Optimal. Leaf size=37 \[ \frac {x \sqrt {1-\frac {1}{a^2 x^2}} \log (a x+1)}{\sqrt {c-a^2 c x^2}} \]
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Rubi [A] time = 0.15, antiderivative size = 37, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.125, Rules used = {6192, 6193, 31} \[ \frac {x \sqrt {1-\frac {1}{a^2 x^2}} \log (a x+1)}{\sqrt {c-a^2 c x^2}} \]
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 = 37, normalized size = 1.00 \[ \frac {x \sqrt {1-\frac {1}{a^2 x^2}} \log (a x+1)}{\sqrt {c-a^2 c x^2}} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.49, size = 22, normalized size = 0.59 \[ -\frac {\sqrt {-a^{2} c} \log \left (a x + 1\right )}{a^{2} c} \]
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 \[ \int \frac {\sqrt {\frac {a x - 1}{a x + 1}}}{\sqrt {-a^{2} c x^{2} + c}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.05, size = 51, normalized size = 1.38 \[ -\frac {\ln \left (a x +1\right ) \sqrt {-c \left (a^{2} x^{2}-1\right )}\, \sqrt {\frac {a x -1}{a x +1}}}{a c \left (a x -1\right )} \]
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 \[ \int \frac {\sqrt {\frac {a x - 1}{a x + 1}}}{\sqrt {-a^{2} c x^{2} + c}}\,{d x} \]
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 \[ \int \frac {\sqrt {\frac {a\,x-1}{a\,x+1}}}{\sqrt {c-a^2\,c\,x^2}} \,d x \]
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 \[ \int \frac {\sqrt {\frac {a x - 1}{a x + 1}}}{\sqrt {- c \left (a x - 1\right ) \left (a x + 1\right )}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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