Optimal. Leaf size=22 \[ \frac {x+4 (4+2 x)}{3-4 e^{-4 x} x} \]
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Rubi [F]
time = 0.99, antiderivative size = 0, normalized size of antiderivative = 0.00, number of steps
used = 0, number of rules used = 0, integrand size = 0, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.000, Rules used = {}
\begin {gather*} \int \frac {27 e^{8 x}+e^{4 x} \left (64-256 x-144 x^2\right )}{9 e^{8 x}-24 e^{4 x} x+16 x^2} \, dx \end {gather*}
Verification is not applicable to the result.
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Rubi steps
\begin {gather*} \begin {aligned} \text {integral} &=\int \frac {e^{4 x} \left (64+27 e^{4 x}-256 x-144 x^2\right )}{\left (3 e^{4 x}-4 x\right )^2} \, dx\\ &=\int \left (\frac {9 e^{4 x}}{3 e^{4 x}-4 x}-\frac {4 e^{4 x} \left (-16+55 x+36 x^2\right )}{\left (3 e^{4 x}-4 x\right )^2}\right ) \, dx\\ &=-\left (4 \int \frac {e^{4 x} \left (-16+55 x+36 x^2\right )}{\left (3 e^{4 x}-4 x\right )^2} \, dx\right )+9 \int \frac {e^{4 x}}{3 e^{4 x}-4 x} \, dx\\ &=\frac {9}{4} \text {Subst}\left (\int \frac {e^x}{3 e^x-x} \, dx,x,4 x\right )-4 \int \left (-\frac {16 e^{4 x}}{\left (3 e^{4 x}-4 x\right )^2}+\frac {55 e^{4 x} x}{\left (3 e^{4 x}-4 x\right )^2}+\frac {36 e^{4 x} x^2}{\left (3 e^{4 x}-4 x\right )^2}\right ) \, dx\\ &=\frac {9}{4} \text {Subst}\left (\int \frac {e^x}{3 e^x-x} \, dx,x,4 x\right )+64 \int \frac {e^{4 x}}{\left (3 e^{4 x}-4 x\right )^2} \, dx-144 \int \frac {e^{4 x} x^2}{\left (3 e^{4 x}-4 x\right )^2} \, dx-220 \int \frac {e^{4 x} x}{\left (3 e^{4 x}-4 x\right )^2} \, dx\\ &=-\frac {16}{3 \left (3 e^{4 x}-4 x\right )}+\frac {55 x}{3 \left (3 e^{4 x}-4 x\right )}+\frac {12 x^2}{3 e^{4 x}-4 x}+\frac {9}{4} \text {Subst}\left (\int \frac {e^x}{3 e^x-x} \, dx,x,4 x\right )-\frac {55}{3} \int \frac {1}{3 e^{4 x}-4 x} \, dx+\frac {64}{3} \int \frac {1}{\left (3 e^{4 x}-4 x\right )^2} \, dx-24 \int \frac {x}{3 e^{4 x}-4 x} \, dx-48 \int \frac {x^2}{\left (3 e^{4 x}-4 x\right )^2} \, dx-\frac {220}{3} \int \frac {x}{\left (3 e^{4 x}-4 x\right )^2} \, dx\\ &=-\frac {16}{3 \left (3 e^{4 x}-4 x\right )}+\frac {55 x}{3 \left (3 e^{4 x}-4 x\right )}+\frac {12 x^2}{3 e^{4 x}-4 x}+\frac {9}{4} \text {Subst}\left (\int \frac {e^x}{3 e^x-x} \, dx,x,4 x\right )-\frac {55}{12} \text {Subst}\left (\int \frac {1}{3 e^x-x} \, dx,x,4 x\right )+\frac {16}{3} \text {Subst}\left (\int \frac {1}{\left (3 e^x-x\right )^2} \, dx,x,4 x\right )-24 \int \frac {x}{3 e^{4 x}-4 x} \, dx-48 \int \frac {x^2}{\left (3 e^{4 x}-4 x\right )^2} \, dx-\frac {220}{3} \int \frac {x}{\left (3 e^{4 x}-4 x\right )^2} \, dx\\ \end {aligned} \end {gather*}
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Mathematica [A]
time = 0.37, size = 24, normalized size = 1.09 \begin {gather*} \frac {\left (64+27 e^{4 x}\right ) x}{9 e^{4 x}-12 x} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.25, size = 25, normalized size = 1.14
method | result | size |
norman | \(\frac {-\frac {64 x}{3}-9 x \,{\mathrm e}^{4 x}}{-3 \,{\mathrm e}^{4 x}+4 x}\) | \(25\) |
risch | \(3 x -\frac {4 \left (9 x +16\right ) x}{3 \left (-3 \,{\mathrm e}^{4 x}+4 x \right )}\) | \(25\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.29, size = 25, normalized size = 1.14 \begin {gather*} -\frac {27 \, x e^{\left (4 \, x\right )} + 64 \, x}{3 \, {\left (4 \, x - 3 \, e^{\left (4 \, x\right )}\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.39, size = 25, normalized size = 1.14 \begin {gather*} -\frac {27 \, x e^{\left (4 \, x\right )} + 64 \, x}{3 \, {\left (4 \, x - 3 \, e^{\left (4 \, x\right )}\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 0.05, size = 20, normalized size = 0.91 \begin {gather*} 3 x + \frac {36 x^{2} + 64 x}{- 12 x + 9 e^{4 x}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 0.42, size = 25, normalized size = 1.14 \begin {gather*} -\frac {27 \, x e^{\left (4 \, x\right )} + 64 \, x}{3 \, {\left (4 \, x - 3 \, e^{\left (4 \, x\right )}\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.09, size = 23, normalized size = 1.05 \begin {gather*} -\frac {x\,\left (27\,{\mathrm {e}}^{4\,x}+64\right )}{3\,\left (4\,x-3\,{\mathrm {e}}^{4\,x}\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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