Optimal. Leaf size=93 \[ \frac {3}{2} a^2 \text {Li}_2\left (e^{2 \sinh ^{-1}(a x)}\right )-\frac {3 a \sqrt {a^2 x^2+1} \sinh ^{-1}(a x)^2}{2 x}-\frac {3}{2} a^2 \sinh ^{-1}(a x)^2+3 a^2 \sinh ^{-1}(a x) \log \left (1-e^{2 \sinh ^{-1}(a x)}\right )-\frac {\sinh ^{-1}(a x)^3}{2 x^2} \]
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Rubi [A] time = 0.17, antiderivative size = 93, 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 = {5661, 5723, 5659, 3716, 2190, 2279, 2391} \[ \frac {3}{2} a^2 \text {PolyLog}\left (2,e^{2 \sinh ^{-1}(a x)}\right )-\frac {3 a \sqrt {a^2 x^2+1} \sinh ^{-1}(a x)^2}{2 x}-\frac {3}{2} a^2 \sinh ^{-1}(a x)^2+3 a^2 \sinh ^{-1}(a x) \log \left (1-e^{2 \sinh ^{-1}(a x)}\right )-\frac {\sinh ^{-1}(a x)^3}{2 x^2} \]
Antiderivative was successfully verified.
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Rule 2190
Rule 2279
Rule 2391
Rule 3716
Rule 5659
Rule 5661
Rule 5723
Rubi steps
\begin {align*} \int \frac {\sinh ^{-1}(a x)^3}{x^3} \, dx &=-\frac {\sinh ^{-1}(a x)^3}{2 x^2}+\frac {1}{2} (3 a) \int \frac {\sinh ^{-1}(a x)^2}{x^2 \sqrt {1+a^2 x^2}} \, dx\\ &=-\frac {3 a \sqrt {1+a^2 x^2} \sinh ^{-1}(a x)^2}{2 x}-\frac {\sinh ^{-1}(a x)^3}{2 x^2}+\left (3 a^2\right ) \int \frac {\sinh ^{-1}(a x)}{x} \, dx\\ &=-\frac {3 a \sqrt {1+a^2 x^2} \sinh ^{-1}(a x)^2}{2 x}-\frac {\sinh ^{-1}(a x)^3}{2 x^2}+\left (3 a^2\right ) \operatorname {Subst}\left (\int x \coth (x) \, dx,x,\sinh ^{-1}(a x)\right )\\ &=-\frac {3}{2} a^2 \sinh ^{-1}(a x)^2-\frac {3 a \sqrt {1+a^2 x^2} \sinh ^{-1}(a x)^2}{2 x}-\frac {\sinh ^{-1}(a x)^3}{2 x^2}-\left (6 a^2\right ) \operatorname {Subst}\left (\int \frac {e^{2 x} x}{1-e^{2 x}} \, dx,x,\sinh ^{-1}(a x)\right )\\ &=-\frac {3}{2} a^2 \sinh ^{-1}(a x)^2-\frac {3 a \sqrt {1+a^2 x^2} \sinh ^{-1}(a x)^2}{2 x}-\frac {\sinh ^{-1}(a x)^3}{2 x^2}+3 a^2 \sinh ^{-1}(a x) \log \left (1-e^{2 \sinh ^{-1}(a x)}\right )-\left (3 a^2\right ) \operatorname {Subst}\left (\int \log \left (1-e^{2 x}\right ) \, dx,x,\sinh ^{-1}(a x)\right )\\ &=-\frac {3}{2} a^2 \sinh ^{-1}(a x)^2-\frac {3 a \sqrt {1+a^2 x^2} \sinh ^{-1}(a x)^2}{2 x}-\frac {\sinh ^{-1}(a x)^3}{2 x^2}+3 a^2 \sinh ^{-1}(a x) \log \left (1-e^{2 \sinh ^{-1}(a x)}\right )-\frac {1}{2} \left (3 a^2\right ) \operatorname {Subst}\left (\int \frac {\log (1-x)}{x} \, dx,x,e^{2 \sinh ^{-1}(a x)}\right )\\ &=-\frac {3}{2} a^2 \sinh ^{-1}(a x)^2-\frac {3 a \sqrt {1+a^2 x^2} \sinh ^{-1}(a x)^2}{2 x}-\frac {\sinh ^{-1}(a x)^3}{2 x^2}+3 a^2 \sinh ^{-1}(a x) \log \left (1-e^{2 \sinh ^{-1}(a x)}\right )+\frac {3}{2} a^2 \text {Li}_2\left (e^{2 \sinh ^{-1}(a x)}\right )\\ \end {align*}
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Mathematica [A] time = 0.35, size = 80, normalized size = 0.86 \[ -\frac {\sinh ^{-1}(a x)^3-3 a x \left (\sinh ^{-1}(a x) \left (\left (a x-\sqrt {a^2 x^2+1}\right ) \sinh ^{-1}(a x)+2 a x \log \left (1-e^{-2 \sinh ^{-1}(a x)}\right )\right )-a x \text {Li}_2\left (e^{-2 \sinh ^{-1}(a x)}\right )\right )}{2 x^2} \]
Warning: Unable to verify antiderivative.
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fricas [F] time = 0.46, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {\operatorname {arsinh}\left (a x\right )^{3}}{x^{3}}, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [F(-2)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Exception raised: TypeError} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.37, size = 149, normalized size = 1.60 \[ -\frac {3 a^{2} \arcsinh \left (a x \right )^{2}}{2}-\frac {3 a \arcsinh \left (a x \right )^{2} \sqrt {a^{2} x^{2}+1}}{2 x}-\frac {\arcsinh \left (a x \right )^{3}}{2 x^{2}}+3 a^{2} \arcsinh \left (a x \right ) \ln \left (1-a x -\sqrt {a^{2} x^{2}+1}\right )+3 a^{2} \polylog \left (2, a x +\sqrt {a^{2} x^{2}+1}\right )+3 a^{2} \arcsinh \left (a x \right ) \ln \left (a x +\sqrt {a^{2} x^{2}+1}+1\right )+3 a^{2} \polylog \left (2, -a x -\sqrt {a^{2} x^{2}+1}\right ) \]
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 \[ -\frac {\log \left (a x + \sqrt {a^{2} x^{2} + 1}\right )^{3}}{2 \, x^{2}} + \int \frac {3 \, {\left (a^{3} x^{2} + \sqrt {a^{2} x^{2} + 1} a^{2} x + a\right )} \log \left (a x + \sqrt {a^{2} x^{2} + 1}\right )^{2}}{2 \, {\left (a^{3} x^{5} + a x^{3} + {\left (a^{2} x^{4} + x^{2}\right )} \sqrt {a^{2} x^{2} + 1}\right )}}\,{d x} \]
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 \[ \int \frac {{\mathrm {asinh}\left (a\,x\right )}^3}{x^3} \,d x \]
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 \[ \int \frac {\operatorname {asinh}^{3}{\left (a x \right )}}{x^{3}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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