Integrand size = 8, antiderivative size = 33 \[ \int \frac {\text {arcsinh}(a x)}{x^3} \, dx=-\frac {a \sqrt {1+a^2 x^2}}{2 x}-\frac {\text {arcsinh}(a x)}{2 x^2} \]
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Time = 0.01 (sec) , antiderivative size = 33, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {5776, 270} \[ \int \frac {\text {arcsinh}(a x)}{x^3} \, dx=-\frac {a \sqrt {a^2 x^2+1}}{2 x}-\frac {\text {arcsinh}(a x)}{2 x^2} \]
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Rule 270
Rule 5776
Rubi steps \begin{align*} \text {integral}& = -\frac {\text {arcsinh}(a x)}{2 x^2}+\frac {1}{2} a \int \frac {1}{x^2 \sqrt {1+a^2 x^2}} \, dx \\ & = -\frac {a \sqrt {1+a^2 x^2}}{2 x}-\frac {\text {arcsinh}(a x)}{2 x^2} \\ \end{align*}
Time = 0.01 (sec) , antiderivative size = 28, normalized size of antiderivative = 0.85 \[ \int \frac {\text {arcsinh}(a x)}{x^3} \, dx=-\frac {a x \sqrt {1+a^2 x^2}+\text {arcsinh}(a x)}{2 x^2} \]
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Time = 0.01 (sec) , antiderivative size = 28, normalized size of antiderivative = 0.85
method | result | size |
parts | \(-\frac {\operatorname {arcsinh}\left (a x \right )}{2 x^{2}}-\frac {a \sqrt {a^{2} x^{2}+1}}{2 x}\) | \(28\) |
derivativedivides | \(a^{2} \left (-\frac {\operatorname {arcsinh}\left (a x \right )}{2 a^{2} x^{2}}-\frac {\sqrt {a^{2} x^{2}+1}}{2 a x}\right )\) | \(37\) |
default | \(a^{2} \left (-\frac {\operatorname {arcsinh}\left (a x \right )}{2 a^{2} x^{2}}-\frac {\sqrt {a^{2} x^{2}+1}}{2 a x}\right )\) | \(37\) |
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none
Time = 0.26 (sec) , antiderivative size = 36, normalized size of antiderivative = 1.09 \[ \int \frac {\text {arcsinh}(a x)}{x^3} \, dx=-\frac {\sqrt {a^{2} x^{2} + 1} a x + \log \left (a x + \sqrt {a^{2} x^{2} + 1}\right )}{2 \, x^{2}} \]
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\[ \int \frac {\text {arcsinh}(a x)}{x^3} \, dx=\int \frac {\operatorname {asinh}{\left (a x \right )}}{x^{3}}\, dx \]
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none
Time = 0.18 (sec) , antiderivative size = 27, normalized size of antiderivative = 0.82 \[ \int \frac {\text {arcsinh}(a x)}{x^3} \, dx=-\frac {\sqrt {a^{2} x^{2} + 1} a}{2 \, x} - \frac {\operatorname {arsinh}\left (a x\right )}{2 \, x^{2}} \]
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Time = 0.27 (sec) , antiderivative size = 50, normalized size of antiderivative = 1.52 \[ \int \frac {\text {arcsinh}(a x)}{x^3} \, dx=\frac {a {\left | a \right |}}{{\left (x {\left | a \right |} - \sqrt {a^{2} x^{2} + 1}\right )}^{2} - 1} - \frac {\log \left (a x + \sqrt {a^{2} x^{2} + 1}\right )}{2 \, x^{2}} \]
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Timed out. \[ \int \frac {\text {arcsinh}(a x)}{x^3} \, dx=\int \frac {\mathrm {asinh}\left (a\,x\right )}{x^3} \,d x \]
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