Optimal. Leaf size=111 \[ -\frac {2 \cosh \left (a+b \log \left (c x^n\right )\right )}{3 b n \sinh ^{\frac {3}{2}}\left (a+b \log \left (c x^n\right )\right )}+\frac {2 i \sqrt {i \sinh \left (a+b \log \left (c x^n\right )\right )} F\left (\left .\frac {1}{2} \left (i a+i b \log \left (c x^n\right )-\frac {\pi }{2}\right )\right |2\right )}{3 b n \sqrt {\sinh \left (a+b \log \left (c x^n\right )\right )}} \]
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Rubi [A] time = 0.06, antiderivative size = 111, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, integrand size = 19, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.158, Rules used = {2636, 2642, 2641} \[ -\frac {2 \cosh \left (a+b \log \left (c x^n\right )\right )}{3 b n \sinh ^{\frac {3}{2}}\left (a+b \log \left (c x^n\right )\right )}+\frac {2 i \sqrt {i \sinh \left (a+b \log \left (c x^n\right )\right )} F\left (\left .\frac {1}{2} \left (i a+i b \log \left (c x^n\right )-\frac {\pi }{2}\right )\right |2\right )}{3 b n \sqrt {\sinh \left (a+b \log \left (c x^n\right )\right )}} \]
Antiderivative was successfully verified.
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Rule 2636
Rule 2641
Rule 2642
Rubi steps
\begin {align*} \int \frac {1}{x \sinh ^{\frac {5}{2}}\left (a+b \log \left (c x^n\right )\right )} \, dx &=\frac {\operatorname {Subst}\left (\int \frac {1}{\sinh ^{\frac {5}{2}}(a+b x)} \, dx,x,\log \left (c x^n\right )\right )}{n}\\ &=-\frac {2 \cosh \left (a+b \log \left (c x^n\right )\right )}{3 b n \sinh ^{\frac {3}{2}}\left (a+b \log \left (c x^n\right )\right )}-\frac {\operatorname {Subst}\left (\int \frac {1}{\sqrt {\sinh (a+b x)}} \, dx,x,\log \left (c x^n\right )\right )}{3 n}\\ &=-\frac {2 \cosh \left (a+b \log \left (c x^n\right )\right )}{3 b n \sinh ^{\frac {3}{2}}\left (a+b \log \left (c x^n\right )\right )}-\frac {\sqrt {i \sinh \left (a+b \log \left (c x^n\right )\right )} \operatorname {Subst}\left (\int \frac {1}{\sqrt {i \sinh (a+b x)}} \, dx,x,\log \left (c x^n\right )\right )}{3 n \sqrt {\sinh \left (a+b \log \left (c x^n\right )\right )}}\\ &=-\frac {2 \cosh \left (a+b \log \left (c x^n\right )\right )}{3 b n \sinh ^{\frac {3}{2}}\left (a+b \log \left (c x^n\right )\right )}+\frac {2 i 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 )}}{3 b n \sqrt {\sinh \left (a+b \log \left (c x^n\right )\right )}}\\ \end {align*}
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Mathematica [C] time = 0.13, size = 122, normalized size = 1.10 \[ -\frac {2 \left (\sinh \left (a+b \log \left (c x^n\right )\right ) \sqrt {-\sinh \left (2 \left (a+b \log \left (c x^n\right )\right )\right )-\cosh \left (2 \left (a+b \log \left (c x^n\right )\right )\right )+1} \, _2F_1\left (\frac {1}{4},\frac {1}{2};\frac {5}{4};\cosh \left (2 \left (a+b \log \left (c x^n\right )\right )\right )+\sinh \left (2 \left (a+b \log \left (c x^n\right )\right )\right )\right )+\cosh \left (a+b \log \left (c x^n\right )\right )\right )}{3 b n \sinh ^{\frac {3}{2}}\left (a+b \log \left (c x^n\right )\right )} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.75, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {1}{x \sinh \left (b \log \left (c x^{n}\right ) + a\right )^{\frac {5}{2}}}, x\right ) \]
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 \[ \int \frac {1}{x \sinh \left (b \log \left (c x^{n}\right ) + a\right )^{\frac {5}{2}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.08, size = 144, normalized size = 1.30 \[ -\frac {i \sqrt {1-i \sinh \left (a +b \ln \left (c \,x^{n}\right )\right )}\, \sqrt {2}\, \sqrt {i \sinh \left (a +b \ln \left (c \,x^{n}\right )\right )+1}\, \sqrt {i \sinh \left (a +b \ln \left (c \,x^{n}\right )\right )}\, \EllipticF \left (\sqrt {1-i \sinh \left (a +b \ln \left (c \,x^{n}\right )\right )}, \frac {\sqrt {2}}{2}\right ) \sinh \left (a +b \ln \left (c \,x^{n}\right )\right )+2 \left (\cosh ^{2}\left (a +b \ln \left (c \,x^{n}\right )\right )\right )}{3 n \sinh \left (a +b \ln \left (c \,x^{n}\right )\right )^{\frac {3}{2}} \cosh \left (a +b \ln \left (c \,x^{n}\right )\right ) b} \]
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 \[ \int \frac {1}{x \sinh \left (b \log \left (c x^{n}\right ) + a\right )^{\frac {5}{2}}}\,{d x} \]
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 \[ \int \frac {1}{x\,{\mathrm {sinh}\left (a+b\,\ln \left (c\,x^n\right )\right )}^{5/2}} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [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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