Optimal. Leaf size=77 \[ \frac{\sqrt{1-a^2 x^2}}{a \sqrt{c-\frac{c}{a^2 x^2}}}-\frac{\sqrt{1-a^2 x^2} \log (a x+1)}{a^2 x \sqrt{c-\frac{c}{a^2 x^2}}} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.127999, antiderivative size = 77, 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 = {6160, 6150, 43} \[ \frac{\sqrt{1-a^2 x^2}}{a \sqrt{c-\frac{c}{a^2 x^2}}}-\frac{\sqrt{1-a^2 x^2} \log (a x+1)}{a^2 x \sqrt{c-\frac{c}{a^2 x^2}}} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 6160
Rule 6150
Rule 43
Rubi steps
\begin{align*} \int \frac{e^{-\tanh ^{-1}(a x)}}{\sqrt{c-\frac{c}{a^2 x^2}}} \, dx &=\frac{\sqrt{1-a^2 x^2} \int \frac{e^{-\tanh ^{-1}(a x)} x}{\sqrt{1-a^2 x^2}} \, dx}{\sqrt{c-\frac{c}{a^2 x^2}} x}\\ &=\frac{\sqrt{1-a^2 x^2} \int \frac{x}{1+a x} \, dx}{\sqrt{c-\frac{c}{a^2 x^2}} x}\\ &=\frac{\sqrt{1-a^2 x^2} \int \left (\frac{1}{a}-\frac{1}{a (1+a x)}\right ) \, dx}{\sqrt{c-\frac{c}{a^2 x^2}} x}\\ &=\frac{\sqrt{1-a^2 x^2}}{a \sqrt{c-\frac{c}{a^2 x^2}}}-\frac{\sqrt{1-a^2 x^2} \log (1+a x)}{a^2 \sqrt{c-\frac{c}{a^2 x^2}} x}\\ \end{align*}
Mathematica [A] time = 0.029466, size = 50, normalized size = 0.65 \[ \frac{\sqrt{1-a^2 x^2} \left (\frac{x}{a}-\frac{\log (a x+1)}{a^2}\right )}{x \sqrt{c-\frac{c}{a^2 x^2}}} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [A] time = 0.148, size = 51, normalized size = 0.7 \begin{align*} -{\frac{-ax+\ln \left ( ax+1 \right ) }{{a}^{2}x}\sqrt{-{a}^{2}{x}^{2}+1}{\frac{1}{\sqrt{{\frac{c \left ({a}^{2}{x}^{2}-1 \right ) }{{a}^{2}{x}^{2}}}}}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [C] time = 1.11338, size = 28, normalized size = 0.36 \begin{align*} \frac{i \, x}{\sqrt{c}} - \frac{i \, \log \left (a x + 1\right )}{a \sqrt{c}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [B] time = 2.24806, size = 763, normalized size = 9.91 \begin{align*} \left [\frac{2 \, \sqrt{-a^{2} x^{2} + 1} a^{2} x^{2} \sqrt{\frac{a^{2} c x^{2} - c}{a^{2} x^{2}}} -{\left (a^{2} x^{2} - 1\right )} \sqrt{-c} \log \left (\frac{a^{6} c x^{6} + 4 \, a^{5} c x^{5} + 5 \, a^{4} c x^{4} - 4 \, a^{2} c x^{2} - 4 \, a c x -{\left (a^{5} x^{5} + 4 \, a^{4} x^{4} + 6 \, a^{3} x^{3} + 4 \, a^{2} x^{2}\right )} \sqrt{-a^{2} x^{2} + 1} \sqrt{-c} \sqrt{\frac{a^{2} c x^{2} - c}{a^{2} x^{2}}} - 2 \, c}{a^{4} x^{4} + 2 \, a^{3} x^{3} - 2 \, a x - 1}\right )}{2 \,{\left (a^{3} c x^{2} - a c\right )}}, \frac{\sqrt{-a^{2} x^{2} + 1} a^{2} x^{2} \sqrt{\frac{a^{2} c x^{2} - c}{a^{2} x^{2}}} -{\left (a^{2} x^{2} - 1\right )} \sqrt{c} \arctan \left (\frac{{\left (a^{2} x^{2} + 2 \, a x + 2\right )} \sqrt{-a^{2} x^{2} + 1} \sqrt{c} \sqrt{\frac{a^{2} c x^{2} - c}{a^{2} x^{2}}}}{a^{3} c x^{3} + 2 \, a^{2} c x^{2} - a c x - 2 \, c}\right )}{a^{3} c x^{2} - a c}\right ] \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Sympy [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\sqrt{- \left (a x - 1\right ) \left (a x + 1\right )}}{\sqrt{- c \left (-1 + \frac{1}{a x}\right ) \left (1 + \frac{1}{a x}\right )} \left (a x + 1\right )}\, dx \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} x^{2} + 1}}{{\left (a x + 1\right )} \sqrt{c - \frac{c}{a^{2} x^{2}}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]