Optimal. Leaf size=60 \[ \frac{2 (a x+1)}{a \sqrt{c-a^2 c x^2}}-\frac{\tan ^{-1}\left (\frac{a \sqrt{c} x}{\sqrt{c-a^2 c x^2}}\right )}{a \sqrt{c}} \]
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Rubi [A] time = 0.0660823, antiderivative size = 60, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.167, Rules used = {6141, 653, 217, 203} \[ \frac{2 (a x+1)}{a \sqrt{c-a^2 c x^2}}-\frac{\tan ^{-1}\left (\frac{a \sqrt{c} x}{\sqrt{c-a^2 c x^2}}\right )}{a \sqrt{c}} \]
Antiderivative was successfully verified.
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Rule 6141
Rule 653
Rule 217
Rule 203
Rubi steps
\begin{align*} \int \frac{e^{2 \tanh ^{-1}(a x)}}{\sqrt{c-a^2 c x^2}} \, dx &=c \int \frac{(1+a x)^2}{\left (c-a^2 c x^2\right )^{3/2}} \, dx\\ &=\frac{2 (1+a x)}{a \sqrt{c-a^2 c x^2}}-\int \frac{1}{\sqrt{c-a^2 c x^2}} \, dx\\ &=\frac{2 (1+a x)}{a \sqrt{c-a^2 c x^2}}-\operatorname{Subst}\left (\int \frac{1}{1+a^2 c x^2} \, dx,x,\frac{x}{\sqrt{c-a^2 c x^2}}\right )\\ &=\frac{2 (1+a x)}{a \sqrt{c-a^2 c x^2}}-\frac{\tan ^{-1}\left (\frac{a \sqrt{c} x}{\sqrt{c-a^2 c x^2}}\right )}{a \sqrt{c}}\\ \end{align*}
Mathematica [A] time = 0.0332282, size = 82, normalized size = 1.37 \[ \frac{2 \sqrt{1-a^2 x^2} \left (\sqrt{a x+1}+\sqrt{1-a x} \sin ^{-1}\left (\frac{\sqrt{1-a x}}{\sqrt{2}}\right )\right )}{a \sqrt{1-a x} \sqrt{c-a^2 c x^2}} \]
Warning: Unable to verify antiderivative.
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Maple [A] time = 0.034, size = 80, normalized size = 1.3 \begin{align*} -{\arctan \left ({x\sqrt{{a}^{2}c}{\frac{1}{\sqrt{-{a}^{2}c{x}^{2}+c}}}} \right ){\frac{1}{\sqrt{{a}^{2}c}}}}-2\,{\frac{1}{{a}^{2}c}\sqrt{-c{a}^{2} \left ( x-{a}^{-1} \right ) ^{2}-2\,ac \left ( x-{a}^{-1} \right ) } \left ( x-{a}^{-1} \right ) ^{-1}} \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 = 2.79638, size = 338, normalized size = 5.63 \begin{align*} \left [-\frac{{\left (a x - 1\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 ) + 4 \, \sqrt{-a^{2} c x^{2} + c}}{2 \,{\left (a^{2} c x - a c\right )}}, \frac{{\left (a x - 1\right )} \sqrt{c} \arctan \left (\frac{\sqrt{-a^{2} c x^{2} + c} a \sqrt{c} x}{a^{2} c x^{2} - c}\right ) - 2 \, \sqrt{-a^{2} c x^{2} + c}}{a^{2} c x - a c}\right ] \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*} - \int \frac{a x}{a x \sqrt{- a^{2} c x^{2} + c} - \sqrt{- a^{2} c x^{2} + c}}\, dx - \int \frac{1}{a x \sqrt{- a^{2} c x^{2} + c} - \sqrt{- a^{2} c x^{2} + c}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F] time = 0., size = 0, normalized size = 0. \begin{align*} \mathit{undef} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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