Optimal. Leaf size=17 \[ \frac {c^2 (a x+1)^3}{3 a} \]
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Rubi [A] time = 0.05, antiderivative size = 17, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, integrand size = 18, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.167, Rules used = {6167, 6129, 32} \[ \frac {c^2 (a x+1)^3}{3 a} \]
Antiderivative was successfully verified.
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Rule 32
Rule 6129
Rule 6167
Rubi steps
\begin {align*} \int e^{4 \coth ^{-1}(a x)} (c-a c x)^2 \, dx &=\int e^{4 \tanh ^{-1}(a x)} (c-a c x)^2 \, dx\\ &=c^2 \int (1+a x)^2 \, dx\\ &=\frac {c^2 (1+a x)^3}{3 a}\\ \end {align*}
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Mathematica [A] time = 0.01, size = 21, normalized size = 1.24 \[ c^2 \left (\frac {a^2 x^3}{3}+a x^2+x\right ) \]
Antiderivative was successfully verified.
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fricas [A] time = 0.59, size = 25, normalized size = 1.47 \[ \frac {1}{3} \, a^{2} c^{2} x^{3} + a c^{2} x^{2} + c^{2} x \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 0.14, size = 40, normalized size = 2.35 \[ \frac {{\left (c^{2} + \frac {6 \, c^{2}}{a x - 1} + \frac {12 \, c^{2}}{{\left (a x - 1\right )}^{2}}\right )} {\left (a x - 1\right )}^{3}}{3 \, a} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.03, size = 16, normalized size = 0.94 \[ \frac {c^{2} \left (a x +1\right )^{3}}{3 a} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.31, size = 25, normalized size = 1.47 \[ \frac {1}{3} \, a^{2} c^{2} x^{3} + a c^{2} x^{2} + c^{2} x \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.03, size = 19, normalized size = 1.12 \[ \frac {c^2\,x\,\left (a^2\,x^2+3\,a\,x+3\right )}{3} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.07, size = 24, normalized size = 1.41 \[ \frac {a^{2} c^{2} x^{3}}{3} + a c^{2} x^{2} + c^{2} x \]
Verification of antiderivative is not currently implemented for this CAS.
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