Optimal. Leaf size=75 \[ \sqrt{c-a^2 c x^2}-2 \sqrt{c} \tan ^{-1}\left (\frac{a \sqrt{c} x}{\sqrt{c-a^2 c x^2}}\right )+\sqrt{c} \tanh ^{-1}\left (\frac{\sqrt{c-a^2 c x^2}}{\sqrt{c}}\right ) \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.344603, antiderivative size = 75, normalized size of antiderivative = 1., number of steps used = 9, number of rules used = 9, integrand size = 27, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.333, Rules used = {6167, 6151, 1809, 844, 217, 203, 266, 63, 208} \[ \sqrt{c-a^2 c x^2}-2 \sqrt{c} \tan ^{-1}\left (\frac{a \sqrt{c} x}{\sqrt{c-a^2 c x^2}}\right )+\sqrt{c} \tanh ^{-1}\left (\frac{\sqrt{c-a^2 c x^2}}{\sqrt{c}}\right ) \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 6167
Rule 6151
Rule 1809
Rule 844
Rule 217
Rule 203
Rule 266
Rule 63
Rule 208
Rubi steps
\begin{align*} \int \frac{e^{2 \coth ^{-1}(a x)} \sqrt{c-a^2 c x^2}}{x} \, dx &=-\int \frac{e^{2 \tanh ^{-1}(a x)} \sqrt{c-a^2 c x^2}}{x} \, dx\\ &=-\left (c \int \frac{(1+a x)^2}{x \sqrt{c-a^2 c x^2}} \, dx\right )\\ &=\sqrt{c-a^2 c x^2}+\frac{\int \frac{-a^2 c-2 a^3 c x}{x \sqrt{c-a^2 c x^2}} \, dx}{a^2}\\ &=\sqrt{c-a^2 c x^2}-c \int \frac{1}{x \sqrt{c-a^2 c x^2}} \, dx-(2 a c) \int \frac{1}{\sqrt{c-a^2 c x^2}} \, dx\\ &=\sqrt{c-a^2 c x^2}-\frac{1}{2} c \operatorname{Subst}\left (\int \frac{1}{x \sqrt{c-a^2 c x}} \, dx,x,x^2\right )-(2 a c) \operatorname{Subst}\left (\int \frac{1}{1+a^2 c x^2} \, dx,x,\frac{x}{\sqrt{c-a^2 c x^2}}\right )\\ &=\sqrt{c-a^2 c x^2}-2 \sqrt{c} \tan ^{-1}\left (\frac{a \sqrt{c} x}{\sqrt{c-a^2 c x^2}}\right )+\frac{\operatorname{Subst}\left (\int \frac{1}{\frac{1}{a^2}-\frac{x^2}{a^2 c}} \, dx,x,\sqrt{c-a^2 c x^2}\right )}{a^2}\\ &=\sqrt{c-a^2 c x^2}-2 \sqrt{c} \tan ^{-1}\left (\frac{a \sqrt{c} x}{\sqrt{c-a^2 c x^2}}\right )+\sqrt{c} \tanh ^{-1}\left (\frac{\sqrt{c-a^2 c x^2}}{\sqrt{c}}\right )\\ \end{align*}
Mathematica [A] time = 0.0767689, size = 97, normalized size = 1.29 \[ \sqrt{c-a^2 c x^2}+\sqrt{c} \log \left (\sqrt{c} \sqrt{c-a^2 c x^2}+c\right )+2 \sqrt{c} \tan ^{-1}\left (\frac{a x \sqrt{c-a^2 c x^2}}{\sqrt{c} \left (a^2 x^2-1\right )}\right )-\sqrt{c} \log (x) \]
Warning: Unable to verify antiderivative.
[In]
[Out]
________________________________________________________________________________________
Maple [B] time = 0.055, size = 129, normalized size = 1.7 \begin{align*} -\sqrt{-{a}^{2}c{x}^{2}+c}+\sqrt{c}\ln \left ({\frac{1}{x} \left ( 2\,c+2\,\sqrt{c}\sqrt{-{a}^{2}c{x}^{2}+c} \right ) } \right ) +2\,\sqrt{-c{a}^{2} \left ( x-{a}^{-1} \right ) ^{2}-2\,ac \left ( x-{a}^{-1} \right ) }-2\,{\frac{ac}{\sqrt{{a}^{2}c}}\arctan \left ({\sqrt{{a}^{2}c}x{\frac{1}{\sqrt{-c{a}^{2} \left ( x-{a}^{-1} \right ) ^{2}-2\,ac \left ( x-{a}^{-1} \right ) }}}} \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 [A] time = 1.66141, size = 441, normalized size = 5.88 \begin{align*} \left [2 \, \sqrt{c} \arctan \left (\frac{\sqrt{-a^{2} c x^{2} + c} a \sqrt{c} x}{a^{2} c x^{2} - c}\right ) + \frac{1}{2} \, \sqrt{c} \log \left (-\frac{a^{2} c x^{2} - 2 \, \sqrt{-a^{2} c x^{2} + c} \sqrt{c} - 2 \, c}{x^{2}}\right ) + \sqrt{-a^{2} c x^{2} + c}, \sqrt{-c} \arctan \left (\frac{\sqrt{-a^{2} c x^{2} + c} \sqrt{-c}}{a^{2} c x^{2} - c}\right ) + \sqrt{-c} \log \left (2 \, a^{2} c x^{2} - 2 \, \sqrt{-a^{2} c x^{2} + c} a \sqrt{-c} x - c\right ) + \sqrt{-a^{2} c x^{2} + 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{- c \left (a x - 1\right ) \left (a x + 1\right )} \left (a x + 1\right )}{x \left (a x - 1\right )}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Giac [A] time = 1.17935, size = 128, normalized size = 1.71 \begin{align*} -\frac{2 \, c \arctan \left (-\frac{\sqrt{-a^{2} c} x - \sqrt{-a^{2} c x^{2} + c}}{\sqrt{-c}}\right )}{\sqrt{-c}} - \frac{2 \, a \sqrt{-c} \log \left ({\left | -\sqrt{-a^{2} c} x + \sqrt{-a^{2} c x^{2} + c} \right |}\right )}{{\left | a \right |}} + \sqrt{-a^{2} c x^{2} + c} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]