Optimal. Leaf size=39 \[ 2 \sqrt {c-a c x}+2 \sqrt {c} \tanh ^{-1}\left (\frac {\sqrt {c-a c x}}{\sqrt {c}}\right ) \]
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Rubi [A] time = 0.20, antiderivative size = 39, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 6, integrand size = 23, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.261, Rules used = {6167, 6130, 21, 80, 63, 208} \[ 2 \sqrt {c-a c x}+2 \sqrt {c} \tanh ^{-1}\left (\frac {\sqrt {c-a c x}}{\sqrt {c}}\right ) \]
Antiderivative was successfully verified.
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Rule 21
Rule 63
Rule 80
Rule 208
Rule 6130
Rule 6167
Rubi steps
\begin {align*} \int \frac {e^{2 \coth ^{-1}(a x)} \sqrt {c-a c x}}{x} \, dx &=-\int \frac {e^{2 \tanh ^{-1}(a x)} \sqrt {c-a c x}}{x} \, dx\\ &=-\int \frac {(1+a x) \sqrt {c-a c x}}{x (1-a x)} \, dx\\ &=-\left (c \int \frac {1+a x}{x \sqrt {c-a c x}} \, dx\right )\\ &=2 \sqrt {c-a c x}-c \int \frac {1}{x \sqrt {c-a c x}} \, dx\\ &=2 \sqrt {c-a c x}+\frac {2 \operatorname {Subst}\left (\int \frac {1}{\frac {1}{a}-\frac {x^2}{a c}} \, dx,x,\sqrt {c-a c x}\right )}{a}\\ &=2 \sqrt {c-a c x}+2 \sqrt {c} \tanh ^{-1}\left (\frac {\sqrt {c-a c x}}{\sqrt {c}}\right )\\ \end {align*}
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Mathematica [A] time = 0.03, size = 39, normalized size = 1.00 \[ 2 \sqrt {c-a c x}+2 \sqrt {c} \tanh ^{-1}\left (\frac {\sqrt {c-a c x}}{\sqrt {c}}\right ) \]
Antiderivative was successfully verified.
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fricas [A] time = 0.46, size = 82, normalized size = 2.10 \[ \left [\sqrt {c} \log \left (\frac {a c x - 2 \, \sqrt {-a c x + c} \sqrt {c} - 2 \, c}{x}\right ) + 2 \, \sqrt {-a c x + c}, -2 \, \sqrt {-c} \arctan \left (\frac {\sqrt {-a c x + c} \sqrt {-c}}{c}\right ) + 2 \, \sqrt {-a c x + c}\right ] \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.14, size = 40, normalized size = 1.03 \[ -2 \, c {\left (\frac {\arctan \left (\frac {\sqrt {-a c x + c}}{\sqrt {-c}}\right )}{\sqrt {-c}} - \frac {\sqrt {-a c x + c}}{c}\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.04, size = 32, normalized size = 0.82 \[ 2 \arctanh \left (\frac {\sqrt {-a c x +c}}{\sqrt {c}}\right ) \sqrt {c}+2 \sqrt {-a c x +c} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.40, size = 49, normalized size = 1.26 \[ -\sqrt {c} \log \left (\frac {\sqrt {-a c x + c} - \sqrt {c}}{\sqrt {-a c x + c} + \sqrt {c}}\right ) + 2 \, \sqrt {-a c x + c} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 1.19, size = 31, normalized size = 0.79 \[ 2\,\sqrt {c}\,\mathrm {atanh}\left (\frac {\sqrt {c-a\,c\,x}}{\sqrt {c}}\right )+2\,\sqrt {c-a\,c\,x} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 4.09, size = 39, normalized size = 1.00 \[ - \frac {2 c \operatorname {atan}{\left (\frac {\sqrt {- a c x + c}}{\sqrt {- c}} \right )}}{\sqrt {- c}} + 2 \sqrt {- a c x + c} \]
Verification of antiderivative is not currently implemented for this CAS.
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