Optimal. Leaf size=74 \[ 2 \sqrt {c-a c x}+2 \sqrt {c} \tanh ^{-1}\left (\frac {\sqrt {c-a c x}}{\sqrt {c}}\right )-4 \sqrt {2} \sqrt {c} \tanh ^{-1}\left (\frac {\sqrt {c-a c x}}{\sqrt {2} \sqrt {c}}\right ) \]
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Rubi [A] time = 0.23, antiderivative size = 74, normalized size of antiderivative = 1.00, number of steps used = 9, number of rules used = 8, integrand size = 23, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.348, Rules used = {6167, 6130, 21, 84, 156, 63, 208, 206} \[ 2 \sqrt {c-a c x}+2 \sqrt {c} \tanh ^{-1}\left (\frac {\sqrt {c-a c x}}{\sqrt {c}}\right )-4 \sqrt {2} \sqrt {c} \tanh ^{-1}\left (\frac {\sqrt {c-a c x}}{\sqrt {2} \sqrt {c}}\right ) \]
Antiderivative was successfully verified.
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Rule 21
Rule 63
Rule 84
Rule 156
Rule 206
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\\ &=-\frac {\int \frac {(c-a c x)^{3/2}}{x (1+a x)} \, dx}{c}\\ &=2 \sqrt {c-a c x}-\frac {\int \frac {a c^2-3 a^2 c^2 x}{x (1+a x) \sqrt {c-a c x}} \, dx}{a c}\\ &=2 \sqrt {c-a c x}-c \int \frac {1}{x \sqrt {c-a c x}} \, dx+(4 a c) \int \frac {1}{(1+a x) \sqrt {c-a c x}} \, dx\\ &=2 \sqrt {c-a c x}-8 \operatorname {Subst}\left (\int \frac {1}{2-\frac {x^2}{c}} \, dx,x,\sqrt {c-a c x}\right )+\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 )-4 \sqrt {2} \sqrt {c} \tanh ^{-1}\left (\frac {\sqrt {c-a c x}}{\sqrt {2} \sqrt {c}}\right )\\ \end {align*}
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Mathematica [A] time = 0.03, size = 74, normalized size = 1.00 \[ 2 \sqrt {c-a c x}+2 \sqrt {c} \tanh ^{-1}\left (\frac {\sqrt {c-a c x}}{\sqrt {c}}\right )-4 \sqrt {2} \sqrt {c} \tanh ^{-1}\left (\frac {\sqrt {c-a c x}}{\sqrt {2} \sqrt {c}}\right ) \]
Antiderivative was successfully verified.
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fricas [A] time = 0.51, size = 157, normalized size = 2.12 \[ \left [2 \, \sqrt {2} \sqrt {c} \log \left (\frac {a c x + 2 \, \sqrt {2} \sqrt {-a c x + c} \sqrt {c} - 3 \, c}{a x + 1}\right ) + \sqrt {c} \log \left (\frac {a c x - 2 \, \sqrt {-a c x + c} \sqrt {c} - 2 \, c}{x}\right ) + 2 \, \sqrt {-a c x + c}, 4 \, \sqrt {2} \sqrt {-c} \arctan \left (\frac {\sqrt {2} \sqrt {-a c x + c} \sqrt {-c}}{2 \, c}\right ) - 2 \, \sqrt {-c} \arctan \left (\frac {\sqrt {-a c x + c} \sqrt {-c}}{c}\right ) + 2 \, \sqrt {-a c x + c}\right ] \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.13, size = 67, normalized size = 0.91 \[ \frac {4 \, \sqrt {2} c \arctan \left (\frac {\sqrt {2} \sqrt {-a c x + c}}{2 \, \sqrt {-c}}\right )}{\sqrt {-c}} - \frac {2 \, c \arctan \left (\frac {\sqrt {-a c x + c}}{\sqrt {-c}}\right )}{\sqrt {-c}} + 2 \, \sqrt {-a c x + c} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.05, size = 58, normalized size = 0.78 \[ 2 \arctanh \left (\frac {\sqrt {-a c x +c}}{\sqrt {c}}\right ) \sqrt {c}-4 \arctanh \left (\frac {\sqrt {-a c x +c}\, \sqrt {2}}{2 \sqrt {c}}\right ) \sqrt {2}\, \sqrt {c}+2 \sqrt {-a c x +c} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.41, size = 98, normalized size = 1.32 \[ 2 \, \sqrt {2} \sqrt {c} \log \left (-\frac {\sqrt {2} \sqrt {c} - \sqrt {-a c x + c}}{\sqrt {2} \sqrt {c} + \sqrt {-a c x + c}}\right ) - \sqrt {c} \log \left (\frac {\sqrt {-a c x + c} - \sqrt {c}}{\sqrt {-a c x + c} + \sqrt {c}}\right ) + 2 \, \sqrt {-a c x + c} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.08, size = 57, normalized size = 0.77 \[ 2\,\sqrt {c}\,\mathrm {atanh}\left (\frac {\sqrt {c-a\,c\,x}}{\sqrt {c}}\right )+2\,\sqrt {c-a\,c\,x}-4\,\sqrt {2}\,\sqrt {c}\,\mathrm {atanh}\left (\frac {\sqrt {2}\,\sqrt {c-a\,c\,x}}{2\,\sqrt {c}}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 5.49, size = 80, normalized size = 1.08 \[ - \frac {2 c \operatorname {atan}{\left (\frac {\sqrt {- a c x + c}}{\sqrt {- c}} \right )}}{\sqrt {- c}} + \frac {4 \sqrt {2} c \operatorname {atan}{\left (\frac {\sqrt {2} \sqrt {- a c x + c}}{2 \sqrt {- c}} \right )}}{\sqrt {- c}} + 2 \sqrt {- a c x + c} \]
Verification of antiderivative is not currently implemented for this CAS.
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