Optimal. Leaf size=171 \[ \frac{a \left (16 c^2 d+3 c^3+12 c d^2+4 d^3\right ) \tan (e+f x)}{6 f}+\frac{a \left (12 c^2 d+8 c^3+12 c d^2+3 d^3\right ) \tanh ^{-1}(\sin (e+f x))}{8 f}+\frac{a d \left (6 c^2+20 c d+9 d^2\right ) \tan (e+f x) \sec (e+f x)}{24 f}+\frac{a \tan (e+f x) (c+d \sec (e+f x))^3}{4 f}+\frac{a (3 c+4 d) \tan (e+f x) (c+d \sec (e+f x))^2}{12 f} \]
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Rubi [A] time = 0.295138, antiderivative size = 171, normalized size of antiderivative = 1., number of steps used = 7, 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} \[ \frac{a \left (16 c^2 d+3 c^3+12 c d^2+4 d^3\right ) \tan (e+f x)}{6 f}+\frac{a \left (12 c^2 d+8 c^3+12 c d^2+3 d^3\right ) \tanh ^{-1}(\sin (e+f x))}{8 f}+\frac{a d \left (6 c^2+20 c d+9 d^2\right ) \tan (e+f x) \sec (e+f x)}{24 f}+\frac{a \tan (e+f x) (c+d \sec (e+f x))^3}{4 f}+\frac{a (3 c+4 d) \tan (e+f x) (c+d \sec (e+f x))^2}{12 f} \]
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))^3 \, dx &=\frac{a (c+d \sec (e+f x))^3 \tan (e+f x)}{4 f}+\frac{1}{4} \int \sec (e+f x) (c+d \sec (e+f x))^2 (a (4 c+3 d)+a (3 c+4 d) \sec (e+f x)) \, dx\\ &=\frac{a (3 c+4 d) (c+d \sec (e+f x))^2 \tan (e+f x)}{12 f}+\frac{a (c+d \sec (e+f x))^3 \tan (e+f x)}{4 f}+\frac{1}{12} \int \sec (e+f x) (c+d \sec (e+f x)) \left (a \left (12 c^2+15 c d+8 d^2\right )+a \left (6 c^2+20 c d+9 d^2\right ) \sec (e+f x)\right ) \, dx\\ &=\frac{a d \left (6 c^2+20 c d+9 d^2\right ) \sec (e+f x) \tan (e+f x)}{24 f}+\frac{a (3 c+4 d) (c+d \sec (e+f x))^2 \tan (e+f x)}{12 f}+\frac{a (c+d \sec (e+f x))^3 \tan (e+f x)}{4 f}+\frac{1}{24} \int \sec (e+f x) \left (3 a \left (8 c^3+12 c^2 d+12 c d^2+3 d^3\right )+4 a \left (3 c^3+16 c^2 d+12 c d^2+4 d^3\right ) \sec (e+f x)\right ) \, dx\\ &=\frac{a d \left (6 c^2+20 c d+9 d^2\right ) \sec (e+f x) \tan (e+f x)}{24 f}+\frac{a (3 c+4 d) (c+d \sec (e+f x))^2 \tan (e+f x)}{12 f}+\frac{a (c+d \sec (e+f x))^3 \tan (e+f x)}{4 f}+\frac{1}{8} \left (a \left (8 c^3+12 c^2 d+12 c d^2+3 d^3\right )\right ) \int \sec (e+f x) \, dx+\frac{1}{6} \left (a \left (3 c^3+16 c^2 d+12 c d^2+4 d^3\right )\right ) \int \sec ^2(e+f x) \, dx\\ &=\frac{a \left (8 c^3+12 c^2 d+12 c d^2+3 d^3\right ) \tanh ^{-1}(\sin (e+f x))}{8 f}+\frac{a d \left (6 c^2+20 c d+9 d^2\right ) \sec (e+f x) \tan (e+f x)}{24 f}+\frac{a (3 c+4 d) (c+d \sec (e+f x))^2 \tan (e+f x)}{12 f}+\frac{a (c+d \sec (e+f x))^3 \tan (e+f x)}{4 f}-\frac{\left (a \left (3 c^3+16 c^2 d+12 c d^2+4 d^3\right )\right ) \operatorname{Subst}(\int 1 \, dx,x,-\tan (e+f x))}{6 f}\\ &=\frac{a \left (8 c^3+12 c^2 d+12 c d^2+3 d^3\right ) \tanh ^{-1}(\sin (e+f x))}{8 f}+\frac{a \left (3 c^3+16 c^2 d+12 c d^2+4 d^3\right ) \tan (e+f x)}{6 f}+\frac{a d \left (6 c^2+20 c d+9 d^2\right ) \sec (e+f x) \tan (e+f x)}{24 f}+\frac{a (3 c+4 d) (c+d \sec (e+f x))^2 \tan (e+f x)}{12 f}+\frac{a (c+d \sec (e+f x))^3 \tan (e+f x)}{4 f}\\ \end{align*}
Mathematica [A] time = 0.677162, size = 103, normalized size = 0.6 \[ \frac{a \left (3 \left (12 c^2 d+8 c^3+12 c d^2+3 d^3\right ) \tanh ^{-1}(\sin (e+f x))+\tan (e+f x) \left (8 d^2 (3 c+d) \tan ^2(e+f x)+9 d (2 c+d)^2 \sec (e+f x)+24 (c+d)^3+6 d^3 \sec ^3(e+f x)\right )\right )}{24 f} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.049, size = 290, normalized size = 1.7 \begin{align*}{\frac{a{c}^{3}\ln \left ( \sec \left ( fx+e \right ) +\tan \left ( fx+e \right ) \right ) }{f}}+3\,{\frac{a{c}^{2}d\tan \left ( fx+e \right ) }{f}}+{\frac{3\,a{d}^{2}c\sec \left ( fx+e \right ) \tan \left ( fx+e \right ) }{2\,f}}+{\frac{3\,a{d}^{2}c\ln \left ( \sec \left ( fx+e \right ) +\tan \left ( fx+e \right ) \right ) }{2\,f}}+{\frac{2\,a{d}^{3}\tan \left ( fx+e \right ) }{3\,f}}+{\frac{a{d}^{3}\tan \left ( fx+e \right ) \left ( \sec \left ( fx+e \right ) \right ) ^{2}}{3\,f}}+{\frac{a{c}^{3}\tan \left ( fx+e \right ) }{f}}+{\frac{3\,a{c}^{2}d\sec \left ( fx+e \right ) \tan \left ( fx+e \right ) }{2\,f}}+{\frac{3\,a{c}^{2}d\ln \left ( \sec \left ( fx+e \right ) +\tan \left ( fx+e \right ) \right ) }{2\,f}}+2\,{\frac{a{d}^{2}c\tan \left ( fx+e \right ) }{f}}+{\frac{a{d}^{2}c\tan \left ( fx+e \right ) \left ( \sec \left ( fx+e \right ) \right ) ^{2}}{f}}+{\frac{a{d}^{3}\tan \left ( fx+e \right ) \left ( \sec \left ( fx+e \right ) \right ) ^{3}}{4\,f}}+{\frac{3\,a{d}^{3}\sec \left ( fx+e \right ) \tan \left ( fx+e \right ) }{8\,f}}+{\frac{3\,a{d}^{3}\ln \left ( \sec \left ( fx+e \right ) +\tan \left ( fx+e \right ) \right ) }{8\,f}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 0.98131, size = 359, normalized size = 2.1 \begin{align*} \frac{48 \,{\left (\tan \left (f x + e\right )^{3} + 3 \, \tan \left (f x + e\right )\right )} a c d^{2} + 16 \,{\left (\tan \left (f x + e\right )^{3} + 3 \, \tan \left (f x + e\right )\right )} a d^{3} - 3 \, a d^{3}{\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 )} - 36 \, a c^{2} 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 )} - 36 \, a c 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 )} + 48 \, a c^{3} \log \left (\sec \left (f x + e\right ) + \tan \left (f x + e\right )\right ) + 48 \, a c^{3} \tan \left (f x + e\right ) + 144 \, a c^{2} d \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 = 0.514157, size = 510, normalized size = 2.98 \begin{align*} \frac{3 \,{\left (8 \, a c^{3} + 12 \, a c^{2} d + 12 \, a c d^{2} + 3 \, a d^{3}\right )} \cos \left (f x + e\right )^{4} \log \left (\sin \left (f x + e\right ) + 1\right ) - 3 \,{\left (8 \, a c^{3} + 12 \, a c^{2} d + 12 \, a c d^{2} + 3 \, a d^{3}\right )} \cos \left (f x + e\right )^{4} \log \left (-\sin \left (f x + e\right ) + 1\right ) + 2 \,{\left (6 \, a d^{3} + 8 \,{\left (3 \, a c^{3} + 9 \, a c^{2} d + 6 \, a c d^{2} + 2 \, a d^{3}\right )} \cos \left (f x + e\right )^{3} + 9 \,{\left (4 \, a c^{2} d + 4 \, a c d^{2} + a d^{3}\right )} \cos \left (f x + e\right )^{2} + 8 \,{\left (3 \, a c d^{2} + a d^{3}\right )} \cos \left (f x + e\right )\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 \left (\int c^{3} \sec{\left (e + f x \right )}\, dx + \int c^{3} \sec ^{2}{\left (e + f x \right )}\, dx + \int d^{3} \sec ^{4}{\left (e + f x \right )}\, dx + \int d^{3} \sec ^{5}{\left (e + f x \right )}\, dx + \int 3 c d^{2} \sec ^{3}{\left (e + f x \right )}\, dx + \int 3 c d^{2} \sec ^{4}{\left (e + f x \right )}\, dx + \int 3 c^{2} d \sec ^{2}{\left (e + f x \right )}\, dx + \int 3 c^{2} 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 = 1.62895, size = 539, normalized size = 3.15 \begin{align*} \frac{3 \,{\left (8 \, a c^{3} + 12 \, a c^{2} d + 12 \, a c d^{2} + 3 \, a d^{3}\right )} \log \left ({\left | \tan \left (\frac{1}{2} \, f x + \frac{1}{2} \, e\right ) + 1 \right |}\right ) - 3 \,{\left (8 \, a c^{3} + 12 \, a c^{2} d + 12 \, a c d^{2} + 3 \, a d^{3}\right )} \log \left ({\left | \tan \left (\frac{1}{2} \, f x + \frac{1}{2} \, e\right ) - 1 \right |}\right ) - \frac{2 \,{\left (24 \, a c^{3} \tan \left (\frac{1}{2} \, f x + \frac{1}{2} \, e\right )^{7} + 36 \, a c^{2} d \tan \left (\frac{1}{2} \, f x + \frac{1}{2} \, e\right )^{7} + 36 \, a c d^{2} \tan \left (\frac{1}{2} \, f x + \frac{1}{2} \, e\right )^{7} + 9 \, a d^{3} \tan \left (\frac{1}{2} \, f x + \frac{1}{2} \, e\right )^{7} - 72 \, a c^{3} \tan \left (\frac{1}{2} \, f x + \frac{1}{2} \, e\right )^{5} - 180 \, a c^{2} d \tan \left (\frac{1}{2} \, f x + \frac{1}{2} \, e\right )^{5} - 84 \, a c d^{2} \tan \left (\frac{1}{2} \, f x + \frac{1}{2} \, e\right )^{5} - 49 \, a d^{3} \tan \left (\frac{1}{2} \, f x + \frac{1}{2} \, e\right )^{5} + 72 \, a c^{3} \tan \left (\frac{1}{2} \, f x + \frac{1}{2} \, e\right )^{3} + 252 \, a c^{2} d \tan \left (\frac{1}{2} \, f x + \frac{1}{2} \, e\right )^{3} + 156 \, a c d^{2} \tan \left (\frac{1}{2} \, f x + \frac{1}{2} \, e\right )^{3} + 31 \, a d^{3} \tan \left (\frac{1}{2} \, f x + \frac{1}{2} \, e\right )^{3} - 24 \, a c^{3} \tan \left (\frac{1}{2} \, f x + \frac{1}{2} \, e\right ) - 108 \, a c^{2} d \tan \left (\frac{1}{2} \, f x + \frac{1}{2} \, e\right ) - 108 \, a c d^{2} \tan \left (\frac{1}{2} \, f x + \frac{1}{2} \, e\right ) - 39 \, a d^{3} \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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