Optimal. Leaf size=70 \[ -\frac {2 \sqrt {\coth \left (a+b \log \left (c x^n\right )\right )}}{b n}+\frac {\tan ^{-1}\left (\sqrt {\coth \left (a+b \log \left (c x^n\right )\right )}\right )}{b n}+\frac {\tanh ^{-1}\left (\sqrt {\coth \left (a+b \log \left (c x^n\right )\right )}\right )}{b n} \]
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Rubi [A] time = 0.05, antiderivative size = 70, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 6, integrand size = 19, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.316, Rules used = {3473, 3476, 329, 212, 206, 203} \[ -\frac {2 \sqrt {\coth \left (a+b \log \left (c x^n\right )\right )}}{b n}+\frac {\tan ^{-1}\left (\sqrt {\coth \left (a+b \log \left (c x^n\right )\right )}\right )}{b n}+\frac {\tanh ^{-1}\left (\sqrt {\coth \left (a+b \log \left (c x^n\right )\right )}\right )}{b n} \]
Antiderivative was successfully verified.
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Rule 203
Rule 206
Rule 212
Rule 329
Rule 3473
Rule 3476
Rubi steps
\begin {align*} \int \frac {\coth ^{\frac {3}{2}}\left (a+b \log \left (c x^n\right )\right )}{x} \, dx &=\frac {\operatorname {Subst}\left (\int \coth ^{\frac {3}{2}}(a+b x) \, dx,x,\log \left (c x^n\right )\right )}{n}\\ &=-\frac {2 \sqrt {\coth \left (a+b \log \left (c x^n\right )\right )}}{b n}+\frac {\operatorname {Subst}\left (\int \frac {1}{\sqrt {\coth (a+b x)}} \, dx,x,\log \left (c x^n\right )\right )}{n}\\ &=-\frac {2 \sqrt {\coth \left (a+b \log \left (c x^n\right )\right )}}{b n}-\frac {\operatorname {Subst}\left (\int \frac {1}{\sqrt {x} \left (-1+x^2\right )} \, dx,x,\coth \left (a+b \log \left (c x^n\right )\right )\right )}{b n}\\ &=-\frac {2 \sqrt {\coth \left (a+b \log \left (c x^n\right )\right )}}{b n}-\frac {2 \operatorname {Subst}\left (\int \frac {1}{-1+x^4} \, dx,x,\sqrt {\coth \left (a+b \log \left (c x^n\right )\right )}\right )}{b n}\\ &=-\frac {2 \sqrt {\coth \left (a+b \log \left (c x^n\right )\right )}}{b n}+\frac {\operatorname {Subst}\left (\int \frac {1}{1-x^2} \, dx,x,\sqrt {\coth \left (a+b \log \left (c x^n\right )\right )}\right )}{b n}+\frac {\operatorname {Subst}\left (\int \frac {1}{1+x^2} \, dx,x,\sqrt {\coth \left (a+b \log \left (c x^n\right )\right )}\right )}{b n}\\ &=\frac {\tan ^{-1}\left (\sqrt {\coth \left (a+b \log \left (c x^n\right )\right )}\right )}{b n}+\frac {\tanh ^{-1}\left (\sqrt {\coth \left (a+b \log \left (c x^n\right )\right )}\right )}{b n}-\frac {2 \sqrt {\coth \left (a+b \log \left (c x^n\right )\right )}}{b n}\\ \end {align*}
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Mathematica [A] time = 0.15, size = 57, normalized size = 0.81 \[ \frac {-2 \sqrt {\coth \left (a+b \log \left (c x^n\right )\right )}+\tan ^{-1}\left (\sqrt {\coth \left (a+b \log \left (c x^n\right )\right )}\right )+\tanh ^{-1}\left (\sqrt {\coth \left (a+b \log \left (c x^n\right )\right )}\right )}{b n} \]
Antiderivative was successfully verified.
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fricas [B] time = 0.47, size = 334, normalized size = 4.77 \[ -\frac {4 \, \sqrt {\frac {\cosh \left (b n \log \relax (x) + b \log \relax (c) + a\right )}{\sinh \left (b n \log \relax (x) + b \log \relax (c) + a\right )}} + 2 \, \arctan \left (-\cosh \left (b n \log \relax (x) + b \log \relax (c) + a\right )^{2} - 2 \, \cosh \left (b n \log \relax (x) + b \log \relax (c) + a\right ) \sinh \left (b n \log \relax (x) + b \log \relax (c) + a\right ) - \sinh \left (b n \log \relax (x) + b \log \relax (c) + a\right )^{2} + {\left (\cosh \left (b n \log \relax (x) + b \log \relax (c) + a\right )^{2} + 2 \, \cosh \left (b n \log \relax (x) + b \log \relax (c) + a\right ) \sinh \left (b n \log \relax (x) + b \log \relax (c) + a\right ) + \sinh \left (b n \log \relax (x) + b \log \relax (c) + a\right )^{2} - 1\right )} \sqrt {\frac {\cosh \left (b n \log \relax (x) + b \log \relax (c) + a\right )}{\sinh \left (b n \log \relax (x) + b \log \relax (c) + a\right )}}\right ) + \log \left (-\cosh \left (b n \log \relax (x) + b \log \relax (c) + a\right )^{2} - 2 \, \cosh \left (b n \log \relax (x) + b \log \relax (c) + a\right ) \sinh \left (b n \log \relax (x) + b \log \relax (c) + a\right ) - \sinh \left (b n \log \relax (x) + b \log \relax (c) + a\right )^{2} + {\left (\cosh \left (b n \log \relax (x) + b \log \relax (c) + a\right )^{2} + 2 \, \cosh \left (b n \log \relax (x) + b \log \relax (c) + a\right ) \sinh \left (b n \log \relax (x) + b \log \relax (c) + a\right ) + \sinh \left (b n \log \relax (x) + b \log \relax (c) + a\right )^{2} - 1\right )} \sqrt {\frac {\cosh \left (b n \log \relax (x) + b \log \relax (c) + a\right )}{\sinh \left (b n \log \relax (x) + b \log \relax (c) + a\right )}}\right )}{2 \, b n} \]
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 = 0.13, size = 92, normalized size = 1.31 \[ -\frac {2 \left (\sqrt {\coth }\left (a +b \ln \left (c \,x^{n}\right )\right )\right )}{b n}-\frac {\ln \left (\sqrt {\coth }\left (a +b \ln \left (c \,x^{n}\right )\right )-1\right )}{2 b n}+\frac {\ln \left (\sqrt {\coth }\left (a +b \ln \left (c \,x^{n}\right )\right )+1\right )}{2 b n}+\frac {\arctan \left (\sqrt {\coth }\left (a +b \ln \left (c \,x^{n}\right )\right )\right )}{b n} \]
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 \[ \int \frac {\coth \left (b \log \left (c x^{n}\right ) + a\right )^{\frac {3}{2}}}{x}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 1.86, size = 51, normalized size = 0.73 \[ \frac {\mathrm {atan}\left (\sqrt {\mathrm {coth}\left (a+b\,\ln \left (c\,x^n\right )\right )}\right )+\mathrm {atanh}\left (\sqrt {\mathrm {coth}\left (a+b\,\ln \left (c\,x^n\right )\right )}\right )-2\,\sqrt {\mathrm {coth}\left (a+b\,\ln \left (c\,x^n\right )\right )}}{b\,n} \]
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 \[ \int \frac {\coth ^{\frac {3}{2}}{\left (a + b \log {\left (c x^{n} \right )} \right )}}{x}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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