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 = 0.0865184, antiderivative size = 108, 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 = {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 6142
Rule 671
Rule 641
Rule 195
Rule 217
Rule 203
Rubi steps
\begin{align*} \int e^{-2 \tanh ^{-1}(a x)} \left (c-a^2 c x^2\right )^{3/2} \, dx &=c \int (1-a x)^2 \sqrt{c-a^2 c x^2} \, dx\\ &=\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 = 0.0970476, 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 = 0.035, size = 174, 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 ( -c{a}^{2} \left ( x+{a}^{-1} \right ) ^{2}+2\,ac \left ( x+{a}^{-1} \right ) \right ) ^{{\frac{3}{2}}}}+c\sqrt{-c{a}^{2} \left ( x+{a}^{-1} \right ) ^{2}+2\,ac \left ( x+{a}^{-1} \right ) }x+{{c}^{2}\arctan \left ({x\sqrt{{a}^{2}c}{\frac{1}{\sqrt{-c{a}^{2} \left ( x+{a}^{-1} \right ) ^{2}+2\,ac \left ( x+{a}^{-1} \right ) }}}} \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 = 2.45324, size = 413, normalized size = 3.82 \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 = 8.39055, 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 [B] time = 1.25445, size = 302, normalized size = 2.8 \begin{align*} -\frac{{\left (240 \, a^{5} c^{\frac{3}{2}} \arctan \left (\frac{\sqrt{-c + \frac{2 \, c}{a x + 1}}}{\sqrt{c}}\right ) \mathrm{sgn}\left (\frac{1}{a x + 1}\right ) \mathrm{sgn}\left (a\right ) - \frac{{\left (15 \, a^{5}{\left (c - \frac{2 \, c}{a x + 1}\right )}^{3} c^{2} \sqrt{-c + \frac{2 \, c}{a x + 1}} \mathrm{sgn}\left (\frac{1}{a x + 1}\right ) \mathrm{sgn}\left (a\right ) + 73 \, a^{5}{\left (c - \frac{2 \, c}{a x + 1}\right )}^{2} c^{3} \sqrt{-c + \frac{2 \, c}{a x + 1}} \mathrm{sgn}\left (\frac{1}{a x + 1}\right ) \mathrm{sgn}\left (a\right ) + 15 \, a^{5} c^{5} \sqrt{-c + \frac{2 \, c}{a x + 1}} \mathrm{sgn}\left (\frac{1}{a x + 1}\right ) \mathrm{sgn}\left (a\right ) + 55 \, a^{5} c^{4}{\left (-c + \frac{2 \, c}{a x + 1}\right )}^{\frac{3}{2}} \mathrm{sgn}\left (\frac{1}{a x + 1}\right ) \mathrm{sgn}\left (a\right )\right )}{\left (a x + 1\right )}^{4}}{c^{4}}\right )}{\left | a \right |}}{192 \, a^{7}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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