Optimal. Leaf size=113 \[ \frac{x \sqrt{c-a^2 c x^2}}{2 \sqrt{1-\frac{1}{a^2 x^2}}}+\frac{3 \sqrt{c-a^2 c x^2}}{a \sqrt{1-\frac{1}{a^2 x^2}}}+\frac{4 \sqrt{c-a^2 c x^2} \log (1-a x)}{a^2 x \sqrt{1-\frac{1}{a^2 x^2}}} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.133175, antiderivative size = 113, 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} \[ \frac{x \sqrt{c-a^2 c x^2}}{2 \sqrt{1-\frac{1}{a^2 x^2}}}+\frac{3 \sqrt{c-a^2 c x^2}}{a \sqrt{1-\frac{1}{a^2 x^2}}}+\frac{4 \sqrt{c-a^2 c x^2} \log (1-a x)}{a^2 x \sqrt{1-\frac{1}{a^2 x^2}}} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 6192
Rule 6193
Rule 43
Rubi steps
\begin{align*} \int e^{3 \coth ^{-1}(a x)} \sqrt{c-a^2 c x^2} \, dx &=\frac{\sqrt{c-a^2 c x^2} \int e^{3 \coth ^{-1}(a x)} \sqrt{1-\frac{1}{a^2 x^2}} x \, dx}{\sqrt{1-\frac{1}{a^2 x^2}} x}\\ &=\frac{\sqrt{c-a^2 c x^2} \int \frac{(1+a x)^2}{-1+a x} \, dx}{a \sqrt{1-\frac{1}{a^2 x^2}} x}\\ &=\frac{\sqrt{c-a^2 c x^2} \int \left (3+a x+\frac{4}{-1+a x}\right ) \, dx}{a \sqrt{1-\frac{1}{a^2 x^2}} x}\\ &=\frac{3 \sqrt{c-a^2 c x^2}}{a \sqrt{1-\frac{1}{a^2 x^2}}}+\frac{x \sqrt{c-a^2 c x^2}}{2 \sqrt{1-\frac{1}{a^2 x^2}}}+\frac{4 \sqrt{c-a^2 c x^2} \log (1-a x)}{a^2 \sqrt{1-\frac{1}{a^2 x^2}} x}\\ \end{align*}
Mathematica [A] time = 0.0244481, size = 57, normalized size = 0.5 \[ \frac{\sqrt{c-a^2 c x^2} (a x (a x+6)+8 \log (1-a x))}{2 a^2 x \sqrt{1-\frac{1}{a^2 x^2}}} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [A] time = 0.174, size = 67, normalized size = 0.6 \begin{align*}{\frac{ \left ({a}^{2}{x}^{2}+6\,ax+8\,\ln \left ( ax-1 \right ) \right ) \left ( ax-1 \right ) }{2\,a \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{\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 = 1.53231, size = 77, normalized size = 0.68 \begin{align*} \frac{{\left (a^{2} x^{2} + 6 \, a x + 8 \, \log \left (a x - 1\right )\right )} \sqrt{-a^{2} c}}{2 \, a^{2}} \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{\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]