Optimal. Leaf size=129 \[ -\frac{3 a^3 \cos ^5(e+f x)}{5 f}+\frac{7 a^3 \cos ^3(e+f x)}{3 f}-\frac{4 a^3 \cos (e+f x)}{f}-\frac{a^3 \sin ^5(e+f x) \cos (e+f x)}{6 f}-\frac{23 a^3 \sin ^3(e+f x) \cos (e+f x)}{24 f}-\frac{23 a^3 \sin (e+f x) \cos (e+f x)}{16 f}+\frac{23 a^3 x}{16} \]
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Rubi [A] time = 0.144826, antiderivative size = 129, normalized size of antiderivative = 1., number of steps used = 13, number of rules used = 4, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.19, Rules used = {2757, 2633, 2635, 8} \[ -\frac{3 a^3 \cos ^5(e+f x)}{5 f}+\frac{7 a^3 \cos ^3(e+f x)}{3 f}-\frac{4 a^3 \cos (e+f x)}{f}-\frac{a^3 \sin ^5(e+f x) \cos (e+f x)}{6 f}-\frac{23 a^3 \sin ^3(e+f x) \cos (e+f x)}{24 f}-\frac{23 a^3 \sin (e+f x) \cos (e+f x)}{16 f}+\frac{23 a^3 x}{16} \]
Antiderivative was successfully verified.
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Rule 2757
Rule 2633
Rule 2635
Rule 8
Rubi steps
\begin{align*} \int \sin ^3(e+f x) (a+a \sin (e+f x))^3 \, dx &=\int \left (a^3 \sin ^3(e+f x)+3 a^3 \sin ^4(e+f x)+3 a^3 \sin ^5(e+f x)+a^3 \sin ^6(e+f x)\right ) \, dx\\ &=a^3 \int \sin ^3(e+f x) \, dx+a^3 \int \sin ^6(e+f x) \, dx+\left (3 a^3\right ) \int \sin ^4(e+f x) \, dx+\left (3 a^3\right ) \int \sin ^5(e+f x) \, dx\\ &=-\frac{3 a^3 \cos (e+f x) \sin ^3(e+f x)}{4 f}-\frac{a^3 \cos (e+f x) \sin ^5(e+f x)}{6 f}+\frac{1}{6} \left (5 a^3\right ) \int \sin ^4(e+f x) \, dx+\frac{1}{4} \left (9 a^3\right ) \int \sin ^2(e+f x) \, dx-\frac{a^3 \operatorname{Subst}\left (\int \left (1-x^2\right ) \, dx,x,\cos (e+f x)\right )}{f}-\frac{\left (3 a^3\right ) \operatorname{Subst}\left (\int \left (1-2 x^2+x^4\right ) \, dx,x,\cos (e+f x)\right )}{f}\\ &=-\frac{4 a^3 \cos (e+f x)}{f}+\frac{7 a^3 \cos ^3(e+f x)}{3 f}-\frac{3 a^3 \cos ^5(e+f x)}{5 f}-\frac{9 a^3 \cos (e+f x) \sin (e+f x)}{8 f}-\frac{23 a^3 \cos (e+f x) \sin ^3(e+f x)}{24 f}-\frac{a^3 \cos (e+f x) \sin ^5(e+f x)}{6 f}+\frac{1}{8} \left (5 a^3\right ) \int \sin ^2(e+f x) \, dx+\frac{1}{8} \left (9 a^3\right ) \int 1 \, dx\\ &=\frac{9 a^3 x}{8}-\frac{4 a^3 \cos (e+f x)}{f}+\frac{7 a^3 \cos ^3(e+f x)}{3 f}-\frac{3 a^3 \cos ^5(e+f x)}{5 f}-\frac{23 a^3 \cos (e+f x) \sin (e+f x)}{16 f}-\frac{23 a^3 \cos (e+f x) \sin ^3(e+f x)}{24 f}-\frac{a^3 \cos (e+f x) \sin ^5(e+f x)}{6 f}+\frac{1}{16} \left (5 a^3\right ) \int 1 \, dx\\ &=\frac{23 a^3 x}{16}-\frac{4 a^3 \cos (e+f x)}{f}+\frac{7 a^3 \cos ^3(e+f x)}{3 f}-\frac{3 a^3 \cos ^5(e+f x)}{5 f}-\frac{23 a^3 \cos (e+f x) \sin (e+f x)}{16 f}-\frac{23 a^3 \cos (e+f x) \sin ^3(e+f x)}{24 f}-\frac{a^3 \cos (e+f x) \sin ^5(e+f x)}{6 f}\\ \end{align*}
Mathematica [A] time = 0.516368, size = 115, normalized size = 0.89 \[ -\frac{a^3 \cos (e+f x) \left (690 \sin ^{-1}\left (\frac{\sqrt{1-\sin (e+f x)}}{\sqrt{2}}\right )+\left (40 \sin ^5(e+f x)+144 \sin ^4(e+f x)+230 \sin ^3(e+f x)+272 \sin ^2(e+f x)+345 \sin (e+f x)+544\right ) \sqrt{\cos ^2(e+f x)}\right )}{240 f \sqrt{\cos ^2(e+f x)}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.034, size = 143, normalized size = 1.1 \begin{align*}{\frac{1}{f} \left ({a}^{3} \left ( -{\frac{\cos \left ( fx+e \right ) }{6} \left ( \left ( \sin \left ( fx+e \right ) \right ) ^{5}+{\frac{5\, \left ( \sin \left ( fx+e \right ) \right ) ^{3}}{4}}+{\frac{15\,\sin \left ( fx+e \right ) }{8}} \right ) }+{\frac{5\,fx}{16}}+{\frac{5\,e}{16}} \right ) -{\frac{3\,{a}^{3}\cos \left ( fx+e \right ) }{5} \left ({\frac{8}{3}}+ \left ( \sin \left ( fx+e \right ) \right ) ^{4}+{\frac{4\, \left ( \sin \left ( fx+e \right ) \right ) ^{2}}{3}} \right ) }+3\,{a}^{3} \left ( -1/4\, \left ( \left ( \sin \left ( fx+e \right ) \right ) ^{3}+3/2\,\sin \left ( fx+e \right ) \right ) \cos \left ( fx+e \right ) +3/8\,fx+3/8\,e \right ) -{\frac{{a}^{3} \left ( 2+ \left ( \sin \left ( fx+e \right ) \right ) ^{2} \right ) \cos \left ( fx+e \right ) }{3}} \right ) } \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.8007, size = 193, normalized size = 1.5 \begin{align*} -\frac{192 \,{\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^{3} - 320 \,{\left (\cos \left (f x + e\right )^{3} - 3 \, \cos \left (f x + e\right )\right )} a^{3} - 5 \,{\left (4 \, \sin \left (2 \, f x + 2 \, e\right )^{3} + 60 \, f x + 60 \, e + 9 \, \sin \left (4 \, f x + 4 \, e\right ) - 48 \, \sin \left (2 \, f x + 2 \, e\right )\right )} a^{3} - 90 \,{\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^{3}}{960 \, f} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.75889, size = 248, normalized size = 1.92 \begin{align*} -\frac{144 \, a^{3} \cos \left (f x + e\right )^{5} - 560 \, a^{3} \cos \left (f x + e\right )^{3} - 345 \, a^{3} f x + 960 \, a^{3} \cos \left (f x + e\right ) + 5 \,{\left (8 \, a^{3} \cos \left (f x + e\right )^{5} - 62 \, a^{3} \cos \left (f x + e\right )^{3} + 123 \, a^{3} \cos \left (f x + e\right )\right )} \sin \left (f x + e\right )}{240 \, f} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 4.45884, size = 379, normalized size = 2.94 \begin{align*} \begin{cases} \frac{5 a^{3} x \sin ^{6}{\left (e + f x \right )}}{16} + \frac{15 a^{3} x \sin ^{4}{\left (e + f x \right )} \cos ^{2}{\left (e + f x \right )}}{16} + \frac{9 a^{3} x \sin ^{4}{\left (e + f x \right )}}{8} + \frac{15 a^{3} x \sin ^{2}{\left (e + f x \right )} \cos ^{4}{\left (e + f x \right )}}{16} + \frac{9 a^{3} x \sin ^{2}{\left (e + f x \right )} \cos ^{2}{\left (e + f x \right )}}{4} + \frac{5 a^{3} x \cos ^{6}{\left (e + f x \right )}}{16} + \frac{9 a^{3} x \cos ^{4}{\left (e + f x \right )}}{8} - \frac{11 a^{3} \sin ^{5}{\left (e + f x \right )} \cos{\left (e + f x \right )}}{16 f} - \frac{3 a^{3} \sin ^{4}{\left (e + f x \right )} \cos{\left (e + f x \right )}}{f} - \frac{5 a^{3} \sin ^{3}{\left (e + f x \right )} \cos ^{3}{\left (e + f x \right )}}{6 f} - \frac{15 a^{3} \sin ^{3}{\left (e + f x \right )} \cos{\left (e + f x \right )}}{8 f} - \frac{4 a^{3} \sin ^{2}{\left (e + f x \right )} \cos ^{3}{\left (e + f x \right )}}{f} - \frac{a^{3} \sin ^{2}{\left (e + f x \right )} \cos{\left (e + f x \right )}}{f} - \frac{5 a^{3} \sin{\left (e + f x \right )} \cos ^{5}{\left (e + f x \right )}}{16 f} - \frac{9 a^{3} \sin{\left (e + f x \right )} \cos ^{3}{\left (e + f x \right )}}{8 f} - \frac{8 a^{3} \cos ^{5}{\left (e + f x \right )}}{5 f} - \frac{2 a^{3} \cos ^{3}{\left (e + f x \right )}}{3 f} & \text{for}\: f \neq 0 \\x \left (a \sin{\left (e \right )} + a\right )^{3} \sin ^{3}{\left (e \right )} & \text{otherwise} \end{cases} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 2.10458, size = 151, normalized size = 1.17 \begin{align*} \frac{23}{16} \, a^{3} x - \frac{3 \, a^{3} \cos \left (5 \, f x + 5 \, e\right )}{80 \, f} + \frac{19 \, a^{3} \cos \left (3 \, f x + 3 \, e\right )}{48 \, f} - \frac{21 \, a^{3} \cos \left (f x + e\right )}{8 \, f} - \frac{a^{3} \sin \left (6 \, f x + 6 \, e\right )}{192 \, f} + \frac{9 \, a^{3} \sin \left (4 \, f x + 4 \, e\right )}{64 \, f} - \frac{63 \, a^{3} \sin \left (2 \, f x + 2 \, e\right )}{64 \, f} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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