Optimal. Leaf size=120 \[ \left (1-e^{2 i a} x^{6 i}\right )^p \left (1+e^{2 i a} x^{6 i}\right )^{-p} \left (-\frac {i \left (1+e^{2 i a} x^{6 i}\right )}{1-e^{2 i a} x^{6 i}}\right )^p x F_1\left (-\frac {i}{6};p,-p;1-\frac {i}{6};e^{2 i a} x^{6 i},-e^{2 i a} x^{6 i}\right ) \]
[Out]
________________________________________________________________________________________
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 = {4588, 1986, 441,
440} \begin {gather*} x \left (1-e^{2 i a} x^{6 i}\right )^p \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 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.
[In]
[Out]
Rule 440
Rule 441
Rule 1986
Rule 4588
Rubi steps
\begin {align*} \int \cot ^p(a+3 \log (x)) \, dx &=\int \cot ^p(a+3 \log (x)) \, dx\\ \end {align*}
________________________________________________________________________________________
Mathematica [A]
time = 0.54, size = 238, normalized size = 1.98 \begin {gather*} \frac {(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 )}{(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 e^{2 i a} p x^{6 i} \left (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 )+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 )\right )} \end {gather*}
Warning: Unable to verify antiderivative.
[In]
[Out]
________________________________________________________________________________________
Maple [F]
time = 0.07, size = 0, normalized size = 0.00 \[\int \cot ^{p}\left (a +3 \ln \left (x \right )\right )\, dx\]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
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.
[In]
[Out]
________________________________________________________________________________________
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.
[In]
[Out]
________________________________________________________________________________________
Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \cot ^{p}{\left (a + 3 \log {\left (x \right )} \right )}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
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.
[In]
[Out]
________________________________________________________________________________________
Mupad [F]
time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \int {\mathrm {cot}\left (a+3\,\ln \left (x\right )\right )}^p \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________