Integrand size = 21, antiderivative size = 160 \[ \int \sin ^2(e+f x) (a+b \sin (e+f x))^3 \, dx=\frac {1}{8} a \left (4 a^2+9 b^2\right ) x-\frac {b \left (15 a^2+4 b^2\right ) \cos (e+f x)}{5 f}+\frac {b \left (15 a^2+4 b^2\right ) \cos ^3(e+f x)}{15 f}-\frac {a \left (4 a^2+9 b^2\right ) \cos (e+f x) \sin (e+f x)}{8 f}-\frac {11 a b^2 \cos (e+f x) \sin ^3(e+f x)}{20 f}-\frac {b^2 \cos (e+f x) \sin ^3(e+f x) (a+b \sin (e+f x))}{5 f} \]
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Time = 0.15 (sec) , antiderivative size = 180, normalized size of antiderivative = 1.12, number of steps used = 4, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.143, Rules used = {2870, 2832, 2813} \[ \int \sin ^2(e+f x) (a+b \sin (e+f x))^3 \, dx=\frac {\left (3 a^2-16 b^2\right ) \cos (e+f x) (a+b \sin (e+f x))^2}{60 b f}+\frac {a \left (6 a^2-71 b^2\right ) \sin (e+f x) \cos (e+f x)}{120 f}+\frac {1}{8} a x \left (4 a^2+9 b^2\right )+\frac {\left (3 a^4-52 a^2 b^2-16 b^4\right ) \cos (e+f x)}{30 b f}-\frac {\cos (e+f x) (a+b \sin (e+f x))^4}{5 b f}+\frac {a \cos (e+f x) (a+b \sin (e+f x))^3}{20 b f} \]
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Rule 2813
Rule 2832
Rule 2870
Rubi steps \begin{align*} \text {integral}& = -\frac {\cos (e+f x) (a+b \sin (e+f x))^4}{5 b f}+\frac {\int (4 b-a \sin (e+f x)) (a+b \sin (e+f x))^3 \, dx}{5 b} \\ & = \frac {a \cos (e+f x) (a+b \sin (e+f x))^3}{20 b f}-\frac {\cos (e+f x) (a+b \sin (e+f x))^4}{5 b f}+\frac {\int (a+b \sin (e+f x))^2 \left (13 a b-\left (3 a^2-16 b^2\right ) \sin (e+f x)\right ) \, dx}{20 b} \\ & = \frac {\left (3 a^2-16 b^2\right ) \cos (e+f x) (a+b \sin (e+f x))^2}{60 b f}+\frac {a \cos (e+f x) (a+b \sin (e+f x))^3}{20 b f}-\frac {\cos (e+f x) (a+b \sin (e+f x))^4}{5 b f}+\frac {\int (a+b \sin (e+f x)) \left (b \left (33 a^2+32 b^2\right )-a \left (6 a^2-71 b^2\right ) \sin (e+f x)\right ) \, dx}{60 b} \\ & = \frac {1}{8} a \left (4 a^2+9 b^2\right ) x+\frac {\left (3 a^4-52 a^2 b^2-16 b^4\right ) \cos (e+f x)}{30 b f}+\frac {a \left (6 a^2-71 b^2\right ) \cos (e+f x) \sin (e+f x)}{120 f}+\frac {\left (3 a^2-16 b^2\right ) \cos (e+f x) (a+b \sin (e+f x))^2}{60 b f}+\frac {a \cos (e+f x) (a+b \sin (e+f x))^3}{20 b f}-\frac {\cos (e+f x) (a+b \sin (e+f x))^4}{5 b f} \\ \end{align*}
Time = 1.23 (sec) , antiderivative size = 117, normalized size of antiderivative = 0.73 \[ \int \sin ^2(e+f x) (a+b \sin (e+f x))^3 \, dx=\frac {-60 b \left (18 a^2+5 b^2\right ) \cos (e+f x)+10 \left (12 a^2 b+5 b^3\right ) \cos (3 (e+f x))-6 b^3 \cos (5 (e+f x))+15 a \left (4 \left (4 a^2+9 b^2\right ) (e+f x)-8 \left (a^2+3 b^2\right ) \sin (2 (e+f x))+3 b^2 \sin (4 (e+f x))\right )}{480 f} \]
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Time = 2.00 (sec) , antiderivative size = 124, normalized size of antiderivative = 0.78
method | result | size |
derivativedivides | \(\frac {a^{3} \left (-\frac {\sin \left (f x +e \right ) \cos \left (f x +e \right )}{2}+\frac {f x}{2}+\frac {e}{2}\right )-a^{2} b \left (2+\sin ^{2}\left (f x +e \right )\right ) \cos \left (f x +e \right )+3 a \,b^{2} \left (-\frac {\left (\sin ^{3}\left (f x +e \right )+\frac {3 \sin \left (f x +e \right )}{2}\right ) \cos \left (f x +e \right )}{4}+\frac {3 f x}{8}+\frac {3 e}{8}\right )-\frac {b^{3} \left (\frac {8}{3}+\sin ^{4}\left (f x +e \right )+\frac {4 \left (\sin ^{2}\left (f x +e \right )\right )}{3}\right ) \cos \left (f x +e \right )}{5}}{f}\) | \(124\) |
default | \(\frac {a^{3} \left (-\frac {\sin \left (f x +e \right ) \cos \left (f x +e \right )}{2}+\frac {f x}{2}+\frac {e}{2}\right )-a^{2} b \left (2+\sin ^{2}\left (f x +e \right )\right ) \cos \left (f x +e \right )+3 a \,b^{2} \left (-\frac {\left (\sin ^{3}\left (f x +e \right )+\frac {3 \sin \left (f x +e \right )}{2}\right ) \cos \left (f x +e \right )}{4}+\frac {3 f x}{8}+\frac {3 e}{8}\right )-\frac {b^{3} \left (\frac {8}{3}+\sin ^{4}\left (f x +e \right )+\frac {4 \left (\sin ^{2}\left (f x +e \right )\right )}{3}\right ) \cos \left (f x +e \right )}{5}}{f}\) | \(124\) |
parallelrisch | \(\frac {\left (120 a^{2} b +50 b^{3}\right ) \cos \left (3 f x +3 e \right )+\left (-120 a^{3}-360 a \,b^{2}\right ) \sin \left (2 f x +2 e \right )-6 b^{3} \cos \left (5 f x +5 e \right )+45 a \,b^{2} \sin \left (4 f x +4 e \right )+\left (-1080 a^{2} b -300 b^{3}\right ) \cos \left (f x +e \right )+240 a^{3} f x +540 a \,b^{2} f x -960 a^{2} b -256 b^{3}}{480 f}\) | \(125\) |
parts | \(\frac {a^{3} \left (-\frac {\sin \left (f x +e \right ) \cos \left (f x +e \right )}{2}+\frac {f x}{2}+\frac {e}{2}\right )}{f}-\frac {b^{3} \left (\frac {8}{3}+\sin ^{4}\left (f x +e \right )+\frac {4 \left (\sin ^{2}\left (f x +e \right )\right )}{3}\right ) \cos \left (f x +e \right )}{5 f}+\frac {3 a \,b^{2} \left (-\frac {\left (\sin ^{3}\left (f x +e \right )+\frac {3 \sin \left (f x +e \right )}{2}\right ) \cos \left (f x +e \right )}{4}+\frac {3 f x}{8}+\frac {3 e}{8}\right )}{f}-\frac {a^{2} b \left (2+\sin ^{2}\left (f x +e \right )\right ) \cos \left (f x +e \right )}{f}\) | \(132\) |
risch | \(\frac {a^{3} x}{2}+\frac {9 a \,b^{2} x}{8}-\frac {9 b \cos \left (f x +e \right ) a^{2}}{4 f}-\frac {5 b^{3} \cos \left (f x +e \right )}{8 f}-\frac {b^{3} \cos \left (5 f x +5 e \right )}{80 f}+\frac {3 a \,b^{2} \sin \left (4 f x +4 e \right )}{32 f}+\frac {b \cos \left (3 f x +3 e \right ) a^{2}}{4 f}+\frac {5 b^{3} \cos \left (3 f x +3 e \right )}{48 f}-\frac {\sin \left (2 f x +2 e \right ) a^{3}}{4 f}-\frac {3 \sin \left (2 f x +2 e \right ) a \,b^{2}}{4 f}\) | \(149\) |
norman | \(\frac {-\frac {60 a^{2} b +16 b^{3}}{15 f}+\frac {a \left (4 a^{2}+9 b^{2}\right ) x}{8}-\frac {12 a^{2} b \left (\tan ^{6}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{f}-\frac {2 \left (42 a^{2} b +16 b^{3}\right ) \left (\tan ^{4}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{3 f}-\frac {\left (60 a^{2} b +16 b^{3}\right ) \left (\tan ^{2}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{3 f}-\frac {a \left (4 a^{2}+9 b^{2}\right ) \tan \left (\frac {f x}{2}+\frac {e}{2}\right )}{4 f}+\frac {a \left (4 a^{2}+9 b^{2}\right ) \left (\tan ^{9}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{4 f}+\frac {5 a \left (4 a^{2}+9 b^{2}\right ) x \left (\tan ^{2}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{8}+\frac {5 a \left (4 a^{2}+9 b^{2}\right ) x \left (\tan ^{4}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{4}+\frac {5 a \left (4 a^{2}+9 b^{2}\right ) x \left (\tan ^{6}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{4}+\frac {5 a \left (4 a^{2}+9 b^{2}\right ) x \left (\tan ^{8}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{8}+\frac {a \left (4 a^{2}+9 b^{2}\right ) x \left (\tan ^{10}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{8}-\frac {a \left (4 a^{2}+21 b^{2}\right ) \left (\tan ^{3}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{2 f}+\frac {a \left (4 a^{2}+21 b^{2}\right ) \left (\tan ^{7}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{2 f}}{\left (1+\tan ^{2}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )^{5}}\) | \(366\) |
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Time = 0.31 (sec) , antiderivative size = 118, normalized size of antiderivative = 0.74 \[ \int \sin ^2(e+f x) (a+b \sin (e+f x))^3 \, dx=-\frac {24 \, b^{3} \cos \left (f x + e\right )^{5} - 40 \, {\left (3 \, a^{2} b + 2 \, b^{3}\right )} \cos \left (f x + e\right )^{3} - 15 \, {\left (4 \, a^{3} + 9 \, a b^{2}\right )} f x + 120 \, {\left (3 \, a^{2} b + b^{3}\right )} \cos \left (f x + e\right ) - 15 \, {\left (6 \, a b^{2} \cos \left (f x + e\right )^{3} - {\left (4 \, a^{3} + 15 \, a b^{2}\right )} \cos \left (f x + e\right )\right )} \sin \left (f x + e\right )}{120 \, f} \]
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Time = 0.26 (sec) , antiderivative size = 284, normalized size of antiderivative = 1.78 \[ \int \sin ^2(e+f x) (a+b \sin (e+f x))^3 \, dx=\begin {cases} \frac {a^{3} x \sin ^{2}{\left (e + f x \right )}}{2} + \frac {a^{3} x \cos ^{2}{\left (e + f x \right )}}{2} - \frac {a^{3} \sin {\left (e + f x \right )} \cos {\left (e + f x \right )}}{2 f} - \frac {3 a^{2} b \sin ^{2}{\left (e + f x \right )} \cos {\left (e + f x \right )}}{f} - \frac {2 a^{2} b \cos ^{3}{\left (e + f x \right )}}{f} + \frac {9 a b^{2} x \sin ^{4}{\left (e + f x \right )}}{8} + \frac {9 a b^{2} x \sin ^{2}{\left (e + f x \right )} \cos ^{2}{\left (e + f x \right )}}{4} + \frac {9 a b^{2} x \cos ^{4}{\left (e + f x \right )}}{8} - \frac {15 a b^{2} \sin ^{3}{\left (e + f x \right )} \cos {\left (e + f x \right )}}{8 f} - \frac {9 a b^{2} \sin {\left (e + f x \right )} \cos ^{3}{\left (e + f x \right )}}{8 f} - \frac {b^{3} \sin ^{4}{\left (e + f x \right )} \cos {\left (e + f x \right )}}{f} - \frac {4 b^{3} \sin ^{2}{\left (e + f x \right )} \cos ^{3}{\left (e + f x \right )}}{3 f} - \frac {8 b^{3} \cos ^{5}{\left (e + f x \right )}}{15 f} & \text {for}\: f \neq 0 \\x \left (a + b \sin {\left (e \right )}\right )^{3} \sin ^{2}{\left (e \right )} & \text {otherwise} \end {cases} \]
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Time = 0.20 (sec) , antiderivative size = 121, normalized size of antiderivative = 0.76 \[ \int \sin ^2(e+f x) (a+b \sin (e+f x))^3 \, dx=\frac {120 \, {\left (2 \, f x + 2 \, e - \sin \left (2 \, f x + 2 \, e\right )\right )} a^{3} + 480 \, {\left (\cos \left (f x + e\right )^{3} - 3 \, \cos \left (f x + e\right )\right )} a^{2} b + 45 \, {\left (12 \, f x + 12 \, e + \sin \left (4 \, f x + 4 \, e\right ) - 8 \, \sin \left (2 \, f x + 2 \, e\right )\right )} a b^{2} - 32 \, {\left (3 \, \cos \left (f x + e\right )^{5} - 10 \, \cos \left (f x + e\right )^{3} + 15 \, \cos \left (f x + e\right )\right )} b^{3}}{480 \, f} \]
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Time = 0.30 (sec) , antiderivative size = 124, normalized size of antiderivative = 0.78 \[ \int \sin ^2(e+f x) (a+b \sin (e+f x))^3 \, dx=-\frac {b^{3} \cos \left (5 \, f x + 5 \, e\right )}{80 \, f} + \frac {3 \, a b^{2} \sin \left (4 \, f x + 4 \, e\right )}{32 \, f} + \frac {1}{8} \, {\left (4 \, a^{3} + 9 \, a b^{2}\right )} x + \frac {{\left (12 \, a^{2} b + 5 \, b^{3}\right )} \cos \left (3 \, f x + 3 \, e\right )}{48 \, f} - \frac {{\left (18 \, a^{2} b + 5 \, b^{3}\right )} \cos \left (f x + e\right )}{8 \, f} - \frac {{\left (a^{3} + 3 \, a b^{2}\right )} \sin \left (2 \, f x + 2 \, e\right )}{4 \, f} \]
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Time = 7.70 (sec) , antiderivative size = 328, normalized size of antiderivative = 2.05 \[ \int \sin ^2(e+f x) (a+b \sin (e+f x))^3 \, dx=\frac {a\,\mathrm {atan}\left (\frac {a\,\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )\,\left (4\,a^2+9\,b^2\right )}{4\,\left (a^3+\frac {9\,a\,b^2}{4}\right )}\right )\,\left (4\,a^2+9\,b^2\right )}{4\,f}-\frac {4\,a^2\,b-{\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^9\,\left (a^3+\frac {9\,a\,b^2}{4}\right )+{\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^3\,\left (2\,a^3+\frac {21\,a\,b^2}{2}\right )-{\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^7\,\left (2\,a^3+\frac {21\,a\,b^2}{2}\right )+{\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^2\,\left (20\,a^2\,b+\frac {16\,b^3}{3}\right )+{\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^4\,\left (28\,a^2\,b+\frac {32\,b^3}{3}\right )+\frac {16\,b^3}{15}+\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )\,\left (a^3+\frac {9\,a\,b^2}{4}\right )+12\,a^2\,b\,{\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^6}{f\,\left ({\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^{10}+5\,{\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^8+10\,{\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^6+10\,{\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^4+5\,{\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^2+1\right )}-\frac {a\,\left (4\,a^2+9\,b^2\right )\,\left (\mathrm {atan}\left (\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )\right )-\frac {f\,x}{2}\right )}{4\,f} \]
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