Optimal. Leaf size=82 \[ \frac{c^2 \sqrt{1-a^2 x^2}}{a}-\frac{c^2 \sqrt{1-a^2 x^2}}{a^2 x}+\frac{3 c^2 \tanh ^{-1}\left (\sqrt{1-a^2 x^2}\right )}{a}+\frac{3 c^2 \sin ^{-1}(a x)}{a} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.237808, antiderivative size = 82, normalized size of antiderivative = 1., number of steps used = 9, number of rules used = 9, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.409, Rules used = {6131, 6128, 1807, 1809, 844, 216, 266, 63, 208} \[ \frac{c^2 \sqrt{1-a^2 x^2}}{a}-\frac{c^2 \sqrt{1-a^2 x^2}}{a^2 x}+\frac{3 c^2 \tanh ^{-1}\left (\sqrt{1-a^2 x^2}\right )}{a}+\frac{3 c^2 \sin ^{-1}(a x)}{a} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 6131
Rule 6128
Rule 1807
Rule 1809
Rule 844
Rule 216
Rule 266
Rule 63
Rule 208
Rubi steps
\begin{align*} \int e^{-\tanh ^{-1}(a x)} \left (c-\frac{c}{a x}\right )^2 \, dx &=\frac{c^2 \int \frac{e^{-\tanh ^{-1}(a x)} (1-a x)^2}{x^2} \, dx}{a^2}\\ &=\frac{c^2 \int \frac{(1-a x)^3}{x^2 \sqrt{1-a^2 x^2}} \, dx}{a^2}\\ &=-\frac{c^2 \sqrt{1-a^2 x^2}}{a^2 x}-\frac{c^2 \int \frac{3 a-3 a^2 x+a^3 x^2}{x \sqrt{1-a^2 x^2}} \, dx}{a^2}\\ &=\frac{c^2 \sqrt{1-a^2 x^2}}{a}-\frac{c^2 \sqrt{1-a^2 x^2}}{a^2 x}+\frac{c^2 \int \frac{-3 a^3+3 a^4 x}{x \sqrt{1-a^2 x^2}} \, dx}{a^4}\\ &=\frac{c^2 \sqrt{1-a^2 x^2}}{a}-\frac{c^2 \sqrt{1-a^2 x^2}}{a^2 x}+\left (3 c^2\right ) \int \frac{1}{\sqrt{1-a^2 x^2}} \, dx-\frac{\left (3 c^2\right ) \int \frac{1}{x \sqrt{1-a^2 x^2}} \, dx}{a}\\ &=\frac{c^2 \sqrt{1-a^2 x^2}}{a}-\frac{c^2 \sqrt{1-a^2 x^2}}{a^2 x}+\frac{3 c^2 \sin ^{-1}(a x)}{a}-\frac{\left (3 c^2\right ) \operatorname{Subst}\left (\int \frac{1}{x \sqrt{1-a^2 x}} \, dx,x,x^2\right )}{2 a}\\ &=\frac{c^2 \sqrt{1-a^2 x^2}}{a}-\frac{c^2 \sqrt{1-a^2 x^2}}{a^2 x}+\frac{3 c^2 \sin ^{-1}(a x)}{a}+\frac{\left (3 c^2\right ) \operatorname{Subst}\left (\int \frac{1}{\frac{1}{a^2}-\frac{x^2}{a^2}} \, dx,x,\sqrt{1-a^2 x^2}\right )}{a^3}\\ &=\frac{c^2 \sqrt{1-a^2 x^2}}{a}-\frac{c^2 \sqrt{1-a^2 x^2}}{a^2 x}+\frac{3 c^2 \sin ^{-1}(a x)}{a}+\frac{3 c^2 \tanh ^{-1}\left (\sqrt{1-a^2 x^2}\right )}{a}\\ \end{align*}
Mathematica [A] time = 0.116056, size = 83, normalized size = 1.01 \[ \frac{\sqrt{1-a^2 x^2} \left (c^2-\frac{c^2}{a x}\right )}{a}+\frac{3 c^2 \log \left (\sqrt{1-a^2 x^2}+1\right )}{a}-\frac{3 c^2 \log (a x)}{a}+\frac{3 c^2 \sin ^{-1}(a x)}{a} \]
Warning: Unable to verify antiderivative.
[In]
[Out]
________________________________________________________________________________________
Maple [B] time = 0.044, size = 186, normalized size = 2.3 \begin{align*} -{\frac{{c}^{2}}{{a}^{2}x} \left ( -{a}^{2}{x}^{2}+1 \right ) ^{{\frac{3}{2}}}}-{c}^{2}x\sqrt{-{a}^{2}{x}^{2}+1}-{{c}^{2}\arctan \left ({x\sqrt{{a}^{2}}{\frac{1}{\sqrt{-{a}^{2}{x}^{2}+1}}}} \right ){\frac{1}{\sqrt{{a}^{2}}}}}+3\,{\frac{{c}^{2}{\it Artanh} \left ({\frac{1}{\sqrt{-{a}^{2}{x}^{2}+1}}} \right ) }{a}}-3\,{\frac{{c}^{2}\sqrt{-{a}^{2}{x}^{2}+1}}{a}}+4\,{\frac{{c}^{2}\sqrt{-{a}^{2} \left ( x+{a}^{-1} \right ) ^{2}+2\,a \left ( x+{a}^{-1} \right ) }}{a}}+4\,{\frac{{c}^{2}}{\sqrt{{a}^{2}}}\arctan \left ({\frac{\sqrt{{a}^{2}}x}{\sqrt{-{a}^{2} \left ( x+{a}^{-1} \right ) ^{2}+2\,a \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 = 2.18273, size = 205, normalized size = 2.5 \begin{align*} -\frac{6 \, a c^{2} x \arctan \left (\frac{\sqrt{-a^{2} x^{2} + 1} - 1}{a x}\right ) + 3 \, a c^{2} x \log \left (\frac{\sqrt{-a^{2} x^{2} + 1} - 1}{x}\right ) - a c^{2} x -{\left (a c^{2} x - c^{2}\right )} \sqrt{-a^{2} x^{2} + 1}}{a^{2} x} \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*} \frac{c^{2} \left (\int \frac{\sqrt{- a^{2} x^{2} + 1}}{a x^{3} + x^{2}}\, dx + \int - \frac{2 a x \sqrt{- a^{2} x^{2} + 1}}{a x^{3} + x^{2}}\, dx + \int \frac{a^{2} x^{2} \sqrt{- a^{2} x^{2} + 1}}{a x^{3} + x^{2}}\, dx\right )}{a^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Giac [A] time = 1.14011, size = 188, normalized size = 2.29 \begin{align*} \frac{a^{2} c^{2} x}{2 \,{\left (\sqrt{-a^{2} x^{2} + 1}{\left | a \right |} + a\right )}{\left | a \right |}} + \frac{3 \, c^{2} \arcsin \left (a x\right ) \mathrm{sgn}\left (a\right )}{{\left | a \right |}} + \frac{3 \, c^{2} \log \left (\frac{{\left | -2 \, \sqrt{-a^{2} x^{2} + 1}{\left | a \right |} - 2 \, a \right |}}{2 \, a^{2}{\left | x \right |}}\right )}{{\left | a \right |}} + \frac{\sqrt{-a^{2} x^{2} + 1} c^{2}}{a} - \frac{{\left (\sqrt{-a^{2} x^{2} + 1}{\left | a \right |} + a\right )} c^{2}}{2 \, a^{2} x{\left | a \right |}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]