Optimal. Leaf size=43 \[ -\frac {\sqrt {c-a c x}}{x}-3 a \sqrt {c} \tanh ^{-1}\left (\frac {\sqrt {c-a c x}}{\sqrt {c}}\right ) \]
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Rubi [A] time = 0.11, antiderivative size = 43, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 5, integrand size = 23, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.217, Rules used = {6130, 21, 78, 63, 208} \[ -\frac {\sqrt {c-a c x}}{x}-3 a \sqrt {c} \tanh ^{-1}\left (\frac {\sqrt {c-a c x}}{\sqrt {c}}\right ) \]
Antiderivative was successfully verified.
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Rule 21
Rule 63
Rule 78
Rule 208
Rule 6130
Rubi steps
\begin {align*} \int \frac {e^{2 \tanh ^{-1}(a x)} \sqrt {c-a c x}}{x^2} \, dx &=\int \frac {(1+a x) \sqrt {c-a c x}}{x^2 (1-a x)} \, dx\\ &=c \int \frac {1+a x}{x^2 \sqrt {c-a c x}} \, dx\\ &=-\frac {\sqrt {c-a c x}}{x}+\frac {1}{2} (3 a c) \int \frac {1}{x \sqrt {c-a c x}} \, dx\\ &=-\frac {\sqrt {c-a c x}}{x}-3 \operatorname {Subst}\left (\int \frac {1}{\frac {1}{a}-\frac {x^2}{a c}} \, dx,x,\sqrt {c-a c x}\right )\\ &=-\frac {\sqrt {c-a c x}}{x}-3 a \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 = 43, normalized size = 1.00 \[ -\frac {\sqrt {c-a c x}}{x}-3 a \sqrt {c} \tanh ^{-1}\left (\frac {\sqrt {c-a c x}}{\sqrt {c}}\right ) \]
Antiderivative was successfully verified.
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fricas [A] time = 0.50, size = 96, normalized size = 2.23 \[ \left [\frac {3 \, a \sqrt {c} x \log \left (\frac {a c x + 2 \, \sqrt {-a c x + c} \sqrt {c} - 2 \, c}{x}\right ) - 2 \, \sqrt {-a c x + c}}{2 \, x}, \frac {3 \, a \sqrt {-c} x \arctan \left (\frac {\sqrt {-a c x + c} \sqrt {-c}}{c}\right ) - \sqrt {-a c x + c}}{x}\right ] \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.17, size = 47, normalized size = 1.09 \[ \frac {\frac {3 \, a^{2} c \arctan \left (\frac {\sqrt {-a c x + c}}{\sqrt {-c}}\right )}{\sqrt {-c}} - \frac {\sqrt {-a c x + c} a}{x}}{a} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.04, size = 45, normalized size = 1.05 \[ 2 a c \left (-\frac {\sqrt {-a c x +c}}{2 x a c}-\frac {3 \arctanh \left (\frac {\sqrt {-a c x +c}}{\sqrt {c}}\right )}{2 \sqrt {c}}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.40, size = 62, normalized size = 1.44 \[ \frac {1}{2} \, a c {\left (\frac {3 \, \log \left (\frac {\sqrt {-a c x + c} - \sqrt {c}}{\sqrt {-a c x + c} + \sqrt {c}}\right )}{\sqrt {c}} - \frac {2 \, \sqrt {-a c x + c}}{a c x}\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.84, size = 35, normalized size = 0.81 \[ -\frac {\sqrt {c-a\,c\,x}}{x}-3\,a\,\sqrt {c}\,\mathrm {atanh}\left (\frac {\sqrt {c-a\,c\,x}}{\sqrt {c}}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [B] time = 19.59, size = 119, normalized size = 2.77 \[ \frac {a c^{2} \sqrt {\frac {1}{c^{3}}} \log {\left (- c^{2} \sqrt {\frac {1}{c^{3}}} + \sqrt {- a c x + c} \right )}}{2} - \frac {a c^{2} \sqrt {\frac {1}{c^{3}}} \log {\left (c^{2} \sqrt {\frac {1}{c^{3}}} + \sqrt {- a c x + c} \right )}}{2} + \frac {2 a c \operatorname {atan}{\left (\frac {\sqrt {- a c x + c}}{\sqrt {- c}} \right )}}{\sqrt {- c}} - \frac {\sqrt {- a c x + c}}{x} \]
Verification of antiderivative is not currently implemented for this CAS.
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