Optimal. Leaf size=131 \[ \frac{\left (6 a^2-b^2\right ) \tanh ^{-1}(\sin (c+d x))}{16 d}+\frac{\left (6 a^2-b^2\right ) \tan (c+d x) \sec ^3(c+d x)}{24 d}+\frac{\left (6 a^2-b^2\right ) \tan (c+d x) \sec (c+d x)}{16 d}+\frac{7 a b \sec ^5(c+d x)}{30 d}+\frac{b \sec ^5(c+d x) (a+b \tan (c+d x))}{6 d} \]
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Rubi [A] time = 0.115188, antiderivative size = 131, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 4, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.19, Rules used = {3508, 3486, 3768, 3770} \[ \frac{\left (6 a^2-b^2\right ) \tanh ^{-1}(\sin (c+d x))}{16 d}+\frac{\left (6 a^2-b^2\right ) \tan (c+d x) \sec ^3(c+d x)}{24 d}+\frac{\left (6 a^2-b^2\right ) \tan (c+d x) \sec (c+d x)}{16 d}+\frac{7 a b \sec ^5(c+d x)}{30 d}+\frac{b \sec ^5(c+d x) (a+b \tan (c+d x))}{6 d} \]
Antiderivative was successfully verified.
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Rule 3508
Rule 3486
Rule 3768
Rule 3770
Rubi steps
\begin{align*} \int \sec ^5(c+d x) (a+b \tan (c+d x))^2 \, dx &=\frac{b \sec ^5(c+d x) (a+b \tan (c+d x))}{6 d}+\frac{1}{6} \int \sec ^5(c+d x) \left (6 a^2-b^2+7 a b \tan (c+d x)\right ) \, dx\\ &=\frac{7 a b \sec ^5(c+d x)}{30 d}+\frac{b \sec ^5(c+d x) (a+b \tan (c+d x))}{6 d}+\frac{1}{6} \left (6 a^2-b^2\right ) \int \sec ^5(c+d x) \, dx\\ &=\frac{7 a b \sec ^5(c+d x)}{30 d}+\frac{\left (6 a^2-b^2\right ) \sec ^3(c+d x) \tan (c+d x)}{24 d}+\frac{b \sec ^5(c+d x) (a+b \tan (c+d x))}{6 d}+\frac{1}{8} \left (6 a^2-b^2\right ) \int \sec ^3(c+d x) \, dx\\ &=\frac{7 a b \sec ^5(c+d x)}{30 d}+\frac{\left (6 a^2-b^2\right ) \sec (c+d x) \tan (c+d x)}{16 d}+\frac{\left (6 a^2-b^2\right ) \sec ^3(c+d x) \tan (c+d x)}{24 d}+\frac{b \sec ^5(c+d x) (a+b \tan (c+d x))}{6 d}+\frac{1}{16} \left (6 a^2-b^2\right ) \int \sec (c+d x) \, dx\\ &=\frac{\left (6 a^2-b^2\right ) \tanh ^{-1}(\sin (c+d x))}{16 d}+\frac{7 a b \sec ^5(c+d x)}{30 d}+\frac{\left (6 a^2-b^2\right ) \sec (c+d x) \tan (c+d x)}{16 d}+\frac{\left (6 a^2-b^2\right ) \sec ^3(c+d x) \tan (c+d x)}{24 d}+\frac{b \sec ^5(c+d x) (a+b \tan (c+d x))}{6 d}\\ \end{align*}
Mathematica [A] time = 0.504837, size = 104, normalized size = 0.79 \[ \frac{15 \left (6 a^2-b^2\right ) \tanh ^{-1}(\sin (c+d x))+10 \left (6 a^2-b^2\right ) \tan (c+d x) \sec ^3(c+d x)+15 \left (6 a^2-b^2\right ) \tan (c+d x) \sec (c+d x)+8 b \sec ^5(c+d x) (12 a+5 b \tan (c+d x))}{240 d} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.052, size = 189, normalized size = 1.4 \begin{align*}{\frac{{b}^{2} \left ( \sin \left ( dx+c \right ) \right ) ^{3}}{6\,d \left ( \cos \left ( dx+c \right ) \right ) ^{6}}}+{\frac{{b}^{2} \left ( \sin \left ( dx+c \right ) \right ) ^{3}}{8\,d \left ( \cos \left ( dx+c \right ) \right ) ^{4}}}+{\frac{{b}^{2} \left ( \sin \left ( dx+c \right ) \right ) ^{3}}{16\,d \left ( \cos \left ( dx+c \right ) \right ) ^{2}}}+{\frac{\sin \left ( dx+c \right ){b}^{2}}{16\,d}}-{\frac{{b}^{2}\ln \left ( \sec \left ( dx+c \right ) +\tan \left ( dx+c \right ) \right ) }{16\,d}}+{\frac{2\,ab}{5\,d \left ( \cos \left ( dx+c \right ) \right ) ^{5}}}+{\frac{{a}^{2} \left ( \sec \left ( dx+c \right ) \right ) ^{3}\tan \left ( dx+c \right ) }{4\,d}}+{\frac{3\,{a}^{2}\sec \left ( dx+c \right ) \tan \left ( dx+c \right ) }{8\,d}}+{\frac{3\,{a}^{2}\ln \left ( \sec \left ( dx+c \right ) +\tan \left ( dx+c \right ) \right ) }{8\,d}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.41491, size = 243, normalized size = 1.85 \begin{align*} \frac{5 \, b^{2}{\left (\frac{2 \,{\left (3 \, \sin \left (d x + c\right )^{5} - 8 \, \sin \left (d x + c\right )^{3} - 3 \, \sin \left (d x + c\right )\right )}}{\sin \left (d x + c\right )^{6} - 3 \, \sin \left (d x + c\right )^{4} + 3 \, \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 )} - 30 \, a^{2}{\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 )} + \frac{192 \, a b}{\cos \left (d x + c\right )^{5}}}{480 \, d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.93548, size = 343, normalized size = 2.62 \begin{align*} \frac{15 \,{\left (6 \, a^{2} - b^{2}\right )} \cos \left (d x + c\right )^{6} \log \left (\sin \left (d x + c\right ) + 1\right ) - 15 \,{\left (6 \, a^{2} - b^{2}\right )} \cos \left (d x + c\right )^{6} \log \left (-\sin \left (d x + c\right ) + 1\right ) + 192 \, a b \cos \left (d x + c\right ) + 10 \,{\left (3 \,{\left (6 \, a^{2} - b^{2}\right )} \cos \left (d x + c\right )^{4} + 2 \,{\left (6 \, a^{2} - b^{2}\right )} \cos \left (d x + c\right )^{2} + 8 \, b^{2}\right )} \sin \left (d x + c\right )}{480 \, d \cos \left (d x + c\right )^{6}} \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 \tan{\left (c + d x \right )}\right )^{2} \sec ^{5}{\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 = 1.55437, size = 463, normalized size = 3.53 \begin{align*} \frac{15 \,{\left (6 \, a^{2} - b^{2}\right )} \log \left ({\left | \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right ) + 1 \right |}\right ) - 15 \,{\left (6 \, a^{2} - b^{2}\right )} \log \left ({\left | \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right ) - 1 \right |}\right ) + \frac{2 \,{\left (150 \, a^{2} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{11} + 15 \, b^{2} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{11} - 480 \, a b \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{10} - 210 \, a^{2} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{9} + 235 \, b^{2} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{9} + 480 \, a b \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{8} + 60 \, a^{2} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{7} + 390 \, b^{2} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{7} - 960 \, a b \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{6} + 60 \, a^{2} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{5} + 390 \, b^{2} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{5} + 960 \, a b \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{4} - 210 \, a^{2} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{3} + 235 \, b^{2} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{3} - 96 \, a b \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{2} + 150 \, a^{2} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right ) + 15 \, b^{2} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right ) + 96 \, a b\right )}}{{\left (\tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{2} - 1\right )}^{6}}}{240 \, d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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