Optimal. Leaf size=38 \[ \frac {x \sqrt {1-\frac {1}{a^2 x^2}} \log (1-a x)}{\sqrt {c-a^2 c x^2}} \]
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Rubi [A] time = 0.15, antiderivative size = 38, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, integrand size = 22, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.136, Rules used = {6192, 6193, 31} \[ \frac {x \sqrt {1-\frac {1}{a^2 x^2}} \log (1-a x)}{\sqrt {c-a^2 c x^2}} \]
Antiderivative was successfully verified.
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Rule 31
Rule 6192
Rule 6193
Rubi steps
\begin {align*} \int \frac {e^{\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^{\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}{-1+a x} \, dx}{\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 = 0.02, size = 38, normalized size = 1.00 \[ \frac {x \sqrt {1-\frac {1}{a^2 x^2}} \log (1-a x)}{\sqrt {c-a^2 c x^2}} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.63, size = 22, normalized size = 0.58 \[ -\frac {\sqrt {-a^{2} c} \log \left (a x - 1\right )}{a^{2} c} \]
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 \[ \int \frac {1}{\sqrt {-a^{2} c x^{2} + c} \sqrt {\frac {a x - 1}{a x + 1}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.06, size = 51, normalized size = 1.34 \[ -\frac {\ln \left (a x -1\right ) \sqrt {-c \left (a^{2} x^{2}-1\right )}}{a c \left (a x +1\right ) \sqrt {\frac {a x -1}{a x +1}}} \]
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 \[ \int \frac {1}{\sqrt {-a^{2} c x^{2} + c} \sqrt {\frac {a x - 1}{a x + 1}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.03 \[ \int \frac {1}{\sqrt {c-a^2\,c\,x^2}\,\sqrt {\frac {a\,x-1}{a\,x+1}}} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {1}{\sqrt {\frac {a x - 1}{a x + 1}} \sqrt {- c \left (a x - 1\right ) \left (a x + 1\right )}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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