Optimal. Leaf size=90 \[ -\frac {e^{2 a} (2-p) x \left (c x^n\right )^{-\frac {2}{n (2-p)}} \left (1-e^{-2 a} \left (c x^n\right )^{\frac {2}{n (2-p)}}\right ) \text {csch}^p\left (a-\frac {\log \left (c x^n\right )}{n (2-p)}\right )}{2 (1-p)} \]
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Rubi [A] time = 0.09, antiderivative size = 90, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, integrand size = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.150, Rules used = {5546, 5550, 261} \[ -\frac {e^{2 a} (2-p) x \left (c x^n\right )^{-\frac {2}{n (2-p)}} \left (1-e^{-2 a} \left (c x^n\right )^{\frac {2}{n (2-p)}}\right ) \text {csch}^p\left (a-\frac {\log \left (c x^n\right )}{n (2-p)}\right )}{2 (1-p)} \]
Antiderivative was successfully verified.
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Rule 261
Rule 5546
Rule 5550
Rubi steps
\begin {align*} \int \text {csch}^p\left (a+\frac {\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}} \text {csch}^p\left (a+\frac {\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 a} \left (c x^n\right )^{-\frac {2}{n (-2+p)}}\right )^p \text {csch}^p\left (a+\frac {\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 a} x^{-\frac {2}{n (-2+p)}}\right )^{-p} \, dx,x,c x^n\right )}{n}\\ &=-\frac {e^{2 a} (2-p) x \left (c x^n\right )^{-\frac {2}{n (2-p)}} \left (1-e^{-2 a} \left (c x^n\right )^{\frac {2}{n (2-p)}}\right ) \text {csch}^p\left (a-\frac {\log \left (c x^n\right )}{n (2-p)}\right )}{2 (1-p)}\\ \end {align*}
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Mathematica [A] time = 6.76, size = 115, normalized size = 1.28 \[ \frac {2^{p-1} (p-2) x \left (\frac {e^a \left (c x^n\right )^{\frac {1}{n (p-2)}}}{e^{2 a} \left (c x^n\right )^{\frac {2}{n (p-2)}}-1}\right )^p \left (e^{2 a} \left (c x^n\right )^{\frac {2}{n (p-2)}} \left (\left (1-e^{-2 a} \left (c x^n\right )^{-\frac {2}{n (p-2)}}\right )^p-1\right )+1\right )}{p-1} \]
Warning: Unable to verify antiderivative.
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fricas [B] time = 0.49, size = 475, normalized size = 5.28 \[ -\frac {{\left (p - 2\right )} x \cosh \left (p \log \left (\frac {2 \, {\left (\cosh \left (\frac {a n p - 2 \, a n + n \log \relax (x) + \log \relax (c)}{n p - 2 \, n}\right ) + \sinh \left (\frac {a n p - 2 \, a n + n \log \relax (x) + \log \relax (c)}{n p - 2 \, n}\right )\right )}}{\cosh \left (\frac {a n p - 2 \, a n + n \log \relax (x) + \log \relax (c)}{n p - 2 \, n}\right )^{2} + 2 \, \cosh \left (\frac {a n p - 2 \, a n + n \log \relax (x) + \log \relax (c)}{n p - 2 \, n}\right ) \sinh \left (\frac {a n p - 2 \, a n + n \log \relax (x) + \log \relax (c)}{n p - 2 \, n}\right ) + \sinh \left (\frac {a n p - 2 \, a n + n \log \relax (x) + \log \relax (c)}{n p - 2 \, n}\right )^{2} - 1}\right )\right ) \sinh \left (\frac {a n p - 2 \, a n + n \log \relax (x) + \log \relax (c)}{n p - 2 \, n}\right ) + {\left (p - 2\right )} x \sinh \left (p \log \left (\frac {2 \, {\left (\cosh \left (\frac {a n p - 2 \, a n + n \log \relax (x) + \log \relax (c)}{n p - 2 \, n}\right ) + \sinh \left (\frac {a n p - 2 \, a n + n \log \relax (x) + \log \relax (c)}{n p - 2 \, n}\right )\right )}}{\cosh \left (\frac {a n p - 2 \, a n + n \log \relax (x) + \log \relax (c)}{n p - 2 \, n}\right )^{2} + 2 \, \cosh \left (\frac {a n p - 2 \, a n + n \log \relax (x) + \log \relax (c)}{n p - 2 \, n}\right ) \sinh \left (\frac {a n p - 2 \, a n + n \log \relax (x) + \log \relax (c)}{n p - 2 \, n}\right ) + \sinh \left (\frac {a n p - 2 \, a n + n \log \relax (x) + \log \relax (c)}{n p - 2 \, n}\right )^{2} - 1}\right )\right ) \sinh \left (\frac {a n p - 2 \, a n + n \log \relax (x) + \log \relax (c)}{n p - 2 \, n}\right )}{{\left (p - 1\right )} \cosh \left (\frac {a n p - 2 \, a n + n \log \relax (x) + \log \relax (c)}{n p - 2 \, n}\right ) - {\left (p - 1\right )} \sinh \left (\frac {a n p - 2 \, a n + n \log \relax (x) + \log \relax (c)}{n p - 2 \, n}\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.51, size = 0, normalized size = 0.00 \[ \int \mathrm {csch}\left (a +\frac {\ln \left (c \,x^{n}\right )}{n \left (-2+p \right )}\right )^{p}\, dx \]
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 \[ \int \operatorname {csch}\left (a + \frac {\log \left (c x^{n}\right )}{n {\left (p - 2\right )}}\right )^{p}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.01 \[ \int {\left (\frac {1}{\mathrm {sinh}\left (a+\frac {\ln \left (c\,x^n\right )}{n\,\left (p-2\right )}\right )}\right )}^p \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \operatorname {csch}^{p}{\left (a + \frac {\log {\left (c x^{n} \right )}}{n \left (p - 2\right )} \right )}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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