Optimal. Leaf size=30 \[ -\frac {\log ^2\left (\frac {\sqrt {1-a x}}{\sqrt {a x+1}}\right )}{2 a} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.02, antiderivative size = 30, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, integrand size = 34, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.029, Rules used = {2505} \[ -\frac {\log ^2\left (\frac {\sqrt {1-a x}}{\sqrt {a x+1}}\right )}{2 a} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 2505
Rubi steps
\begin {align*} \int \frac {\log \left (\frac {\sqrt {1-a x}}{\sqrt {1+a x}}\right )}{1-a^2 x^2} \, dx &=-\frac {\log ^2\left (\frac {\sqrt {1-a x}}{\sqrt {1+a x}}\right )}{2 a}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A] time = 0.01, size = 30, normalized size = 1.00 \[ -\frac {\log ^2\left (\frac {\sqrt {1-a x}}{\sqrt {a x+1}}\right )}{2 a} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
fricas [A] time = 0.61, size = 24, normalized size = 0.80 \[ -\frac {\log \left (\frac {\sqrt {-a x + 1}}{\sqrt {a x + 1}}\right )^{2}}{2 \, a} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [B] time = 0.28, size = 58, normalized size = 1.93 \[ \frac {1}{4} \, {\left (\frac {\log \left (a x + 1\right )}{a} - \frac {\log \left (a x - 1\right )}{a}\right )} \log \left (-a x + 1\right ) - \frac {\log \left (a x + 1\right )^{2}}{8 \, a} + \frac {\log \left (a x - 1\right )^{2}}{8 \, a} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [F] time = 0.32, size = 0, normalized size = 0.00 \[ \int \frac {\ln \left (\frac {\sqrt {-a x +1}}{\sqrt {a x +1}}\right )}{-a^{2} x^{2}+1}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [B] time = 2.53, size = 83, normalized size = 2.77 \[ \frac {1}{2} \, {\left (\frac {\log \left (a x + 1\right )}{a} - \frac {\log \left (a x - 1\right )}{a}\right )} \log \left (\frac {\sqrt {-a x + 1}}{\sqrt {a x + 1}}\right ) + \frac {\log \left (a x - 1\right )^{2}}{8 \, a} + \frac {\log \left (a x + 1\right )^{2} - 2 \, \log \left (a x + 1\right ) \log \left (a x - 1\right )}{8 \, a} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [F] time = 0.00, size = -1, normalized size = -0.03 \[ -\int \frac {\ln \left (\frac {\sqrt {1-a\,x}}{\sqrt {a\,x+1}}\right )}{a^2\,x^2-1} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
sympy [B] time = 6.28, size = 65, normalized size = 2.17 \[ - \frac {\operatorname {atan}^{2}{\left (\frac {x}{\sqrt {- \frac {1}{a^{2}}}} \right )}}{2 a} - \frac {\log {\left (\frac {\sqrt {- a x + 1}}{\sqrt {a x + 1}} \right )} \operatorname {atan}{\left (\frac {x}{\sqrt {- \frac {1}{a^{2}}}} \right )}}{a^{2} \sqrt {- \frac {1}{a^{2}}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________