Optimal. Leaf size=86 \[ \frac{(a x+1) \sqrt{c-a^2 c x^2}}{2 a}+\frac{3 \sqrt{c-a^2 c x^2}}{2 a}-\frac{3 \sqrt{c} \tan ^{-1}\left (\frac{a \sqrt{c} x}{\sqrt{c-a^2 c x^2}}\right )}{2 a} \]
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Rubi [A] time = 0.113928, antiderivative size = 86, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 6, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.25, Rules used = {6167, 6141, 671, 641, 217, 203} \[ \frac{(a x+1) \sqrt{c-a^2 c x^2}}{2 a}+\frac{3 \sqrt{c-a^2 c x^2}}{2 a}-\frac{3 \sqrt{c} \tan ^{-1}\left (\frac{a \sqrt{c} x}{\sqrt{c-a^2 c x^2}}\right )}{2 a} \]
Antiderivative was successfully verified.
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Rule 6167
Rule 6141
Rule 671
Rule 641
Rule 217
Rule 203
Rubi steps
\begin{align*} \int e^{2 \coth ^{-1}(a x)} \sqrt{c-a^2 c x^2} \, dx &=-\int e^{2 \tanh ^{-1}(a x)} \sqrt{c-a^2 c x^2} \, dx\\ &=-\left (c \int \frac{(1+a x)^2}{\sqrt{c-a^2 c x^2}} \, dx\right )\\ &=\frac{(1+a x) \sqrt{c-a^2 c x^2}}{2 a}-\frac{1}{2} (3 c) \int \frac{1+a x}{\sqrt{c-a^2 c x^2}} \, dx\\ &=\frac{3 \sqrt{c-a^2 c x^2}}{2 a}+\frac{(1+a x) \sqrt{c-a^2 c x^2}}{2 a}-\frac{1}{2} (3 c) \int \frac{1}{\sqrt{c-a^2 c x^2}} \, dx\\ &=\frac{3 \sqrt{c-a^2 c x^2}}{2 a}+\frac{(1+a x) \sqrt{c-a^2 c x^2}}{2 a}-\frac{1}{2} (3 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 )\\ &=\frac{3 \sqrt{c-a^2 c x^2}}{2 a}+\frac{(1+a x) \sqrt{c-a^2 c x^2}}{2 a}-\frac{3 \sqrt{c} \tan ^{-1}\left (\frac{a \sqrt{c} x}{\sqrt{c-a^2 c x^2}}\right )}{2 a}\\ \end{align*}
Mathematica [A] time = 0.0532475, size = 76, normalized size = 0.88 \[ \frac{\sqrt{c-a^2 c x^2} \left (\sqrt{1-a^2 x^2} (a x+4)+6 \sin ^{-1}\left (\frac{\sqrt{1-a x}}{\sqrt{2}}\right )\right )}{2 a \sqrt{1-a^2 x^2}} \]
Warning: Unable to verify antiderivative.
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Maple [A] time = 0.049, size = 134, normalized size = 1.6 \begin{align*}{\frac{x}{2}\sqrt{-{a}^{2}c{x}^{2}+c}}+{\frac{c}{2}\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}\sqrt{-c{a}^{2} \left ( x-{a}^{-1} \right ) ^{2}-2\,ac \left ( x-{a}^{-1} \right ) }}-2\,{\frac{c}{\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.
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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 = 1.64674, size = 306, normalized size = 3.56 \begin{align*} \left [\frac{2 \, \sqrt{-a^{2} c x^{2} + c}{\left (a x + 4\right )} + 3 \, \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 \, a}, \frac{\sqrt{-a^{2} c x^{2} + c}{\left (a x + 4\right )} + 3 \, \sqrt{c} \arctan \left (\frac{\sqrt{-a^{2} c x^{2} + c} a \sqrt{c} x}{a^{2} c x^{2} - c}\right )}{2 \, a}\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{\sqrt{- c \left (a x - 1\right ) \left (a x + 1\right )} \left (a x + 1\right )}{a x - 1}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.15314, size = 84, normalized size = 0.98 \begin{align*} \frac{1}{2} \, \sqrt{-a^{2} c x^{2} + c}{\left (x + \frac{4}{a}\right )} + \frac{3 \, c \log \left ({\left | -\sqrt{-a^{2} c} x + \sqrt{-a^{2} c x^{2} + c} \right |}\right )}{2 \, \sqrt{-c}{\left | a \right |}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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