Optimal. Leaf size=72 \[ \frac{x^2 \sqrt{c-\frac{c}{a^2 x^2}}}{2 \sqrt{1-\frac{1}{a^2 x^2}}}-\frac{x \sqrt{c-\frac{c}{a^2 x^2}}}{a \sqrt{1-\frac{1}{a^2 x^2}}} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.176238, antiderivative size = 72, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 25, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.08, Rules used = {6197, 6193} \[ \frac{x^2 \sqrt{c-\frac{c}{a^2 x^2}}}{2 \sqrt{1-\frac{1}{a^2 x^2}}}-\frac{x \sqrt{c-\frac{c}{a^2 x^2}}}{a \sqrt{1-\frac{1}{a^2 x^2}}} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 6197
Rule 6193
Rubi steps
\begin{align*} \int e^{-\coth ^{-1}(a x)} \sqrt{c-\frac{c}{a^2 x^2}} x \, dx &=\frac{\sqrt{c-\frac{c}{a^2 x^2}} \int e^{-\coth ^{-1}(a x)} \sqrt{1-\frac{1}{a^2 x^2}} x \, dx}{\sqrt{1-\frac{1}{a^2 x^2}}}\\ &=\frac{\sqrt{c-\frac{c}{a^2 x^2}} \int (-1+a x) \, dx}{a \sqrt{1-\frac{1}{a^2 x^2}}}\\ &=-\frac{\sqrt{c-\frac{c}{a^2 x^2}} x}{a \sqrt{1-\frac{1}{a^2 x^2}}}+\frac{\sqrt{c-\frac{c}{a^2 x^2}} x^2}{2 \sqrt{1-\frac{1}{a^2 x^2}}}\\ \end{align*}
Mathematica [A] time = 0.0234619, size = 42, normalized size = 0.58 \[ \frac{x (a x-2) \sqrt{c-\frac{c}{a^2 x^2}}}{2 a \sqrt{1-\frac{1}{a^2 x^2}}} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [A] time = 0.116, size = 52, normalized size = 0.7 \begin{align*}{\frac{ \left ( ax-2 \right ){x}^{2}}{2\,ax-2}\sqrt{{\frac{c \left ({a}^{2}{x}^{2}-1 \right ) }{{a}^{2}{x}^{2}}}}\sqrt{{\frac{ax-1}{ax+1}}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \sqrt{c - \frac{c}{a^{2} x^{2}}} x \sqrt{\frac{a x - 1}{a x + 1}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [A] time = 1.51924, size = 47, normalized size = 0.65 \begin{align*} \frac{\sqrt{a^{2} c}{\left (a x^{2} - 2 \, x\right )}}{2 \, a^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Sympy [F(-1)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \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 \sqrt{c - \frac{c}{a^{2} x^{2}}} x \sqrt{\frac{a x - 1}{a x + 1}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]