Optimal. Leaf size=72 \[ -\frac {2 i \sqrt {i \sinh \left (a+b \log \left (c x^n\right )\right )} \sqrt {\text {csch}\left (a+b \log \left (c x^n\right )\right )} F\left (\left .\frac {1}{2} \left (i a+i b \log \left (c x^n\right )-\frac {\pi }{2}\right )\right |2\right )}{b n} \]
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Rubi [A] time = 0.04, antiderivative size = 72, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, integrand size = 19, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.105, Rules used = {3771, 2641} \[ -\frac {2 i \sqrt {i \sinh \left (a+b \log \left (c x^n\right )\right )} \sqrt {\text {csch}\left (a+b \log \left (c x^n\right )\right )} F\left (\left .\frac {1}{2} \left (i a+i b \log \left (c x^n\right )-\frac {\pi }{2}\right )\right |2\right )}{b n} \]
Antiderivative was successfully verified.
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Rule 2641
Rule 3771
Rubi steps
\begin {align*} \int \frac {\sqrt {\text {csch}\left (a+b \log \left (c x^n\right )\right )}}{x} \, dx &=\frac {\operatorname {Subst}\left (\int \sqrt {\text {csch}(a+b x)} \, dx,x,\log \left (c x^n\right )\right )}{n}\\ &=\frac {\left (\sqrt {\text {csch}\left (a+b \log \left (c x^n\right )\right )} \sqrt {i \sinh \left (a+b \log \left (c x^n\right )\right )}\right ) \operatorname {Subst}\left (\int \frac {1}{\sqrt {i \sinh (a+b x)}} \, dx,x,\log \left (c x^n\right )\right )}{n}\\ &=-\frac {2 i \sqrt {\text {csch}\left (a+b \log \left (c x^n\right )\right )} F\left (\left .\frac {1}{2} \left (i a-\frac {\pi }{2}+i b \log \left (c x^n\right )\right )\right |2\right ) \sqrt {i \sinh \left (a+b \log \left (c x^n\right )\right )}}{b n}\\ \end {align*}
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Mathematica [A] time = 0.10, size = 66, normalized size = 0.92 \[ \frac {2 \left (i \sinh \left (a+b \log \left (c x^n\right )\right )\right )^{3/2} \text {csch}^{\frac {3}{2}}\left (a+b \log \left (c x^n\right )\right ) F\left (\left .\frac {1}{4} \left (-2 i a-2 i b \log \left (c x^n\right )+\pi \right )\right |2\right )}{b n} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.43, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {\sqrt {\operatorname {csch}\left (b \log \left (c x^{n}\right ) + a\right )}}{x}, x\right ) \]
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 [A] time = 0.34, size = 120, normalized size = 1.67 \[ \frac {i \sqrt {-i \left (\sinh \left (a +b \ln \left (c \,x^{n}\right )\right )+i\right )}\, \sqrt {2}\, \sqrt {-i \left (-\sinh \left (a +b \ln \left (c \,x^{n}\right )\right )+i\right )}\, \sqrt {i \sinh \left (a +b \ln \left (c \,x^{n}\right )\right )}\, \EllipticF \left (\sqrt {-i \left (\sinh \left (a +b \ln \left (c \,x^{n}\right )\right )+i\right )}, \frac {\sqrt {2}}{2}\right )}{n \cosh \left (a +b \ln \left (c \,x^{n}\right )\right ) \sqrt {\sinh \left (a +b \ln \left (c \,x^{n}\right )\right )}\, b} \]
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 \[ \int \frac {\sqrt {\operatorname {csch}\left (b \log \left (c x^{n}\right ) + a\right )}}{x}\,{d x} \]
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 \[ \int \frac {\sqrt {\frac {1}{\mathrm {sinh}\left (a+b\,\ln \left (c\,x^n\right )\right )}}}{x} \,d x \]
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 \[ \int \frac {\sqrt {\operatorname {csch}{\left (a + b \log {\left (c x^{n} \right )} \right )}}}{x}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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