Integrand size = 10, antiderivative size = 159 \[ \int \frac {\text {arcsinh}(a x)^3}{x^5} \, dx=-\frac {a^3 \sqrt {1+a^2 x^2}}{4 x}-\frac {a^2 \text {arcsinh}(a x)}{4 x^2}+\frac {1}{2} a^4 \text {arcsinh}(a x)^2-\frac {a \sqrt {1+a^2 x^2} \text {arcsinh}(a x)^2}{4 x^3}+\frac {a^3 \sqrt {1+a^2 x^2} \text {arcsinh}(a x)^2}{2 x}-\frac {\text {arcsinh}(a x)^3}{4 x^4}-a^4 \text {arcsinh}(a x) \log \left (1-e^{2 \text {arcsinh}(a x)}\right )-\frac {1}{2} a^4 \operatorname {PolyLog}\left (2,e^{2 \text {arcsinh}(a x)}\right ) \]
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Time = 0.19 (sec) , antiderivative size = 159, normalized size of antiderivative = 1.00, number of steps used = 10, number of rules used = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.900, Rules used = {5776, 5809, 5800, 5775, 3797, 2221, 2317, 2438, 270} \[ \int \frac {\text {arcsinh}(a x)^3}{x^5} \, dx=-\frac {1}{2} a^4 \operatorname {PolyLog}\left (2,e^{2 \text {arcsinh}(a x)}\right )+\frac {1}{2} a^4 \text {arcsinh}(a x)^2-a^4 \text {arcsinh}(a x) \log \left (1-e^{2 \text {arcsinh}(a x)}\right )-\frac {a^2 \text {arcsinh}(a x)}{4 x^2}-\frac {a \sqrt {a^2 x^2+1} \text {arcsinh}(a x)^2}{4 x^3}+\frac {a^3 \sqrt {a^2 x^2+1} \text {arcsinh}(a x)^2}{2 x}-\frac {a^3 \sqrt {a^2 x^2+1}}{4 x}-\frac {\text {arcsinh}(a x)^3}{4 x^4} \]
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Rule 270
Rule 2221
Rule 2317
Rule 2438
Rule 3797
Rule 5775
Rule 5776
Rule 5800
Rule 5809
Rubi steps \begin{align*} \text {integral}& = -\frac {\text {arcsinh}(a x)^3}{4 x^4}+\frac {1}{4} (3 a) \int \frac {\text {arcsinh}(a x)^2}{x^4 \sqrt {1+a^2 x^2}} \, dx \\ & = -\frac {a \sqrt {1+a^2 x^2} \text {arcsinh}(a x)^2}{4 x^3}-\frac {\text {arcsinh}(a x)^3}{4 x^4}+\frac {1}{2} a^2 \int \frac {\text {arcsinh}(a x)}{x^3} \, dx-\frac {1}{2} a^3 \int \frac {\text {arcsinh}(a x)^2}{x^2 \sqrt {1+a^2 x^2}} \, dx \\ & = -\frac {a^2 \text {arcsinh}(a x)}{4 x^2}-\frac {a \sqrt {1+a^2 x^2} \text {arcsinh}(a x)^2}{4 x^3}+\frac {a^3 \sqrt {1+a^2 x^2} \text {arcsinh}(a x)^2}{2 x}-\frac {\text {arcsinh}(a x)^3}{4 x^4}+\frac {1}{4} a^3 \int \frac {1}{x^2 \sqrt {1+a^2 x^2}} \, dx-a^4 \int \frac {\text {arcsinh}(a x)}{x} \, dx \\ & = -\frac {a^3 \sqrt {1+a^2 x^2}}{4 x}-\frac {a^2 \text {arcsinh}(a x)}{4 x^2}-\frac {a \sqrt {1+a^2 x^2} \text {arcsinh}(a x)^2}{4 x^3}+\frac {a^3 \sqrt {1+a^2 x^2} \text {arcsinh}(a x)^2}{2 x}-\frac {\text {arcsinh}(a x)^3}{4 x^4}-a^4 \text {Subst}(\int x \coth (x) \, dx,x,\text {arcsinh}(a x)) \\ & = -\frac {a^3 \sqrt {1+a^2 x^2}}{4 x}-\frac {a^2 \text {arcsinh}(a x)}{4 x^2}+\frac {1}{2} a^4 \text {arcsinh}(a x)^2-\frac {a \sqrt {1+a^2 x^2} \text {arcsinh}(a x)^2}{4 x^3}+\frac {a^3 \sqrt {1+a^2 x^2} \text {arcsinh}(a x)^2}{2 x}-\frac {\text {arcsinh}(a x)^3}{4 x^4}+\left (2 a^4\right ) \text {Subst}\left (\int \frac {e^{2 x} x}{1-e^{2 x}} \, dx,x,\text {arcsinh}(a x)\right ) \\ & = -\frac {a^3 \sqrt {1+a^2 x^2}}{4 x}-\frac {a^2 \text {arcsinh}(a x)}{4 x^2}+\frac {1}{2} a^4 \text {arcsinh}(a x)^2-\frac {a \sqrt {1+a^2 x^2} \text {arcsinh}(a x)^2}{4 x^3}+\frac {a^3 \sqrt {1+a^2 x^2} \text {arcsinh}(a x)^2}{2 x}-\frac {\text {arcsinh}(a x)^3}{4 x^4}-a^4 \text {arcsinh}(a x) \log \left (1-e^{2 \text {arcsinh}(a x)}\right )+a^4 \text {Subst}\left (\int \log \left (1-e^{2 x}\right ) \, dx,x,\text {arcsinh}(a x)\right ) \\ & = -\frac {a^3 \sqrt {1+a^2 x^2}}{4 x}-\frac {a^2 \text {arcsinh}(a x)}{4 x^2}+\frac {1}{2} a^4 \text {arcsinh}(a x)^2-\frac {a \sqrt {1+a^2 x^2} \text {arcsinh}(a x)^2}{4 x^3}+\frac {a^3 \sqrt {1+a^2 x^2} \text {arcsinh}(a x)^2}{2 x}-\frac {\text {arcsinh}(a x)^3}{4 x^4}-a^4 \text {arcsinh}(a x) \log \left (1-e^{2 \text {arcsinh}(a x)}\right )+\frac {1}{2} a^4 \text {Subst}\left (\int \frac {\log (1-x)}{x} \, dx,x,e^{2 \text {arcsinh}(a x)}\right ) \\ & = -\frac {a^3 \sqrt {1+a^2 x^2}}{4 x}-\frac {a^2 \text {arcsinh}(a x)}{4 x^2}+\frac {1}{2} a^4 \text {arcsinh}(a x)^2-\frac {a \sqrt {1+a^2 x^2} \text {arcsinh}(a x)^2}{4 x^3}+\frac {a^3 \sqrt {1+a^2 x^2} \text {arcsinh}(a x)^2}{2 x}-\frac {\text {arcsinh}(a x)^3}{4 x^4}-a^4 \text {arcsinh}(a x) \log \left (1-e^{2 \text {arcsinh}(a x)}\right )-\frac {1}{2} a^4 \operatorname {PolyLog}\left (2,e^{2 \text {arcsinh}(a x)}\right ) \\ \end{align*}
Time = 0.56 (sec) , antiderivative size = 107, normalized size of antiderivative = 0.67 \[ \int \frac {\text {arcsinh}(a x)^3}{x^5} \, dx=\frac {1}{4} \left (-\frac {\text {arcsinh}(a x)^3}{x^4}+a^4 \left (-\frac {\sqrt {1+a^2 x^2} \left (1+\left (-2+\frac {1}{a^2 x^2}\right ) \text {arcsinh}(a x)^2\right )}{a x}-\text {arcsinh}(a x) \left (\frac {1}{a^2 x^2}+2 \text {arcsinh}(a x)+4 \log \left (1-e^{-2 \text {arcsinh}(a x)}\right )\right )+2 \operatorname {PolyLog}\left (2,e^{-2 \text {arcsinh}(a x)}\right )\right )\right ) \]
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Time = 0.07 (sec) , antiderivative size = 213, normalized size of antiderivative = 1.34
method | result | size |
derivativedivides | \(a^{4} \left (-\frac {-2 a^{3} x^{3} \operatorname {arcsinh}\left (a x \right )^{2} \sqrt {a^{2} x^{2}+1}+2 a^{4} x^{4} \operatorname {arcsinh}\left (a x \right )^{2}+\operatorname {arcsinh}\left (a x \right )^{2} \sqrt {a^{2} x^{2}+1}\, a x +a^{3} x^{3} \sqrt {a^{2} x^{2}+1}-a^{4} x^{4}+\operatorname {arcsinh}\left (a x \right )^{3}+a^{2} x^{2} \operatorname {arcsinh}\left (a x \right )}{4 a^{4} x^{4}}+\operatorname {arcsinh}\left (a x \right )^{2}-\operatorname {arcsinh}\left (a x \right ) \ln \left (1+a x +\sqrt {a^{2} x^{2}+1}\right )-\operatorname {polylog}\left (2, -a x -\sqrt {a^{2} x^{2}+1}\right )-\operatorname {arcsinh}\left (a x \right ) \ln \left (1-a x -\sqrt {a^{2} x^{2}+1}\right )-\operatorname {polylog}\left (2, a x +\sqrt {a^{2} x^{2}+1}\right )\right )\) | \(213\) |
default | \(a^{4} \left (-\frac {-2 a^{3} x^{3} \operatorname {arcsinh}\left (a x \right )^{2} \sqrt {a^{2} x^{2}+1}+2 a^{4} x^{4} \operatorname {arcsinh}\left (a x \right )^{2}+\operatorname {arcsinh}\left (a x \right )^{2} \sqrt {a^{2} x^{2}+1}\, a x +a^{3} x^{3} \sqrt {a^{2} x^{2}+1}-a^{4} x^{4}+\operatorname {arcsinh}\left (a x \right )^{3}+a^{2} x^{2} \operatorname {arcsinh}\left (a x \right )}{4 a^{4} x^{4}}+\operatorname {arcsinh}\left (a x \right )^{2}-\operatorname {arcsinh}\left (a x \right ) \ln \left (1+a x +\sqrt {a^{2} x^{2}+1}\right )-\operatorname {polylog}\left (2, -a x -\sqrt {a^{2} x^{2}+1}\right )-\operatorname {arcsinh}\left (a x \right ) \ln \left (1-a x -\sqrt {a^{2} x^{2}+1}\right )-\operatorname {polylog}\left (2, a x +\sqrt {a^{2} x^{2}+1}\right )\right )\) | \(213\) |
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\[ \int \frac {\text {arcsinh}(a x)^3}{x^5} \, dx=\int { \frac {\operatorname {arsinh}\left (a x\right )^{3}}{x^{5}} \,d x } \]
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\[ \int \frac {\text {arcsinh}(a x)^3}{x^5} \, dx=\int \frac {\operatorname {asinh}^{3}{\left (a x \right )}}{x^{5}}\, dx \]
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\[ \int \frac {\text {arcsinh}(a x)^3}{x^5} \, dx=\int { \frac {\operatorname {arsinh}\left (a x\right )^{3}}{x^{5}} \,d x } \]
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Exception generated. \[ \int \frac {\text {arcsinh}(a x)^3}{x^5} \, dx=\text {Exception raised: TypeError} \]
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Timed out. \[ \int \frac {\text {arcsinh}(a x)^3}{x^5} \, dx=\int \frac {{\mathrm {asinh}\left (a\,x\right )}^3}{x^5} \,d x \]
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