Optimal. Leaf size=108 \[ -\frac{5 c^{3/2} \tan ^{-1}\left (\frac{a \sqrt{c} x}{\sqrt{c-a^2 c x^2}}\right )}{8 a}-\frac{5}{8} c x \sqrt{c-a^2 c x^2}-\frac{(1-a x) \left (c-a^2 c x^2\right )^{3/2}}{4 a}-\frac{5 \left (c-a^2 c x^2\right )^{3/2}}{12 a} \]
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Rubi [A] time = 0.127054, antiderivative size = 108, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 7, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.292, Rules used = {6167, 6142, 671, 641, 195, 217, 203} \[ -\frac{5 c^{3/2} \tan ^{-1}\left (\frac{a \sqrt{c} x}{\sqrt{c-a^2 c x^2}}\right )}{8 a}-\frac{5}{8} c x \sqrt{c-a^2 c x^2}-\frac{(1-a x) \left (c-a^2 c x^2\right )^{3/2}}{4 a}-\frac{5 \left (c-a^2 c x^2\right )^{3/2}}{12 a} \]
Antiderivative was successfully verified.
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Rule 6167
Rule 6142
Rule 671
Rule 641
Rule 195
Rule 217
Rule 203
Rubi steps
\begin{align*} \int e^{-2 \coth ^{-1}(a x)} \left (c-a^2 c x^2\right )^{3/2} \, dx &=-\int e^{-2 \tanh ^{-1}(a x)} \left (c-a^2 c x^2\right )^{3/2} \, dx\\ &=-\left (c \int (1-a x)^2 \sqrt{c-a^2 c x^2} \, dx\right )\\ &=-\frac{(1-a x) \left (c-a^2 c x^2\right )^{3/2}}{4 a}-\frac{1}{4} (5 c) \int (1-a x) \sqrt{c-a^2 c x^2} \, dx\\ &=-\frac{5 \left (c-a^2 c x^2\right )^{3/2}}{12 a}-\frac{(1-a x) \left (c-a^2 c x^2\right )^{3/2}}{4 a}-\frac{1}{4} (5 c) \int \sqrt{c-a^2 c x^2} \, dx\\ &=-\frac{5}{8} c x \sqrt{c-a^2 c x^2}-\frac{5 \left (c-a^2 c x^2\right )^{3/2}}{12 a}-\frac{(1-a x) \left (c-a^2 c x^2\right )^{3/2}}{4 a}-\frac{1}{8} \left (5 c^2\right ) \int \frac{1}{\sqrt{c-a^2 c x^2}} \, dx\\ &=-\frac{5}{8} c x \sqrt{c-a^2 c x^2}-\frac{5 \left (c-a^2 c x^2\right )^{3/2}}{12 a}-\frac{(1-a x) \left (c-a^2 c x^2\right )^{3/2}}{4 a}-\frac{1}{8} \left (5 c^2\right ) \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{5}{8} c x \sqrt{c-a^2 c x^2}-\frac{5 \left (c-a^2 c x^2\right )^{3/2}}{12 a}-\frac{(1-a x) \left (c-a^2 c x^2\right )^{3/2}}{4 a}-\frac{5 c^{3/2} \tan ^{-1}\left (\frac{a \sqrt{c} x}{\sqrt{c-a^2 c x^2}}\right )}{8 a}\\ \end{align*}
Mathematica [A] time = 0.08938, size = 117, normalized size = 1.08 \[ \frac{c \sqrt{c-a^2 c x^2} \left (\sqrt{a x+1} \left (6 a^4 x^4-22 a^3 x^3+25 a^2 x^2+7 a x-16\right )+30 \sqrt{1-a x} \sin ^{-1}\left (\frac{\sqrt{1-a x}}{\sqrt{2}}\right )\right )}{24 a \sqrt{1-a x} \sqrt{1-a^2 x^2}} \]
Warning: Unable to verify antiderivative.
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Maple [A] time = 0.049, size = 176, normalized size = 1.6 \begin{align*}{\frac{x}{4} \left ( -{a}^{2}c{x}^{2}+c \right ) ^{{\frac{3}{2}}}}+{\frac{3\,cx}{8}\sqrt{-{a}^{2}c{x}^{2}+c}}+{\frac{3\,{c}^{2}}{8}\arctan \left ({x\sqrt{{a}^{2}c}{\frac{1}{\sqrt{-{a}^{2}c{x}^{2}+c}}}} \right ){\frac{1}{\sqrt{{a}^{2}c}}}}-{\frac{2}{3\,a} \left ( -{a}^{2}c \left ( x+{a}^{-1} \right ) ^{2}+2\, \left ( x+{a}^{-1} \right ) ac \right ) ^{{\frac{3}{2}}}}-c\sqrt{-{a}^{2}c \left ( x+{a}^{-1} \right ) ^{2}+2\, \left ( x+{a}^{-1} \right ) ac}x-{{c}^{2}\arctan \left ({x\sqrt{{a}^{2}c}{\frac{1}{\sqrt{-{a}^{2}c \left ( x+{a}^{-1} \right ) ^{2}+2\, \left ( x+{a}^{-1} \right ) ac}}}} \right ){\frac{1}{\sqrt{{a}^{2}c}}}} \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 = 1.72989, size = 412, normalized size = 3.81 \begin{align*} \left [\frac{15 \, \sqrt{-c} c \log \left (2 \, a^{2} c x^{2} - 2 \, \sqrt{-a^{2} c x^{2} + c} a \sqrt{-c} x - c\right ) - 2 \,{\left (6 \, a^{3} c x^{3} - 16 \, a^{2} c x^{2} + 9 \, a c x + 16 \, c\right )} \sqrt{-a^{2} c x^{2} + c}}{48 \, a}, \frac{15 \, c^{\frac{3}{2}} \arctan \left (\frac{\sqrt{-a^{2} c x^{2} + c} a \sqrt{c} x}{a^{2} c x^{2} - c}\right ) -{\left (6 \, a^{3} c x^{3} - 16 \, a^{2} c x^{2} + 9 \, a c x + 16 \, c\right )} \sqrt{-a^{2} c x^{2} + c}}{24 \, a}\right ] \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [C] time = 9.25963, size = 340, normalized size = 3.15 \begin{align*} - a^{2} c \left (\begin{cases} \frac{i a^{2} \sqrt{c} x^{5}}{4 \sqrt{a^{2} x^{2} - 1}} - \frac{3 i \sqrt{c} x^{3}}{8 \sqrt{a^{2} x^{2} - 1}} + \frac{i \sqrt{c} x}{8 a^{2} \sqrt{a^{2} x^{2} - 1}} - \frac{i \sqrt{c} \operatorname{acosh}{\left (a x \right )}}{8 a^{3}} & \text{for}\: \left |{a^{2} x^{2}}\right | > 1 \\- \frac{a^{2} \sqrt{c} x^{5}}{4 \sqrt{- a^{2} x^{2} + 1}} + \frac{3 \sqrt{c} x^{3}}{8 \sqrt{- a^{2} x^{2} + 1}} - \frac{\sqrt{c} x}{8 a^{2} \sqrt{- a^{2} x^{2} + 1}} + \frac{\sqrt{c} \operatorname{asin}{\left (a x \right )}}{8 a^{3}} & \text{otherwise} \end{cases}\right ) + 2 a c \left (\begin{cases} 0 & \text{for}\: c = 0 \\\frac{\sqrt{c} x^{2}}{2} & \text{for}\: a^{2} = 0 \\- \frac{\left (- a^{2} c x^{2} + c\right )^{\frac{3}{2}}}{3 a^{2} c} & \text{otherwise} \end{cases}\right ) - c \left (\begin{cases} \frac{i a^{2} \sqrt{c} x^{3}}{2 \sqrt{a^{2} x^{2} - 1}} - \frac{i \sqrt{c} x}{2 \sqrt{a^{2} x^{2} - 1}} - \frac{i \sqrt{c} \operatorname{acosh}{\left (a x \right )}}{2 a} & \text{for}\: \left |{a^{2} x^{2}}\right | > 1 \\\frac{\sqrt{c} x \sqrt{- a^{2} x^{2} + 1}}{2} + \frac{\sqrt{c} \operatorname{asin}{\left (a x \right )}}{2 a} & \text{otherwise} \end{cases}\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.14766, size = 115, normalized size = 1.06 \begin{align*} -\frac{1}{24} \, \sqrt{-a^{2} c x^{2} + c}{\left ({\left (2 \,{\left (3 \, a^{2} c x - 8 \, a c\right )} x + 9 \, c\right )} x + \frac{16 \, c}{a}\right )} + \frac{5 \, c^{2} \log \left ({\left | -\sqrt{-a^{2} c} x + \sqrt{-a^{2} c x^{2} + c} \right |}\right )}{8 \, \sqrt{-c}{\left | a \right |}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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