Optimal. Leaf size=93 \[ -\frac {3}{2} a^2 \sinh ^{-1}(a x)^2-\frac {3 a \sqrt {1+a^2 x^2} \sinh ^{-1}(a x)^2}{2 x}-\frac {\sinh ^{-1}(a x)^3}{2 x^2}+3 a^2 \sinh ^{-1}(a x) \log \left (1-e^{2 \sinh ^{-1}(a x)}\right )+\frac {3}{2} a^2 \text {PolyLog}\left (2,e^{2 \sinh ^{-1}(a x)}\right ) \]
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Rubi [A]
time = 0.11, 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 = {5776, 5800,
5775, 3797, 2221, 2317, 2438} \begin {gather*} \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} \end {gather*}
Antiderivative was successfully verified.
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Rule 2221
Rule 2317
Rule 2438
Rule 3797
Rule 5775
Rule 5776
Rule 5800
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 ) \text {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 ) \text {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 ) \text {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 ) \text {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 = 0.23, size = 80, normalized size = 0.86 \begin {gather*} -\frac {\sinh ^{-1}(a x)^3-3 a x \left (\sinh ^{-1}(a x) \left (\left (a x-\sqrt {1+a^2 x^2}\right ) \sinh ^{-1}(a x)+2 a x \log \left (1-e^{-2 \sinh ^{-1}(a x)}\right )\right )-a x \text {PolyLog}\left (2,e^{-2 \sinh ^{-1}(a x)}\right )\right )}{2 x^2} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 2.48, size = 146, normalized size = 1.57
method | result | size |
derivativedivides | \(a^{2} \left (-\frac {\arcsinh \left (a x \right )^{2} \left (3 a x \sqrt {a^{2} x^{2}+1}-3 a^{2} x^{2}+\arcsinh \left (a x \right )\right )}{2 a^{2} x^{2}}-3 \arcsinh \left (a x \right )^{2}+3 \arcsinh \left (a x \right ) \ln \left (1+a x +\sqrt {a^{2} x^{2}+1}\right )+3 \polylog \left (2, -a x -\sqrt {a^{2} x^{2}+1}\right )+3 \arcsinh \left (a x \right ) \ln \left (1-a x -\sqrt {a^{2} x^{2}+1}\right )+3 \polylog \left (2, a x +\sqrt {a^{2} x^{2}+1}\right )\right )\) | \(146\) |
default | \(a^{2} \left (-\frac {\arcsinh \left (a x \right )^{2} \left (3 a x \sqrt {a^{2} x^{2}+1}-3 a^{2} x^{2}+\arcsinh \left (a x \right )\right )}{2 a^{2} x^{2}}-3 \arcsinh \left (a x \right )^{2}+3 \arcsinh \left (a x \right ) \ln \left (1+a x +\sqrt {a^{2} x^{2}+1}\right )+3 \polylog \left (2, -a x -\sqrt {a^{2} x^{2}+1}\right )+3 \arcsinh \left (a x \right ) \ln \left (1-a x -\sqrt {a^{2} x^{2}+1}\right )+3 \polylog \left (2, a x +\sqrt {a^{2} x^{2}+1}\right )\right )\) | \(146\) |
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 \begin {gather*} \text {Failed to integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}
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 \begin {gather*} \int \frac {\operatorname {asinh}^{3}{\left (a x \right )}}{x^{3}}\, dx \end {gather*}
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 \begin {gather*} \text {Exception raised: TypeError} \end {gather*}
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 \begin {gather*} \int \frac {{\mathrm {asinh}\left (a\,x\right )}^3}{x^3} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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