Integrand size = 8, antiderivative size = 77 \[ \int x^3 \text {arccosh}(a x) \, dx=-\frac {3 x \sqrt {-1+a x} \sqrt {1+a x}}{32 a^3}-\frac {x^3 \sqrt {-1+a x} \sqrt {1+a x}}{16 a}-\frac {3 \text {arccosh}(a x)}{32 a^4}+\frac {1}{4} x^4 \text {arccosh}(a x) \]
[Out]
Time = 0.02 (sec) , antiderivative size = 77, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.625, Rules used = {5883, 102, 12, 92, 54} \[ \int x^3 \text {arccosh}(a x) \, dx=-\frac {3 \text {arccosh}(a x)}{32 a^4}-\frac {3 x \sqrt {a x-1} \sqrt {a x+1}}{32 a^3}+\frac {1}{4} x^4 \text {arccosh}(a x)-\frac {x^3 \sqrt {a x-1} \sqrt {a x+1}}{16 a} \]
[In]
[Out]
Rule 12
Rule 54
Rule 92
Rule 102
Rule 5883
Rubi steps \begin{align*} \text {integral}& = \frac {1}{4} x^4 \text {arccosh}(a x)-\frac {1}{4} a \int \frac {x^4}{\sqrt {-1+a x} \sqrt {1+a x}} \, dx \\ & = -\frac {x^3 \sqrt {-1+a x} \sqrt {1+a x}}{16 a}+\frac {1}{4} x^4 \text {arccosh}(a x)-\frac {\int \frac {3 x^2}{\sqrt {-1+a x} \sqrt {1+a x}} \, dx}{16 a} \\ & = -\frac {x^3 \sqrt {-1+a x} \sqrt {1+a x}}{16 a}+\frac {1}{4} x^4 \text {arccosh}(a x)-\frac {3 \int \frac {x^2}{\sqrt {-1+a x} \sqrt {1+a x}} \, dx}{16 a} \\ & = -\frac {3 x \sqrt {-1+a x} \sqrt {1+a x}}{32 a^3}-\frac {x^3 \sqrt {-1+a x} \sqrt {1+a x}}{16 a}+\frac {1}{4} x^4 \text {arccosh}(a x)-\frac {3 \int \frac {1}{\sqrt {-1+a x} \sqrt {1+a x}} \, dx}{32 a^3} \\ & = -\frac {3 x \sqrt {-1+a x} \sqrt {1+a x}}{32 a^3}-\frac {x^3 \sqrt {-1+a x} \sqrt {1+a x}}{16 a}-\frac {3 \text {arccosh}(a x)}{32 a^4}+\frac {1}{4} x^4 \text {arccosh}(a x) \\ \end{align*}
Time = 0.04 (sec) , antiderivative size = 71, normalized size of antiderivative = 0.92 \[ \int x^3 \text {arccosh}(a x) \, dx=-\frac {a x \sqrt {-1+a x} \sqrt {1+a x} \left (3+2 a^2 x^2\right )-8 a^4 x^4 \text {arccosh}(a x)+6 \text {arctanh}\left (\sqrt {\frac {-1+a x}{1+a x}}\right )}{32 a^4} \]
[In]
[Out]
Time = 0.04 (sec) , antiderivative size = 98, normalized size of antiderivative = 1.27
method | result | size |
derivativedivides | \(\frac {\frac {a^{4} x^{4} \operatorname {arccosh}\left (a x \right )}{4}-\frac {\sqrt {a x -1}\, \sqrt {a x +1}\, \left (2 a^{3} x^{3} \sqrt {a^{2} x^{2}-1}+3 a x \sqrt {a^{2} x^{2}-1}+3 \ln \left (a x +\sqrt {a^{2} x^{2}-1}\right )\right )}{32 \sqrt {a^{2} x^{2}-1}}}{a^{4}}\) | \(98\) |
default | \(\frac {\frac {a^{4} x^{4} \operatorname {arccosh}\left (a x \right )}{4}-\frac {\sqrt {a x -1}\, \sqrt {a x +1}\, \left (2 a^{3} x^{3} \sqrt {a^{2} x^{2}-1}+3 a x \sqrt {a^{2} x^{2}-1}+3 \ln \left (a x +\sqrt {a^{2} x^{2}-1}\right )\right )}{32 \sqrt {a^{2} x^{2}-1}}}{a^{4}}\) | \(98\) |
parts | \(\frac {x^{4} \operatorname {arccosh}\left (a x \right )}{4}-\frac {\sqrt {a x -1}\, \sqrt {a x +1}\, \left (2 \,\operatorname {csgn}\left (a \right ) a^{3} x^{3} \sqrt {a^{2} x^{2}-1}+3 x \sqrt {a^{2} x^{2}-1}\, \operatorname {csgn}\left (a \right ) a +3 \ln \left (\left (\operatorname {csgn}\left (a \right ) \sqrt {a^{2} x^{2}-1}+a x \right ) \operatorname {csgn}\left (a \right )\right )\right ) \operatorname {csgn}\left (a \right )}{32 a^{4} \sqrt {a^{2} x^{2}-1}}\) | \(106\) |
[In]
[Out]
none
Time = 0.26 (sec) , antiderivative size = 59, normalized size of antiderivative = 0.77 \[ \int x^3 \text {arccosh}(a x) \, dx=\frac {{\left (8 \, a^{4} x^{4} - 3\right )} \log \left (a x + \sqrt {a^{2} x^{2} - 1}\right ) - {\left (2 \, a^{3} x^{3} + 3 \, a x\right )} \sqrt {a^{2} x^{2} - 1}}{32 \, a^{4}} \]
[In]
[Out]
\[ \int x^3 \text {arccosh}(a x) \, dx=\int x^{3} \operatorname {acosh}{\left (a x \right )}\, dx \]
[In]
[Out]
none
Time = 0.27 (sec) , antiderivative size = 77, normalized size of antiderivative = 1.00 \[ \int x^3 \text {arccosh}(a x) \, dx=\frac {1}{4} \, x^{4} \operatorname {arcosh}\left (a x\right ) - \frac {1}{32} \, {\left (\frac {2 \, \sqrt {a^{2} x^{2} - 1} x^{3}}{a^{2}} + \frac {3 \, \sqrt {a^{2} x^{2} - 1} x}{a^{4}} + \frac {3 \, \log \left (2 \, a^{2} x + 2 \, \sqrt {a^{2} x^{2} - 1} a\right )}{a^{5}}\right )} a \]
[In]
[Out]
Exception generated. \[ \int x^3 \text {arccosh}(a x) \, dx=\text {Exception raised: TypeError} \]
[In]
[Out]
Timed out. \[ \int x^3 \text {arccosh}(a x) \, dx=\int x^3\,\mathrm {acosh}\left (a\,x\right ) \,d x \]
[In]
[Out]