{2}}-\frac {2 \left (2 A c -2 d A -2 B c +2 d B \right )}{3 \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )+1\right )^{3}}+2 d B \arctan \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{a^{2} f}\) | \(110\) |
default | \(\frac {-\frac {2 \left (A c -d B \right )}{\tan \left (\frac {f x}{2}+\frac {e}{2}\right )+1}-\frac {-2 A c +2 d A +2 B c -2 d B}{\left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )+1\right )^{2}}-\frac {2 \left (2 A c -2 d A -2 B c +2 d B \right )}{3 \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )+1\right )^{3}}+2 d B \arctan \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{a^{2} f}\) | \(110\) |
parallelrisch | \(\frac {3 B x \left (\tan ^{3}\left (\frac {f x}{2}+\frac {e}{2}\right )\right ) d f +\left (\left (9 f x +6\right ) d B -6 A c \right ) \left (\tan ^{2}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )+\left (\left (9 d f x -6 c +18 d \right ) B -6 A \left (c +d \right )\right ) \tan \left (\frac {f x}{2}+\frac {e}{2}\right )+\left (3 d f x -2 c +8 d \right ) B -4 \left (c +\frac {d}{2}\right ) A}{3 f \,a^{2} \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )+1\right )^{3}}\) | \(119\) |
risch | \(\frac {B d x}{a^{2}}-\frac {2 \left (-A c -2 d A +3 i A c \,{\mathrm e}^{i \left (f x +e \right )}+3 i d \,{\mathrm e}^{i \left (f x +e \right )} A -2 B c +5 d B +3 i B c \,{\mathrm e}^{i \left (f x +e \right )}-9 i B d \,{\mathrm e}^{i \left (f x +e \right )}+3 A d \,{\mathrm e}^{2 i \left (f x +e \right )}+3 B c \,{\mathrm e}^{2 i \left (f x +e \right )}-6 B d \,{\mathrm e}^{2 i \left (f x +e \right )}\right )}{3 f \,a^{2} \left ({\mathrm e}^{i \left (f x +e \right )}+i\right )^{3}}\) | \(143\) |
norman | \(\frac {\frac {x d B}{a}+\frac {x d B \left (\tan ^{7}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{a}-\frac {4 A c +2 d A +2 B c -8 d B}{3 a f}-\frac {\left (2 A c -2 d B \right ) \left (\tan ^{6}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{a f}-\frac {\left (16 A c +2 d A +2 B c -20 d B \right ) \left (\tan ^{4}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{3 a f}-\frac {\left (14 A c +4 d A +4 B c -22 d B \right ) \left (\tan ^{2}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{3 a f}-\frac {\left (4 A c +4 d A +4 B c -12 d B \right ) \left (\tan ^{3}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{a f}-\frac {\left (2 A c +2 d A +2 B c -6 d B \right ) \left (\tan ^{5}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{a f}-\frac {\left (2 A c +2 d A +2 B c -6 d B \right ) \tan \left (\frac {f x}{2}+\frac {e}{2}\right )}{a f}+\frac {3 x d B \tan \left (\frac {f x}{2}+\frac {e}{2}\right )}{a}+\frac {5 x d B \left (\tan ^{2}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{a}+\frac {7 x d B \left (\tan ^{3}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{a}+\frac {7 x d B \left (\tan ^{4}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{a}+\frac {5 x d B \left (\tan ^{5}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{a}+\frac {3 x d B \left (\tan ^{6}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{a}}{\left (1+\tan ^{2}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )^{2} a \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )+1\right )^{3}}\) | \(402\) |
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Leaf count of result is larger than twice the leaf count of optimal. 208 vs. \(2 (81) = 162\).
Time = 0.27 (sec) , antiderivative size = 208, normalized size of antiderivative = 2.45 \[ \int \frac {(A+B \sin (e+f x)) (c+d \sin (e+f x))}{(a+a \sin (e+f x))^2} \, dx=-\frac {6 \, B d f x - {\left (3 \, B d f x + {\left (A + 2 \, B\right )} c + {\left (2 \, A - 5 \, B\right )} d\right )} \cos \left (f x + e\right )^{2} - {\left (A - B\right )} c + {\left (A - B\right )} d + {\left (3 \, B d f x - {\left (2 \, A + B\right )} c - {\left (A - 4 \, B\right )} d\right )} \cos \left (f x + e\right ) + {\left (6 \, B d f x + {\left (A - B\right )} c - {\left (A - B\right )} d + {\left (3 \, B d f x - {\left (A + 2 \, B\right )} c - {\left (2 \, A - 5 \, B\right )} d\right )} \cos \left (f x + e\right )\right )} \sin \left (f x + e\right )}{3 \, {\left (a^{2} f \cos \left (f x + e\right )^{2} - a^{2} f \cos \left (f x + e\right ) - 2 \, a^{2} f - {\left (a^{2} f \cos \left (f x + e\right ) + 2 \, a^{2} f\right )} \sin \left (f x + e\right )\right )}} \]
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Leaf count of result is larger than twice the leaf count of optimal. 1062 vs. \(2 (83) = 166\).
Time = 2.11 (sec) , antiderivative size = 1062, normalized size of antiderivative = 12.49 \[ \int \frac {(A+B \sin (e+f x)) (c+d \sin (e+f x))}{(a+a \sin (e+f x))^2} \, dx=\text {Too large to display} \]
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Leaf count of result is larger than twice the leaf count of optimal. 454 vs. \(2 (81) = 162\).
Time = 0.31 (sec) , antiderivative size = 454, normalized size of antiderivative = 5.34 \[ \int \frac {(A+B \sin (e+f x)) (c+d \sin (e+f x))}{(a+a \sin (e+f x))^2} \, dx=\frac {2 \, {\left (B d {\left (\frac {\frac {9 \, \sin \left (f x + e\right )}{\cos \left (f x + e\right ) + 1} + \frac {3 \, \sin \left (f x + e\right )^{2}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{2}} + 4}{a^{2} + \frac {3 \, a^{2} \sin \left (f x + e\right )}{\cos \left (f x + e\right ) + 1} + \frac {3 \, a^{2} \sin \left (f x + e\right )^{2}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{2}} + \frac {a^{2} \sin \left (f x + e\right )^{3}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{3}}} + \frac {3 \, \arctan \left (\frac {\sin \left (f x + e\right )}{\cos \left (f x + e\right ) + 1}\right )}{a^{2}}\right )} - \frac {A c {\left (\frac {3 \, \sin \left (f x + e\right )}{\cos \left (f x + e\right ) + 1} + \frac {3 \, \sin \left (f x + e\right )^{2}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{2}} + 2\right )}}{a^{2} + \frac {3 \, a^{2} \sin \left (f x + e\right )}{\cos \left (f x + e\right ) + 1} + \frac {3 \, a^{2} \sin \left (f x + e\right )^{2}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{2}} + \frac {a^{2} \sin \left (f x + e\right )^{3}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{3}}} - \frac {B c {\left (\frac {3 \, \sin \left (f x + e\right )}{\cos \left (f x + e\right ) + 1} + 1\right )}}{a^{2} + \frac {3 \, a^{2} \sin \left (f x + e\right )}{\cos \left (f x + e\right ) + 1} + \frac {3 \, a^{2} \sin \left (f x + e\right )^{2}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{2}} + \frac {a^{2} \sin \left (f x + e\right )^{3}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{3}}} - \frac {A d {\left (\frac {3 \, \sin \left (f x + e\right )}{\cos \left (f x + e\right ) + 1} + 1\right )}}{a^{2} + \frac {3 \, a^{2} \sin \left (f x + e\right )}{\cos \left (f x + e\right ) + 1} + \frac {3 \, a^{2} \sin \left (f x + e\right )^{2}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{2}} + \frac {a^{2} \sin \left (f x + e\right )^{3}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{3}}}\right )}}{3 \, f} \]
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Time = 0.31 (sec) , antiderivative size = 133, normalized size of antiderivative = 1.56 \[ \int \frac {(A+B \sin (e+f x)) (c+d \sin (e+f x))}{(a+a \sin (e+f x))^2} \, dx=\frac {\frac {3 \, {\left (f x + e\right )} B d}{a^{2}} - \frac {2 \, {\left (3 \, A c \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{2} - 3 \, B d \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{2} + 3 \, A c \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right ) + 3 \, B c \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right ) + 3 \, A d \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right ) - 9 \, B d \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right ) + 2 \, A c + B c + A d - 4 \, B d\right )}}{a^{2} {\left (\tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right ) + 1\right )}^{3}}}{3 \, f} \]
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Time = 14.08 (sec) , antiderivative size = 94, normalized size of antiderivative = 1.11 \[ \int \frac {(A+B \sin (e+f x)) (c+d \sin (e+f x))}{(a+a \sin (e+f x))^2} \, dx=\frac {B\,d\,x}{a^2}-\frac {\left (2\,A\,c-2\,B\,d\right )\,{\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^2+\left (2\,A\,c+2\,A\,d+2\,B\,c-6\,B\,d\right )\,\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )+\frac {4\,A\,c}{3}+\frac {2\,A\,d}{3}+\frac {2\,B\,c}{3}-\frac {8\,B\,d}{3}}{a^2\,f\,{\left (\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )+1\right )}^3} \]
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