Optimal. Leaf size=76 \[ -\frac {2 (c-a c x)^{3/2}}{3 a c}-\frac {4 \sqrt {c-a c x}}{a}+\frac {4 \sqrt {2} \sqrt {c} \tanh ^{-1}\left (\frac {\sqrt {c-a c x}}{\sqrt {2} \sqrt {c}}\right )}{a} \]
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Rubi [A] time = 0.10, antiderivative size = 76, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 6, integrand size = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.300, Rules used = {6167, 6130, 21, 50, 63, 206} \[ -\frac {2 (c-a c x)^{3/2}}{3 a c}-\frac {4 \sqrt {c-a c x}}{a}+\frac {4 \sqrt {2} \sqrt {c} \tanh ^{-1}\left (\frac {\sqrt {c-a c x}}{\sqrt {2} \sqrt {c}}\right )}{a} \]
Antiderivative was successfully verified.
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Rule 21
Rule 50
Rule 63
Rule 206
Rule 6130
Rule 6167
Rubi steps
\begin {align*} \int e^{-2 \coth ^{-1}(a x)} \sqrt {c-a c x} \, dx &=-\int e^{-2 \tanh ^{-1}(a x)} \sqrt {c-a c x} \, dx\\ &=-\int \frac {(1-a x) \sqrt {c-a c x}}{1+a x} \, dx\\ &=-\frac {\int \frac {(c-a c x)^{3/2}}{1+a x} \, dx}{c}\\ &=-\frac {2 (c-a c x)^{3/2}}{3 a c}-2 \int \frac {\sqrt {c-a c x}}{1+a x} \, dx\\ &=-\frac {4 \sqrt {c-a c x}}{a}-\frac {2 (c-a c x)^{3/2}}{3 a c}-(4 c) \int \frac {1}{(1+a x) \sqrt {c-a c x}} \, dx\\ &=-\frac {4 \sqrt {c-a c x}}{a}-\frac {2 (c-a c x)^{3/2}}{3 a c}+\frac {8 \operatorname {Subst}\left (\int \frac {1}{2-\frac {x^2}{c}} \, dx,x,\sqrt {c-a c x}\right )}{a}\\ &=-\frac {4 \sqrt {c-a c x}}{a}-\frac {2 (c-a c x)^{3/2}}{3 a c}+\frac {4 \sqrt {2} \sqrt {c} \tanh ^{-1}\left (\frac {\sqrt {c-a c x}}{\sqrt {2} \sqrt {c}}\right )}{a}\\ \end {align*}
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Mathematica [A] time = 0.03, size = 61, normalized size = 0.80 \[ \frac {2 (a x-7) \sqrt {c-a c x}+12 \sqrt {2} \sqrt {c} \tanh ^{-1}\left (\frac {\sqrt {c-a c x}}{\sqrt {2} \sqrt {c}}\right )}{3 a} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.50, size = 119, normalized size = 1.57 \[ \left [\frac {2 \, {\left (3 \, \sqrt {2} \sqrt {c} \log \left (\frac {a c x - 2 \, \sqrt {2} \sqrt {-a c x + c} \sqrt {c} - 3 \, c}{a x + 1}\right ) + \sqrt {-a c x + c} {\left (a x - 7\right )}\right )}}{3 \, a}, -\frac {2 \, {\left (6 \, \sqrt {2} \sqrt {-c} \arctan \left (\frac {\sqrt {2} \sqrt {-a c x + c} \sqrt {-c}}{2 \, c}\right ) - \sqrt {-a c x + c} {\left (a x - 7\right )}\right )}}{3 \, a}\right ] \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.14, size = 77, normalized size = 1.01 \[ -\frac {4 \, \sqrt {2} c \arctan \left (\frac {\sqrt {2} \sqrt {-a c x + c}}{2 \, \sqrt {-c}}\right )}{a \sqrt {-c}} - \frac {2 \, {\left ({\left (-a c x + c\right )}^{\frac {3}{2}} a^{2} c^{2} + 6 \, \sqrt {-a c x + c} a^{2} c^{3}\right )}}{3 \, a^{3} c^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.04, size = 59, normalized size = 0.78 \[ -\frac {2 \left (\frac {\left (-a c x +c \right )^{\frac {3}{2}}}{3}+2 c \sqrt {-a c x +c}-2 c^{\frac {3}{2}} \sqrt {2}\, \arctanh \left (\frac {\sqrt {-a c x +c}\, \sqrt {2}}{2 \sqrt {c}}\right )\right )}{c a} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.42, size = 79, normalized size = 1.04 \[ -\frac {2 \, {\left (3 \, \sqrt {2} c^{\frac {3}{2}} \log \left (-\frac {\sqrt {2} \sqrt {c} - \sqrt {-a c x + c}}{\sqrt {2} \sqrt {c} + \sqrt {-a c x + c}}\right ) + {\left (-a c x + c\right )}^{\frac {3}{2}} + 6 \, \sqrt {-a c x + c} c\right )}}{3 \, a c} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.00, size = 61, normalized size = 0.80 \[ \frac {4\,\sqrt {2}\,\sqrt {c}\,\mathrm {atanh}\left (\frac {\sqrt {2}\,\sqrt {c-a\,c\,x}}{2\,\sqrt {c}}\right )}{a}-\frac {2\,{\left (c-a\,c\,x\right )}^{3/2}}{3\,a\,c}-\frac {4\,\sqrt {c-a\,c\,x}}{a} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 3.85, size = 73, normalized size = 0.96 \[ - \frac {2 \left (\frac {2 \sqrt {2} c^{2} \operatorname {atan}{\left (\frac {\sqrt {2} \sqrt {- a c x + c}}{2 \sqrt {- c}} \right )}}{\sqrt {- c}} + 2 c \sqrt {- a c x + c} + \frac {\left (- a c x + c\right )^{\frac {3}{2}}}{3}\right )}{a c} \]
Verification of antiderivative is not currently implemented for this CAS.
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