Integrand size = 10, antiderivative size = 33 \[ \int \frac {\text {arcsinh}\left (a x^2\right )}{x^3} \, dx=-\frac {\text {arcsinh}\left (a x^2\right )}{2 x^2}-\frac {1}{2} a \text {arctanh}\left (\sqrt {1+a^2 x^4}\right ) \]
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Time = 0.02 (sec) , antiderivative size = 33, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.500, Rules used = {5875, 12, 272, 65, 214} \[ \int \frac {\text {arcsinh}\left (a x^2\right )}{x^3} \, dx=-\frac {1}{2} a \text {arctanh}\left (\sqrt {a^2 x^4+1}\right )-\frac {\text {arcsinh}\left (a x^2\right )}{2 x^2} \]
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Rule 12
Rule 65
Rule 214
Rule 272
Rule 5875
Rubi steps \begin{align*} \text {integral}& = -\frac {\text {arcsinh}\left (a x^2\right )}{2 x^2}+\frac {1}{2} \int \frac {2 a}{x \sqrt {1+a^2 x^4}} \, dx \\ & = -\frac {\text {arcsinh}\left (a x^2\right )}{2 x^2}+a \int \frac {1}{x \sqrt {1+a^2 x^4}} \, dx \\ & = -\frac {\text {arcsinh}\left (a x^2\right )}{2 x^2}+\frac {1}{4} a \text {Subst}\left (\int \frac {1}{x \sqrt {1+a^2 x}} \, dx,x,x^4\right ) \\ & = -\frac {\text {arcsinh}\left (a x^2\right )}{2 x^2}+\frac {\text {Subst}\left (\int \frac {1}{-\frac {1}{a^2}+\frac {x^2}{a^2}} \, dx,x,\sqrt {1+a^2 x^4}\right )}{2 a} \\ & = -\frac {\text {arcsinh}\left (a x^2\right )}{2 x^2}-\frac {1}{2} a \text {arctanh}\left (\sqrt {1+a^2 x^4}\right ) \\ \end{align*}
Time = 0.01 (sec) , antiderivative size = 33, normalized size of antiderivative = 1.00 \[ \int \frac {\text {arcsinh}\left (a x^2\right )}{x^3} \, dx=-\frac {\text {arcsinh}\left (a x^2\right )}{2 x^2}-\frac {1}{2} a \text {arctanh}\left (\sqrt {1+a^2 x^4}\right ) \]
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Time = 0.01 (sec) , antiderivative size = 28, normalized size of antiderivative = 0.85
method | result | size |
default | \(-\frac {\operatorname {arcsinh}\left (a \,x^{2}\right )}{2 x^{2}}-\frac {a \,\operatorname {arctanh}\left (\frac {1}{\sqrt {a^{2} x^{4}+1}}\right )}{2}\) | \(28\) |
parts | \(-\frac {\operatorname {arcsinh}\left (a \,x^{2}\right )}{2 x^{2}}-\frac {a \,\operatorname {arctanh}\left (\frac {1}{\sqrt {a^{2} x^{4}+1}}\right )}{2}\) | \(28\) |
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Leaf count of result is larger than twice the leaf count of optimal. 106 vs. \(2 (27) = 54\).
Time = 0.26 (sec) , antiderivative size = 106, normalized size of antiderivative = 3.21 \[ \int \frac {\text {arcsinh}\left (a x^2\right )}{x^3} \, dx=-\frac {a x^{2} \log \left (-a x^{2} + \sqrt {a^{2} x^{4} + 1} + 1\right ) - a x^{2} \log \left (-a x^{2} + \sqrt {a^{2} x^{4} + 1} - 1\right ) - x^{2} \log \left (-a x^{2} + \sqrt {a^{2} x^{4} + 1}\right ) - {\left (x^{2} - 1\right )} \log \left (a x^{2} + \sqrt {a^{2} x^{4} + 1}\right )}{2 \, x^{2}} \]
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\[ \int \frac {\text {arcsinh}\left (a x^2\right )}{x^3} \, dx=\int \frac {\operatorname {asinh}{\left (a x^{2} \right )}}{x^{3}}\, dx \]
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none
Time = 0.23 (sec) , antiderivative size = 46, normalized size of antiderivative = 1.39 \[ \int \frac {\text {arcsinh}\left (a x^2\right )}{x^3} \, dx=-\frac {1}{4} \, a {\left (\log \left (\sqrt {a^{2} x^{4} + 1} + 1\right ) - \log \left (\sqrt {a^{2} x^{4} + 1} - 1\right )\right )} - \frac {\operatorname {arsinh}\left (a x^{2}\right )}{2 \, x^{2}} \]
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Leaf count of result is larger than twice the leaf count of optimal. 58 vs. \(2 (27) = 54\).
Time = 0.28 (sec) , antiderivative size = 58, normalized size of antiderivative = 1.76 \[ \int \frac {\text {arcsinh}\left (a x^2\right )}{x^3} \, dx=-\frac {1}{4} \, a {\left (\log \left (\sqrt {a^{2} x^{4} + 1} + 1\right ) - \log \left (\sqrt {a^{2} x^{4} + 1} - 1\right )\right )} - \frac {\log \left (a x^{2} + \sqrt {a^{2} x^{4} + 1}\right )}{2 \, x^{2}} \]
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Timed out. \[ \int \frac {\text {arcsinh}\left (a x^2\right )}{x^3} \, dx=\int \frac {\mathrm {asinh}\left (a\,x^2\right )}{x^3} \,d x \]
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