Integrand size = 8, antiderivative size = 56 \[ \int \frac {\text {arcsinh}(a x)}{x^5} \, dx=-\frac {a \sqrt {1+a^2 x^2}}{12 x^3}+\frac {a^3 \sqrt {1+a^2 x^2}}{6 x}-\frac {\text {arcsinh}(a x)}{4 x^4} \]
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Time = 0.02 (sec) , antiderivative size = 56, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.375, Rules used = {5776, 277, 270} \[ \int \frac {\text {arcsinh}(a x)}{x^5} \, dx=-\frac {a \sqrt {a^2 x^2+1}}{12 x^3}+\frac {a^3 \sqrt {a^2 x^2+1}}{6 x}-\frac {\text {arcsinh}(a x)}{4 x^4} \]
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Rule 270
Rule 277
Rule 5776
Rubi steps \begin{align*} \text {integral}& = -\frac {\text {arcsinh}(a x)}{4 x^4}+\frac {1}{4} a \int \frac {1}{x^4 \sqrt {1+a^2 x^2}} \, dx \\ & = -\frac {a \sqrt {1+a^2 x^2}}{12 x^3}-\frac {\text {arcsinh}(a x)}{4 x^4}-\frac {1}{6} a^3 \int \frac {1}{x^2 \sqrt {1+a^2 x^2}} \, dx \\ & = -\frac {a \sqrt {1+a^2 x^2}}{12 x^3}+\frac {a^3 \sqrt {1+a^2 x^2}}{6 x}-\frac {\text {arcsinh}(a x)}{4 x^4} \\ \end{align*}
Time = 0.01 (sec) , antiderivative size = 40, normalized size of antiderivative = 0.71 \[ \int \frac {\text {arcsinh}(a x)}{x^5} \, dx=\frac {a x \sqrt {1+a^2 x^2} \left (-1+2 a^2 x^2\right )-3 \text {arcsinh}(a x)}{12 x^4} \]
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Time = 0.01 (sec) , antiderivative size = 50, normalized size of antiderivative = 0.89
method | result | size |
parts | \(-\frac {\operatorname {arcsinh}\left (a x \right )}{4 x^{4}}+\frac {a \left (-\frac {\sqrt {a^{2} x^{2}+1}}{3 x^{3}}+\frac {2 a^{2} \sqrt {a^{2} x^{2}+1}}{3 x}\right )}{4}\) | \(50\) |
derivativedivides | \(a^{4} \left (-\frac {\operatorname {arcsinh}\left (a x \right )}{4 a^{4} x^{4}}-\frac {\sqrt {a^{2} x^{2}+1}}{12 a^{3} x^{3}}+\frac {\sqrt {a^{2} x^{2}+1}}{6 a x}\right )\) | \(56\) |
default | \(a^{4} \left (-\frac {\operatorname {arcsinh}\left (a x \right )}{4 a^{4} x^{4}}-\frac {\sqrt {a^{2} x^{2}+1}}{12 a^{3} x^{3}}+\frac {\sqrt {a^{2} x^{2}+1}}{6 a x}\right )\) | \(56\) |
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Time = 0.27 (sec) , antiderivative size = 49, normalized size of antiderivative = 0.88 \[ \int \frac {\text {arcsinh}(a x)}{x^5} \, dx=\frac {{\left (2 \, a^{3} x^{3} - a x\right )} \sqrt {a^{2} x^{2} + 1} - 3 \, \log \left (a x + \sqrt {a^{2} x^{2} + 1}\right )}{12 \, x^{4}} \]
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\[ \int \frac {\text {arcsinh}(a x)}{x^5} \, dx=\int \frac {\operatorname {asinh}{\left (a x \right )}}{x^{5}}\, dx \]
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Time = 0.20 (sec) , antiderivative size = 49, normalized size of antiderivative = 0.88 \[ \int \frac {\text {arcsinh}(a x)}{x^5} \, dx=\frac {1}{12} \, {\left (\frac {2 \, \sqrt {a^{2} x^{2} + 1} a^{2}}{x} - \frac {\sqrt {a^{2} x^{2} + 1}}{x^{3}}\right )} a - \frac {\operatorname {arsinh}\left (a x\right )}{4 \, x^{4}} \]
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Time = 0.26 (sec) , antiderivative size = 77, normalized size of antiderivative = 1.38 \[ \int \frac {\text {arcsinh}(a x)}{x^5} \, dx=\frac {{\left (3 \, {\left (x {\left | a \right |} - \sqrt {a^{2} x^{2} + 1}\right )}^{2} - 1\right )} a^{3} {\left | a \right |}}{3 \, {\left ({\left (x {\left | a \right |} - \sqrt {a^{2} x^{2} + 1}\right )}^{2} - 1\right )}^{3}} - \frac {\log \left (a x + \sqrt {a^{2} x^{2} + 1}\right )}{4 \, x^{4}} \]
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Timed out. \[ \int \frac {\text {arcsinh}(a x)}{x^5} \, dx=\int \frac {\mathrm {asinh}\left (a\,x\right )}{x^5} \,d x \]
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