Integrand size = 8, antiderivative size = 38 \[ \int \frac {\text {arccosh}(a x)}{x^3} \, dx=\frac {a \sqrt {-1+a x} \sqrt {1+a x}}{2 x}-\frac {\text {arccosh}(a x)}{2 x^2} \]
[Out]
Time = 0.01 (sec) , antiderivative size = 38, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {5883, 97} \[ \int \frac {\text {arccosh}(a x)}{x^3} \, dx=\frac {a \sqrt {a x-1} \sqrt {a x+1}}{2 x}-\frac {\text {arccosh}(a x)}{2 x^2} \]
[In]
[Out]
Rule 97
Rule 5883
Rubi steps \begin{align*} \text {integral}& = -\frac {\text {arccosh}(a x)}{2 x^2}+\frac {1}{2} a \int \frac {1}{x^2 \sqrt {-1+a x} \sqrt {1+a x}} \, dx \\ & = \frac {a \sqrt {-1+a x} \sqrt {1+a x}}{2 x}-\frac {\text {arccosh}(a x)}{2 x^2} \\ \end{align*}
Time = 0.01 (sec) , antiderivative size = 35, normalized size of antiderivative = 0.92 \[ \int \frac {\text {arccosh}(a x)}{x^3} \, dx=\frac {a x \sqrt {-1+a x} \sqrt {1+a x}-\text {arccosh}(a x)}{2 x^2} \]
[In]
[Out]
Result contains higher order function than in optimal. Order 9 vs. order 3.
Time = 0.03 (sec) , antiderivative size = 35, normalized size of antiderivative = 0.92
method | result | size |
parts | \(-\frac {\operatorname {arccosh}\left (a x \right )}{2 x^{2}}+\frac {a \sqrt {a x -1}\, \sqrt {a x +1}\, \operatorname {csgn}\left (a \right )^{2}}{2 x}\) | \(35\) |
derivativedivides | \(a^{2} \left (-\frac {\operatorname {arccosh}\left (a x \right )}{2 a^{2} x^{2}}+\frac {\sqrt {a x -1}\, \sqrt {a x +1}}{2 a x}\right )\) | \(40\) |
default | \(a^{2} \left (-\frac {\operatorname {arccosh}\left (a x \right )}{2 a^{2} x^{2}}+\frac {\sqrt {a x -1}\, \sqrt {a x +1}}{2 a x}\right )\) | \(40\) |
[In]
[Out]
none
Time = 0.25 (sec) , antiderivative size = 38, normalized size of antiderivative = 1.00 \[ \int \frac {\text {arccosh}(a x)}{x^3} \, dx=\frac {\sqrt {a^{2} x^{2} - 1} a x - \log \left (a x + \sqrt {a^{2} x^{2} - 1}\right )}{2 \, x^{2}} \]
[In]
[Out]
\[ \int \frac {\text {arccosh}(a x)}{x^3} \, dx=\int \frac {\operatorname {acosh}{\left (a x \right )}}{x^{3}}\, dx \]
[In]
[Out]
none
Time = 0.36 (sec) , antiderivative size = 27, normalized size of antiderivative = 0.71 \[ \int \frac {\text {arccosh}(a x)}{x^3} \, dx=\frac {\sqrt {a^{2} x^{2} - 1} a}{2 \, x} - \frac {\operatorname {arcosh}\left (a x\right )}{2 \, x^{2}} \]
[In]
[Out]
none
Time = 0.30 (sec) , antiderivative size = 50, normalized size of antiderivative = 1.32 \[ \int \frac {\text {arccosh}(a x)}{x^3} \, dx=\frac {a {\left | a \right |}}{{\left (x {\left | a \right |} - \sqrt {a^{2} x^{2} - 1}\right )}^{2} + 1} - \frac {\log \left (a x + \sqrt {a^{2} x^{2} - 1}\right )}{2 \, x^{2}} \]
[In]
[Out]
Timed out. \[ \int \frac {\text {arccosh}(a x)}{x^3} \, dx=\int \frac {\mathrm {acosh}\left (a\,x\right )}{x^3} \,d x \]
[In]
[Out]