Optimal. Leaf size=79 \[ \frac{2 x \sqrt{1-\frac{1}{a^2 x^2}}}{(1-a x) \sqrt{c-a^2 c x^2}}+\frac{x \sqrt{1-\frac{1}{a^2 x^2}} \log (1-a x)}{\sqrt{c-a^2 c x^2}} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.167164, antiderivative size = 79, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.125, Rules used = {6192, 6193, 43} \[ \frac{2 x \sqrt{1-\frac{1}{a^2 x^2}}}{(1-a x) \sqrt{c-a^2 c x^2}}+\frac{x \sqrt{1-\frac{1}{a^2 x^2}} \log (1-a x)}{\sqrt{c-a^2 c x^2}} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 6192
Rule 6193
Rule 43
Rubi steps
\begin{align*} \int \frac{e^{3 \coth ^{-1}(a x)}}{\sqrt{c-a^2 c x^2}} \, dx &=\frac{\left (\sqrt{1-\frac{1}{a^2 x^2}} x\right ) \int \frac{e^{3 \coth ^{-1}(a x)}}{\sqrt{1-\frac{1}{a^2 x^2}} x} \, dx}{\sqrt{c-a^2 c x^2}}\\ &=\frac{\left (a \sqrt{1-\frac{1}{a^2 x^2}} x\right ) \int \frac{1+a x}{(-1+a x)^2} \, dx}{\sqrt{c-a^2 c x^2}}\\ &=\frac{\left (a \sqrt{1-\frac{1}{a^2 x^2}} x\right ) \int \left (\frac{2}{(-1+a x)^2}+\frac{1}{-1+a x}\right ) \, dx}{\sqrt{c-a^2 c x^2}}\\ &=\frac{2 \sqrt{1-\frac{1}{a^2 x^2}} x}{(1-a x) \sqrt{c-a^2 c x^2}}+\frac{\sqrt{1-\frac{1}{a^2 x^2}} x \log (1-a x)}{\sqrt{c-a^2 c x^2}}\\ \end{align*}
Mathematica [A] time = 0.0323132, size = 53, normalized size = 0.67 \[ \frac{x \sqrt{1-\frac{1}{a^2 x^2}} ((a x-1) \log (1-a x)-2)}{(a x-1) \sqrt{c-a^2 c x^2}} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [A] time = 0.17, size = 64, normalized size = 0.8 \begin{align*} -{\frac{\ln \left ( ax-1 \right ) xa-\ln \left ( ax-1 \right ) -2}{ac \left ( ax+1 \right ) ^{2}}\sqrt{-c \left ({a}^{2}{x}^{2}-1 \right ) } \left ({\frac{ax-1}{ax+1}} \right ) ^{-{\frac{3}{2}}}} \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 \frac{1}{\sqrt{-a^{2} c x^{2} + c} \left (\frac{a x - 1}{a x + 1}\right )^{\frac{3}{2}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [A] time = 1.89787, size = 84, normalized size = 1.06 \begin{align*} -\frac{\sqrt{-a^{2} c}{\left ({\left (a x - 1\right )} \log \left (a x - 1\right ) - 2\right )}}{a^{3} c x - a^{2} 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 [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{\sqrt{-a^{2} c x^{2} + c} \left (\frac{a x - 1}{a x + 1}\right )^{\frac{3}{2}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]