Integrand size = 10, antiderivative size = 132 \[ \int x^2 \text {arcsinh}(a x)^3 \, dx=\frac {14 \sqrt {1+a^2 x^2}}{9 a^3}-\frac {2 \left (1+a^2 x^2\right )^{3/2}}{27 a^3}-\frac {4 x \text {arcsinh}(a x)}{3 a^2}+\frac {2}{9} x^3 \text {arcsinh}(a x)+\frac {2 \sqrt {1+a^2 x^2} \text {arcsinh}(a x)^2}{3 a^3}-\frac {x^2 \sqrt {1+a^2 x^2} \text {arcsinh}(a x)^2}{3 a}+\frac {1}{3} x^3 \text {arcsinh}(a x)^3 \]
[Out]
Time = 0.16 (sec) , antiderivative size = 132, normalized size of antiderivative = 1.00, number of steps used = 9, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.700, Rules used = {5776, 5812, 5798, 5772, 267, 272, 45} \[ \int x^2 \text {arcsinh}(a x)^3 \, dx=-\frac {x^2 \sqrt {a^2 x^2+1} \text {arcsinh}(a x)^2}{3 a}-\frac {4 x \text {arcsinh}(a x)}{3 a^2}+\frac {2 \sqrt {a^2 x^2+1} \text {arcsinh}(a x)^2}{3 a^3}-\frac {2 \left (a^2 x^2+1\right )^{3/2}}{27 a^3}+\frac {14 \sqrt {a^2 x^2+1}}{9 a^3}+\frac {1}{3} x^3 \text {arcsinh}(a x)^3+\frac {2}{9} x^3 \text {arcsinh}(a x) \]
[In]
[Out]
Rule 45
Rule 267
Rule 272
Rule 5772
Rule 5776
Rule 5798
Rule 5812
Rubi steps \begin{align*} \text {integral}& = \frac {1}{3} x^3 \text {arcsinh}(a x)^3-a \int \frac {x^3 \text {arcsinh}(a x)^2}{\sqrt {1+a^2 x^2}} \, dx \\ & = -\frac {x^2 \sqrt {1+a^2 x^2} \text {arcsinh}(a x)^2}{3 a}+\frac {1}{3} x^3 \text {arcsinh}(a x)^3+\frac {2}{3} \int x^2 \text {arcsinh}(a x) \, dx+\frac {2 \int \frac {x \text {arcsinh}(a x)^2}{\sqrt {1+a^2 x^2}} \, dx}{3 a} \\ & = \frac {2}{9} x^3 \text {arcsinh}(a x)+\frac {2 \sqrt {1+a^2 x^2} \text {arcsinh}(a x)^2}{3 a^3}-\frac {x^2 \sqrt {1+a^2 x^2} \text {arcsinh}(a x)^2}{3 a}+\frac {1}{3} x^3 \text {arcsinh}(a x)^3-\frac {4 \int \text {arcsinh}(a x) \, dx}{3 a^2}-\frac {1}{9} (2 a) \int \frac {x^3}{\sqrt {1+a^2 x^2}} \, dx \\ & = -\frac {4 x \text {arcsinh}(a x)}{3 a^2}+\frac {2}{9} x^3 \text {arcsinh}(a x)+\frac {2 \sqrt {1+a^2 x^2} \text {arcsinh}(a x)^2}{3 a^3}-\frac {x^2 \sqrt {1+a^2 x^2} \text {arcsinh}(a x)^2}{3 a}+\frac {1}{3} x^3 \text {arcsinh}(a x)^3+\frac {4 \int \frac {x}{\sqrt {1+a^2 x^2}} \, dx}{3 a}-\frac {1}{9} a \text {Subst}\left (\int \frac {x}{\sqrt {1+a^2 x}} \, dx,x,x^2\right ) \\ & = \frac {4 \sqrt {1+a^2 x^2}}{3 a^3}-\frac {4 x \text {arcsinh}(a x)}{3 a^2}+\frac {2}{9} x^3 \text {arcsinh}(a x)+\frac {2 \sqrt {1+a^2 x^2} \text {arcsinh}(a x)^2}{3 a^3}-\frac {x^2 \sqrt {1+a^2 x^2} \text {arcsinh}(a x)^2}{3 a}+\frac {1}{3} x^3 \text {arcsinh}(a x)^3-\frac {1}{9} 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 {14 \sqrt {1+a^2 x^2}}{9 a^3}-\frac {2 \left (1+a^2 x^2\right )^{3/2}}{27 a^3}-\frac {4 x \text {arcsinh}(a x)}{3 a^2}+\frac {2}{9} x^3 \text {arcsinh}(a x)+\frac {2 \sqrt {1+a^2 x^2} \text {arcsinh}(a x)^2}{3 a^3}-\frac {x^2 \sqrt {1+a^2 x^2} \text {arcsinh}(a x)^2}{3 a}+\frac {1}{3} x^3 \text {arcsinh}(a x)^3 \\ \end{align*}
Time = 0.05 (sec) , antiderivative size = 93, normalized size of antiderivative = 0.70 \[ \int x^2 \text {arcsinh}(a x)^3 \, dx=\frac {-2 \left (-20+a^2 x^2\right ) \sqrt {1+a^2 x^2}+6 a x \left (-6+a^2 x^2\right ) \text {arcsinh}(a x)-9 \left (-2+a^2 x^2\right ) \sqrt {1+a^2 x^2} \text {arcsinh}(a x)^2+9 a^3 x^3 \text {arcsinh}(a x)^3}{27 a^3} \]
[In]
[Out]
Time = 0.03 (sec) , antiderivative size = 116, normalized size of antiderivative = 0.88
method | result | size |
derivativedivides | \(\frac {\frac {a^{3} x^{3} \operatorname {arcsinh}\left (a x \right )^{3}}{3}+\frac {2 \operatorname {arcsinh}\left (a x \right )^{2} \sqrt {a^{2} x^{2}+1}}{3}-\frac {a^{2} x^{2} \operatorname {arcsinh}\left (a x \right )^{2} \sqrt {a^{2} x^{2}+1}}{3}-\frac {4 a x \,\operatorname {arcsinh}\left (a x \right )}{3}+\frac {40 \sqrt {a^{2} x^{2}+1}}{27}+\frac {2 a^{3} x^{3} \operatorname {arcsinh}\left (a x \right )}{9}-\frac {2 a^{2} x^{2} \sqrt {a^{2} x^{2}+1}}{27}}{a^{3}}\) | \(116\) |
default | \(\frac {\frac {a^{3} x^{3} \operatorname {arcsinh}\left (a x \right )^{3}}{3}+\frac {2 \operatorname {arcsinh}\left (a x \right )^{2} \sqrt {a^{2} x^{2}+1}}{3}-\frac {a^{2} x^{2} \operatorname {arcsinh}\left (a x \right )^{2} \sqrt {a^{2} x^{2}+1}}{3}-\frac {4 a x \,\operatorname {arcsinh}\left (a x \right )}{3}+\frac {40 \sqrt {a^{2} x^{2}+1}}{27}+\frac {2 a^{3} x^{3} \operatorname {arcsinh}\left (a x \right )}{9}-\frac {2 a^{2} x^{2} \sqrt {a^{2} x^{2}+1}}{27}}{a^{3}}\) | \(116\) |
[In]
[Out]
none
Time = 0.26 (sec) , antiderivative size = 124, normalized size of antiderivative = 0.94 \[ \int x^2 \text {arcsinh}(a x)^3 \, dx=\frac {9 \, a^{3} x^{3} \log \left (a x + \sqrt {a^{2} x^{2} + 1}\right )^{3} - 9 \, \sqrt {a^{2} x^{2} + 1} {\left (a^{2} x^{2} - 2\right )} \log \left (a x + \sqrt {a^{2} x^{2} + 1}\right )^{2} + 6 \, {\left (a^{3} x^{3} - 6 \, a x\right )} \log \left (a x + \sqrt {a^{2} x^{2} + 1}\right ) - 2 \, \sqrt {a^{2} x^{2} + 1} {\left (a^{2} x^{2} - 20\right )}}{27 \, a^{3}} \]
[In]
[Out]
Time = 0.39 (sec) , antiderivative size = 128, normalized size of antiderivative = 0.97 \[ \int x^2 \text {arcsinh}(a x)^3 \, dx=\begin {cases} \frac {x^{3} \operatorname {asinh}^{3}{\left (a x \right )}}{3} + \frac {2 x^{3} \operatorname {asinh}{\left (a x \right )}}{9} - \frac {x^{2} \sqrt {a^{2} x^{2} + 1} \operatorname {asinh}^{2}{\left (a x \right )}}{3 a} - \frac {2 x^{2} \sqrt {a^{2} x^{2} + 1}}{27 a} - \frac {4 x \operatorname {asinh}{\left (a x \right )}}{3 a^{2}} + \frac {2 \sqrt {a^{2} x^{2} + 1} \operatorname {asinh}^{2}{\left (a x \right )}}{3 a^{3}} + \frac {40 \sqrt {a^{2} x^{2} + 1}}{27 a^{3}} & \text {for}\: a \neq 0 \\0 & \text {otherwise} \end {cases} \]
[In]
[Out]
none
Time = 0.20 (sec) , antiderivative size = 116, normalized size of antiderivative = 0.88 \[ \int x^2 \text {arcsinh}(a x)^3 \, dx=\frac {1}{3} \, x^{3} \operatorname {arsinh}\left (a x\right )^{3} - \frac {1}{3} \, a {\left (\frac {\sqrt {a^{2} x^{2} + 1} x^{2}}{a^{2}} - \frac {2 \, \sqrt {a^{2} x^{2} + 1}}{a^{4}}\right )} \operatorname {arsinh}\left (a x\right )^{2} - \frac {2}{27} \, a {\left (\frac {\sqrt {a^{2} x^{2} + 1} x^{2} - \frac {20 \, \sqrt {a^{2} x^{2} + 1}}{a^{2}}}{a^{2}} - \frac {3 \, {\left (a^{2} x^{3} - 6 \, x\right )} \operatorname {arsinh}\left (a x\right )}{a^{3}}\right )} \]
[In]
[Out]
Exception generated. \[ \int x^2 \text {arcsinh}(a x)^3 \, dx=\text {Exception raised: TypeError} \]
[In]
[Out]
Timed out. \[ \int x^2 \text {arcsinh}(a x)^3 \, dx=\int x^2\,{\mathrm {asinh}\left (a\,x\right )}^3 \,d x \]
[In]
[Out]