Integrand size = 10, antiderivative size = 151 \[ \int \frac {\text {arcsinh}(a x)^3}{x^4} \, dx=-\frac {a^2 \text {arcsinh}(a x)}{x}-\frac {a \sqrt {1+a^2 x^2} \text {arcsinh}(a x)^2}{2 x^2}-\frac {\text {arcsinh}(a x)^3}{3 x^3}+a^3 \text {arcsinh}(a x)^2 \text {arctanh}\left (e^{\text {arcsinh}(a x)}\right )-a^3 \text {arctanh}\left (\sqrt {1+a^2 x^2}\right )+a^3 \text {arcsinh}(a x) \operatorname {PolyLog}\left (2,-e^{\text {arcsinh}(a x)}\right )-a^3 \text {arcsinh}(a x) \operatorname {PolyLog}\left (2,e^{\text {arcsinh}(a x)}\right )-a^3 \operatorname {PolyLog}\left (3,-e^{\text {arcsinh}(a x)}\right )+a^3 \operatorname {PolyLog}\left (3,e^{\text {arcsinh}(a x)}\right ) \]
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Time = 0.20 (sec) , antiderivative size = 151, normalized size of antiderivative = 1.00, number of steps used = 14, number of rules used = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 1.000, Rules used = {5776, 5809, 5816, 4267, 2611, 2320, 6724, 272, 65, 214} \[ \int \frac {\text {arcsinh}(a x)^3}{x^4} \, dx=a^3 \text {arcsinh}(a x)^2 \text {arctanh}\left (e^{\text {arcsinh}(a x)}\right )+a^3 \text {arcsinh}(a x) \operatorname {PolyLog}\left (2,-e^{\text {arcsinh}(a x)}\right )-a^3 \text {arcsinh}(a x) \operatorname {PolyLog}\left (2,e^{\text {arcsinh}(a x)}\right )-a^3 \operatorname {PolyLog}\left (3,-e^{\text {arcsinh}(a x)}\right )+a^3 \operatorname {PolyLog}\left (3,e^{\text {arcsinh}(a x)}\right )-\frac {a \sqrt {a^2 x^2+1} \text {arcsinh}(a x)^2}{2 x^2}-\frac {a^2 \text {arcsinh}(a x)}{x}-a^3 \text {arctanh}\left (\sqrt {a^2 x^2+1}\right )-\frac {\text {arcsinh}(a x)^3}{3 x^3} \]
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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*} \text {integral}& = -\frac {\text {arcsinh}(a x)^3}{3 x^3}+a \int \frac {\text {arcsinh}(a x)^2}{x^3 \sqrt {1+a^2 x^2}} \, dx \\ & = -\frac {a \sqrt {1+a^2 x^2} \text {arcsinh}(a x)^2}{2 x^2}-\frac {\text {arcsinh}(a x)^3}{3 x^3}+a^2 \int \frac {\text {arcsinh}(a x)}{x^2} \, dx-\frac {1}{2} a^3 \int \frac {\text {arcsinh}(a x)^2}{x \sqrt {1+a^2 x^2}} \, dx \\ & = -\frac {a^2 \text {arcsinh}(a x)}{x}-\frac {a \sqrt {1+a^2 x^2} \text {arcsinh}(a x)^2}{2 x^2}-\frac {\text {arcsinh}(a x)^3}{3 x^3}-\frac {1}{2} a^3 \text {Subst}\left (\int x^2 \text {csch}(x) \, dx,x,\text {arcsinh}(a x)\right )+a^3 \int \frac {1}{x \sqrt {1+a^2 x^2}} \, dx \\ & = -\frac {a^2 \text {arcsinh}(a x)}{x}-\frac {a \sqrt {1+a^2 x^2} \text {arcsinh}(a x)^2}{2 x^2}-\frac {\text {arcsinh}(a x)^3}{3 x^3}+a^3 \text {arcsinh}(a x)^2 \text {arctanh}\left (e^{\text {arcsinh}(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,\text {arcsinh}(a x)\right )-a^3 \text {Subst}\left (\int x \log \left (1+e^x\right ) \, dx,x,\text {arcsinh}(a x)\right ) \\ & = -\frac {a^2 \text {arcsinh}(a x)}{x}-\frac {a \sqrt {1+a^2 x^2} \text {arcsinh}(a x)^2}{2 x^2}-\frac {\text {arcsinh}(a x)^3}{3 x^3}+a^3 \text {arcsinh}(a x)^2 \text {arctanh}\left (e^{\text {arcsinh}(a x)}\right )+a^3 \text {arcsinh}(a x) \operatorname {PolyLog}\left (2,-e^{\text {arcsinh}(a x)}\right )-a^3 \text {arcsinh}(a x) \operatorname {PolyLog}\left (2,e^{\text {arcsinh}(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 \operatorname {PolyLog}\left (2,-e^x\right ) \, dx,x,\text {arcsinh}(a x)\right )+a^3 \text {Subst}\left (\int \operatorname {PolyLog}\left (2,e^x\right ) \, dx,x,\text {arcsinh}(a x)\right ) \\ & = -\frac {a^2 \text {arcsinh}(a x)}{x}-\frac {a \sqrt {1+a^2 x^2} \text {arcsinh}(a x)^2}{2 x^2}-\frac {\text {arcsinh}(a x)^3}{3 x^3}+a^3 \text {arcsinh}(a x)^2 \text {arctanh}\left (e^{\text {arcsinh}(a x)}\right )-a^3 \text {arctanh}\left (\sqrt {1+a^2 x^2}\right )+a^3 \text {arcsinh}(a x) \operatorname {PolyLog}\left (2,-e^{\text {arcsinh}(a x)}\right )-a^3 \text {arcsinh}(a x) \operatorname {PolyLog}\left (2,e^{\text {arcsinh}(a x)}\right )-a^3 \text {Subst}\left (\int \frac {\operatorname {PolyLog}(2,-x)}{x} \, dx,x,e^{\text {arcsinh}(a x)}\right )+a^3 \text {Subst}\left (\int \frac {\operatorname {PolyLog}(2,x)}{x} \, dx,x,e^{\text {arcsinh}(a x)}\right ) \\ & = -\frac {a^2 \text {arcsinh}(a x)}{x}-\frac {a \sqrt {1+a^2 x^2} \text {arcsinh}(a x)^2}{2 x^2}-\frac {\text {arcsinh}(a x)^3}{3 x^3}+a^3 \text {arcsinh}(a x)^2 \text {arctanh}\left (e^{\text {arcsinh}(a x)}\right )-a^3 \text {arctanh}\left (\sqrt {1+a^2 x^2}\right )+a^3 \text {arcsinh}(a x) \operatorname {PolyLog}\left (2,-e^{\text {arcsinh}(a x)}\right )-a^3 \text {arcsinh}(a x) \operatorname {PolyLog}\left (2,e^{\text {arcsinh}(a x)}\right )-a^3 \operatorname {PolyLog}\left (3,-e^{\text {arcsinh}(a x)}\right )+a^3 \operatorname {PolyLog}\left (3,e^{\text {arcsinh}(a x)}\right ) \\ \end{align*}
Time = 1.69 (sec) , antiderivative size = 268, normalized size of antiderivative = 1.77 \[ \int \frac {\text {arcsinh}(a x)^3}{x^4} \, dx=\frac {1}{48} a^3 \left (-24 \text {arcsinh}(a x) \coth \left (\frac {1}{2} \text {arcsinh}(a x)\right )+4 \text {arcsinh}(a x)^3 \coth \left (\frac {1}{2} \text {arcsinh}(a x)\right )-6 \text {arcsinh}(a x)^2 \text {csch}^2\left (\frac {1}{2} \text {arcsinh}(a x)\right )-a x \text {arcsinh}(a x)^3 \text {csch}^4\left (\frac {1}{2} \text {arcsinh}(a x)\right )-24 \text {arcsinh}(a x)^2 \log \left (1-e^{-\text {arcsinh}(a x)}\right )+24 \text {arcsinh}(a x)^2 \log \left (1+e^{-\text {arcsinh}(a x)}\right )+48 \log \left (\tanh \left (\frac {1}{2} \text {arcsinh}(a x)\right )\right )-48 \text {arcsinh}(a x) \operatorname {PolyLog}\left (2,-e^{-\text {arcsinh}(a x)}\right )+48 \text {arcsinh}(a x) \operatorname {PolyLog}\left (2,e^{-\text {arcsinh}(a x)}\right )-48 \operatorname {PolyLog}\left (3,-e^{-\text {arcsinh}(a x)}\right )+48 \operatorname {PolyLog}\left (3,e^{-\text {arcsinh}(a x)}\right )-6 \text {arcsinh}(a x)^2 \text {sech}^2\left (\frac {1}{2} \text {arcsinh}(a x)\right )-\frac {16 \text {arcsinh}(a x)^3 \sinh ^4\left (\frac {1}{2} \text {arcsinh}(a x)\right )}{a^3 x^3}+24 \text {arcsinh}(a x) \tanh \left (\frac {1}{2} \text {arcsinh}(a x)\right )-4 \text {arcsinh}(a x)^3 \tanh \left (\frac {1}{2} \text {arcsinh}(a x)\right )\right ) \]
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Time = 0.07 (sec) , antiderivative size = 212, normalized size of antiderivative = 1.40
method | result | size |
derivativedivides | \(a^{3} \left (-\frac {\operatorname {arcsinh}\left (a x \right ) \left (3 \,\operatorname {arcsinh}\left (a x \right ) \sqrt {a^{2} x^{2}+1}\, a x +2 \operatorname {arcsinh}\left (a x \right )^{2}+6 a^{2} x^{2}\right )}{6 a^{3} x^{3}}+\frac {\operatorname {arcsinh}\left (a x \right )^{2} \ln \left (1+a x +\sqrt {a^{2} x^{2}+1}\right )}{2}+\operatorname {arcsinh}\left (a x \right ) \operatorname {polylog}\left (2, -a x -\sqrt {a^{2} x^{2}+1}\right )-\operatorname {polylog}\left (3, -a x -\sqrt {a^{2} x^{2}+1}\right )-\frac {\operatorname {arcsinh}\left (a x \right )^{2} \ln \left (1-a x -\sqrt {a^{2} x^{2}+1}\right )}{2}-\operatorname {arcsinh}\left (a x \right ) \operatorname {polylog}\left (2, a x +\sqrt {a^{2} x^{2}+1}\right )+\operatorname {polylog}\left (3, a x +\sqrt {a^{2} x^{2}+1}\right )-2 \,\operatorname {arctanh}\left (a x +\sqrt {a^{2} x^{2}+1}\right )\right )\) | \(212\) |
default | \(a^{3} \left (-\frac {\operatorname {arcsinh}\left (a x \right ) \left (3 \,\operatorname {arcsinh}\left (a x \right ) \sqrt {a^{2} x^{2}+1}\, a x +2 \operatorname {arcsinh}\left (a x \right )^{2}+6 a^{2} x^{2}\right )}{6 a^{3} x^{3}}+\frac {\operatorname {arcsinh}\left (a x \right )^{2} \ln \left (1+a x +\sqrt {a^{2} x^{2}+1}\right )}{2}+\operatorname {arcsinh}\left (a x \right ) \operatorname {polylog}\left (2, -a x -\sqrt {a^{2} x^{2}+1}\right )-\operatorname {polylog}\left (3, -a x -\sqrt {a^{2} x^{2}+1}\right )-\frac {\operatorname {arcsinh}\left (a x \right )^{2} \ln \left (1-a x -\sqrt {a^{2} x^{2}+1}\right )}{2}-\operatorname {arcsinh}\left (a x \right ) \operatorname {polylog}\left (2, a x +\sqrt {a^{2} x^{2}+1}\right )+\operatorname {polylog}\left (3, a x +\sqrt {a^{2} x^{2}+1}\right )-2 \,\operatorname {arctanh}\left (a x +\sqrt {a^{2} x^{2}+1}\right )\right )\) | \(212\) |
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\[ \int \frac {\text {arcsinh}(a x)^3}{x^4} \, dx=\int { \frac {\operatorname {arsinh}\left (a x\right )^{3}}{x^{4}} \,d x } \]
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\[ \int \frac {\text {arcsinh}(a x)^3}{x^4} \, dx=\int \frac {\operatorname {asinh}^{3}{\left (a x \right )}}{x^{4}}\, dx \]
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\[ \int \frac {\text {arcsinh}(a x)^3}{x^4} \, dx=\int { \frac {\operatorname {arsinh}\left (a x\right )^{3}}{x^{4}} \,d x } \]
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\[ \int \frac {\text {arcsinh}(a x)^3}{x^4} \, dx=\int { \frac {\operatorname {arsinh}\left (a x\right )^{3}}{x^{4}} \,d x } \]
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Timed out. \[ \int \frac {\text {arcsinh}(a x)^3}{x^4} \, dx=\int \frac {{\mathrm {asinh}\left (a\,x\right )}^3}{x^4} \,d x \]
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