Optimal. Leaf size=21 \[ \frac {\sqrt {1-a^2 x^2}}{a c} \]
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Rubi [A] time = 0.07, antiderivative size = 21, 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 = {6131, 6128, 261} \[ \frac {\sqrt {1-a^2 x^2}}{a c} \]
Antiderivative was successfully verified.
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Rule 261
Rule 6128
Rule 6131
Rubi steps
\begin {align*} \int \frac {e^{-\tanh ^{-1}(a x)}}{c-\frac {c}{a x}} \, dx &=-\frac {a \int \frac {e^{-\tanh ^{-1}(a x)} x}{1-a x} \, dx}{c}\\ &=-\frac {a \int \frac {x}{\sqrt {1-a^2 x^2}} \, dx}{c}\\ &=\frac {\sqrt {1-a^2 x^2}}{a c}\\ \end {align*}
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Mathematica [A] time = 0.01, size = 21, normalized size = 1.00 \[ \frac {\sqrt {1-a^2 x^2}}{a c} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.55, size = 19, normalized size = 0.90 \[ \frac {\sqrt {-a^{2} x^{2} + 1}}{a c} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.16, size = 19, normalized size = 0.90 \[ \frac {\sqrt {-a^{2} x^{2} + 1}}{a c} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.03, size = 20, normalized size = 0.95 \[ \frac {\sqrt {-a^{2} x^{2}+1}}{a c} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.33, size = 22, normalized size = 1.05 \[ \frac {\sqrt {a x + 1} \sqrt {-a x + 1}}{a c} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.04, size = 19, normalized size = 0.90 \[ \frac {\sqrt {1-a^2\,x^2}}{a\,c} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 9.02, size = 53, normalized size = 2.52 \[ a \left (\begin {cases} \tilde {\infty } \left (\begin {cases} \frac {x^{2}}{2} & \text {for}\: a^{2} = 0 \\- \frac {\left (- a^{2} x^{2} + 1\right )^{\frac {3}{2}}}{3 a^{2}} & \text {otherwise} \end {cases}\right ) & \text {for}\: c = 0 \\- \frac {x^{2}}{2 c} & \text {for}\: a^{2} = 0 \\\frac {\sqrt {- a^{2} x^{2} + 1}}{a^{2} c} & \text {otherwise} \end {cases}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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