Integrand size = 21, antiderivative size = 80 \[ \int \frac {\text {arcsinh}(a x)}{x^3 \sqrt {1+a^2 x^2}} \, dx=-\frac {a}{2 x}-\frac {\sqrt {1+a^2 x^2} \text {arcsinh}(a x)}{2 x^2}+a^2 \text {arcsinh}(a x) \text {arctanh}\left (e^{\text {arcsinh}(a x)}\right )+\frac {1}{2} a^2 \operatorname {PolyLog}\left (2,-e^{\text {arcsinh}(a x)}\right )-\frac {1}{2} a^2 \operatorname {PolyLog}\left (2,e^{\text {arcsinh}(a x)}\right ) \]
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Time = 0.10 (sec) , antiderivative size = 80, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.286, Rules used = {5809, 5816, 4267, 2317, 2438, 30} \[ \int \frac {\text {arcsinh}(a x)}{x^3 \sqrt {1+a^2 x^2}} \, dx=a^2 \text {arcsinh}(a x) \text {arctanh}\left (e^{\text {arcsinh}(a x)}\right )+\frac {1}{2} a^2 \operatorname {PolyLog}\left (2,-e^{\text {arcsinh}(a x)}\right )-\frac {1}{2} a^2 \operatorname {PolyLog}\left (2,e^{\text {arcsinh}(a x)}\right )-\frac {\sqrt {a^2 x^2+1} \text {arcsinh}(a x)}{2 x^2}-\frac {a}{2 x} \]
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Rule 30
Rule 2317
Rule 2438
Rule 4267
Rule 5809
Rule 5816
Rubi steps \begin{align*} \text {integral}& = -\frac {\sqrt {1+a^2 x^2} \text {arcsinh}(a x)}{2 x^2}+\frac {1}{2} a \int \frac {1}{x^2} \, dx-\frac {1}{2} a^2 \int \frac {\text {arcsinh}(a x)}{x \sqrt {1+a^2 x^2}} \, dx \\ & = -\frac {a}{2 x}-\frac {\sqrt {1+a^2 x^2} \text {arcsinh}(a x)}{2 x^2}-\frac {1}{2} a^2 \text {Subst}(\int x \text {csch}(x) \, dx,x,\text {arcsinh}(a x)) \\ & = -\frac {a}{2 x}-\frac {\sqrt {1+a^2 x^2} \text {arcsinh}(a x)}{2 x^2}+a^2 \text {arcsinh}(a x) \text {arctanh}\left (e^{\text {arcsinh}(a x)}\right )+\frac {1}{2} a^2 \text {Subst}\left (\int \log \left (1-e^x\right ) \, dx,x,\text {arcsinh}(a x)\right )-\frac {1}{2} a^2 \text {Subst}\left (\int \log \left (1+e^x\right ) \, dx,x,\text {arcsinh}(a x)\right ) \\ & = -\frac {a}{2 x}-\frac {\sqrt {1+a^2 x^2} \text {arcsinh}(a x)}{2 x^2}+a^2 \text {arcsinh}(a x) \text {arctanh}\left (e^{\text {arcsinh}(a x)}\right )+\frac {1}{2} a^2 \text {Subst}\left (\int \frac {\log (1-x)}{x} \, dx,x,e^{\text {arcsinh}(a x)}\right )-\frac {1}{2} a^2 \text {Subst}\left (\int \frac {\log (1+x)}{x} \, dx,x,e^{\text {arcsinh}(a x)}\right ) \\ & = -\frac {a}{2 x}-\frac {\sqrt {1+a^2 x^2} \text {arcsinh}(a x)}{2 x^2}+a^2 \text {arcsinh}(a x) \text {arctanh}\left (e^{\text {arcsinh}(a x)}\right )+\frac {1}{2} a^2 \operatorname {PolyLog}\left (2,-e^{\text {arcsinh}(a x)}\right )-\frac {1}{2} a^2 \operatorname {PolyLog}\left (2,e^{\text {arcsinh}(a x)}\right ) \\ \end{align*}
Time = 0.55 (sec) , antiderivative size = 126, normalized size of antiderivative = 1.58 \[ \int \frac {\text {arcsinh}(a x)}{x^3 \sqrt {1+a^2 x^2}} \, dx=\frac {1}{8} a^2 \left (-2 \coth \left (\frac {1}{2} \text {arcsinh}(a x)\right )-\text {arcsinh}(a x) \text {csch}^2\left (\frac {1}{2} \text {arcsinh}(a x)\right )-4 \text {arcsinh}(a x) \log \left (1-e^{-\text {arcsinh}(a x)}\right )+4 \text {arcsinh}(a x) \log \left (1+e^{-\text {arcsinh}(a x)}\right )-4 \operatorname {PolyLog}\left (2,-e^{-\text {arcsinh}(a x)}\right )+4 \operatorname {PolyLog}\left (2,e^{-\text {arcsinh}(a x)}\right )-\text {arcsinh}(a x) \text {sech}^2\left (\frac {1}{2} \text {arcsinh}(a x)\right )+2 \tanh \left (\frac {1}{2} \text {arcsinh}(a x)\right )\right ) \]
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Time = 0.20 (sec) , antiderivative size = 150, normalized size of antiderivative = 1.88
method | result | size |
default | \(-\frac {a^{2} x^{2} \operatorname {arcsinh}\left (a x \right )+a x \sqrt {a^{2} x^{2}+1}+\operatorname {arcsinh}\left (a x \right )}{2 \sqrt {a^{2} x^{2}+1}\, x^{2}}-\frac {a^{2} \operatorname {arcsinh}\left (a x \right ) \ln \left (1-a x -\sqrt {a^{2} x^{2}+1}\right )}{2}-\frac {a^{2} \operatorname {polylog}\left (2, a x +\sqrt {a^{2} x^{2}+1}\right )}{2}+\frac {a^{2} \operatorname {arcsinh}\left (a x \right ) \ln \left (1+a x +\sqrt {a^{2} x^{2}+1}\right )}{2}+\frac {a^{2} \operatorname {polylog}\left (2, -a x -\sqrt {a^{2} x^{2}+1}\right )}{2}\) | \(150\) |
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\[ \int \frac {\text {arcsinh}(a x)}{x^3 \sqrt {1+a^2 x^2}} \, dx=\int { \frac {\operatorname {arsinh}\left (a x\right )}{\sqrt {a^{2} x^{2} + 1} x^{3}} \,d x } \]
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\[ \int \frac {\text {arcsinh}(a x)}{x^3 \sqrt {1+a^2 x^2}} \, dx=\int \frac {\operatorname {asinh}{\left (a x \right )}}{x^{3} \sqrt {a^{2} x^{2} + 1}}\, dx \]
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\[ \int \frac {\text {arcsinh}(a x)}{x^3 \sqrt {1+a^2 x^2}} \, dx=\int { \frac {\operatorname {arsinh}\left (a x\right )}{\sqrt {a^{2} x^{2} + 1} x^{3}} \,d x } \]
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\[ \int \frac {\text {arcsinh}(a x)}{x^3 \sqrt {1+a^2 x^2}} \, dx=\int { \frac {\operatorname {arsinh}\left (a x\right )}{\sqrt {a^{2} x^{2} + 1} x^{3}} \,d x } \]
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Timed out. \[ \int \frac {\text {arcsinh}(a x)}{x^3 \sqrt {1+a^2 x^2}} \, dx=\int \frac {\mathrm {asinh}\left (a\,x\right )}{x^3\,\sqrt {a^2\,x^2+1}} \,d x \]
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