Optimal. Leaf size=104 \[ -\frac{a x \sqrt{c-a^2 c x^2}}{\sqrt{1-a^2 x^2}}+\frac{\log (x) \sqrt{c-a^2 c x^2}}{\sqrt{1-a^2 x^2}}-\frac{4 \sqrt{c-a^2 c x^2} \log (1-a x)}{\sqrt{1-a^2 x^2}} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.193995, antiderivative size = 104, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 27, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.111, Rules used = {6153, 6150, 72} \[ -\frac{a x \sqrt{c-a^2 c x^2}}{\sqrt{1-a^2 x^2}}+\frac{\log (x) \sqrt{c-a^2 c x^2}}{\sqrt{1-a^2 x^2}}-\frac{4 \sqrt{c-a^2 c x^2} \log (1-a x)}{\sqrt{1-a^2 x^2}} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 6153
Rule 6150
Rule 72
Rubi steps
\begin{align*} \int \frac{e^{3 \tanh ^{-1}(a x)} \sqrt{c-a^2 c x^2}}{x} \, dx &=\frac{\sqrt{c-a^2 c x^2} \int \frac{e^{3 \tanh ^{-1}(a x)} \sqrt{1-a^2 x^2}}{x} \, dx}{\sqrt{1-a^2 x^2}}\\ &=\frac{\sqrt{c-a^2 c x^2} \int \frac{(1+a x)^2}{x (1-a x)} \, dx}{\sqrt{1-a^2 x^2}}\\ &=\frac{\sqrt{c-a^2 c x^2} \int \left (-a+\frac{1}{x}-\frac{4 a}{-1+a x}\right ) \, dx}{\sqrt{1-a^2 x^2}}\\ &=-\frac{a x \sqrt{c-a^2 c x^2}}{\sqrt{1-a^2 x^2}}+\frac{\sqrt{c-a^2 c x^2} \log (x)}{\sqrt{1-a^2 x^2}}-\frac{4 \sqrt{c-a^2 c x^2} \log (1-a x)}{\sqrt{1-a^2 x^2}}\\ \end{align*}
Mathematica [A] time = 0.030042, size = 46, normalized size = 0.44 \[ \frac{\sqrt{c-a^2 c x^2} (-a x-4 \log (1-a x)+\log (x))}{\sqrt{1-a^2 x^2}} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [A] time = 0.096, size = 55, normalized size = 0.5 \begin{align*}{\frac{ax-\ln \left ( x \right ) +4\,\ln \left ( ax-1 \right ) }{{a}^{2}{x}^{2}-1}\sqrt{-{a}^{2}{x}^{2}+1}\sqrt{-c \left ({a}^{2}{x}^{2}-1 \right ) }} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [F(-2)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: ValueError} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [F] time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (\frac{\sqrt{-a^{2} c x^{2} + c} \sqrt{-a^{2} x^{2} + 1}{\left (a x + 1\right )}}{a^{2} x^{3} - 2 \, a x^{2} + x}, x\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{- c \left (a x - 1\right ) \left (a x + 1\right )} \left (a x + 1\right )^{3}}{x \left (- \left (a x - 1\right ) \left (a x + 1\right )\right )^{\frac{3}{2}}}\, 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} c x^{2} + c}{\left (a x + 1\right )}^{3}}{{\left (-a^{2} x^{2} + 1\right )}^{\frac{3}{2}} x}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]