Optimal. Leaf size=86 \[ -\frac{2 a^3 c \tan ^3(e+f x)}{3 f}+\frac{5 a^3 c \tanh ^{-1}(\sin (e+f x))}{8 f}-\frac{a^3 c \tan (e+f x) \sec ^3(e+f x)}{4 f}-\frac{3 a^3 c \tan (e+f x) \sec (e+f x)}{8 f} \]
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Rubi [A] time = 0.15574, antiderivative size = 86, normalized size of antiderivative = 1., number of steps used = 9, number of rules used = 6, integrand size = 30, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.2, Rules used = {3958, 2611, 3770, 2607, 30, 3768} \[ -\frac{2 a^3 c \tan ^3(e+f x)}{3 f}+\frac{5 a^3 c \tanh ^{-1}(\sin (e+f x))}{8 f}-\frac{a^3 c \tan (e+f x) \sec ^3(e+f x)}{4 f}-\frac{3 a^3 c \tan (e+f x) \sec (e+f x)}{8 f} \]
Antiderivative was successfully verified.
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Rule 3958
Rule 2611
Rule 3770
Rule 2607
Rule 30
Rule 3768
Rubi steps
\begin{align*} \int \sec (e+f x) (a+a \sec (e+f x))^3 (c-c \sec (e+f x)) \, dx &=-\left ((a c) \int \left (a^2 \sec (e+f x) \tan ^2(e+f x)+2 a^2 \sec ^2(e+f x) \tan ^2(e+f x)+a^2 \sec ^3(e+f x) \tan ^2(e+f x)\right ) \, dx\right )\\ &=-\left (\left (a^3 c\right ) \int \sec (e+f x) \tan ^2(e+f x) \, dx\right )-\left (a^3 c\right ) \int \sec ^3(e+f x) \tan ^2(e+f x) \, dx-\left (2 a^3 c\right ) \int \sec ^2(e+f x) \tan ^2(e+f x) \, dx\\ &=-\frac{a^3 c \sec (e+f x) \tan (e+f x)}{2 f}-\frac{a^3 c \sec ^3(e+f x) \tan (e+f x)}{4 f}+\frac{1}{4} \left (a^3 c\right ) \int \sec ^3(e+f x) \, dx+\frac{1}{2} \left (a^3 c\right ) \int \sec (e+f x) \, dx-\frac{\left (2 a^3 c\right ) \operatorname{Subst}\left (\int x^2 \, dx,x,\tan (e+f x)\right )}{f}\\ &=\frac{a^3 c \tanh ^{-1}(\sin (e+f x))}{2 f}-\frac{3 a^3 c \sec (e+f x) \tan (e+f x)}{8 f}-\frac{a^3 c \sec ^3(e+f x) \tan (e+f x)}{4 f}-\frac{2 a^3 c \tan ^3(e+f x)}{3 f}+\frac{1}{8} \left (a^3 c\right ) \int \sec (e+f x) \, dx\\ &=\frac{5 a^3 c \tanh ^{-1}(\sin (e+f x))}{8 f}-\frac{3 a^3 c \sec (e+f x) \tan (e+f x)}{8 f}-\frac{a^3 c \sec ^3(e+f x) \tan (e+f x)}{4 f}-\frac{2 a^3 c \tan ^3(e+f x)}{3 f}\\ \end{align*}
Mathematica [A] time = 0.616023, size = 70, normalized size = 0.81 \[ \frac{a^3 c \left (60 \tanh ^{-1}(\sin (e+f x))-(33 \sin (e+f x)+16 \sin (2 (e+f x))+9 \sin (3 (e+f x))-8 \sin (4 (e+f x))) \sec ^4(e+f x)\right )}{96 f} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.024, size = 107, normalized size = 1.2 \begin{align*}{\frac{2\,{a}^{3}c\tan \left ( fx+e \right ) }{3\,f}}+{\frac{5\,{a}^{3}c\ln \left ( \sec \left ( fx+e \right ) +\tan \left ( fx+e \right ) \right ) }{8\,f}}-{\frac{2\,{a}^{3}c\tan \left ( fx+e \right ) \left ( \sec \left ( fx+e \right ) \right ) ^{2}}{3\,f}}-{\frac{{a}^{3}c \left ( \sec \left ( fx+e \right ) \right ) ^{3}\tan \left ( fx+e \right ) }{4\,f}}-{\frac{3\,{a}^{3}c\sec \left ( fx+e \right ) \tan \left ( fx+e \right ) }{8\,f}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.01203, size = 180, normalized size = 2.09 \begin{align*} -\frac{32 \,{\left (\tan \left (f x + e\right )^{3} + 3 \, \tan \left (f x + e\right )\right )} a^{3} c - 3 \, a^{3} c{\left (\frac{2 \,{\left (3 \, \sin \left (f x + e\right )^{3} - 5 \, \sin \left (f x + e\right )\right )}}{\sin \left (f x + e\right )^{4} - 2 \, \sin \left (f x + e\right )^{2} + 1} - 3 \, \log \left (\sin \left (f x + e\right ) + 1\right ) + 3 \, \log \left (\sin \left (f x + e\right ) - 1\right )\right )} - 48 \, a^{3} c \log \left (\sec \left (f x + e\right ) + \tan \left (f x + e\right )\right ) - 96 \, a^{3} c \tan \left (f x + e\right )}{48 \, f} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 0.487776, size = 302, normalized size = 3.51 \begin{align*} \frac{15 \, a^{3} c \cos \left (f x + e\right )^{4} \log \left (\sin \left (f x + e\right ) + 1\right ) - 15 \, a^{3} c \cos \left (f x + e\right )^{4} \log \left (-\sin \left (f x + e\right ) + 1\right ) + 2 \,{\left (16 \, a^{3} c \cos \left (f x + e\right )^{3} - 9 \, a^{3} c \cos \left (f x + e\right )^{2} - 16 \, a^{3} c \cos \left (f x + e\right ) - 6 \, a^{3} c\right )} \sin \left (f x + e\right )}{48 \, f \cos \left (f x + e\right )^{4}} \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*} - a^{3} c \left (\int - \sec{\left (e + f x \right )}\, dx + \int - 2 \sec ^{2}{\left (e + f x \right )}\, dx + \int 2 \sec ^{4}{\left (e + f x \right )}\, dx + \int \sec ^{5}{\left (e + f x \right )}\, dx\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.31749, size = 182, normalized size = 2.12 \begin{align*} \frac{15 \, a^{3} c \log \left ({\left | \tan \left (\frac{1}{2} \, f x + \frac{1}{2} \, e\right ) + 1 \right |}\right ) - 15 \, a^{3} c \log \left ({\left | \tan \left (\frac{1}{2} \, f x + \frac{1}{2} \, e\right ) - 1 \right |}\right ) - \frac{2 \,{\left (15 \, a^{3} c \tan \left (\frac{1}{2} \, f x + \frac{1}{2} \, e\right )^{7} - 55 \, a^{3} c \tan \left (\frac{1}{2} \, f x + \frac{1}{2} \, e\right )^{5} + 73 \, a^{3} c \tan \left (\frac{1}{2} \, f x + \frac{1}{2} \, e\right )^{3} + 15 \, a^{3} c \tan \left (\frac{1}{2} \, f x + \frac{1}{2} \, e\right )\right )}}{{\left (\tan \left (\frac{1}{2} \, f x + \frac{1}{2} \, e\right )^{2} - 1\right )}^{4}}}{24 \, f} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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