Optimal. Leaf size=29 \[ \frac {2 (a x+1) e^{-\coth ^{-1}(a x)}}{a \sqrt {c-a c x}} \]
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Rubi [A] time = 0.04, antiderivative size = 29, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, integrand size = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.050, Rules used = {6174} \[ \frac {2 (a x+1) e^{-\coth ^{-1}(a x)}}{a \sqrt {c-a c x}} \]
Antiderivative was successfully verified.
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Rule 6174
Rubi steps
\begin {align*} \int \frac {e^{-\coth ^{-1}(a x)}}{\sqrt {c-a c x}} \, dx &=\frac {2 e^{-\coth ^{-1}(a x)} (1+a x)}{a \sqrt {c-a c x}}\\ \end {align*}
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Mathematica [A] time = 0.03, size = 28, normalized size = 0.97 \[ \frac {2 x \sqrt {1-\frac {1}{a^2 x^2}}}{\sqrt {c-a c x}} \]
Warning: Unable to verify antiderivative.
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fricas [A] time = 0.72, size = 44, normalized size = 1.52 \[ -\frac {2 \, \sqrt {-a c x + c} {\left (a x + 1\right )} \sqrt {\frac {a x - 1}{a x + 1}}}{a^{2} c x - a c} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [F(-2)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Exception raised: TypeError} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.04, size = 35, normalized size = 1.21 \[ \frac {2 \left (a x +1\right ) \sqrt {\frac {a x -1}{a x +1}}}{a \sqrt {-a c x +c}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.33, size = 29, normalized size = 1.00 \[ -\frac {2 \, {\left (a \sqrt {-c} x + \sqrt {-c}\right )}}{\sqrt {a x + 1} a c} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 1.25, size = 34, normalized size = 1.17 \[ \frac {\left (2\,x+\frac {2}{a}\right )\,\sqrt {\frac {a\,x-1}{a\,x+1}}}{\sqrt {c-a\,c\,x}} \]
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 \[ \int \frac {\sqrt {\frac {a x - 1}{a x + 1}}}{\sqrt {- c \left (a x - 1\right )}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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