Optimal. Leaf size=32 \[ -\frac{\cot (c+d x) \log (\cos (c+d x)) \left (b \tan ^p(c+d x)\right )^{\frac{1}{p}}}{d} \]
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Rubi [A] time = 0.0191736, antiderivative size = 32, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 14, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.143, Rules used = {3659, 3475} \[ -\frac{\cot (c+d x) \log (\cos (c+d x)) \left (b \tan ^p(c+d x)\right )^{\frac{1}{p}}}{d} \]
Antiderivative was successfully verified.
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Rule 3659
Rule 3475
Rubi steps
\begin{align*} \int \left (b \tan ^p(c+d x)\right )^{\frac{1}{p}} \, dx &=\left (\cot (c+d x) \left (b \tan ^p(c+d x)\right )^{\frac{1}{p}}\right ) \int \tan (c+d x) \, dx\\ &=-\frac{\cot (c+d x) \log (\cos (c+d x)) \left (b \tan ^p(c+d x)\right )^{\frac{1}{p}}}{d}\\ \end{align*}
Mathematica [A] time = 0.0220582, size = 32, normalized size = 1. \[ -\frac{\cot (c+d x) \log (\cos (c+d x)) \left (b \tan ^p(c+d x)\right )^{\frac{1}{p}}}{d} \]
Antiderivative was successfully verified.
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Maple [C] time = 3.563, size = 18076, normalized size = 564.9 \begin{align*} \text{output too large to display} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \left (b \tan \left (d x + c\right )^{p}\right )^{\left (\frac{1}{p}\right )}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.09594, size = 59, normalized size = 1.84 \begin{align*} -\frac{b^{\left (\frac{1}{p}\right )} \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 [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \left (b \tan ^{p}{\left (c + d x \right )}\right )^{\frac{1}{p}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \left (b \tan \left (d x + c\right )^{p}\right )^{\left (\frac{1}{p}\right )}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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