Optimal. Leaf size=88 \[ -\frac{x \sqrt{1-a^2 x^2} \cosh ^{-1}(a x)}{2 a^2}+\frac{\sqrt{a x-1} \cosh ^{-1}(a x)^2}{4 a^3 \sqrt{1-a x}}-\frac{x^2 \sqrt{a x-1}}{4 a \sqrt{1-a x}} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.324336, antiderivative size = 125, normalized size of antiderivative = 1.42, number of steps used = 4, number of rules used = 4, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.182, Rules used = {5798, 5759, 5676, 30} \[ -\frac{x^2 \sqrt{a x-1} \sqrt{a x+1}}{4 a \sqrt{1-a^2 x^2}}-\frac{x (1-a x) (a x+1) \cosh ^{-1}(a x)}{2 a^2 \sqrt{1-a^2 x^2}}+\frac{\sqrt{a x-1} \sqrt{a x+1} \cosh ^{-1}(a x)^2}{4 a^3 \sqrt{1-a^2 x^2}} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 5798
Rule 5759
Rule 5676
Rule 30
Rubi steps
\begin{align*} \int \frac{x^2 \cosh ^{-1}(a x)}{\sqrt{1-a^2 x^2}} \, dx &=\frac{\left (\sqrt{-1+a x} \sqrt{1+a x}\right ) \int \frac{x^2 \cosh ^{-1}(a x)}{\sqrt{-1+a x} \sqrt{1+a x}} \, dx}{\sqrt{1-a^2 x^2}}\\ &=-\frac{x (1-a x) (1+a x) \cosh ^{-1}(a x)}{2 a^2 \sqrt{1-a^2 x^2}}+\frac{\left (\sqrt{-1+a x} \sqrt{1+a x}\right ) \int \frac{\cosh ^{-1}(a x)}{\sqrt{-1+a x} \sqrt{1+a x}} \, dx}{2 a^2 \sqrt{1-a^2 x^2}}-\frac{\left (\sqrt{-1+a x} \sqrt{1+a x}\right ) \int x \, dx}{2 a \sqrt{1-a^2 x^2}}\\ &=-\frac{x^2 \sqrt{-1+a x} \sqrt{1+a x}}{4 a \sqrt{1-a^2 x^2}}-\frac{x (1-a x) (1+a x) \cosh ^{-1}(a x)}{2 a^2 \sqrt{1-a^2 x^2}}+\frac{\sqrt{-1+a x} \sqrt{1+a x} \cosh ^{-1}(a x)^2}{4 a^3 \sqrt{1-a^2 x^2}}\\ \end{align*}
Mathematica [A] time = 0.155927, size = 75, normalized size = 0.85 \[ -\frac{\sqrt{-(a x-1) (a x+1)} \left (2 \cosh ^{-1}(a x) \left (\cosh ^{-1}(a x)+\sinh \left (2 \cosh ^{-1}(a x)\right )\right )-\cosh \left (2 \cosh ^{-1}(a x)\right )\right )}{8 a^3 \sqrt{\frac{a x-1}{a x+1}} (a x+1)} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [B] time = 0.186, size = 223, normalized size = 2.5 \begin{align*} -{\frac{ \left ({\rm arccosh} \left (ax\right ) \right ) ^{2}}{4\,{a}^{3} \left ({a}^{2}{x}^{2}-1 \right ) }\sqrt{-{a}^{2}{x}^{2}+1}\sqrt{ax-1}\sqrt{ax+1}}-{\frac{-1+2\,{\rm arccosh} \left (ax\right )}{16\,{a}^{3} \left ({a}^{2}{x}^{2}-1 \right ) }\sqrt{-{a}^{2}{x}^{2}+1} \left ( 2\,{x}^{3}{a}^{3}-2\,ax+2\,\sqrt{ax+1}\sqrt{ax-1}{x}^{2}{a}^{2}-\sqrt{ax-1}\sqrt{ax+1} \right ) }-{\frac{1+2\,{\rm arccosh} \left (ax\right )}{16\,{a}^{3} \left ({a}^{2}{x}^{2}-1 \right ) }\sqrt{-{a}^{2}{x}^{2}+1} \left ( 2\,{x}^{3}{a}^{3}-2\,ax-2\,\sqrt{ax+1}\sqrt{ax-1}{x}^{2}{a}^{2}+\sqrt{ax-1}\sqrt{ax+1} \right ) } \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [F(-2)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: RuntimeError} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [F] time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (-\frac{\sqrt{-a^{2} x^{2} + 1} x^{2} \operatorname{arcosh}\left (a x\right )}{a^{2} x^{2} - 1}, x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Sympy [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{x^{2} \operatorname{acosh}{\left (a x \right )}}{\sqrt{- \left (a x - 1\right ) \left (a x + 1\right )}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Giac [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{x^{2} \operatorname{arcosh}\left (a x\right )}{\sqrt{-a^{2} x^{2} + 1}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]