Integrand size = 8, antiderivative size = 52 \[ \int x^2 \text {arcsinh}(a x) \, dx=\frac {\sqrt {1+a^2 x^2}}{3 a^3}-\frac {\left (1+a^2 x^2\right )^{3/2}}{9 a^3}+\frac {1}{3} x^3 \text {arcsinh}(a x) \]
[Out]
Time = 0.03 (sec) , antiderivative size = 52, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.375, Rules used = {5776, 272, 45} \[ \int x^2 \text {arcsinh}(a x) \, dx=-\frac {\left (a^2 x^2+1\right )^{3/2}}{9 a^3}+\frac {\sqrt {a^2 x^2+1}}{3 a^3}+\frac {1}{3} x^3 \text {arcsinh}(a x) \]
[In]
[Out]
Rule 45
Rule 272
Rule 5776
Rubi steps \begin{align*} \text {integral}& = \frac {1}{3} x^3 \text {arcsinh}(a x)-\frac {1}{3} a \int \frac {x^3}{\sqrt {1+a^2 x^2}} \, dx \\ & = \frac {1}{3} x^3 \text {arcsinh}(a x)-\frac {1}{6} a \text {Subst}\left (\int \frac {x}{\sqrt {1+a^2 x}} \, dx,x,x^2\right ) \\ & = \frac {1}{3} x^3 \text {arcsinh}(a x)-\frac {1}{6} a \text {Subst}\left (\int \left (-\frac {1}{a^2 \sqrt {1+a^2 x}}+\frac {\sqrt {1+a^2 x}}{a^2}\right ) \, dx,x,x^2\right ) \\ & = \frac {\sqrt {1+a^2 x^2}}{3 a^3}-\frac {\left (1+a^2 x^2\right )^{3/2}}{9 a^3}+\frac {1}{3} x^3 \text {arcsinh}(a x) \\ \end{align*}
Time = 0.02 (sec) , antiderivative size = 41, normalized size of antiderivative = 0.79 \[ \int x^2 \text {arcsinh}(a x) \, dx=\frac {1}{9} \left (\frac {\left (2-a^2 x^2\right ) \sqrt {1+a^2 x^2}}{a^3}+3 x^3 \text {arcsinh}(a x)\right ) \]
[In]
[Out]
Time = 0.01 (sec) , antiderivative size = 50, normalized size of antiderivative = 0.96
method | result | size |
derivativedivides | \(\frac {\frac {a^{3} x^{3} \operatorname {arcsinh}\left (a x \right )}{3}-\frac {a^{2} x^{2} \sqrt {a^{2} x^{2}+1}}{9}+\frac {2 \sqrt {a^{2} x^{2}+1}}{9}}{a^{3}}\) | \(50\) |
default | \(\frac {\frac {a^{3} x^{3} \operatorname {arcsinh}\left (a x \right )}{3}-\frac {a^{2} x^{2} \sqrt {a^{2} x^{2}+1}}{9}+\frac {2 \sqrt {a^{2} x^{2}+1}}{9}}{a^{3}}\) | \(50\) |
parts | \(\frac {x^{3} \operatorname {arcsinh}\left (a x \right )}{3}-\frac {a \left (\frac {x^{2} \sqrt {a^{2} x^{2}+1}}{3 a^{2}}-\frac {2 \sqrt {a^{2} x^{2}+1}}{3 a^{4}}\right )}{3}\) | \(50\) |
[In]
[Out]
none
Time = 0.26 (sec) , antiderivative size = 52, normalized size of antiderivative = 1.00 \[ \int x^2 \text {arcsinh}(a x) \, dx=\frac {3 \, a^{3} x^{3} \log \left (a x + \sqrt {a^{2} x^{2} + 1}\right ) - \sqrt {a^{2} x^{2} + 1} {\left (a^{2} x^{2} - 2\right )}}{9 \, a^{3}} \]
[In]
[Out]
Time = 0.22 (sec) , antiderivative size = 48, normalized size of antiderivative = 0.92 \[ \int x^2 \text {arcsinh}(a x) \, dx=\begin {cases} \frac {x^{3} \operatorname {asinh}{\left (a x \right )}}{3} - \frac {x^{2} \sqrt {a^{2} x^{2} + 1}}{9 a} + \frac {2 \sqrt {a^{2} x^{2} + 1}}{9 a^{3}} & \text {for}\: a \neq 0 \\0 & \text {otherwise} \end {cases} \]
[In]
[Out]
none
Time = 0.20 (sec) , antiderivative size = 48, normalized size of antiderivative = 0.92 \[ \int x^2 \text {arcsinh}(a x) \, dx=\frac {1}{3} \, x^{3} \operatorname {arsinh}\left (a x\right ) - \frac {1}{9} \, a {\left (\frac {\sqrt {a^{2} x^{2} + 1} x^{2}}{a^{2}} - \frac {2 \, \sqrt {a^{2} x^{2} + 1}}{a^{4}}\right )} \]
[In]
[Out]
Exception generated. \[ \int x^2 \text {arcsinh}(a x) \, dx=\text {Exception raised: TypeError} \]
[In]
[Out]
Timed out. \[ \int x^2 \text {arcsinh}(a x) \, dx=\int x^2\,\mathrm {asinh}\left (a\,x\right ) \,d x \]
[In]
[Out]