Optimal. Leaf size=72 \[ \frac {x \sqrt {1-\frac {1}{a^2 x^2}}}{\sqrt {c-\frac {c}{a^2 x^2}}}+\frac {\sqrt {1-\frac {1}{a^2 x^2}} \log (1-a x)}{a \sqrt {c-\frac {c}{a^2 x^2}}} \]
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Rubi [A] time = 0.11, antiderivative size = 72, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, integrand size = 22, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.136, Rules used = {6197, 6193, 43} \[ \frac {x \sqrt {1-\frac {1}{a^2 x^2}}}{\sqrt {c-\frac {c}{a^2 x^2}}}+\frac {\sqrt {1-\frac {1}{a^2 x^2}} \log (1-a x)}{a \sqrt {c-\frac {c}{a^2 x^2}}} \]
Antiderivative was successfully verified.
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Rule 43
Rule 6193
Rule 6197
Rubi steps
\begin {align*} \int \frac {e^{\coth ^{-1}(a x)}}{\sqrt {c-\frac {c}{a^2 x^2}}} \, dx &=\frac {\sqrt {1-\frac {1}{a^2 x^2}} \int \frac {e^{\coth ^{-1}(a x)}}{\sqrt {1-\frac {1}{a^2 x^2}}} \, dx}{\sqrt {c-\frac {c}{a^2 x^2}}}\\ &=\frac {\left (a \sqrt {1-\frac {1}{a^2 x^2}}\right ) \int \frac {x}{-1+a x} \, dx}{\sqrt {c-\frac {c}{a^2 x^2}}}\\ &=\frac {\left (a \sqrt {1-\frac {1}{a^2 x^2}}\right ) \int \left (\frac {1}{a}+\frac {1}{a (-1+a x)}\right ) \, dx}{\sqrt {c-\frac {c}{a^2 x^2}}}\\ &=\frac {\sqrt {1-\frac {1}{a^2 x^2}} x}{\sqrt {c-\frac {c}{a^2 x^2}}}+\frac {\sqrt {1-\frac {1}{a^2 x^2}} \log (1-a x)}{a \sqrt {c-\frac {c}{a^2 x^2}}}\\ \end {align*}
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Mathematica [A] time = 0.03, size = 44, normalized size = 0.61 \[ \frac {\sqrt {1-\frac {1}{a^2 x^2}} (a x+\log (1-a x))}{a \sqrt {c-\frac {c}{a^2 x^2}}} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.64, size = 24, normalized size = 0.33 \[ \frac {\sqrt {a^{2} c} {\left (a x + \log \left (a x - 1\right )\right )}}{a^{2} c} \]
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 \[ \int \frac {1}{\sqrt {c - \frac {c}{a^{2} x^{2}}} \sqrt {\frac {a x - 1}{a x + 1}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.05, size = 57, normalized size = 0.79 \[ \frac {\left (a x -1\right ) \left (a x +\ln \left (a x -1\right )\right )}{\sqrt {\frac {a x -1}{a x +1}}\, \sqrt {\frac {c \left (a^{2} x^{2}-1\right )}{a^{2} x^{2}}}\, x \,a^{2}} \]
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 \[ \int \frac {1}{\sqrt {c - \frac {c}{a^{2} x^{2}}} \sqrt {\frac {a x - 1}{a x + 1}}}\,{d x} \]
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 \[ \int \frac {1}{\sqrt {c-\frac {c}{a^2\,x^2}}\,\sqrt {\frac {a\,x-1}{a\,x+1}}} \,d x \]
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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