Optimal. Leaf size=37 \[ -\frac {c^2 (1-a x)^4}{2 a}+\frac {c^2 (1-a x)^5}{5 a} \]
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Rubi [A]
time = 0.03, antiderivative size = 37, normalized size of antiderivative = 1.00, number of steps
used = 3, number of rules used = 2, integrand size = 22, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.091, Rules used = {6275, 45}
\begin {gather*} \frac {c^2 (1-a x)^5}{5 a}-\frac {c^2 (1-a x)^4}{2 a} \end {gather*}
Antiderivative was successfully verified.
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Rule 45
Rule 6275
Rubi steps
\begin {align*} \int e^{-2 \tanh ^{-1}(a x)} \left (c-a^2 c x^2\right )^2 \, dx &=c^2 \int (1-a x)^3 (1+a x) \, dx\\ &=c^2 \int \left (2 (1-a x)^3-(1-a x)^4\right ) \, dx\\ &=-\frac {c^2 (1-a x)^4}{2 a}+\frac {c^2 (1-a x)^5}{5 a}\\ \end {align*}
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Mathematica [A]
time = 0.01, size = 32, normalized size = 0.86 \begin {gather*} c^2 \left (x-a x^2+\frac {a^3 x^4}{2}-\frac {a^4 x^5}{5}\right ) \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.06, size = 29, normalized size = 0.78
method | result | size |
gosper | \(-\frac {c^{2} x \left (2 a^{4} x^{4}-5 a^{3} x^{3}+10 a x -10\right )}{10}\) | \(29\) |
default | \(c^{2} \left (-\frac {1}{5} a^{4} x^{5}+\frac {1}{2} a^{3} x^{4}-x^{2} a +x \right )\) | \(29\) |
risch | \(-\frac {1}{5} a^{4} c^{2} x^{5}+\frac {1}{2} a^{3} c^{2} x^{4}-a \,c^{2} x^{2}+c^{2} x\) | \(38\) |
norman | \(\frac {c^{2} x -a^{2} c^{2} x^{3}+\frac {1}{2} a^{3} c^{2} x^{4}+\frac {3}{10} a^{4} c^{2} x^{5}-\frac {1}{5} a^{5} c^{2} x^{6}}{a x +1}\) | \(59\) |
meijerg | \(-\frac {c^{2} \left (\frac {a x \left (14 x^{5} a^{5}-21 a^{4} x^{4}+35 a^{3} x^{3}-70 a^{2} x^{2}+210 a x +420\right )}{70 a x +70}-6 \ln \left (a x +1\right )\right )}{a}+\frac {3 c^{2} \left (\frac {x a \left (5 a^{3} x^{3}-10 a^{2} x^{2}+30 a x +60\right )}{15 a x +15}-4 \ln \left (a x +1\right )\right )}{a}-\frac {3 c^{2} \left (\frac {a x \left (3 a x +6\right )}{3 a x +3}-2 \ln \left (a x +1\right )\right )}{a}+\frac {c^{2} x}{a x +1}\) | \(164\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.25, size = 37, normalized size = 1.00 \begin {gather*} -\frac {1}{5} \, a^{4} c^{2} x^{5} + \frac {1}{2} \, a^{3} c^{2} x^{4} - a c^{2} x^{2} + c^{2} x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.34, size = 37, normalized size = 1.00 \begin {gather*} -\frac {1}{5} \, a^{4} c^{2} x^{5} + \frac {1}{2} \, a^{3} c^{2} x^{4} - a c^{2} x^{2} + c^{2} x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 0.03, size = 36, normalized size = 0.97 \begin {gather*} - \frac {a^{4} c^{2} x^{5}}{5} + \frac {a^{3} c^{2} x^{4}}{2} - a c^{2} x^{2} + c^{2} x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 0.42, size = 54, normalized size = 1.46 \begin {gather*} -\frac {{\left (2 \, c^{2} - \frac {15 \, c^{2}}{a x + 1} + \frac {40 \, c^{2}}{{\left (a x + 1\right )}^{2}} - \frac {40 \, c^{2}}{{\left (a x + 1\right )}^{3}}\right )} {\left (a x + 1\right )}^{5}}{10 \, a} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.05, size = 37, normalized size = 1.00 \begin {gather*} -\frac {a^4\,c^2\,x^5}{5}+\frac {a^3\,c^2\,x^4}{2}-a\,c^2\,x^2+c^2\,x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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