Optimal. Leaf size=151 \[ a^3 \sinh ^{-1}(a x) \text {Li}_2\left (-e^{\sinh ^{-1}(a x)}\right )-a^3 \sinh ^{-1}(a x) \text {Li}_2\left (e^{\sinh ^{-1}(a x)}\right )-a^3 \text {Li}_3\left (-e^{\sinh ^{-1}(a x)}\right )+a^3 \text {Li}_3\left (e^{\sinh ^{-1}(a x)}\right )+a^3 \sinh ^{-1}(a x)^2 \tanh ^{-1}\left (e^{\sinh ^{-1}(a x)}\right )-\frac {a \sqrt {a^2 x^2+1} \sinh ^{-1}(a x)^2}{2 x^2}-\frac {a^2 \sinh ^{-1}(a x)}{x}-a^3 \tanh ^{-1}\left (\sqrt {a^2 x^2+1}\right )-\frac {\sinh ^{-1}(a x)^3}{3 x^3} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.28, 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 = {5661, 5747, 5760, 4182, 2531, 2282, 6589, 266, 63, 208} \[ a^3 \sinh ^{-1}(a x) \text {PolyLog}\left (2,-e^{\sinh ^{-1}(a x)}\right )-a^3 \sinh ^{-1}(a x) \text {PolyLog}\left (2,e^{\sinh ^{-1}(a x)}\right )-a^3 \text {PolyLog}\left (3,-e^{\sinh ^{-1}(a x)}\right )+a^3 \text {PolyLog}\left (3,e^{\sinh ^{-1}(a x)}\right )-\frac {a \sqrt {a^2 x^2+1} \sinh ^{-1}(a x)^2}{2 x^2}-a^3 \tanh ^{-1}\left (\sqrt {a^2 x^2+1}\right )-\frac {a^2 \sinh ^{-1}(a x)}{x}+a^3 \sinh ^{-1}(a x)^2 \tanh ^{-1}\left (e^{\sinh ^{-1}(a x)}\right )-\frac {\sinh ^{-1}(a x)^3}{3 x^3} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 63
Rule 208
Rule 266
Rule 2282
Rule 2531
Rule 4182
Rule 5661
Rule 5747
Rule 5760
Rule 6589
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 \operatorname {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 \operatorname {Subst}\left (\int \frac {1}{x \sqrt {1+a^2 x}} \, dx,x,x^2\right )+a^3 \operatorname {Subst}\left (\int x \log \left (1-e^x\right ) \, dx,x,\sinh ^{-1}(a x)\right )-a^3 \operatorname {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 \operatorname {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 \operatorname {Subst}\left (\int \text {Li}_2\left (-e^x\right ) \, dx,x,\sinh ^{-1}(a x)\right )+a^3 \operatorname {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 \operatorname {Subst}\left (\int \frac {\text {Li}_2(-x)}{x} \, dx,x,e^{\sinh ^{-1}(a x)}\right )+a^3 \operatorname {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*}
________________________________________________________________________________________
Mathematica [A] time = 2.25, size = 268, normalized size = 1.77 \[ \frac {1}{48} a^3 \left (-\frac {16 \sinh ^{-1}(a x)^3 \sinh ^4\left (\frac {1}{2} \sinh ^{-1}(a x)\right )}{a^3 x^3}-48 \sinh ^{-1}(a x) \text {Li}_2\left (-e^{-\sinh ^{-1}(a x)}\right )+48 \sinh ^{-1}(a x) \text {Li}_2\left (e^{-\sinh ^{-1}(a x)}\right )-48 \text {Li}_3\left (-e^{-\sinh ^{-1}(a x)}\right )+48 \text {Li}_3\left (e^{-\sinh ^{-1}(a x)}\right )-24 \sinh ^{-1}(a x)^2 \log \left (1-e^{-\sinh ^{-1}(a x)}\right )+24 \sinh ^{-1}(a x)^2 \log \left (e^{-\sinh ^{-1}(a x)}+1\right )-4 \sinh ^{-1}(a x)^3 \tanh \left (\frac {1}{2} \sinh ^{-1}(a x)\right )+24 \sinh ^{-1}(a x) \tanh \left (\frac {1}{2} \sinh ^{-1}(a x)\right )+4 \sinh ^{-1}(a x)^3 \coth \left (\frac {1}{2} \sinh ^{-1}(a x)\right )-24 \sinh ^{-1}(a x) \coth \left (\frac {1}{2} \sinh ^{-1}(a x)\right )-a x \sinh ^{-1}(a x)^3 \text {csch}^4\left (\frac {1}{2} \sinh ^{-1}(a x)\right )-6 \sinh ^{-1}(a x)^2 \text {csch}^2\left (\frac {1}{2} \sinh ^{-1}(a x)\right )-6 \sinh ^{-1}(a x)^2 \text {sech}^2\left (\frac {1}{2} \sinh ^{-1}(a x)\right )+48 \log \left (\tanh \left (\frac {1}{2} \sinh ^{-1}(a x)\right )\right )\right ) \]
Warning: Unable to verify antiderivative.
[In]
[Out]
________________________________________________________________________________________
fricas [F] time = 0.43, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {\operatorname {arsinh}\left (a x\right )^{3}}{x^{4}}, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\operatorname {arsinh}\left (a x\right )^{3}}{x^{4}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [A] time = 0.54, size = 228, normalized size = 1.51 \[ -\frac {a \arcsinh \left (a x \right )^{2} \sqrt {a^{2} x^{2}+1}}{2 x^{2}}-\frac {a^{2} \arcsinh \left (a x \right )}{x}-\frac {\arcsinh \left (a x \right )^{3}}{3 x^{3}}-\frac {a^{3} \arcsinh \left (a x \right )^{2} \ln \left (1-a x -\sqrt {a^{2} x^{2}+1}\right )}{2}-a^{3} \arcsinh \left (a x \right ) \polylog \left (2, a x +\sqrt {a^{2} x^{2}+1}\right )+a^{3} \polylog \left (3, a x +\sqrt {a^{2} x^{2}+1}\right )+\frac {a^{3} \arcsinh \left (a x \right )^{2} \ln \left (a x +\sqrt {a^{2} x^{2}+1}+1\right )}{2}+a^{3} \arcsinh \left (a x \right ) \polylog \left (2, -a x -\sqrt {a^{2} x^{2}+1}\right )-a^{3} \polylog \left (3, -a x -\sqrt {a^{2} x^{2}+1}\right )-2 a^{3} \arctanh \left (a x +\sqrt {a^{2} x^{2}+1}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [F] time = 0.00, size = 0, normalized size = 0.00 \[ -\frac {\log \left (a x + \sqrt {a^{2} x^{2} + 1}\right )^{3}}{3 \, x^{3}} + \int \frac {{\left (a^{3} x^{2} + \sqrt {a^{2} x^{2} + 1} a^{2} x + a\right )} \log \left (a x + \sqrt {a^{2} x^{2} + 1}\right )^{2}}{a^{3} x^{6} + a x^{4} + {\left (a^{2} x^{5} + x^{3}\right )} \sqrt {a^{2} x^{2} + 1}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [F] time = 0.00, size = -1, normalized size = -0.01 \[ \int \frac {{\mathrm {asinh}\left (a\,x\right )}^3}{x^4} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\operatorname {asinh}^{3}{\left (a x \right )}}{x^{4}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________