Optimal. Leaf size=121 \[ -\frac{a \left (a^2+4 b^2\right ) \cos (e+f x)}{2 f}-\frac{b \left (2 a^2+3 b^2\right ) \sin (e+f x) \cos (e+f x)}{8 f}+\frac{3}{8} b x \left (4 a^2+b^2\right )-\frac{\cos (e+f x) (a+b \sin (e+f x))^3}{4 f}-\frac{a \cos (e+f x) (a+b \sin (e+f x))^2}{4 f} \]
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Rubi [A] time = 0.11403, antiderivative size = 121, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.105, Rules used = {2753, 2734} \[ -\frac{a \left (a^2+4 b^2\right ) \cos (e+f x)}{2 f}-\frac{b \left (2 a^2+3 b^2\right ) \sin (e+f x) \cos (e+f x)}{8 f}+\frac{3}{8} b x \left (4 a^2+b^2\right )-\frac{\cos (e+f x) (a+b \sin (e+f x))^3}{4 f}-\frac{a \cos (e+f x) (a+b \sin (e+f x))^2}{4 f} \]
Antiderivative was successfully verified.
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Rule 2753
Rule 2734
Rubi steps
\begin{align*} \int \sin (e+f x) (a+b \sin (e+f x))^3 \, dx &=-\frac{\cos (e+f x) (a+b \sin (e+f x))^3}{4 f}+\frac{1}{4} \int (3 b+3 a \sin (e+f x)) (a+b \sin (e+f x))^2 \, dx\\ &=-\frac{a \cos (e+f x) (a+b \sin (e+f x))^2}{4 f}-\frac{\cos (e+f x) (a+b \sin (e+f x))^3}{4 f}+\frac{1}{12} \int (a+b \sin (e+f x)) \left (15 a b+3 \left (2 a^2+3 b^2\right ) \sin (e+f x)\right ) \, dx\\ &=\frac{3}{8} b \left (4 a^2+b^2\right ) x-\frac{a \left (a^2+4 b^2\right ) \cos (e+f x)}{2 f}-\frac{b \left (2 a^2+3 b^2\right ) \cos (e+f x) \sin (e+f x)}{8 f}-\frac{a \cos (e+f x) (a+b \sin (e+f x))^2}{4 f}-\frac{\cos (e+f x) (a+b \sin (e+f x))^3}{4 f}\\ \end{align*}
Mathematica [A] time = 0.338307, size = 100, normalized size = 0.83 \[ \frac{b \left (-8 \left (3 a^2+b^2\right ) \sin (2 (e+f x))+48 a^2 e+48 a^2 f x+8 a b \cos (3 (e+f x))+b^2 \sin (4 (e+f x))+12 b^2 e+12 b^2 f x\right )-8 a \left (4 a^2+9 b^2\right ) \cos (e+f x)}{32 f} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.022, size = 104, normalized size = 0.9 \begin{align*}{\frac{1}{f} \left ({b}^{3} \left ( -{\frac{\cos \left ( fx+e \right ) }{4} \left ( \left ( \sin \left ( fx+e \right ) \right ) ^{3}+{\frac{3\,\sin \left ( fx+e \right ) }{2}} \right ) }+{\frac{3\,fx}{8}}+{\frac{3\,e}{8}} \right ) -a{b}^{2} \left ( 2+ \left ( \sin \left ( fx+e \right ) \right ) ^{2} \right ) \cos \left ( fx+e \right ) +3\,{a}^{2}b \left ( -1/2\,\sin \left ( fx+e \right ) \cos \left ( fx+e \right ) +1/2\,fx+e/2 \right ) -{a}^{3}\cos \left ( fx+e \right ) \right ) } \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.42315, size = 131, normalized size = 1.08 \begin{align*} \frac{24 \,{\left (2 \, f x + 2 \, e - \sin \left (2 \, f x + 2 \, e\right )\right )} a^{2} b + 32 \,{\left (\cos \left (f x + e\right )^{3} - 3 \, \cos \left (f x + e\right )\right )} a b^{2} +{\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^{3} - 32 \, a^{3} \cos \left (f x + e\right )}{32 \, f} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.6102, size = 217, normalized size = 1.79 \begin{align*} \frac{8 \, a b^{2} \cos \left (f x + e\right )^{3} + 3 \,{\left (4 \, a^{2} b + b^{3}\right )} f x - 8 \,{\left (a^{3} + 3 \, a b^{2}\right )} \cos \left (f x + e\right ) +{\left (2 \, b^{3} \cos \left (f x + e\right )^{3} -{\left (12 \, a^{2} b + 5 \, b^{3}\right )} \cos \left (f x + e\right )\right )} \sin \left (f x + e\right )}{8 \, f} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 2.01214, size = 233, normalized size = 1.93 \begin{align*} \begin{cases} - \frac{a^{3} \cos{\left (e + f x \right )}}{f} + \frac{3 a^{2} b x \sin ^{2}{\left (e + f x \right )}}{2} + \frac{3 a^{2} b x \cos ^{2}{\left (e + f x \right )}}{2} - \frac{3 a^{2} b \sin{\left (e + f x \right )} \cos{\left (e + f x \right )}}{2 f} - \frac{3 a b^{2} \sin ^{2}{\left (e + f x \right )} \cos{\left (e + f x \right )}}{f} - \frac{2 a b^{2} \cos ^{3}{\left (e + f x \right )}}{f} + \frac{3 b^{3} x \sin ^{4}{\left (e + f x \right )}}{8} + \frac{3 b^{3} x \sin ^{2}{\left (e + f x \right )} \cos ^{2}{\left (e + f x \right )}}{4} + \frac{3 b^{3} x \cos ^{4}{\left (e + f x \right )}}{8} - \frac{5 b^{3} \sin ^{3}{\left (e + f x \right )} \cos{\left (e + f x \right )}}{8 f} - \frac{3 b^{3} \sin{\left (e + f x \right )} \cos ^{3}{\left (e + f x \right )}}{8 f} & \text{for}\: f \neq 0 \\x \left (a + b \sin{\left (e \right )}\right )^{3} \sin{\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 = 1.6002, size = 157, normalized size = 1.3 \begin{align*} \frac{a b^{2} \cos \left (3 \, f x + 3 \, e\right )}{4 \, f} - \frac{3 \, a b^{2} \cos \left (f x + e\right )}{4 \, f} + \frac{b^{3} \sin \left (4 \, f x + 4 \, e\right )}{32 \, f} + \frac{3}{8} \,{\left (4 \, a^{2} b + b^{3}\right )} x - \frac{{\left (2 \, a^{3} + 3 \, a b^{2}\right )} \cos \left (f x + e\right )}{2 \, f} - \frac{{\left (3 \, a^{2} b + b^{3}\right )} \sin \left (2 \, f x + 2 \, e\right )}{4 \, f} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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