Optimal. Leaf size=94 \[ \frac{a \sqrt{a x-1} \sqrt{a x+1} \text{Unintegrable}\left (\frac{x}{\left (1-a^2 x^2\right ) \sqrt{\cosh ^{-1}(a x)}},x\right )}{2 c \sqrt{c-a^2 c x^2}}+\frac{x \sqrt{\cosh ^{-1}(a x)}}{c \sqrt{c-a^2 c x^2}} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.243686, antiderivative size = 0, normalized size of antiderivative = 0., number of steps used = 0, number of rules used = 0, integrand size = 0, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0., Rules used = {} \[ \int \frac{\sqrt{\cosh ^{-1}(a x)}}{\left (c-a^2 c x^2\right )^{3/2}} \, dx \]
Verification is Not applicable to the result.
[In]
[Out]
Rubi steps
\begin{align*} \int \frac{\sqrt{\cosh ^{-1}(a x)}}{\left (c-a^2 c x^2\right )^{3/2}} \, dx &=-\frac{\left (\sqrt{-1+a x} \sqrt{1+a x}\right ) \int \frac{\sqrt{\cosh ^{-1}(a x)}}{(-1+a x)^{3/2} (1+a x)^{3/2}} \, dx}{c \sqrt{c-a^2 c x^2}}\\ &=\frac{x \sqrt{\cosh ^{-1}(a x)}}{c \sqrt{c-a^2 c x^2}}+\frac{\left (a \sqrt{-1+a x} \sqrt{1+a x}\right ) \int \frac{x}{\left (1-a^2 x^2\right ) \sqrt{\cosh ^{-1}(a x)}} \, dx}{2 c \sqrt{c-a^2 c x^2}}\\ \end{align*}
Mathematica [A] time = 1.58432, size = 0, normalized size = 0. \[ \int \frac{\sqrt{\cosh ^{-1}(a x)}}{\left (c-a^2 c x^2\right )^{3/2}} \, dx \]
Verification is Not applicable to the result.
[In]
[Out]
________________________________________________________________________________________
Maple [A] time = 0.336, size = 0, normalized size = 0. \begin{align*} \int{\sqrt{{\rm arccosh} \left (ax\right )} \left ( -{a}^{2}c{x}^{2}+c \right ) ^{-{\frac{3}{2}}}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [A] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\sqrt{\operatorname{arcosh}\left (a x\right )}}{{\left (-a^{2} c x^{2} + c\right )}^{\frac{3}{2}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [F(-2)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: UnboundLocalError} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Sympy [A] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\sqrt{\operatorname{acosh}{\left (a x \right )}}}{\left (- c \left (a x - 1\right ) \left (a x + 1\right )\right )^{\frac{3}{2}}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Giac [A] time = 0., size = 0, normalized size = 0. \begin{align*} \mathit{sage}_{0} x \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]