Optimal. Leaf size=78 \[ \frac {\sqrt {c-a c x}}{x}-5 a \sqrt {c} \tanh ^{-1}\left (\frac {\sqrt {c-a c x}}{\sqrt {c}}\right )+4 \sqrt {2} a \sqrt {c} \tanh ^{-1}\left (\frac {\sqrt {c-a c x}}{\sqrt {2} \sqrt {c}}\right ) \]
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Rubi [A] time = 0.23, antiderivative size = 78, 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, 98, 156, 63, 208, 206} \[ \frac {\sqrt {c-a c x}}{x}-5 a \sqrt {c} \tanh ^{-1}\left (\frac {\sqrt {c-a c x}}{\sqrt {c}}\right )+4 \sqrt {2} a \sqrt {c} \tanh ^{-1}\left (\frac {\sqrt {c-a c x}}{\sqrt {2} \sqrt {c}}\right ) \]
Antiderivative was successfully verified.
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Rule 21
Rule 63
Rule 98
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^2} \, dx &=-\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\\ &=-\frac {\int \frac {(c-a c x)^{3/2}}{x^2 (1+a x)} \, dx}{c}\\ &=\frac {\sqrt {c-a c x}}{x}+\frac {\int \frac {\frac {5 a c^2}{2}-\frac {3}{2} a^2 c^2 x}{x (1+a x) \sqrt {c-a c x}} \, dx}{c}\\ &=\frac {\sqrt {c-a c x}}{x}+\frac {1}{2} (5 a c) \int \frac {1}{x \sqrt {c-a c x}} \, dx-\left (4 a^2 c\right ) \int \frac {1}{(1+a x) \sqrt {c-a c x}} \, dx\\ &=\frac {\sqrt {c-a c x}}{x}-5 \operatorname {Subst}\left (\int \frac {1}{\frac {1}{a}-\frac {x^2}{a c}} \, dx,x,\sqrt {c-a c x}\right )+(8 a) \operatorname {Subst}\left (\int \frac {1}{2-\frac {x^2}{c}} \, dx,x,\sqrt {c-a c x}\right )\\ &=\frac {\sqrt {c-a c x}}{x}-5 a \sqrt {c} \tanh ^{-1}\left (\frac {\sqrt {c-a c x}}{\sqrt {c}}\right )+4 \sqrt {2} a \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 = 0.04, size = 78, normalized size = 1.00 \[ \frac {\sqrt {c-a c x}}{x}-5 a \sqrt {c} \tanh ^{-1}\left (\frac {\sqrt {c-a c x}}{\sqrt {c}}\right )+4 \sqrt {2} a \sqrt {c} \tanh ^{-1}\left (\frac {\sqrt {c-a c x}}{\sqrt {2} \sqrt {c}}\right ) \]
Antiderivative was successfully verified.
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fricas [A] time = 0.60, size = 176, normalized size = 2.26 \[ \left [\frac {4 \, \sqrt {2} a \sqrt {c} x \log \left (\frac {a c x - 2 \, \sqrt {2} \sqrt {-a c x + c} \sqrt {c} - 3 \, c}{a x + 1}\right ) + 5 \, a \sqrt {c} x \log \left (\frac {a c x + 2 \, \sqrt {-a c x + c} \sqrt {c} - 2 \, c}{x}\right ) + 2 \, \sqrt {-a c x + c}}{2 \, x}, -\frac {4 \, \sqrt {2} a \sqrt {-c} x \arctan \left (\frac {\sqrt {2} \sqrt {-a c x + c} \sqrt {-c}}{2 \, c}\right ) - 5 \, a \sqrt {-c} x \arctan \left (\frac {\sqrt {-a c x + c} \sqrt {-c}}{c}\right ) - \sqrt {-a c x + c}}{x}\right ] \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.14, size = 71, normalized size = 0.91 \[ -\frac {4 \, \sqrt {2} a c \arctan \left (\frac {\sqrt {2} \sqrt {-a c x + c}}{2 \, \sqrt {-c}}\right )}{\sqrt {-c}} + \frac {5 \, a c \arctan \left (\frac {\sqrt {-a c x + c}}{\sqrt {-c}}\right )}{\sqrt {-c}} + \frac {\sqrt {-a c x + c}}{x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.05, size = 71, normalized size = 0.91 \[ -2 a c \left (-\frac {2 \sqrt {2}\, \arctanh \left (\frac {\sqrt {-a c x +c}\, \sqrt {2}}{2 \sqrt {c}}\right )}{\sqrt {c}}-\frac {\sqrt {-a c x +c}}{2 x a c}+\frac {5 \arctanh \left (\frac {\sqrt {-a c x +c}}{\sqrt {c}}\right )}{2 \sqrt {c}}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.42, size = 111, normalized size = 1.42 \[ -\frac {1}{2} \, a c {\left (\frac {4 \, \sqrt {2} \log \left (-\frac {\sqrt {2} \sqrt {c} - \sqrt {-a c x + c}}{\sqrt {2} \sqrt {c} + \sqrt {-a c x + c}}\right )}{\sqrt {c}} - \frac {5 \, \log \left (\frac {\sqrt {-a c x + c} - \sqrt {c}}{\sqrt {-a c x + c} + \sqrt {c}}\right )}{\sqrt {c}} - \frac {2 \, \sqrt {-a c x + c}}{a c x}\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 1.24, size = 61, normalized size = 0.78 \[ \frac {\sqrt {c-a\,c\,x}}{x}-5\,a\,\sqrt {c}\,\mathrm {atanh}\left (\frac {\sqrt {c-a\,c\,x}}{\sqrt {c}}\right )+4\,\sqrt {2}\,a\,\sqrt {c}\,\mathrm {atanh}\left (\frac {\sqrt {2}\,\sqrt {c-a\,c\,x}}{2\,\sqrt {c}}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [B] time = 8.10, size = 162, normalized size = 2.08 \[ - \frac {a c^{2} \sqrt {\frac {1}{c^{3}}} \log {\left (- c^{2} \sqrt {\frac {1}{c^{3}}} + \sqrt {- a c x + c} \right )}}{2} + \frac {a c^{2} \sqrt {\frac {1}{c^{3}}} \log {\left (c^{2} \sqrt {\frac {1}{c^{3}}} + \sqrt {- a c x + c} \right )}}{2} + \frac {6 a c \operatorname {atan}{\left (\frac {\sqrt {- a c x + c}}{\sqrt {- c}} \right )}}{\sqrt {- c}} - \frac {4 \sqrt {2} a c \operatorname {atan}{\left (\frac {\sqrt {2} \sqrt {- a c x + c}}{2 \sqrt {- c}} \right )}}{\sqrt {- c}} + \frac {\sqrt {- a c x + c}}{x} \]
Verification of antiderivative is not currently implemented for this CAS.
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