Integrand size = 24, antiderivative size = 18 \[ \int \frac {1}{\sqrt {1+c^2 x^2} (a+b \text {arcsinh}(c x))^2} \, dx=-\frac {1}{b c (a+b \text {arcsinh}(c x))} \]
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Time = 0.03 (sec) , antiderivative size = 18, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.042, Rules used = {5783} \[ \int \frac {1}{\sqrt {1+c^2 x^2} (a+b \text {arcsinh}(c x))^2} \, dx=-\frac {1}{b c (a+b \text {arcsinh}(c x))} \]
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Rule 5783
Rubi steps \begin{align*} \text {integral}& = -\frac {1}{b c (a+b \text {arcsinh}(c x))} \\ \end{align*}
Time = 0.01 (sec) , antiderivative size = 18, normalized size of antiderivative = 1.00 \[ \int \frac {1}{\sqrt {1+c^2 x^2} (a+b \text {arcsinh}(c x))^2} \, dx=-\frac {1}{b c (a+b \text {arcsinh}(c x))} \]
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Time = 0.25 (sec) , antiderivative size = 19, normalized size of antiderivative = 1.06
method | result | size |
derivativedivides | \(-\frac {1}{b c \left (a +b \,\operatorname {arcsinh}\left (c x \right )\right )}\) | \(19\) |
default | \(-\frac {1}{b c \left (a +b \,\operatorname {arcsinh}\left (c x \right )\right )}\) | \(19\) |
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none
Time = 0.26 (sec) , antiderivative size = 30, normalized size of antiderivative = 1.67 \[ \int \frac {1}{\sqrt {1+c^2 x^2} (a+b \text {arcsinh}(c x))^2} \, dx=-\frac {1}{b^{2} c \log \left (c x + \sqrt {c^{2} x^{2} + 1}\right ) + a b c} \]
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Leaf count of result is larger than twice the leaf count of optimal. 36 vs. \(2 (14) = 28\).
Time = 2.02 (sec) , antiderivative size = 36, normalized size of antiderivative = 2.00 \[ \int \frac {1}{\sqrt {1+c^2 x^2} (a+b \text {arcsinh}(c x))^2} \, dx=\begin {cases} \frac {x}{a^{2}} & \text {for}\: b = 0 \wedge c = 0 \\\frac {\operatorname {asinh}{\left (c x \right )}}{a^{2} c} & \text {for}\: b = 0 \\\frac {x}{a^{2}} & \text {for}\: c = 0 \\- \frac {1}{a b c + b^{2} c \operatorname {asinh}{\left (c x \right )}} & \text {otherwise} \end {cases} \]
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none
Time = 0.20 (sec) , antiderivative size = 18, normalized size of antiderivative = 1.00 \[ \int \frac {1}{\sqrt {1+c^2 x^2} (a+b \text {arcsinh}(c x))^2} \, dx=-\frac {1}{{\left (b \operatorname {arsinh}\left (c x\right ) + a\right )} b c} \]
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\[ \int \frac {1}{\sqrt {1+c^2 x^2} (a+b \text {arcsinh}(c x))^2} \, dx=\int { \frac {1}{\sqrt {c^{2} x^{2} + 1} {\left (b \operatorname {arsinh}\left (c x\right ) + a\right )}^{2}} \,d x } \]
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Time = 2.65 (sec) , antiderivative size = 18, normalized size of antiderivative = 1.00 \[ \int \frac {1}{\sqrt {1+c^2 x^2} (a+b \text {arcsinh}(c x))^2} \, dx=-\frac {1}{c\,\mathrm {asinh}\left (c\,x\right )\,b^2+a\,c\,b} \]
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