Integrand size = 23, antiderivative size = 102 \[ \int \frac {A+B \sin (e+f x)}{(a+a \sin (e+f x))^3} \, dx=-\frac {(A-B) \cos (e+f x)}{5 f (a+a \sin (e+f x))^3}-\frac {(2 A+3 B) \cos (e+f x)}{15 a f (a+a \sin (e+f x))^2}-\frac {(2 A+3 B) \cos (e+f x)}{15 f \left (a^3+a^3 \sin (e+f x)\right )} \]
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Time = 0.06 (sec) , antiderivative size = 102, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.130, Rules used = {2829, 2729, 2727} \[ \int \frac {A+B \sin (e+f x)}{(a+a \sin (e+f x))^3} \, dx=-\frac {(2 A+3 B) \cos (e+f x)}{15 f \left (a^3 \sin (e+f x)+a^3\right )}-\frac {(2 A+3 B) \cos (e+f x)}{15 a f (a \sin (e+f x)+a)^2}-\frac {(A-B) \cos (e+f x)}{5 f (a \sin (e+f x)+a)^3} \]
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Rule 2727
Rule 2729
Rule 2829
Rubi steps \begin{align*} \text {integral}& = -\frac {(A-B) \cos (e+f x)}{5 f (a+a \sin (e+f x))^3}+\frac {(2 A+3 B) \int \frac {1}{(a+a \sin (e+f x))^2} \, dx}{5 a} \\ & = -\frac {(A-B) \cos (e+f x)}{5 f (a+a \sin (e+f x))^3}-\frac {(2 A+3 B) \cos (e+f x)}{15 a f (a+a \sin (e+f x))^2}+\frac {(2 A+3 B) \int \frac {1}{a+a \sin (e+f x)} \, dx}{15 a^2} \\ & = -\frac {(A-B) \cos (e+f x)}{5 f (a+a \sin (e+f x))^3}-\frac {(2 A+3 B) \cos (e+f x)}{15 a f (a+a \sin (e+f x))^2}-\frac {(2 A+3 B) \cos (e+f x)}{15 f \left (a^3+a^3 \sin (e+f x)\right )} \\ \end{align*}
Time = 0.06 (sec) , antiderivative size = 63, normalized size of antiderivative = 0.62 \[ \int \frac {A+B \sin (e+f x)}{(a+a \sin (e+f x))^3} \, dx=-\frac {\cos (e+f x) \left (7 A+3 B+(6 A+9 B) \sin (e+f x)+(2 A+3 B) \sin ^2(e+f x)\right )}{15 a^3 f (1+\sin (e+f x))^3} \]
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Result contains complex when optimal does not.
Time = 0.56 (sec) , antiderivative size = 95, normalized size of antiderivative = 0.93
method | result | size |
risch | \(-\frac {2 i \left (20 i A \,{\mathrm e}^{2 i \left (f x +e \right )}+15 i B \,{\mathrm e}^{2 i \left (f x +e \right )}+15 B \,{\mathrm e}^{3 i \left (f x +e \right )}-2 i A -10 A \,{\mathrm e}^{i \left (f x +e \right )}-3 i B -15 B \,{\mathrm e}^{i \left (f x +e \right )}\right )}{15 f \,a^{3} \left ({\mathrm e}^{i \left (f x +e \right )}+i\right )^{5}}\) | \(95\) |
parallelrisch | \(\frac {-30 A \left (\tan ^{4}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )+\left (-60 A -30 B \right ) \left (\tan ^{3}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )+\left (-80 A -30 B \right ) \left (\tan ^{2}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )+\left (-40 A -30 B \right ) \tan \left (\frac {f x}{2}+\frac {e}{2}\right )-14 A -6 B}{15 f \,a^{3} \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )+1\right )^{5}}\) | \(98\) |
derivativedivides | \(\frac {-\frac {2 A}{\tan \left (\frac {f x}{2}+\frac {e}{2}\right )+1}-\frac {2 \left (8 A -6 B \right )}{3 \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )+1\right )^{3}}-\frac {-8 A +8 B}{2 \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )+1\right )^{4}}-\frac {2 \left (4 A -4 B \right )}{5 \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )+1\right )^{5}}-\frac {-4 A +2 B}{\left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )+1\right )^{2}}}{a^{3} f}\) | \(114\) |
default | \(\frac {-\frac {2 A}{\tan \left (\frac {f x}{2}+\frac {e}{2}\right )+1}-\frac {2 \left (8 A -6 B \right )}{3 \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )+1\right )^{3}}-\frac {-8 A +8 B}{2 \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )+1\right )^{4}}-\frac {2 \left (4 A -4 B \right )}{5 \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )+1\right )^{5}}-\frac {-4 A +2 B}{\left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )+1\right )^{2}}}{a^{3} f}\) | \(114\) |
norman | \(\frac {-\frac {14 A +6 B}{15 f a}-\frac {2 A \left (\tan ^{6}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{f a}-\frac {\left (94 A +36 B \right ) \left (\tan ^{2}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{15 f a}-\frac {\left (20 A +12 B \right ) \left (\tan ^{3}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{3 f a}-\frac {\left (22 A +6 B \right ) \left (\tan ^{4}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{3 f a}-\frac {\left (8 A +6 B \right ) \tan \left (\frac {f x}{2}+\frac {e}{2}\right )}{3 f a}-\frac {\left (4 A +2 B \right ) \left (\tan ^{5}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{f a}}{\left (1+\tan ^{2}\left (\frac {f x}{2}+\frac {e}{2}\right )\right ) a^{2} \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )+1\right )^{5}}\) | \(197\) |
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Time = 0.24 (sec) , antiderivative size = 190, normalized size of antiderivative = 1.86 \[ \int \frac {A+B \sin (e+f x)}{(a+a \sin (e+f x))^3} \, dx=-\frac {{\left (2 \, A + 3 \, B\right )} \cos \left (f x + e\right )^{3} - 2 \, {\left (2 \, A + 3 \, B\right )} \cos \left (f x + e\right )^{2} - 3 \, {\left (3 \, A + 2 \, B\right )} \cos \left (f x + e\right ) - {\left ({\left (2 \, A + 3 \, B\right )} \cos \left (f x + e\right )^{2} + 3 \, {\left (2 \, A + 3 \, B\right )} \cos \left (f x + e\right ) - 3 \, A + 3 \, B\right )} \sin \left (f x + e\right ) - 3 \, A + 3 \, B}{15 \, {\left (a^{3} f \cos \left (f x + e\right )^{3} + 3 \, a^{3} f \cos \left (f x + e\right )^{2} - 2 \, a^{3} f \cos \left (f x + e\right ) - 4 \, a^{3} f + {\left (a^{3} f \cos \left (f x + e\right )^{2} - 2 \, a^{3} f \cos \left (f x + e\right ) - 4 \, a^{3} f\right )} \sin \left (f x + e\right )\right )}} \]
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Leaf count of result is larger than twice the leaf count of optimal. 1015 vs. \(2 (87) = 174\).
Time = 2.37 (sec) , antiderivative size = 1015, normalized size of antiderivative = 9.95 \[ \int \frac {A+B \sin (e+f x)}{(a+a \sin (e+f x))^3} \, dx=\text {Too large to display} \]
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Leaf count of result is larger than twice the leaf count of optimal. 387 vs. \(2 (96) = 192\).
Time = 0.30 (sec) , antiderivative size = 387, normalized size of antiderivative = 3.79 \[ \int \frac {A+B \sin (e+f x)}{(a+a \sin (e+f x))^3} \, dx=-\frac {2 \, {\left (\frac {A {\left (\frac {20 \, \sin \left (f x + e\right )}{\cos \left (f x + e\right ) + 1} + \frac {40 \, \sin \left (f x + e\right )^{2}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{2}} + \frac {30 \, \sin \left (f x + e\right )^{3}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{3}} + \frac {15 \, \sin \left (f x + e\right )^{4}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{4}} + 7\right )}}{a^{3} + \frac {5 \, a^{3} \sin \left (f x + e\right )}{\cos \left (f x + e\right ) + 1} + \frac {10 \, a^{3} \sin \left (f x + e\right )^{2}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{2}} + \frac {10 \, a^{3} \sin \left (f x + e\right )^{3}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{3}} + \frac {5 \, a^{3} \sin \left (f x + e\right )^{4}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{4}} + \frac {a^{3} \sin \left (f x + e\right )^{5}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{5}}} + \frac {3 \, B {\left (\frac {5 \, \sin \left (f x + e\right )}{\cos \left (f x + e\right ) + 1} + \frac {5 \, \sin \left (f x + e\right )^{2}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{2}} + \frac {5 \, \sin \left (f x + e\right )^{3}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{3}} + 1\right )}}{a^{3} + \frac {5 \, a^{3} \sin \left (f x + e\right )}{\cos \left (f x + e\right ) + 1} + \frac {10 \, a^{3} \sin \left (f x + e\right )^{2}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{2}} + \frac {10 \, a^{3} \sin \left (f x + e\right )^{3}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{3}} + \frac {5 \, a^{3} \sin \left (f x + e\right )^{4}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{4}} + \frac {a^{3} \sin \left (f x + e\right )^{5}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{5}}}\right )}}{15 \, f} \]
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Time = 0.29 (sec) , antiderivative size = 122, normalized size of antiderivative = 1.20 \[ \int \frac {A+B \sin (e+f x)}{(a+a \sin (e+f x))^3} \, dx=-\frac {2 \, {\left (15 \, A \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{4} + 30 \, A \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{3} + 15 \, B \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{3} + 40 \, A \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{2} + 15 \, B \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{2} + 20 \, A \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right ) + 15 \, B \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right ) + 7 \, A + 3 \, B\right )}}{15 \, a^{3} f {\left (\tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right ) + 1\right )}^{5}} \]
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Time = 13.02 (sec) , antiderivative size = 150, normalized size of antiderivative = 1.47 \[ \int \frac {A+B \sin (e+f x)}{(a+a \sin (e+f x))^3} \, dx=\frac {2\,\cos \left (\frac {e}{2}+\frac {f\,x}{2}\right )\,\left (\frac {53\,A}{4}+3\,B-4\,A\,\cos \left (e+f\,x\right )+\frac {3\,B\,\cos \left (e+f\,x\right )}{2}+\frac {25\,A\,\sin \left (e+f\,x\right )}{2}+\frac {15\,B\,\sin \left (e+f\,x\right )}{2}-\frac {9\,A\,\cos \left (2\,e+2\,f\,x\right )}{4}-\frac {3\,B\,\cos \left (2\,e+2\,f\,x\right )}{2}-\frac {5\,A\,\sin \left (2\,e+2\,f\,x\right )}{4}\right )}{15\,a^3\,f\,\left (\frac {5\,\sqrt {2}\,\cos \left (\frac {3\,e}{2}+\frac {\pi }{4}+\frac {3\,f\,x}{2}\right )}{4}-\frac {5\,\sqrt {2}\,\cos \left (\frac {e}{2}-\frac {\pi }{4}+\frac {f\,x}{2}\right )}{2}+\frac {\sqrt {2}\,\cos \left (\frac {5\,e}{2}-\frac {\pi }{4}+\frac {5\,f\,x}{2}\right )}{4}\right )} \]
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