Optimal. Leaf size=39 \[ \frac {\sqrt {1-a^2 x^2} \log (a x+1)}{a \sqrt {c-a^2 c x^2}} \]
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Rubi [A] time = 0.07, antiderivative size = 39, 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 = {6143, 6140, 31} \[ \frac {\sqrt {1-a^2 x^2} \log (a x+1)}{a \sqrt {c-a^2 c x^2}} \]
Antiderivative was successfully verified.
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Rule 31
Rule 6140
Rule 6143
Rubi steps
\begin {align*} \int \frac {e^{-\tanh ^{-1}(a x)}}{\sqrt {c-a^2 c x^2}} \, dx &=\frac {\sqrt {1-a^2 x^2} \int \frac {e^{-\tanh ^{-1}(a x)}}{\sqrt {1-a^2 x^2}} \, dx}{\sqrt {c-a^2 c x^2}}\\ &=\frac {\sqrt {1-a^2 x^2} \int \frac {1}{1+a x} \, dx}{\sqrt {c-a^2 c x^2}}\\ &=\frac {\sqrt {1-a^2 x^2} \log (1+a x)}{a \sqrt {c-a^2 c x^2}}\\ \end {align*}
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Mathematica [A] time = 0.02, size = 39, normalized size = 1.00 \[ \frac {\sqrt {1-a^2 x^2} \log (a x+1)}{a \sqrt {c-a^2 c x^2}} \]
Antiderivative was successfully verified.
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fricas [B] time = 0.60, size = 227, normalized size = 5.82 \[ \left [\frac {\log \left (\frac {a^{6} c x^{6} + 4 \, a^{5} c x^{5} + 5 \, a^{4} c x^{4} - 4 \, a^{2} c x^{2} - 4 \, a c x - {\left (a^{4} x^{4} + 4 \, a^{3} x^{3} + 6 \, a^{2} x^{2} + 4 \, a x\right )} \sqrt {-a^{2} c x^{2} + c} \sqrt {-a^{2} x^{2} + 1} \sqrt {c} - 2 \, c}{a^{4} x^{4} + 2 \, a^{3} x^{3} - 2 \, a x - 1}\right )}{2 \, a \sqrt {c}}, \frac {\sqrt {-c} \arctan \left (\frac {\sqrt {-a^{2} c x^{2} + c} {\left (a^{2} x^{2} + 2 \, a x + 2\right )} \sqrt {-a^{2} x^{2} + 1} \sqrt {-c}}{a^{4} c x^{4} + 2 \, a^{3} c x^{3} - a^{2} c x^{2} - 2 \, a c x}\right )}{a c}\right ] \]
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 \[ \int \frac {\sqrt {-a^{2} x^{2} + 1}}{\sqrt {-a^{2} c x^{2} + c} {\left (a x + 1\right )}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.03, size = 40, normalized size = 1.03 \[ \frac {\sqrt {-c \left (a^{2} x^{2}-1\right )}\, \ln \left (a x +1\right )}{\sqrt {-a^{2} x^{2}+1}\, c a} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.33, size = 13, normalized size = 0.33 \[ \frac {\log \left (a x + 1\right )}{a \sqrt {c}} \]
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 \[ \int \frac {\sqrt {1-a^2\,x^2}}{\sqrt {c-a^2\,c\,x^2}\,\left (a\,x+1\right )} \,d x \]
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 \[ \int \frac {\sqrt {- \left (a x - 1\right ) \left (a x + 1\right )}}{\sqrt {- c \left (a x - 1\right ) \left (a x + 1\right )} \left (a x + 1\right )}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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