Optimal. Leaf size=60 \[ \frac {\text {Li}_2\left (e^{2 \sinh ^{-1}\left (a x^n\right )}\right )}{2 n}-\frac {\sinh ^{-1}\left (a x^n\right )^2}{2 n}+\frac {\sinh ^{-1}\left (a x^n\right ) \log \left (1-e^{2 \sinh ^{-1}\left (a x^n\right )}\right )}{n} \]
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Rubi [A] time = 0.07, antiderivative size = 60, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 5, integrand size = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.500, Rules used = {5890, 3716, 2190, 2279, 2391} \[ \frac {\text {PolyLog}\left (2,e^{2 \sinh ^{-1}\left (a x^n\right )}\right )}{2 n}-\frac {\sinh ^{-1}\left (a x^n\right )^2}{2 n}+\frac {\sinh ^{-1}\left (a x^n\right ) \log \left (1-e^{2 \sinh ^{-1}\left (a x^n\right )}\right )}{n} \]
Antiderivative was successfully verified.
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Rule 2190
Rule 2279
Rule 2391
Rule 3716
Rule 5890
Rubi steps
\begin {align*} \int \frac {\sinh ^{-1}\left (a x^n\right )}{x} \, dx &=\frac {\operatorname {Subst}\left (\int x \coth (x) \, dx,x,\sinh ^{-1}\left (a x^n\right )\right )}{n}\\ &=-\frac {\sinh ^{-1}\left (a x^n\right )^2}{2 n}-\frac {2 \operatorname {Subst}\left (\int \frac {e^{2 x} x}{1-e^{2 x}} \, dx,x,\sinh ^{-1}\left (a x^n\right )\right )}{n}\\ &=-\frac {\sinh ^{-1}\left (a x^n\right )^2}{2 n}+\frac {\sinh ^{-1}\left (a x^n\right ) \log \left (1-e^{2 \sinh ^{-1}\left (a x^n\right )}\right )}{n}-\frac {\operatorname {Subst}\left (\int \log \left (1-e^{2 x}\right ) \, dx,x,\sinh ^{-1}\left (a x^n\right )\right )}{n}\\ &=-\frac {\sinh ^{-1}\left (a x^n\right )^2}{2 n}+\frac {\sinh ^{-1}\left (a x^n\right ) \log \left (1-e^{2 \sinh ^{-1}\left (a x^n\right )}\right )}{n}-\frac {\operatorname {Subst}\left (\int \frac {\log (1-x)}{x} \, dx,x,e^{2 \sinh ^{-1}\left (a x^n\right )}\right )}{2 n}\\ &=-\frac {\sinh ^{-1}\left (a x^n\right )^2}{2 n}+\frac {\sinh ^{-1}\left (a x^n\right ) \log \left (1-e^{2 \sinh ^{-1}\left (a x^n\right )}\right )}{n}+\frac {\text {Li}_2\left (e^{2 \sinh ^{-1}\left (a x^n\right )}\right )}{2 n}\\ \end {align*}
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Mathematica [A] time = 0.01, size = 60, normalized size = 1.00 \[ \frac {\text {Li}_2\left (e^{2 \sinh ^{-1}\left (a x^n\right )}\right )}{2 n}-\frac {\sinh ^{-1}\left (a x^n\right )^2}{2 n}+\frac {\sinh ^{-1}\left (a x^n\right ) \log \left (1-e^{2 \sinh ^{-1}\left (a x^n\right )}\right )}{n} \]
Antiderivative was successfully verified.
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fricas [F(-2)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Exception raised: TypeError} \]
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 \[ \int \frac {\operatorname {arsinh}\left (a x^{n}\right )}{x}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.00, size = 133, normalized size = 2.22 \[ -\frac {\arcsinh \left (a \,x^{n}\right )^{2}}{2 n}+\frac {\arcsinh \left (a \,x^{n}\right ) \ln \left (1+a \,x^{n}+\sqrt {1+a^{2} x^{2 n}}\right )}{n}+\frac {\polylog \left (2, -a \,x^{n}-\sqrt {1+a^{2} x^{2 n}}\right )}{n}+\frac {\arcsinh \left (a \,x^{n}\right ) \ln \left (1-a \,x^{n}-\sqrt {1+a^{2} x^{2 n}}\right )}{n}+\frac {\polylog \left (2, a \,x^{n}+\sqrt {1+a^{2} x^{2 n}}\right )}{n} \]
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 \[ -a n \int \frac {x^{n} \log \relax (x)}{a^{3} x x^{3 \, n} + a x x^{n} + {\left (a^{2} x x^{2 \, n} + x\right )} \sqrt {a^{2} x^{2 \, n} + 1}}\,{d x} - \frac {1}{2} \, n \log \relax (x)^{2} + n \int \frac {\log \relax (x)}{a^{2} x x^{2 \, n} + x}\,{d x} + \log \left (a x^{n} + \sqrt {a^{2} x^{2 \, n} + 1}\right ) \log \relax (x) \]
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 \[ \int \frac {\mathrm {asinh}\left (a\,x^n\right )}{x} \,d x \]
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 \[ \int \frac {\operatorname {asinh}{\left (a x^{n} \right )}}{x}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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