Integrand size = 36, antiderivative size = 109 \[ \int \frac {(a+a \sin (e+f x))^2 (A+B \sin (e+f x))}{(c-c \sin (e+f x))^2} \, dx=\frac {a^2 (A+4 B) x}{c^2}-\frac {a^2 (A+4 B) \cos (e+f x)}{c^2 f}+\frac {a^2 (A+B) c^2 \cos ^5(e+f x)}{3 f (c-c \sin (e+f x))^4}-\frac {2 a^2 (A+4 B) \cos ^3(e+f x)}{3 f (c-c \sin (e+f x))^2} \]
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Time = 0.19 (sec) , antiderivative size = 109, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.139, Rules used = {3046, 2938, 2759, 2761, 8} \[ \int \frac {(a+a \sin (e+f x))^2 (A+B \sin (e+f x))}{(c-c \sin (e+f x))^2} \, dx=-\frac {a^2 (A+4 B) \cos (e+f x)}{c^2 f}+\frac {a^2 c^2 (A+B) \cos ^5(e+f x)}{3 f (c-c \sin (e+f x))^4}+\frac {a^2 x (A+4 B)}{c^2}-\frac {2 a^2 (A+4 B) \cos ^3(e+f x)}{3 f (c-c \sin (e+f x))^2} \]
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Rule 8
Rule 2759
Rule 2761
Rule 2938
Rule 3046
Rubi steps \begin{align*} \text {integral}& = \left (a^2 c^2\right ) \int \frac {\cos ^4(e+f x) (A+B \sin (e+f x))}{(c-c \sin (e+f x))^4} \, dx \\ & = \frac {a^2 (A+B) c^2 \cos ^5(e+f x)}{3 f (c-c \sin (e+f x))^4}-\frac {1}{3} \left (a^2 (A+4 B) c\right ) \int \frac {\cos ^4(e+f x)}{(c-c \sin (e+f x))^3} \, dx \\ & = \frac {a^2 (A+B) c^2 \cos ^5(e+f x)}{3 f (c-c \sin (e+f x))^4}-\frac {2 a^2 (A+4 B) \cos ^3(e+f x)}{3 f (c-c \sin (e+f x))^2}+\frac {\left (a^2 (A+4 B)\right ) \int \frac {\cos ^2(e+f x)}{c-c \sin (e+f x)} \, dx}{c} \\ & = -\frac {a^2 (A+4 B) \cos (e+f x)}{c^2 f}+\frac {a^2 (A+B) c^2 \cos ^5(e+f x)}{3 f (c-c \sin (e+f x))^4}-\frac {2 a^2 (A+4 B) \cos ^3(e+f x)}{3 f (c-c \sin (e+f x))^2}+\frac {\left (a^2 (A+4 B)\right ) \int 1 \, dx}{c^2} \\ & = \frac {a^2 (A+4 B) x}{c^2}-\frac {a^2 (A+4 B) \cos (e+f x)}{c^2 f}+\frac {a^2 (A+B) c^2 \cos ^5(e+f x)}{3 f (c-c \sin (e+f x))^4}-\frac {2 a^2 (A+4 B) \cos ^3(e+f x)}{3 f (c-c \sin (e+f x))^2} \\ \end{align*}
Leaf count is larger than twice the leaf count of optimal. \(238\) vs. \(2(109)=218\).
Time = 11.23 (sec) , antiderivative size = 238, normalized size of antiderivative = 2.18 \[ \int \frac {(a+a \sin (e+f x))^2 (A+B \sin (e+f x))}{(c-c \sin (e+f x))^2} \, dx=\frac {a^2 \left (\cos \left (\frac {1}{2} (e+f x)\right )-\sin \left (\frac {1}{2} (e+f x)\right )\right ) \left (4 (A+B) \left (\cos \left (\frac {1}{2} (e+f x)\right )-\sin \left (\frac {1}{2} (e+f x)\right )\right )+3 (A+4 B) (e+f x) \left (\cos \left (\frac {1}{2} (e+f x)\right )-\sin \left (\frac {1}{2} (e+f x)\right )\right )^3-3 B \cos (e+f x) \left (\cos \left (\frac {1}{2} (e+f x)\right )-\sin \left (\frac {1}{2} (e+f x)\right )\right )^3+8 (A+B) \sin \left (\frac {1}{2} (e+f x)\right )-8 (2 A+5 B) \left (\cos \left (\frac {1}{2} (e+f x)\right )-\sin \left (\frac {1}{2} (e+f x)\right )\right )^2 \sin \left (\frac {1}{2} (e+f x)\right )\right ) (1+\sin (e+f x))^2}{3 f \left (\cos \left (\frac {1}{2} (e+f x)\right )+\sin \left (\frac {1}{2} (e+f x)\right )\right )^4 (c-c \sin (e+f x))^2} \]
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Time = 0.77 (sec) , antiderivative size = 107, normalized size of antiderivative = 0.98
method | result | size |
derivativedivides | \(\frac {2 a^{2} \left (-\frac {B}{1+\tan ^{2}\left (\frac {f x}{2}+\frac {e}{2}\right )}+\left (A +4 B \right ) \arctan \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )\right )-\frac {8 A +8 B}{3 \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )-1\right )^{3}}-\frac {8 A +8 B}{2 \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )-1\right )^{2}}+\frac {4 B}{\tan \left (\frac {f x}{2}+\frac {e}{2}\right )-1}\right )}{f \,c^{2}}\) | \(107\) |
default | \(\frac {2 a^{2} \left (-\frac {B}{1+\tan ^{2}\left (\frac {f x}{2}+\frac {e}{2}\right )}+\left (A +4 B \right ) \arctan \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )\right )-\frac {8 A +8 B}{3 \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )-1\right )^{3}}-\frac {8 A +8 B}{2 \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )-1\right )^{2}}+\frac {4 B}{\tan \left (\frac {f x}{2}+\frac {e}{2}\right )-1}\right )}{f \,c^{2}}\) | \(107\) |
risch | \(\frac {a^{2} x A}{c^{2}}+\frac {4 a^{2} x B}{c^{2}}-\frac {B \,a^{2} {\mathrm e}^{i \left (f x +e \right )}}{2 c^{2} f}-\frac {B \,a^{2} {\mathrm e}^{-i \left (f x +e \right )}}{2 c^{2} f}-\frac {8 \left (-3 i A \,a^{2} {\mathrm e}^{i \left (f x +e \right )}+3 A \,a^{2} {\mathrm e}^{2 i \left (f x +e \right )}-9 i B \,a^{2} {\mathrm e}^{i \left (f x +e \right )}+6 B \,a^{2} {\mathrm e}^{2 i \left (f x +e \right )}-2 A \,a^{2}-5 B \,a^{2}\right )}{3 \left ({\mathrm e}^{i \left (f x +e \right )}-i\right )^{3} f \,c^{2}}\) | \(160\) |
parallelrisch | \(\frac {3 \left (\left (4 \left (-f x +2\right ) B -f x A +\frac {4 A}{3}\right ) \cos \left (\frac {f x}{2}+\frac {e}{2}\right )+\frac {\left (\left (\frac {11}{6}+4 f x \right ) B +f x A +\frac {4 A}{3}\right ) \cos \left (\frac {3 f x}{2}+\frac {3 e}{2}\right )}{3}+\left (f x A +4 f x B -\frac {4}{3} A -\frac {14}{3} B \right ) \sin \left (\frac {f x}{2}+\frac {e}{2}\right )+\frac {\left (f x A +4 f x B -4 A -\frac {29}{2} B \right ) \sin \left (\frac {3 f x}{2}+\frac {3 e}{2}\right )}{3}-\frac {B \left (\cos \left (\frac {5 f x}{2}+\frac {5 e}{2}\right )+\sin \left (\frac {5 f x}{2}+\frac {5 e}{2}\right )\right )}{6}\right ) a^{2}}{f \,c^{2} \left (-3 \cos \left (\frac {f x}{2}+\frac {e}{2}\right )+\cos \left (\frac {3 f x}{2}+\frac {3 e}{2}\right )+\sin \left (\frac {3 f x}{2}+\frac {3 e}{2}\right )+3 \sin \left (\frac {f x}{2}+\frac {e}{2}\right )\right )}\) | \(186\) |
norman | \(\frac {\frac {8 B \,a^{2} \left (\tan ^{8}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{c f}+\frac {a^{2} \left (A +4 B \right ) x \left (\tan ^{9}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{c}+\frac {8 A \,a^{2}+38 B \,a^{2}}{3 c f}-\frac {a^{2} \left (A +4 B \right ) x}{c}-\frac {2 \left (4 A \,a^{2}+13 B \,a^{2}\right ) \left (\tan ^{7}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{c f}+\frac {2 \left (4 A \,a^{2}+25 B \,a^{2}\right ) \left (\tan ^{2}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{c f}+\frac {2 \left (4 A \,a^{2}+35 B \,a^{2}\right ) \left (\tan ^{4}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{c f}+\frac {2 \left (4 A \,a^{2}+61 B \,a^{2}\right ) \left (\tan ^{6}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{3 c f}-\frac {\left (8 A \,a^{2}+30 B \,a^{2}\right ) \tan \left (\frac {f x}{2}+\frac {e}{2}\right )}{c f}-\frac {2 \left (12 A \,a^{2}+41 B \,a^{2}\right ) \left (\tan ^{5}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{c f}-\frac {2 \left (12 A \,a^{2}+43 B \,a^{2}\right ) \left (\tan ^{3}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{c f}+\frac {3 a^{2} \left (A +4 B \right ) x \tan \left (\frac {f x}{2}+\frac {e}{2}\right )}{c}-\frac {6 a^{2} \left (A +4 B \right ) x \left (\tan ^{2}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{c}+\frac {10 a^{2} \left (A +4 B \right ) x \left (\tan ^{3}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{c}-\frac {12 a^{2} \left (A +4 B \right ) x \left (\tan ^{4}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{c}+\frac {12 a^{2} \left (A +4 B \right ) x \left (\tan ^{5}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{c}-\frac {10 a^{2} \left (A +4 B \right ) x \left (\tan ^{6}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{c}+\frac {6 a^{2} \left (A +4 B \right ) x \left (\tan ^{7}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{c}-\frac {3 a^{2} \left (A +4 B \right ) x \left (\tan ^{8}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{c}}{\left (1+\tan ^{2}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )^{3} c \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )-1\right )^{3}}\) | \(536\) |
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Leaf count of result is larger than twice the leaf count of optimal. 237 vs. \(2 (107) = 214\).
Time = 0.27 (sec) , antiderivative size = 237, normalized size of antiderivative = 2.17 \[ \int \frac {(a+a \sin (e+f x))^2 (A+B \sin (e+f x))}{(c-c \sin (e+f x))^2} \, dx=-\frac {3 \, B a^{2} \cos \left (f x + e\right )^{3} + 6 \, {\left (A + 4 \, B\right )} a^{2} f x + 4 \, {\left (A + B\right )} a^{2} - {\left (3 \, {\left (A + 4 \, B\right )} a^{2} f x + {\left (8 \, A + 23 \, B\right )} a^{2}\right )} \cos \left (f x + e\right )^{2} + {\left (3 \, {\left (A + 4 \, B\right )} a^{2} f x - 2 \, {\left (2 \, A + 11 \, B\right )} a^{2}\right )} \cos \left (f x + e\right ) - {\left (6 \, {\left (A + 4 \, B\right )} a^{2} f x - 3 \, B a^{2} \cos \left (f x + e\right )^{2} - 4 \, {\left (A + B\right )} a^{2} + {\left (3 \, {\left (A + 4 \, B\right )} a^{2} f x - 2 \, {\left (4 \, A + 13 \, B\right )} a^{2}\right )} \cos \left (f x + e\right )\right )} \sin \left (f x + e\right )}{3 \, {\left (c^{2} f \cos \left (f x + e\right )^{2} - c^{2} f \cos \left (f x + e\right ) - 2 \, c^{2} f + {\left (c^{2} f \cos \left (f x + e\right ) + 2 \, c^{2} f\right )} \sin \left (f x + e\right )\right )}} \]
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Leaf count of result is larger than twice the leaf count of optimal. 2474 vs. \(2 (100) = 200\).
Time = 3.93 (sec) , antiderivative size = 2474, normalized size of antiderivative = 22.70 \[ \int \frac {(a+a \sin (e+f x))^2 (A+B \sin (e+f x))}{(c-c \sin (e+f x))^2} \, dx=\text {Too large to display} \]
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Leaf count of result is larger than twice the leaf count of optimal. 839 vs. \(2 (107) = 214\).
Time = 0.31 (sec) , antiderivative size = 839, normalized size of antiderivative = 7.70 \[ \int \frac {(a+a \sin (e+f x))^2 (A+B \sin (e+f x))}{(c-c \sin (e+f x))^2} \, dx=\text {Too large to display} \]
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Time = 0.32 (sec) , antiderivative size = 129, normalized size of antiderivative = 1.18 \[ \int \frac {(a+a \sin (e+f x))^2 (A+B \sin (e+f x))}{(c-c \sin (e+f x))^2} \, dx=\frac {\frac {3 \, {\left (A a^{2} + 4 \, B a^{2}\right )} {\left (f x + e\right )}}{c^{2}} - \frac {6 \, B a^{2}}{{\left (\tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{2} + 1\right )} c^{2}} + \frac {8 \, {\left (3 \, B a^{2} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{2} - 3 \, A a^{2} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right ) - 9 \, B a^{2} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right ) + A a^{2} + 4 \, B a^{2}\right )}}{c^{2} {\left (\tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right ) - 1\right )}^{3}}}{3 \, f} \]
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Time = 15.25 (sec) , antiderivative size = 246, normalized size of antiderivative = 2.26 \[ \int \frac {(a+a \sin (e+f x))^2 (A+B \sin (e+f x))}{(c-c \sin (e+f x))^2} \, dx=\frac {2\,a^2\,\mathrm {atan}\left (\frac {2\,a^2\,\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )\,\left (A+4\,B\right )}{2\,A\,a^2+8\,B\,a^2}\right )\,\left (A+4\,B\right )}{c^2\,f}-\frac {\frac {8\,A\,a^2}{3}-\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )\,\left (8\,A\,a^2+30\,B\,a^2\right )+\frac {38\,B\,a^2}{3}-{\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^3\,\left (8\,A\,a^2+26\,B\,a^2\right )+{\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^2\,\left (\frac {8\,A\,a^2}{3}+\frac {74\,B\,a^2}{3}\right )+8\,B\,a^2\,{\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^4}{f\,\left (-c^2\,{\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^5+3\,c^2\,{\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^4-4\,c^2\,{\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^3+4\,c^2\,{\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^2-3\,c^2\,\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )+c^2\right )} \]
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