Optimal. Leaf size=67 \[ \frac {2 \sinh \left (a+b \log \left (c x^n\right )\right )}{3 b n \cosh ^{\frac {3}{2}}\left (a+b \log \left (c x^n\right )\right )}-\frac {2 i F\left (\left .\frac {1}{2} i \left (a+b \log \left (c x^n\right )\right )\right |2\right )}{3 b n} \]
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Rubi [A] time = 0.04, antiderivative size = 67, 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, 2641} \[ \frac {2 \sinh \left (a+b \log \left (c x^n\right )\right )}{3 b n \cosh ^{\frac {3}{2}}\left (a+b \log \left (c x^n\right )\right )}-\frac {2 i F\left (\left .\frac {1}{2} i \left (a+b \log \left (c x^n\right )\right )\right |2\right )}{3 b n} \]
Antiderivative was successfully verified.
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Rule 2636
Rule 2641
Rubi steps
\begin {align*} \int \frac {1}{x \cosh ^{\frac {5}{2}}\left (a+b \log \left (c x^n\right )\right )} \, dx &=\frac {\operatorname {Subst}\left (\int \frac {1}{\cosh ^{\frac {5}{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 )}{3 b n \cosh ^{\frac {3}{2}}\left (a+b \log \left (c x^n\right )\right )}+\frac {\operatorname {Subst}\left (\int \frac {1}{\sqrt {\cosh (a+b x)}} \, dx,x,\log \left (c x^n\right )\right )}{3 n}\\ &=-\frac {2 i F\left (\left .\frac {1}{2} i \left (a+b \log \left (c x^n\right )\right )\right |2\right )}{3 b n}+\frac {2 \sinh \left (a+b \log \left (c x^n\right )\right )}{3 b n \cosh ^{\frac {3}{2}}\left (a+b \log \left (c x^n\right )\right )}\\ \end {align*}
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Mathematica [C] time = 0.09, size = 122, normalized size = 1.82 \[ \frac {2 \left (\cosh \left (a+b \log \left (c x^n\right )\right ) \sqrt {\sinh \left (2 \left (a+b \log \left (c x^n\right )\right )\right )+\cosh \left (2 \left (a+b \log \left (c x^n\right )\right )\right )+1} \, _2F_1\left (\frac {1}{4},\frac {1}{2};\frac {5}{4};-\cosh \left (2 \left (a+b \log \left (c x^n\right )\right )\right )-\sinh \left (2 \left (a+b \log \left (c x^n\right )\right )\right )\right )+\sinh \left (a+b \log \left (c x^n\right )\right )\right )}{3 b n \cosh ^{\frac {3}{2}}\left (a+b \log \left (c x^n\right )\right )} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.45, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {1}{x \cosh \left (b \log \left (c x^{n}\right ) + a\right )^{\frac {5}{2}}}, x\right ) \]
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 \[ \int \frac {1}{x \cosh \left (b \log \left (c x^{n}\right ) + a\right )^{\frac {5}{2}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.40, size = 295, normalized size = 4.40 \[ \frac {2 \left (2 \sqrt {-\left (\sinh ^{2}\left (\frac {a}{2}+\frac {b \ln \left (c \,x^{n}\right )}{2}\right )\right )}\, \sqrt {-2 \left (\sinh ^{2}\left (\frac {a}{2}+\frac {b \ln \left (c \,x^{n}\right )}{2}\right )\right )-1}\, \EllipticF \left (\cosh \left (\frac {a}{2}+\frac {b \ln \left (c \,x^{n}\right )}{2}\right ), \sqrt {2}\right ) \left (\sinh ^{2}\left (\frac {a}{2}+\frac {b \ln \left (c \,x^{n}\right )}{2}\right )\right )+\sqrt {-\left (\sinh ^{2}\left (\frac {a}{2}+\frac {b \ln \left (c \,x^{n}\right )}{2}\right )\right )}\, \sqrt {-2 \left (\sinh ^{2}\left (\frac {a}{2}+\frac {b \ln \left (c \,x^{n}\right )}{2}\right )\right )-1}\, \EllipticF \left (\cosh \left (\frac {a}{2}+\frac {b \ln \left (c \,x^{n}\right )}{2}\right ), \sqrt {2}\right )+2 \cosh \left (\frac {a}{2}+\frac {b \ln \left (c \,x^{n}\right )}{2}\right ) \left (\sinh ^{2}\left (\frac {a}{2}+\frac {b \ln \left (c \,x^{n}\right )}{2}\right )\right )\right ) \sqrt {\left (2 \left (\cosh ^{2}\left (\frac {a}{2}+\frac {b \ln \left (c \,x^{n}\right )}{2}\right )\right )-1\right ) \left (\sinh ^{2}\left (\frac {a}{2}+\frac {b \ln \left (c \,x^{n}\right )}{2}\right )\right )}}{3 n \sqrt {2 \left (\sinh ^{4}\left (\frac {a}{2}+\frac {b \ln \left (c \,x^{n}\right )}{2}\right )\right )+\sinh ^{2}\left (\frac {a}{2}+\frac {b \ln \left (c \,x^{n}\right )}{2}\right )}\, \left (2 \left (\cosh ^{2}\left (\frac {a}{2}+\frac {b \ln \left (c \,x^{n}\right )}{2}\right )\right )-1\right )^{\frac {3}{2}} \sinh \left (\frac {a}{2}+\frac {b \ln \left (c \,x^{n}\right )}{2}\right ) b} \]
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 \[ \int \frac {1}{x \cosh \left (b \log \left (c x^{n}\right ) + a\right )^{\frac {5}{2}}}\,{d x} \]
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 \[ \int \frac {1}{x\,{\mathrm {cosh}\left (a+b\,\ln \left (c\,x^n\right )\right )}^{5/2}} \,d x \]
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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