Optimal. Leaf size=88 \[ -\frac {\sqrt {a^2 x^2+1} \sinh ^{-1}(a x)^3}{x}+3 a \sinh ^{-1}(a x) \text {Li}_2\left (e^{2 \sinh ^{-1}(a x)}\right )-\frac {3}{2} a \text {Li}_3\left (e^{2 \sinh ^{-1}(a x)}\right )-a \sinh ^{-1}(a x)^3+3 a \sinh ^{-1}(a x)^2 \log \left (1-e^{2 \sinh ^{-1}(a x)}\right ) \]
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Rubi [A] time = 0.19, antiderivative size = 88, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 7, integrand size = 23, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.304, Rules used = {5723, 5659, 3716, 2190, 2531, 2282, 6589} \[ 3 a \sinh ^{-1}(a x) \text {PolyLog}\left (2,e^{2 \sinh ^{-1}(a x)}\right )-\frac {3}{2} a \text {PolyLog}\left (3,e^{2 \sinh ^{-1}(a x)}\right )-\frac {\sqrt {a^2 x^2+1} \sinh ^{-1}(a x)^3}{x}-a \sinh ^{-1}(a x)^3+3 a \sinh ^{-1}(a x)^2 \log \left (1-e^{2 \sinh ^{-1}(a x)}\right ) \]
Antiderivative was successfully verified.
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Rule 2190
Rule 2282
Rule 2531
Rule 3716
Rule 5659
Rule 5723
Rule 6589
Rubi steps
\begin {align*} \int \frac {\sinh ^{-1}(a x)^3}{x^2 \sqrt {1+a^2 x^2}} \, dx &=-\frac {\sqrt {1+a^2 x^2} \sinh ^{-1}(a x)^3}{x}+(3 a) \int \frac {\sinh ^{-1}(a x)^2}{x} \, dx\\ &=-\frac {\sqrt {1+a^2 x^2} \sinh ^{-1}(a x)^3}{x}+(3 a) \operatorname {Subst}\left (\int x^2 \coth (x) \, dx,x,\sinh ^{-1}(a x)\right )\\ &=-a \sinh ^{-1}(a x)^3-\frac {\sqrt {1+a^2 x^2} \sinh ^{-1}(a x)^3}{x}-(6 a) \operatorname {Subst}\left (\int \frac {e^{2 x} x^2}{1-e^{2 x}} \, dx,x,\sinh ^{-1}(a x)\right )\\ &=-a \sinh ^{-1}(a x)^3-\frac {\sqrt {1+a^2 x^2} \sinh ^{-1}(a x)^3}{x}+3 a \sinh ^{-1}(a x)^2 \log \left (1-e^{2 \sinh ^{-1}(a x)}\right )-(6 a) \operatorname {Subst}\left (\int x \log \left (1-e^{2 x}\right ) \, dx,x,\sinh ^{-1}(a x)\right )\\ &=-a \sinh ^{-1}(a x)^3-\frac {\sqrt {1+a^2 x^2} \sinh ^{-1}(a x)^3}{x}+3 a \sinh ^{-1}(a x)^2 \log \left (1-e^{2 \sinh ^{-1}(a x)}\right )+3 a \sinh ^{-1}(a x) \text {Li}_2\left (e^{2 \sinh ^{-1}(a x)}\right )-(3 a) \operatorname {Subst}\left (\int \text {Li}_2\left (e^{2 x}\right ) \, dx,x,\sinh ^{-1}(a x)\right )\\ &=-a \sinh ^{-1}(a x)^3-\frac {\sqrt {1+a^2 x^2} \sinh ^{-1}(a x)^3}{x}+3 a \sinh ^{-1}(a x)^2 \log \left (1-e^{2 \sinh ^{-1}(a x)}\right )+3 a \sinh ^{-1}(a x) \text {Li}_2\left (e^{2 \sinh ^{-1}(a x)}\right )-\frac {1}{2} (3 a) \operatorname {Subst}\left (\int \frac {\text {Li}_2(x)}{x} \, dx,x,e^{2 \sinh ^{-1}(a x)}\right )\\ &=-a \sinh ^{-1}(a x)^3-\frac {\sqrt {1+a^2 x^2} \sinh ^{-1}(a x)^3}{x}+3 a \sinh ^{-1}(a x)^2 \log \left (1-e^{2 \sinh ^{-1}(a x)}\right )+3 a \sinh ^{-1}(a x) \text {Li}_2\left (e^{2 \sinh ^{-1}(a x)}\right )-\frac {3}{2} a \text {Li}_3\left (e^{2 \sinh ^{-1}(a x)}\right )\\ \end {align*}
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Mathematica [C] time = 0.21, size = 97, normalized size = 1.10 \[ \frac {1}{8} a \left (-\frac {8 \sqrt {a^2 x^2+1} \sinh ^{-1}(a x)^3}{a x}+24 \sinh ^{-1}(a x) \text {Li}_2\left (e^{2 \sinh ^{-1}(a x)}\right )-12 \text {Li}_3\left (e^{2 \sinh ^{-1}(a x)}\right )-8 \sinh ^{-1}(a x)^3+24 \sinh ^{-1}(a x)^2 \log \left (1-e^{2 \sinh ^{-1}(a x)}\right )+i \pi ^3\right ) \]
Antiderivative was successfully verified.
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fricas [F] time = 0.45, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {\sqrt {a^{2} x^{2} + 1} \operatorname {arsinh}\left (a x\right )^{3}}{a^{2} x^{4} + x^{2}}, x\right ) \]
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 = 0.18, size = 187, normalized size = 2.12 \[ \frac {\left (a x -\sqrt {a^{2} x^{2}+1}\right ) \arcsinh \left (a x \right )^{3}}{x}-2 a \arcsinh \left (a x \right )^{3}+3 a \arcsinh \left (a x \right )^{2} \ln \left (1+a x +\sqrt {a^{2} x^{2}+1}\right )+6 a \arcsinh \left (a x \right ) \polylog \left (2, -a x -\sqrt {a^{2} x^{2}+1}\right )-6 a \polylog \left (3, -a x -\sqrt {a^{2} x^{2}+1}\right )+3 a \arcsinh \left (a x \right )^{2} \ln \left (1-a x -\sqrt {a^{2} x^{2}+1}\right )+6 a \arcsinh \left (a x \right ) \polylog \left (2, a x +\sqrt {a^{2} x^{2}+1}\right )-6 a \polylog \left (3, a x +\sqrt {a^{2} x^{2}+1}\right ) \]
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 \[ -\frac {\sqrt {a^{2} x^{2} + 1} \log \left (a x + \sqrt {a^{2} x^{2} + 1}\right )^{3}}{x} + \int \frac {3 \, {\left (a^{3} x^{2} + \sqrt {a^{2} x^{2} + 1} a^{2} x + a\right )} \log \left (a x + \sqrt {a^{2} x^{2} + 1}\right )^{2}}{\sqrt {a^{2} x^{2} + 1} a x^{2} + {\left (a^{2} x^{2} + 1\right )} x}\,{d x} \]
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 \[ \int \frac {{\mathrm {asinh}\left (a\,x\right )}^3}{x^2\,\sqrt {a^2\,x^2+1}} \,d x \]
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 \[ \int \frac {\operatorname {asinh}^{3}{\left (a x \right )}}{x^{2} \sqrt {a^{2} x^{2} + 1}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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