Optimal. Leaf size=50 \[ \frac{c \sqrt{1-a^2 x^2}}{a}+\frac{c \tanh ^{-1}\left (\sqrt{1-a^2 x^2}\right )}{a}+\frac{2 c \sin ^{-1}(a x)}{a} \]
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Rubi [A] time = 0.151384, antiderivative size = 50, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 8, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.4, Rules used = {6131, 6128, 1809, 844, 216, 266, 63, 208} \[ \frac{c \sqrt{1-a^2 x^2}}{a}+\frac{c \tanh ^{-1}\left (\sqrt{1-a^2 x^2}\right )}{a}+\frac{2 c \sin ^{-1}(a x)}{a} \]
Antiderivative was successfully verified.
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Rule 6131
Rule 6128
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 ) \, dx &=-\frac{c \int \frac{e^{-\tanh ^{-1}(a x)} (1-a x)}{x} \, dx}{a}\\ &=-\frac{c \int \frac{(1-a x)^2}{x \sqrt{1-a^2 x^2}} \, dx}{a}\\ &=\frac{c \sqrt{1-a^2 x^2}}{a}+\frac{c \int \frac{-a^2+2 a^3 x}{x \sqrt{1-a^2 x^2}} \, dx}{a^3}\\ &=\frac{c \sqrt{1-a^2 x^2}}{a}+(2 c) \int \frac{1}{\sqrt{1-a^2 x^2}} \, dx-\frac{c \int \frac{1}{x \sqrt{1-a^2 x^2}} \, dx}{a}\\ &=\frac{c \sqrt{1-a^2 x^2}}{a}+\frac{2 c \sin ^{-1}(a x)}{a}-\frac{c \operatorname{Subst}\left (\int \frac{1}{x \sqrt{1-a^2 x}} \, dx,x,x^2\right )}{2 a}\\ &=\frac{c \sqrt{1-a^2 x^2}}{a}+\frac{2 c \sin ^{-1}(a x)}{a}+\frac{c \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 \sqrt{1-a^2 x^2}}{a}+\frac{2 c \sin ^{-1}(a x)}{a}+\frac{c \tanh ^{-1}\left (\sqrt{1-a^2 x^2}\right )}{a}\\ \end{align*}
Mathematica [A] time = 0.0528544, size = 47, normalized size = 0.94 \[ \frac{c \left (\sqrt{1-a^2 x^2}+\log \left (\sqrt{1-a^2 x^2}+1\right )+2 \sin ^{-1}(a x)-\log (x)\right )}{a} \]
Warning: Unable to verify antiderivative.
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Maple [B] time = 0.045, size = 106, normalized size = 2.1 \begin{align*} -{\frac{c}{a}\sqrt{-{a}^{2}{x}^{2}+1}}+{\frac{c}{a}{\it Artanh} \left ({\frac{1}{\sqrt{-{a}^{2}{x}^{2}+1}}} \right ) }+2\,{\frac{c\sqrt{-{a}^{2} \left ( x+{a}^{-1} \right ) ^{2}+2\,a \left ( x+{a}^{-1} \right ) }}{a}}+2\,{\frac{c}{\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.
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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.
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Fricas [A] time = 2.12852, size = 146, normalized size = 2.92 \begin{align*} -\frac{4 \, c \arctan \left (\frac{\sqrt{-a^{2} x^{2} + 1} - 1}{a x}\right ) + c \log \left (\frac{\sqrt{-a^{2} x^{2} + 1} - 1}{x}\right ) - \sqrt{-a^{2} x^{2} + 1} c}{a} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F] time = 0., size = 0, normalized size = 0. \begin{align*} \frac{c \left (\int - \frac{\sqrt{- a^{2} x^{2} + 1}}{a x^{2} + x}\, dx + \int \frac{a x \sqrt{- a^{2} x^{2} + 1}}{a x^{2} + x}\, dx\right )}{a} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.1359, size = 92, normalized size = 1.84 \begin{align*} \frac{2 \, c \arcsin \left (a x\right ) \mathrm{sgn}\left (a\right )}{{\left | a \right |}} + \frac{c \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}{a} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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