Optimal. Leaf size=60 \[ -\frac {1}{3} \sinh ^{-1}(a x)^3+\sinh ^{-1}(a x)^2 \log \left (1-e^{2 \sinh ^{-1}(a x)}\right )+\sinh ^{-1}(a x) \text {PolyLog}\left (2,e^{2 \sinh ^{-1}(a x)}\right )-\frac {1}{2} \text {PolyLog}\left (3,e^{2 \sinh ^{-1}(a x)}\right ) \]
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Rubi [A]
time = 0.07, antiderivative size = 60, normalized size of antiderivative = 1.00, number of steps
used = 6, number of rules used = 6, integrand size = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.600, Rules used = {5775, 3797,
2221, 2611, 2320, 6724} \begin {gather*} \sinh ^{-1}(a x) \text {Li}_2\left (e^{2 \sinh ^{-1}(a x)}\right )-\frac {1}{2} \text {Li}_3\left (e^{2 \sinh ^{-1}(a x)}\right )-\frac {1}{3} \sinh ^{-1}(a x)^3+\sinh ^{-1}(a x)^2 \log \left (1-e^{2 \sinh ^{-1}(a x)}\right ) \end {gather*}
Antiderivative was successfully verified.
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Rule 2221
Rule 2320
Rule 2611
Rule 3797
Rule 5775
Rule 6724
Rubi steps
\begin {align*} \int \frac {\sinh ^{-1}(a x)^2}{x} \, dx &=\text {Subst}\left (\int x^2 \coth (x) \, dx,x,\sinh ^{-1}(a x)\right )\\ &=-\frac {1}{3} \sinh ^{-1}(a x)^3-2 \text {Subst}\left (\int \frac {e^{2 x} x^2}{1-e^{2 x}} \, dx,x,\sinh ^{-1}(a x)\right )\\ &=-\frac {1}{3} \sinh ^{-1}(a x)^3+\sinh ^{-1}(a x)^2 \log \left (1-e^{2 \sinh ^{-1}(a x)}\right )-2 \text {Subst}\left (\int x \log \left (1-e^{2 x}\right ) \, dx,x,\sinh ^{-1}(a x)\right )\\ &=-\frac {1}{3} \sinh ^{-1}(a x)^3+\sinh ^{-1}(a x)^2 \log \left (1-e^{2 \sinh ^{-1}(a x)}\right )+\sinh ^{-1}(a x) \text {Li}_2\left (e^{2 \sinh ^{-1}(a x)}\right )-\text {Subst}\left (\int \text {Li}_2\left (e^{2 x}\right ) \, dx,x,\sinh ^{-1}(a x)\right )\\ &=-\frac {1}{3} \sinh ^{-1}(a x)^3+\sinh ^{-1}(a x)^2 \log \left (1-e^{2 \sinh ^{-1}(a x)}\right )+\sinh ^{-1}(a x) \text {Li}_2\left (e^{2 \sinh ^{-1}(a x)}\right )-\frac {1}{2} \text {Subst}\left (\int \frac {\text {Li}_2(x)}{x} \, dx,x,e^{2 \sinh ^{-1}(a x)}\right )\\ &=-\frac {1}{3} \sinh ^{-1}(a x)^3+\sinh ^{-1}(a x)^2 \log \left (1-e^{2 \sinh ^{-1}(a x)}\right )+\sinh ^{-1}(a x) \text {Li}_2\left (e^{2 \sinh ^{-1}(a x)}\right )-\frac {1}{2} \text {Li}_3\left (e^{2 \sinh ^{-1}(a x)}\right )\\ \end {align*}
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Mathematica [A]
time = 0.01, size = 60, normalized size = 1.00 \begin {gather*} -\frac {1}{3} \sinh ^{-1}(a x)^3+\sinh ^{-1}(a x)^2 \log \left (1-e^{2 \sinh ^{-1}(a x)}\right )+\sinh ^{-1}(a x) \text {PolyLog}\left (2,e^{2 \sinh ^{-1}(a x)}\right )-\frac {1}{2} \text {PolyLog}\left (3,e^{2 \sinh ^{-1}(a x)}\right ) \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 1.28, size = 151, normalized size = 2.52
method | result | size |
derivativedivides | \(-\frac {\arcsinh \left (a x \right )^{3}}{3}+\arcsinh \left (a x \right )^{2} \ln \left (1+a x +\sqrt {a^{2} x^{2}+1}\right )+2 \arcsinh \left (a x \right ) \polylog \left (2, -a x -\sqrt {a^{2} x^{2}+1}\right )-2 \polylog \left (3, -a x -\sqrt {a^{2} x^{2}+1}\right )+\arcsinh \left (a x \right )^{2} \ln \left (1-a x -\sqrt {a^{2} x^{2}+1}\right )+2 \arcsinh \left (a x \right ) \polylog \left (2, a x +\sqrt {a^{2} x^{2}+1}\right )-2 \polylog \left (3, a x +\sqrt {a^{2} x^{2}+1}\right )\) | \(151\) |
default | \(-\frac {\arcsinh \left (a x \right )^{3}}{3}+\arcsinh \left (a x \right )^{2} \ln \left (1+a x +\sqrt {a^{2} x^{2}+1}\right )+2 \arcsinh \left (a x \right ) \polylog \left (2, -a x -\sqrt {a^{2} x^{2}+1}\right )-2 \polylog \left (3, -a x -\sqrt {a^{2} x^{2}+1}\right )+\arcsinh \left (a x \right )^{2} \ln \left (1-a x -\sqrt {a^{2} x^{2}+1}\right )+2 \arcsinh \left (a x \right ) \polylog \left (2, a x +\sqrt {a^{2} x^{2}+1}\right )-2 \polylog \left (3, a x +\sqrt {a^{2} x^{2}+1}\right )\) | \(151\) |
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}^{2}{\left (a x \right )}}{x}\, dx \end {gather*}
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 \begin {gather*} \text {could not integrate} \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 = -0.02 \begin {gather*} \int \frac {{\mathrm {asinh}\left (a\,x\right )}^2}{x} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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