Integrand size = 10, antiderivative size = 75 \[ \int \frac {\text {arcsinh}\left (a x^2\right )}{x^2} \, dx=-\frac {\text {arcsinh}\left (a x^2\right )}{x}+\frac {\sqrt {a} \left (1+a x^2\right ) \sqrt {\frac {1+a^2 x^4}{\left (1+a x^2\right )^2}} \operatorname {EllipticF}\left (2 \arctan \left (\sqrt {a} x\right ),\frac {1}{2}\right )}{\sqrt {1+a^2 x^4}} \]
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Time = 0.02 (sec) , antiderivative size = 75, 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 = {5875, 12, 226} \[ \int \frac {\text {arcsinh}\left (a x^2\right )}{x^2} \, dx=\frac {\sqrt {a} \left (a x^2+1\right ) \sqrt {\frac {a^2 x^4+1}{\left (a x^2+1\right )^2}} \operatorname {EllipticF}\left (2 \arctan \left (\sqrt {a} x\right ),\frac {1}{2}\right )}{\sqrt {a^2 x^4+1}}-\frac {\text {arcsinh}\left (a x^2\right )}{x} \]
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Rule 12
Rule 226
Rule 5875
Rubi steps \begin{align*} \text {integral}& = -\frac {\text {arcsinh}\left (a x^2\right )}{x}+\int \frac {2 a}{\sqrt {1+a^2 x^4}} \, dx \\ & = -\frac {\text {arcsinh}\left (a x^2\right )}{x}+(2 a) \int \frac {1}{\sqrt {1+a^2 x^4}} \, dx \\ & = -\frac {\text {arcsinh}\left (a x^2\right )}{x}+\frac {\sqrt {a} \left (1+a x^2\right ) \sqrt {\frac {1+a^2 x^4}{\left (1+a x^2\right )^2}} \operatorname {EllipticF}\left (2 \arctan \left (\sqrt {a} x\right ),\frac {1}{2}\right )}{\sqrt {1+a^2 x^4}} \\ \end{align*}
Result contains complex when optimal does not.
Time = 0.03 (sec) , antiderivative size = 42, normalized size of antiderivative = 0.56 \[ \int \frac {\text {arcsinh}\left (a x^2\right )}{x^2} \, dx=-\frac {\text {arcsinh}\left (a x^2\right )+2 \sqrt {i a} x \operatorname {EllipticF}\left (i \text {arcsinh}\left (\sqrt {i a} x\right ),-1\right )}{x} \]
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Result contains complex when optimal does not.
Time = 0.08 (sec) , antiderivative size = 66, normalized size of antiderivative = 0.88
method | result | size |
default | \(-\frac {\operatorname {arcsinh}\left (a \,x^{2}\right )}{x}+\frac {2 a \sqrt {-i a \,x^{2}+1}\, \sqrt {i a \,x^{2}+1}\, \operatorname {EllipticF}\left (x \sqrt {i a}, i\right )}{\sqrt {i a}\, \sqrt {a^{2} x^{4}+1}}\) | \(66\) |
parts | \(-\frac {\operatorname {arcsinh}\left (a \,x^{2}\right )}{x}+\frac {2 a \sqrt {-i a \,x^{2}+1}\, \sqrt {i a \,x^{2}+1}\, \operatorname {EllipticF}\left (x \sqrt {i a}, i\right )}{\sqrt {i a}\, \sqrt {a^{2} x^{4}+1}}\) | \(66\) |
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none
Time = 0.10 (sec) , antiderivative size = 76, normalized size of antiderivative = 1.01 \[ \int \frac {\text {arcsinh}\left (a x^2\right )}{x^2} \, dx=\frac {2 \, a^{2} x \left (-\frac {1}{a^{2}}\right )^{\frac {3}{4}} F(\arcsin \left (\frac {\left (-\frac {1}{a^{2}}\right )^{\frac {1}{4}}}{x}\right )\,|\,-1) + {\left (x - 1\right )} \log \left (a x^{2} + \sqrt {a^{2} x^{4} + 1}\right ) + x \log \left (a x^{2} - \sqrt {a^{2} x^{4} + 1}\right )}{x} \]
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\[ \int \frac {\text {arcsinh}\left (a x^2\right )}{x^2} \, dx=\int \frac {\operatorname {asinh}{\left (a x^{2} \right )}}{x^{2}}\, dx \]
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\[ \int \frac {\text {arcsinh}\left (a x^2\right )}{x^2} \, dx=\int { \frac {\operatorname {arsinh}\left (a x^{2}\right )}{x^{2}} \,d x } \]
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\[ \int \frac {\text {arcsinh}\left (a x^2\right )}{x^2} \, dx=\int { \frac {\operatorname {arsinh}\left (a x^{2}\right )}{x^{2}} \,d x } \]
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Timed out. \[ \int \frac {\text {arcsinh}\left (a x^2\right )}{x^2} \, dx=\int \frac {\mathrm {asinh}\left (a\,x^2\right )}{x^2} \,d x \]
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