Integrand size = 12, antiderivative size = 80 \[ \int (3+b \sin (e+f x))^3 \, dx=\frac {3}{2} \left (18+3 b^2\right ) x-\frac {2 b \left (36+b^2\right ) \cos (e+f x)}{3 f}-\frac {5 b^2 \cos (e+f x) \sin (e+f x)}{2 f}-\frac {b \cos (e+f x) (3+b \sin (e+f x))^2}{3 f} \]
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Time = 0.05 (sec) , antiderivative size = 90, normalized size of antiderivative = 1.12, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.167, Rules used = {2735, 2813} \[ \int (3+b \sin (e+f x))^3 \, dx=-\frac {2 b \left (4 a^2+b^2\right ) \cos (e+f x)}{3 f}+\frac {1}{2} a x \left (2 a^2+3 b^2\right )-\frac {5 a b^2 \sin (e+f x) \cos (e+f x)}{6 f}-\frac {b \cos (e+f x) (a+b \sin (e+f x))^2}{3 f} \]
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Rule 2735
Rule 2813
Rubi steps \begin{align*} \text {integral}& = -\frac {b \cos (e+f x) (a+b \sin (e+f x))^2}{3 f}+\frac {1}{3} \int (a+b \sin (e+f x)) \left (3 a^2+2 b^2+5 a b \sin (e+f x)\right ) \, dx \\ & = \frac {1}{2} a \left (2 a^2+3 b^2\right ) x-\frac {2 b \left (4 a^2+b^2\right ) \cos (e+f x)}{3 f}-\frac {5 a b^2 \cos (e+f x) \sin (e+f x)}{6 f}-\frac {b \cos (e+f x) (a+b \sin (e+f x))^2}{3 f} \\ \end{align*}
Time = 0.18 (sec) , antiderivative size = 59, normalized size of antiderivative = 0.74 \[ \int (3+b \sin (e+f x))^3 \, dx=\frac {54 \left (6+b^2\right ) (e+f x)-9 b \left (36+b^2\right ) \cos (e+f x)+b^3 \cos (3 (e+f x))-27 b^2 \sin (2 (e+f x))}{12 f} \]
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Time = 1.43 (sec) , antiderivative size = 76, normalized size of antiderivative = 0.95
method | result | size |
derivativedivides | \(\frac {-\frac {b^{3} \left (2+\sin ^{2}\left (f x +e \right )\right ) \cos \left (f x +e \right )}{3}+3 a \,b^{2} \left (-\frac {\sin \left (f x +e \right ) \cos \left (f x +e \right )}{2}+\frac {f x}{2}+\frac {e}{2}\right )-3 a^{2} b \cos \left (f x +e \right )+a^{3} \left (f x +e \right )}{f}\) | \(76\) |
default | \(\frac {-\frac {b^{3} \left (2+\sin ^{2}\left (f x +e \right )\right ) \cos \left (f x +e \right )}{3}+3 a \,b^{2} \left (-\frac {\sin \left (f x +e \right ) \cos \left (f x +e \right )}{2}+\frac {f x}{2}+\frac {e}{2}\right )-3 a^{2} b \cos \left (f x +e \right )+a^{3} \left (f x +e \right )}{f}\) | \(76\) |
parts | \(a^{3} x -\frac {b^{3} \left (2+\sin ^{2}\left (f x +e \right )\right ) \cos \left (f x +e \right )}{3 f}-\frac {3 b \cos \left (f x +e \right ) a^{2}}{f}+\frac {3 a \,b^{2} \left (-\frac {\sin \left (f x +e \right ) \cos \left (f x +e \right )}{2}+\frac {f x}{2}+\frac {e}{2}\right )}{f}\) | \(77\) |
risch | \(a^{3} x +\frac {3 a \,b^{2} x}{2}-\frac {3 b \cos \left (f x +e \right ) a^{2}}{f}-\frac {3 b^{3} \cos \left (f x +e \right )}{4 f}+\frac {b^{3} \cos \left (3 f x +3 e \right )}{12 f}-\frac {3 a \,b^{2} \sin \left (2 f x +2 e \right )}{4 f}\) | \(78\) |
parallelrisch | \(\frac {b^{3} \cos \left (3 f x +3 e \right )-9 a \,b^{2} \sin \left (2 f x +2 e \right )+\left (-36 a^{2} b -9 b^{3}\right ) \cos \left (f x +e \right )+12 a^{3} f x +18 a \,b^{2} f x -36 a^{2} b -8 b^{3}}{12 f}\) | \(80\) |
norman | \(\frac {\left (a^{3}+\frac {3}{2} a \,b^{2}\right ) x +\left (a^{3}+\frac {3}{2} a \,b^{2}\right ) x \left (\tan ^{6}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )+\left (3 a^{3}+\frac {9}{2} a \,b^{2}\right ) x \left (\tan ^{2}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )+\left (3 a^{3}+\frac {9}{2} a \,b^{2}\right ) x \left (\tan ^{4}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )-\frac {18 a^{2} b +4 b^{3}}{3 f}-\frac {6 a^{2} b \left (\tan ^{4}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{f}-\frac {\left (12 a^{2} b +4 b^{3}\right ) \left (\tan ^{2}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{f}-\frac {3 a \,b^{2} \tan \left (\frac {f x}{2}+\frac {e}{2}\right )}{f}+\frac {3 a \,b^{2} \left (\tan ^{5}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{f}}{\left (1+\tan ^{2}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )^{3}}\) | \(206\) |
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Time = 0.26 (sec) , antiderivative size = 71, normalized size of antiderivative = 0.89 \[ \int (3+b \sin (e+f x))^3 \, dx=\frac {2 \, b^{3} \cos \left (f x + e\right )^{3} - 9 \, a b^{2} \cos \left (f x + e\right ) \sin \left (f x + e\right ) + 3 \, {\left (2 \, a^{3} + 3 \, a b^{2}\right )} f x - 6 \, {\left (3 \, a^{2} b + b^{3}\right )} \cos \left (f x + e\right )}{6 \, f} \]
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Time = 0.13 (sec) , antiderivative size = 128, normalized size of antiderivative = 1.60 \[ \int (3+b \sin (e+f x))^3 \, dx=\begin {cases} a^{3} x - \frac {3 a^{2} b \cos {\left (e + f x \right )}}{f} + \frac {3 a b^{2} x \sin ^{2}{\left (e + f x \right )}}{2} + \frac {3 a b^{2} x \cos ^{2}{\left (e + f x \right )}}{2} - \frac {3 a b^{2} \sin {\left (e + f x \right )} \cos {\left (e + f x \right )}}{2 f} - \frac {b^{3} \sin ^{2}{\left (e + f x \right )} \cos {\left (e + f x \right )}}{f} - \frac {2 b^{3} \cos ^{3}{\left (e + f x \right )}}{3 f} & \text {for}\: f \neq 0 \\x \left (a + b \sin {\left (e \right )}\right )^{3} & \text {otherwise} \end {cases} \]
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Time = 0.21 (sec) , antiderivative size = 74, normalized size of antiderivative = 0.92 \[ \int (3+b \sin (e+f x))^3 \, dx=a^{3} x + \frac {3 \, {\left (2 \, f x + 2 \, e - \sin \left (2 \, f x + 2 \, e\right )\right )} a b^{2}}{4 \, f} + \frac {{\left (\cos \left (f x + e\right )^{3} - 3 \, \cos \left (f x + e\right )\right )} b^{3}}{3 \, f} - \frac {3 \, a^{2} b \cos \left (f x + e\right )}{f} \]
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Time = 0.28 (sec) , antiderivative size = 72, normalized size of antiderivative = 0.90 \[ \int (3+b \sin (e+f x))^3 \, dx=\frac {b^{3} \cos \left (3 \, f x + 3 \, e\right )}{12 \, f} - \frac {3 \, a b^{2} \sin \left (2 \, f x + 2 \, e\right )}{4 \, f} + \frac {1}{2} \, {\left (2 \, a^{3} + 3 \, a b^{2}\right )} x - \frac {3 \, {\left (4 \, a^{2} b + b^{3}\right )} \cos \left (f x + e\right )}{4 \, f} \]
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Time = 8.14 (sec) , antiderivative size = 127, normalized size of antiderivative = 1.59 \[ \int (3+b \sin (e+f x))^3 \, dx=a^3\,x-\frac {4\,b^3\,{\cos \left (\frac {e}{2}+\frac {f\,x}{2}\right )}^4}{f}+\frac {8\,b^3\,{\cos \left (\frac {e}{2}+\frac {f\,x}{2}\right )}^6}{3\,f}+\frac {3\,a\,b^2\,x}{2}-\frac {6\,a^2\,b\,{\cos \left (\frac {e}{2}+\frac {f\,x}{2}\right )}^2}{f}-\frac {6\,a\,b^2\,{\cos \left (\frac {e}{2}+\frac {f\,x}{2}\right )}^3\,\sin \left (\frac {e}{2}+\frac {f\,x}{2}\right )}{f}+\frac {3\,a\,b^2\,\cos \left (\frac {e}{2}+\frac {f\,x}{2}\right )\,\sin \left (\frac {e}{2}+\frac {f\,x}{2}\right )}{f} \]
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