Integrand size = 10, antiderivative size = 46 \[ \int (a+b \text {arcsinh}(c x))^2 \, dx=2 b^2 x-\frac {2 b \sqrt {1+c^2 x^2} (a+b \text {arcsinh}(c x))}{c}+x (a+b \text {arcsinh}(c x))^2 \]
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Time = 0.05 (sec) , antiderivative size = 46, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.300, Rules used = {5772, 5798, 8} \[ \int (a+b \text {arcsinh}(c x))^2 \, dx=-\frac {2 b \sqrt {c^2 x^2+1} (a+b \text {arcsinh}(c x))}{c}+x (a+b \text {arcsinh}(c x))^2+2 b^2 x \]
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Rule 8
Rule 5772
Rule 5798
Rubi steps \begin{align*} \text {integral}& = x (a+b \text {arcsinh}(c x))^2-(2 b c) \int \frac {x (a+b \text {arcsinh}(c x))}{\sqrt {1+c^2 x^2}} \, dx \\ & = -\frac {2 b \sqrt {1+c^2 x^2} (a+b \text {arcsinh}(c x))}{c}+x (a+b \text {arcsinh}(c x))^2+\left (2 b^2\right ) \int 1 \, dx \\ & = 2 b^2 x-\frac {2 b \sqrt {1+c^2 x^2} (a+b \text {arcsinh}(c x))}{c}+x (a+b \text {arcsinh}(c x))^2 \\ \end{align*}
Time = 0.09 (sec) , antiderivative size = 74, normalized size of antiderivative = 1.61 \[ \int (a+b \text {arcsinh}(c x))^2 \, dx=\left (a^2+2 b^2\right ) x-\frac {2 a b \sqrt {1+c^2 x^2}}{c}+\frac {2 b \left (a c x-b \sqrt {1+c^2 x^2}\right ) \text {arcsinh}(c x)}{c}+b^2 x \text {arcsinh}(c x)^2 \]
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Time = 0.11 (sec) , antiderivative size = 72, normalized size of antiderivative = 1.57
method | result | size |
derivativedivides | \(\frac {c x \,a^{2}+b^{2} \left (\operatorname {arcsinh}\left (c x \right )^{2} x c -2 \,\operatorname {arcsinh}\left (c x \right ) \sqrt {c^{2} x^{2}+1}+2 c x \right )+2 a b \left (\operatorname {arcsinh}\left (c x \right ) c x -\sqrt {c^{2} x^{2}+1}\right )}{c}\) | \(72\) |
default | \(\frac {c x \,a^{2}+b^{2} \left (\operatorname {arcsinh}\left (c x \right )^{2} x c -2 \,\operatorname {arcsinh}\left (c x \right ) \sqrt {c^{2} x^{2}+1}+2 c x \right )+2 a b \left (\operatorname {arcsinh}\left (c x \right ) c x -\sqrt {c^{2} x^{2}+1}\right )}{c}\) | \(72\) |
parts | \(a^{2} x +\frac {b^{2} \left (\operatorname {arcsinh}\left (c x \right )^{2} x c -2 \,\operatorname {arcsinh}\left (c x \right ) \sqrt {c^{2} x^{2}+1}+2 c x \right )}{c}+\frac {2 a b \left (\operatorname {arcsinh}\left (c x \right ) c x -\sqrt {c^{2} x^{2}+1}\right )}{c}\) | \(73\) |
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Leaf count of result is larger than twice the leaf count of optimal. 96 vs. \(2 (44) = 88\).
Time = 0.26 (sec) , antiderivative size = 96, normalized size of antiderivative = 2.09 \[ \int (a+b \text {arcsinh}(c x))^2 \, dx=\frac {b^{2} c x \log \left (c x + \sqrt {c^{2} x^{2} + 1}\right )^{2} + {\left (a^{2} + 2 \, b^{2}\right )} c x - 2 \, \sqrt {c^{2} x^{2} + 1} a b + 2 \, {\left (a b c x - \sqrt {c^{2} x^{2} + 1} b^{2}\right )} \log \left (c x + \sqrt {c^{2} x^{2} + 1}\right )}{c} \]
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Time = 0.11 (sec) , antiderivative size = 82, normalized size of antiderivative = 1.78 \[ \int (a+b \text {arcsinh}(c x))^2 \, dx=\begin {cases} a^{2} x + 2 a b x \operatorname {asinh}{\left (c x \right )} - \frac {2 a b \sqrt {c^{2} x^{2} + 1}}{c} + b^{2} x \operatorname {asinh}^{2}{\left (c x \right )} + 2 b^{2} x - \frac {2 b^{2} \sqrt {c^{2} x^{2} + 1} \operatorname {asinh}{\left (c x \right )}}{c} & \text {for}\: c \neq 0 \\a^{2} x & \text {otherwise} \end {cases} \]
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none
Time = 0.18 (sec) , antiderivative size = 72, normalized size of antiderivative = 1.57 \[ \int (a+b \text {arcsinh}(c x))^2 \, dx=b^{2} x \operatorname {arsinh}\left (c x\right )^{2} + 2 \, b^{2} {\left (x - \frac {\sqrt {c^{2} x^{2} + 1} \operatorname {arsinh}\left (c x\right )}{c}\right )} + a^{2} x + \frac {2 \, {\left (c x \operatorname {arsinh}\left (c x\right ) - \sqrt {c^{2} x^{2} + 1}\right )} a b}{c} \]
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Leaf count of result is larger than twice the leaf count of optimal. 111 vs. \(2 (44) = 88\).
Time = 0.40 (sec) , antiderivative size = 111, normalized size of antiderivative = 2.41 \[ \int (a+b \text {arcsinh}(c x))^2 \, dx=2 \, {\left (x \log \left (c x + \sqrt {c^{2} x^{2} + 1}\right ) - \frac {\sqrt {c^{2} x^{2} + 1}}{c}\right )} a b + {\left (x \log \left (c x + \sqrt {c^{2} x^{2} + 1}\right )^{2} + 2 \, c {\left (\frac {x}{c} - \frac {\sqrt {c^{2} x^{2} + 1} \log \left (c x + \sqrt {c^{2} x^{2} + 1}\right )}{c^{2}}\right )}\right )} b^{2} + a^{2} x \]
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Timed out. \[ \int (a+b \text {arcsinh}(c x))^2 \, dx=\int {\left (a+b\,\mathrm {asinh}\left (c\,x\right )\right )}^2 \,d x \]
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