Optimal. Leaf size=71 \[ \frac {(2-p) x \left (1-e^{2 i a} \left (c x^n\right )^{-\frac {2}{n (2-p)}}\right ) \csc ^p\left (a+\frac {i \log \left (c x^n\right )}{n (2-p)}\right )}{2 (1-p)} \]
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Rubi [A] time = 0.08, antiderivative size = 71, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, integrand size = 23, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.130, Rules used = {4504, 4508, 264} \[ \frac {(2-p) x \left (1-e^{2 i a} \left (c x^n\right )^{-\frac {2}{n (2-p)}}\right ) \csc ^p\left (a+\frac {i \log \left (c x^n\right )}{n (2-p)}\right )}{2 (1-p)} \]
Antiderivative was successfully verified.
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Rule 264
Rule 4504
Rule 4508
Rubi steps
\begin {align*} \int \csc ^p\left (a-\frac {i \log \left (c x^n\right )}{n (-2+p)}\right ) \, dx &=\frac {\left (x \left (c x^n\right )^{-1/n}\right ) \operatorname {Subst}\left (\int x^{-1+\frac {1}{n}} \csc ^p\left (a-\frac {i \log (x)}{n (-2+p)}\right ) \, dx,x,c x^n\right )}{n}\\ &=\frac {\left (x \left (c x^n\right )^{-\frac {1}{n}-\frac {p}{n (-2+p)}} \left (1-e^{2 i a} \left (c x^n\right )^{\frac {2}{n (-2+p)}}\right )^p \csc ^p\left (a-\frac {i \log \left (c x^n\right )}{n (-2+p)}\right )\right ) \operatorname {Subst}\left (\int x^{-1+\frac {1}{n}+\frac {p}{n (-2+p)}} \left (1-e^{2 i a} x^{\frac {2}{n (-2+p)}}\right )^{-p} \, dx,x,c x^n\right )}{n}\\ &=\frac {(2-p) x \left (1-e^{2 i a} \left (c x^n\right )^{-\frac {2}{n (2-p)}}\right ) \csc ^p\left (a+\frac {i \log \left (c x^n\right )}{n (2-p)}\right )}{2 (1-p)}\\ \end {align*}
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Mathematica [A] time = 3.10, size = 128, normalized size = 1.80 \[ \frac {2^{p-1} (p-2) x \left (\frac {i e^{i a} \left (c x^n\right )^{\frac {1}{n (p-2)}}}{-1+e^{2 i a} \left (c x^n\right )^{\frac {2}{n (p-2)}}}\right )^p \left (1+e^{2 i a} \left (c x^n\right )^{\frac {2}{n (p-2)}} \left (-1+\left (1-e^{-2 i a} \left (c x^n\right )^{-\frac {2}{n (p-2)}}\right )^p\right )\right )}{p-1} \]
Warning: Unable to verify antiderivative.
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fricas [B] time = 0.83, size = 150, normalized size = 2.11 \[ \frac {{\left ({\left (p - 2\right )} x e^{\left (\frac {2 \, {\left (-i \, a n p + 2 i \, a n - n \log \relax (x) - \log \relax (c)\right )}}{n p - 2 \, n}\right )} - {\left (p - 2\right )} x\right )} \left (-\frac {2 i \, e^{\left (\frac {-i \, a n p + 2 i \, a n - n \log \relax (x) - \log \relax (c)}{n p - 2 \, n}\right )}}{e^{\left (\frac {2 \, {\left (-i \, a n p + 2 i \, a n - n \log \relax (x) - \log \relax (c)\right )}}{n p - 2 \, n}\right )} - 1}\right )^{p} e^{\left (-\frac {2 \, {\left (-i \, a n p + 2 i \, a n - n \log \relax (x) - \log \relax (c)\right )}}{n p - 2 \, n}\right )}}{2 \, {\left (p - 1\right )}} \]
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 \[ \int \csc \left (a - \frac {i \, \log \left (c x^{n}\right )}{n {\left (p - 2\right )}}\right )^{p}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.24, size = 0, normalized size = 0.00 \[ \int \csc ^{p}\left (a -\frac {i \ln \left (c \,x^{n}\right )}{n \left (-2+p \right )}\right )\, dx \]
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 \[ \int \left (-\csc \left (-a + \frac {i \, \log \left (c x^{n}\right )}{n {\left (p - 2\right )}}\right )\right )^{p}\,{d x} \]
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 \[ \int {\left (\frac {1}{\sin \left (a-\frac {\ln \left (c\,x^n\right )\,1{}\mathrm {i}}{n\,\left (p-2\right )}\right )}\right )}^p \,d x \]
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 \[ \int \csc ^{p}{\left (a - \frac {i \log {\left (c x^{n} \right )}}{n \left (p - 2\right )} \right )}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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