Optimal. Leaf size=120 \[ \left (1-e^{2 i a} x^{6 i}\right )^{-p} \left (\frac {i \left (1-e^{2 i a} x^{6 i}\right )}{1+e^{2 i a} x^{6 i}}\right )^p \left (1+e^{2 i a} x^{6 i}\right )^p x F_1\left (-\frac {i}{6};-p,p;1-\frac {i}{6};e^{2 i a} x^{6 i},-e^{2 i a} x^{6 i}\right ) \]
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Rubi [A]
time = 0.04, antiderivative size = 120, 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^{6 i}\right )^{-p} \left (\frac {i \left (1-e^{2 i a} x^{6 i}\right )}{1+e^{2 i a} x^{6 i}}\right )^p \left (1+e^{2 i a} x^{6 i}\right )^p F_1\left (-\frac {i}{6};-p,p;1-\frac {i}{6};e^{2 i a} x^{6 i},-e^{2 i a} x^{6 i}\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+3 \log (x)) \, dx &=\int \tan ^p(a+3 \log (x)) \, dx\\ \end {align*}
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Mathematica [A]
time = 0.56, size = 240, normalized size = 2.00 \begin {gather*} \frac {(1+6 i) \left (-\frac {i \left (-1+e^{2 i a} x^{6 i}\right )}{1+e^{2 i a} x^{6 i}}\right )^p x F_1\left (-\frac {i}{6};-p,p;1-\frac {i}{6};e^{2 i a} x^{6 i},-e^{2 i a} x^{6 i}\right )}{(1+6 i) F_1\left (-\frac {i}{6};-p,p;1-\frac {i}{6};e^{2 i a} x^{6 i},-e^{2 i a} x^{6 i}\right )-6 i e^{2 i a} p x^{6 i} \left (F_1\left (1-\frac {i}{6};1-p,p;2-\frac {i}{6};e^{2 i a} x^{6 i},-e^{2 i a} x^{6 i}\right )+F_1\left (1-\frac {i}{6};-p,1+p;2-\frac {i}{6};e^{2 i a} x^{6 i},-e^{2 i a} x^{6 i}\right )\right )} \end {gather*}
Warning: Unable to verify antiderivative.
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Maple [F]
time = 0.04, size = 0, normalized size = 0.00 \[\int \tan ^{p}\left (a +3 \ln \left (x \right )\right )\, dx\]
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 + 3 \log {\left (x \right )} \right )}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F(-1)] Timed out
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \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+3\,\ln \left (x\right )\right )}^p \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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