Optimal. Leaf size=54 \[ -\frac {1}{4} \sinh ^{-1}\left (a x^2\right )^2+\frac {1}{2} \sinh ^{-1}\left (a x^2\right ) \log \left (1-e^{2 \sinh ^{-1}\left (a x^2\right )}\right )+\frac {1}{4} \text {PolyLog}\left (2,e^{2 \sinh ^{-1}\left (a x^2\right )}\right ) \]
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Rubi [A]
time = 0.04, antiderivative size = 54, 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 = {5869, 3797,
2221, 2317, 2438} \begin {gather*} \frac {1}{4} \text {Li}_2\left (e^{2 \sinh ^{-1}\left (a x^2\right )}\right )-\frac {1}{4} \sinh ^{-1}\left (a x^2\right )^2+\frac {1}{2} \sinh ^{-1}\left (a x^2\right ) \log \left (1-e^{2 \sinh ^{-1}\left (a x^2\right )}\right ) \end {gather*}
Antiderivative was successfully verified.
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Rule 2221
Rule 2317
Rule 2438
Rule 3797
Rule 5869
Rubi steps
\begin {align*} \int \frac {\sinh ^{-1}\left (a x^2\right )}{x} \, dx &=\frac {1}{2} \text {Subst}\left (\int x \coth (x) \, dx,x,\sinh ^{-1}\left (a x^2\right )\right )\\ &=-\frac {1}{4} \sinh ^{-1}\left (a x^2\right )^2-\text {Subst}\left (\int \frac {e^{2 x} x}{1-e^{2 x}} \, dx,x,\sinh ^{-1}\left (a x^2\right )\right )\\ &=-\frac {1}{4} \sinh ^{-1}\left (a x^2\right )^2+\frac {1}{2} \sinh ^{-1}\left (a x^2\right ) \log \left (1-e^{2 \sinh ^{-1}\left (a x^2\right )}\right )-\frac {1}{2} \text {Subst}\left (\int \log \left (1-e^{2 x}\right ) \, dx,x,\sinh ^{-1}\left (a x^2\right )\right )\\ &=-\frac {1}{4} \sinh ^{-1}\left (a x^2\right )^2+\frac {1}{2} \sinh ^{-1}\left (a x^2\right ) \log \left (1-e^{2 \sinh ^{-1}\left (a x^2\right )}\right )-\frac {1}{4} \text {Subst}\left (\int \frac {\log (1-x)}{x} \, dx,x,e^{2 \sinh ^{-1}\left (a x^2\right )}\right )\\ &=-\frac {1}{4} \sinh ^{-1}\left (a x^2\right )^2+\frac {1}{2} \sinh ^{-1}\left (a x^2\right ) \log \left (1-e^{2 \sinh ^{-1}\left (a x^2\right )}\right )+\frac {1}{4} \text {Li}_2\left (e^{2 \sinh ^{-1}\left (a x^2\right )}\right )\\ \end {align*}
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Mathematica [A]
time = 0.01, size = 54, normalized size = 1.00 \begin {gather*} -\frac {1}{4} \sinh ^{-1}\left (a x^2\right )^2+\frac {1}{2} \sinh ^{-1}\left (a x^2\right ) \log \left (1-e^{2 \sinh ^{-1}\left (a x^2\right )}\right )+\frac {1}{4} \text {PolyLog}\left (2,e^{2 \sinh ^{-1}\left (a x^2\right )}\right ) \end {gather*}
Antiderivative was successfully verified.
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Maple [F]
time = 0.41, size = 0, normalized size = 0.00 \[\int \frac {\arcsinh \left (a \,x^{2}\right )}{x}\, dx\]
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 \begin {gather*} \text {Failed to integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}
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 \begin {gather*} \int \frac {\operatorname {asinh}{\left (a x^{2} \right )}}{x}\, dx \end {gather*}
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 \begin {gather*} \text {could not integrate} \end {gather*}
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 \begin {gather*} \int \frac {\mathrm {asinh}\left (a\,x^2\right )}{x} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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