Optimal. Leaf size=95 \[ \frac {x \sqrt {c-\frac {c}{a x}}}{c}-\frac {3 \tanh ^{-1}\left (\frac {\sqrt {c-\frac {c}{a x}}}{\sqrt {c}}\right )}{a \sqrt {c}}+\frac {2 \sqrt {2} \tanh ^{-1}\left (\frac {\sqrt {c-\frac {c}{a x}}}{\sqrt {2} \sqrt {c}}\right )}{a \sqrt {c}} \]
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Rubi [A] time = 0.21, antiderivative size = 95, normalized size of antiderivative = 1.00, number of steps used = 11, number of rules used = 9, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.375, Rules used = {6167, 6133, 25, 514, 375, 99, 156, 63, 208} \[ \frac {x \sqrt {c-\frac {c}{a x}}}{c}-\frac {3 \tanh ^{-1}\left (\frac {\sqrt {c-\frac {c}{a x}}}{\sqrt {c}}\right )}{a \sqrt {c}}+\frac {2 \sqrt {2} \tanh ^{-1}\left (\frac {\sqrt {c-\frac {c}{a x}}}{\sqrt {2} \sqrt {c}}\right )}{a \sqrt {c}} \]
Antiderivative was successfully verified.
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Rule 25
Rule 63
Rule 99
Rule 156
Rule 208
Rule 375
Rule 514
Rule 6133
Rule 6167
Rubi steps
\begin {align*} \int \frac {e^{-2 \coth ^{-1}(a x)}}{\sqrt {c-\frac {c}{a x}}} \, dx &=-\int \frac {e^{-2 \tanh ^{-1}(a x)}}{\sqrt {c-\frac {c}{a x}}} \, dx\\ &=-\int \frac {1-a x}{\sqrt {c-\frac {c}{a x}} (1+a x)} \, dx\\ &=\frac {a \int \frac {\sqrt {c-\frac {c}{a x}} x}{1+a x} \, dx}{c}\\ &=\frac {a \int \frac {\sqrt {c-\frac {c}{a x}}}{a+\frac {1}{x}} \, dx}{c}\\ &=-\frac {a \operatorname {Subst}\left (\int \frac {\sqrt {c-\frac {c x}{a}}}{x^2 (a+x)} \, dx,x,\frac {1}{x}\right )}{c}\\ &=\frac {\sqrt {c-\frac {c}{a x}} x}{c}-\frac {\operatorname {Subst}\left (\int \frac {-\frac {3 c}{2}+\frac {c x}{2 a}}{x (a+x) \sqrt {c-\frac {c x}{a}}} \, dx,x,\frac {1}{x}\right )}{c}\\ &=\frac {\sqrt {c-\frac {c}{a x}} x}{c}+\frac {3 \operatorname {Subst}\left (\int \frac {1}{x \sqrt {c-\frac {c x}{a}}} \, dx,x,\frac {1}{x}\right )}{2 a}-\frac {2 \operatorname {Subst}\left (\int \frac {1}{(a+x) \sqrt {c-\frac {c x}{a}}} \, dx,x,\frac {1}{x}\right )}{a}\\ &=\frac {\sqrt {c-\frac {c}{a x}} x}{c}-\frac {3 \operatorname {Subst}\left (\int \frac {1}{a-\frac {a x^2}{c}} \, dx,x,\sqrt {c-\frac {c}{a x}}\right )}{c}+\frac {4 \operatorname {Subst}\left (\int \frac {1}{2 a-\frac {a x^2}{c}} \, dx,x,\sqrt {c-\frac {c}{a x}}\right )}{c}\\ &=\frac {\sqrt {c-\frac {c}{a x}} x}{c}-\frac {3 \tanh ^{-1}\left (\frac {\sqrt {c-\frac {c}{a x}}}{\sqrt {c}}\right )}{a \sqrt {c}}+\frac {2 \sqrt {2} \tanh ^{-1}\left (\frac {\sqrt {c-\frac {c}{a x}}}{\sqrt {2} \sqrt {c}}\right )}{a \sqrt {c}}\\ \end {align*}
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Mathematica [A] time = 0.05, size = 95, normalized size = 1.00 \[ \frac {x \sqrt {c-\frac {c}{a x}}}{c}-\frac {3 \tanh ^{-1}\left (\frac {\sqrt {c-\frac {c}{a x}}}{\sqrt {c}}\right )}{a \sqrt {c}}+\frac {2 \sqrt {2} \tanh ^{-1}\left (\frac {\sqrt {c-\frac {c}{a x}}}{\sqrt {2} \sqrt {c}}\right )}{a \sqrt {c}} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.61, size = 234, normalized size = 2.46 \[ \left [\frac {2 \, a x \sqrt {\frac {a c x - c}{a x}} + 2 \, \sqrt {2} \sqrt {c} \log \left (-\frac {\frac {2 \, \sqrt {2} a x \sqrt {\frac {a c x - c}{a x}}}{\sqrt {c}} + 3 \, a x - 1}{a x + 1}\right ) + 3 \, \sqrt {c} \log \left (-2 \, a c x + 2 \, a \sqrt {c} x \sqrt {\frac {a c x - c}{a x}} + c\right )}{2 \, a c}, -\frac {2 \, \sqrt {2} c \sqrt {-\frac {1}{c}} \arctan \left (\frac {\sqrt {2} a x \sqrt {-\frac {1}{c}} \sqrt {\frac {a c x - c}{a x}}}{a x - 1}\right ) - a x \sqrt {\frac {a c x - c}{a x}} - 3 \, \sqrt {-c} \arctan \left (\frac {\sqrt {-c} \sqrt {\frac {a c x - c}{a x}}}{c}\right )}{a c}\right ] \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.16, size = 131, normalized size = 1.38 \[ -a c {\left (\frac {2 \, \sqrt {2} \arctan \left (\frac {\sqrt {2} \sqrt {\frac {a c x - c}{a x}}}{2 \, \sqrt {-c}}\right )}{a^{2} \sqrt {-c} c} - \frac {3 \, \arctan \left (\frac {\sqrt {\frac {a c x - c}{a x}}}{\sqrt {-c}}\right )}{a^{2} \sqrt {-c} c} - \frac {\sqrt {\frac {a c x - c}{a x}}}{a^{2} {\left (c - \frac {a c x - c}{a x}\right )} c}\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.04, size = 136, normalized size = 1.43 \[ -\frac {\sqrt {\frac {c \left (a x -1\right )}{a x}}\, x \left (-2 \sqrt {\left (a x -1\right ) x}\, a^{\frac {3}{2}} \sqrt {\frac {1}{a}}+3 \ln \left (\frac {2 \sqrt {\left (a x -1\right ) x}\, \sqrt {a}+2 a x -1}{2 \sqrt {a}}\right ) a \sqrt {\frac {1}{a}}+2 \sqrt {2}\, \ln \left (\frac {2 \sqrt {2}\, \sqrt {\frac {1}{a}}\, \sqrt {\left (a x -1\right ) x}\, a -3 a x +1}{a x +1}\right ) \sqrt {a}\right )}{2 \sqrt {\left (a x -1\right ) x}\, c \,a^{\frac {3}{2}} \sqrt {\frac {1}{a}}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {a x - 1}{{\left (a x + 1\right )} \sqrt {c - \frac {c}{a x}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.01 \[ \int \frac {a\,x-1}{\sqrt {c-\frac {c}{a\,x}}\,\left (a\,x+1\right )} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {a x - 1}{\sqrt {- c \left (-1 + \frac {1}{a x}\right )} \left (a x + 1\right )}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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