Optimal. Leaf size=99 \[ -\frac {a^2}{3 x}-\frac {a \sqrt {1+a^2 x^2} \sinh ^{-1}(a x)}{3 x^2}-\frac {\sinh ^{-1}(a x)^2}{3 x^3}+\frac {2}{3} a^3 \sinh ^{-1}(a x) \tanh ^{-1}\left (e^{\sinh ^{-1}(a x)}\right )+\frac {1}{3} a^3 \text {PolyLog}\left (2,-e^{\sinh ^{-1}(a x)}\right )-\frac {1}{3} a^3 \text {PolyLog}\left (2,e^{\sinh ^{-1}(a x)}\right ) \]
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Rubi [A]
time = 0.11, antiderivative size = 99, normalized size of antiderivative = 1.00, number of steps
used = 9, number of rules used = 7, integrand size = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.700, Rules used = {5776, 5809,
5816, 4267, 2317, 2438, 30} \begin {gather*} \frac {1}{3} a^3 \text {Li}_2\left (-e^{\sinh ^{-1}(a x)}\right )-\frac {1}{3} a^3 \text {Li}_2\left (e^{\sinh ^{-1}(a x)}\right )+\frac {2}{3} a^3 \sinh ^{-1}(a x) \tanh ^{-1}\left (e^{\sinh ^{-1}(a x)}\right )-\frac {a \sqrt {a^2 x^2+1} \sinh ^{-1}(a x)}{3 x^2}-\frac {a^2}{3 x}-\frac {\sinh ^{-1}(a x)^2}{3 x^3} \end {gather*}
Antiderivative was successfully verified.
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Rule 30
Rule 2317
Rule 2438
Rule 4267
Rule 5776
Rule 5809
Rule 5816
Rubi steps
\begin {align*} \int \frac {\sinh ^{-1}(a x)^2}{x^4} \, dx &=-\frac {\sinh ^{-1}(a x)^2}{3 x^3}+\frac {1}{3} (2 a) \int \frac {\sinh ^{-1}(a x)}{x^3 \sqrt {1+a^2 x^2}} \, dx\\ &=-\frac {a \sqrt {1+a^2 x^2} \sinh ^{-1}(a x)}{3 x^2}-\frac {\sinh ^{-1}(a x)^2}{3 x^3}+\frac {1}{3} a^2 \int \frac {1}{x^2} \, dx-\frac {1}{3} a^3 \int \frac {\sinh ^{-1}(a x)}{x \sqrt {1+a^2 x^2}} \, dx\\ &=-\frac {a^2}{3 x}-\frac {a \sqrt {1+a^2 x^2} \sinh ^{-1}(a x)}{3 x^2}-\frac {\sinh ^{-1}(a x)^2}{3 x^3}-\frac {1}{3} a^3 \text {Subst}\left (\int x \text {csch}(x) \, dx,x,\sinh ^{-1}(a x)\right )\\ &=-\frac {a^2}{3 x}-\frac {a \sqrt {1+a^2 x^2} \sinh ^{-1}(a x)}{3 x^2}-\frac {\sinh ^{-1}(a x)^2}{3 x^3}+\frac {2}{3} a^3 \sinh ^{-1}(a x) \tanh ^{-1}\left (e^{\sinh ^{-1}(a x)}\right )+\frac {1}{3} a^3 \text {Subst}\left (\int \log \left (1-e^x\right ) \, dx,x,\sinh ^{-1}(a x)\right )-\frac {1}{3} a^3 \text {Subst}\left (\int \log \left (1+e^x\right ) \, dx,x,\sinh ^{-1}(a x)\right )\\ &=-\frac {a^2}{3 x}-\frac {a \sqrt {1+a^2 x^2} \sinh ^{-1}(a x)}{3 x^2}-\frac {\sinh ^{-1}(a x)^2}{3 x^3}+\frac {2}{3} a^3 \sinh ^{-1}(a x) \tanh ^{-1}\left (e^{\sinh ^{-1}(a x)}\right )+\frac {1}{3} a^3 \text {Subst}\left (\int \frac {\log (1-x)}{x} \, dx,x,e^{\sinh ^{-1}(a x)}\right )-\frac {1}{3} a^3 \text {Subst}\left (\int \frac {\log (1+x)}{x} \, dx,x,e^{\sinh ^{-1}(a x)}\right )\\ &=-\frac {a^2}{3 x}-\frac {a \sqrt {1+a^2 x^2} \sinh ^{-1}(a x)}{3 x^2}-\frac {\sinh ^{-1}(a x)^2}{3 x^3}+\frac {2}{3} a^3 \sinh ^{-1}(a x) \tanh ^{-1}\left (e^{\sinh ^{-1}(a x)}\right )+\frac {1}{3} a^3 \text {Li}_2\left (-e^{\sinh ^{-1}(a x)}\right )-\frac {1}{3} a^3 \text {Li}_2\left (e^{\sinh ^{-1}(a x)}\right )\\ \end {align*}
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Mathematica [A]
time = 0.35, size = 125, normalized size = 1.26 \begin {gather*} -\frac {a^2 x^2+a x \sqrt {1+a^2 x^2} \sinh ^{-1}(a x)+\sinh ^{-1}(a x)^2+a^3 x^3 \sinh ^{-1}(a x) \log \left (1-e^{-\sinh ^{-1}(a x)}\right )-a^3 x^3 \sinh ^{-1}(a x) \log \left (1+e^{-\sinh ^{-1}(a x)}\right )+a^3 x^3 \text {PolyLog}\left (2,-e^{-\sinh ^{-1}(a x)}\right )-a^3 x^3 \text {PolyLog}\left (2,e^{-\sinh ^{-1}(a x)}\right )}{3 x^3} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 3.23, size = 136, normalized size = 1.37
method | result | size |
derivativedivides | \(a^{3} \left (-\frac {a x \arcsinh \left (a x \right ) \sqrt {a^{2} x^{2}+1}+\arcsinh \left (a x \right )^{2}+a^{2} x^{2}}{3 a^{3} x^{3}}+\frac {\arcsinh \left (a x \right ) \ln \left (1+a x +\sqrt {a^{2} x^{2}+1}\right )}{3}+\frac {\polylog \left (2, -a x -\sqrt {a^{2} x^{2}+1}\right )}{3}-\frac {\arcsinh \left (a x \right ) \ln \left (1-a x -\sqrt {a^{2} x^{2}+1}\right )}{3}-\frac {\polylog \left (2, a x +\sqrt {a^{2} x^{2}+1}\right )}{3}\right )\) | \(136\) |
default | \(a^{3} \left (-\frac {a x \arcsinh \left (a x \right ) \sqrt {a^{2} x^{2}+1}+\arcsinh \left (a x \right )^{2}+a^{2} x^{2}}{3 a^{3} x^{3}}+\frac {\arcsinh \left (a x \right ) \ln \left (1+a x +\sqrt {a^{2} x^{2}+1}\right )}{3}+\frac {\polylog \left (2, -a x -\sqrt {a^{2} x^{2}+1}\right )}{3}-\frac {\arcsinh \left (a x \right ) \ln \left (1-a x -\sqrt {a^{2} x^{2}+1}\right )}{3}-\frac {\polylog \left (2, a x +\sqrt {a^{2} x^{2}+1}\right )}{3}\right )\) | \(136\) |
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^{4}}\, 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 )}^2}{x^4} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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