Optimal. Leaf size=80 \[ -\frac {a}{2 x}-\frac {\sqrt {1+a^2 x^2} \sinh ^{-1}(a x)}{2 x^2}+a^2 \sinh ^{-1}(a x) \tanh ^{-1}\left (e^{\sinh ^{-1}(a x)}\right )+\frac {1}{2} a^2 \text {PolyLog}\left (2,-e^{\sinh ^{-1}(a x)}\right )-\frac {1}{2} a^2 \text {PolyLog}\left (2,e^{\sinh ^{-1}(a x)}\right ) \]
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Rubi [A]
time = 0.10, antiderivative size = 80, normalized size of antiderivative = 1.00, number of steps
used = 8, number of rules used = 6, integrand size = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.286, Rules used = {5809, 5816,
4267, 2317, 2438, 30} \begin {gather*} \frac {1}{2} a^2 \text {Li}_2\left (-e^{\sinh ^{-1}(a x)}\right )-\frac {1}{2} a^2 \text {Li}_2\left (e^{\sinh ^{-1}(a x)}\right )-\frac {\sqrt {a^2 x^2+1} \sinh ^{-1}(a x)}{2 x^2}+a^2 \sinh ^{-1}(a x) \tanh ^{-1}\left (e^{\sinh ^{-1}(a x)}\right )-\frac {a}{2 x} \end {gather*}
Antiderivative was successfully verified.
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Rule 30
Rule 2317
Rule 2438
Rule 4267
Rule 5809
Rule 5816
Rubi steps
\begin {align*} \int \frac {\sinh ^{-1}(a x)}{x^3 \sqrt {1+a^2 x^2}} \, dx &=-\frac {\sqrt {1+a^2 x^2} \sinh ^{-1}(a x)}{2 x^2}+\frac {1}{2} a \int \frac {1}{x^2} \, dx-\frac {1}{2} a^2 \int \frac {\sinh ^{-1}(a x)}{x \sqrt {1+a^2 x^2}} \, dx\\ &=-\frac {a}{2 x}-\frac {\sqrt {1+a^2 x^2} \sinh ^{-1}(a x)}{2 x^2}-\frac {1}{2} a^2 \text {Subst}\left (\int x \text {csch}(x) \, dx,x,\sinh ^{-1}(a x)\right )\\ &=-\frac {a}{2 x}-\frac {\sqrt {1+a^2 x^2} \sinh ^{-1}(a x)}{2 x^2}+a^2 \sinh ^{-1}(a x) \tanh ^{-1}\left (e^{\sinh ^{-1}(a x)}\right )+\frac {1}{2} a^2 \text {Subst}\left (\int \log \left (1-e^x\right ) \, dx,x,\sinh ^{-1}(a x)\right )-\frac {1}{2} a^2 \text {Subst}\left (\int \log \left (1+e^x\right ) \, dx,x,\sinh ^{-1}(a x)\right )\\ &=-\frac {a}{2 x}-\frac {\sqrt {1+a^2 x^2} \sinh ^{-1}(a x)}{2 x^2}+a^2 \sinh ^{-1}(a x) \tanh ^{-1}\left (e^{\sinh ^{-1}(a x)}\right )+\frac {1}{2} a^2 \text {Subst}\left (\int \frac {\log (1-x)}{x} \, dx,x,e^{\sinh ^{-1}(a x)}\right )-\frac {1}{2} a^2 \text {Subst}\left (\int \frac {\log (1+x)}{x} \, dx,x,e^{\sinh ^{-1}(a x)}\right )\\ &=-\frac {a}{2 x}-\frac {\sqrt {1+a^2 x^2} \sinh ^{-1}(a x)}{2 x^2}+a^2 \sinh ^{-1}(a x) \tanh ^{-1}\left (e^{\sinh ^{-1}(a x)}\right )+\frac {1}{2} a^2 \text {Li}_2\left (-e^{\sinh ^{-1}(a x)}\right )-\frac {1}{2} a^2 \text {Li}_2\left (e^{\sinh ^{-1}(a x)}\right )\\ \end {align*}
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Mathematica [A]
time = 0.10, size = 126, normalized size = 1.58 \begin {gather*} \frac {1}{8} a^2 \left (-2 \coth \left (\frac {1}{2} \sinh ^{-1}(a x)\right )-\sinh ^{-1}(a x) \text {csch}^2\left (\frac {1}{2} \sinh ^{-1}(a x)\right )-4 \sinh ^{-1}(a x) \log \left (1-e^{-\sinh ^{-1}(a x)}\right )+4 \sinh ^{-1}(a x) \log \left (1+e^{-\sinh ^{-1}(a x)}\right )-4 \text {PolyLog}\left (2,-e^{-\sinh ^{-1}(a x)}\right )+4 \text {PolyLog}\left (2,e^{-\sinh ^{-1}(a x)}\right )-\sinh ^{-1}(a x) \text {sech}^2\left (\frac {1}{2} \sinh ^{-1}(a x)\right )+2 \tanh \left (\frac {1}{2} \sinh ^{-1}(a x)\right )\right ) \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 5.02, size = 150, normalized size = 1.88
method | result | size |
default | \(-\frac {\arcsinh \left (a x \right ) a^{2} x^{2}+\sqrt {a^{2} x^{2}+1}\, a x +\arcsinh \left (a x \right )}{2 \sqrt {a^{2} x^{2}+1}\, x^{2}}-\frac {a^{2} \arcsinh \left (a x \right ) \ln \left (1-a x -\sqrt {a^{2} x^{2}+1}\right )}{2}-\frac {a^{2} \polylog \left (2, a x +\sqrt {a^{2} x^{2}+1}\right )}{2}+\frac {a^{2} \arcsinh \left (a x \right ) \ln \left (1+a x +\sqrt {a^{2} x^{2}+1}\right )}{2}+\frac {a^{2} \polylog \left (2, -a x -\sqrt {a^{2} x^{2}+1}\right )}{2}\) | \(150\) |
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}{\left (a x \right )}}{x^{3} \sqrt {a^{2} x^{2} + 1}}\, 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 = -0.01 \begin {gather*} \int \frac {\mathrm {asinh}\left (a\,x\right )}{x^3\,\sqrt {a^2\,x^2+1}} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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