Optimal. Leaf size=190 \[ x \left (1-e^{2 i a d} \left (c x^n\right )^{2 i b d}\right )^p \left (1+e^{2 i a d} \left (c x^n\right )^{2 i b d}\right )^{-p} \left (-\frac {i \left (1+e^{2 i a d} \left (c x^n\right )^{2 i b d}\right )}{1-e^{2 i a d} \left (c x^n\right )^{2 i b d}}\right )^p F_1\left (-\frac {i}{2 b d n};p,-p;1-\frac {i}{2 b d n};e^{2 i a d} \left (c x^n\right )^{2 i b d},-e^{2 i a d} \left (c x^n\right )^{2 i b d}\right ) \]
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Rubi [A]
time = 0.11, antiderivative size = 190, normalized size of antiderivative = 1.00, number of steps
used = 5, number of rules used = 5, integrand size = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.333, Rules used = {4590, 4592,
1986, 525, 524} \begin {gather*} x \left (1-e^{2 i a d} \left (c x^n\right )^{2 i b d}\right )^p \left (1+e^{2 i a d} \left (c x^n\right )^{2 i b d}\right )^{-p} \left (-\frac {i \left (1+e^{2 i a d} \left (c x^n\right )^{2 i b d}\right )}{1-e^{2 i a d} \left (c x^n\right )^{2 i b d}}\right )^p F_1\left (-\frac {i}{2 b d n};p,-p;1-\frac {i}{2 b d n};e^{2 i a d} \left (c x^n\right )^{2 i b d},-e^{2 i a d} \left (c x^n\right )^{2 i b d}\right ) \end {gather*}
Antiderivative was successfully verified.
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Rule 524
Rule 525
Rule 1986
Rule 4590
Rule 4592
Rubi steps
\begin {align*} \int \cot ^p\left (d \left (a+b \log \left (c x^n\right )\right )\right ) \, dx &=\int \cot ^p\left (d \left (a+b \log \left (c x^n\right )\right )\right ) \, 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. \(458\) vs. \(2(190)=380\).
time = 1.36, size = 458, normalized size = 2.41 \begin {gather*} \frac {(-i+2 b d n) x \left (\frac {i \left (1+e^{2 i a d} \left (c x^n\right )^{2 i b d}\right )}{-1+e^{2 i a d} \left (c x^n\right )^{2 i b d}}\right )^p F_1\left (-\frac {i}{2 b d n};p,-p;1-\frac {i}{2 b d n};e^{2 i a d} \left (c x^n\right )^{2 i b d},-e^{2 i a d} \left (c x^n\right )^{2 i b d}\right )}{2 b d e^{2 i a d} n p \left (c x^n\right )^{2 i b d} F_1\left (1-\frac {i}{2 b d n};p,1-p;2-\frac {i}{2 b d n};e^{2 i a d} \left (c x^n\right )^{2 i b d},-e^{2 i a d} \left (c x^n\right )^{2 i b d}\right )+2 b d e^{2 i a d} n p \left (c x^n\right )^{2 i b d} F_1\left (1-\frac {i}{2 b d n};1+p,-p;2-\frac {i}{2 b d n};e^{2 i a d} \left (c x^n\right )^{2 i b d},-e^{2 i a d} \left (c x^n\right )^{2 i b d}\right )+(-i+2 b d n) F_1\left (-\frac {i}{2 b d n};p,-p;1-\frac {i}{2 b d n};e^{2 i a d} \left (c x^n\right )^{2 i b d},-e^{2 i a d} \left (c x^n\right )^{2 i b d}\right )} \end {gather*}
Warning: Unable to verify antiderivative.
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Maple [F]
time = 0.06, size = 0, normalized size = 0.00 \[\int \cot ^{p}\left (d \left (a +b \ln \left (c \,x^{n}\right )\right )\right )\, dx\]
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 \cot ^{p}{\left (d \left (a + b \log {\left (c x^{n} \right )}\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 {cot}\left (d\,\left (a+b\,\ln \left (c\,x^n\right )\right )\right )}^p \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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