Optimal. Leaf size=223 \[ -\frac {2 a^2 \sinh ^{-1}(a x)^2}{x}-\frac {2 a \sqrt {1+a^2 x^2} \sinh ^{-1}(a x)^3}{3 x^2}-\frac {\sinh ^{-1}(a x)^4}{3 x^3}-8 a^3 \sinh ^{-1}(a x) \tanh ^{-1}\left (e^{\sinh ^{-1}(a x)}\right )+\frac {4}{3} a^3 \sinh ^{-1}(a x)^3 \tanh ^{-1}\left (e^{\sinh ^{-1}(a x)}\right )-4 a^3 \text {PolyLog}\left (2,-e^{\sinh ^{-1}(a x)}\right )+2 a^3 \sinh ^{-1}(a x)^2 \text {PolyLog}\left (2,-e^{\sinh ^{-1}(a x)}\right )+4 a^3 \text {PolyLog}\left (2,e^{\sinh ^{-1}(a x)}\right )-2 a^3 \sinh ^{-1}(a x)^2 \text {PolyLog}\left (2,e^{\sinh ^{-1}(a x)}\right )-4 a^3 \sinh ^{-1}(a x) \text {PolyLog}\left (3,-e^{\sinh ^{-1}(a x)}\right )+4 a^3 \sinh ^{-1}(a x) \text {PolyLog}\left (3,e^{\sinh ^{-1}(a x)}\right )+4 a^3 \text {PolyLog}\left (4,-e^{\sinh ^{-1}(a x)}\right )-4 a^3 \text {PolyLog}\left (4,e^{\sinh ^{-1}(a x)}\right ) \]
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Rubi [A]
time = 0.25, antiderivative size = 223, normalized size of antiderivative = 1.00, number of steps
used = 19, 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, 6744, 2320, 6724, 2317, 2438} \begin {gather*} 2 a^3 \sinh ^{-1}(a x)^2 \text {Li}_2\left (-e^{\sinh ^{-1}(a x)}\right )-2 a^3 \sinh ^{-1}(a x)^2 \text {Li}_2\left (e^{\sinh ^{-1}(a x)}\right )-4 a^3 \sinh ^{-1}(a x) \text {Li}_3\left (-e^{\sinh ^{-1}(a x)}\right )+4 a^3 \sinh ^{-1}(a x) \text {Li}_3\left (e^{\sinh ^{-1}(a x)}\right )-4 a^3 \text {Li}_2\left (-e^{\sinh ^{-1}(a x)}\right )+4 a^3 \text {Li}_2\left (e^{\sinh ^{-1}(a x)}\right )+4 a^3 \text {Li}_4\left (-e^{\sinh ^{-1}(a x)}\right )-4 a^3 \text {Li}_4\left (e^{\sinh ^{-1}(a x)}\right )+\frac {4}{3} a^3 \sinh ^{-1}(a x)^3 \tanh ^{-1}\left (e^{\sinh ^{-1}(a x)}\right )-8 a^3 \sinh ^{-1}(a x) \tanh ^{-1}\left (e^{\sinh ^{-1}(a x)}\right )-\frac {2 a \sqrt {a^2 x^2+1} \sinh ^{-1}(a x)^3}{3 x^2}-\frac {2 a^2 \sinh ^{-1}(a x)^2}{x}-\frac {\sinh ^{-1}(a x)^4}{3 x^3} \end {gather*}
Antiderivative was successfully verified.
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Rule 2317
Rule 2320
Rule 2438
Rule 2611
Rule 4267
Rule 5776
Rule 5809
Rule 5816
Rule 6724
Rule 6744
Rubi steps
\begin {align*} \int \frac {\sinh ^{-1}(a x)^4}{x^4} \, dx &=-\frac {\sinh ^{-1}(a x)^4}{3 x^3}+\frac {1}{3} (4 a) \int \frac {\sinh ^{-1}(a x)^3}{x^3 \sqrt {1+a^2 x^2}} \, dx\\ &=-\frac {2 a \sqrt {1+a^2 x^2} \sinh ^{-1}(a x)^3}{3 x^2}-\frac {\sinh ^{-1}(a x)^4}{3 x^3}+\left (2 a^2\right ) \int \frac {\sinh ^{-1}(a x)^2}{x^2} \, dx-\frac {1}{3} \left (2 a^3\right ) \int \frac {\sinh ^{-1}(a x)^3}{x \sqrt {1+a^2 x^2}} \, dx\\ &=-\frac {2 a^2 \sinh ^{-1}(a x)^2}{x}-\frac {2 a \sqrt {1+a^2 x^2} \sinh ^{-1}(a x)^3}{3 x^2}-\frac {\sinh ^{-1}(a x)^4}{3 x^3}-\frac {1}{3} \left (2 a^3\right ) \text {Subst}\left (\int x^3 \text {csch}(x) \, dx,x,\sinh ^{-1}(a x)\right )+\left (4 a^3\right ) \int \frac {\sinh ^{-1}(a x)}{x \sqrt {1+a^2 x^2}} \, dx\\ &=-\frac {2 a^2 \sinh ^{-1}(a x)^2}{x}-\frac {2 a \sqrt {1+a^2 x^2} \sinh ^{-1}(a x)^3}{3 x^2}-\frac {\sinh ^{-1}(a x)^4}{3 x^3}+\frac {4}{3} a^3 \sinh ^{-1}(a x)^3 \tanh ^{-1}\left (e^{\sinh ^{-1}(a x)}\right )+\left (2 a^3\right ) \text {Subst}\left (\int x^2 \log \left (1-e^x\right ) \, dx,x,\sinh ^{-1}(a x)\right )-\left (2 a^3\right ) \text {Subst}\left (\int x^2 \log \left (1+e^x\right ) \, dx,x,\sinh ^{-1}(a x)\right )+\left (4 a^3\right ) \text {Subst}\left (\int x \text {csch}(x) \, dx,x,\sinh ^{-1}(a x)\right )\\ &=-\frac {2 a^2 \sinh ^{-1}(a x)^2}{x}-\frac {2 a \sqrt {1+a^2 x^2} \sinh ^{-1}(a x)^3}{3 x^2}-\frac {\sinh ^{-1}(a x)^4}{3 x^3}-8 a^3 \sinh ^{-1}(a x) \tanh ^{-1}\left (e^{\sinh ^{-1}(a x)}\right )+\frac {4}{3} a^3 \sinh ^{-1}(a x)^3 \tanh ^{-1}\left (e^{\sinh ^{-1}(a x)}\right )+2 a^3 \sinh ^{-1}(a x)^2 \text {Li}_2\left (-e^{\sinh ^{-1}(a x)}\right )-2 a^3 \sinh ^{-1}(a x)^2 \text {Li}_2\left (e^{\sinh ^{-1}(a x)}\right )-\left (4 a^3\right ) \text {Subst}\left (\int \log \left (1-e^x\right ) \, dx,x,\sinh ^{-1}(a x)\right )+\left (4 a^3\right ) \text {Subst}\left (\int \log \left (1+e^x\right ) \, dx,x,\sinh ^{-1}(a x)\right )-\left (4 a^3\right ) \text {Subst}\left (\int x \text {Li}_2\left (-e^x\right ) \, dx,x,\sinh ^{-1}(a x)\right )+\left (4 a^3\right ) \text {Subst}\left (\int x \text {Li}_2\left (e^x\right ) \, dx,x,\sinh ^{-1}(a x)\right )\\ &=-\frac {2 a^2 \sinh ^{-1}(a x)^2}{x}-\frac {2 a \sqrt {1+a^2 x^2} \sinh ^{-1}(a x)^3}{3 x^2}-\frac {\sinh ^{-1}(a x)^4}{3 x^3}-8 a^3 \sinh ^{-1}(a x) \tanh ^{-1}\left (e^{\sinh ^{-1}(a x)}\right )+\frac {4}{3} a^3 \sinh ^{-1}(a x)^3 \tanh ^{-1}\left (e^{\sinh ^{-1}(a x)}\right )+2 a^3 \sinh ^{-1}(a x)^2 \text {Li}_2\left (-e^{\sinh ^{-1}(a x)}\right )-2 a^3 \sinh ^{-1}(a x)^2 \text {Li}_2\left (e^{\sinh ^{-1}(a x)}\right )-4 a^3 \sinh ^{-1}(a x) \text {Li}_3\left (-e^{\sinh ^{-1}(a x)}\right )+4 a^3 \sinh ^{-1}(a x) \text {Li}_3\left (e^{\sinh ^{-1}(a x)}\right )-\left (4 a^3\right ) \text {Subst}\left (\int \frac {\log (1-x)}{x} \, dx,x,e^{\sinh ^{-1}(a x)}\right )+\left (4 a^3\right ) \text {Subst}\left (\int \frac {\log (1+x)}{x} \, dx,x,e^{\sinh ^{-1}(a x)}\right )+\left (4 a^3\right ) \text {Subst}\left (\int \text {Li}_3\left (-e^x\right ) \, dx,x,\sinh ^{-1}(a x)\right )-\left (4 a^3\right ) \text {Subst}\left (\int \text {Li}_3\left (e^x\right ) \, dx,x,\sinh ^{-1}(a x)\right )\\ &=-\frac {2 a^2 \sinh ^{-1}(a x)^2}{x}-\frac {2 a \sqrt {1+a^2 x^2} \sinh ^{-1}(a x)^3}{3 x^2}-\frac {\sinh ^{-1}(a x)^4}{3 x^3}-8 a^3 \sinh ^{-1}(a x) \tanh ^{-1}\left (e^{\sinh ^{-1}(a x)}\right )+\frac {4}{3} a^3 \sinh ^{-1}(a x)^3 \tanh ^{-1}\left (e^{\sinh ^{-1}(a x)}\right )-4 a^3 \text {Li}_2\left (-e^{\sinh ^{-1}(a x)}\right )+2 a^3 \sinh ^{-1}(a x)^2 \text {Li}_2\left (-e^{\sinh ^{-1}(a x)}\right )+4 a^3 \text {Li}_2\left (e^{\sinh ^{-1}(a x)}\right )-2 a^3 \sinh ^{-1}(a x)^2 \text {Li}_2\left (e^{\sinh ^{-1}(a x)}\right )-4 a^3 \sinh ^{-1}(a x) \text {Li}_3\left (-e^{\sinh ^{-1}(a x)}\right )+4 a^3 \sinh ^{-1}(a x) \text {Li}_3\left (e^{\sinh ^{-1}(a x)}\right )+\left (4 a^3\right ) \text {Subst}\left (\int \frac {\text {Li}_3(-x)}{x} \, dx,x,e^{\sinh ^{-1}(a x)}\right )-\left (4 a^3\right ) \text {Subst}\left (\int \frac {\text {Li}_3(x)}{x} \, dx,x,e^{\sinh ^{-1}(a x)}\right )\\ &=-\frac {2 a^2 \sinh ^{-1}(a x)^2}{x}-\frac {2 a \sqrt {1+a^2 x^2} \sinh ^{-1}(a x)^3}{3 x^2}-\frac {\sinh ^{-1}(a x)^4}{3 x^3}-8 a^3 \sinh ^{-1}(a x) \tanh ^{-1}\left (e^{\sinh ^{-1}(a x)}\right )+\frac {4}{3} a^3 \sinh ^{-1}(a x)^3 \tanh ^{-1}\left (e^{\sinh ^{-1}(a x)}\right )-4 a^3 \text {Li}_2\left (-e^{\sinh ^{-1}(a x)}\right )+2 a^3 \sinh ^{-1}(a x)^2 \text {Li}_2\left (-e^{\sinh ^{-1}(a x)}\right )+4 a^3 \text {Li}_2\left (e^{\sinh ^{-1}(a x)}\right )-2 a^3 \sinh ^{-1}(a x)^2 \text {Li}_2\left (e^{\sinh ^{-1}(a x)}\right )-4 a^3 \sinh ^{-1}(a x) \text {Li}_3\left (-e^{\sinh ^{-1}(a x)}\right )+4 a^3 \sinh ^{-1}(a x) \text {Li}_3\left (e^{\sinh ^{-1}(a x)}\right )+4 a^3 \text {Li}_4\left (-e^{\sinh ^{-1}(a x)}\right )-4 a^3 \text {Li}_4\left (e^{\sinh ^{-1}(a x)}\right )\\ \end {align*}
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Mathematica [A]
time = 1.63, size = 355, normalized size = 1.59 \begin {gather*} \frac {1}{24} a^3 \left (-2 \pi ^4+4 \sinh ^{-1}(a x)^4-24 \sinh ^{-1}(a x)^2 \coth \left (\frac {1}{2} \sinh ^{-1}(a x)\right )+2 \sinh ^{-1}(a x)^4 \coth \left (\frac {1}{2} \sinh ^{-1}(a x)\right )-4 \sinh ^{-1}(a x)^3 \text {csch}^2\left (\frac {1}{2} \sinh ^{-1}(a x)\right )-\frac {1}{2} a x \sinh ^{-1}(a x)^4 \text {csch}^4\left (\frac {1}{2} \sinh ^{-1}(a x)\right )+96 \sinh ^{-1}(a x) \log \left (1-e^{-\sinh ^{-1}(a x)}\right )-96 \sinh ^{-1}(a x) \log \left (1+e^{-\sinh ^{-1}(a x)}\right )+16 \sinh ^{-1}(a x)^3 \log \left (1+e^{-\sinh ^{-1}(a x)}\right )-16 \sinh ^{-1}(a x)^3 \log \left (1-e^{\sinh ^{-1}(a x)}\right )-48 \left (-2+\sinh ^{-1}(a x)^2\right ) \text {PolyLog}\left (2,-e^{-\sinh ^{-1}(a x)}\right )-96 \text {PolyLog}\left (2,e^{-\sinh ^{-1}(a x)}\right )-48 \sinh ^{-1}(a x)^2 \text {PolyLog}\left (2,e^{\sinh ^{-1}(a x)}\right )-96 \sinh ^{-1}(a x) \text {PolyLog}\left (3,-e^{-\sinh ^{-1}(a x)}\right )+96 \sinh ^{-1}(a x) \text {PolyLog}\left (3,e^{\sinh ^{-1}(a x)}\right )-96 \text {PolyLog}\left (4,-e^{-\sinh ^{-1}(a x)}\right )-96 \text {PolyLog}\left (4,e^{\sinh ^{-1}(a x)}\right )-4 \sinh ^{-1}(a x)^3 \text {sech}^2\left (\frac {1}{2} \sinh ^{-1}(a x)\right )-\frac {8 \sinh ^{-1}(a x)^4 \sinh ^4\left (\frac {1}{2} \sinh ^{-1}(a x)\right )}{a^3 x^3}+24 \sinh ^{-1}(a x)^2 \tanh \left (\frac {1}{2} \sinh ^{-1}(a x)\right )-2 \sinh ^{-1}(a x)^4 \tanh \left (\frac {1}{2} \sinh ^{-1}(a x)\right )\right ) \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 2.98, size = 340, normalized size = 1.52
method | result | size |
derivativedivides | \(a^{3} \left (-\frac {\arcsinh \left (a x \right )^{2} \left (2 a x \arcsinh \left (a x \right ) \sqrt {a^{2} x^{2}+1}+\arcsinh \left (a x \right )^{2}+6 a^{2} x^{2}\right )}{3 a^{3} x^{3}}+\frac {2 \arcsinh \left (a x \right )^{3} \ln \left (1+a x +\sqrt {a^{2} x^{2}+1}\right )}{3}+2 \arcsinh \left (a x \right )^{2} \polylog \left (2, -a x -\sqrt {a^{2} x^{2}+1}\right )-4 \arcsinh \left (a x \right ) \polylog \left (3, -a x -\sqrt {a^{2} x^{2}+1}\right )+4 \polylog \left (4, -a x -\sqrt {a^{2} x^{2}+1}\right )-\frac {2 \arcsinh \left (a x \right )^{3} \ln \left (1-a x -\sqrt {a^{2} x^{2}+1}\right )}{3}-2 \arcsinh \left (a x \right )^{2} \polylog \left (2, a x +\sqrt {a^{2} x^{2}+1}\right )+4 \arcsinh \left (a x \right ) \polylog \left (3, a x +\sqrt {a^{2} x^{2}+1}\right )-4 \polylog \left (4, a x +\sqrt {a^{2} x^{2}+1}\right )-4 \arcsinh \left (a x \right ) \ln \left (1+a x +\sqrt {a^{2} x^{2}+1}\right )-4 \polylog \left (2, -a x -\sqrt {a^{2} x^{2}+1}\right )+4 \arcsinh \left (a x \right ) \ln \left (1-a x -\sqrt {a^{2} x^{2}+1}\right )+4 \polylog \left (2, a x +\sqrt {a^{2} x^{2}+1}\right )\right )\) | \(340\) |
default | \(a^{3} \left (-\frac {\arcsinh \left (a x \right )^{2} \left (2 a x \arcsinh \left (a x \right ) \sqrt {a^{2} x^{2}+1}+\arcsinh \left (a x \right )^{2}+6 a^{2} x^{2}\right )}{3 a^{3} x^{3}}+\frac {2 \arcsinh \left (a x \right )^{3} \ln \left (1+a x +\sqrt {a^{2} x^{2}+1}\right )}{3}+2 \arcsinh \left (a x \right )^{2} \polylog \left (2, -a x -\sqrt {a^{2} x^{2}+1}\right )-4 \arcsinh \left (a x \right ) \polylog \left (3, -a x -\sqrt {a^{2} x^{2}+1}\right )+4 \polylog \left (4, -a x -\sqrt {a^{2} x^{2}+1}\right )-\frac {2 \arcsinh \left (a x \right )^{3} \ln \left (1-a x -\sqrt {a^{2} x^{2}+1}\right )}{3}-2 \arcsinh \left (a x \right )^{2} \polylog \left (2, a x +\sqrt {a^{2} x^{2}+1}\right )+4 \arcsinh \left (a x \right ) \polylog \left (3, a x +\sqrt {a^{2} x^{2}+1}\right )-4 \polylog \left (4, a x +\sqrt {a^{2} x^{2}+1}\right )-4 \arcsinh \left (a x \right ) \ln \left (1+a x +\sqrt {a^{2} x^{2}+1}\right )-4 \polylog \left (2, -a x -\sqrt {a^{2} x^{2}+1}\right )+4 \arcsinh \left (a x \right ) \ln \left (1-a x -\sqrt {a^{2} x^{2}+1}\right )+4 \polylog \left (2, a x +\sqrt {a^{2} x^{2}+1}\right )\right )\) | \(340\) |
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}^{4}{\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 = -0.00 \begin {gather*} \int \frac {{\mathrm {asinh}\left (a\,x\right )}^4}{x^4} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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