Integrand size = 6, antiderivative size = 34 \[ \int \text {arcsinh}(a x)^2 \, dx=2 x-\frac {2 \sqrt {1+a^2 x^2} \text {arcsinh}(a x)}{a}+x \text {arcsinh}(a x)^2 \]
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Time = 0.03 (sec) , antiderivative size = 34, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.500, Rules used = {5772, 5798, 8} \[ \int \text {arcsinh}(a x)^2 \, dx=-\frac {2 \sqrt {a^2 x^2+1} \text {arcsinh}(a x)}{a}+x \text {arcsinh}(a x)^2+2 x \]
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Rule 8
Rule 5772
Rule 5798
Rubi steps \begin{align*} \text {integral}& = x \text {arcsinh}(a x)^2-(2 a) \int \frac {x \text {arcsinh}(a x)}{\sqrt {1+a^2 x^2}} \, dx \\ & = -\frac {2 \sqrt {1+a^2 x^2} \text {arcsinh}(a x)}{a}+x \text {arcsinh}(a x)^2+2 \int 1 \, dx \\ & = 2 x-\frac {2 \sqrt {1+a^2 x^2} \text {arcsinh}(a x)}{a}+x \text {arcsinh}(a x)^2 \\ \end{align*}
Time = 0.02 (sec) , antiderivative size = 34, normalized size of antiderivative = 1.00 \[ \int \text {arcsinh}(a x)^2 \, dx=2 x-\frac {2 \sqrt {1+a^2 x^2} \text {arcsinh}(a x)}{a}+x \text {arcsinh}(a x)^2 \]
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Time = 0.03 (sec) , antiderivative size = 36, normalized size of antiderivative = 1.06
method | result | size |
derivativedivides | \(\frac {a x \operatorname {arcsinh}\left (a x \right )^{2}-2 \,\operatorname {arcsinh}\left (a x \right ) \sqrt {a^{2} x^{2}+1}+2 a x}{a}\) | \(36\) |
default | \(\frac {a x \operatorname {arcsinh}\left (a x \right )^{2}-2 \,\operatorname {arcsinh}\left (a x \right ) \sqrt {a^{2} x^{2}+1}+2 a x}{a}\) | \(36\) |
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Time = 0.25 (sec) , antiderivative size = 59, normalized size of antiderivative = 1.74 \[ \int \text {arcsinh}(a x)^2 \, dx=\frac {a x \log \left (a x + \sqrt {a^{2} x^{2} + 1}\right )^{2} + 2 \, a x - 2 \, \sqrt {a^{2} x^{2} + 1} \log \left (a x + \sqrt {a^{2} x^{2} + 1}\right )}{a} \]
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Time = 0.10 (sec) , antiderivative size = 32, normalized size of antiderivative = 0.94 \[ \int \text {arcsinh}(a x)^2 \, dx=\begin {cases} x \operatorname {asinh}^{2}{\left (a x \right )} + 2 x - \frac {2 \sqrt {a^{2} x^{2} + 1} \operatorname {asinh}{\left (a x \right )}}{a} & \text {for}\: a \neq 0 \\0 & \text {otherwise} \end {cases} \]
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Time = 0.20 (sec) , antiderivative size = 32, normalized size of antiderivative = 0.94 \[ \int \text {arcsinh}(a x)^2 \, dx=x \operatorname {arsinh}\left (a x\right )^{2} + 2 \, x - \frac {2 \, \sqrt {a^{2} x^{2} + 1} \operatorname {arsinh}\left (a x\right )}{a} \]
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Time = 0.27 (sec) , antiderivative size = 62, normalized size of antiderivative = 1.82 \[ \int \text {arcsinh}(a x)^2 \, dx=x \log \left (a x + \sqrt {a^{2} x^{2} + 1}\right )^{2} + 2 \, a {\left (\frac {x}{a} - \frac {\sqrt {a^{2} x^{2} + 1} \log \left (a x + \sqrt {a^{2} x^{2} + 1}\right )}{a^{2}}\right )} \]
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Timed out. \[ \int \text {arcsinh}(a x)^2 \, dx=\int {\mathrm {asinh}\left (a\,x\right )}^2 \,d x \]
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