Optimal. Leaf size=75 \[ \frac{4 \tan ^3(e+f x)}{15 a^3 c^2 f}+\frac{4 \tan (e+f x)}{5 a^3 c^2 f}-\frac{\sec ^3(e+f x)}{5 c^2 f \left (a^3 \sin (e+f x)+a^3\right )} \]
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Rubi [A] time = 0.110116, antiderivative size = 75, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 26, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.115, Rules used = {2736, 2672, 3767} \[ \frac{4 \tan ^3(e+f x)}{15 a^3 c^2 f}+\frac{4 \tan (e+f x)}{5 a^3 c^2 f}-\frac{\sec ^3(e+f x)}{5 c^2 f \left (a^3 \sin (e+f x)+a^3\right )} \]
Antiderivative was successfully verified.
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Rule 2736
Rule 2672
Rule 3767
Rubi steps
\begin{align*} \int \frac{1}{(a+a \sin (e+f x))^3 (c-c \sin (e+f x))^2} \, dx &=\frac{\int \frac{\sec ^4(e+f x)}{a+a \sin (e+f x)} \, dx}{a^2 c^2}\\ &=-\frac{\sec ^3(e+f x)}{5 c^2 f \left (a^3+a^3 \sin (e+f x)\right )}+\frac{4 \int \sec ^4(e+f x) \, dx}{5 a^3 c^2}\\ &=-\frac{\sec ^3(e+f x)}{5 c^2 f \left (a^3+a^3 \sin (e+f x)\right )}-\frac{4 \operatorname{Subst}\left (\int \left (1+x^2\right ) \, dx,x,-\tan (e+f x)\right )}{5 a^3 c^2 f}\\ &=-\frac{\sec ^3(e+f x)}{5 c^2 f \left (a^3+a^3 \sin (e+f x)\right )}+\frac{4 \tan (e+f x)}{5 a^3 c^2 f}+\frac{4 \tan ^3(e+f x)}{15 a^3 c^2 f}\\ \end{align*}
Mathematica [A] time = 0.743909, size = 131, normalized size = 1.75 \[ \frac{18 \sin (e+f x)+512 \sin (2 (e+f x))+27 \sin (3 (e+f x))+128 \sin (4 (e+f x))+9 \sin (5 (e+f x))-128 \cos (e+f x)+72 \cos (2 (e+f x))-192 \cos (3 (e+f x))+18 \cos (4 (e+f x))-64 \cos (5 (e+f x))+54}{1920 a^3 c^2 f (\sin (e+f x)-1)^2 (\sin (e+f x)+1)^3} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.055, size = 133, normalized size = 1.8 \begin{align*} 2\,{\frac{1}{f{a}^{3}{c}^{2}} \left ( -1/12\, \left ( \tan \left ( 1/2\,fx+e/2 \right ) -1 \right ) ^{-3}-1/8\, \left ( \tan \left ( 1/2\,fx+e/2 \right ) -1 \right ) ^{-2}-{\frac{5}{16\,\tan \left ( 1/2\,fx+e/2 \right ) -16}}-1/5\, \left ( \tan \left ( 1/2\,fx+e/2 \right ) +1 \right ) ^{-5}+1/2\, \left ( \tan \left ( 1/2\,fx+e/2 \right ) +1 \right ) ^{-4}-5/6\, \left ( \tan \left ( 1/2\,fx+e/2 \right ) +1 \right ) ^{-3}+3/4\, \left ( \tan \left ( 1/2\,fx+e/2 \right ) +1 \right ) ^{-2}-{\frac{11}{16\,\tan \left ( 1/2\,fx+e/2 \right ) +16}} \right ) } \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [B] time = 1.10226, size = 452, normalized size = 6.03 \begin{align*} \frac{2 \,{\left (\frac{9 \, \sin \left (f x + e\right )}{\cos \left (f x + e\right ) + 1} + \frac{21 \, \sin \left (f x + e\right )^{2}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{2}} + \frac{13 \, \sin \left (f x + e\right )^{3}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{3}} - \frac{25 \, \sin \left (f x + e\right )^{4}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{4}} - \frac{5 \, \sin \left (f x + e\right )^{5}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{5}} + \frac{15 \, \sin \left (f x + e\right )^{6}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{6}} + \frac{15 \, \sin \left (f x + e\right )^{7}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{7}} - 3\right )}}{15 \,{\left (a^{3} c^{2} + \frac{2 \, a^{3} c^{2} \sin \left (f x + e\right )}{\cos \left (f x + e\right ) + 1} - \frac{2 \, a^{3} c^{2} \sin \left (f x + e\right )^{2}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{2}} - \frac{6 \, a^{3} c^{2} \sin \left (f x + e\right )^{3}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{3}} + \frac{6 \, a^{3} c^{2} \sin \left (f x + e\right )^{5}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{5}} + \frac{2 \, a^{3} c^{2} \sin \left (f x + e\right )^{6}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{6}} - \frac{2 \, a^{3} c^{2} \sin \left (f x + e\right )^{7}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{7}} - \frac{a^{3} c^{2} \sin \left (f x + e\right )^{8}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{8}}\right )} f} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.53963, size = 211, normalized size = 2.81 \begin{align*} -\frac{8 \, \cos \left (f x + e\right )^{4} - 4 \, \cos \left (f x + e\right )^{2} - 4 \,{\left (2 \, \cos \left (f x + e\right )^{2} + 1\right )} \sin \left (f x + e\right ) - 1}{15 \,{\left (a^{3} c^{2} f \cos \left (f x + e\right )^{3} \sin \left (f x + e\right ) + a^{3} c^{2} f \cos \left (f x + e\right )^{3}\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F(-1)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 2.09315, size = 180, normalized size = 2.4 \begin{align*} -\frac{\frac{5 \,{\left (15 \, \tan \left (\frac{1}{2} \, f x + \frac{1}{2} \, e\right )^{2} - 24 \, \tan \left (\frac{1}{2} \, f x + \frac{1}{2} \, e\right ) + 13\right )}}{a^{3} c^{2}{\left (\tan \left (\frac{1}{2} \, f x + \frac{1}{2} \, e\right ) - 1\right )}^{3}} + \frac{165 \, \tan \left (\frac{1}{2} \, f x + \frac{1}{2} \, e\right )^{4} + 480 \, \tan \left (\frac{1}{2} \, f x + \frac{1}{2} \, e\right )^{3} + 650 \, \tan \left (\frac{1}{2} \, f x + \frac{1}{2} \, e\right )^{2} + 400 \, \tan \left (\frac{1}{2} \, f x + \frac{1}{2} \, e\right ) + 113}{a^{3} c^{2}{\left (\tan \left (\frac{1}{2} \, f x + \frac{1}{2} \, e\right ) + 1\right )}^{5}}}{120 \, f} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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