Optimal. Leaf size=84 \[ -\frac {\sinh ^{-1}(a x)^3}{x}-6 a \sinh ^{-1}(a x)^2 \tanh ^{-1}\left (e^{\sinh ^{-1}(a x)}\right )-6 a \sinh ^{-1}(a x) \text {PolyLog}\left (2,-e^{\sinh ^{-1}(a x)}\right )+6 a \sinh ^{-1}(a x) \text {PolyLog}\left (2,e^{\sinh ^{-1}(a x)}\right )+6 a \text {PolyLog}\left (3,-e^{\sinh ^{-1}(a x)}\right )-6 a \text {PolyLog}\left (3,e^{\sinh ^{-1}(a x)}\right ) \]
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Rubi [A]
time = 0.11, antiderivative size = 84, normalized size of antiderivative = 1.00, number of steps
used = 9, number of rules used = 6, integrand size = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.600, Rules used = {5776, 5816,
4267, 2611, 2320, 6724} \begin {gather*} -6 a \sinh ^{-1}(a x) \text {Li}_2\left (-e^{\sinh ^{-1}(a x)}\right )+6 a \sinh ^{-1}(a x) \text {Li}_2\left (e^{\sinh ^{-1}(a x)}\right )+6 a \text {Li}_3\left (-e^{\sinh ^{-1}(a x)}\right )-6 a \text {Li}_3\left (e^{\sinh ^{-1}(a x)}\right )-\frac {\sinh ^{-1}(a x)^3}{x}-6 a \sinh ^{-1}(a x)^2 \tanh ^{-1}\left (e^{\sinh ^{-1}(a x)}\right ) \end {gather*}
Antiderivative was successfully verified.
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Rule 2320
Rule 2611
Rule 4267
Rule 5776
Rule 5816
Rule 6724
Rubi steps
\begin {align*} \int \frac {\sinh ^{-1}(a x)^3}{x^2} \, dx &=-\frac {\sinh ^{-1}(a x)^3}{x}+(3 a) \int \frac {\sinh ^{-1}(a x)^2}{x \sqrt {1+a^2 x^2}} \, dx\\ &=-\frac {\sinh ^{-1}(a x)^3}{x}+(3 a) \text {Subst}\left (\int x^2 \text {csch}(x) \, dx,x,\sinh ^{-1}(a x)\right )\\ &=-\frac {\sinh ^{-1}(a x)^3}{x}-6 a \sinh ^{-1}(a x)^2 \tanh ^{-1}\left (e^{\sinh ^{-1}(a x)}\right )-(6 a) \text {Subst}\left (\int x \log \left (1-e^x\right ) \, dx,x,\sinh ^{-1}(a x)\right )+(6 a) \text {Subst}\left (\int x \log \left (1+e^x\right ) \, dx,x,\sinh ^{-1}(a x)\right )\\ &=-\frac {\sinh ^{-1}(a x)^3}{x}-6 a \sinh ^{-1}(a x)^2 \tanh ^{-1}\left (e^{\sinh ^{-1}(a x)}\right )-6 a \sinh ^{-1}(a x) \text {Li}_2\left (-e^{\sinh ^{-1}(a x)}\right )+6 a \sinh ^{-1}(a x) \text {Li}_2\left (e^{\sinh ^{-1}(a x)}\right )+(6 a) \text {Subst}\left (\int \text {Li}_2\left (-e^x\right ) \, dx,x,\sinh ^{-1}(a x)\right )-(6 a) \text {Subst}\left (\int \text {Li}_2\left (e^x\right ) \, dx,x,\sinh ^{-1}(a x)\right )\\ &=-\frac {\sinh ^{-1}(a x)^3}{x}-6 a \sinh ^{-1}(a x)^2 \tanh ^{-1}\left (e^{\sinh ^{-1}(a x)}\right )-6 a \sinh ^{-1}(a x) \text {Li}_2\left (-e^{\sinh ^{-1}(a x)}\right )+6 a \sinh ^{-1}(a x) \text {Li}_2\left (e^{\sinh ^{-1}(a x)}\right )+(6 a) \text {Subst}\left (\int \frac {\text {Li}_2(-x)}{x} \, dx,x,e^{\sinh ^{-1}(a x)}\right )-(6 a) \text {Subst}\left (\int \frac {\text {Li}_2(x)}{x} \, dx,x,e^{\sinh ^{-1}(a x)}\right )\\ &=-\frac {\sinh ^{-1}(a x)^3}{x}-6 a \sinh ^{-1}(a x)^2 \tanh ^{-1}\left (e^{\sinh ^{-1}(a x)}\right )-6 a \sinh ^{-1}(a x) \text {Li}_2\left (-e^{\sinh ^{-1}(a x)}\right )+6 a \sinh ^{-1}(a x) \text {Li}_2\left (e^{\sinh ^{-1}(a x)}\right )+6 a \text {Li}_3\left (-e^{\sinh ^{-1}(a x)}\right )-6 a \text {Li}_3\left (e^{\sinh ^{-1}(a x)}\right )\\ \end {align*}
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Mathematica [A]
time = 0.08, size = 117, normalized size = 1.39 \begin {gather*} a \left (-\frac {\sinh ^{-1}(a x)^3}{a x}+3 \sinh ^{-1}(a x)^2 \log \left (1-e^{-\sinh ^{-1}(a x)}\right )-3 \sinh ^{-1}(a x)^2 \log \left (1+e^{-\sinh ^{-1}(a x)}\right )+6 \sinh ^{-1}(a x) \text {PolyLog}\left (2,-e^{-\sinh ^{-1}(a x)}\right )-6 \sinh ^{-1}(a x) \text {PolyLog}\left (2,e^{-\sinh ^{-1}(a x)}\right )+6 \text {PolyLog}\left (3,-e^{-\sinh ^{-1}(a x)}\right )-6 \text {PolyLog}\left (3,e^{-\sinh ^{-1}(a x)}\right )\right ) \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 1.79, size = 161, normalized size = 1.92
method | result | size |
derivativedivides | \(a \left (-\frac {\arcsinh \left (a x \right )^{3}}{a x}-3 \arcsinh \left (a x \right )^{2} \ln \left (1+a x +\sqrt {a^{2} x^{2}+1}\right )-6 \arcsinh \left (a x \right ) \polylog \left (2, -a x -\sqrt {a^{2} x^{2}+1}\right )+6 \polylog \left (3, -a x -\sqrt {a^{2} x^{2}+1}\right )+3 \arcsinh \left (a x \right )^{2} \ln \left (1-a x -\sqrt {a^{2} x^{2}+1}\right )+6 \arcsinh \left (a x \right ) \polylog \left (2, a x +\sqrt {a^{2} x^{2}+1}\right )-6 \polylog \left (3, a x +\sqrt {a^{2} x^{2}+1}\right )\right )\) | \(161\) |
default | \(a \left (-\frac {\arcsinh \left (a x \right )^{3}}{a x}-3 \arcsinh \left (a x \right )^{2} \ln \left (1+a x +\sqrt {a^{2} x^{2}+1}\right )-6 \arcsinh \left (a x \right ) \polylog \left (2, -a x -\sqrt {a^{2} x^{2}+1}\right )+6 \polylog \left (3, -a x -\sqrt {a^{2} x^{2}+1}\right )+3 \arcsinh \left (a x \right )^{2} \ln \left (1-a x -\sqrt {a^{2} x^{2}+1}\right )+6 \arcsinh \left (a x \right ) \polylog \left (2, a x +\sqrt {a^{2} x^{2}+1}\right )-6 \polylog \left (3, a x +\sqrt {a^{2} x^{2}+1}\right )\right )\) | \(161\) |
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^{2}}\, 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 )}^3}{x^2} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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