Optimal. Leaf size=142 \[ x \left (1-e^{2 i a} x^{2 i b}\right )^{-p} \left (\frac{i \left (1-e^{2 i a} x^{2 i b}\right )}{1+e^{2 i a} x^{2 i b}}\right )^p \left (1+e^{2 i a} x^{2 i b}\right )^p F_1\left (-\frac{i}{2 b};-p,p;1-\frac{i}{2 b};e^{2 i a} x^{2 i b},-e^{2 i a} x^{2 i b}\right ) \]
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Rubi [F] time = 0.0242006, antiderivative size = 0, normalized size of antiderivative = 0., number of steps used = 0, number of rules used = 0, integrand size = 0, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0., Rules used = {} \[ \int \tan ^p(a+b \log (x)) \, dx \]
Verification is Not applicable to the result.
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Rubi steps
\begin{align*} \int \tan ^p(a+b \log (x)) \, dx &=\int \tan ^p(a+b \log (x)) \, dx\\ \end{align*}
Mathematica [B] time = 0.665116, size = 330, normalized size = 2.32 \[ \frac{(2 b-i) x \left (-\frac{i \left (-1+e^{2 i a} x^{2 i b}\right )}{1+e^{2 i a} x^{2 i b}}\right )^p F_1\left (-\frac{i}{2 b};-p,p;1-\frac{i}{2 b};e^{2 i a} x^{2 i b},-e^{2 i a} x^{2 i b}\right )}{-2 e^{2 i a} b p x^{2 i b} F_1\left (1-\frac{i}{2 b};1-p,p;2-\frac{i}{2 b};e^{2 i a} x^{2 i b},-e^{2 i a} x^{2 i b}\right )-2 e^{2 i a} b p x^{2 i b} F_1\left (1-\frac{i}{2 b};-p,p+1;2-\frac{i}{2 b};e^{2 i a} x^{2 i b},-e^{2 i a} x^{2 i b}\right )+(2 b-i) F_1\left (-\frac{i}{2 b};-p,p;1-\frac{i}{2 b};e^{2 i a} x^{2 i b},-e^{2 i a} x^{2 i b}\right )} \]
Warning: Unable to verify antiderivative.
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Maple [F] time = 0.374, size = 0, normalized size = 0. \begin{align*} \int \left ( \tan \left ( a+b\ln \left ( x \right ) \right ) \right ) ^{p}\, dx \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 \tan \left (b \log \left (x\right ) + a\right )^{p}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [F] time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (\tan \left (b \log \left (x\right ) + a\right )^{p}, x\right ) \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 \tan ^{p}{\left (a + b \log{\left (x \right )} \right )}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F(-1)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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