Optimal. Leaf size=29 \[ x \left (6-e^{2 x (2+x)}+\frac {e^x}{(-4+x)^{\frac {1}{x}}+x}\right ) \]
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Rubi [F]
time = 10.97, 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 {-24 x^3+6 x^4+e^x \left (-4 x^3+x^4\right )+e^{4 x+2 x^2} \left (4 x^3+15 x^4+12 x^5-4 x^6\right )+(-4+x)^{2/x} \left (-24 x+6 x^2+e^{4 x+2 x^2} \left (4 x+15 x^2+12 x^3-4 x^4\right )\right )+(-4+x)^{\frac {1}{x}} \left (-48 x^2+12 x^3+e^x \left (-5 x-3 x^2+x^3\right )+e^{4 x+2 x^2} \left (8 x^2+30 x^3+24 x^4-8 x^5\right )+e^x (-4+x) \log (-4+x)\right )}{-4 x^3+x^4+(-4+x)^{2/x} \left (-4 x+x^2\right )+(-4+x)^{\frac {1}{x}} \left (-8 x^2+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 {-x \left (6 (-4+x) \left ((-4+x)^{\frac {1}{x}}+x\right )^2-e^{2 x (2+x)} (-4+x) \left ((-4+x)^{\frac {1}{x}}+x\right )^2 (1+2 x)^2+e^x \left (-5 (-4+x)^{\frac {1}{x}}-3 (-4+x)^{\frac {1}{x}} x+\left (-4+(-4+x)^{\frac {1}{x}}\right ) x^2+x^3\right )\right )-e^x (-4+x)^{1+\frac {1}{x}} \log (-4+x)}{(4-x) x \left ((-4+x)^{\frac {1}{x}}+x\right )^2} \, dx\\ &=\int \left (6-e^{2 x (2+x)}-4 e^{2 x (2+x)} x-4 e^{2 x (2+x)} x^2-\frac {e^x \left (-5 x+x^2-4 \log (-4+x)+x \log (-4+x)\right )}{(-4+x) \left ((-4+x)^{\frac {1}{x}}+x\right )^2}+\frac {e^x \left (-5 x-3 x^2+x^3-4 \log (-4+x)+x \log (-4+x)\right )}{(-4+x) x \left ((-4+x)^{\frac {1}{x}}+x\right )}\right ) \, dx\\ &=6 x-4 \int e^{2 x (2+x)} x \, dx-4 \int e^{2 x (2+x)} x^2 \, dx-\int e^{2 x (2+x)} \, dx-\int \frac {e^x \left (-5 x+x^2-4 \log (-4+x)+x \log (-4+x)\right )}{(-4+x) \left ((-4+x)^{\frac {1}{x}}+x\right )^2} \, dx+\int \frac {e^x \left (-5 x-3 x^2+x^3-4 \log (-4+x)+x \log (-4+x)\right )}{(-4+x) x \left ((-4+x)^{\frac {1}{x}}+x\right )} \, dx\\ &=6 x-4 \int e^{4 x+2 x^2} x \, dx-4 \int e^{4 x+2 x^2} x^2 \, dx-\int e^{4 x+2 x^2} \, dx-\int \left (-\frac {5 e^x x}{(-4+x) \left ((-4+x)^{\frac {1}{x}}+x\right )^2}+\frac {e^x x^2}{(-4+x) \left ((-4+x)^{\frac {1}{x}}+x\right )^2}-\frac {4 e^x \log (-4+x)}{(-4+x) \left ((-4+x)^{\frac {1}{x}}+x\right )^2}+\frac {e^x x \log (-4+x)}{(-4+x) \left ((-4+x)^{\frac {1}{x}}+x\right )^2}\right ) \, dx+\int \left (\frac {e^x \left (-5 x-3 x^2+x^3-4 \log (-4+x)+x \log (-4+x)\right )}{4 (-4+x) \left ((-4+x)^{\frac {1}{x}}+x\right )}-\frac {e^x \left (-5 x-3 x^2+x^3-4 \log (-4+x)+x \log (-4+x)\right )}{4 x \left ((-4+x)^{\frac {1}{x}}+x\right )}\right ) \, dx\\ &=-e^{4 x+2 x^2}+6 x-e^{4 x+2 x^2} x+\frac {1}{4} \int \frac {e^x \left (-5 x-3 x^2+x^3-4 \log (-4+x)+x \log (-4+x)\right )}{(-4+x) \left ((-4+x)^{\frac {1}{x}}+x\right )} \, dx-\frac {1}{4} \int \frac {e^x \left (-5 x-3 x^2+x^3-4 \log (-4+x)+x \log (-4+x)\right )}{x \left ((-4+x)^{\frac {1}{x}}+x\right )} \, dx+4 \int e^{4 x+2 x^2} \, dx+4 \int e^{4 x+2 x^2} x \, dx+4 \int \frac {e^x \log (-4+x)}{(-4+x) \left ((-4+x)^{\frac {1}{x}}+x\right )^2} \, dx+5 \int \frac {e^x x}{(-4+x) \left ((-4+x)^{\frac {1}{x}}+x\right )^2} \, dx-\frac {\int e^{\frac {1}{8} (4+4 x)^2} \, dx}{e^2}+\int e^{4 x+2 x^2} \, dx-\int \frac {e^x x^2}{(-4+x) \left ((-4+x)^{\frac {1}{x}}+x\right )^2} \, dx-\int \frac {e^x x \log (-4+x)}{(-4+x) \left ((-4+x)^{\frac {1}{x}}+x\right )^2} \, dx\\ &=6 x-e^{4 x+2 x^2} x-\frac {\sqrt {\frac {\pi }{2}} \text {erfi}\left (\sqrt {2} (1+x)\right )}{2 e^2}-\frac {1}{4} \int \left (-\frac {5 e^x}{(-4+x)^{\frac {1}{x}}+x}-\frac {3 e^x x}{(-4+x)^{\frac {1}{x}}+x}+\frac {e^x x^2}{(-4+x)^{\frac {1}{x}}+x}+\frac {e^x \log (-4+x)}{(-4+x)^{\frac {1}{x}}+x}-\frac {4 e^x \log (-4+x)}{x \left ((-4+x)^{\frac {1}{x}}+x\right )}\right ) \, dx+\frac {1}{4} \int \left (-\frac {5 e^x x}{(-4+x) \left ((-4+x)^{\frac {1}{x}}+x\right )}-\frac {3 e^x x^2}{(-4+x) \left ((-4+x)^{\frac {1}{x}}+x\right )}+\frac {e^x x^3}{(-4+x) \left ((-4+x)^{\frac {1}{x}}+x\right )}-\frac {4 e^x \log (-4+x)}{(-4+x) \left ((-4+x)^{\frac {1}{x}}+x\right )}+\frac {e^x x \log (-4+x)}{(-4+x) \left ((-4+x)^{\frac {1}{x}}+x\right )}\right ) \, dx-4 \int e^{4 x+2 x^2} \, dx-4 \int \frac {\int \frac {e^x}{(-4+x) \left ((-4+x)^{\frac {1}{x}}+x\right )^2} \, dx}{-4+x} \, dx+5 \int \left (\frac {e^x}{\left ((-4+x)^{\frac {1}{x}}+x\right )^2}+\frac {4 e^x}{(-4+x) \left ((-4+x)^{\frac {1}{x}}+x\right )^2}\right ) \, dx+\frac {\int e^{\frac {1}{8} (4+4 x)^2} \, dx}{e^2}+\frac {4 \int e^{\frac {1}{8} (4+4 x)^2} \, dx}{e^2}-\log (-4+x) \int \frac {e^x}{\left ((-4+x)^{\frac {1}{x}}+x\right )^2} \, dx-\int \left (\frac {4 e^x}{\left ((-4+x)^{\frac {1}{x}}+x\right )^2}+\frac {16 e^x}{(-4+x) \left ((-4+x)^{\frac {1}{x}}+x\right )^2}+\frac {e^x x}{\left ((-4+x)^{\frac {1}{x}}+x\right )^2}\right ) \, dx+\int \frac {\int \frac {e^x}{\left ((-4+x)^{\frac {1}{x}}+x\right )^2} \, dx+4 \int \frac {e^x}{(-4+x) \left ((-4+x)^{\frac {1}{x}}+x\right )^2} \, dx}{-4+x} \, dx\\ &=6 x-e^{4 x+2 x^2} x+\frac {\sqrt {2 \pi } \text {erfi}\left (\sqrt {2} (1+x)\right )}{e^2}-\frac {1}{4} \int \frac {e^x x^2}{(-4+x)^{\frac {1}{x}}+x} \, dx+\frac {1}{4} \int \frac {e^x x^3}{(-4+x) \left ((-4+x)^{\frac {1}{x}}+x\right )} \, dx-\frac {1}{4} \int \frac {e^x \log (-4+x)}{(-4+x)^{\frac {1}{x}}+x} \, dx+\frac {1}{4} \int \frac {e^x x \log (-4+x)}{(-4+x) \left ((-4+x)^{\frac {1}{x}}+x\right )} \, dx+\frac {3}{4} \int \frac {e^x x}{(-4+x)^{\frac {1}{x}}+x} \, dx-\frac {3}{4} \int \frac {e^x x^2}{(-4+x) \left ((-4+x)^{\frac {1}{x}}+x\right )} \, dx+\frac {5}{4} \int \frac {e^x}{(-4+x)^{\frac {1}{x}}+x} \, dx-\frac {5}{4} \int \frac {e^x x}{(-4+x) \left ((-4+x)^{\frac {1}{x}}+x\right )} \, dx-4 \int \frac {e^x}{\left ((-4+x)^{\frac {1}{x}}+x\right )^2} \, dx-4 \int \frac {\int \frac {e^x}{(-4+x) \left ((-4+x)^{\frac {1}{x}}+x\right )^2} \, dx}{-4+x} \, dx+5 \int \frac {e^x}{\left ((-4+x)^{\frac {1}{x}}+x\right )^2} \, dx-16 \int \frac {e^x}{(-4+x) \left ((-4+x)^{\frac {1}{x}}+x\right )^2} \, dx+20 \int \frac {e^x}{(-4+x) \left ((-4+x)^{\frac {1}{x}}+x\right )^2} \, dx-\frac {4 \int e^{\frac {1}{8} (4+4 x)^2} \, dx}{e^2}-\log (-4+x) \int \frac {e^x}{\left ((-4+x)^{\frac {1}{x}}+x\right )^2} \, dx-\int \frac {e^x x}{\left ((-4+x)^{\frac {1}{x}}+x\right )^2} \, dx-\int \frac {e^x \log (-4+x)}{(-4+x) \left ((-4+x)^{\frac {1}{x}}+x\right )} \, dx+\int \frac {e^x \log (-4+x)}{x \left ((-4+x)^{\frac {1}{x}}+x\right )} \, dx+\int \left (\frac {\int \frac {e^x}{\left ((-4+x)^{\frac {1}{x}}+x\right )^2} \, dx}{-4+x}+\frac {4 \int \frac {e^x}{(-4+x) \left ((-4+x)^{\frac {1}{x}}+x\right )^2} \, dx}{-4+x}\right ) \, dx\\ &=6 x-e^{4 x+2 x^2} x-\frac {1}{4} \int \frac {e^x x^2}{(-4+x)^{\frac {1}{x}}+x} \, dx+\frac {1}{4} \int \left (\frac {16 e^x}{(-4+x)^{\frac {1}{x}}+x}+\frac {64 e^x}{(-4+x) \left ((-4+x)^{\frac {1}{x}}+x\right )}+\frac {4 e^x x}{(-4+x)^{\frac {1}{x}}+x}+\frac {e^x x^2}{(-4+x)^{\frac {1}{x}}+x}\right ) \, dx+\frac {1}{4} \int \frac {\int \frac {e^x}{(-4+x)^{\frac {1}{x}}+x} \, dx}{-4+x} \, dx-\frac {1}{4} \int \frac {\int \frac {e^x}{(-4+x)^{\frac {1}{x}}+x} \, dx+4 \int \frac {e^x}{(-4+x) \left ((-4+x)^{\frac {1}{x}}+x\right )} \, dx}{-4+x} \, dx+\frac {3}{4} \int \frac {e^x x}{(-4+x)^{\frac {1}{x}}+x} \, dx-\frac {3}{4} \int \left (\frac {4 e^x}{(-4+x)^{\frac {1}{x}}+x}+\frac {16 e^x}{(-4+x) \left ((-4+x)^{\frac {1}{x}}+x\right )}+\frac {e^x x}{(-4+x)^{\frac {1}{x}}+x}\right ) \, dx+\frac {5}{4} \int \frac {e^x}{(-4+x)^{\frac {1}{x}}+x} \, dx-\frac {5}{4} \int \left (\frac {e^x}{(-4+x)^{\frac {1}{x}}+x}+\frac {4 e^x}{(-4+x) \left ((-4+x)^{\frac {1}{x}}+x\right )}\right ) \, dx-4 \int \frac {e^x}{\left ((-4+x)^{\frac {1}{x}}+x\right )^2} \, dx+5 \int \frac {e^x}{\left ((-4+x)^{\frac {1}{x}}+x\right )^2} \, dx-16 \int \frac {e^x}{(-4+x) \left ((-4+x)^{\frac {1}{x}}+x\right )^2} \, dx+20 \int \frac {e^x}{(-4+x) \left ((-4+x)^{\frac {1}{x}}+x\right )^2} \, dx-\log (-4+x) \int \frac {e^x}{\left ((-4+x)^{\frac {1}{x}}+x\right )^2} \, dx+\log (-4+x) \int \frac {e^x}{x \left ((-4+x)^{\frac {1}{x}}+x\right )} \, dx-\int \frac {e^x x}{\left ((-4+x)^{\frac {1}{x}}+x\right )^2} \, dx+\int \frac {\int \frac {e^x}{\left ((-4+x)^{\frac {1}{x}}+x\right )^2} \, dx}{-4+x} \, dx+\int \frac {\int \frac {e^x}{(-4+x) \left ((-4+x)^{\frac {1}{x}}+x\right )} \, dx}{-4+x} \, dx-\int \frac {\int \frac {e^x}{x \left ((-4+x)^{\frac {1}{x}}+x\right )} \, dx}{-4+x} \, dx\\ &=6 x-e^{4 x+2 x^2} x+\frac {1}{4} \int \frac {\int \frac {e^x}{(-4+x)^{\frac {1}{x}}+x} \, dx}{-4+x} \, dx-\frac {1}{4} \int \left (\frac {\int \frac {e^x}{(-4+x)^{\frac {1}{x}}+x} \, dx}{-4+x}+\frac {4 \int \frac {e^x}{(-4+x) \left ((-4+x)^{\frac {1}{x}}+x\right )} \, dx}{-4+x}\right ) \, dx-3 \int \frac {e^x}{(-4+x)^{\frac {1}{x}}+x} \, dx-4 \int \frac {e^x}{\left ((-4+x)^{\frac {1}{x}}+x\right )^2} \, dx+4 \int \frac {e^x}{(-4+x)^{\frac {1}{x}}+x} \, dx+5 \int \frac {e^x}{\left ((-4+x)^{\frac {1}{x}}+x\right )^2} \, dx-5 \int \frac {e^x}{(-4+x) \left ((-4+x)^{\frac {1}{x}}+x\right )} \, dx-12 \int \frac {e^x}{(-4+x) \left ((-4+x)^{\frac {1}{x}}+x\right )} \, dx-16 \int \frac {e^x}{(-4+x) \left ((-4+x)^{\frac {1}{x}}+x\right )^2} \, dx+16 \int \frac {e^x}{(-4+x) \left ((-4+x)^{\frac {1}{x}}+x\right )} \, dx+20 \int \frac {e^x}{(-4+x) \left ((-4+x)^{\frac {1}{x}}+x\right )^2} \, dx-\log (-4+x) \int \frac {e^x}{\left ((-4+x)^{\frac {1}{x}}+x\right )^2} \, dx+\log (-4+x) \int \frac {e^x}{x \left ((-4+x)^{\frac {1}{x}}+x\right )} \, dx-\int \frac {e^x x}{\left ((-4+x)^{\frac {1}{x}}+x\right )^2} \, dx+\int \frac {e^x x}{(-4+x)^{\frac {1}{x}}+x} \, dx+\int \frac {\int \frac {e^x}{\left ((-4+x)^{\frac {1}{x}}+x\right )^2} \, dx}{-4+x} \, dx+\int \frac {\int \frac {e^x}{(-4+x) \left ((-4+x)^{\frac {1}{x}}+x\right )} \, dx}{-4+x} \, dx-\int \frac {\int \frac {e^x}{x \left ((-4+x)^{\frac {1}{x}}+x\right )} \, dx}{-4+x} \, dx\\ &=6 x-e^{4 x+2 x^2} x-3 \int \frac {e^x}{(-4+x)^{\frac {1}{x}}+x} \, dx-4 \int \frac {e^x}{\left ((-4+x)^{\frac {1}{x}}+x\right )^2} \, dx+4 \int \frac {e^x}{(-4+x)^{\frac {1}{x}}+x} \, dx+5 \int \frac {e^x}{\left ((-4+x)^{\frac {1}{x}}+x\right )^2} \, dx-5 \int \frac {e^x}{(-4+x) \left ((-4+x)^{\frac {1}{x}}+x\right )} \, dx-12 \int \frac {e^x}{(-4+x) \left ((-4+x)^{\frac {1}{x}}+x\right )} \, dx-16 \int \frac {e^x}{(-4+x) \left ((-4+x)^{\frac {1}{x}}+x\right )^2} \, dx+16 \int \frac {e^x}{(-4+x) \left ((-4+x)^{\frac {1}{x}}+x\right )} \, dx+20 \int \frac {e^x}{(-4+x) \left ((-4+x)^{\frac {1}{x}}+x\right )^2} \, dx-\log (-4+x) \int \frac {e^x}{\left ((-4+x)^{\frac {1}{x}}+x\right )^2} \, dx+\log (-4+x) \int \frac {e^x}{x \left ((-4+x)^{\frac {1}{x}}+x\right )} \, dx-\int \frac {e^x x}{\left ((-4+x)^{\frac {1}{x}}+x\right )^2} \, dx+\int \frac {e^x x}{(-4+x)^{\frac {1}{x}}+x} \, dx+\int \frac {\int \frac {e^x}{\left ((-4+x)^{\frac {1}{x}}+x\right )^2} \, dx}{-4+x} \, dx-\int \frac {\int \frac {e^x}{x \left ((-4+x)^{\frac {1}{x}}+x\right )} \, dx}{-4+x} \, dx\\ \end {aligned} \end {gather*}
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Mathematica [A]
time = 0.24, size = 29, normalized size = 1.00 \begin {gather*} x \left (6-e^{2 x (2+x)}+\frac {e^x}{(-4+x)^{\frac {1}{x}}+x}\right ) \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 1.37, size = 30, normalized size = 1.03
method | result | size |
risch | \(-x \,{\mathrm e}^{2 x \left (2+x \right )}+6 x +\frac {x \,{\mathrm e}^{x}}{\left (x -4\right )^{\frac {1}{x}}+x}\) | \(30\) |
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. 62 vs.
\(2 (27) = 54\).
time = 0.35, size = 62, normalized size = 2.14 \begin {gather*} -\frac {x^{2} e^{\left (2 \, x^{2} + 4 \, x\right )} + {\left (x e^{\left (2 \, x^{2} + 4 \, x\right )} - 6 \, x\right )} {\left (x - 4\right )}^{\left (\frac {1}{x}\right )} - 6 \, x^{2} - x e^{x}}{{\left (x - 4\right )}^{\left (\frac {1}{x}\right )} + x} \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.
\(2 (27) = 54\).
time = 0.38, size = 62, normalized size = 2.14 \begin {gather*} -\frac {x^{2} e^{\left (2 \, x^{2} + 4 \, x\right )} + {\left (x e^{\left (2 \, x^{2} + 4 \, x\right )} - 6 \, x\right )} {\left (x - 4\right )}^{\left (\frac {1}{x}\right )} - 6 \, x^{2} - x e^{x}}{{\left (x - 4\right )}^{\left (\frac {1}{x}\right )} + x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 1.00, size = 29, normalized size = 1.00 \begin {gather*} - x e^{2 x^{2} + 4 x} + 6 x + \frac {x e^{x}}{x + e^{\frac {\log {\left (x - 4 \right )}}{x}}} \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. 6706 vs.
\(2 (27) = 54\).
time = 1.17, size = 6706, normalized size = 231.24 \begin {gather*} \text {Too large to display} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [F]
time = 0.00, size = -1, normalized size = -0.03 \begin {gather*} \int -\frac {{\mathrm {e}}^{2\,x^2+4\,x}\,\left (-4\,x^6+12\,x^5+15\,x^4+4\,x^3\right )-{\mathrm {e}}^x\,\left (4\,x^3-x^4\right )+{\mathrm {e}}^{\frac {\ln \left (x-4\right )}{x}}\,\left ({\mathrm {e}}^{2\,x^2+4\,x}\,\left (-8\,x^5+24\,x^4+30\,x^3+8\,x^2\right )-48\,x^2+12\,x^3-{\mathrm {e}}^x\,\left (-x^3+3\,x^2+5\,x\right )+\ln \left (x-4\right )\,{\mathrm {e}}^x\,\left (x-4\right )\right )+{\mathrm {e}}^{\frac {2\,\ln \left (x-4\right )}{x}}\,\left ({\mathrm {e}}^{2\,x^2+4\,x}\,\left (-4\,x^4+12\,x^3+15\,x^2+4\,x\right )-24\,x+6\,x^2\right )-24\,x^3+6\,x^4}{{\mathrm {e}}^{\frac {\ln \left (x-4\right )}{x}}\,\left (8\,x^2-2\,x^3\right )+{\mathrm {e}}^{\frac {2\,\ln \left (x-4\right )}{x}}\,\left (4\,x-x^2\right )+4\,x^3-x^4} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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