Integrand size = 10, antiderivative size = 101 \[ \int x^2 \text {arcsinh}\left (a x^2\right ) \, dx=-\frac {2 x \sqrt {1+a^2 x^4}}{9 a}+\frac {1}{3} x^3 \text {arcsinh}\left (a x^2\right )+\frac {\left (1+a x^2\right ) \sqrt {\frac {1+a^2 x^4}{\left (1+a x^2\right )^2}} \operatorname {EllipticF}\left (2 \arctan \left (\sqrt {a} x\right ),\frac {1}{2}\right )}{9 a^{3/2} \sqrt {1+a^2 x^4}} \]
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Time = 0.04 (sec) , antiderivative size = 101, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.400, Rules used = {5875, 12, 327, 226} \[ \int x^2 \text {arcsinh}\left (a x^2\right ) \, dx=-\frac {2 x \sqrt {a^2 x^4+1}}{9 a}+\frac {\left (a x^2+1\right ) \sqrt {\frac {a^2 x^4+1}{\left (a x^2+1\right )^2}} \operatorname {EllipticF}\left (2 \arctan \left (\sqrt {a} x\right ),\frac {1}{2}\right )}{9 a^{3/2} \sqrt {a^2 x^4+1}}+\frac {1}{3} x^3 \text {arcsinh}\left (a x^2\right ) \]
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Rule 12
Rule 226
Rule 327
Rule 5875
Rubi steps \begin{align*} \text {integral}& = \frac {1}{3} x^3 \text {arcsinh}\left (a x^2\right )-\frac {1}{3} \int \frac {2 a x^4}{\sqrt {1+a^2 x^4}} \, dx \\ & = \frac {1}{3} x^3 \text {arcsinh}\left (a x^2\right )-\frac {1}{3} (2 a) \int \frac {x^4}{\sqrt {1+a^2 x^4}} \, dx \\ & = -\frac {2 x \sqrt {1+a^2 x^4}}{9 a}+\frac {1}{3} x^3 \text {arcsinh}\left (a x^2\right )+\frac {2 \int \frac {1}{\sqrt {1+a^2 x^4}} \, dx}{9 a} \\ & = -\frac {2 x \sqrt {1+a^2 x^4}}{9 a}+\frac {1}{3} x^3 \text {arcsinh}\left (a x^2\right )+\frac {\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 )}{9 a^{3/2} \sqrt {1+a^2 x^4}} \\ \end{align*}
Result contains complex when optimal does not.
Time = 0.09 (sec) , antiderivative size = 75, normalized size of antiderivative = 0.74 \[ \int x^2 \text {arcsinh}\left (a x^2\right ) \, dx=\frac {1}{9} \left (-\frac {2 \left (x+a^2 x^5\right )}{a \sqrt {1+a^2 x^4}}+3 x^3 \text {arcsinh}\left (a x^2\right )-\frac {2 \sqrt {i a} \operatorname {EllipticF}\left (i \text {arcsinh}\left (\sqrt {i a} x\right ),-1\right )}{a^2}\right ) \]
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Result contains complex when optimal does not.
Time = 0.45 (sec) , antiderivative size = 89, normalized size of antiderivative = 0.88
method | result | size |
default | \(\frac {x^{3} \operatorname {arcsinh}\left (a \,x^{2}\right )}{3}-\frac {2 a \left (\frac {x \sqrt {a^{2} x^{4}+1}}{3 a^{2}}-\frac {\sqrt {-i a \,x^{2}+1}\, \sqrt {i a \,x^{2}+1}\, \operatorname {EllipticF}\left (x \sqrt {i a}, i\right )}{3 a^{2} \sqrt {i a}\, \sqrt {a^{2} x^{4}+1}}\right )}{3}\) | \(89\) |
parts | \(\frac {x^{3} \operatorname {arcsinh}\left (a \,x^{2}\right )}{3}-\frac {2 a \left (\frac {x \sqrt {a^{2} x^{4}+1}}{3 a^{2}}-\frac {\sqrt {-i a \,x^{2}+1}\, \sqrt {i a \,x^{2}+1}\, \operatorname {EllipticF}\left (x \sqrt {i a}, i\right )}{3 a^{2} \sqrt {i a}\, \sqrt {a^{2} x^{4}+1}}\right )}{3}\) | \(89\) |
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Time = 0.09 (sec) , antiderivative size = 68, normalized size of antiderivative = 0.67 \[ \int x^2 \text {arcsinh}\left (a x^2\right ) \, dx=\frac {3 \, a x^{3} \log \left (a x^{2} + \sqrt {a^{2} x^{4} + 1}\right ) + 2 \, a \left (-\frac {1}{a^{2}}\right )^{\frac {3}{4}} F(\arcsin \left (\frac {\left (-\frac {1}{a^{2}}\right )^{\frac {1}{4}}}{x}\right )\,|\,-1) - 2 \, \sqrt {a^{2} x^{4} + 1} x}{9 \, a} \]
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\[ \int x^2 \text {arcsinh}\left (a x^2\right ) \, dx=\int x^{2} \operatorname {asinh}{\left (a x^{2} \right )}\, dx \]
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\[ \int x^2 \text {arcsinh}\left (a x^2\right ) \, dx=\int { x^{2} \operatorname {arsinh}\left (a x^{2}\right ) \,d x } \]
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\[ \int x^2 \text {arcsinh}\left (a x^2\right ) \, dx=\int { x^{2} \operatorname {arsinh}\left (a x^{2}\right ) \,d x } \]
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Timed out. \[ \int x^2 \text {arcsinh}\left (a x^2\right ) \, dx=\int x^2\,\mathrm {asinh}\left (a\,x^2\right ) \,d x \]
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