Integrand size = 22, antiderivative size = 28 \[ \int \frac {x}{\sqrt {1-a^2 x^2} \text {arccosh}(a x)} \, dx=\frac {\sqrt {-1+a x} \text {Chi}(\text {arccosh}(a x))}{a^2 \sqrt {1-a x}} \]
[Out]
Time = 0.06 (sec) , antiderivative size = 28, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.091, Rules used = {5952, 3382} \[ \int \frac {x}{\sqrt {1-a^2 x^2} \text {arccosh}(a x)} \, dx=\frac {\sqrt {a x-1} \text {Chi}(\text {arccosh}(a x))}{a^2 \sqrt {1-a x}} \]
[In]
[Out]
Rule 3382
Rule 5952
Rubi steps \begin{align*} \text {integral}& = \frac {\sqrt {-1+a x} \text {Subst}\left (\int \frac {\cosh (x)}{x} \, dx,x,\text {arccosh}(a x)\right )}{a^2 \sqrt {1-a x}} \\ & = \frac {\sqrt {-1+a x} \text {Chi}(\text {arccosh}(a x))}{a^2 \sqrt {1-a x}} \\ \end{align*}
Time = 0.07 (sec) , antiderivative size = 50, normalized size of antiderivative = 1.79 \[ \int \frac {x}{\sqrt {1-a^2 x^2} \text {arccosh}(a x)} \, dx=-\frac {\sqrt {-((-1+a x) (1+a x))} \text {Chi}(\text {arccosh}(a x))}{a^2 \sqrt {\frac {-1+a x}{1+a x}} (1+a x)} \]
[In]
[Out]
Leaf count of result is larger than twice the leaf count of optimal. \(57\) vs. \(2(24)=48\).
Time = 0.78 (sec) , antiderivative size = 58, normalized size of antiderivative = 2.07
method | result | size |
default | \(\frac {\sqrt {-a^{2} x^{2}+1}\, \sqrt {a x -1}\, \sqrt {a x +1}\, \left (\operatorname {Ei}_{1}\left (\operatorname {arccosh}\left (a x \right )\right )+\operatorname {Ei}_{1}\left (-\operatorname {arccosh}\left (a x \right )\right )\right )}{2 a^{2} \left (a^{2} x^{2}-1\right )}\) | \(58\) |
[In]
[Out]
\[ \int \frac {x}{\sqrt {1-a^2 x^2} \text {arccosh}(a x)} \, dx=\int { \frac {x}{\sqrt {-a^{2} x^{2} + 1} \operatorname {arcosh}\left (a x\right )} \,d x } \]
[In]
[Out]
\[ \int \frac {x}{\sqrt {1-a^2 x^2} \text {arccosh}(a x)} \, dx=\int \frac {x}{\sqrt {- \left (a x - 1\right ) \left (a x + 1\right )} \operatorname {acosh}{\left (a x \right )}}\, dx \]
[In]
[Out]
\[ \int \frac {x}{\sqrt {1-a^2 x^2} \text {arccosh}(a x)} \, dx=\int { \frac {x}{\sqrt {-a^{2} x^{2} + 1} \operatorname {arcosh}\left (a x\right )} \,d x } \]
[In]
[Out]
\[ \int \frac {x}{\sqrt {1-a^2 x^2} \text {arccosh}(a x)} \, dx=\int { \frac {x}{\sqrt {-a^{2} x^{2} + 1} \operatorname {arcosh}\left (a x\right )} \,d x } \]
[In]
[Out]
Timed out. \[ \int \frac {x}{\sqrt {1-a^2 x^2} \text {arccosh}(a x)} \, dx=\int \frac {x}{\mathrm {acosh}\left (a\,x\right )\,\sqrt {1-a^2\,x^2}} \,d x \]
[In]
[Out]