Integrand size = 12, antiderivative size = 32 \[ \int \sqrt {1+x^2} \text {arcsinh}(x) \, dx=-\frac {x^2}{4}+\frac {1}{2} x \sqrt {1+x^2} \text {arcsinh}(x)+\frac {\text {arcsinh}(x)^2}{4} \]
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Time = 0.02 (sec) , antiderivative size = 32, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {5785, 5783, 30} \[ \int \sqrt {1+x^2} \text {arcsinh}(x) \, dx=\frac {1}{2} \sqrt {x^2+1} x \text {arcsinh}(x)+\frac {\text {arcsinh}(x)^2}{4}-\frac {x^2}{4} \]
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Rule 30
Rule 5783
Rule 5785
Rubi steps \begin{align*} \text {integral}& = \frac {1}{2} x \sqrt {1+x^2} \text {arcsinh}(x)-\frac {\int x \, dx}{2}+\frac {1}{2} \int \frac {\text {arcsinh}(x)}{\sqrt {1+x^2}} \, dx \\ & = -\frac {x^2}{4}+\frac {1}{2} x \sqrt {1+x^2} \text {arcsinh}(x)+\frac {\text {arcsinh}(x)^2}{4} \\ \end{align*}
Time = 0.01 (sec) , antiderivative size = 28, normalized size of antiderivative = 0.88 \[ \int \sqrt {1+x^2} \text {arcsinh}(x) \, dx=\frac {1}{4} \left (-x^2+2 x \sqrt {1+x^2} \text {arcsinh}(x)+\text {arcsinh}(x)^2\right ) \]
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Time = 0.23 (sec) , antiderivative size = 26, normalized size of antiderivative = 0.81
method | result | size |
default | \(\frac {x \,\operatorname {arcsinh}\left (x \right ) \sqrt {x^{2}+1}}{2}+\frac {\operatorname {arcsinh}\left (x \right )^{2}}{4}-\frac {x^{2}}{4}-\frac {1}{4}\) | \(26\) |
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Time = 0.26 (sec) , antiderivative size = 40, normalized size of antiderivative = 1.25 \[ \int \sqrt {1+x^2} \text {arcsinh}(x) \, dx=\frac {1}{2} \, \sqrt {x^{2} + 1} x \log \left (x + \sqrt {x^{2} + 1}\right ) - \frac {1}{4} \, x^{2} + \frac {1}{4} \, \log \left (x + \sqrt {x^{2} + 1}\right )^{2} \]
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\[ \int \sqrt {1+x^2} \text {arcsinh}(x) \, dx=\int \sqrt {x^{2} + 1} \operatorname {asinh}{\left (x \right )}\, dx \]
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Time = 0.28 (sec) , antiderivative size = 28, normalized size of antiderivative = 0.88 \[ \int \sqrt {1+x^2} \text {arcsinh}(x) \, dx=-\frac {1}{4} \, x^{2} + \frac {1}{2} \, {\left (\sqrt {x^{2} + 1} x + \operatorname {arsinh}\left (x\right )\right )} \operatorname {arsinh}\left (x\right ) - \frac {1}{4} \, \operatorname {arsinh}\left (x\right )^{2} \]
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\[ \int \sqrt {1+x^2} \text {arcsinh}(x) \, dx=\int { \sqrt {x^{2} + 1} \operatorname {arsinh}\left (x\right ) \,d x } \]
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Timed out. \[ \int \sqrt {1+x^2} \text {arcsinh}(x) \, dx=\int \mathrm {asinh}\left (x\right )\,\sqrt {x^2+1} \,d x \]
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