Optimal. Leaf size=15 \[ -\frac {c (a x+1)^3}{3 a} \]
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Rubi [A] time = 0.03, antiderivative size = 15, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, integrand size = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.150, Rules used = {6167, 6140, 32} \[ -\frac {c (a x+1)^3}{3 a} \]
Antiderivative was successfully verified.
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Rule 32
Rule 6140
Rule 6167
Rubi steps
\begin {align*} \int e^{2 \coth ^{-1}(a x)} \left (c-a^2 c x^2\right ) \, dx &=-\int e^{2 \tanh ^{-1}(a x)} \left (c-a^2 c x^2\right ) \, dx\\ &=-\left (c \int (1+a x)^2 \, dx\right )\\ &=-\frac {c (1+a x)^3}{3 a}\\ \end {align*}
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Mathematica [A] time = 0.01, size = 20, normalized size = 1.33 \[ -c \left (\frac {a^2 x^3}{3}+a x^2+x\right ) \]
Antiderivative was successfully verified.
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fricas [A] time = 0.52, size = 21, normalized size = 1.40 \[ -\frac {1}{3} \, a^{2} c x^{3} - a c x^{2} - c x \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.12, size = 21, normalized size = 1.40 \[ -\frac {1}{3} \, a^{2} c x^{3} - a c x^{2} - c x \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.03, size = 14, normalized size = 0.93 \[ -\frac {c \left (a x +1\right )^{3}}{3 a} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.31, size = 21, normalized size = 1.40 \[ -\frac {1}{3} \, a^{2} c x^{3} - a c x^{2} - c x \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.03, size = 17, normalized size = 1.13 \[ -\frac {c\,x\,\left (a^2\,x^2+3\,a\,x+3\right )}{3} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.06, size = 20, normalized size = 1.33 \[ - \frac {a^{2} c x^{3}}{3} - a c x^{2} - c x \]
Verification of antiderivative is not currently implemented for this CAS.
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