Optimal. Leaf size=27 \[ \frac{\tan ^2(c+d x)}{2 d}+\frac{\log (\cos (c+d x))}{d} \]
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Rubi [A] time = 0.0114313, antiderivative size = 27, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 8, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.25, Rules used = {3473, 3475} \[ \frac{\tan ^2(c+d x)}{2 d}+\frac{\log (\cos (c+d x))}{d} \]
Antiderivative was successfully verified.
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Rule 3473
Rule 3475
Rubi steps
\begin{align*} \int \tan ^3(c+d x) \, dx &=\frac{\tan ^2(c+d x)}{2 d}-\int \tan (c+d x) \, dx\\ &=\frac{\log (\cos (c+d x))}{d}+\frac{\tan ^2(c+d x)}{2 d}\\ \end{align*}
Mathematica [A] time = 0.0240356, size = 25, normalized size = 0.93 \[ \frac{\tan ^2(c+d x)+2 \log (\cos (c+d x))}{2 d} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.003, size = 31, normalized size = 1.2 \begin{align*}{\frac{ \left ( \tan \left ( dx+c \right ) \right ) ^{2}}{2\,d}}-{\frac{\ln \left ( 1+ \left ( \tan \left ( dx+c \right ) \right ) ^{2} \right ) }{2\,d}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 0.872092, size = 42, normalized size = 1.56 \begin{align*} -\frac{\frac{1}{\sin \left (d x + c\right )^{2} - 1} - \log \left (\sin \left (d x + c\right )^{2} - 1\right )}{2 \, d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.77591, size = 73, normalized size = 2.7 \begin{align*} \frac{\tan \left (d x + c\right )^{2} + \log \left (\frac{1}{\tan \left (d x + c\right )^{2} + 1}\right )}{2 \, d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0.211822, size = 32, normalized size = 1.19 \begin{align*} \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}{\left (c \right )} & \text{otherwise} \end{cases} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] time = 1.92996, size = 332, normalized size = 12.3 \begin{align*} \frac{\log \left (\frac{4 \,{\left (\tan \left (c\right )^{2} + 1\right )}}{\tan \left (d x\right )^{4} \tan \left (c\right )^{2} - 2 \, \tan \left (d x\right )^{3} \tan \left (c\right ) + \tan \left (d x\right )^{2} \tan \left (c\right )^{2} + \tan \left (d x\right )^{2} - 2 \, \tan \left (d x\right ) \tan \left (c\right ) + 1}\right ) \tan \left (d x\right )^{2} \tan \left (c\right )^{2} + \tan \left (d x\right )^{2} \tan \left (c\right )^{2} - 2 \, \log \left (\frac{4 \,{\left (\tan \left (c\right )^{2} + 1\right )}}{\tan \left (d x\right )^{4} \tan \left (c\right )^{2} - 2 \, \tan \left (d x\right )^{3} \tan \left (c\right ) + \tan \left (d x\right )^{2} \tan \left (c\right )^{2} + \tan \left (d x\right )^{2} - 2 \, \tan \left (d x\right ) \tan \left (c\right ) + 1}\right ) \tan \left (d x\right ) \tan \left (c\right ) + \tan \left (d x\right )^{2} + \tan \left (c\right )^{2} + \log \left (\frac{4 \,{\left (\tan \left (c\right )^{2} + 1\right )}}{\tan \left (d x\right )^{4} \tan \left (c\right )^{2} - 2 \, \tan \left (d x\right )^{3} \tan \left (c\right ) + \tan \left (d x\right )^{2} \tan \left (c\right )^{2} + \tan \left (d x\right )^{2} - 2 \, \tan \left (d x\right ) \tan \left (c\right ) + 1}\right ) + 1}{2 \,{\left (d \tan \left (d x\right )^{2} \tan \left (c\right )^{2} - 2 \, d \tan \left (d x\right ) \tan \left (c\right ) + d\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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