Optimal. Leaf size=58 \[ \frac {2 \sqrt {2} \tanh ^{-1}\left (\frac {\sqrt {c-a c x}}{\sqrt {2} \sqrt {c}}\right )}{a \sqrt {c}}-\frac {2 \sqrt {c-a c x}}{a c} \]
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Rubi [A] time = 0.10, antiderivative size = 58, normalized size of antiderivative = 1.00, number of steps used = 6, 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} \[ \frac {2 \sqrt {2} \tanh ^{-1}\left (\frac {\sqrt {c-a c x}}{\sqrt {2} \sqrt {c}}\right )}{a \sqrt {c}}-\frac {2 \sqrt {c-a c x}}{a c} \]
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 \frac {e^{-2 \coth ^{-1}(a x)}}{\sqrt {c-a c x}} \, dx &=-\int \frac {e^{-2 \tanh ^{-1}(a x)}}{\sqrt {c-a c x}} \, dx\\ &=-\int \frac {1-a x}{(1+a x) \sqrt {c-a c x}} \, dx\\ &=-\frac {\int \frac {\sqrt {c-a c x}}{1+a x} \, dx}{c}\\ &=-\frac {2 \sqrt {c-a c x}}{a c}-2 \int \frac {1}{(1+a x) \sqrt {c-a c x}} \, dx\\ &=-\frac {2 \sqrt {c-a c x}}{a c}+\frac {4 \operatorname {Subst}\left (\int \frac {1}{2-\frac {x^2}{c}} \, dx,x,\sqrt {c-a c x}\right )}{a c}\\ &=-\frac {2 \sqrt {c-a c x}}{a c}+\frac {2 \sqrt {2} \tanh ^{-1}\left (\frac {\sqrt {c-a c x}}{\sqrt {2} \sqrt {c}}\right )}{a \sqrt {c}}\\ \end {align*}
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Mathematica [A] time = 0.03, size = 58, normalized size = 1.00 \[ \frac {2 \sqrt {2} \tanh ^{-1}\left (\frac {\sqrt {c-a c x}}{\sqrt {2} \sqrt {c}}\right )}{a \sqrt {c}}-\frac {2 \sqrt {c-a c x}}{a c} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.56, size = 118, normalized size = 2.03 \[ \left [\frac {\sqrt {2} \sqrt {c} \log \left (\frac {a x - \frac {2 \, \sqrt {2} \sqrt {-a c x + c}}{\sqrt {c}} - 3}{a x + 1}\right ) - 2 \, \sqrt {-a c x + c}}{a c}, \frac {2 \, {\left (\sqrt {2} c \sqrt {-\frac {1}{c}} \arctan \left (\frac {\sqrt {2} \sqrt {-a c x + c} \sqrt {-\frac {1}{c}}}{a x - 1}\right ) - \sqrt {-a c x + c}\right )}}{a c}\right ] \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.16, size = 51, normalized size = 0.88 \[ -\frac {2 \, \sqrt {2} \arctan \left (\frac {\sqrt {2} \sqrt {-a c x + c}}{2 \, \sqrt {-c}}\right )}{a \sqrt {-c}} - \frac {2 \, \sqrt {-a c x + c}}{a c} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.04, size = 45, normalized size = 0.78 \[ -\frac {2 \left (\sqrt {-a c x +c}-\arctanh \left (\frac {\sqrt {-a c x +c}\, \sqrt {2}}{2 \sqrt {c}}\right ) \sqrt {2}\, \sqrt {c}\right )}{c a} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.41, size = 68, normalized size = 1.17 \[ -\frac {\sqrt {2} \sqrt {c} \log \left (-\frac {\sqrt {2} \sqrt {c} - \sqrt {-a c x + c}}{\sqrt {2} \sqrt {c} + \sqrt {-a c x + c}}\right ) + 2 \, \sqrt {-a c x + c}}{a c} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.07, size = 47, normalized size = 0.81 \[ \frac {2\,\sqrt {2}\,\mathrm {atanh}\left (\frac {\sqrt {2}\,\sqrt {c-a\,c\,x}}{2\,\sqrt {c}}\right )}{a\,\sqrt {c}}-\frac {2\,\sqrt {c-a\,c\,x}}{a\,c} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 18.12, size = 60, normalized size = 1.03 \[ - \frac {2 \sqrt {- a c x + c}}{a c} - \frac {2 \sqrt {2} \operatorname {atan}{\left (\frac {\sqrt {2}}{\sqrt {- \frac {1}{c}} \sqrt {- a c x + c}} \right )}}{a c \sqrt {- \frac {1}{c}}} \]
Verification of antiderivative is not currently implemented for this CAS.
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