Optimal. Leaf size=112 \[ \frac {\sqrt {c-a^2 c x^2}}{\sqrt {1-\frac {1}{a^2 x^2}}}+\frac {\log (x) \sqrt {c-a^2 c x^2}}{a x \sqrt {1-\frac {1}{a^2 x^2}}}-\frac {4 \sqrt {c-a^2 c x^2} \log (a x+1)}{a x \sqrt {1-\frac {1}{a^2 x^2}}} \]
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Rubi [A] time = 0.16, antiderivative size = 112, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, integrand size = 27, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.111, Rules used = {6192, 6193, 72} \[ \frac {\sqrt {c-a^2 c x^2}}{\sqrt {1-\frac {1}{a^2 x^2}}}+\frac {\log (x) \sqrt {c-a^2 c x^2}}{a x \sqrt {1-\frac {1}{a^2 x^2}}}-\frac {4 \sqrt {c-a^2 c x^2} \log (a x+1)}{a x \sqrt {1-\frac {1}{a^2 x^2}}} \]
Antiderivative was successfully verified.
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Rule 72
Rule 6192
Rule 6193
Rubi steps
\begin {align*} \int \frac {e^{-3 \coth ^{-1}(a x)} \sqrt {c-a^2 c x^2}}{x} \, dx &=\frac {\sqrt {c-a^2 c x^2} \int e^{-3 \coth ^{-1}(a x)} \sqrt {1-\frac {1}{a^2 x^2}} \, dx}{\sqrt {1-\frac {1}{a^2 x^2}} x}\\ &=\frac {\sqrt {c-a^2 c x^2} \int \frac {(-1+a x)^2}{x (1+a x)} \, dx}{a \sqrt {1-\frac {1}{a^2 x^2}} x}\\ &=\frac {\sqrt {c-a^2 c x^2} \int \left (a+\frac {1}{x}-\frac {4 a}{1+a x}\right ) \, dx}{a \sqrt {1-\frac {1}{a^2 x^2}} x}\\ &=\frac {\sqrt {c-a^2 c x^2}}{\sqrt {1-\frac {1}{a^2 x^2}}}+\frac {\sqrt {c-a^2 c x^2} \log (x)}{a \sqrt {1-\frac {1}{a^2 x^2}} x}-\frac {4 \sqrt {c-a^2 c x^2} \log (1+a x)}{a \sqrt {1-\frac {1}{a^2 x^2}} x}\\ \end {align*}
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Mathematica [A] time = 0.03, size = 50, normalized size = 0.45 \[ \frac {\sqrt {c-a^2 c x^2} (a x-4 \log (a x+1)+\log (x))}{a x \sqrt {1-\frac {1}{a^2 x^2}}} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.64, size = 26, normalized size = 0.23 \[ \frac {\sqrt {-a^{2} c} {\left (a x - 4 \, \log \left (a x + 1\right ) + \log \relax (x)\right )}}{a} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\sqrt {-a^{2} c x^{2} + c} \left (\frac {a x - 1}{a x + 1}\right )^{\frac {3}{2}}}{x}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.06, size = 57, normalized size = 0.51 \[ \frac {\left (a x +\ln \relax (x )-4 \ln \left (a x +1\right )\right ) \sqrt {-c \left (a^{2} x^{2}-1\right )}\, \left (a x +1\right ) \left (\frac {a x -1}{a x +1}\right )^{\frac {3}{2}}}{\left (a x -1\right )^{2}} \]
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 \[ \int \frac {\sqrt {-a^{2} c x^{2} + c} \left (\frac {a x - 1}{a x + 1}\right )^{\frac {3}{2}}}{x}\,{d x} \]
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 \[ \int \frac {\sqrt {c-a^2\,c\,x^2}\,{\left (\frac {a\,x-1}{a\,x+1}\right )}^{3/2}}{x} \,d x \]
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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