Optimal. Leaf size=30 \[ \frac {2 \sqrt {1-a^2 x^2}}{a \sqrt {c-a c x}} \]
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Rubi [A] time = 0.05, antiderivative size = 30, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.100, Rules used = {6127, 649} \[ \frac {2 \sqrt {1-a^2 x^2}}{a \sqrt {c-a c x}} \]
Antiderivative was successfully verified.
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Rule 649
Rule 6127
Rubi steps
\begin {align*} \int \frac {e^{-\tanh ^{-1}(a x)}}{\sqrt {c-a c x}} \, dx &=\frac {\int \frac {\sqrt {c-a c x}}{\sqrt {1-a^2 x^2}} \, dx}{c}\\ &=\frac {2 \sqrt {1-a^2 x^2}}{a \sqrt {c-a c x}}\\ \end {align*}
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Mathematica [A] time = 0.02, size = 30, normalized size = 1.00 \[ \frac {2 \sqrt {1-a^2 x^2}}{a \sqrt {c-a c x}} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.59, size = 36, normalized size = 1.20 \[ -\frac {2 \, \sqrt {-a^{2} x^{2} + 1} \sqrt {-a c x + c}}{a^{2} c x - a c} \]
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 = 0.03, size = 27, normalized size = 0.90 \[ \frac {2 \sqrt {-a^{2} x^{2}+1}}{a \sqrt {-a c x +c}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.37, size = 15, normalized size = 0.50 \[ \frac {2 \, \sqrt {a x + 1}}{a \sqrt {c}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.96, size = 26, normalized size = 0.87 \[ \frac {2\,\sqrt {1-a^2\,x^2}}{a\,\sqrt {c-a\,c\,x}} \]
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 \[ \int \frac {\sqrt {- \left (a x - 1\right ) \left (a x + 1\right )}}{\sqrt {- c \left (a x - 1\right )} \left (a x + 1\right )}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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