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 [A]
time = 0.04, antiderivative size = 142, normalized size of antiderivative = 1.00, number of steps
used = 4, number of rules used = 4, integrand size = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.444, Rules used = {4587, 1986, 441,
440} \begin {gather*} 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 ) \end {gather*}
Antiderivative was successfully verified.
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Rule 440
Rule 441
Rule 1986
Rule 4587
Rubi steps
\begin {align*} \int \tan ^p(a+b \log (x)) \, dx &=\int \tan ^p(a+b \log (x)) \, dx\\ \end {align*}
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Mathematica [B] Both result and optimal contain complex but leaf count is larger than twice
the leaf count of optimal. \(330\) vs. \(2(142)=284\).
time = 0.74, size = 330, normalized size = 2.32 \begin {gather*} \frac {(-i+2 b) 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 b e^{2 i a} 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 b e^{2 i a} p x^{2 i b} F_1\left (1-\frac {i}{2 b};-p,1+p;2-\frac {i}{2 b};e^{2 i a} x^{2 i b},-e^{2 i a} x^{2 i b}\right )+(-i+2 b) 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 )} \end {gather*}
Warning: Unable to verify antiderivative.
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Maple [F]
time = 0.04, size = 0, normalized size = 0.00 \[\int \tan ^{p}\left (a +b \ln \left (x \right )\right )\, dx\]
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \tan ^{p}{\left (a + b \log {\left (x \right )} \right )}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [F]
time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \int {\mathrm {tan}\left (a+b\,\ln \left (x\right )\right )}^p \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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