Optimal. Leaf size=54 \[ \frac {c^2 (1-a x)^3}{3 a}+\frac {c^2 (1-a x)^2}{a}+\frac {8 c^2 \log (a x+1)}{a}-4 c^2 x \]
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Rubi [A] time = 0.04, antiderivative size = 54, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, integrand size = 18, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.111, Rules used = {6129, 43} \[ \frac {c^2 (1-a x)^3}{3 a}+\frac {c^2 (1-a x)^2}{a}+\frac {8 c^2 \log (a x+1)}{a}-4 c^2 x \]
Antiderivative was successfully verified.
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Rule 43
Rule 6129
Rubi steps
\begin {align*} \int e^{-2 \tanh ^{-1}(a x)} (c-a c x)^2 \, dx &=c^2 \int \frac {(1-a x)^3}{1+a x} \, dx\\ &=c^2 \int \left (-4-2 (1-a x)-(1-a x)^2+\frac {8}{1+a x}\right ) \, dx\\ &=-4 c^2 x+\frac {c^2 (1-a x)^2}{a}+\frac {c^2 (1-a x)^3}{3 a}+\frac {8 c^2 \log (1+a x)}{a}\\ \end {align*}
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Mathematica [A] time = 0.01, size = 39, normalized size = 0.72 \[ -\frac {c^2 \left (a^3 x^3-6 a^2 x^2+21 a x-24 \log (a x+1)-4\right )}{3 a} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.44, size = 45, normalized size = 0.83 \[ -\frac {a^{3} c^{2} x^{3} - 6 \, a^{2} c^{2} x^{2} + 21 \, a c^{2} x - 24 \, c^{2} \log \left (a x + 1\right )}{3 \, a} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.19, size = 68, normalized size = 1.26 \[ -\frac {{\left (c^{2} - \frac {9 \, c^{2}}{a x + 1} + \frac {36 \, c^{2}}{{\left (a x + 1\right )}^{2}}\right )} {\left (a x + 1\right )}^{3}}{3 \, a} - \frac {8 \, c^{2} \log \left (\frac {{\left | a x + 1 \right |}}{{\left (a x + 1\right )}^{2} {\left | a \right |}}\right )}{a} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.03, size = 42, normalized size = 0.78 \[ -\frac {a^{2} c^{2} x^{3}}{3}+2 c^{2} x^{2} a -7 c^{2} x +\frac {8 c^{2} \ln \left (a x +1\right )}{a} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.35, size = 41, normalized size = 0.76 \[ -\frac {1}{3} \, a^{2} c^{2} x^{3} + 2 \, a c^{2} x^{2} - 7 \, c^{2} x + \frac {8 \, c^{2} \log \left (a x + 1\right )}{a} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.05, size = 41, normalized size = 0.76 \[ 2\,a\,c^2\,x^2-7\,c^2\,x-\frac {a^2\,c^2\,x^3}{3}+\frac {8\,c^2\,\ln \left (a\,x+1\right )}{a} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.13, size = 41, normalized size = 0.76 \[ - \frac {a^{2} c^{2} x^{3}}{3} + 2 a c^{2} x^{2} - 7 c^{2} x + \frac {8 c^{2} \log {\left (a x + 1 \right )}}{a} \]
Verification of antiderivative is not currently implemented for this CAS.
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