Optimal. Leaf size=99 \[ \frac {1}{3} a^3 \text {Li}_2\left (-e^{\sinh ^{-1}(a x)}\right )-\frac {1}{3} a^3 \text {Li}_2\left (e^{\sinh ^{-1}(a x)}\right )+\frac {2}{3} a^3 \sinh ^{-1}(a x) \tanh ^{-1}\left (e^{\sinh ^{-1}(a x)}\right )-\frac {a \sqrt {a^2 x^2+1} \sinh ^{-1}(a x)}{3 x^2}-\frac {a^2}{3 x}-\frac {\sinh ^{-1}(a x)^2}{3 x^3} \]
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Rubi [A] time = 0.16, antiderivative size = 99, normalized size of antiderivative = 1.00, number of steps used = 9, number of rules used = 7, integrand size = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.700, Rules used = {5661, 5747, 5760, 4182, 2279, 2391, 30} \[ \frac {1}{3} a^3 \text {PolyLog}\left (2,-e^{\sinh ^{-1}(a x)}\right )-\frac {1}{3} a^3 \text {PolyLog}\left (2,e^{\sinh ^{-1}(a x)}\right )-\frac {a \sqrt {a^2 x^2+1} \sinh ^{-1}(a x)}{3 x^2}-\frac {a^2}{3 x}+\frac {2}{3} a^3 \sinh ^{-1}(a x) \tanh ^{-1}\left (e^{\sinh ^{-1}(a x)}\right )-\frac {\sinh ^{-1}(a x)^2}{3 x^3} \]
Antiderivative was successfully verified.
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Rule 30
Rule 2279
Rule 2391
Rule 4182
Rule 5661
Rule 5747
Rule 5760
Rubi steps
\begin {align*} \int \frac {\sinh ^{-1}(a x)^2}{x^4} \, dx &=-\frac {\sinh ^{-1}(a x)^2}{3 x^3}+\frac {1}{3} (2 a) \int \frac {\sinh ^{-1}(a x)}{x^3 \sqrt {1+a^2 x^2}} \, dx\\ &=-\frac {a \sqrt {1+a^2 x^2} \sinh ^{-1}(a x)}{3 x^2}-\frac {\sinh ^{-1}(a x)^2}{3 x^3}+\frac {1}{3} a^2 \int \frac {1}{x^2} \, dx-\frac {1}{3} a^3 \int \frac {\sinh ^{-1}(a x)}{x \sqrt {1+a^2 x^2}} \, dx\\ &=-\frac {a^2}{3 x}-\frac {a \sqrt {1+a^2 x^2} \sinh ^{-1}(a x)}{3 x^2}-\frac {\sinh ^{-1}(a x)^2}{3 x^3}-\frac {1}{3} a^3 \operatorname {Subst}\left (\int x \text {csch}(x) \, dx,x,\sinh ^{-1}(a x)\right )\\ &=-\frac {a^2}{3 x}-\frac {a \sqrt {1+a^2 x^2} \sinh ^{-1}(a x)}{3 x^2}-\frac {\sinh ^{-1}(a x)^2}{3 x^3}+\frac {2}{3} a^3 \sinh ^{-1}(a x) \tanh ^{-1}\left (e^{\sinh ^{-1}(a x)}\right )+\frac {1}{3} a^3 \operatorname {Subst}\left (\int \log \left (1-e^x\right ) \, dx,x,\sinh ^{-1}(a x)\right )-\frac {1}{3} a^3 \operatorname {Subst}\left (\int \log \left (1+e^x\right ) \, dx,x,\sinh ^{-1}(a x)\right )\\ &=-\frac {a^2}{3 x}-\frac {a \sqrt {1+a^2 x^2} \sinh ^{-1}(a x)}{3 x^2}-\frac {\sinh ^{-1}(a x)^2}{3 x^3}+\frac {2}{3} a^3 \sinh ^{-1}(a x) \tanh ^{-1}\left (e^{\sinh ^{-1}(a x)}\right )+\frac {1}{3} a^3 \operatorname {Subst}\left (\int \frac {\log (1-x)}{x} \, dx,x,e^{\sinh ^{-1}(a x)}\right )-\frac {1}{3} a^3 \operatorname {Subst}\left (\int \frac {\log (1+x)}{x} \, dx,x,e^{\sinh ^{-1}(a x)}\right )\\ &=-\frac {a^2}{3 x}-\frac {a \sqrt {1+a^2 x^2} \sinh ^{-1}(a x)}{3 x^2}-\frac {\sinh ^{-1}(a x)^2}{3 x^3}+\frac {2}{3} a^3 \sinh ^{-1}(a x) \tanh ^{-1}\left (e^{\sinh ^{-1}(a x)}\right )+\frac {1}{3} a^3 \text {Li}_2\left (-e^{\sinh ^{-1}(a x)}\right )-\frac {1}{3} a^3 \text {Li}_2\left (e^{\sinh ^{-1}(a x)}\right )\\ \end {align*}
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Mathematica [A] time = 0.53, size = 125, normalized size = 1.26 \[ -\frac {a^3 x^3 \text {Li}_2\left (-e^{-\sinh ^{-1}(a x)}\right )-a^3 x^3 \text {Li}_2\left (e^{-\sinh ^{-1}(a x)}\right )+a^3 x^3 \sinh ^{-1}(a x) \log \left (1-e^{-\sinh ^{-1}(a x)}\right )-a^3 x^3 \sinh ^{-1}(a x) \log \left (e^{-\sinh ^{-1}(a x)}+1\right )+a^2 x^2+a x \sqrt {a^2 x^2+1} \sinh ^{-1}(a x)+\sinh ^{-1}(a x)^2}{3 x^3} \]
Warning: Unable to verify antiderivative.
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fricas [F] time = 0.42, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {\operatorname {arsinh}\left (a x\right )^{2}}{x^{4}}, x\right ) \]
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 \[ \int \frac {\operatorname {arsinh}\left (a x\right )^{2}}{x^{4}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.48, size = 144, normalized size = 1.45 \[ -\frac {a \arcsinh \left (a x \right ) \sqrt {a^{2} x^{2}+1}}{3 x^{2}}-\frac {a^{2}}{3 x}-\frac {\arcsinh \left (a x \right )^{2}}{3 x^{3}}-\frac {a^{3} \arcsinh \left (a x \right ) \ln \left (1-a x -\sqrt {a^{2} x^{2}+1}\right )}{3}-\frac {a^{3} \polylog \left (2, a x +\sqrt {a^{2} x^{2}+1}\right )}{3}+\frac {a^{3} \arcsinh \left (a x \right ) \ln \left (a x +\sqrt {a^{2} x^{2}+1}+1\right )}{3}+\frac {a^{3} \polylog \left (2, -a x -\sqrt {a^{2} x^{2}+1}\right )}{3} \]
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 \[ -\frac {\log \left (a x + \sqrt {a^{2} x^{2} + 1}\right )^{2}}{3 \, x^{3}} + \int \frac {2 \, {\left (a^{3} x^{2} + \sqrt {a^{2} x^{2} + 1} a^{2} x + a\right )} \log \left (a x + \sqrt {a^{2} x^{2} + 1}\right )}{3 \, {\left (a^{3} x^{6} + a x^{4} + {\left (a^{2} x^{5} + x^{3}\right )} \sqrt {a^{2} x^{2} + 1}\right )}}\,{d x} \]
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 \[ \int \frac {{\mathrm {asinh}\left (a\,x\right )}^2}{x^4} \,d x \]
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 \[ \int \frac {\operatorname {asinh}^{2}{\left (a x \right )}}{x^{4}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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