Optimal. Leaf size=151 \[ -\frac {a^2 \sinh ^{-1}(a x)}{x}-\frac {a \sqrt {1+a^2 x^2} \sinh ^{-1}(a x)^2}{2 x^2}-\frac {\sinh ^{-1}(a x)^3}{3 x^3}+a^3 \sinh ^{-1}(a x)^2 \tanh ^{-1}\left (e^{\sinh ^{-1}(a x)}\right )-a^3 \tanh ^{-1}\left (\sqrt {1+a^2 x^2}\right )+a^3 \sinh ^{-1}(a x) \text {PolyLog}\left (2,-e^{\sinh ^{-1}(a x)}\right )-a^3 \sinh ^{-1}(a x) \text {PolyLog}\left (2,e^{\sinh ^{-1}(a x)}\right )-a^3 \text {PolyLog}\left (3,-e^{\sinh ^{-1}(a x)}\right )+a^3 \text {PolyLog}\left (3,e^{\sinh ^{-1}(a x)}\right ) \]
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Rubi [A]
time = 0.19, antiderivative size = 151, normalized size of antiderivative = 1.00, number of steps
used = 14, number of rules used = 10, integrand size = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 1.000, Rules used = {5776, 5809,
5816, 4267, 2611, 2320, 6724, 272, 65, 214} \begin {gather*} a^3 \sinh ^{-1}(a x) \text {Li}_2\left (-e^{\sinh ^{-1}(a x)}\right )-a^3 \sinh ^{-1}(a x) \text {Li}_2\left (e^{\sinh ^{-1}(a x)}\right )-a^3 \text {Li}_3\left (-e^{\sinh ^{-1}(a x)}\right )+a^3 \text {Li}_3\left (e^{\sinh ^{-1}(a x)}\right )+a^3 \sinh ^{-1}(a x)^2 \tanh ^{-1}\left (e^{\sinh ^{-1}(a x)}\right )-\frac {a \sqrt {a^2 x^2+1} \sinh ^{-1}(a x)^2}{2 x^2}-\frac {a^2 \sinh ^{-1}(a x)}{x}-a^3 \tanh ^{-1}\left (\sqrt {a^2 x^2+1}\right )-\frac {\sinh ^{-1}(a x)^3}{3 x^3} \end {gather*}
Antiderivative was successfully verified.
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Rule 65
Rule 214
Rule 272
Rule 2320
Rule 2611
Rule 4267
Rule 5776
Rule 5809
Rule 5816
Rule 6724
Rubi steps
\begin {align*} \int \frac {\sinh ^{-1}(a x)^3}{x^4} \, dx &=-\frac {\sinh ^{-1}(a x)^3}{3 x^3}+a \int \frac {\sinh ^{-1}(a x)^2}{x^3 \sqrt {1+a^2 x^2}} \, dx\\ &=-\frac {a \sqrt {1+a^2 x^2} \sinh ^{-1}(a x)^2}{2 x^2}-\frac {\sinh ^{-1}(a x)^3}{3 x^3}+a^2 \int \frac {\sinh ^{-1}(a x)}{x^2} \, dx-\frac {1}{2} a^3 \int \frac {\sinh ^{-1}(a x)^2}{x \sqrt {1+a^2 x^2}} \, dx\\ &=-\frac {a^2 \sinh ^{-1}(a x)}{x}-\frac {a \sqrt {1+a^2 x^2} \sinh ^{-1}(a x)^2}{2 x^2}-\frac {\sinh ^{-1}(a x)^3}{3 x^3}-\frac {1}{2} a^3 \text {Subst}\left (\int x^2 \text {csch}(x) \, dx,x,\sinh ^{-1}(a x)\right )+a^3 \int \frac {1}{x \sqrt {1+a^2 x^2}} \, dx\\ &=-\frac {a^2 \sinh ^{-1}(a x)}{x}-\frac {a \sqrt {1+a^2 x^2} \sinh ^{-1}(a x)^2}{2 x^2}-\frac {\sinh ^{-1}(a x)^3}{3 x^3}+a^3 \sinh ^{-1}(a x)^2 \tanh ^{-1}\left (e^{\sinh ^{-1}(a x)}\right )+\frac {1}{2} a^3 \text {Subst}\left (\int \frac {1}{x \sqrt {1+a^2 x}} \, dx,x,x^2\right )+a^3 \text {Subst}\left (\int x \log \left (1-e^x\right ) \, dx,x,\sinh ^{-1}(a x)\right )-a^3 \text {Subst}\left (\int x \log \left (1+e^x\right ) \, dx,x,\sinh ^{-1}(a x)\right )\\ &=-\frac {a^2 \sinh ^{-1}(a x)}{x}-\frac {a \sqrt {1+a^2 x^2} \sinh ^{-1}(a x)^2}{2 x^2}-\frac {\sinh ^{-1}(a x)^3}{3 x^3}+a^3 \sinh ^{-1}(a x)^2 \tanh ^{-1}\left (e^{\sinh ^{-1}(a x)}\right )+a^3 \sinh ^{-1}(a x) \text {Li}_2\left (-e^{\sinh ^{-1}(a x)}\right )-a^3 \sinh ^{-1}(a x) \text {Li}_2\left (e^{\sinh ^{-1}(a x)}\right )+a \text {Subst}\left (\int \frac {1}{-\frac {1}{a^2}+\frac {x^2}{a^2}} \, dx,x,\sqrt {1+a^2 x^2}\right )-a^3 \text {Subst}\left (\int \text {Li}_2\left (-e^x\right ) \, dx,x,\sinh ^{-1}(a x)\right )+a^3 \text {Subst}\left (\int \text {Li}_2\left (e^x\right ) \, dx,x,\sinh ^{-1}(a x)\right )\\ &=-\frac {a^2 \sinh ^{-1}(a x)}{x}-\frac {a \sqrt {1+a^2 x^2} \sinh ^{-1}(a x)^2}{2 x^2}-\frac {\sinh ^{-1}(a x)^3}{3 x^3}+a^3 \sinh ^{-1}(a x)^2 \tanh ^{-1}\left (e^{\sinh ^{-1}(a x)}\right )-a^3 \tanh ^{-1}\left (\sqrt {1+a^2 x^2}\right )+a^3 \sinh ^{-1}(a x) \text {Li}_2\left (-e^{\sinh ^{-1}(a x)}\right )-a^3 \sinh ^{-1}(a x) \text {Li}_2\left (e^{\sinh ^{-1}(a x)}\right )-a^3 \text {Subst}\left (\int \frac {\text {Li}_2(-x)}{x} \, dx,x,e^{\sinh ^{-1}(a x)}\right )+a^3 \text {Subst}\left (\int \frac {\text {Li}_2(x)}{x} \, dx,x,e^{\sinh ^{-1}(a x)}\right )\\ &=-\frac {a^2 \sinh ^{-1}(a x)}{x}-\frac {a \sqrt {1+a^2 x^2} \sinh ^{-1}(a x)^2}{2 x^2}-\frac {\sinh ^{-1}(a x)^3}{3 x^3}+a^3 \sinh ^{-1}(a x)^2 \tanh ^{-1}\left (e^{\sinh ^{-1}(a x)}\right )-a^3 \tanh ^{-1}\left (\sqrt {1+a^2 x^2}\right )+a^3 \sinh ^{-1}(a x) \text {Li}_2\left (-e^{\sinh ^{-1}(a x)}\right )-a^3 \sinh ^{-1}(a x) \text {Li}_2\left (e^{\sinh ^{-1}(a x)}\right )-a^3 \text {Li}_3\left (-e^{\sinh ^{-1}(a x)}\right )+a^3 \text {Li}_3\left (e^{\sinh ^{-1}(a x)}\right )\\ \end {align*}
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Mathematica [A]
time = 1.49, size = 268, normalized size = 1.77 \begin {gather*} \frac {1}{48} a^3 \left (-24 \sinh ^{-1}(a x) \coth \left (\frac {1}{2} \sinh ^{-1}(a x)\right )+4 \sinh ^{-1}(a x)^3 \coth \left (\frac {1}{2} \sinh ^{-1}(a x)\right )-6 \sinh ^{-1}(a x)^2 \text {csch}^2\left (\frac {1}{2} \sinh ^{-1}(a x)\right )-a x \sinh ^{-1}(a x)^3 \text {csch}^4\left (\frac {1}{2} \sinh ^{-1}(a x)\right )-24 \sinh ^{-1}(a x)^2 \log \left (1-e^{-\sinh ^{-1}(a x)}\right )+24 \sinh ^{-1}(a x)^2 \log \left (1+e^{-\sinh ^{-1}(a x)}\right )+48 \log \left (\tanh \left (\frac {1}{2} \sinh ^{-1}(a x)\right )\right )-48 \sinh ^{-1}(a x) \text {PolyLog}\left (2,-e^{-\sinh ^{-1}(a x)}\right )+48 \sinh ^{-1}(a x) \text {PolyLog}\left (2,e^{-\sinh ^{-1}(a x)}\right )-48 \text {PolyLog}\left (3,-e^{-\sinh ^{-1}(a x)}\right )+48 \text {PolyLog}\left (3,e^{-\sinh ^{-1}(a x)}\right )-6 \sinh ^{-1}(a x)^2 \text {sech}^2\left (\frac {1}{2} \sinh ^{-1}(a x)\right )-\frac {16 \sinh ^{-1}(a x)^3 \sinh ^4\left (\frac {1}{2} \sinh ^{-1}(a x)\right )}{a^3 x^3}+24 \sinh ^{-1}(a x) \tanh \left (\frac {1}{2} \sinh ^{-1}(a x)\right )-4 \sinh ^{-1}(a x)^3 \tanh \left (\frac {1}{2} \sinh ^{-1}(a x)\right )\right ) \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 3.21, size = 212, normalized size = 1.40
method | result | size |
derivativedivides | \(a^{3} \left (-\frac {\arcsinh \left (a x \right ) \left (3 a x \arcsinh \left (a x \right ) \sqrt {a^{2} x^{2}+1}+2 \arcsinh \left (a x \right )^{2}+6 a^{2} x^{2}\right )}{6 a^{3} x^{3}}+\frac {\arcsinh \left (a x \right )^{2} \ln \left (1+a x +\sqrt {a^{2} x^{2}+1}\right )}{2}+\arcsinh \left (a x \right ) \polylog \left (2, -a x -\sqrt {a^{2} x^{2}+1}\right )-\polylog \left (3, -a x -\sqrt {a^{2} x^{2}+1}\right )-\frac {\arcsinh \left (a x \right )^{2} \ln \left (1-a x -\sqrt {a^{2} x^{2}+1}\right )}{2}-\arcsinh \left (a x \right ) \polylog \left (2, a x +\sqrt {a^{2} x^{2}+1}\right )+\polylog \left (3, a x +\sqrt {a^{2} x^{2}+1}\right )-2 \arctanh \left (a x +\sqrt {a^{2} x^{2}+1}\right )\right )\) | \(212\) |
default | \(a^{3} \left (-\frac {\arcsinh \left (a x \right ) \left (3 a x \arcsinh \left (a x \right ) \sqrt {a^{2} x^{2}+1}+2 \arcsinh \left (a x \right )^{2}+6 a^{2} x^{2}\right )}{6 a^{3} x^{3}}+\frac {\arcsinh \left (a x \right )^{2} \ln \left (1+a x +\sqrt {a^{2} x^{2}+1}\right )}{2}+\arcsinh \left (a x \right ) \polylog \left (2, -a x -\sqrt {a^{2} x^{2}+1}\right )-\polylog \left (3, -a x -\sqrt {a^{2} x^{2}+1}\right )-\frac {\arcsinh \left (a x \right )^{2} \ln \left (1-a x -\sqrt {a^{2} x^{2}+1}\right )}{2}-\arcsinh \left (a x \right ) \polylog \left (2, a x +\sqrt {a^{2} x^{2}+1}\right )+\polylog \left (3, a x +\sqrt {a^{2} x^{2}+1}\right )-2 \arctanh \left (a x +\sqrt {a^{2} x^{2}+1}\right )\right )\) | \(212\) |
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^{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 )}^3}{x^4} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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