Optimal. Leaf size=79 \[ \frac {2 x \sqrt {1-\frac {1}{a^2 x^2}}}{(1-a x) \sqrt {c-a^2 c x^2}}+\frac {x \sqrt {1-\frac {1}{a^2 x^2}} \log (1-a x)}{\sqrt {c-a^2 c x^2}} \]
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Rubi [A] time = 0.17, antiderivative size = 79, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.125, Rules used = {6192, 6193, 43} \[ \frac {2 x \sqrt {1-\frac {1}{a^2 x^2}}}{(1-a x) \sqrt {c-a^2 c x^2}}+\frac {x \sqrt {1-\frac {1}{a^2 x^2}} \log (1-a x)}{\sqrt {c-a^2 c x^2}} \]
Antiderivative was successfully verified.
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Rule 43
Rule 6192
Rule 6193
Rubi steps
\begin {align*} \int \frac {e^{3 \coth ^{-1}(a x)}}{\sqrt {c-a^2 c x^2}} \, dx &=\frac {\left (\sqrt {1-\frac {1}{a^2 x^2}} x\right ) \int \frac {e^{3 \coth ^{-1}(a x)}}{\sqrt {1-\frac {1}{a^2 x^2}} x} \, dx}{\sqrt {c-a^2 c x^2}}\\ &=\frac {\left (a \sqrt {1-\frac {1}{a^2 x^2}} x\right ) \int \frac {1+a x}{(-1+a x)^2} \, dx}{\sqrt {c-a^2 c x^2}}\\ &=\frac {\left (a \sqrt {1-\frac {1}{a^2 x^2}} x\right ) \int \left (\frac {2}{(-1+a x)^2}+\frac {1}{-1+a x}\right ) \, dx}{\sqrt {c-a^2 c x^2}}\\ &=\frac {2 \sqrt {1-\frac {1}{a^2 x^2}} x}{(1-a x) \sqrt {c-a^2 c x^2}}+\frac {\sqrt {1-\frac {1}{a^2 x^2}} x \log (1-a x)}{\sqrt {c-a^2 c x^2}}\\ \end {align*}
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Mathematica [A] time = 0.03, size = 53, normalized size = 0.67 \[ \frac {x \sqrt {1-\frac {1}{a^2 x^2}} ((a x-1) \log (1-a x)-2)}{(a x-1) \sqrt {c-a^2 c x^2}} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.56, size = 39, normalized size = 0.49 \[ -\frac {\sqrt {-a^{2} c} {\left ({\left (a x - 1\right )} \log \left (a x - 1\right ) - 2\right )}}{a^{3} c x - a^{2} c} \]
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 \[ \int \frac {1}{\sqrt {-a^{2} c x^{2} + c} \left (\frac {a x - 1}{a x + 1}\right )^{\frac {3}{2}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.06, size = 64, normalized size = 0.81 \[ -\frac {\sqrt {-c \left (a^{2} x^{2}-1\right )}\, \left (\ln \left (a x -1\right ) x a -\ln \left (a x -1\right )-2\right )}{a c \left (a x +1\right )^{2} \left (\frac {a x -1}{a x +1}\right )^{\frac {3}{2}}} \]
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 \[ \int \frac {1}{\sqrt {-a^{2} c x^{2} + c} \left (\frac {a x - 1}{a x + 1}\right )^{\frac {3}{2}}}\,{d x} \]
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 \[ \int \frac {1}{\sqrt {c-a^2\,c\,x^2}\,{\left (\frac {a\,x-1}{a\,x+1}\right )}^{3/2}} \,d x \]
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 \[ \int \frac {1}{\left (\frac {a x - 1}{a x + 1}\right )^{\frac {3}{2}} \sqrt {- c \left (a x - 1\right ) \left (a x + 1\right )}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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