Integrand size = 20, antiderivative size = 17 \[ \int \frac {\text {arcsinh}(a x)^n}{\sqrt {1+a^2 x^2}} \, dx=\frac {\text {arcsinh}(a x)^{1+n}}{a (1+n)} \]
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Time = 0.03 (sec) , antiderivative size = 17, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.050, Rules used = {5783} \[ \int \frac {\text {arcsinh}(a x)^n}{\sqrt {1+a^2 x^2}} \, dx=\frac {\text {arcsinh}(a x)^{n+1}}{a (n+1)} \]
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Rule 5783
Rubi steps \begin{align*} \text {integral}& = \frac {\text {arcsinh}(a x)^{1+n}}{a (1+n)} \\ \end{align*}
Time = 0.01 (sec) , antiderivative size = 17, normalized size of antiderivative = 1.00 \[ \int \frac {\text {arcsinh}(a x)^n}{\sqrt {1+a^2 x^2}} \, dx=\frac {\text {arcsinh}(a x)^{1+n}}{a (1+n)} \]
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Time = 0.27 (sec) , antiderivative size = 18, normalized size of antiderivative = 1.06
method | result | size |
derivativedivides | \(\frac {\operatorname {arcsinh}\left (a x \right )^{1+n}}{a \left (1+n \right )}\) | \(18\) |
default | \(\frac {\operatorname {arcsinh}\left (a x \right )^{1+n}}{a \left (1+n \right )}\) | \(18\) |
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Leaf count of result is larger than twice the leaf count of optimal. 83 vs. \(2 (17) = 34\).
Time = 0.27 (sec) , antiderivative size = 83, normalized size of antiderivative = 4.88 \[ \int \frac {\text {arcsinh}(a x)^n}{\sqrt {1+a^2 x^2}} \, dx=\frac {\cosh \left (n \log \left (\log \left (a x + \sqrt {a^{2} x^{2} + 1}\right )\right )\right ) \log \left (a x + \sqrt {a^{2} x^{2} + 1}\right ) + \log \left (a x + \sqrt {a^{2} x^{2} + 1}\right ) \sinh \left (n \log \left (\log \left (a x + \sqrt {a^{2} x^{2} + 1}\right )\right )\right )}{a n + a} \]
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Leaf count of result is larger than twice the leaf count of optimal. 34 vs. \(2 (12) = 24\).
Time = 0.36 (sec) , antiderivative size = 34, normalized size of antiderivative = 2.00 \[ \int \frac {\text {arcsinh}(a x)^n}{\sqrt {1+a^2 x^2}} \, dx=\begin {cases} \tilde {\infty } x & \text {for}\: a = 0 \wedge n = -1 \\0^{n} x & \text {for}\: a = 0 \\\frac {\log {\left (\operatorname {asinh}{\left (a x \right )} \right )}}{a} & \text {for}\: n = -1 \\\frac {\operatorname {asinh}{\left (a x \right )} \operatorname {asinh}^{n}{\left (a x \right )}}{a n + a} & \text {otherwise} \end {cases} \]
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none
Time = 0.19 (sec) , antiderivative size = 17, normalized size of antiderivative = 1.00 \[ \int \frac {\text {arcsinh}(a x)^n}{\sqrt {1+a^2 x^2}} \, dx=\frac {\operatorname {arsinh}\left (a x\right )^{n + 1}}{a {\left (n + 1\right )}} \]
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none
Time = 0.28 (sec) , antiderivative size = 29, normalized size of antiderivative = 1.71 \[ \int \frac {\text {arcsinh}(a x)^n}{\sqrt {1+a^2 x^2}} \, dx=\frac {\log \left (a x + \sqrt {a^{2} x^{2} + 1}\right )^{n + 1}}{a {\left (n + 1\right )}} \]
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Time = 2.93 (sec) , antiderivative size = 33, normalized size of antiderivative = 1.94 \[ \int \frac {\text {arcsinh}(a x)^n}{\sqrt {1+a^2 x^2}} \, dx=\left \{\begin {array}{cl} \frac {\ln \left (\mathrm {asinh}\left (a\,x\right )\right )}{a} & \text {\ if\ \ }n=-1\\ \frac {{\mathrm {asinh}\left (a\,x\right )}^{n+1}}{a\,\left (n+1\right )} & \text {\ if\ \ }n\neq -1 \end {array}\right . \]
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