Optimal. Leaf size=134 \[ -\frac {2 \sqrt {\frac {1}{a x}+1} \sqrt {c-\frac {c}{a x}}}{\sqrt {1-\frac {1}{a x}}}-\frac {8 \sqrt {c-\frac {c}{a x}}}{\sqrt {1-\frac {1}{a x}} \sqrt {\frac {1}{a x}+1}}+\frac {2 \sqrt {c-\frac {c}{a x}} \tanh ^{-1}\left (\sqrt {\frac {1}{a x}+1}\right )}{\sqrt {1-\frac {1}{a x}}} \]
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Rubi [A] time = 0.26, antiderivative size = 134, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 5, integrand size = 27, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.185, Rules used = {6182, 6180, 87, 63, 208} \[ -\frac {2 \sqrt {\frac {1}{a x}+1} \sqrt {c-\frac {c}{a x}}}{\sqrt {1-\frac {1}{a x}}}-\frac {8 \sqrt {c-\frac {c}{a x}}}{\sqrt {1-\frac {1}{a x}} \sqrt {\frac {1}{a x}+1}}+\frac {2 \sqrt {c-\frac {c}{a x}} \tanh ^{-1}\left (\sqrt {\frac {1}{a x}+1}\right )}{\sqrt {1-\frac {1}{a x}}} \]
Antiderivative was successfully verified.
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Rule 63
Rule 87
Rule 208
Rule 6180
Rule 6182
Rubi steps
\begin {align*} \int \frac {e^{-3 \coth ^{-1}(a x)} \sqrt {c-\frac {c}{a x}}}{x} \, dx &=\frac {\sqrt {c-\frac {c}{a x}} \int \frac {e^{-3 \coth ^{-1}(a x)} \sqrt {1-\frac {1}{a x}}}{x} \, dx}{\sqrt {1-\frac {1}{a x}}}\\ &=-\frac {\sqrt {c-\frac {c}{a x}} \operatorname {Subst}\left (\int \frac {\left (1-\frac {x}{a}\right )^2}{x \left (1+\frac {x}{a}\right )^{3/2}} \, dx,x,\frac {1}{x}\right )}{\sqrt {1-\frac {1}{a x}}}\\ &=-\frac {\sqrt {c-\frac {c}{a x}} \operatorname {Subst}\left (\int \left (-\frac {4}{a \left (1+\frac {x}{a}\right )^{3/2}}+\frac {1}{a \sqrt {1+\frac {x}{a}}}+\frac {1}{x \sqrt {1+\frac {x}{a}}}\right ) \, dx,x,\frac {1}{x}\right )}{\sqrt {1-\frac {1}{a x}}}\\ &=-\frac {8 \sqrt {c-\frac {c}{a x}}}{\sqrt {1-\frac {1}{a x}} \sqrt {1+\frac {1}{a x}}}-\frac {2 \sqrt {1+\frac {1}{a x}} \sqrt {c-\frac {c}{a x}}}{\sqrt {1-\frac {1}{a x}}}-\frac {\sqrt {c-\frac {c}{a x}} \operatorname {Subst}\left (\int \frac {1}{x \sqrt {1+\frac {x}{a}}} \, dx,x,\frac {1}{x}\right )}{\sqrt {1-\frac {1}{a x}}}\\ &=-\frac {8 \sqrt {c-\frac {c}{a x}}}{\sqrt {1-\frac {1}{a x}} \sqrt {1+\frac {1}{a x}}}-\frac {2 \sqrt {1+\frac {1}{a x}} \sqrt {c-\frac {c}{a x}}}{\sqrt {1-\frac {1}{a x}}}-\frac {\left (2 a \sqrt {c-\frac {c}{a x}}\right ) \operatorname {Subst}\left (\int \frac {1}{-a+a x^2} \, dx,x,\sqrt {1+\frac {1}{a x}}\right )}{\sqrt {1-\frac {1}{a x}}}\\ &=-\frac {8 \sqrt {c-\frac {c}{a x}}}{\sqrt {1-\frac {1}{a x}} \sqrt {1+\frac {1}{a x}}}-\frac {2 \sqrt {1+\frac {1}{a x}} \sqrt {c-\frac {c}{a x}}}{\sqrt {1-\frac {1}{a x}}}+\frac {2 \sqrt {c-\frac {c}{a x}} \tanh ^{-1}\left (\sqrt {1+\frac {1}{a x}}\right )}{\sqrt {1-\frac {1}{a x}}}\\ \end {align*}
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Mathematica [A] time = 0.41, size = 131, normalized size = 0.98 \[ -\frac {2 a x \sqrt {1-\frac {1}{a^2 x^2}} (5 a x+1) \sqrt {c-\frac {c}{a x}}}{a^2 x^2-1}+\sqrt {c} \log \left (2 a^2 \sqrt {c} x^2 \sqrt {1-\frac {1}{a^2 x^2}} \sqrt {c-\frac {c}{a x}}+c \left (2 a^2 x^2-a x-1\right )\right )-\sqrt {c} \log (1-a x) \]
Warning: Unable to verify antiderivative.
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fricas [A] time = 0.65, size = 277, normalized size = 2.07 \[ \left [\frac {{\left (a x - 1\right )} \sqrt {c} \log \left (-\frac {8 \, a^{3} c x^{3} - 7 \, a c x + 4 \, {\left (2 \, a^{3} x^{3} + 3 \, a^{2} x^{2} + a x\right )} \sqrt {c} \sqrt {\frac {a x - 1}{a x + 1}} \sqrt {\frac {a c x - c}{a x}} - c}{a x - 1}\right ) - 4 \, {\left (5 \, a x + 1\right )} \sqrt {\frac {a x - 1}{a x + 1}} \sqrt {\frac {a c x - c}{a x}}}{2 \, {\left (a x - 1\right )}}, -\frac {{\left (a x - 1\right )} \sqrt {-c} \arctan \left (\frac {2 \, {\left (a^{2} x^{2} + a x\right )} \sqrt {-c} \sqrt {\frac {a x - 1}{a x + 1}} \sqrt {\frac {a c x - c}{a x}}}{2 \, a^{2} c x^{2} - a c x - c}\right ) + 2 \, {\left (5 \, a x + 1\right )} \sqrt {\frac {a x - 1}{a x + 1}} \sqrt {\frac {a c x - c}{a x}}}{a x - 1}\right ] \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [F(-2)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Exception raised: TypeError} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.07, size = 151, normalized size = 1.13 \[ -\frac {\left (\frac {a x -1}{a x +1}\right )^{\frac {3}{2}} \left (a x +1\right ) \sqrt {\frac {c \left (a x -1\right )}{a x}}\, \left (10 a^{\frac {3}{2}} x \sqrt {\left (a x +1\right ) x}-\ln \left (\frac {2 \sqrt {\left (a x +1\right ) x}\, \sqrt {a}+2 a x +1}{2 \sqrt {a}}\right ) x^{2} a^{2}-\ln \left (\frac {2 \sqrt {\left (a x +1\right ) x}\, \sqrt {a}+2 a x +1}{2 \sqrt {a}}\right ) x a +2 \sqrt {\left (a x +1\right ) x}\, \sqrt {a}\right )}{\left (a x -1\right )^{2} \sqrt {a}\, \sqrt {\left (a x +1\right ) x}} \]
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 \[ \int \frac {\sqrt {c - \frac {c}{a x}} \left (\frac {a x - 1}{a x + 1}\right )^{\frac {3}{2}}}{x}\,{d x} \]
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 \[ \int \frac {\sqrt {c-\frac {c}{a\,x}}\,{\left (\frac {a\,x-1}{a\,x+1}\right )}^{3/2}}{x} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
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