Optimal. Leaf size=135 \[ \frac{\left (5 a^2+4 b^2\right ) \tan ^3(c+d x)}{15 d}+\frac{\left (5 a^2+4 b^2\right ) \tan (c+d x)}{5 d}+\frac{3 a b \tanh ^{-1}(\sin (c+d x))}{4 d}+\frac{a b \tan (c+d x) \sec ^3(c+d x)}{2 d}+\frac{3 a b \tan (c+d x) \sec (c+d x)}{4 d}+\frac{b^2 \tan (c+d x) \sec ^4(c+d x)}{5 d} \]
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Rubi [A] time = 0.106355, antiderivative size = 135, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 5, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.238, Rules used = {3788, 3768, 3770, 4046, 3767} \[ \frac{\left (5 a^2+4 b^2\right ) \tan ^3(c+d x)}{15 d}+\frac{\left (5 a^2+4 b^2\right ) \tan (c+d x)}{5 d}+\frac{3 a b \tanh ^{-1}(\sin (c+d x))}{4 d}+\frac{a b \tan (c+d x) \sec ^3(c+d x)}{2 d}+\frac{3 a b \tan (c+d x) \sec (c+d x)}{4 d}+\frac{b^2 \tan (c+d x) \sec ^4(c+d x)}{5 d} \]
Antiderivative was successfully verified.
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Rule 3788
Rule 3768
Rule 3770
Rule 4046
Rule 3767
Rubi steps
\begin{align*} \int \sec ^4(c+d x) (a+b \sec (c+d x))^2 \, dx &=(2 a b) \int \sec ^5(c+d x) \, dx+\int \sec ^4(c+d x) \left (a^2+b^2 \sec ^2(c+d x)\right ) \, dx\\ &=\frac{a b \sec ^3(c+d x) \tan (c+d x)}{2 d}+\frac{b^2 \sec ^4(c+d x) \tan (c+d x)}{5 d}+\frac{1}{2} (3 a b) \int \sec ^3(c+d x) \, dx+\frac{1}{5} \left (5 a^2+4 b^2\right ) \int \sec ^4(c+d x) \, dx\\ &=\frac{3 a b \sec (c+d x) \tan (c+d x)}{4 d}+\frac{a b \sec ^3(c+d x) \tan (c+d x)}{2 d}+\frac{b^2 \sec ^4(c+d x) \tan (c+d x)}{5 d}+\frac{1}{4} (3 a b) \int \sec (c+d x) \, dx-\frac{\left (5 a^2+4 b^2\right ) \operatorname{Subst}\left (\int \left (1+x^2\right ) \, dx,x,-\tan (c+d x)\right )}{5 d}\\ &=\frac{3 a b \tanh ^{-1}(\sin (c+d x))}{4 d}+\frac{\left (5 a^2+4 b^2\right ) \tan (c+d x)}{5 d}+\frac{3 a b \sec (c+d x) \tan (c+d x)}{4 d}+\frac{a b \sec ^3(c+d x) \tan (c+d x)}{2 d}+\frac{b^2 \sec ^4(c+d x) \tan (c+d x)}{5 d}+\frac{\left (5 a^2+4 b^2\right ) \tan ^3(c+d x)}{15 d}\\ \end{align*}
Mathematica [A] time = 0.599066, size = 118, normalized size = 0.87 \[ \frac{a^2 \left (\frac{1}{3} \tan ^3(c+d x)+\tan (c+d x)\right )}{d}+\frac{a b \tan (c+d x) \sec ^3(c+d x)}{2 d}+\frac{3 a b \left (\tanh ^{-1}(\sin (c+d x))+\tan (c+d x) \sec (c+d x)\right )}{4 d}+\frac{b^2 \left (\frac{1}{5} \tan ^5(c+d x)+\frac{2}{3} \tan ^3(c+d x)+\tan (c+d x)\right )}{d} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.031, size = 157, normalized size = 1.2 \begin{align*}{\frac{2\,{a}^{2}\tan \left ( dx+c \right ) }{3\,d}}+{\frac{{a}^{2} \left ( \sec \left ( dx+c \right ) \right ) ^{2}\tan \left ( dx+c \right ) }{3\,d}}+{\frac{ab \left ( \sec \left ( dx+c \right ) \right ) ^{3}\tan \left ( dx+c \right ) }{2\,d}}+{\frac{3\,ab\sec \left ( dx+c \right ) \tan \left ( dx+c \right ) }{4\,d}}+{\frac{3\,ab\ln \left ( \sec \left ( dx+c \right ) +\tan \left ( dx+c \right ) \right ) }{4\,d}}+{\frac{8\,{b}^{2}\tan \left ( dx+c \right ) }{15\,d}}+{\frac{{b}^{2} \left ( \sec \left ( dx+c \right ) \right ) ^{4}\tan \left ( dx+c \right ) }{5\,d}}+{\frac{4\,{b}^{2} \left ( \sec \left ( dx+c \right ) \right ) ^{2}\tan \left ( dx+c \right ) }{15\,d}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.20733, size = 178, normalized size = 1.32 \begin{align*} \frac{40 \,{\left (\tan \left (d x + c\right )^{3} + 3 \, \tan \left (d x + c\right )\right )} a^{2} + 8 \,{\left (3 \, \tan \left (d x + c\right )^{5} + 10 \, \tan \left (d x + c\right )^{3} + 15 \, \tan \left (d x + c\right )\right )} b^{2} - 15 \, a b{\left (\frac{2 \,{\left (3 \, \sin \left (d x + c\right )^{3} - 5 \, \sin \left (d x + c\right )\right )}}{\sin \left (d x + c\right )^{4} - 2 \, \sin \left (d x + c\right )^{2} + 1} - 3 \, \log \left (\sin \left (d x + c\right ) + 1\right ) + 3 \, \log \left (\sin \left (d x + c\right ) - 1\right )\right )}}{120 \, d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.82317, size = 352, normalized size = 2.61 \begin{align*} \frac{45 \, a b \cos \left (d x + c\right )^{5} \log \left (\sin \left (d x + c\right ) + 1\right ) - 45 \, a b \cos \left (d x + c\right )^{5} \log \left (-\sin \left (d x + c\right ) + 1\right ) + 2 \,{\left (45 \, a b \cos \left (d x + c\right )^{3} + 8 \,{\left (5 \, a^{2} + 4 \, b^{2}\right )} \cos \left (d x + c\right )^{4} + 30 \, a b \cos \left (d x + c\right ) + 4 \,{\left (5 \, a^{2} + 4 \, b^{2}\right )} \cos \left (d x + c\right )^{2} + 12 \, b^{2}\right )} \sin \left (d x + c\right )}{120 \, d \cos \left (d x + c\right )^{5}} \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*} \int \left (a + b \sec{\left (c + d x \right )}\right )^{2} \sec ^{4}{\left (c + d x \right )}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] time = 1.39754, size = 367, normalized size = 2.72 \begin{align*} \frac{45 \, a b \log \left ({\left | \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right ) + 1 \right |}\right ) - 45 \, a b \log \left ({\left | \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right ) - 1 \right |}\right ) - \frac{2 \,{\left (60 \, a^{2} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{9} - 75 \, a b \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{9} + 60 \, b^{2} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{9} - 160 \, a^{2} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{7} + 30 \, a b \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{7} - 80 \, b^{2} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{7} + 200 \, a^{2} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{5} + 232 \, b^{2} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{5} - 160 \, a^{2} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{3} - 30 \, a b \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{3} - 80 \, b^{2} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{3} + 60 \, a^{2} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right ) + 75 \, a b \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right ) + 60 \, b^{2} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )\right )}}{{\left (\tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{2} - 1\right )}^{5}}}{60 \, d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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