Optimal. Leaf size=37 \[ -\frac {c^4 (1-a x)^4}{2 a}+\frac {c^4 (1-a x)^5}{5 a} \]
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Rubi [A]
time = 0.02, 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^4 (1-a x)^5}{5 a}-\frac {c^4 (1-a x)^4}{2 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)^4 \, dx &=c^4 \int (1-a x)^3 (1+a x) \, dx\\ &=c^4 \int \left (2 (1-a x)^3-(1-a x)^4\right ) \, dx\\ &=-\frac {c^4 (1-a x)^4}{2 a}+\frac {c^4 (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^4 \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 = 1.08, size = 29, normalized size = 0.78
method | result | size |
gosper | \(-\frac {\left (2 a^{4} x^{4}-5 a^{3} x^{3}+10 a x -10\right ) c^{4} x}{10}\) | \(29\) |
default | \(c^{4} \left (-\frac {1}{5} a^{4} x^{5}+\frac {1}{2} a^{3} x^{4}-a \,x^{2}+x \right )\) | \(29\) |
norman | \(c^{4} x -\frac {1}{5} a^{4} c^{4} x^{5}-c^{4} a \,x^{2}+\frac {1}{2} c^{4} x^{4} a^{3}\) | \(38\) |
risch | \(c^{4} x -\frac {1}{5} a^{4} c^{4} x^{5}-c^{4} a \,x^{2}+\frac {1}{2} c^{4} x^{4} a^{3}\) | \(38\) |
meijerg | \(-\frac {c^{4} \left (-\frac {2 x \left (-a^{2}\right )^{\frac {7}{2}} \left (21 a^{4} x^{4}+35 a^{2} x^{2}+105\right )}{105 a^{6}}+\frac {2 \left (-a^{2}\right )^{\frac {7}{2}} \arctanh \left (a x \right )}{a^{7}}\right )}{2 \sqrt {-a^{2}}}+\frac {c^{4} \left (\frac {a^{2} x^{2} \left (3 a^{2} x^{2}+6\right )}{6}+\ln \left (-a^{2} x^{2}+1\right )\right )}{a}-\frac {c^{4} \left (-\frac {2 x \left (-a^{2}\right )^{\frac {5}{2}} \left (5 a^{2} x^{2}+15\right )}{15 a^{4}}+\frac {2 \left (-a^{2}\right )^{\frac {5}{2}} \arctanh \left (a x \right )}{a^{5}}\right )}{2 \sqrt {-a^{2}}}+\frac {2 c^{4} \left (-a^{2} x^{2}-\ln \left (-a^{2} x^{2}+1\right )\right )}{a}+\frac {c^{4} \left (-\frac {2 x \left (-a^{2}\right )^{\frac {3}{2}}}{a^{2}}+\frac {2 \left (-a^{2}\right )^{\frac {3}{2}} \arctanh \left (a x \right )}{a^{3}}\right )}{2 \sqrt {-a^{2}}}+\frac {c^{4} \ln \left (-a^{2} x^{2}+1\right )}{a}+\frac {c^{4} \arctanh \left (a x \right )}{a}\) | \(252\) |
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}{5} \, a^{4} c^{4} x^{5} + \frac {1}{2} \, a^{3} c^{4} x^{4} - a c^{4} x^{2} + c^{4} 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}{5} \, a^{4} c^{4} x^{5} + \frac {1}{2} \, a^{3} c^{4} x^{4} - a c^{4} x^{2} + c^{4} x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 0.02, size = 36, normalized size = 0.97 \begin {gather*} - \frac {a^{4} c^{4} x^{5}}{5} + \frac {a^{3} c^{4} x^{4}}{2} - a c^{4} x^{2} + c^{4} x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 0.42, size = 37, normalized size = 1.00 \begin {gather*} -\frac {1}{5} \, a^{4} c^{4} x^{5} + \frac {1}{2} \, a^{3} c^{4} x^{4} - a c^{4} x^{2} + c^{4} 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^4\,c^4\,x^5}{5}+\frac {a^3\,c^4\,x^4}{2}-a\,c^4\,x^2+c^4\,x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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