Optimal. Leaf size=120 \[ -12 a \sinh ^{-1}(a x)^2 \text {Li}_2\left (-e^{\sinh ^{-1}(a x)}\right )+12 a \sinh ^{-1}(a x)^2 \text {Li}_2\left (e^{\sinh ^{-1}(a x)}\right )+24 a \sinh ^{-1}(a x) \text {Li}_3\left (-e^{\sinh ^{-1}(a x)}\right )-24 a \sinh ^{-1}(a x) \text {Li}_3\left (e^{\sinh ^{-1}(a x)}\right )-24 a \text {Li}_4\left (-e^{\sinh ^{-1}(a x)}\right )+24 a \text {Li}_4\left (e^{\sinh ^{-1}(a x)}\right )-\frac {\sinh ^{-1}(a x)^4}{x}-8 a \sinh ^{-1}(a x)^3 \tanh ^{-1}\left (e^{\sinh ^{-1}(a x)}\right ) \]
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Rubi [A] time = 0.19, antiderivative size = 120, normalized size of antiderivative = 1.00, number of steps used = 11, number of rules used = 7, integrand size = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.700, Rules used = {5661, 5760, 4182, 2531, 6609, 2282, 6589} \[ -12 a \sinh ^{-1}(a x)^2 \text {PolyLog}\left (2,-e^{\sinh ^{-1}(a x)}\right )+12 a \sinh ^{-1}(a x)^2 \text {PolyLog}\left (2,e^{\sinh ^{-1}(a x)}\right )+24 a \sinh ^{-1}(a x) \text {PolyLog}\left (3,-e^{\sinh ^{-1}(a x)}\right )-24 a \sinh ^{-1}(a x) \text {PolyLog}\left (3,e^{\sinh ^{-1}(a x)}\right )-24 a \text {PolyLog}\left (4,-e^{\sinh ^{-1}(a x)}\right )+24 a \text {PolyLog}\left (4,e^{\sinh ^{-1}(a x)}\right )-\frac {\sinh ^{-1}(a x)^4}{x}-8 a \sinh ^{-1}(a x)^3 \tanh ^{-1}\left (e^{\sinh ^{-1}(a x)}\right ) \]
Antiderivative was successfully verified.
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Rule 2282
Rule 2531
Rule 4182
Rule 5661
Rule 5760
Rule 6589
Rule 6609
Rubi steps
\begin {align*} \int \frac {\sinh ^{-1}(a x)^4}{x^2} \, dx &=-\frac {\sinh ^{-1}(a x)^4}{x}+(4 a) \int \frac {\sinh ^{-1}(a x)^3}{x \sqrt {1+a^2 x^2}} \, dx\\ &=-\frac {\sinh ^{-1}(a x)^4}{x}+(4 a) \operatorname {Subst}\left (\int x^3 \text {csch}(x) \, dx,x,\sinh ^{-1}(a x)\right )\\ &=-\frac {\sinh ^{-1}(a x)^4}{x}-8 a \sinh ^{-1}(a x)^3 \tanh ^{-1}\left (e^{\sinh ^{-1}(a x)}\right )-(12 a) \operatorname {Subst}\left (\int x^2 \log \left (1-e^x\right ) \, dx,x,\sinh ^{-1}(a x)\right )+(12 a) \operatorname {Subst}\left (\int x^2 \log \left (1+e^x\right ) \, dx,x,\sinh ^{-1}(a x)\right )\\ &=-\frac {\sinh ^{-1}(a x)^4}{x}-8 a \sinh ^{-1}(a x)^3 \tanh ^{-1}\left (e^{\sinh ^{-1}(a x)}\right )-12 a \sinh ^{-1}(a x)^2 \text {Li}_2\left (-e^{\sinh ^{-1}(a x)}\right )+12 a \sinh ^{-1}(a x)^2 \text {Li}_2\left (e^{\sinh ^{-1}(a x)}\right )+(24 a) \operatorname {Subst}\left (\int x \text {Li}_2\left (-e^x\right ) \, dx,x,\sinh ^{-1}(a x)\right )-(24 a) \operatorname {Subst}\left (\int x \text {Li}_2\left (e^x\right ) \, dx,x,\sinh ^{-1}(a x)\right )\\ &=-\frac {\sinh ^{-1}(a x)^4}{x}-8 a \sinh ^{-1}(a x)^3 \tanh ^{-1}\left (e^{\sinh ^{-1}(a x)}\right )-12 a \sinh ^{-1}(a x)^2 \text {Li}_2\left (-e^{\sinh ^{-1}(a x)}\right )+12 a \sinh ^{-1}(a x)^2 \text {Li}_2\left (e^{\sinh ^{-1}(a x)}\right )+24 a \sinh ^{-1}(a x) \text {Li}_3\left (-e^{\sinh ^{-1}(a x)}\right )-24 a \sinh ^{-1}(a x) \text {Li}_3\left (e^{\sinh ^{-1}(a x)}\right )-(24 a) \operatorname {Subst}\left (\int \text {Li}_3\left (-e^x\right ) \, dx,x,\sinh ^{-1}(a x)\right )+(24 a) \operatorname {Subst}\left (\int \text {Li}_3\left (e^x\right ) \, dx,x,\sinh ^{-1}(a x)\right )\\ &=-\frac {\sinh ^{-1}(a x)^4}{x}-8 a \sinh ^{-1}(a x)^3 \tanh ^{-1}\left (e^{\sinh ^{-1}(a x)}\right )-12 a \sinh ^{-1}(a x)^2 \text {Li}_2\left (-e^{\sinh ^{-1}(a x)}\right )+12 a \sinh ^{-1}(a x)^2 \text {Li}_2\left (e^{\sinh ^{-1}(a x)}\right )+24 a \sinh ^{-1}(a x) \text {Li}_3\left (-e^{\sinh ^{-1}(a x)}\right )-24 a \sinh ^{-1}(a x) \text {Li}_3\left (e^{\sinh ^{-1}(a x)}\right )-(24 a) \operatorname {Subst}\left (\int \frac {\text {Li}_3(-x)}{x} \, dx,x,e^{\sinh ^{-1}(a x)}\right )+(24 a) \operatorname {Subst}\left (\int \frac {\text {Li}_3(x)}{x} \, dx,x,e^{\sinh ^{-1}(a x)}\right )\\ &=-\frac {\sinh ^{-1}(a x)^4}{x}-8 a \sinh ^{-1}(a x)^3 \tanh ^{-1}\left (e^{\sinh ^{-1}(a x)}\right )-12 a \sinh ^{-1}(a x)^2 \text {Li}_2\left (-e^{\sinh ^{-1}(a x)}\right )+12 a \sinh ^{-1}(a x)^2 \text {Li}_2\left (e^{\sinh ^{-1}(a x)}\right )+24 a \sinh ^{-1}(a x) \text {Li}_3\left (-e^{\sinh ^{-1}(a x)}\right )-24 a \sinh ^{-1}(a x) \text {Li}_3\left (e^{\sinh ^{-1}(a x)}\right )-24 a \text {Li}_4\left (-e^{\sinh ^{-1}(a x)}\right )+24 a \text {Li}_4\left (e^{\sinh ^{-1}(a x)}\right )\\ \end {align*}
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Mathematica [A] time = 0.25, size = 161, normalized size = 1.34 \[ \frac {1}{2} a \left (24 \sinh ^{-1}(a x)^2 \text {Li}_2\left (-e^{-\sinh ^{-1}(a x)}\right )+24 \sinh ^{-1}(a x)^2 \text {Li}_2\left (e^{\sinh ^{-1}(a x)}\right )+48 \sinh ^{-1}(a x) \text {Li}_3\left (-e^{-\sinh ^{-1}(a x)}\right )-48 \sinh ^{-1}(a x) \text {Li}_3\left (e^{\sinh ^{-1}(a x)}\right )+48 \text {Li}_4\left (-e^{-\sinh ^{-1}(a x)}\right )+48 \text {Li}_4\left (e^{\sinh ^{-1}(a x)}\right )-\frac {2 \sinh ^{-1}(a x)^4}{a x}-2 \sinh ^{-1}(a x)^4-8 \sinh ^{-1}(a x)^3 \log \left (e^{-\sinh ^{-1}(a x)}+1\right )+8 \sinh ^{-1}(a x)^3 \log \left (1-e^{\sinh ^{-1}(a x)}\right )+\pi ^4\right ) \]
Warning: Unable to verify antiderivative.
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fricas [F] time = 0.43, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {\operatorname {arsinh}\left (a x\right )^{4}}{x^{2}}, x\right ) \]
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 \[ \int \frac {\operatorname {arsinh}\left (a x\right )^{4}}{x^{2}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.24, size = 217, normalized size = 1.81 \[ -\frac {\arcsinh \left (a x \right )^{4}}{x}+4 a \arcsinh \left (a x \right )^{3} \ln \left (1-a x -\sqrt {a^{2} x^{2}+1}\right )+12 a \arcsinh \left (a x \right )^{2} \polylog \left (2, a x +\sqrt {a^{2} x^{2}+1}\right )-24 a \arcsinh \left (a x \right ) \polylog \left (3, a x +\sqrt {a^{2} x^{2}+1}\right )+24 a \polylog \left (4, a x +\sqrt {a^{2} x^{2}+1}\right )-4 a \arcsinh \left (a x \right )^{3} \ln \left (a x +\sqrt {a^{2} x^{2}+1}+1\right )-12 a \arcsinh \left (a x \right )^{2} \polylog \left (2, -a x -\sqrt {a^{2} x^{2}+1}\right )+24 a \arcsinh \left (a x \right ) \polylog \left (3, -a x -\sqrt {a^{2} x^{2}+1}\right )-24 a \polylog \left (4, -a x -\sqrt {a^{2} x^{2}+1}\right ) \]
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 \[ -\frac {\log \left (a x + \sqrt {a^{2} x^{2} + 1}\right )^{4}}{x} + \int \frac {4 \, {\left (a^{3} x^{2} + \sqrt {a^{2} x^{2} + 1} a^{2} x + a\right )} \log \left (a x + \sqrt {a^{2} x^{2} + 1}\right )^{3}}{a^{3} x^{4} + a x^{2} + {\left (a^{2} x^{3} + x\right )} \sqrt {a^{2} x^{2} + 1}}\,{d x} \]
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 \[ \int \frac {{\mathrm {asinh}\left (a\,x\right )}^4}{x^2} \,d x \]
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 \[ \int \frac {\operatorname {asinh}^{4}{\left (a x \right )}}{x^{2}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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