Optimal. Leaf size=36 \[ -\frac {2 \sqrt {c-a c x}}{a c}-\frac {4}{a \sqrt {c-a c x}} \]
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Rubi [A] time = 0.08, antiderivative size = 36, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 4, integrand size = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.200, Rules used = {6167, 6130, 21, 43} \[ -\frac {2 \sqrt {c-a c x}}{a c}-\frac {4}{a \sqrt {c-a c x}} \]
Antiderivative was successfully verified.
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Rule 21
Rule 43
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\\ &=-\left (c \int \frac {1+a x}{(c-a c x)^{3/2}} \, dx\right )\\ &=-\left (c \int \left (\frac {2}{(c-a c x)^{3/2}}-\frac {1}{c \sqrt {c-a c x}}\right ) \, dx\right )\\ &=-\frac {4}{a \sqrt {c-a c x}}-\frac {2 \sqrt {c-a c x}}{a c}\\ \end {align*}
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Mathematica [A] time = 0.03, size = 21, normalized size = 0.58 \[ \frac {2 a x-6}{a \sqrt {c-a c x}} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.62, size = 29, normalized size = 0.81 \[ -\frac {2 \, \sqrt {-a c x + c} {\left (a x - 3\right )}}{a^{2} c x - a c} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.17, size = 32, normalized size = 0.89 \[ -\frac {4}{\sqrt {-a c x + c} a} - \frac {2 \, \sqrt {-a c x + c}}{a c} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.04, size = 20, normalized size = 0.56 \[ \frac {2 a x -6}{a \sqrt {-a c x +c}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.31, size = 30, normalized size = 0.83 \[ -\frac {2 \, {\left (\sqrt {-a c x + c} + \frac {2 \, c}{\sqrt {-a c x + c}}\right )}}{a c} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 1.21, size = 19, normalized size = 0.53 \[ \frac {2\,a\,x-6}{a\,\sqrt {c-a\,c\,x}} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 14.07, size = 49, normalized size = 1.36 \[ \begin {cases} \frac {- \frac {2}{\sqrt {- a c x + c}} + \frac {2 \left (- \frac {c}{\sqrt {- a c x + c}} - \sqrt {- a c x + c}\right )}{c}}{a} & \text {for}\: a \neq 0 \\- \frac {x}{\sqrt {c}} & \text {otherwise} \end {cases} \]
Verification of antiderivative is not currently implemented for this CAS.
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