Integrand size = 32, antiderivative size = 96 \[ \int \sin ^2(e+f x) (a+a \sin (e+f x))^2 (c-c \sin (e+f x)) \, dx=\frac {1}{8} a^2 c x-\frac {a^2 c \cos ^3(e+f x)}{3 f}+\frac {a^2 c \cos ^5(e+f x)}{5 f}-\frac {a^2 c \cos (e+f x) \sin (e+f x)}{8 f}+\frac {a^2 c \cos (e+f x) \sin ^3(e+f x)}{4 f} \]
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Time = 0.11 (sec) , antiderivative size = 96, normalized size of antiderivative = 1.00, number of steps used = 11, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.125, Rules used = {3045, 2715, 8, 2713} \[ \int \sin ^2(e+f x) (a+a \sin (e+f x))^2 (c-c \sin (e+f x)) \, dx=\frac {a^2 c \cos ^5(e+f x)}{5 f}-\frac {a^2 c \cos ^3(e+f x)}{3 f}+\frac {a^2 c \sin ^3(e+f x) \cos (e+f x)}{4 f}-\frac {a^2 c \sin (e+f x) \cos (e+f x)}{8 f}+\frac {1}{8} a^2 c x \]
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Rule 8
Rule 2713
Rule 2715
Rule 3045
Rubi steps \begin{align*} \text {integral}& = \int \left (a^2 c \sin ^2(e+f x)+a^2 c \sin ^3(e+f x)-a^2 c \sin ^4(e+f x)-a^2 c \sin ^5(e+f x)\right ) \, dx \\ & = \left (a^2 c\right ) \int \sin ^2(e+f x) \, dx+\left (a^2 c\right ) \int \sin ^3(e+f x) \, dx-\left (a^2 c\right ) \int \sin ^4(e+f x) \, dx-\left (a^2 c\right ) \int \sin ^5(e+f x) \, dx \\ & = -\frac {a^2 c \cos (e+f x) \sin (e+f x)}{2 f}+\frac {a^2 c \cos (e+f x) \sin ^3(e+f x)}{4 f}+\frac {1}{2} \left (a^2 c\right ) \int 1 \, dx-\frac {1}{4} \left (3 a^2 c\right ) \int \sin ^2(e+f x) \, dx-\frac {\left (a^2 c\right ) \text {Subst}\left (\int \left (1-x^2\right ) \, dx,x,\cos (e+f x)\right )}{f}+\frac {\left (a^2 c\right ) \text {Subst}\left (\int \left (1-2 x^2+x^4\right ) \, dx,x,\cos (e+f x)\right )}{f} \\ & = \frac {1}{2} a^2 c x-\frac {a^2 c \cos ^3(e+f x)}{3 f}+\frac {a^2 c \cos ^5(e+f x)}{5 f}-\frac {a^2 c \cos (e+f x) \sin (e+f x)}{8 f}+\frac {a^2 c \cos (e+f x) \sin ^3(e+f x)}{4 f}-\frac {1}{8} \left (3 a^2 c\right ) \int 1 \, dx \\ & = \frac {1}{8} a^2 c x-\frac {a^2 c \cos ^3(e+f x)}{3 f}+\frac {a^2 c \cos ^5(e+f x)}{5 f}-\frac {a^2 c \cos (e+f x) \sin (e+f x)}{8 f}+\frac {a^2 c \cos (e+f x) \sin ^3(e+f x)}{4 f} \\ \end{align*}
Time = 0.13 (sec) , antiderivative size = 57, normalized size of antiderivative = 0.59 \[ \int \sin ^2(e+f x) (a+a \sin (e+f x))^2 (c-c \sin (e+f x)) \, dx=\frac {a^2 c (60 e+60 f x-60 \cos (e+f x)-10 \cos (3 (e+f x))+6 \cos (5 (e+f x))-15 \sin (4 (e+f x)))}{480 f} \]
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Time = 1.06 (sec) , antiderivative size = 57, normalized size of antiderivative = 0.59
method | result | size |
parallelrisch | \(-\frac {a^{2} c \left (-60 f x +60 \cos \left (f x +e \right )-6 \cos \left (5 f x +5 e \right )+15 \sin \left (4 f x +4 e \right )+10 \cos \left (3 f x +3 e \right )+64\right )}{480 f}\) | \(57\) |
risch | \(\frac {a^{2} c x}{8}-\frac {a^{2} c \cos \left (f x +e \right )}{8 f}+\frac {a^{2} c \cos \left (5 f x +5 e \right )}{80 f}-\frac {a^{2} c \sin \left (4 f x +4 e \right )}{32 f}-\frac {a^{2} c \cos \left (3 f x +3 e \right )}{48 f}\) | \(78\) |
derivativedivides | \(\frac {\frac {a^{2} c \left (\frac {8}{3}+\sin ^{4}\left (f x +e \right )+\frac {4 \left (\sin ^{2}\left (f x +e \right )\right )}{3}\right ) \cos \left (f x +e \right )}{5}-a^{2} c \left (-\frac {\left (\sin ^{3}\left (f x +e \right )+\frac {3 \sin \left (f x +e \right )}{2}\right ) \cos \left (f x +e \right )}{4}+\frac {3 f x}{8}+\frac {3 e}{8}\right )-\frac {a^{2} c \left (2+\sin ^{2}\left (f x +e \right )\right ) \cos \left (f x +e \right )}{3}+a^{2} c \left (-\frac {\cos \left (f x +e \right ) \sin \left (f x +e \right )}{2}+\frac {f x}{2}+\frac {e}{2}\right )}{f}\) | \(126\) |
default | \(\frac {\frac {a^{2} c \left (\frac {8}{3}+\sin ^{4}\left (f x +e \right )+\frac {4 \left (\sin ^{2}\left (f x +e \right )\right )}{3}\right ) \cos \left (f x +e \right )}{5}-a^{2} c \left (-\frac {\left (\sin ^{3}\left (f x +e \right )+\frac {3 \sin \left (f x +e \right )}{2}\right ) \cos \left (f x +e \right )}{4}+\frac {3 f x}{8}+\frac {3 e}{8}\right )-\frac {a^{2} c \left (2+\sin ^{2}\left (f x +e \right )\right ) \cos \left (f x +e \right )}{3}+a^{2} c \left (-\frac {\cos \left (f x +e \right ) \sin \left (f x +e \right )}{2}+\frac {f x}{2}+\frac {e}{2}\right )}{f}\) | \(126\) |
parts | \(\frac {a^{2} c \left (-\frac {\cos \left (f x +e \right ) \sin \left (f x +e \right )}{2}+\frac {f x}{2}+\frac {e}{2}\right )}{f}-\frac {a^{2} c \left (2+\sin ^{2}\left (f x +e \right )\right ) \cos \left (f x +e \right )}{3 f}-\frac {a^{2} c \left (-\frac {\left (\sin ^{3}\left (f x +e \right )+\frac {3 \sin \left (f x +e \right )}{2}\right ) \cos \left (f x +e \right )}{4}+\frac {3 f x}{8}+\frac {3 e}{8}\right )}{f}+\frac {a^{2} c \left (\frac {8}{3}+\sin ^{4}\left (f x +e \right )+\frac {4 \left (\sin ^{2}\left (f x +e \right )\right )}{3}\right ) \cos \left (f x +e \right )}{5 f}\) | \(134\) |
norman | \(\frac {-\frac {4 a^{2} c}{15 f}+\frac {a^{2} c x}{8}-\frac {4 a^{2} c \left (\tan ^{6}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{f}+\frac {4 a^{2} c \left (\tan ^{4}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{3 f}-\frac {4 a^{2} c \left (\tan ^{2}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{3 f}-\frac {a^{2} c \tan \left (\frac {f x}{2}+\frac {e}{2}\right )}{4 f}+\frac {3 a^{2} c \left (\tan ^{3}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{2 f}-\frac {3 a^{2} c \left (\tan ^{7}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{2 f}+\frac {a^{2} c \left (\tan ^{9}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{4 f}+\frac {5 a^{2} c x \left (\tan ^{2}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{8}+\frac {5 a^{2} c x \left (\tan ^{4}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{4}+\frac {5 a^{2} c x \left (\tan ^{6}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{4}+\frac {5 a^{2} c x \left (\tan ^{8}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{8}+\frac {a^{2} c x \left (\tan ^{10}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{8}}{\left (1+\tan ^{2}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )^{5}}\) | \(262\) |
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Time = 0.27 (sec) , antiderivative size = 77, normalized size of antiderivative = 0.80 \[ \int \sin ^2(e+f x) (a+a \sin (e+f x))^2 (c-c \sin (e+f x)) \, dx=\frac {24 \, a^{2} c \cos \left (f x + e\right )^{5} - 40 \, a^{2} c \cos \left (f x + e\right )^{3} + 15 \, a^{2} c f x - 15 \, {\left (2 \, a^{2} c \cos \left (f x + e\right )^{3} - a^{2} c \cos \left (f x + e\right )\right )} \sin \left (f x + e\right )}{120 \, f} \]
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Leaf count of result is larger than twice the leaf count of optimal. 301 vs. \(2 (87) = 174\).
Time = 0.26 (sec) , antiderivative size = 301, normalized size of antiderivative = 3.14 \[ \int \sin ^2(e+f x) (a+a \sin (e+f x))^2 (c-c \sin (e+f x)) \, dx=\begin {cases} - \frac {3 a^{2} c x \sin ^{4}{\left (e + f x \right )}}{8} - \frac {3 a^{2} c x \sin ^{2}{\left (e + f x \right )} \cos ^{2}{\left (e + f x \right )}}{4} + \frac {a^{2} c x \sin ^{2}{\left (e + f x \right )}}{2} - \frac {3 a^{2} c x \cos ^{4}{\left (e + f x \right )}}{8} + \frac {a^{2} c x \cos ^{2}{\left (e + f x \right )}}{2} + \frac {a^{2} c \sin ^{4}{\left (e + f x \right )} \cos {\left (e + f x \right )}}{f} + \frac {5 a^{2} c \sin ^{3}{\left (e + f x \right )} \cos {\left (e + f x \right )}}{8 f} + \frac {4 a^{2} c \sin ^{2}{\left (e + f x \right )} \cos ^{3}{\left (e + f x \right )}}{3 f} - \frac {a^{2} c \sin ^{2}{\left (e + f x \right )} \cos {\left (e + f x \right )}}{f} + \frac {3 a^{2} c \sin {\left (e + f x \right )} \cos ^{3}{\left (e + f x \right )}}{8 f} - \frac {a^{2} c \sin {\left (e + f x \right )} \cos {\left (e + f x \right )}}{2 f} + \frac {8 a^{2} c \cos ^{5}{\left (e + f x \right )}}{15 f} - \frac {2 a^{2} c \cos ^{3}{\left (e + f x \right )}}{3 f} & \text {for}\: f \neq 0 \\x \left (a \sin {\left (e \right )} + a\right )^{2} \left (- c \sin {\left (e \right )} + c\right ) \sin ^{2}{\left (e \right )} & \text {otherwise} \end {cases} \]
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Time = 0.20 (sec) , antiderivative size = 123, normalized size of antiderivative = 1.28 \[ \int \sin ^2(e+f x) (a+a \sin (e+f x))^2 (c-c \sin (e+f x)) \, dx=\frac {32 \, {\left (3 \, \cos \left (f x + e\right )^{5} - 10 \, \cos \left (f x + e\right )^{3} + 15 \, \cos \left (f x + e\right )\right )} a^{2} c + 160 \, {\left (\cos \left (f x + e\right )^{3} - 3 \, \cos \left (f x + e\right )\right )} a^{2} c - 15 \, {\left (12 \, f x + 12 \, e + \sin \left (4 \, f x + 4 \, e\right ) - 8 \, \sin \left (2 \, f x + 2 \, e\right )\right )} a^{2} c + 120 \, {\left (2 \, f x + 2 \, e - \sin \left (2 \, f x + 2 \, e\right )\right )} a^{2} c}{480 \, f} \]
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Time = 0.38 (sec) , antiderivative size = 77, normalized size of antiderivative = 0.80 \[ \int \sin ^2(e+f x) (a+a \sin (e+f x))^2 (c-c \sin (e+f x)) \, dx=\frac {1}{8} \, a^{2} c x + \frac {a^{2} c \cos \left (5 \, f x + 5 \, e\right )}{80 \, f} - \frac {a^{2} c \cos \left (3 \, f x + 3 \, e\right )}{48 \, f} - \frac {a^{2} c \cos \left (f x + e\right )}{8 \, f} - \frac {a^{2} c \sin \left (4 \, f x + 4 \, e\right )}{32 \, f} \]
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Time = 14.24 (sec) , antiderivative size = 212, normalized size of antiderivative = 2.21 \[ \int \sin ^2(e+f x) (a+a \sin (e+f x))^2 (c-c \sin (e+f x)) \, dx=\frac {a^2\,c\,\left (15\,e-30\,\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )-160\,{\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^2+180\,{\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^3+160\,{\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^4-480\,{\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^6-180\,{\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^7+30\,{\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^9+15\,f\,x+75\,{\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^2\,\left (e+f\,x\right )+150\,{\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^4\,\left (e+f\,x\right )+150\,{\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^6\,\left (e+f\,x\right )+75\,{\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^8\,\left (e+f\,x\right )+15\,{\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^{10}\,\left (e+f\,x\right )-32\right )}{120\,f\,{\left ({\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^2+1\right )}^5} \]
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