Optimal. Leaf size=17 \[ -\frac {\log \left (1-2 \coth ^{-1}(x)\right )}{2 a b} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.04, antiderivative size = 17, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, integrand size = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.050, Rules used = {5947} \[ -\frac {\log \left (1-2 \coth ^{-1}(x)\right )}{2 a b} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 5947
Rubi steps
\begin {align*} \int \frac {1}{\left (a-a x^2\right ) \left (b-2 b \coth ^{-1}(x)\right )} \, dx &=-\frac {\log \left (1-2 \coth ^{-1}(x)\right )}{2 a b}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A] time = 0.06, size = 17, normalized size = 1.00 \[ -\frac {\log \left (2 \coth ^{-1}(x)-1\right )}{2 a b} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
fricas [A] time = 0.73, size = 21, normalized size = 1.24 \[ -\frac {\log \left (\log \left (\frac {x + 1}{x - 1}\right ) - 1\right )}{2 \, a b} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {1}{{\left (a x^{2} - a\right )} {\left (2 \, b \operatorname {arcoth}\relax (x) - b\right )}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [A] time = 0.07, size = 19, normalized size = 1.12 \[ -\frac {\ln \left (2 b \,\mathrm {arccoth}\relax (x )-b \right )}{2 a b} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [A] time = 0.33, size = 21, normalized size = 1.24 \[ -\frac {\log \left (\log \left (x + 1\right ) - \log \left (x - 1\right ) - 1\right )}{2 \, a b} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [B] time = 1.40, size = 15, normalized size = 0.88 \[ -\frac {\ln \left (2\,\mathrm {acoth}\relax (x)-1\right )}{2\,a\,b} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
sympy [A] time = 0.57, size = 14, normalized size = 0.82 \[ - \frac {\log {\left (\operatorname {acoth}{\relax (x )} - \frac {1}{2} \right )}}{2 a b} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________