Optimal. Leaf size=83 \[ -\frac{2 \cos (e+f x)}{15 f \left (a^3 \sin (e+f x)+a^3\right )}-\frac{2 \cos (e+f x)}{15 a f (a \sin (e+f x)+a)^2}-\frac{\cos (e+f x)}{5 f (a \sin (e+f x)+a)^3} \]
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Rubi [A] time = 0.0469928, antiderivative size = 83, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 12, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.167, Rules used = {2650, 2648} \[ -\frac{2 \cos (e+f x)}{15 f \left (a^3 \sin (e+f x)+a^3\right )}-\frac{2 \cos (e+f x)}{15 a f (a \sin (e+f x)+a)^2}-\frac{\cos (e+f x)}{5 f (a \sin (e+f x)+a)^3} \]
Antiderivative was successfully verified.
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Rule 2650
Rule 2648
Rubi steps
\begin{align*} \int \frac{1}{(a+a \sin (e+f x))^3} \, dx &=-\frac{\cos (e+f x)}{5 f (a+a \sin (e+f x))^3}+\frac{2 \int \frac{1}{(a+a \sin (e+f x))^2} \, dx}{5 a}\\ &=-\frac{\cos (e+f x)}{5 f (a+a \sin (e+f x))^3}-\frac{2 \cos (e+f x)}{15 a f (a+a \sin (e+f x))^2}+\frac{2 \int \frac{1}{a+a \sin (e+f x)} \, dx}{15 a^2}\\ &=-\frac{\cos (e+f x)}{5 f (a+a \sin (e+f x))^3}-\frac{2 \cos (e+f x)}{15 a f (a+a \sin (e+f x))^2}-\frac{2 \cos (e+f x)}{15 f \left (a^3+a^3 \sin (e+f x)\right )}\\ \end{align*}
Mathematica [A] time = 0.122553, size = 76, normalized size = 0.92 \[ \frac{15 \sin (e+f x)-6 \sin (2 (e+f x))-\sin (3 (e+f x))-15 \cos (e+f x)-6 \cos (2 (e+f x))+\cos (3 (e+f x))+10}{30 a^3 f (\sin (e+f x)+1)^3} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.045, size = 85, normalized size = 1. \begin{align*} 2\,{\frac{1}{f{a}^{3}} \left ( -4/5\, \left ( \tan \left ( 1/2\,fx+e/2 \right ) +1 \right ) ^{-5}+2\, \left ( \tan \left ( 1/2\,fx+e/2 \right ) +1 \right ) ^{-4}- \left ( \tan \left ( 1/2\,fx+e/2 \right ) +1 \right ) ^{-1}+2\, \left ( \tan \left ( 1/2\,fx+e/2 \right ) +1 \right ) ^{-2}-8/3\, \left ( \tan \left ( 1/2\,fx+e/2 \right ) +1 \right ) ^{-3} \right ) } \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [B] time = 1.7579, size = 274, normalized size = 3.3 \begin{align*} -\frac{2 \,{\left (\frac{20 \, \sin \left (f x + e\right )}{\cos \left (f x + e\right ) + 1} + \frac{40 \, \sin \left (f x + e\right )^{2}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{2}} + \frac{30 \, \sin \left (f x + e\right )^{3}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{3}} + \frac{15 \, \sin \left (f x + e\right )^{4}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{4}} + 7\right )}}{15 \,{\left (a^{3} + \frac{5 \, a^{3} \sin \left (f x + e\right )}{\cos \left (f x + e\right ) + 1} + \frac{10 \, a^{3} \sin \left (f x + e\right )^{2}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{2}} + \frac{10 \, a^{3} \sin \left (f x + e\right )^{3}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{3}} + \frac{5 \, a^{3} \sin \left (f x + e\right )^{4}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{4}} + \frac{a^{3} \sin \left (f x + e\right )^{5}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{5}}\right )} f} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.4969, size = 369, normalized size = 4.45 \begin{align*} -\frac{2 \, \cos \left (f x + e\right )^{3} - 4 \, \cos \left (f x + e\right )^{2} -{\left (2 \, \cos \left (f x + e\right )^{2} + 6 \, \cos \left (f x + e\right ) - 3\right )} \sin \left (f x + e\right ) - 9 \, \cos \left (f x + e\right ) - 3}{15 \,{\left (a^{3} f \cos \left (f x + e\right )^{3} + 3 \, a^{3} f \cos \left (f x + e\right )^{2} - 2 \, a^{3} f \cos \left (f x + e\right ) - 4 \, a^{3} f +{\left (a^{3} f \cos \left (f x + e\right )^{2} - 2 \, a^{3} f \cos \left (f x + e\right ) - 4 \, a^{3} f\right )} \sin \left (f x + e\right )\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 4.77631, size = 558, normalized size = 6.72 \begin{align*} \text{result too large to display} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.2323, size = 105, normalized size = 1.27 \begin{align*} -\frac{2 \,{\left (15 \, \tan \left (\frac{1}{2} \, f x + \frac{1}{2} \, e\right )^{4} + 30 \, \tan \left (\frac{1}{2} \, f x + \frac{1}{2} \, e\right )^{3} + 40 \, \tan \left (\frac{1}{2} \, f x + \frac{1}{2} \, e\right )^{2} + 20 \, \tan \left (\frac{1}{2} \, f x + \frac{1}{2} \, e\right ) + 7\right )}}{15 \, a^{3} f{\left (\tan \left (\frac{1}{2} \, f x + \frac{1}{2} \, e\right ) + 1\right )}^{5}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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