Optimal. Leaf size=37 \[ -\frac {2 c^3 (1-a x)^3}{3 a}+\frac {c^3 (1-a x)^4}{4 a} \]
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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 = 18, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.111, Rules used = {6264, 45}
\begin {gather*} \frac {c^3 (1-a x)^4}{4 a}-\frac {2 c^3 (1-a x)^3}{3 a} \end {gather*}
Antiderivative was successfully verified.
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Rule 45
Rule 6264
Rubi steps
\begin {align*} \int e^{2 \tanh ^{-1}(a x)} (c-a c x)^3 \, dx &=c^3 \int (1-a x)^2 (1+a x) \, dx\\ &=c^3 \int \left (2 (1-a x)^2-(1-a x)^3\right ) \, dx\\ &=-\frac {2 c^3 (1-a x)^3}{3 a}+\frac {c^3 (1-a x)^4}{4 a}\\ \end {align*}
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Mathematica [A]
time = 0.01, size = 30, normalized size = 0.81 \begin {gather*} \frac {1}{12} c^3 x \left (12-6 a x-4 a^2 x^2+3 a^3 x^3\right ) \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 1.07, size = 29, normalized size = 0.78
method | result | size |
gosper | \(\frac {\left (3 a^{3} x^{3}-4 a^{2} x^{2}-6 a x +12\right ) x \,c^{3}}{12}\) | \(29\) |
default | \(c^{3} \left (\frac {1}{4} a^{3} x^{4}-\frac {1}{3} a^{2} x^{3}-\frac {1}{2} a \,x^{2}+x \right )\) | \(29\) |
norman | \(x \,c^{3}-\frac {1}{2} a \,c^{3} x^{2}-\frac {1}{3} a^{2} c^{3} x^{3}+\frac {1}{4} a^{3} c^{3} x^{4}\) | \(38\) |
risch | \(x \,c^{3}-\frac {1}{2} a \,c^{3} x^{2}-\frac {1}{3} a^{2} c^{3} x^{3}+\frac {1}{4} a^{3} c^{3} x^{4}\) | \(38\) |
meijerg | \(\frac {c^{3} \left (\frac {a^{2} x^{2} \left (3 a^{2} x^{2}+6\right )}{6}+\ln \left (-a^{2} x^{2}+1\right )\right )}{2 a}+\frac {c^{3} \left (-\frac {2 x \left (-a^{2}\right )^{\frac {5}{2}} \left (5 a^{2} x^{2}+15\right )}{15 a^{4}}+\frac {2 \left (-a^{2}\right )^{\frac {5}{2}} \arctanh \left (a x \right )}{a^{5}}\right )}{2 \sqrt {-a^{2}}}+\frac {c^{3} \left (-a^{2} x^{2}-\ln \left (-a^{2} x^{2}+1\right )\right )}{a}+\frac {c^{3} \left (-\frac {2 x \left (-a^{2}\right )^{\frac {3}{2}}}{a^{2}}+\frac {2 \left (-a^{2}\right )^{\frac {3}{2}} \arctanh \left (a x \right )}{a^{3}}\right )}{\sqrt {-a^{2}}}+\frac {c^{3} \ln \left (-a^{2} x^{2}+1\right )}{2 a}+\frac {c^{3} \arctanh \left (a x \right )}{a}\) | \(192\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.26, size = 37, normalized size = 1.00 \begin {gather*} \frac {1}{4} \, a^{3} c^{3} x^{4} - \frac {1}{3} \, a^{2} c^{3} x^{3} - \frac {1}{2} \, a c^{3} x^{2} + c^{3} x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.35, size = 37, normalized size = 1.00 \begin {gather*} \frac {1}{4} \, a^{3} c^{3} x^{4} - \frac {1}{3} \, a^{2} c^{3} x^{3} - \frac {1}{2} \, a c^{3} x^{2} + c^{3} x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 0.03, size = 37, normalized size = 1.00 \begin {gather*} \frac {a^{3} c^{3} x^{4}}{4} - \frac {a^{2} c^{3} x^{3}}{3} - \frac {a c^{3} x^{2}}{2} + c^{3} x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 0.40, size = 37, normalized size = 1.00 \begin {gather*} \frac {1}{4} \, a^{3} c^{3} x^{4} - \frac {1}{3} \, a^{2} c^{3} x^{3} - \frac {1}{2} \, a c^{3} x^{2} + c^{3} x \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^3\,c^3\,x^4}{4}-\frac {a^2\,c^3\,x^3}{3}-\frac {a\,c^3\,x^2}{2}+c^3\,x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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