Optimal. Leaf size=132 \[ -\frac{a^3 \tanh ^{-1}(\sin (e+f x))}{c^3 f}-\frac{2 a^3 \tan (e+f x)}{f \left (c^3-c^3 \sec (e+f x)\right )}+\frac{2 \tan (e+f x) \left (a^3 \sec (e+f x)+a^3\right )}{3 c f (c-c \sec (e+f x))^2}-\frac{2 a \tan (e+f x) (a \sec (e+f x)+a)^2}{5 f (c-c \sec (e+f x))^3} \]
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Rubi [A] time = 0.21442, antiderivative size = 132, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 2, integrand size = 32, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.062, Rules used = {3957, 3770} \[ -\frac{a^3 \tanh ^{-1}(\sin (e+f x))}{c^3 f}-\frac{2 a^3 \tan (e+f x)}{f \left (c^3-c^3 \sec (e+f x)\right )}+\frac{2 \tan (e+f x) \left (a^3 \sec (e+f x)+a^3\right )}{3 c f (c-c \sec (e+f x))^2}-\frac{2 a \tan (e+f x) (a \sec (e+f x)+a)^2}{5 f (c-c \sec (e+f x))^3} \]
Antiderivative was successfully verified.
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Rule 3957
Rule 3770
Rubi steps
\begin{align*} \int \frac{\sec (e+f x) (a+a \sec (e+f x))^3}{(c-c \sec (e+f x))^3} \, dx &=-\frac{2 a (a+a \sec (e+f x))^2 \tan (e+f x)}{5 f (c-c \sec (e+f x))^3}-\frac{a \int \frac{\sec (e+f x) (a+a \sec (e+f x))^2}{(c-c \sec (e+f x))^2} \, dx}{c}\\ &=-\frac{2 a (a+a \sec (e+f x))^2 \tan (e+f x)}{5 f (c-c \sec (e+f x))^3}+\frac{2 \left (a^3+a^3 \sec (e+f x)\right ) \tan (e+f x)}{3 c f (c-c \sec (e+f x))^2}+\frac{a^2 \int \frac{\sec (e+f x) (a+a \sec (e+f x))}{c-c \sec (e+f x)} \, dx}{c^2}\\ &=-\frac{2 a (a+a \sec (e+f x))^2 \tan (e+f x)}{5 f (c-c \sec (e+f x))^3}+\frac{2 \left (a^3+a^3 \sec (e+f x)\right ) \tan (e+f x)}{3 c f (c-c \sec (e+f x))^2}-\frac{2 a^3 \tan (e+f x)}{f \left (c^3-c^3 \sec (e+f x)\right )}-\frac{a^3 \int \sec (e+f x) \, dx}{c^3}\\ &=-\frac{a^3 \tanh ^{-1}(\sin (e+f x))}{c^3 f}-\frac{2 a (a+a \sec (e+f x))^2 \tan (e+f x)}{5 f (c-c \sec (e+f x))^3}+\frac{2 \left (a^3+a^3 \sec (e+f x)\right ) \tan (e+f x)}{3 c f (c-c \sec (e+f x))^2}-\frac{2 a^3 \tan (e+f x)}{f \left (c^3-c^3 \sec (e+f x)\right )}\\ \end{align*}
Mathematica [A] time = 0.108118, size = 139, normalized size = 1.05 \[ -\frac{a^3 \left (-\frac{26 \cot \left (\frac{1}{2} (e+f x)\right )}{15 f}-\frac{2 \cot \left (\frac{1}{2} (e+f x)\right ) \csc ^4\left (\frac{1}{2} (e+f x)\right )}{5 f}+\frac{2 \cot \left (\frac{1}{2} (e+f x)\right ) \csc ^2\left (\frac{1}{2} (e+f x)\right )}{15 f}-\frac{\log \left (\cos \left (\frac{1}{2} (e+f x)\right )-\sin \left (\frac{1}{2} (e+f x)\right )\right )}{f}+\frac{\log \left (\sin \left (\frac{1}{2} (e+f x)\right )+\cos \left (\frac{1}{2} (e+f x)\right )\right )}{f}\right )}{c^3} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.09, size = 113, normalized size = 0.9 \begin{align*} -{\frac{{a}^{3}}{f{c}^{3}}\ln \left ( \tan \left ({\frac{fx}{2}}+{\frac{e}{2}} \right ) +1 \right ) }+{\frac{{a}^{3}}{f{c}^{3}}\ln \left ( \tan \left ({\frac{fx}{2}}+{\frac{e}{2}} \right ) -1 \right ) }+{\frac{2\,{a}^{3}}{5\,f{c}^{3}} \left ( \tan \left ({\frac{fx}{2}}+{\frac{e}{2}} \right ) \right ) ^{-5}}+{\frac{2\,{a}^{3}}{3\,f{c}^{3}} \left ( \tan \left ({\frac{fx}{2}}+{\frac{e}{2}} \right ) \right ) ^{-3}}+2\,{\frac{{a}^{3}}{f{c}^{3}\tan \left ( 1/2\,fx+e/2 \right ) }} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [B] time = 1.02943, size = 417, normalized size = 3.16 \begin{align*} -\frac{a^{3}{\left (\frac{60 \, \log \left (\frac{\sin \left (f x + e\right )}{\cos \left (f x + e\right ) + 1} + 1\right )}{c^{3}} - \frac{60 \, \log \left (\frac{\sin \left (f x + e\right )}{\cos \left (f x + e\right ) + 1} - 1\right )}{c^{3}} - \frac{{\left (\frac{20 \, \sin \left (f x + e\right )^{2}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{2}} + \frac{105 \, \sin \left (f x + e\right )^{4}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{4}} + 3\right )}{\left (\cos \left (f x + e\right ) + 1\right )}^{5}}{c^{3} \sin \left (f x + e\right )^{5}}\right )} - \frac{3 \, a^{3}{\left (\frac{10 \, \sin \left (f x + e\right )^{2}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{2}} + \frac{15 \, \sin \left (f x + e\right )^{4}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{4}} + 3\right )}{\left (\cos \left (f x + e\right ) + 1\right )}^{5}}{c^{3} \sin \left (f x + e\right )^{5}} + \frac{a^{3}{\left (\frac{10 \, \sin \left (f x + e\right )^{2}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{2}} - \frac{15 \, \sin \left (f x + e\right )^{4}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{4}} - 3\right )}{\left (\cos \left (f x + e\right ) + 1\right )}^{5}}{c^{3} \sin \left (f x + e\right )^{5}} + \frac{9 \, a^{3}{\left (\frac{5 \, \sin \left (f x + e\right )^{4}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{4}} - 1\right )}{\left (\cos \left (f x + e\right ) + 1\right )}^{5}}{c^{3} \sin \left (f x + e\right )^{5}}}{60 \, f} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 0.484952, size = 446, normalized size = 3.38 \begin{align*} \frac{52 \, a^{3} \cos \left (f x + e\right )^{3} - 44 \, a^{3} \cos \left (f x + e\right )^{2} - 4 \, a^{3} \cos \left (f x + e\right ) + 92 \, a^{3} - 15 \,{\left (a^{3} \cos \left (f x + e\right )^{2} - 2 \, a^{3} \cos \left (f x + e\right ) + a^{3}\right )} \log \left (\sin \left (f x + e\right ) + 1\right ) \sin \left (f x + e\right ) + 15 \,{\left (a^{3} \cos \left (f x + e\right )^{2} - 2 \, a^{3} \cos \left (f x + e\right ) + a^{3}\right )} \log \left (-\sin \left (f x + e\right ) + 1\right ) \sin \left (f x + e\right )}{30 \,{\left (c^{3} f \cos \left (f x + e\right )^{2} - 2 \, c^{3} f \cos \left (f x + e\right ) + c^{3} f\right )} \sin \left (f x + e\right )} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F] time = 0., size = 0, normalized size = 0. \begin{align*} - \frac{a^{3} \left (\int \frac{\sec{\left (e + f x \right )}}{\sec ^{3}{\left (e + f x \right )} - 3 \sec ^{2}{\left (e + f x \right )} + 3 \sec{\left (e + f x \right )} - 1}\, dx + \int \frac{3 \sec ^{2}{\left (e + f x \right )}}{\sec ^{3}{\left (e + f x \right )} - 3 \sec ^{2}{\left (e + f x \right )} + 3 \sec{\left (e + f x \right )} - 1}\, dx + \int \frac{3 \sec ^{3}{\left (e + f x \right )}}{\sec ^{3}{\left (e + f x \right )} - 3 \sec ^{2}{\left (e + f x \right )} + 3 \sec{\left (e + f x \right )} - 1}\, dx + \int \frac{\sec ^{4}{\left (e + f x \right )}}{\sec ^{3}{\left (e + f x \right )} - 3 \sec ^{2}{\left (e + f x \right )} + 3 \sec{\left (e + f x \right )} - 1}\, dx\right )}{c^{3}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.28838, size = 144, normalized size = 1.09 \begin{align*} -\frac{\frac{15 \, a^{3} \log \left ({\left | \tan \left (\frac{1}{2} \, f x + \frac{1}{2} \, e\right ) + 1 \right |}\right )}{c^{3}} - \frac{15 \, a^{3} \log \left ({\left | \tan \left (\frac{1}{2} \, f x + \frac{1}{2} \, e\right ) - 1 \right |}\right )}{c^{3}} - \frac{2 \,{\left (15 \, a^{3} \tan \left (\frac{1}{2} \, f x + \frac{1}{2} \, e\right )^{4} + 5 \, a^{3} \tan \left (\frac{1}{2} \, f x + \frac{1}{2} \, e\right )^{2} + 3 \, a^{3}\right )}}{c^{3} \tan \left (\frac{1}{2} \, f x + \frac{1}{2} \, e\right )^{5}}}{15 \, f} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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