Optimal. Leaf size=223 \[ 2 a^3 \sinh ^{-1}(a x)^2 \text {Li}_2\left (-e^{\sinh ^{-1}(a x)}\right )-2 a^3 \sinh ^{-1}(a x)^2 \text {Li}_2\left (e^{\sinh ^{-1}(a x)}\right )-4 a^3 \sinh ^{-1}(a x) \text {Li}_3\left (-e^{\sinh ^{-1}(a x)}\right )+4 a^3 \sinh ^{-1}(a x) \text {Li}_3\left (e^{\sinh ^{-1}(a x)}\right )-4 a^3 \text {Li}_2\left (-e^{\sinh ^{-1}(a x)}\right )+4 a^3 \text {Li}_2\left (e^{\sinh ^{-1}(a x)}\right )+4 a^3 \text {Li}_4\left (-e^{\sinh ^{-1}(a x)}\right )-4 a^3 \text {Li}_4\left (e^{\sinh ^{-1}(a x)}\right )+\frac {4}{3} a^3 \sinh ^{-1}(a x)^3 \tanh ^{-1}\left (e^{\sinh ^{-1}(a x)}\right )-8 a^3 \sinh ^{-1}(a x) \tanh ^{-1}\left (e^{\sinh ^{-1}(a x)}\right )-\frac {2 a \sqrt {a^2 x^2+1} \sinh ^{-1}(a x)^3}{3 x^2}-\frac {2 a^2 \sinh ^{-1}(a x)^2}{x}-\frac {\sinh ^{-1}(a x)^4}{3 x^3} \]
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Rubi [A] time = 0.39, 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 = {5661, 5747, 5760, 4182, 2531, 6609, 2282, 6589, 2279, 2391} \[ 2 a^3 \sinh ^{-1}(a x)^2 \text {PolyLog}\left (2,-e^{\sinh ^{-1}(a x)}\right )-2 a^3 \sinh ^{-1}(a x)^2 \text {PolyLog}\left (2,e^{\sinh ^{-1}(a x)}\right )-4 a^3 \sinh ^{-1}(a x) \text {PolyLog}\left (3,-e^{\sinh ^{-1}(a x)}\right )+4 a^3 \sinh ^{-1}(a x) \text {PolyLog}\left (3,e^{\sinh ^{-1}(a x)}\right )-4 a^3 \text {PolyLog}\left (2,-e^{\sinh ^{-1}(a x)}\right )+4 a^3 \text {PolyLog}\left (2,e^{\sinh ^{-1}(a x)}\right )+4 a^3 \text {PolyLog}\left (4,-e^{\sinh ^{-1}(a x)}\right )-4 a^3 \text {PolyLog}\left (4,e^{\sinh ^{-1}(a x)}\right )-\frac {2 a \sqrt {a^2 x^2+1} \sinh ^{-1}(a x)^3}{3 x^2}-\frac {2 a^2 \sinh ^{-1}(a x)^2}{x}+\frac {4}{3} a^3 \sinh ^{-1}(a x)^3 \tanh ^{-1}\left (e^{\sinh ^{-1}(a x)}\right )-8 a^3 \sinh ^{-1}(a x) \tanh ^{-1}\left (e^{\sinh ^{-1}(a x)}\right )-\frac {\sinh ^{-1}(a x)^4}{3 x^3} \]
Antiderivative was successfully verified.
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Rule 2279
Rule 2282
Rule 2391
Rule 2531
Rule 4182
Rule 5661
Rule 5747
Rule 5760
Rule 6589
Rule 6609
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 ) \operatorname {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 ) \operatorname {Subst}\left (\int x^2 \log \left (1-e^x\right ) \, dx,x,\sinh ^{-1}(a x)\right )-\left (2 a^3\right ) \operatorname {Subst}\left (\int x^2 \log \left (1+e^x\right ) \, dx,x,\sinh ^{-1}(a x)\right )+\left (4 a^3\right ) \operatorname {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 ) \operatorname {Subst}\left (\int \log \left (1-e^x\right ) \, dx,x,\sinh ^{-1}(a x)\right )+\left (4 a^3\right ) \operatorname {Subst}\left (\int \log \left (1+e^x\right ) \, dx,x,\sinh ^{-1}(a x)\right )-\left (4 a^3\right ) \operatorname {Subst}\left (\int x \text {Li}_2\left (-e^x\right ) \, dx,x,\sinh ^{-1}(a x)\right )+\left (4 a^3\right ) \operatorname {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 ) \operatorname {Subst}\left (\int \frac {\log (1-x)}{x} \, dx,x,e^{\sinh ^{-1}(a x)}\right )+\left (4 a^3\right ) \operatorname {Subst}\left (\int \frac {\log (1+x)}{x} \, dx,x,e^{\sinh ^{-1}(a x)}\right )+\left (4 a^3\right ) \operatorname {Subst}\left (\int \text {Li}_3\left (-e^x\right ) \, dx,x,\sinh ^{-1}(a x)\right )-\left (4 a^3\right ) \operatorname {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 ) \operatorname {Subst}\left (\int \frac {\text {Li}_3(-x)}{x} \, dx,x,e^{\sinh ^{-1}(a x)}\right )-\left (4 a^3\right ) \operatorname {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 = 2.64, size = 355, normalized size = 1.59 \[ \frac {1}{24} a^3 \left (-\frac {8 \sinh ^4\left (\frac {1}{2} \sinh ^{-1}(a x)\right ) \sinh ^{-1}(a x)^4}{a^3 x^3}-48 \sinh ^{-1}(a x)^2 \text {Li}_2\left (e^{\sinh ^{-1}(a x)}\right )-96 \sinh ^{-1}(a x) \text {Li}_3\left (-e^{-\sinh ^{-1}(a x)}\right )+96 \sinh ^{-1}(a x) \text {Li}_3\left (e^{\sinh ^{-1}(a x)}\right )-48 \left (\sinh ^{-1}(a x)^2-2\right ) \text {Li}_2\left (-e^{-\sinh ^{-1}(a x)}\right )-96 \text {Li}_2\left (e^{-\sinh ^{-1}(a x)}\right )-96 \text {Li}_4\left (-e^{-\sinh ^{-1}(a x)}\right )-96 \text {Li}_4\left (e^{\sinh ^{-1}(a x)}\right )+4 \sinh ^{-1}(a x)^4+16 \sinh ^{-1}(a x)^3 \log \left (e^{-\sinh ^{-1}(a x)}+1\right )-16 \sinh ^{-1}(a x)^3 \log \left (1-e^{\sinh ^{-1}(a x)}\right )+96 \sinh ^{-1}(a x) \log \left (1-e^{-\sinh ^{-1}(a x)}\right )-96 \sinh ^{-1}(a x) \log \left (e^{-\sinh ^{-1}(a x)}+1\right )-2 \sinh ^{-1}(a x)^4 \tanh \left (\frac {1}{2} \sinh ^{-1}(a x)\right )+24 \sinh ^{-1}(a x)^2 \tanh \left (\frac {1}{2} \sinh ^{-1}(a x)\right )+2 \sinh ^{-1}(a x)^4 \coth \left (\frac {1}{2} \sinh ^{-1}(a x)\right )-24 \sinh ^{-1}(a x)^2 \coth \left (\frac {1}{2} \sinh ^{-1}(a x)\right )-\frac {1}{2} a x \sinh ^{-1}(a x)^4 \text {csch}^4\left (\frac {1}{2} \sinh ^{-1}(a x)\right )-4 \sinh ^{-1}(a x)^3 \text {csch}^2\left (\frac {1}{2} \sinh ^{-1}(a x)\right )-4 \sinh ^{-1}(a x)^3 \text {sech}^2\left (\frac {1}{2} \sinh ^{-1}(a x)\right )-2 \pi ^4\right ) \]
Warning: Unable to verify antiderivative.
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fricas [F] time = 0.43, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {\operatorname {arsinh}\left (a x\right )^{4}}{x^{4}}, x\right ) \]
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 \[ \int \frac {\operatorname {arsinh}\left (a x\right )^{4}}{x^{4}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.48, size = 372, normalized size = 1.67 \[ -\frac {2 a \arcsinh \left (a x \right )^{3} \sqrt {a^{2} x^{2}+1}}{3 x^{2}}-\frac {2 a^{2} \arcsinh \left (a x \right )^{2}}{x}-\frac {\arcsinh \left (a x \right )^{4}}{3 x^{3}}-\frac {2 a^{3} \arcsinh \left (a x \right )^{3} \ln \left (1-a x -\sqrt {a^{2} x^{2}+1}\right )}{3}-2 a^{3} \arcsinh \left (a x \right )^{2} \polylog \left (2, a x +\sqrt {a^{2} x^{2}+1}\right )+4 a^{3} \arcsinh \left (a x \right ) \polylog \left (3, a x +\sqrt {a^{2} x^{2}+1}\right )-4 a^{3} \polylog \left (4, a x +\sqrt {a^{2} x^{2}+1}\right )+\frac {2 a^{3} \arcsinh \left (a x \right )^{3} \ln \left (a x +\sqrt {a^{2} x^{2}+1}+1\right )}{3}+2 a^{3} \arcsinh \left (a x \right )^{2} \polylog \left (2, -a x -\sqrt {a^{2} x^{2}+1}\right )-4 a^{3} \arcsinh \left (a x \right ) \polylog \left (3, -a x -\sqrt {a^{2} x^{2}+1}\right )+4 a^{3} \polylog \left (4, -a x -\sqrt {a^{2} x^{2}+1}\right )+4 a^{3} \arcsinh \left (a x \right ) \ln \left (1-a x -\sqrt {a^{2} x^{2}+1}\right )+4 a^{3} \polylog \left (2, a x +\sqrt {a^{2} x^{2}+1}\right )-4 a^{3} \arcsinh \left (a x \right ) \ln \left (a x +\sqrt {a^{2} x^{2}+1}+1\right )-4 a^{3} \polylog \left (2, -a x -\sqrt {a^{2} x^{2}+1}\right ) \]
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 \[ -\frac {\log \left (a x + \sqrt {a^{2} x^{2} + 1}\right )^{4}}{3 \, x^{3}} + \int \frac {4 \, {\left (a^{3} x^{2} + \sqrt {a^{2} x^{2} + 1} a^{2} x + a\right )} \log \left (a x + \sqrt {a^{2} x^{2} + 1}\right )^{3}}{3 \, {\left (a^{3} x^{6} + a x^{4} + {\left (a^{2} x^{5} + x^{3}\right )} \sqrt {a^{2} x^{2} + 1}\right )}}\,{d x} \]
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 \[ \int \frac {{\mathrm {asinh}\left (a\,x\right )}^4}{x^4} \,d x \]
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 \[ \int \frac {\operatorname {asinh}^{4}{\left (a x \right )}}{x^{4}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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