Optimal. Leaf size=20 \[ \frac {1}{3} a^2 c^2 x^3-c^2 x \]
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Rubi [A] time = 0.05, antiderivative size = 20, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, integrand size = 18, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.167, Rules used = {6167, 6129, 41} \[ \frac {1}{3} a^2 c^2 x^3-c^2 x \]
Antiderivative was successfully verified.
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Rule 41
Rule 6129
Rule 6167
Rubi steps
\begin {align*} \int e^{2 \coth ^{-1}(a x)} (c-a c x)^2 \, dx &=-\int e^{2 \tanh ^{-1}(a x)} (c-a c x)^2 \, dx\\ &=-\left (c^2 \int (1-a x) (1+a x) \, dx\right )\\ &=-\left (c^2 \int \left (1-a^2 x^2\right ) \, dx\right )\\ &=-c^2 x+\frac {1}{3} a^2 c^2 x^3\\ \end {align*}
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Mathematica [A] time = 0.01, size = 17, normalized size = 0.85 \[ -c^2 \left (x-\frac {a^2 x^3}{3}\right ) \]
Antiderivative was successfully verified.
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fricas [A] time = 0.54, size = 18, normalized size = 0.90 \[ \frac {1}{3} \, a^{2} c^{2} x^{3} - c^{2} x \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.13, size = 18, normalized size = 0.90 \[ \frac {1}{3} \, a^{2} c^{2} x^{3} - c^{2} x \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.03, size = 17, normalized size = 0.85 \[ c^{2} \left (\frac {1}{3} x^{3} a^{2}-x \right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.31, size = 18, normalized size = 0.90 \[ \frac {1}{3} \, a^{2} c^{2} x^{3} - c^{2} x \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.03, size = 15, normalized size = 0.75 \[ \frac {c^2\,x\,\left (a^2\,x^2-3\right )}{3} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.06, size = 15, normalized size = 0.75 \[ \frac {a^{2} c^{2} x^{3}}{3} - c^{2} x \]
Verification of antiderivative is not currently implemented for this CAS.
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