Optimal. Leaf size=61 \[ -\frac{a^2 c \tan ^3(e+f x)}{3 f}+\frac{a^2 c \tanh ^{-1}(\sin (e+f x))}{2 f}-\frac{a^2 c \tan (e+f x) \sec (e+f x)}{2 f} \]
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Rubi [A] time = 0.102527, antiderivative size = 61, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 5, integrand size = 30, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.167, Rules used = {3958, 2611, 3770, 2607, 30} \[ -\frac{a^2 c \tan ^3(e+f x)}{3 f}+\frac{a^2 c \tanh ^{-1}(\sin (e+f x))}{2 f}-\frac{a^2 c \tan (e+f x) \sec (e+f x)}{2 f} \]
Antiderivative was successfully verified.
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Rule 3958
Rule 2611
Rule 3770
Rule 2607
Rule 30
Rubi steps
\begin{align*} \int \sec (e+f x) (a+a \sec (e+f x))^2 (c-c \sec (e+f x)) \, dx &=-\left ((a c) \int \left (a \sec (e+f x) \tan ^2(e+f x)+a \sec ^2(e+f x) \tan ^2(e+f x)\right ) \, dx\right )\\ &=-\left (\left (a^2 c\right ) \int \sec (e+f x) \tan ^2(e+f x) \, dx\right )-\left (a^2 c\right ) \int \sec ^2(e+f x) \tan ^2(e+f x) \, dx\\ &=-\frac{a^2 c \sec (e+f x) \tan (e+f x)}{2 f}+\frac{1}{2} \left (a^2 c\right ) \int \sec (e+f x) \, dx-\frac{\left (a^2 c\right ) \operatorname{Subst}\left (\int x^2 \, dx,x,\tan (e+f x)\right )}{f}\\ &=\frac{a^2 c \tanh ^{-1}(\sin (e+f x))}{2 f}-\frac{a^2 c \sec (e+f x) \tan (e+f x)}{2 f}-\frac{a^2 c \tan ^3(e+f x)}{3 f}\\ \end{align*}
Mathematica [A] time = 0.102027, size = 45, normalized size = 0.74 \[ \frac{a^2 c \left (-2 \tan ^3(e+f x)+3 \tanh ^{-1}(\sin (e+f x))-3 \tan (e+f x) \sec (e+f x)\right )}{6 f} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.02, size = 84, normalized size = 1.4 \begin{align*}{\frac{{a}^{2}c\tan \left ( fx+e \right ) }{3\,f}}+{\frac{{a}^{2}c\ln \left ( \sec \left ( fx+e \right ) +\tan \left ( fx+e \right ) \right ) }{2\,f}}-{\frac{{a}^{2}c\sec \left ( fx+e \right ) \tan \left ( fx+e \right ) }{2\,f}}-{\frac{{a}^{2}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 = 1.00824, size = 146, normalized size = 2.39 \begin{align*} -\frac{4 \,{\left (\tan \left (f x + e\right )^{3} + 3 \, \tan \left (f x + e\right )\right )} a^{2} c - 3 \, a^{2} 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^{2} c \log \left (\sec \left (f x + e\right ) + \tan \left (f x + e\right )\right ) - 12 \, a^{2} c \tan \left (f x + e\right )}{12 \, f} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 0.481272, size = 263, normalized size = 4.31 \begin{align*} \frac{3 \, a^{2} c \cos \left (f x + e\right )^{3} \log \left (\sin \left (f x + e\right ) + 1\right ) - 3 \, a^{2} c \cos \left (f x + e\right )^{3} \log \left (-\sin \left (f x + e\right ) + 1\right ) + 2 \,{\left (2 \, a^{2} c \cos \left (f x + e\right )^{2} - 3 \, a^{2} c \cos \left (f x + e\right ) - 2 \, a^{2} c\right )} \sin \left (f x + e\right )}{12 \, 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^{2} c \left (\int - \sec{\left (e + f x \right )}\, dx + \int - \sec ^{2}{\left (e + f x \right )}\, dx + \int \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 [B] time = 1.33076, size = 158, normalized size = 2.59 \begin{align*} \frac{3 \, a^{2} c \log \left ({\left | \tan \left (\frac{1}{2} \, f x + \frac{1}{2} \, e\right ) + 1 \right |}\right ) - 3 \, a^{2} c \log \left ({\left | \tan \left (\frac{1}{2} \, f x + \frac{1}{2} \, e\right ) - 1 \right |}\right ) - \frac{2 \,{\left (3 \, a^{2} c \tan \left (\frac{1}{2} \, f x + \frac{1}{2} \, e\right )^{5} - 8 \, a^{2} c \tan \left (\frac{1}{2} \, f x + \frac{1}{2} \, e\right )^{3} - 3 \, a^{2} 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}}}{6 \, f} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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