Integrand size = 10, antiderivative size = 43 \[ \int \frac {\text {arcsinh}(a x)^2}{x^3} \, dx=-\frac {a \sqrt {1+a^2 x^2} \text {arcsinh}(a x)}{x}-\frac {\text {arcsinh}(a x)^2}{2 x^2}+a^2 \log (x) \]
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Time = 0.05 (sec) , antiderivative size = 43, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.300, Rules used = {5776, 5800, 29} \[ \int \frac {\text {arcsinh}(a x)^2}{x^3} \, dx=-\frac {a \sqrt {a^2 x^2+1} \text {arcsinh}(a x)}{x}+a^2 \log (x)-\frac {\text {arcsinh}(a x)^2}{2 x^2} \]
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Rule 29
Rule 5776
Rule 5800
Rubi steps \begin{align*} \text {integral}& = -\frac {\text {arcsinh}(a x)^2}{2 x^2}+a \int \frac {\text {arcsinh}(a x)}{x^2 \sqrt {1+a^2 x^2}} \, dx \\ & = -\frac {a \sqrt {1+a^2 x^2} \text {arcsinh}(a x)}{x}-\frac {\text {arcsinh}(a x)^2}{2 x^2}+a^2 \int \frac {1}{x} \, dx \\ & = -\frac {a \sqrt {1+a^2 x^2} \text {arcsinh}(a x)}{x}-\frac {\text {arcsinh}(a x)^2}{2 x^2}+a^2 \log (x) \\ \end{align*}
Time = 0.02 (sec) , antiderivative size = 43, normalized size of antiderivative = 1.00 \[ \int \frac {\text {arcsinh}(a x)^2}{x^3} \, dx=-\frac {a \sqrt {1+a^2 x^2} \text {arcsinh}(a x)}{x}-\frac {\text {arcsinh}(a x)^2}{2 x^2}+a^2 \log (x) \]
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Time = 0.06 (sec) , antiderivative size = 72, normalized size of antiderivative = 1.67
method | result | size |
derivativedivides | \(a^{2} \left (-2 \,\operatorname {arcsinh}\left (a x \right )-\frac {\operatorname {arcsinh}\left (a x \right ) \left (-2 a^{2} x^{2}+2 a x \sqrt {a^{2} x^{2}+1}+\operatorname {arcsinh}\left (a x \right )\right )}{2 a^{2} x^{2}}+\ln \left (\left (a x +\sqrt {a^{2} x^{2}+1}\right )^{2}-1\right )\right )\) | \(72\) |
default | \(a^{2} \left (-2 \,\operatorname {arcsinh}\left (a x \right )-\frac {\operatorname {arcsinh}\left (a x \right ) \left (-2 a^{2} x^{2}+2 a x \sqrt {a^{2} x^{2}+1}+\operatorname {arcsinh}\left (a x \right )\right )}{2 a^{2} x^{2}}+\ln \left (\left (a x +\sqrt {a^{2} x^{2}+1}\right )^{2}-1\right )\right )\) | \(72\) |
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Time = 0.25 (sec) , antiderivative size = 67, normalized size of antiderivative = 1.56 \[ \int \frac {\text {arcsinh}(a x)^2}{x^3} \, dx=\frac {2 \, a^{2} x^{2} \log \left (x\right ) - 2 \, \sqrt {a^{2} x^{2} + 1} a x \log \left (a x + \sqrt {a^{2} x^{2} + 1}\right ) - \log \left (a x + \sqrt {a^{2} x^{2} + 1}\right )^{2}}{2 \, x^{2}} \]
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\[ \int \frac {\text {arcsinh}(a x)^2}{x^3} \, dx=\int \frac {\operatorname {asinh}^{2}{\left (a x \right )}}{x^{3}}\, dx \]
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Time = 0.19 (sec) , antiderivative size = 39, normalized size of antiderivative = 0.91 \[ \int \frac {\text {arcsinh}(a x)^2}{x^3} \, dx=a^{2} \log \left (x\right ) - \frac {\sqrt {a^{2} x^{2} + 1} a \operatorname {arsinh}\left (a x\right )}{x} - \frac {\operatorname {arsinh}\left (a x\right )^{2}}{2 \, x^{2}} \]
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Leaf count of result is larger than twice the leaf count of optimal. 98 vs. \(2 (39) = 78\).
Time = 0.30 (sec) , antiderivative size = 98, normalized size of antiderivative = 2.28 \[ \int \frac {\text {arcsinh}(a x)^2}{x^3} \, dx=-{\left (a \log \left (-x {\left | a \right |} + \sqrt {a^{2} x^{2} + 1}\right ) - a \log \left ({\left | x \right |}\right ) - \frac {2 \, {\left | a \right |} \log \left (a x + \sqrt {a^{2} x^{2} + 1}\right )}{{\left (x {\left | a \right |} - \sqrt {a^{2} x^{2} + 1}\right )}^{2} - 1}\right )} a - \frac {\log \left (a x + \sqrt {a^{2} x^{2} + 1}\right )^{2}}{2 \, x^{2}} \]
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Timed out. \[ \int \frac {\text {arcsinh}(a x)^2}{x^3} \, dx=\int \frac {{\mathrm {asinh}\left (a\,x\right )}^2}{x^3} \,d x \]
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