Integrand size = 19, antiderivative size = 28 \[ \int \frac {x \text {arcsinh}(a x)}{\sqrt {1+a^2 x^2}} \, dx=-\frac {x}{a}+\frac {\sqrt {1+a^2 x^2} \text {arcsinh}(a x)}{a^2} \]
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Time = 0.03 (sec) , antiderivative size = 28, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.105, Rules used = {5798, 8} \[ \int \frac {x \text {arcsinh}(a x)}{\sqrt {1+a^2 x^2}} \, dx=\frac {\sqrt {a^2 x^2+1} \text {arcsinh}(a x)}{a^2}-\frac {x}{a} \]
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Rule 8
Rule 5798
Rubi steps \begin{align*} \text {integral}& = \frac {\sqrt {1+a^2 x^2} \text {arcsinh}(a x)}{a^2}-\frac {\int 1 \, dx}{a} \\ & = -\frac {x}{a}+\frac {\sqrt {1+a^2 x^2} \text {arcsinh}(a x)}{a^2} \\ \end{align*}
Time = 0.04 (sec) , antiderivative size = 28, normalized size of antiderivative = 1.00 \[ \int \frac {x \text {arcsinh}(a x)}{\sqrt {1+a^2 x^2}} \, dx=-\frac {x}{a}+\frac {\sqrt {1+a^2 x^2} \text {arcsinh}(a x)}{a^2} \]
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Time = 0.22 (sec) , antiderivative size = 47, normalized size of antiderivative = 1.68
method | result | size |
default | \(\frac {a^{2} x^{2} \operatorname {arcsinh}\left (a x \right )+\operatorname {arcsinh}\left (a x \right )-a x \sqrt {a^{2} x^{2}+1}}{a^{2} \sqrt {a^{2} x^{2}+1}}\) | \(47\) |
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Time = 0.27 (sec) , antiderivative size = 38, normalized size of antiderivative = 1.36 \[ \int \frac {x \text {arcsinh}(a x)}{\sqrt {1+a^2 x^2}} \, dx=-\frac {a x - \sqrt {a^{2} x^{2} + 1} \log \left (a x + \sqrt {a^{2} x^{2} + 1}\right )}{a^{2}} \]
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Time = 0.24 (sec) , antiderivative size = 24, normalized size of antiderivative = 0.86 \[ \int \frac {x \text {arcsinh}(a x)}{\sqrt {1+a^2 x^2}} \, dx=\begin {cases} - \frac {x}{a} + \frac {\sqrt {a^{2} x^{2} + 1} \operatorname {asinh}{\left (a x \right )}}{a^{2}} & \text {for}\: a \neq 0 \\0 & \text {otherwise} \end {cases} \]
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Time = 0.19 (sec) , antiderivative size = 26, normalized size of antiderivative = 0.93 \[ \int \frac {x \text {arcsinh}(a x)}{\sqrt {1+a^2 x^2}} \, dx=-\frac {x}{a} + \frac {\sqrt {a^{2} x^{2} + 1} \operatorname {arsinh}\left (a x\right )}{a^{2}} \]
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Time = 0.29 (sec) , antiderivative size = 38, normalized size of antiderivative = 1.36 \[ \int \frac {x \text {arcsinh}(a x)}{\sqrt {1+a^2 x^2}} \, dx=-\frac {x}{a} + \frac {\sqrt {a^{2} x^{2} + 1} \log \left (a x + \sqrt {a^{2} x^{2} + 1}\right )}{a^{2}} \]
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Timed out. \[ \int \frac {x \text {arcsinh}(a x)}{\sqrt {1+a^2 x^2}} \, dx=\int \frac {x\,\mathrm {asinh}\left (a\,x\right )}{\sqrt {a^2\,x^2+1}} \,d x \]
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