Integrand size = 23, antiderivative size = 127 \[ \int (a+a \sin (e+f x))^3 (A+B \sin (e+f x)) \, dx=\frac {5}{8} a^3 (4 A+3 B) x-\frac {5 a^3 (4 A+3 B) \cos (e+f x)}{6 f}-\frac {5 a^3 (4 A+3 B) \cos (e+f x) \sin (e+f x)}{24 f}-\frac {a (4 A+3 B) \cos (e+f x) (a+a \sin (e+f x))^2}{12 f}-\frac {B \cos (e+f x) (a+a \sin (e+f x))^3}{4 f} \]
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Time = 0.07 (sec) , antiderivative size = 117, normalized size of antiderivative = 0.92, number of steps used = 8, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.261, Rules used = {2830, 2724, 2718, 2715, 8, 2713} \[ \int (a+a \sin (e+f x))^3 (A+B \sin (e+f x)) \, dx=\frac {a^3 (4 A+3 B) \cos ^3(e+f x)}{12 f}-\frac {a^3 (4 A+3 B) \cos (e+f x)}{f}-\frac {3 a^3 (4 A+3 B) \sin (e+f x) \cos (e+f x)}{8 f}+\frac {5}{8} a^3 x (4 A+3 B)-\frac {B \cos (e+f x) (a \sin (e+f x)+a)^3}{4 f} \]
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Rule 8
Rule 2713
Rule 2715
Rule 2718
Rule 2724
Rule 2830
Rubi steps \begin{align*} \text {integral}& = -\frac {B \cos (e+f x) (a+a \sin (e+f x))^3}{4 f}+\frac {1}{4} (4 A+3 B) \int (a+a \sin (e+f x))^3 \, dx \\ & = -\frac {B \cos (e+f x) (a+a \sin (e+f x))^3}{4 f}+\frac {1}{4} (4 A+3 B) \int \left (a^3+3 a^3 \sin (e+f x)+3 a^3 \sin ^2(e+f x)+a^3 \sin ^3(e+f x)\right ) \, dx \\ & = \frac {1}{4} a^3 (4 A+3 B) x-\frac {B \cos (e+f x) (a+a \sin (e+f x))^3}{4 f}+\frac {1}{4} \left (a^3 (4 A+3 B)\right ) \int \sin ^3(e+f x) \, dx+\frac {1}{4} \left (3 a^3 (4 A+3 B)\right ) \int \sin (e+f x) \, dx+\frac {1}{4} \left (3 a^3 (4 A+3 B)\right ) \int \sin ^2(e+f x) \, dx \\ & = \frac {1}{4} a^3 (4 A+3 B) x-\frac {3 a^3 (4 A+3 B) \cos (e+f x)}{4 f}-\frac {3 a^3 (4 A+3 B) \cos (e+f x) \sin (e+f x)}{8 f}-\frac {B \cos (e+f x) (a+a \sin (e+f x))^3}{4 f}+\frac {1}{8} \left (3 a^3 (4 A+3 B)\right ) \int 1 \, dx-\frac {\left (a^3 (4 A+3 B)\right ) \text {Subst}\left (\int \left (1-x^2\right ) \, dx,x,\cos (e+f x)\right )}{4 f} \\ & = \frac {5}{8} a^3 (4 A+3 B) x-\frac {a^3 (4 A+3 B) \cos (e+f x)}{f}+\frac {a^3 (4 A+3 B) \cos ^3(e+f x)}{12 f}-\frac {3 a^3 (4 A+3 B) \cos (e+f x) \sin (e+f x)}{8 f}-\frac {B \cos (e+f x) (a+a \sin (e+f x))^3}{4 f} \\ \end{align*}
Time = 0.34 (sec) , antiderivative size = 120, normalized size of antiderivative = 0.94 \[ \int (a+a \sin (e+f x))^3 (A+B \sin (e+f x)) \, dx=-\frac {a^3 \cos (e+f x) \left (30 (4 A+3 B) \arcsin \left (\frac {\sqrt {1-\sin (e+f x)}}{\sqrt {2}}\right )+\sqrt {\cos ^2(e+f x)} \left (88 A+72 B+9 (4 A+5 B) \sin (e+f x)+8 (A+3 B) \sin ^2(e+f x)+6 B \sin ^3(e+f x)\right )\right )}{24 f \sqrt {\cos ^2(e+f x)}} \]
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Time = 1.09 (sec) , antiderivative size = 87, normalized size of antiderivative = 0.69
method | result | size |
parallelrisch | \(\frac {a^{3} \left (8 \left (A +3 B \right ) \cos \left (3 f x +3 e \right )+24 \left (-3 A -4 B \right ) \sin \left (2 f x +2 e \right )+3 B \sin \left (4 f x +4 e \right )+24 \left (-15 A -13 B \right ) \cos \left (f x +e \right )+240 f x A +180 f x B -352 A -288 B \right )}{96 f}\) | \(87\) |
risch | \(\frac {5 a^{3} A x}{2}+\frac {15 a^{3} B x}{8}-\frac {15 a^{3} \cos \left (f x +e \right ) A}{4 f}-\frac {13 a^{3} \cos \left (f x +e \right ) B}{4 f}+\frac {B \,a^{3} \sin \left (4 f x +4 e \right )}{32 f}+\frac {a^{3} \cos \left (3 f x +3 e \right ) A}{12 f}+\frac {a^{3} \cos \left (3 f x +3 e \right ) B}{4 f}-\frac {3 \sin \left (2 f x +2 e \right ) A \,a^{3}}{4 f}-\frac {\sin \left (2 f x +2 e \right ) B \,a^{3}}{f}\) | \(136\) |
parts | \(a^{3} A x -\frac {\left (A \,a^{3}+3 B \,a^{3}\right ) \left (2+\sin ^{2}\left (f x +e \right )\right ) \cos \left (f x +e \right )}{3 f}-\frac {\left (3 A \,a^{3}+B \,a^{3}\right ) \cos \left (f x +e \right )}{f}+\frac {\left (3 A \,a^{3}+3 B \,a^{3}\right ) \left (-\frac {\cos \left (f x +e \right ) \sin \left (f x +e \right )}{2}+\frac {f x}{2}+\frac {e}{2}\right )}{f}+\frac {B \,a^{3} \left (-\frac {\left (\sin ^{3}\left (f x +e \right )+\frac {3 \sin \left (f x +e \right )}{2}\right ) \cos \left (f x +e \right )}{4}+\frac {3 f x}{8}+\frac {3 e}{8}\right )}{f}\) | \(144\) |
derivativedivides | \(\frac {-\frac {A \,a^{3} \left (2+\sin ^{2}\left (f x +e \right )\right ) \cos \left (f x +e \right )}{3}+B \,a^{3} \left (-\frac {\left (\sin ^{3}\left (f x +e \right )+\frac {3 \sin \left (f x +e \right )}{2}\right ) \cos \left (f x +e \right )}{4}+\frac {3 f x}{8}+\frac {3 e}{8}\right )+3 A \,a^{3} \left (-\frac {\cos \left (f x +e \right ) \sin \left (f x +e \right )}{2}+\frac {f x}{2}+\frac {e}{2}\right )-B \,a^{3} \left (2+\sin ^{2}\left (f x +e \right )\right ) \cos \left (f x +e \right )-3 A \,a^{3} \cos \left (f x +e \right )+3 B \,a^{3} \left (-\frac {\cos \left (f x +e \right ) \sin \left (f x +e \right )}{2}+\frac {f x}{2}+\frac {e}{2}\right )+A \,a^{3} \left (f x +e \right )-B \,a^{3} \cos \left (f x +e \right )}{f}\) | \(178\) |
default | \(\frac {-\frac {A \,a^{3} \left (2+\sin ^{2}\left (f x +e \right )\right ) \cos \left (f x +e \right )}{3}+B \,a^{3} \left (-\frac {\left (\sin ^{3}\left (f x +e \right )+\frac {3 \sin \left (f x +e \right )}{2}\right ) \cos \left (f x +e \right )}{4}+\frac {3 f x}{8}+\frac {3 e}{8}\right )+3 A \,a^{3} \left (-\frac {\cos \left (f x +e \right ) \sin \left (f x +e \right )}{2}+\frac {f x}{2}+\frac {e}{2}\right )-B \,a^{3} \left (2+\sin ^{2}\left (f x +e \right )\right ) \cos \left (f x +e \right )-3 A \,a^{3} \cos \left (f x +e \right )+3 B \,a^{3} \left (-\frac {\cos \left (f x +e \right ) \sin \left (f x +e \right )}{2}+\frac {f x}{2}+\frac {e}{2}\right )+A \,a^{3} \left (f x +e \right )-B \,a^{3} \cos \left (f x +e \right )}{f}\) | \(178\) |
norman | \(\frac {\left (\frac {5}{2} A \,a^{3}+\frac {15}{8} B \,a^{3}\right ) x +\left (10 A \,a^{3}+\frac {15}{2} B \,a^{3}\right ) x \left (\tan ^{2}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )+\left (10 A \,a^{3}+\frac {15}{2} B \,a^{3}\right ) x \left (\tan ^{6}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )+\left (15 A \,a^{3}+\frac {45}{4} B \,a^{3}\right ) x \left (\tan ^{4}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )+\left (\frac {5}{2} A \,a^{3}+\frac {15}{8} B \,a^{3}\right ) x \left (\tan ^{8}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )-\frac {22 A \,a^{3}+18 B \,a^{3}}{3 f}-\frac {2 \left (3 A \,a^{3}+B \,a^{3}\right ) \left (\tan ^{6}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{f}-\frac {2 \left (11 A \,a^{3}+9 B \,a^{3}\right ) \left (\tan ^{4}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{f}-\frac {2 \left (35 A \,a^{3}+33 B \,a^{3}\right ) \left (\tan ^{2}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{3 f}-\frac {3 a^{3} \left (4 A +5 B \right ) \tan \left (\frac {f x}{2}+\frac {e}{2}\right )}{4 f}+\frac {3 a^{3} \left (4 A +5 B \right ) \left (\tan ^{7}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{4 f}-\frac {a^{3} \left (12 A +23 B \right ) \left (\tan ^{3}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{4 f}+\frac {a^{3} \left (12 A +23 B \right ) \left (\tan ^{5}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{4 f}}{\left (1+\tan ^{2}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )^{4}}\) | \(343\) |
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Time = 0.27 (sec) , antiderivative size = 93, normalized size of antiderivative = 0.73 \[ \int (a+a \sin (e+f x))^3 (A+B \sin (e+f x)) \, dx=\frac {8 \, {\left (A + 3 \, B\right )} a^{3} \cos \left (f x + e\right )^{3} + 15 \, {\left (4 \, A + 3 \, B\right )} a^{3} f x - 96 \, {\left (A + B\right )} a^{3} \cos \left (f x + e\right ) + 3 \, {\left (2 \, B a^{3} \cos \left (f x + e\right )^{3} - {\left (12 \, A + 17 \, B\right )} a^{3} \cos \left (f x + e\right )\right )} \sin \left (f x + e\right )}{24 \, f} \]
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Leaf count of result is larger than twice the leaf count of optimal. 371 vs. \(2 (119) = 238\).
Time = 0.20 (sec) , antiderivative size = 371, normalized size of antiderivative = 2.92 \[ \int (a+a \sin (e+f x))^3 (A+B \sin (e+f x)) \, dx=\begin {cases} \frac {3 A a^{3} x \sin ^{2}{\left (e + f x \right )}}{2} + \frac {3 A a^{3} x \cos ^{2}{\left (e + f x \right )}}{2} + A a^{3} x - \frac {A a^{3} \sin ^{2}{\left (e + f x \right )} \cos {\left (e + f x \right )}}{f} - \frac {3 A a^{3} \sin {\left (e + f x \right )} \cos {\left (e + f x \right )}}{2 f} - \frac {2 A a^{3} \cos ^{3}{\left (e + f x \right )}}{3 f} - \frac {3 A a^{3} \cos {\left (e + f x \right )}}{f} + \frac {3 B a^{3} x \sin ^{4}{\left (e + f x \right )}}{8} + \frac {3 B a^{3} x \sin ^{2}{\left (e + f x \right )} \cos ^{2}{\left (e + f x \right )}}{4} + \frac {3 B a^{3} x \sin ^{2}{\left (e + f x \right )}}{2} + \frac {3 B a^{3} x \cos ^{4}{\left (e + f x \right )}}{8} + \frac {3 B a^{3} x \cos ^{2}{\left (e + f x \right )}}{2} - \frac {5 B a^{3} \sin ^{3}{\left (e + f x \right )} \cos {\left (e + f x \right )}}{8 f} - \frac {3 B a^{3} \sin ^{2}{\left (e + f x \right )} \cos {\left (e + f x \right )}}{f} - \frac {3 B a^{3} \sin {\left (e + f x \right )} \cos ^{3}{\left (e + f x \right )}}{8 f} - \frac {3 B a^{3} \sin {\left (e + f x \right )} \cos {\left (e + f x \right )}}{2 f} - \frac {2 B a^{3} \cos ^{3}{\left (e + f x \right )}}{f} - \frac {B a^{3} \cos {\left (e + f x \right )}}{f} & \text {for}\: f \neq 0 \\x \left (A + B \sin {\left (e \right )}\right ) \left (a \sin {\left (e \right )} + a\right )^{3} & \text {otherwise} \end {cases} \]
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Time = 0.29 (sec) , antiderivative size = 171, normalized size of antiderivative = 1.35 \[ \int (a+a \sin (e+f x))^3 (A+B \sin (e+f x)) \, dx=\frac {32 \, {\left (\cos \left (f x + e\right )^{3} - 3 \, \cos \left (f x + e\right )\right )} A a^{3} + 72 \, {\left (2 \, f x + 2 \, e - \sin \left (2 \, f x + 2 \, e\right )\right )} A a^{3} + 96 \, {\left (f x + e\right )} A a^{3} + 96 \, {\left (\cos \left (f x + e\right )^{3} - 3 \, \cos \left (f x + e\right )\right )} B a^{3} + 3 \, {\left (12 \, f x + 12 \, e + \sin \left (4 \, f x + 4 \, e\right ) - 8 \, \sin \left (2 \, f x + 2 \, e\right )\right )} B a^{3} + 72 \, {\left (2 \, f x + 2 \, e - \sin \left (2 \, f x + 2 \, e\right )\right )} B a^{3} - 288 \, A a^{3} \cos \left (f x + e\right ) - 96 \, B a^{3} \cos \left (f x + e\right )}{96 \, f} \]
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Time = 0.28 (sec) , antiderivative size = 112, normalized size of antiderivative = 0.88 \[ \int (a+a \sin (e+f x))^3 (A+B \sin (e+f x)) \, dx=\frac {B a^{3} \sin \left (4 \, f x + 4 \, e\right )}{32 \, f} + \frac {5}{8} \, {\left (4 \, A a^{3} + 3 \, B a^{3}\right )} x + \frac {{\left (A a^{3} + 3 \, B a^{3}\right )} \cos \left (3 \, f x + 3 \, e\right )}{12 \, f} - \frac {{\left (15 \, A a^{3} + 13 \, B a^{3}\right )} \cos \left (f x + e\right )}{4 \, f} - \frac {{\left (3 \, A a^{3} + 4 \, B a^{3}\right )} \sin \left (2 \, f x + 2 \, e\right )}{4 \, f} \]
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Time = 14.61 (sec) , antiderivative size = 330, normalized size of antiderivative = 2.60 \[ \int (a+a \sin (e+f x))^3 (A+B \sin (e+f x)) \, dx=\frac {5\,a^3\,\mathrm {atan}\left (\frac {5\,a^3\,\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )\,\left (4\,A+3\,B\right )}{4\,\left (5\,A\,a^3+\frac {15\,B\,a^3}{4}\right )}\right )\,\left (4\,A+3\,B\right )}{4\,f}-\frac {\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )\,\left (3\,A\,a^3+\frac {15\,B\,a^3}{4}\right )+\frac {22\,A\,a^3}{3}+6\,B\,a^3+{\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^6\,\left (6\,A\,a^3+2\,B\,a^3\right )-{\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^7\,\left (3\,A\,a^3+\frac {15\,B\,a^3}{4}\right )+{\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^3\,\left (3\,A\,a^3+\frac {23\,B\,a^3}{4}\right )-{\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^5\,\left (3\,A\,a^3+\frac {23\,B\,a^3}{4}\right )+{\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^4\,\left (22\,A\,a^3+18\,B\,a^3\right )+{\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^2\,\left (\frac {70\,A\,a^3}{3}+22\,B\,a^3\right )}{f\,\left ({\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^8+4\,{\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^6+6\,{\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^4+4\,{\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^2+1\right )}-\frac {5\,a^3\,\left (4\,A+3\,B\right )\,\left (\mathrm {atan}\left (\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )\right )-\frac {f\,x}{2}\right )}{4\,f} \]
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