Optimal. Leaf size=108 \[ \frac{2 a \left (c^2+3 c d+d^2\right ) \tan (e+f x)}{3 f}+\frac{a \left (2 c^2+2 c d+d^2\right ) \tanh ^{-1}(\sin (e+f x))}{2 f}+\frac{a \tan (e+f x) (c+d \sec (e+f x))^2}{3 f}+\frac{a d (2 c+3 d) \tan (e+f x) \sec (e+f x)}{6 f} \]
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Rubi [A] time = 0.165014, antiderivative size = 108, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 6, integrand size = 29, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.207, Rules used = {4002, 3997, 3787, 3770, 3767, 8} \[ \frac{2 a \left (c^2+3 c d+d^2\right ) \tan (e+f x)}{3 f}+\frac{a \left (2 c^2+2 c d+d^2\right ) \tanh ^{-1}(\sin (e+f x))}{2 f}+\frac{a \tan (e+f x) (c+d \sec (e+f x))^2}{3 f}+\frac{a d (2 c+3 d) \tan (e+f x) \sec (e+f x)}{6 f} \]
Antiderivative was successfully verified.
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Rule 4002
Rule 3997
Rule 3787
Rule 3770
Rule 3767
Rule 8
Rubi steps
\begin{align*} \int \sec (e+f x) (a+a \sec (e+f x)) (c+d \sec (e+f x))^2 \, dx &=\frac{a (c+d \sec (e+f x))^2 \tan (e+f x)}{3 f}+\frac{1}{3} \int \sec (e+f x) (c+d \sec (e+f x)) (a (3 c+2 d)+a (2 c+3 d) \sec (e+f x)) \, dx\\ &=\frac{a d (2 c+3 d) \sec (e+f x) \tan (e+f x)}{6 f}+\frac{a (c+d \sec (e+f x))^2 \tan (e+f x)}{3 f}+\frac{1}{6} \int \sec (e+f x) \left (3 a \left (2 c^2+2 c d+d^2\right )+4 a \left (c^2+3 c d+d^2\right ) \sec (e+f x)\right ) \, dx\\ &=\frac{a d (2 c+3 d) \sec (e+f x) \tan (e+f x)}{6 f}+\frac{a (c+d \sec (e+f x))^2 \tan (e+f x)}{3 f}+\frac{1}{2} \left (a \left (2 c^2+2 c d+d^2\right )\right ) \int \sec (e+f x) \, dx+\frac{1}{3} \left (2 a \left (c^2+3 c d+d^2\right )\right ) \int \sec ^2(e+f x) \, dx\\ &=\frac{a \left (2 c^2+2 c d+d^2\right ) \tanh ^{-1}(\sin (e+f x))}{2 f}+\frac{a d (2 c+3 d) \sec (e+f x) \tan (e+f x)}{6 f}+\frac{a (c+d \sec (e+f x))^2 \tan (e+f x)}{3 f}-\frac{\left (2 a \left (c^2+3 c d+d^2\right )\right ) \operatorname{Subst}(\int 1 \, dx,x,-\tan (e+f x))}{3 f}\\ &=\frac{a \left (2 c^2+2 c d+d^2\right ) \tanh ^{-1}(\sin (e+f x))}{2 f}+\frac{2 a \left (c^2+3 c d+d^2\right ) \tan (e+f x)}{3 f}+\frac{a d (2 c+3 d) \sec (e+f x) \tan (e+f x)}{6 f}+\frac{a (c+d \sec (e+f x))^2 \tan (e+f x)}{3 f}\\ \end{align*}
Mathematica [A] time = 0.413438, size = 75, normalized size = 0.69 \[ \frac{a \left (3 \left (2 c^2+2 c d+d^2\right ) \tanh ^{-1}(\sin (e+f x))+\tan (e+f x) \left (2 \left (3 (c+d)^2+d^2 \tan ^2(e+f x)\right )+3 d (2 c+d) \sec (e+f x)\right )\right )}{6 f} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.042, size = 174, normalized size = 1.6 \begin{align*}{\frac{{c}^{2}a\ln \left ( \sec \left ( fx+e \right ) +\tan \left ( fx+e \right ) \right ) }{f}}+2\,{\frac{acd\tan \left ( fx+e \right ) }{f}}+{\frac{a{d}^{2}\sec \left ( fx+e \right ) \tan \left ( fx+e \right ) }{2\,f}}+{\frac{a{d}^{2}\ln \left ( \sec \left ( fx+e \right ) +\tan \left ( fx+e \right ) \right ) }{2\,f}}+{\frac{{c}^{2}a\tan \left ( fx+e \right ) }{f}}+{\frac{acd\sec \left ( fx+e \right ) \tan \left ( fx+e \right ) }{f}}+{\frac{acd\ln \left ( \sec \left ( fx+e \right ) +\tan \left ( fx+e \right ) \right ) }{f}}+{\frac{2\,a{d}^{2}\tan \left ( fx+e \right ) }{3\,f}}+{\frac{a{d}^{2}\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 = 0.964594, size = 223, normalized size = 2.06 \begin{align*} \frac{4 \,{\left (\tan \left (f x + e\right )^{3} + 3 \, \tan \left (f x + e\right )\right )} a d^{2} - 6 \, a c d{\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 )} - 3 \, a d^{2}{\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 c^{2} \log \left (\sec \left (f x + e\right ) + \tan \left (f x + e\right )\right ) + 12 \, a c^{2} \tan \left (f x + e\right ) + 24 \, a c d \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 = 0.494363, size = 371, normalized size = 3.44 \begin{align*} \frac{3 \,{\left (2 \, a c^{2} + 2 \, a c d + a d^{2}\right )} \cos \left (f x + e\right )^{3} \log \left (\sin \left (f x + e\right ) + 1\right ) - 3 \,{\left (2 \, a c^{2} + 2 \, a c d + a d^{2}\right )} \cos \left (f x + e\right )^{3} \log \left (-\sin \left (f x + e\right ) + 1\right ) + 2 \,{\left (2 \, a d^{2} + 2 \,{\left (3 \, a c^{2} + 6 \, a c d + 2 \, a d^{2}\right )} \cos \left (f x + e\right )^{2} + 3 \,{\left (2 \, a c d + a d^{2}\right )} \cos \left (f x + e\right )\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 \left (\int c^{2} \sec{\left (e + f x \right )}\, dx + \int c^{2} \sec ^{2}{\left (e + f x \right )}\, dx + \int d^{2} \sec ^{3}{\left (e + f x \right )}\, dx + \int d^{2} \sec ^{4}{\left (e + f x \right )}\, dx + \int 2 c d \sec ^{2}{\left (e + f x \right )}\, dx + \int 2 c d \sec ^{3}{\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 = 1.62735, size = 329, normalized size = 3.05 \begin{align*} \frac{3 \,{\left (2 \, a c^{2} + 2 \, a c d + a d^{2}\right )} \log \left ({\left | \tan \left (\frac{1}{2} \, f x + \frac{1}{2} \, e\right ) + 1 \right |}\right ) - 3 \,{\left (2 \, a c^{2} + 2 \, a c d + a d^{2}\right )} \log \left ({\left | \tan \left (\frac{1}{2} \, f x + \frac{1}{2} \, e\right ) - 1 \right |}\right ) - \frac{2 \,{\left (6 \, a c^{2} \tan \left (\frac{1}{2} \, f x + \frac{1}{2} \, e\right )^{5} + 6 \, a c d \tan \left (\frac{1}{2} \, f x + \frac{1}{2} \, e\right )^{5} + 3 \, a d^{2} \tan \left (\frac{1}{2} \, f x + \frac{1}{2} \, e\right )^{5} - 12 \, a c^{2} \tan \left (\frac{1}{2} \, f x + \frac{1}{2} \, e\right )^{3} - 24 \, a c d \tan \left (\frac{1}{2} \, f x + \frac{1}{2} \, e\right )^{3} - 4 \, a d^{2} \tan \left (\frac{1}{2} \, f x + \frac{1}{2} \, e\right )^{3} + 6 \, a c^{2} \tan \left (\frac{1}{2} \, f x + \frac{1}{2} \, e\right ) + 18 \, a c d \tan \left (\frac{1}{2} \, f x + \frac{1}{2} \, e\right ) + 9 \, a d^{2} \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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