Optimal. Leaf size=23 \[ e^4 \log \left (2-x+\frac {x}{4 e^{-x}+x}\right ) \]
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Rubi [F]
time = 1.42, 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 {16 e^4+e^{4+2 x} x^2+e^{4+x} (-4+4 x)}{-32+16 x+e^x \left (-20 x+8 x^2\right )+e^{2 x} \left (-3 x^2+x^3\right )} \, 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 \left (-16+4 e^x-4 e^x x-e^{2 x} x^2\right )}{32-16 x-e^x \left (-20 x+8 x^2\right )-e^{2 x} \left (-3 x^2+x^3\right )} \, dx\\ &=e^4 \int \frac {-16+4 e^x-4 e^x x-e^{2 x} x^2}{32-16 x-e^x \left (-20 x+8 x^2\right )-e^{2 x} \left (-3 x^2+x^3\right )} \, dx\\ &=e^4 \int \left (\frac {1}{-3+x}+\frac {4 (1+x)}{x \left (4+e^x x\right )}-\frac {4 \left (6+2 x-4 x^2+x^3\right )}{(-3+x) x \left (-8+4 x-3 e^x x+e^x x^2\right )}\right ) \, dx\\ &=e^4 \log (3-x)+\left (4 e^4\right ) \int \frac {1+x}{x \left (4+e^x x\right )} \, dx-\left (4 e^4\right ) \int \frac {6+2 x-4 x^2+x^3}{(-3+x) x \left (-8+4 x-3 e^x x+e^x x^2\right )} \, dx\\ &=e^4 \log (3-x)+\left (4 e^4\right ) \int \left (\frac {1}{4+e^x x}+\frac {1}{x \left (4+e^x x\right )}\right ) \, dx-\left (4 e^4\right ) \int \left (-\frac {1}{-8+4 x-3 e^x x+e^x x^2}+\frac {1}{(-3+x) \left (-8+4 x-3 e^x x+e^x x^2\right )}-\frac {2}{x \left (-8+4 x-3 e^x x+e^x x^2\right )}+\frac {x}{-8+4 x-3 e^x x+e^x x^2}\right ) \, dx\\ &=e^4 \log (3-x)+\left (4 e^4\right ) \int \frac {1}{4+e^x x} \, dx+\left (4 e^4\right ) \int \frac {1}{x \left (4+e^x x\right )} \, dx+\left (4 e^4\right ) \int \frac {1}{-8+4 x-3 e^x x+e^x x^2} \, dx-\left (4 e^4\right ) \int \frac {1}{(-3+x) \left (-8+4 x-3 e^x x+e^x x^2\right )} \, dx-\left (4 e^4\right ) \int \frac {x}{-8+4 x-3 e^x x+e^x x^2} \, dx+\left (8 e^4\right ) \int \frac {1}{x \left (-8+4 x-3 e^x x+e^x x^2\right )} \, dx\\ \end {aligned} \end {gather*}
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Mathematica [A]
time = 1.72, size = 35, normalized size = 1.52 \begin {gather*} e^4 \left (-2 \tanh ^{-1}\left (5-2 x+\frac {3 e^x x}{2}-\frac {e^x x^2}{2}\right )+\log (-3+x)\right ) \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.56, size = 33, normalized size = 1.43
method | result | size |
norman | \({\mathrm e}^{4} \ln \left ({\mathrm e}^{x} x^{2}-3 \,{\mathrm e}^{x} x +4 x -8\right )-{\mathrm e}^{4} \ln \left ({\mathrm e}^{x} x +4\right )\) | \(33\) |
risch | \({\mathrm e}^{4} \ln \left (x -3\right )+{\mathrm e}^{4} \ln \left ({\mathrm e}^{x}+\frac {4 x -8}{x \left (x -3\right )}\right )-{\mathrm e}^{4} \ln \left ({\mathrm e}^{x}+\frac {4}{x}\right )\) | \(42\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [B] Leaf count of result is larger than twice the leaf count of optimal. 52 vs.
\(2 (21) = 42\).
time = 0.30, size = 52, normalized size = 2.26 \begin {gather*} e^{4} \log \left (x - 3\right ) + e^{4} \log \left (\frac {{\left (x^{2} - 3 \, x\right )} e^{x} + 4 \, x - 8}{x^{2} - 3 \, x}\right ) - e^{4} \log \left (\frac {x e^{x} + 4}{x}\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 62 vs.
\(2 (21) = 42\).
time = 0.38, size = 62, normalized size = 2.70 \begin {gather*} e^{4} \log \left (x - 3\right ) + e^{4} \log \left (\frac {4 \, {\left (x - 2\right )} e^{4} + {\left (x^{2} - 3 \, x\right )} e^{\left (x + 4\right )}}{x^{2} - 3 \, x}\right ) - e^{4} \log \left (\frac {x e^{\left (x + 4\right )} + 4 \, e^{4}}{x}\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: PolynomialError} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 66 vs.
\(2 (21) = 42\).
time = 0.43, size = 66, normalized size = 2.87 \begin {gather*} e^{4} \log \left ({\left (x + 4\right )}^{2} e^{\left (x + 4\right )} + 4 \, {\left (x + 4\right )} e^{4} - 11 \, {\left (x + 4\right )} e^{\left (x + 4\right )} - 24 \, e^{4} + 28 \, e^{\left (x + 4\right )}\right ) - e^{4} \log \left ({\left (x + 4\right )} e^{\left (x + 4\right )} + 4 \, e^{4} - 4 \, e^{\left (x + 4\right )}\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.27, size = 30, normalized size = 1.30 \begin {gather*} {\mathrm {e}}^4\,\left (\ln \left (4\,x+x^2\,{\mathrm {e}}^x-3\,x\,{\mathrm {e}}^x-8\right )-\ln \left (x\,{\mathrm {e}}^x+4\right )\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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