Optimal. Leaf size=77 \[ -\frac{a^3 c \tan ^3(e+f x)}{3 f}-\frac{a^3 c \tan (e+f x)}{f}+\frac{a^3 c \tanh ^{-1}(\sin (e+f x))}{f}-\frac{a^3 c \tan (e+f x) \sec (e+f x)}{f}+a^3 c x \]
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Rubi [A] time = 0.147325, antiderivative size = 77, normalized size of antiderivative = 1., number of steps used = 9, number of rules used = 8, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.333, Rules used = {3904, 3886, 3473, 8, 2611, 3770, 2607, 30} \[ -\frac{a^3 c \tan ^3(e+f x)}{3 f}-\frac{a^3 c \tan (e+f x)}{f}+\frac{a^3 c \tanh ^{-1}(\sin (e+f x))}{f}-\frac{a^3 c \tan (e+f x) \sec (e+f x)}{f}+a^3 c x \]
Antiderivative was successfully verified.
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Rule 3904
Rule 3886
Rule 3473
Rule 8
Rule 2611
Rule 3770
Rule 2607
Rule 30
Rubi steps
\begin{align*} \int (a+a \sec (e+f x))^3 (c-c \sec (e+f x)) \, dx &=-\left ((a c) \int (a+a \sec (e+f x))^2 \tan ^2(e+f x) \, dx\right )\\ &=-\left ((a c) \int \left (a^2 \tan ^2(e+f x)+2 a^2 \sec (e+f x) \tan ^2(e+f x)+a^2 \sec ^2(e+f x) \tan ^2(e+f x)\right ) \, dx\right )\\ &=-\left (\left (a^3 c\right ) \int \tan ^2(e+f x) \, dx\right )-\left (a^3 c\right ) \int \sec ^2(e+f x) \tan ^2(e+f x) \, dx-\left (2 a^3 c\right ) \int \sec (e+f x) \tan ^2(e+f x) \, dx\\ &=-\frac{a^3 c \tan (e+f x)}{f}-\frac{a^3 c \sec (e+f x) \tan (e+f x)}{f}+\left (a^3 c\right ) \int 1 \, dx+\left (a^3 c\right ) \int \sec (e+f x) \, dx-\frac{\left (a^3 c\right ) \operatorname{Subst}\left (\int x^2 \, dx,x,\tan (e+f x)\right )}{f}\\ &=a^3 c x+\frac{a^3 c \tanh ^{-1}(\sin (e+f x))}{f}-\frac{a^3 c \tan (e+f x)}{f}-\frac{a^3 c \sec (e+f x) \tan (e+f x)}{f}-\frac{a^3 c \tan ^3(e+f x)}{3 f}\\ \end{align*}
Mathematica [A] time = 0.440177, size = 101, normalized size = 1.31 \[ \frac{a^3 c \sec ^3(e+f x) \left (-6 \sin (e+f x)-6 \sin (2 (e+f x))-2 \sin (3 (e+f x))+9 (e+f x) \cos (e+f x)+3 e \cos (3 (e+f x))+3 f x \cos (3 (e+f x))+12 \cos ^3(e+f x) \tanh ^{-1}(\sin (e+f x))\right )}{12 f} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.023, size = 98, normalized size = 1.3 \begin{align*}{\frac{{a}^{3}c\ln \left ( \sec \left ( fx+e \right ) +\tan \left ( fx+e \right ) \right ) }{f}}+{a}^{3}cx+{\frac{{a}^{3}ce}{f}}-{\frac{{a}^{3}c\sec \left ( fx+e \right ) \tan \left ( fx+e \right ) }{f}}-{\frac{2\,{a}^{3}c\tan \left ( fx+e \right ) }{3\,f}}-{\frac{{a}^{3}c\tan \left ( fx+e \right ) \left ( \sec \left ( fx+e \right ) \right ) ^{2}}{3\,f}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.01755, size = 144, normalized size = 1.87 \begin{align*} -\frac{2 \,{\left (\tan \left (f x + e\right )^{3} + 3 \, \tan \left (f x + e\right )\right )} a^{3} c - 6 \,{\left (f x + e\right )} a^{3} c - 3 \, a^{3} c{\left (\frac{2 \, \sin \left (f x + e\right )}{\sin \left (f x + e\right )^{2} - 1} - \log \left (\sin \left (f x + e\right ) + 1\right ) + \log \left (\sin \left (f x + e\right ) - 1\right )\right )} - 12 \, a^{3} c \log \left (\sec \left (f x + e\right ) + \tan \left (f x + e\right )\right )}{6 \, f} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.10899, size = 298, normalized size = 3.87 \begin{align*} \frac{6 \, a^{3} c f x \cos \left (f x + e\right )^{3} + 3 \, a^{3} c \cos \left (f x + e\right )^{3} \log \left (\sin \left (f x + e\right ) + 1\right ) - 3 \, a^{3} c \cos \left (f x + e\right )^{3} \log \left (-\sin \left (f x + e\right ) + 1\right ) - 2 \,{\left (2 \, a^{3} c \cos \left (f x + e\right )^{2} + 3 \, a^{3} c \cos \left (f x + e\right ) + a^{3} c\right )} \sin \left (f x + e\right )}{6 \, f \cos \left (f x + e\right )^{3}} \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 \left (-1\right )\, dx + \int - 2 \sec{\left (e + f x \right )}\, dx + \int 2 \sec ^{3}{\left (e + f x \right )}\, dx + \int \sec ^{4}{\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 = 1.39224, size = 149, normalized size = 1.94 \begin{align*} \frac{3 \,{\left (f x + e\right )} a^{3} c + 3 \, a^{3} c \log \left ({\left | \tan \left (\frac{1}{2} \, f x + \frac{1}{2} \, e\right ) + 1 \right |}\right ) - 3 \, a^{3} c \log \left ({\left | \tan \left (\frac{1}{2} \, f x + \frac{1}{2} \, e\right ) - 1 \right |}\right ) - \frac{4 \,{\left (a^{3} c \tan \left (\frac{1}{2} \, f x + \frac{1}{2} \, e\right )^{3} - 3 \, 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 )}^{3}}}{3 \, f} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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