Optimal. Leaf size=102 \[ -\frac{a^2 \cos ^5(e+f x)}{5 f}+\frac{a^2 \cos ^3(e+f x)}{f}-\frac{2 a^2 \cos (e+f x)}{f}-\frac{a^2 \sin ^3(e+f x) \cos (e+f x)}{2 f}-\frac{3 a^2 \sin (e+f x) \cos (e+f x)}{4 f}+\frac{3 a^2 x}{4} \]
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Rubi [A] time = 0.101512, antiderivative size = 102, normalized size of antiderivative = 1., number of steps used = 9, number of rules used = 4, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.19, Rules used = {2757, 2633, 2635, 8} \[ -\frac{a^2 \cos ^5(e+f x)}{5 f}+\frac{a^2 \cos ^3(e+f x)}{f}-\frac{2 a^2 \cos (e+f x)}{f}-\frac{a^2 \sin ^3(e+f x) \cos (e+f x)}{2 f}-\frac{3 a^2 \sin (e+f x) \cos (e+f x)}{4 f}+\frac{3 a^2 x}{4} \]
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))^2 \, dx &=\int \left (a^2 \sin ^3(e+f x)+2 a^2 \sin ^4(e+f x)+a^2 \sin ^5(e+f x)\right ) \, dx\\ &=a^2 \int \sin ^3(e+f x) \, dx+a^2 \int \sin ^5(e+f x) \, dx+\left (2 a^2\right ) \int \sin ^4(e+f x) \, dx\\ &=-\frac{a^2 \cos (e+f x) \sin ^3(e+f x)}{2 f}+\frac{1}{2} \left (3 a^2\right ) \int \sin ^2(e+f x) \, dx-\frac{a^2 \operatorname{Subst}\left (\int \left (1-x^2\right ) \, dx,x,\cos (e+f x)\right )}{f}-\frac{a^2 \operatorname{Subst}\left (\int \left (1-2 x^2+x^4\right ) \, dx,x,\cos (e+f x)\right )}{f}\\ &=-\frac{2 a^2 \cos (e+f x)}{f}+\frac{a^2 \cos ^3(e+f x)}{f}-\frac{a^2 \cos ^5(e+f x)}{5 f}-\frac{3 a^2 \cos (e+f x) \sin (e+f x)}{4 f}-\frac{a^2 \cos (e+f x) \sin ^3(e+f x)}{2 f}+\frac{1}{4} \left (3 a^2\right ) \int 1 \, dx\\ &=\frac{3 a^2 x}{4}-\frac{2 a^2 \cos (e+f x)}{f}+\frac{a^2 \cos ^3(e+f x)}{f}-\frac{a^2 \cos ^5(e+f x)}{5 f}-\frac{3 a^2 \cos (e+f x) \sin (e+f x)}{4 f}-\frac{a^2 \cos (e+f x) \sin ^3(e+f x)}{2 f}\\ \end{align*}
Mathematica [A] time = 0.428919, size = 105, normalized size = 1.03 \[ -\frac{a^2 \cos (e+f x) \left (30 \sin ^{-1}\left (\frac{\sqrt{1-\sin (e+f x)}}{\sqrt{2}}\right )+\left (4 \sin ^4(e+f x)+10 \sin ^3(e+f x)+12 \sin ^2(e+f x)+15 \sin (e+f x)+24\right ) \sqrt{\cos ^2(e+f x)}\right )}{20 f \sqrt{\cos ^2(e+f x)}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.031, size = 96, normalized size = 0.9 \begin{align*}{\frac{1}{f} \left ( -{\frac{{a}^{2}\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 ) }+2\,{a}^{2} \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}^{2} \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 = 1.72938, size = 128, normalized size = 1.25 \begin{align*} -\frac{16 \,{\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} - 80 \,{\left (\cos \left (f x + e\right )^{3} - 3 \, \cos \left (f x + e\right )\right )} a^{2} - 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}}{240 \, f} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.8163, size = 205, normalized size = 2.01 \begin{align*} -\frac{4 \, a^{2} \cos \left (f x + e\right )^{5} - 20 \, a^{2} \cos \left (f x + e\right )^{3} - 15 \, a^{2} f x + 40 \, a^{2} \cos \left (f x + e\right ) - 5 \,{\left (2 \, a^{2} \cos \left (f x + e\right )^{3} - 5 \, a^{2} \cos \left (f x + e\right )\right )} \sin \left (f x + e\right )}{20 \, f} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 2.24003, size = 221, normalized size = 2.17 \begin{align*} \begin{cases} \frac{3 a^{2} x \sin ^{4}{\left (e + f x \right )}}{4} + \frac{3 a^{2} x \sin ^{2}{\left (e + f x \right )} \cos ^{2}{\left (e + f x \right )}}{2} + \frac{3 a^{2} x \cos ^{4}{\left (e + f x \right )}}{4} - \frac{a^{2} \sin ^{4}{\left (e + f x \right )} \cos{\left (e + f x \right )}}{f} - \frac{5 a^{2} \sin ^{3}{\left (e + f x \right )} \cos{\left (e + f x \right )}}{4 f} - \frac{4 a^{2} \sin ^{2}{\left (e + f x \right )} \cos ^{3}{\left (e + f x \right )}}{3 f} - \frac{a^{2} \sin ^{2}{\left (e + f x \right )} \cos{\left (e + f x \right )}}{f} - \frac{3 a^{2} \sin{\left (e + f x \right )} \cos ^{3}{\left (e + f x \right )}}{4 f} - \frac{8 a^{2} \cos ^{5}{\left (e + f x \right )}}{15 f} - \frac{2 a^{2} \cos ^{3}{\left (e + f x \right )}}{3 f} & \text{for}\: f \neq 0 \\x \left (a \sin{\left (e \right )} + a\right )^{2} \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 = 2.08549, size = 127, normalized size = 1.25 \begin{align*} \frac{3}{4} \, a^{2} x - \frac{a^{2} \cos \left (5 \, f x + 5 \, e\right )}{80 \, f} + \frac{3 \, a^{2} \cos \left (3 \, f x + 3 \, e\right )}{16 \, f} - \frac{11 \, a^{2} \cos \left (f x + e\right )}{8 \, f} + \frac{a^{2} \sin \left (4 \, f x + 4 \, e\right )}{16 \, f} - \frac{a^{2} \sin \left (2 \, f x + 2 \, e\right )}{2 \, f} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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