Optimal. Leaf size=63 \[ \frac {2 \sinh \left (a+b \log \left (c x^n\right )\right )}{b n \sqrt {\cosh \left (a+b \log \left (c x^n\right )\right )}}+\frac {2 i E\left (\left .\frac {1}{2} i \left (a+b \log \left (c x^n\right )\right )\right |2\right )}{b n} \]
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Rubi [A] time = 0.04, antiderivative size = 63, 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 = {2636, 2639} \[ \frac {2 \sinh \left (a+b \log \left (c x^n\right )\right )}{b n \sqrt {\cosh \left (a+b \log \left (c x^n\right )\right )}}+\frac {2 i E\left (\left .\frac {1}{2} i \left (a+b \log \left (c x^n\right )\right )\right |2\right )}{b n} \]
Antiderivative was successfully verified.
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Rule 2636
Rule 2639
Rubi steps
\begin {align*} \int \frac {1}{x \cosh ^{\frac {3}{2}}\left (a+b \log \left (c x^n\right )\right )} \, dx &=\frac {\operatorname {Subst}\left (\int \frac {1}{\cosh ^{\frac {3}{2}}(a+b x)} \, dx,x,\log \left (c x^n\right )\right )}{n}\\ &=\frac {2 \sinh \left (a+b \log \left (c x^n\right )\right )}{b n \sqrt {\cosh \left (a+b \log \left (c x^n\right )\right )}}-\frac {\operatorname {Subst}\left (\int \sqrt {\cosh (a+b x)} \, dx,x,\log \left (c x^n\right )\right )}{n}\\ &=\frac {2 i E\left (\left .\frac {1}{2} i \left (a+b \log \left (c x^n\right )\right )\right |2\right )}{b n}+\frac {2 \sinh \left (a+b \log \left (c x^n\right )\right )}{b n \sqrt {\cosh \left (a+b \log \left (c x^n\right )\right )}}\\ \end {align*}
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Mathematica [A] time = 0.06, size = 58, normalized size = 0.92 \[ \frac {2 \left (\frac {\sinh \left (a+b \log \left (c x^n\right )\right )}{\sqrt {\cosh \left (a+b \log \left (c x^n\right )\right )}}+i E\left (\left .\frac {1}{2} i \left (a+b \log \left (c x^n\right )\right )\right |2\right )\right )}{b n} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.51, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {1}{x \cosh \left (b \log \left (c x^{n}\right ) + a\right )^{\frac {3}{2}}}, x\right ) \]
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 \[ \int \frac {1}{x \cosh \left (b \log \left (c x^{n}\right ) + a\right )^{\frac {3}{2}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.43, size = 141, normalized size = 2.24 \[ \frac {2 \sqrt {-\left (\sinh ^{2}\left (\frac {a}{2}+\frac {b \ln \left (c \,x^{n}\right )}{2}\right )\right )}\, \sqrt {-2 \left (\sinh ^{2}\left (\frac {a}{2}+\frac {b \ln \left (c \,x^{n}\right )}{2}\right )\right )-1}\, \EllipticE \left (\cosh \left (\frac {a}{2}+\frac {b \ln \left (c \,x^{n}\right )}{2}\right ), \sqrt {2}\right )+4 \cosh \left (\frac {a}{2}+\frac {b \ln \left (c \,x^{n}\right )}{2}\right ) \left (\sinh ^{2}\left (\frac {a}{2}+\frac {b \ln \left (c \,x^{n}\right )}{2}\right )\right )}{n \sinh \left (\frac {a}{2}+\frac {b \ln \left (c \,x^{n}\right )}{2}\right ) \sqrt {2 \left (\cosh ^{2}\left (\frac {a}{2}+\frac {b \ln \left (c \,x^{n}\right )}{2}\right )\right )-1}\, b} \]
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 \[ \int \frac {1}{x \cosh \left (b \log \left (c x^{n}\right ) + a\right )^{\frac {3}{2}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.02 \[ \int \frac {1}{x\,{\mathrm {cosh}\left (a+b\,\ln \left (c\,x^n\right )\right )}^{3/2}} \,d x \]
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 \[ \int \frac {1}{x \cosh ^{\frac {3}{2}}{\left (a + b \log {\left (c x^{n} \right )} \right )}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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