Optimal. Leaf size=45 \[ -\frac{\log \left (1-c^2 x^2\right )}{2 c^3}-\frac{\sqrt{1-c x}}{c^3 \sqrt{\frac{1}{c x+1}}} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.125038, antiderivative size = 45, normalized size of antiderivative = 1., 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 = {6341, 1956, 74, 260} \[ -\frac{\log \left (1-c^2 x^2\right )}{2 c^3}-\frac{\sqrt{1-c x}}{c^3 \sqrt{\frac{1}{c x+1}}} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 6341
Rule 1956
Rule 74
Rule 260
Rubi steps
\begin{align*} \int \frac{e^{\text{sech}^{-1}(c x)} x^2}{1-c^2 x^2} \, dx &=\frac{\int \frac{x \sqrt{\frac{1}{1+c x}}}{\sqrt{1-c x}} \, dx}{c}+\frac{\int \frac{x}{1-c^2 x^2} \, dx}{c}\\ &=-\frac{\log \left (1-c^2 x^2\right )}{2 c^3}+\frac{\left (\sqrt{\frac{1}{1+c x}} \sqrt{1+c x}\right ) \int \frac{x}{\sqrt{1-c x} \sqrt{1+c x}} \, dx}{c}\\ &=-\frac{\sqrt{1-c x}}{c^3 \sqrt{\frac{1}{1+c x}}}-\frac{\log \left (1-c^2 x^2\right )}{2 c^3}\\ \end{align*}
Mathematica [A] time = 0.0903014, size = 44, normalized size = 0.98 \[ -\frac{\log \left (1-c^2 x^2\right )+2 \sqrt{\frac{1-c x}{c x+1}} (c x+1)}{2 c^3} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [A] time = 0.2, size = 52, normalized size = 1.2 \begin{align*} -{\frac{x}{{c}^{2}}\sqrt{-{\frac{cx-1}{cx}}}\sqrt{{\frac{cx+1}{cx}}}}-{\frac{\ln \left ({c}^{2}{x}^{2}-1 \right ) }{2\,{c}^{3}}} \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*} -\frac{\log \left (c x + 1\right )}{2 \, c^{3}} - \frac{\log \left (c x - 1\right )}{2 \, c^{3}} - \int \frac{\sqrt{c x + 1} \sqrt{-c x + 1} x}{c^{3} x^{2} - c}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [A] time = 2.11662, size = 111, normalized size = 2.47 \begin{align*} -\frac{2 \, c x \sqrt{\frac{c x + 1}{c x}} \sqrt{-\frac{c x - 1}{c x}} + \log \left (c^{2} x^{2} - 1\right )}{2 \, c^{3}} \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*} - \frac{\int \frac{x}{c^{2} x^{2} - 1}\, dx + \int \frac{c x^{2} \sqrt{-1 + \frac{1}{c x}} \sqrt{1 + \frac{1}{c x}}}{c^{2} x^{2} - 1}\, dx}{c} \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}{\left (\sqrt{\frac{1}{c x} + 1} \sqrt{\frac{1}{c x} - 1} + \frac{1}{c x}\right )}}{c^{2} x^{2} - 1}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]