Optimal. Leaf size=83 \[ \frac {3}{2} \sinh ^{-1}(a x)^2 \text {Li}_2\left (e^{2 \sinh ^{-1}(a x)}\right )-\frac {3}{2} \sinh ^{-1}(a x) \text {Li}_3\left (e^{2 \sinh ^{-1}(a x)}\right )+\frac {3}{4} \text {Li}_4\left (e^{2 \sinh ^{-1}(a x)}\right )-\frac {1}{4} \sinh ^{-1}(a x)^4+\sinh ^{-1}(a x)^3 \log \left (1-e^{2 \sinh ^{-1}(a x)}\right ) \]
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Rubi [A] time = 0.11, antiderivative size = 83, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 7, integrand size = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.700, Rules used = {5659, 3716, 2190, 2531, 6609, 2282, 6589} \[ \frac {3}{2} \sinh ^{-1}(a x)^2 \text {PolyLog}\left (2,e^{2 \sinh ^{-1}(a x)}\right )-\frac {3}{2} \sinh ^{-1}(a x) \text {PolyLog}\left (3,e^{2 \sinh ^{-1}(a x)}\right )+\frac {3}{4} \text {PolyLog}\left (4,e^{2 \sinh ^{-1}(a x)}\right )-\frac {1}{4} \sinh ^{-1}(a x)^4+\sinh ^{-1}(a x)^3 \log \left (1-e^{2 \sinh ^{-1}(a x)}\right ) \]
Antiderivative was successfully verified.
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Rule 2190
Rule 2282
Rule 2531
Rule 3716
Rule 5659
Rule 6589
Rule 6609
Rubi steps
\begin {align*} \int \frac {\sinh ^{-1}(a x)^3}{x} \, dx &=\operatorname {Subst}\left (\int x^3 \coth (x) \, dx,x,\sinh ^{-1}(a x)\right )\\ &=-\frac {1}{4} \sinh ^{-1}(a x)^4-2 \operatorname {Subst}\left (\int \frac {e^{2 x} x^3}{1-e^{2 x}} \, dx,x,\sinh ^{-1}(a x)\right )\\ &=-\frac {1}{4} \sinh ^{-1}(a x)^4+\sinh ^{-1}(a x)^3 \log \left (1-e^{2 \sinh ^{-1}(a x)}\right )-3 \operatorname {Subst}\left (\int x^2 \log \left (1-e^{2 x}\right ) \, dx,x,\sinh ^{-1}(a x)\right )\\ &=-\frac {1}{4} \sinh ^{-1}(a x)^4+\sinh ^{-1}(a x)^3 \log \left (1-e^{2 \sinh ^{-1}(a x)}\right )+\frac {3}{2} \sinh ^{-1}(a x)^2 \text {Li}_2\left (e^{2 \sinh ^{-1}(a x)}\right )-3 \operatorname {Subst}\left (\int x \text {Li}_2\left (e^{2 x}\right ) \, dx,x,\sinh ^{-1}(a x)\right )\\ &=-\frac {1}{4} \sinh ^{-1}(a x)^4+\sinh ^{-1}(a x)^3 \log \left (1-e^{2 \sinh ^{-1}(a x)}\right )+\frac {3}{2} \sinh ^{-1}(a x)^2 \text {Li}_2\left (e^{2 \sinh ^{-1}(a x)}\right )-\frac {3}{2} \sinh ^{-1}(a x) \text {Li}_3\left (e^{2 \sinh ^{-1}(a x)}\right )+\frac {3}{2} \operatorname {Subst}\left (\int \text {Li}_3\left (e^{2 x}\right ) \, dx,x,\sinh ^{-1}(a x)\right )\\ &=-\frac {1}{4} \sinh ^{-1}(a x)^4+\sinh ^{-1}(a x)^3 \log \left (1-e^{2 \sinh ^{-1}(a x)}\right )+\frac {3}{2} \sinh ^{-1}(a x)^2 \text {Li}_2\left (e^{2 \sinh ^{-1}(a x)}\right )-\frac {3}{2} \sinh ^{-1}(a x) \text {Li}_3\left (e^{2 \sinh ^{-1}(a x)}\right )+\frac {3}{4} \operatorname {Subst}\left (\int \frac {\text {Li}_3(x)}{x} \, dx,x,e^{2 \sinh ^{-1}(a x)}\right )\\ &=-\frac {1}{4} \sinh ^{-1}(a x)^4+\sinh ^{-1}(a x)^3 \log \left (1-e^{2 \sinh ^{-1}(a x)}\right )+\frac {3}{2} \sinh ^{-1}(a x)^2 \text {Li}_2\left (e^{2 \sinh ^{-1}(a x)}\right )-\frac {3}{2} \sinh ^{-1}(a x) \text {Li}_3\left (e^{2 \sinh ^{-1}(a x)}\right )+\frac {3}{4} \text {Li}_4\left (e^{2 \sinh ^{-1}(a x)}\right )\\ \end {align*}
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Mathematica [A] time = 0.01, size = 83, normalized size = 1.00 \[ \frac {3}{2} \sinh ^{-1}(a x)^2 \text {Li}_2\left (e^{2 \sinh ^{-1}(a x)}\right )-\frac {3}{2} \sinh ^{-1}(a x) \text {Li}_3\left (e^{2 \sinh ^{-1}(a x)}\right )+\frac {3}{4} \text {Li}_4\left (e^{2 \sinh ^{-1}(a x)}\right )-\frac {1}{4} \sinh ^{-1}(a x)^4+\sinh ^{-1}(a x)^3 \log \left (1-e^{2 \sinh ^{-1}(a x)}\right ) \]
Antiderivative was successfully verified.
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fricas [F] time = 0.41, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {\operatorname {arsinh}\left (a x\right )^{3}}{x}, x\right ) \]
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 \[ \int \frac {\operatorname {arsinh}\left (a x\right )^{3}}{x}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.16, size = 204, normalized size = 2.46 \[ -\frac {\arcsinh \left (a x \right )^{4}}{4}+\arcsinh \left (a x \right )^{3} \ln \left (1-a x -\sqrt {a^{2} x^{2}+1}\right )+3 \arcsinh \left (a x \right )^{2} \polylog \left (2, a x +\sqrt {a^{2} x^{2}+1}\right )-6 \arcsinh \left (a x \right ) \polylog \left (3, a x +\sqrt {a^{2} x^{2}+1}\right )+6 \polylog \left (4, a x +\sqrt {a^{2} x^{2}+1}\right )+\arcsinh \left (a x \right )^{3} \ln \left (a x +\sqrt {a^{2} x^{2}+1}+1\right )+3 \arcsinh \left (a x \right )^{2} \polylog \left (2, -a x -\sqrt {a^{2} x^{2}+1}\right )-6 \arcsinh \left (a x \right ) \polylog \left (3, -a x -\sqrt {a^{2} x^{2}+1}\right )+6 \polylog \left (4, -a x -\sqrt {a^{2} x^{2}+1}\right ) \]
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 \[ \int \frac {\operatorname {arsinh}\left (a x\right )^{3}}{x}\,{d x} \]
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 \[ \int \frac {{\mathrm {asinh}\left (a\,x\right )}^3}{x} \,d x \]
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 \[ \int \frac {\operatorname {asinh}^{3}{\left (a x \right )}}{x}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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