Optimal. Leaf size=14 \[ -\frac {1}{2} a c x^2-c x \]
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Rubi [C] time = 0.01, antiderivative size = 26, normalized size of antiderivative = 1.86, number of steps used = 1, number of rules used = 1, integrand size = 16, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.062, Rules used = {2288} \[ \frac {c \left (1-a^2 x^2\right ) e^{2 \coth ^{-1}(a x)}}{2 a} \]
Antiderivative was successfully verified.
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Rule 2288
Rubi steps
\begin {align*} \int e^{2 \coth ^{-1}(a x)} (c-a c x) \, dx &=\frac {c e^{2 \coth ^{-1}(a x)} \left (1-a^2 x^2\right )}{2 a}\\ \end {align*}
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Mathematica [C] time = 0.01, size = 26, normalized size = 1.86 \[ \frac {c \left (1-a^2 x^2\right ) e^{2 \coth ^{-1}(a x)}}{2 a} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.54, size = 12, normalized size = 0.86 \[ -\frac {1}{2} \, a c x^{2} - c x \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.12, size = 12, normalized size = 0.86 \[ -\frac {1}{2} \, a c x^{2} - c x \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.03, size = 13, normalized size = 0.93 \[ c \left (-\frac {1}{2} a \,x^{2}-x \right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.32, size = 12, normalized size = 0.86 \[ -\frac {1}{2} \, a c x^{2} - c x \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.02, size = 9, normalized size = 0.64 \[ -\frac {c\,x\,\left (a\,x+2\right )}{2} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.05, size = 12, normalized size = 0.86 \[ - \frac {a c x^{2}}{2} - c x \]
Verification of antiderivative is not currently implemented for this CAS.
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