Optimal. Leaf size=34 \[ \frac{1}{b c \left (a+b \log \left (\frac{\sqrt{1-c x}}{\sqrt{c x+1}}\right )\right )} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.0654521, antiderivative size = 34, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 40, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.075, Rules used = {2512, 2302, 30} \[ \frac{1}{b c \left (a+b \log \left (\frac{\sqrt{1-c x}}{\sqrt{c x+1}}\right )\right )} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 2512
Rule 2302
Rule 30
Rubi steps
\begin{align*} \int \frac{1}{\left (1-c^2 x^2\right ) \left (a+b \log \left (\frac{\sqrt{1-c x}}{\sqrt{1+c x}}\right )\right )^2} \, dx &=-\frac{\operatorname{Subst}\left (\int \frac{1}{x (a+b \log (x))^2} \, dx,x,\frac{\sqrt{1-c x}}{\sqrt{1+c x}}\right )}{c}\\ &=-\frac{\operatorname{Subst}\left (\int \frac{1}{x^2} \, dx,x,a+b \log \left (\frac{\sqrt{1-c x}}{\sqrt{1+c x}}\right )\right )}{b c}\\ &=\frac{1}{b c \left (a+b \log \left (\frac{\sqrt{1-c x}}{\sqrt{1+c x}}\right )\right )}\\ \end{align*}
Mathematica [A] time = 0.0104415, size = 34, normalized size = 1. \[ \frac{1}{b c \left (a+b \log \left (\frac{\sqrt{1-c x}}{\sqrt{c x+1}}\right )\right )} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [F] time = 0.356, size = 0, normalized size = 0. \begin{align*} \int{\frac{1}{-{c}^{2}{x}^{2}+1} \left ( a+b\ln \left ({\sqrt{-cx+1}{\frac{1}{\sqrt{cx+1}}}} \right ) \right ) ^{-2}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [A] time = 1.58297, size = 46, normalized size = 1.35 \begin{align*} -\frac{2}{b^{2} c \log \left (c x + 1\right ) - b^{2} c \log \left (-c x + 1\right ) - 2 \, a b c} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [A] time = 1.99131, size = 72, normalized size = 2.12 \begin{align*} \frac{1}{b^{2} c \log \left (\frac{\sqrt{-c x + 1}}{\sqrt{c x + 1}}\right ) + a b c} \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 [A] time = 1.25946, size = 46, normalized size = 1.35 \begin{align*} -\frac{2}{b^{2} c \log \left (c x + 1\right ) - b^{2} c \log \left (-c x + 1\right ) - 2 \, a b c} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]