Integrand size = 19, antiderivative size = 75 \[ \int \left (d+c^2 d x^2\right ) (a+b \text {arcsinh}(c x)) \, dx=-\frac {2 b d \sqrt {1+c^2 x^2}}{3 c}-\frac {b d \left (1+c^2 x^2\right )^{3/2}}{9 c}+d x (a+b \text {arcsinh}(c x))+\frac {1}{3} c^2 d x^3 (a+b \text {arcsinh}(c x)) \]
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Time = 0.04 (sec) , antiderivative size = 75, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.211, Rules used = {5784, 12, 455, 45} \[ \int \left (d+c^2 d x^2\right ) (a+b \text {arcsinh}(c x)) \, dx=\frac {1}{3} c^2 d x^3 (a+b \text {arcsinh}(c x))+d x (a+b \text {arcsinh}(c x))-\frac {b d \left (c^2 x^2+1\right )^{3/2}}{9 c}-\frac {2 b d \sqrt {c^2 x^2+1}}{3 c} \]
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Rule 12
Rule 45
Rule 455
Rule 5784
Rubi steps \begin{align*} \text {integral}& = d x (a+b \text {arcsinh}(c x))+\frac {1}{3} c^2 d x^3 (a+b \text {arcsinh}(c x))-(b c) \int \frac {d x \left (3+c^2 x^2\right )}{3 \sqrt {1+c^2 x^2}} \, dx \\ & = d x (a+b \text {arcsinh}(c x))+\frac {1}{3} c^2 d x^3 (a+b \text {arcsinh}(c x))-\frac {1}{3} (b c d) \int \frac {x \left (3+c^2 x^2\right )}{\sqrt {1+c^2 x^2}} \, dx \\ & = d x (a+b \text {arcsinh}(c x))+\frac {1}{3} c^2 d x^3 (a+b \text {arcsinh}(c x))-\frac {1}{6} (b c d) \text {Subst}\left (\int \frac {3+c^2 x}{\sqrt {1+c^2 x}} \, dx,x,x^2\right ) \\ & = d x (a+b \text {arcsinh}(c x))+\frac {1}{3} c^2 d x^3 (a+b \text {arcsinh}(c x))-\frac {1}{6} (b c d) \text {Subst}\left (\int \left (\frac {2}{\sqrt {1+c^2 x}}+\sqrt {1+c^2 x}\right ) \, dx,x,x^2\right ) \\ & = -\frac {2 b d \sqrt {1+c^2 x^2}}{3 c}-\frac {b d \left (1+c^2 x^2\right )^{3/2}}{9 c}+d x (a+b \text {arcsinh}(c x))+\frac {1}{3} c^2 d x^3 (a+b \text {arcsinh}(c x)) \\ \end{align*}
Time = 0.03 (sec) , antiderivative size = 86, normalized size of antiderivative = 1.15 \[ \int \left (d+c^2 d x^2\right ) (a+b \text {arcsinh}(c x)) \, dx=a d x+\frac {1}{3} a c^2 d x^3-\frac {7 b d \sqrt {1+c^2 x^2}}{9 c}-\frac {1}{9} b c d x^2 \sqrt {1+c^2 x^2}+b d x \text {arcsinh}(c x)+\frac {1}{3} b c^2 d x^3 \text {arcsinh}(c x) \]
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Time = 0.02 (sec) , antiderivative size = 73, normalized size of antiderivative = 0.97
method | result | size |
parts | \(d a \left (\frac {1}{3} x^{3} c^{2}+x \right )+\frac {d b \left (\frac {\operatorname {arcsinh}\left (c x \right ) c^{3} x^{3}}{3}+\operatorname {arcsinh}\left (c x \right ) c x -\frac {c^{2} x^{2} \sqrt {c^{2} x^{2}+1}}{9}-\frac {7 \sqrt {c^{2} x^{2}+1}}{9}\right )}{c}\) | \(73\) |
derivativedivides | \(\frac {d a \left (\frac {1}{3} c^{3} x^{3}+c x \right )+d b \left (\frac {\operatorname {arcsinh}\left (c x \right ) c^{3} x^{3}}{3}+\operatorname {arcsinh}\left (c x \right ) c x -\frac {c^{2} x^{2} \sqrt {c^{2} x^{2}+1}}{9}-\frac {7 \sqrt {c^{2} x^{2}+1}}{9}\right )}{c}\) | \(76\) |
default | \(\frac {d a \left (\frac {1}{3} c^{3} x^{3}+c x \right )+d b \left (\frac {\operatorname {arcsinh}\left (c x \right ) c^{3} x^{3}}{3}+\operatorname {arcsinh}\left (c x \right ) c x -\frac {c^{2} x^{2} \sqrt {c^{2} x^{2}+1}}{9}-\frac {7 \sqrt {c^{2} x^{2}+1}}{9}\right )}{c}\) | \(76\) |
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Time = 0.27 (sec) , antiderivative size = 83, normalized size of antiderivative = 1.11 \[ \int \left (d+c^2 d x^2\right ) (a+b \text {arcsinh}(c x)) \, dx=\frac {3 \, a c^{3} d x^{3} + 9 \, a c d x + 3 \, {\left (b c^{3} d x^{3} + 3 \, b c d x\right )} \log \left (c x + \sqrt {c^{2} x^{2} + 1}\right ) - {\left (b c^{2} d x^{2} + 7 \, b d\right )} \sqrt {c^{2} x^{2} + 1}}{9 \, c} \]
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Time = 0.17 (sec) , antiderivative size = 90, normalized size of antiderivative = 1.20 \[ \int \left (d+c^2 d x^2\right ) (a+b \text {arcsinh}(c x)) \, dx=\begin {cases} \frac {a c^{2} d x^{3}}{3} + a d x + \frac {b c^{2} d x^{3} \operatorname {asinh}{\left (c x \right )}}{3} - \frac {b c d x^{2} \sqrt {c^{2} x^{2} + 1}}{9} + b d x \operatorname {asinh}{\left (c x \right )} - \frac {7 b d \sqrt {c^{2} x^{2} + 1}}{9 c} & \text {for}\: c \neq 0 \\a d x & \text {otherwise} \end {cases} \]
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Time = 0.19 (sec) , antiderivative size = 97, normalized size of antiderivative = 1.29 \[ \int \left (d+c^2 d x^2\right ) (a+b \text {arcsinh}(c x)) \, dx=\frac {1}{3} \, a c^{2} d x^{3} + \frac {1}{9} \, {\left (3 \, x^{3} \operatorname {arsinh}\left (c x\right ) - c {\left (\frac {\sqrt {c^{2} x^{2} + 1} x^{2}}{c^{2}} - \frac {2 \, \sqrt {c^{2} x^{2} + 1}}{c^{4}}\right )}\right )} b c^{2} d + a d x + \frac {{\left (c x \operatorname {arsinh}\left (c x\right ) - \sqrt {c^{2} x^{2} + 1}\right )} b d}{c} \]
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Exception generated. \[ \int \left (d+c^2 d x^2\right ) (a+b \text {arcsinh}(c x)) \, dx=\text {Exception raised: TypeError} \]
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Timed out. \[ \int \left (d+c^2 d x^2\right ) (a+b \text {arcsinh}(c x)) \, dx=\int \left (a+b\,\mathrm {asinh}\left (c\,x\right )\right )\,\left (d\,c^2\,x^2+d\right ) \,d x \]
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