Optimal. Leaf size=68 \[ -2 \sinh ^{-1}(a x) \text {Li}_2\left (-e^{\sinh ^{-1}(a x)}\right )+2 \sinh ^{-1}(a x) \text {Li}_2\left (e^{\sinh ^{-1}(a x)}\right )+2 \text {Li}_3\left (-e^{\sinh ^{-1}(a x)}\right )-2 \text {Li}_3\left (e^{\sinh ^{-1}(a x)}\right )-2 \sinh ^{-1}(a x)^2 \tanh ^{-1}\left (e^{\sinh ^{-1}(a x)}\right ) \]
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Rubi [A] time = 0.15, antiderivative size = 68, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 5, integrand size = 23, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.217, Rules used = {5760, 4182, 2531, 2282, 6589} \[ -2 \sinh ^{-1}(a x) \text {PolyLog}\left (2,-e^{\sinh ^{-1}(a x)}\right )+2 \sinh ^{-1}(a x) \text {PolyLog}\left (2,e^{\sinh ^{-1}(a x)}\right )+2 \text {PolyLog}\left (3,-e^{\sinh ^{-1}(a x)}\right )-2 \text {PolyLog}\left (3,e^{\sinh ^{-1}(a x)}\right )-2 \sinh ^{-1}(a x)^2 \tanh ^{-1}\left (e^{\sinh ^{-1}(a x)}\right ) \]
Antiderivative was successfully verified.
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Rule 2282
Rule 2531
Rule 4182
Rule 5760
Rule 6589
Rubi steps
\begin {align*} \int \frac {\sinh ^{-1}(a x)^2}{x \sqrt {1+a^2 x^2}} \, dx &=\operatorname {Subst}\left (\int x^2 \text {csch}(x) \, dx,x,\sinh ^{-1}(a x)\right )\\ &=-2 \sinh ^{-1}(a x)^2 \tanh ^{-1}\left (e^{\sinh ^{-1}(a x)}\right )-2 \operatorname {Subst}\left (\int x \log \left (1-e^x\right ) \, dx,x,\sinh ^{-1}(a x)\right )+2 \operatorname {Subst}\left (\int x \log \left (1+e^x\right ) \, dx,x,\sinh ^{-1}(a x)\right )\\ &=-2 \sinh ^{-1}(a x)^2 \tanh ^{-1}\left (e^{\sinh ^{-1}(a x)}\right )-2 \sinh ^{-1}(a x) \text {Li}_2\left (-e^{\sinh ^{-1}(a x)}\right )+2 \sinh ^{-1}(a x) \text {Li}_2\left (e^{\sinh ^{-1}(a x)}\right )+2 \operatorname {Subst}\left (\int \text {Li}_2\left (-e^x\right ) \, dx,x,\sinh ^{-1}(a x)\right )-2 \operatorname {Subst}\left (\int \text {Li}_2\left (e^x\right ) \, dx,x,\sinh ^{-1}(a x)\right )\\ &=-2 \sinh ^{-1}(a x)^2 \tanh ^{-1}\left (e^{\sinh ^{-1}(a x)}\right )-2 \sinh ^{-1}(a x) \text {Li}_2\left (-e^{\sinh ^{-1}(a x)}\right )+2 \sinh ^{-1}(a x) \text {Li}_2\left (e^{\sinh ^{-1}(a x)}\right )+2 \operatorname {Subst}\left (\int \frac {\text {Li}_2(-x)}{x} \, dx,x,e^{\sinh ^{-1}(a x)}\right )-2 \operatorname {Subst}\left (\int \frac {\text {Li}_2(x)}{x} \, dx,x,e^{\sinh ^{-1}(a x)}\right )\\ &=-2 \sinh ^{-1}(a x)^2 \tanh ^{-1}\left (e^{\sinh ^{-1}(a x)}\right )-2 \sinh ^{-1}(a x) \text {Li}_2\left (-e^{\sinh ^{-1}(a x)}\right )+2 \sinh ^{-1}(a x) \text {Li}_2\left (e^{\sinh ^{-1}(a x)}\right )+2 \text {Li}_3\left (-e^{\sinh ^{-1}(a x)}\right )-2 \text {Li}_3\left (e^{\sinh ^{-1}(a x)}\right )\\ \end {align*}
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Mathematica [A] time = 0.11, size = 100, normalized size = 1.47 \[ 2 \sinh ^{-1}(a x) \text {Li}_2\left (-e^{-\sinh ^{-1}(a x)}\right )-2 \sinh ^{-1}(a x) \text {Li}_2\left (e^{-\sinh ^{-1}(a x)}\right )+2 \text {Li}_3\left (-e^{-\sinh ^{-1}(a x)}\right )-2 \text {Li}_3\left (e^{-\sinh ^{-1}(a x)}\right )+\sinh ^{-1}(a x)^2 \log \left (1-e^{-\sinh ^{-1}(a x)}\right )-\sinh ^{-1}(a x)^2 \log \left (e^{-\sinh ^{-1}(a x)}+1\right ) \]
Warning: Unable to verify antiderivative.
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fricas [F] time = 0.45, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {\sqrt {a^{2} x^{2} + 1} \operatorname {arsinh}\left (a x\right )^{2}}{a^{2} x^{3} + x}, x\right ) \]
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 \[ \int \frac {\operatorname {arsinh}\left (a x\right )^{2}}{\sqrt {a^{2} x^{2} + 1} x}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.09, size = 144, normalized size = 2.12 \[ -\arcsinh \left (a x \right )^{2} \ln \left (1+a x +\sqrt {a^{2} x^{2}+1}\right )-2 \arcsinh \left (a x \right ) \polylog \left (2, -a x -\sqrt {a^{2} x^{2}+1}\right )+2 \polylog \left (3, -a x -\sqrt {a^{2} x^{2}+1}\right )+\arcsinh \left (a x \right )^{2} \ln \left (1-a x -\sqrt {a^{2} x^{2}+1}\right )+2 \arcsinh \left (a x \right ) \polylog \left (2, a x +\sqrt {a^{2} x^{2}+1}\right )-2 \polylog \left (3, a x +\sqrt {a^{2} x^{2}+1}\right ) \]
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 \[ \int \frac {\operatorname {arsinh}\left (a x\right )^{2}}{\sqrt {a^{2} x^{2} + 1} x}\,{d x} \]
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 \[ \int \frac {{\mathrm {asinh}\left (a\,x\right )}^2}{x\,\sqrt {a^2\,x^2+1}} \,d x \]
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 \[ \int \frac {\operatorname {asinh}^{2}{\left (a x \right )}}{x \sqrt {a^{2} x^{2} + 1}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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