Integrand size = 10, antiderivative size = 197 \[ \int \frac {\text {arcsinh}\left (a x^2\right )}{x^4} \, dx=-\frac {2 a \sqrt {1+a^2 x^4}}{3 x}+\frac {2 a^2 x \sqrt {1+a^2 x^4}}{3 \left (1+a x^2\right )}-\frac {\text {arcsinh}\left (a x^2\right )}{3 x^3}-\frac {2 a^{3/2} \left (1+a x^2\right ) \sqrt {\frac {1+a^2 x^4}{\left (1+a x^2\right )^2}} E\left (2 \arctan \left (\sqrt {a} x\right )|\frac {1}{2}\right )}{3 \sqrt {1+a^2 x^4}}+\frac {a^{3/2} \left (1+a x^2\right ) \sqrt {\frac {1+a^2 x^4}{\left (1+a x^2\right )^2}} \operatorname {EllipticF}\left (2 \arctan \left (\sqrt {a} x\right ),\frac {1}{2}\right )}{3 \sqrt {1+a^2 x^4}} \]
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Time = 0.08 (sec) , antiderivative size = 197, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.600, Rules used = {5875, 12, 331, 311, 226, 1210} \[ \int \frac {\text {arcsinh}\left (a x^2\right )}{x^4} \, dx=-\frac {2 a \sqrt {a^2 x^4+1}}{3 x}+\frac {2 a^2 x \sqrt {a^2 x^4+1}}{3 \left (a x^2+1\right )}+\frac {a^{3/2} \left (a x^2+1\right ) \sqrt {\frac {a^2 x^4+1}{\left (a x^2+1\right )^2}} \operatorname {EllipticF}\left (2 \arctan \left (\sqrt {a} x\right ),\frac {1}{2}\right )}{3 \sqrt {a^2 x^4+1}}-\frac {2 a^{3/2} \left (a x^2+1\right ) \sqrt {\frac {a^2 x^4+1}{\left (a x^2+1\right )^2}} E\left (2 \arctan \left (\sqrt {a} x\right )|\frac {1}{2}\right )}{3 \sqrt {a^2 x^4+1}}-\frac {\text {arcsinh}\left (a x^2\right )}{3 x^3} \]
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Rule 12
Rule 226
Rule 311
Rule 331
Rule 1210
Rule 5875
Rubi steps \begin{align*} \text {integral}& = -\frac {\text {arcsinh}\left (a x^2\right )}{3 x^3}+\frac {1}{3} \int \frac {2 a}{x^2 \sqrt {1+a^2 x^4}} \, dx \\ & = -\frac {\text {arcsinh}\left (a x^2\right )}{3 x^3}+\frac {1}{3} (2 a) \int \frac {1}{x^2 \sqrt {1+a^2 x^4}} \, dx \\ & = -\frac {2 a \sqrt {1+a^2 x^4}}{3 x}-\frac {\text {arcsinh}\left (a x^2\right )}{3 x^3}+\frac {1}{3} \left (2 a^3\right ) \int \frac {x^2}{\sqrt {1+a^2 x^4}} \, dx \\ & = -\frac {2 a \sqrt {1+a^2 x^4}}{3 x}-\frac {\text {arcsinh}\left (a x^2\right )}{3 x^3}+\frac {1}{3} \left (2 a^2\right ) \int \frac {1}{\sqrt {1+a^2 x^4}} \, dx-\frac {1}{3} \left (2 a^2\right ) \int \frac {1-a x^2}{\sqrt {1+a^2 x^4}} \, dx \\ & = -\frac {2 a \sqrt {1+a^2 x^4}}{3 x}+\frac {2 a^2 x \sqrt {1+a^2 x^4}}{3 \left (1+a x^2\right )}-\frac {\text {arcsinh}\left (a x^2\right )}{3 x^3}-\frac {2 a^{3/2} \left (1+a x^2\right ) \sqrt {\frac {1+a^2 x^4}{\left (1+a x^2\right )^2}} E\left (2 \arctan \left (\sqrt {a} x\right )|\frac {1}{2}\right )}{3 \sqrt {1+a^2 x^4}}+\frac {a^{3/2} \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 )}{3 \sqrt {1+a^2 x^4}} \\ \end{align*}
Result contains complex when optimal does not.
Time = 0.11 (sec) , antiderivative size = 88, normalized size of antiderivative = 0.45 \[ \int \frac {\text {arcsinh}\left (a x^2\right )}{x^4} \, dx=\frac {1}{3} \left (-\frac {2 a \sqrt {1+a^2 x^4}}{x}-\frac {\text {arcsinh}\left (a x^2\right )}{x^3}+\frac {2 a^2 \left (E\left (\left .i \text {arcsinh}\left (\sqrt {i a} x\right )\right |-1\right )-\operatorname {EllipticF}\left (i \text {arcsinh}\left (\sqrt {i a} x\right ),-1\right )\right )}{\sqrt {i a}}\right ) \]
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Result contains complex when optimal does not.
Time = 0.14 (sec) , antiderivative size = 101, normalized size of antiderivative = 0.51
method | result | size |
default | \(-\frac {\operatorname {arcsinh}\left (a \,x^{2}\right )}{3 x^{3}}+\frac {2 a \left (-\frac {\sqrt {a^{2} x^{4}+1}}{x}+\frac {i a \sqrt {-i a \,x^{2}+1}\, \sqrt {i a \,x^{2}+1}\, \left (\operatorname {EllipticF}\left (x \sqrt {i a}, i\right )-\operatorname {EllipticE}\left (x \sqrt {i a}, i\right )\right )}{\sqrt {i a}\, \sqrt {a^{2} x^{4}+1}}\right )}{3}\) | \(101\) |
parts | \(-\frac {\operatorname {arcsinh}\left (a \,x^{2}\right )}{3 x^{3}}+\frac {2 a \left (-\frac {\sqrt {a^{2} x^{4}+1}}{x}+\frac {i a \sqrt {-i a \,x^{2}+1}\, \sqrt {i a \,x^{2}+1}\, \left (\operatorname {EllipticF}\left (x \sqrt {i a}, i\right )-\operatorname {EllipticE}\left (x \sqrt {i a}, i\right )\right )}{\sqrt {i a}\, \sqrt {a^{2} x^{4}+1}}\right )}{3}\) | \(101\) |
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\[ \int \frac {\text {arcsinh}\left (a x^2\right )}{x^4} \, dx=\int { \frac {\operatorname {arsinh}\left (a x^{2}\right )}{x^{4}} \,d x } \]
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\[ \int \frac {\text {arcsinh}\left (a x^2\right )}{x^4} \, dx=\int \frac {\operatorname {asinh}{\left (a x^{2} \right )}}{x^{4}}\, dx \]
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\[ \int \frac {\text {arcsinh}\left (a x^2\right )}{x^4} \, dx=\int { \frac {\operatorname {arsinh}\left (a x^{2}\right )}{x^{4}} \,d x } \]
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\[ \int \frac {\text {arcsinh}\left (a x^2\right )}{x^4} \, dx=\int { \frac {\operatorname {arsinh}\left (a x^{2}\right )}{x^{4}} \,d x } \]
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Timed out. \[ \int \frac {\text {arcsinh}\left (a x^2\right )}{x^4} \, dx=\int \frac {\mathrm {asinh}\left (a\,x^2\right )}{x^4} \,d x \]
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