Integrand size = 8, antiderivative size = 65 \[ \int x^2 \text {arccosh}(a x) \, dx=-\frac {2 \sqrt {-1+a x} \sqrt {1+a x}}{9 a^3}-\frac {x^2 \sqrt {-1+a x} \sqrt {1+a x}}{9 a}+\frac {1}{3} x^3 \text {arccosh}(a x) \]
[Out]
Time = 0.02 (sec) , antiderivative size = 65, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.500, Rules used = {5883, 102, 12, 75} \[ \int x^2 \text {arccosh}(a x) \, dx=-\frac {2 \sqrt {a x-1} \sqrt {a x+1}}{9 a^3}+\frac {1}{3} x^3 \text {arccosh}(a x)-\frac {x^2 \sqrt {a x-1} \sqrt {a x+1}}{9 a} \]
[In]
[Out]
Rule 12
Rule 75
Rule 102
Rule 5883
Rubi steps \begin{align*} \text {integral}& = \frac {1}{3} x^3 \text {arccosh}(a x)-\frac {1}{3} a \int \frac {x^3}{\sqrt {-1+a x} \sqrt {1+a x}} \, dx \\ & = -\frac {x^2 \sqrt {-1+a x} \sqrt {1+a x}}{9 a}+\frac {1}{3} x^3 \text {arccosh}(a x)-\frac {\int \frac {2 x}{\sqrt {-1+a x} \sqrt {1+a x}} \, dx}{9 a} \\ & = -\frac {x^2 \sqrt {-1+a x} \sqrt {1+a x}}{9 a}+\frac {1}{3} x^3 \text {arccosh}(a x)-\frac {2 \int \frac {x}{\sqrt {-1+a x} \sqrt {1+a x}} \, dx}{9 a} \\ & = -\frac {2 \sqrt {-1+a x} \sqrt {1+a x}}{9 a^3}-\frac {x^2 \sqrt {-1+a x} \sqrt {1+a x}}{9 a}+\frac {1}{3} x^3 \text {arccosh}(a x) \\ \end{align*}
Time = 0.02 (sec) , antiderivative size = 46, normalized size of antiderivative = 0.71 \[ \int x^2 \text {arccosh}(a x) \, dx=-\frac {\sqrt {-1+a x} \sqrt {1+a x} \left (2+a^2 x^2\right )}{9 a^3}+\frac {1}{3} x^3 \text {arccosh}(a x) \]
[In]
[Out]
Time = 0.03 (sec) , antiderivative size = 39, normalized size of antiderivative = 0.60
method | result | size |
parts | \(\frac {x^{3} \operatorname {arccosh}\left (a x \right )}{3}-\frac {\sqrt {a x -1}\, \sqrt {a x +1}\, \left (a^{2} x^{2}+2\right )}{9 a^{3}}\) | \(39\) |
derivativedivides | \(\frac {\frac {a^{3} x^{3} \operatorname {arccosh}\left (a x \right )}{3}-\frac {\sqrt {a x -1}\, \sqrt {a x +1}\, \left (a^{2} x^{2}+2\right )}{9}}{a^{3}}\) | \(43\) |
default | \(\frac {\frac {a^{3} x^{3} \operatorname {arccosh}\left (a x \right )}{3}-\frac {\sqrt {a x -1}\, \sqrt {a x +1}\, \left (a^{2} x^{2}+2\right )}{9}}{a^{3}}\) | \(43\) |
[In]
[Out]
none
Time = 0.25 (sec) , antiderivative size = 52, normalized size of antiderivative = 0.80 \[ \int x^2 \text {arccosh}(a x) \, dx=\frac {3 \, a^{3} x^{3} \log \left (a x + \sqrt {a^{2} x^{2} - 1}\right ) - {\left (a^{2} x^{2} + 2\right )} \sqrt {a^{2} x^{2} - 1}}{9 \, a^{3}} \]
[In]
[Out]
\[ \int x^2 \text {arccosh}(a x) \, dx=\int x^{2} \operatorname {acosh}{\left (a x \right )}\, dx \]
[In]
[Out]
none
Time = 0.23 (sec) , antiderivative size = 48, normalized size of antiderivative = 0.74 \[ \int x^2 \text {arccosh}(a x) \, dx=\frac {1}{3} \, x^{3} \operatorname {arcosh}\left (a x\right ) - \frac {1}{9} \, a {\left (\frac {\sqrt {a^{2} x^{2} - 1} x^{2}}{a^{2}} + \frac {2 \, \sqrt {a^{2} x^{2} - 1}}{a^{4}}\right )} \]
[In]
[Out]
Exception generated. \[ \int x^2 \text {arccosh}(a x) \, dx=\text {Exception raised: TypeError} \]
[In]
[Out]
Timed out. \[ \int x^2 \text {arccosh}(a x) \, dx=\int x^2\,\mathrm {acosh}\left (a\,x\right ) \,d x \]
[In]
[Out]