Optimal. Leaf size=87 \[ -\frac{(1-a x) \sqrt{c-a^2 c x^2}}{2 a}-\frac{3 \sqrt{c-a^2 c x^2}}{2 a}-\frac{3 \sqrt{c} \tan ^{-1}\left (\frac{a \sqrt{c} x}{\sqrt{c-a^2 c x^2}}\right )}{2 a} \]
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Rubi [A] time = 0.11069, antiderivative size = 87, 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, 6142, 671, 641, 217, 203} \[ -\frac{(1-a x) \sqrt{c-a^2 c x^2}}{2 a}-\frac{3 \sqrt{c-a^2 c x^2}}{2 a}-\frac{3 \sqrt{c} \tan ^{-1}\left (\frac{a \sqrt{c} x}{\sqrt{c-a^2 c x^2}}\right )}{2 a} \]
Antiderivative was successfully verified.
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Rule 6167
Rule 6142
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 = 0.050338, size = 100, normalized size = 1.15 \[ \frac{\sqrt{c-a^2 c x^2} \left (6 \sqrt{1-a x} \sin ^{-1}\left (\frac{\sqrt{1-a x}}{\sqrt{2}}\right )-\sqrt{a x+1} \left (a^2 x^2-5 a x+4\right )\right )}{2 a \sqrt{1-a x} \sqrt{1-a^2 x^2}} \]
Warning: Unable to verify antiderivative.
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Maple [A] time = 0.05, size = 126, normalized size = 1.5 \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{\sqrt{-{a}^{2}c \left ( x+{a}^{-1} \right ) ^{2}+2\, \left ( x+{a}^{-1} \right ) ac}}{a}}-2\,{\frac{c}{\sqrt{{a}^{2}c}}\arctan \left ({\frac{\sqrt{{a}^{2}c}x}{\sqrt{-{a}^{2}c \left ( x+{a}^{-1} \right ) ^{2}+2\, \left ( x+{a}^{-1} \right ) ac}}} \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 = 1.76703, size = 306, normalized size = 3.52 \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 = 1.17145, size = 84, normalized size = 0.97 \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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