Optimal. Leaf size=32 \[ a x+\frac {b \tan ^2(c+d x)}{2 d}+\frac {b \log (\cos (c+d x))}{d} \]
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Rubi [A] time = 0.02, antiderivative size = 32, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, integrand size = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.167, Rules used = {3473, 3475} \[ a x+\frac {b \tan ^2(c+d x)}{2 d}+\frac {b \log (\cos (c+d x))}{d} \]
Antiderivative was successfully verified.
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Rule 3473
Rule 3475
Rubi steps
\begin {align*} \int \left (a+b \tan ^3(c+d x)\right ) \, dx &=a x+b \int \tan ^3(c+d x) \, dx\\ &=a x+\frac {b \tan ^2(c+d x)}{2 d}-b \int \tan (c+d x) \, dx\\ &=a x+\frac {b \log (\cos (c+d x))}{d}+\frac {b \tan ^2(c+d x)}{2 d}\\ \end {align*}
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Mathematica [A] time = 0.09, size = 30, normalized size = 0.94 \[ a x+\frac {b \left (\tan ^2(c+d x)+2 \log (\cos (c+d x))\right )}{2 d} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.53, size = 36, normalized size = 1.12 \[ \frac {2 \, a d x + b \tan \left (d x + c\right )^{2} + b \log \left (\frac {1}{\tan \left (d x + c\right )^{2} + 1}\right )}{2 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 1.62, size = 251, normalized size = 7.84 \[ a x + \frac {{\left (\log \left (\frac {4 \, {\left (\tan \left (d x\right )^{4} \tan \relax (c)^{2} - 2 \, \tan \left (d x\right )^{3} \tan \relax (c) + \tan \left (d x\right )^{2} \tan \relax (c)^{2} + \tan \left (d x\right )^{2} - 2 \, \tan \left (d x\right ) \tan \relax (c) + 1\right )}}{\tan \relax (c)^{2} + 1}\right ) \tan \left (d x\right )^{2} \tan \relax (c)^{2} + \tan \left (d x\right )^{2} \tan \relax (c)^{2} - 2 \, \log \left (\frac {4 \, {\left (\tan \left (d x\right )^{4} \tan \relax (c)^{2} - 2 \, \tan \left (d x\right )^{3} \tan \relax (c) + \tan \left (d x\right )^{2} \tan \relax (c)^{2} + \tan \left (d x\right )^{2} - 2 \, \tan \left (d x\right ) \tan \relax (c) + 1\right )}}{\tan \relax (c)^{2} + 1}\right ) \tan \left (d x\right ) \tan \relax (c) + \tan \left (d x\right )^{2} + \tan \relax (c)^{2} + \log \left (\frac {4 \, {\left (\tan \left (d x\right )^{4} \tan \relax (c)^{2} - 2 \, \tan \left (d x\right )^{3} \tan \relax (c) + \tan \left (d x\right )^{2} \tan \relax (c)^{2} + \tan \left (d x\right )^{2} - 2 \, \tan \left (d x\right ) \tan \relax (c) + 1\right )}}{\tan \relax (c)^{2} + 1}\right ) + 1\right )} b}{2 \, {\left (d \tan \left (d x\right )^{2} \tan \relax (c)^{2} - 2 \, d \tan \left (d x\right ) \tan \relax (c) + d\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.02, size = 36, normalized size = 1.12 \[ a x +\frac {b \left (\tan ^{2}\left (d x +c \right )\right )}{2 d}-\frac {b \ln \left (1+\tan ^{2}\left (d x +c \right )\right )}{2 d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.30, size = 36, normalized size = 1.12 \[ a x - \frac {b {\left (\frac {1}{\sin \left (d x + c\right )^{2} - 1} - \log \left (\sin \left (d x + c\right )^{2} - 1\right )\right )}}{2 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 11.53, size = 34, normalized size = 1.06 \[ \frac {\frac {b\,{\mathrm {tan}\left (c+d\,x\right )}^2}{2}-\frac {b\,\ln \left ({\mathrm {tan}\left (c+d\,x\right )}^2+1\right )}{2}+a\,d\,x}{d} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.18, size = 37, normalized size = 1.16 \[ a x + b \left (\begin {cases} - \frac {\log {\left (\tan ^{2}{\left (c + d x \right )} + 1 \right )}}{2 d} + \frac {\tan ^{2}{\left (c + d x \right )}}{2 d} & \text {for}\: d \neq 0 \\x \tan ^{3}{\relax (c )} & \text {otherwise} \end {cases}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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