Integrand size = 21, antiderivative size = 52 \[ \int \frac {x \text {arcsinh}(a x)^2}{\sqrt {1+a^2 x^2}} \, dx=\frac {2 \sqrt {1+a^2 x^2}}{a^2}-\frac {2 x \text {arcsinh}(a x)}{a}+\frac {\sqrt {1+a^2 x^2} \text {arcsinh}(a x)^2}{a^2} \]
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Time = 0.05 (sec) , antiderivative size = 52, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.143, Rules used = {5798, 5772, 267} \[ \int \frac {x \text {arcsinh}(a x)^2}{\sqrt {1+a^2 x^2}} \, dx=\frac {\sqrt {a^2 x^2+1} \text {arcsinh}(a x)^2}{a^2}+\frac {2 \sqrt {a^2 x^2+1}}{a^2}-\frac {2 x \text {arcsinh}(a x)}{a} \]
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Rule 267
Rule 5772
Rule 5798
Rubi steps \begin{align*} \text {integral}& = \frac {\sqrt {1+a^2 x^2} \text {arcsinh}(a x)^2}{a^2}-\frac {2 \int \text {arcsinh}(a x) \, dx}{a} \\ & = -\frac {2 x \text {arcsinh}(a x)}{a}+\frac {\sqrt {1+a^2 x^2} \text {arcsinh}(a x)^2}{a^2}+2 \int \frac {x}{\sqrt {1+a^2 x^2}} \, dx \\ & = \frac {2 \sqrt {1+a^2 x^2}}{a^2}-\frac {2 x \text {arcsinh}(a x)}{a}+\frac {\sqrt {1+a^2 x^2} \text {arcsinh}(a x)^2}{a^2} \\ \end{align*}
Time = 0.03 (sec) , antiderivative size = 48, normalized size of antiderivative = 0.92 \[ \int \frac {x \text {arcsinh}(a x)^2}{\sqrt {1+a^2 x^2}} \, dx=\frac {2 \sqrt {1+a^2 x^2}-2 a x \text {arcsinh}(a x)+\sqrt {1+a^2 x^2} \text {arcsinh}(a x)^2}{a^2} \]
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Time = 0.28 (sec) , antiderivative size = 64, normalized size of antiderivative = 1.23
method | result | size |
default | \(\frac {\operatorname {arcsinh}\left (a x \right )^{2} a^{2} x^{2}+\operatorname {arcsinh}\left (a x \right )^{2}-2 \,\operatorname {arcsinh}\left (a x \right ) \sqrt {a^{2} x^{2}+1}\, a x +2 a^{2} x^{2}+2}{a^{2} \sqrt {a^{2} x^{2}+1}}\) | \(64\) |
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Time = 0.25 (sec) , antiderivative size = 70, normalized size of antiderivative = 1.35 \[ \int \frac {x \text {arcsinh}(a x)^2}{\sqrt {1+a^2 x^2}} \, dx=-\frac {2 \, a x \log \left (a x + \sqrt {a^{2} x^{2} + 1}\right ) - \sqrt {a^{2} x^{2} + 1} \log \left (a x + \sqrt {a^{2} x^{2} + 1}\right )^{2} - 2 \, \sqrt {a^{2} x^{2} + 1}}{a^{2}} \]
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Time = 0.33 (sec) , antiderivative size = 49, normalized size of antiderivative = 0.94 \[ \int \frac {x \text {arcsinh}(a x)^2}{\sqrt {1+a^2 x^2}} \, dx=\begin {cases} - \frac {2 x \operatorname {asinh}{\left (a x \right )}}{a} + \frac {\sqrt {a^{2} x^{2} + 1} \operatorname {asinh}^{2}{\left (a x \right )}}{a^{2}} + \frac {2 \sqrt {a^{2} x^{2} + 1}}{a^{2}} & \text {for}\: a \neq 0 \\0 & \text {otherwise} \end {cases} \]
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Time = 0.20 (sec) , antiderivative size = 48, normalized size of antiderivative = 0.92 \[ \int \frac {x \text {arcsinh}(a x)^2}{\sqrt {1+a^2 x^2}} \, dx=\frac {\sqrt {a^{2} x^{2} + 1} \operatorname {arsinh}\left (a x\right )^{2}}{a^{2}} - \frac {2 \, {\left (a x \operatorname {arsinh}\left (a x\right ) - \sqrt {a^{2} x^{2} + 1}\right )}}{a^{2}} \]
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Time = 0.30 (sec) , antiderivative size = 74, normalized size of antiderivative = 1.42 \[ \int \frac {x \text {arcsinh}(a x)^2}{\sqrt {1+a^2 x^2}} \, dx=\frac {\sqrt {a^{2} x^{2} + 1} \log \left (a x + \sqrt {a^{2} x^{2} + 1}\right )^{2}}{a^{2}} - \frac {2 \, {\left (x \log \left (a x + \sqrt {a^{2} x^{2} + 1}\right ) - \frac {\sqrt {a^{2} x^{2} + 1}}{a}\right )}}{a} \]
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Timed out. \[ \int \frac {x \text {arcsinh}(a x)^2}{\sqrt {1+a^2 x^2}} \, dx=\int \frac {x\,{\mathrm {asinh}\left (a\,x\right )}^2}{\sqrt {a^2\,x^2+1}} \,d x \]
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