Optimal. Leaf size=29 \[ 3 x \left (-4+\frac {x}{(-4+2 (x+e x)) (4+x-\log (-4+x))}\right ) \]
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Rubi [F]
time = 2.99, 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 {6144-4416 x-462 x^2+681 x^3-24 x^5+e \left (-6144 x+1488 x^2+1167 x^3-96 x^4-48 x^5\right )+e^2 \left (1536 x^2+384 x^3-96 x^4-24 x^5\right )+\left (-3072+3024 x-552 x^2-195 x^3+48 x^4+e^2 \left (-768 x^2+48 x^4\right )+e \left (3072 x-1524 x^2-195 x^3+96 x^4\right )\right ) \log (-4+x)+\left (384-480 x+192 x^2-24 x^3+e \left (-384 x+288 x^2-48 x^3\right )+e^2 \left (96 x^2-24 x^3\right )\right ) \log ^2(-4+x)}{-512+384 x+32 x^2-56 x^3+2 x^5+e^2 \left (-128 x^2-32 x^3+8 x^4+2 x^5\right )+e \left (512 x-128 x^2-96 x^3+8 x^4+4 x^5\right )+\left (256-256 x+48 x^2+16 x^3-4 x^4+e \left (-256 x+128 x^2+16 x^3-8 x^4\right )+e^2 \left (64 x^2-4 x^4\right )\right ) \log (-4+x)+\left (-32+40 x-16 x^2+2 x^3+e^2 \left (-8 x^2+2 x^3\right )+e \left (32 x-24 x^2+4 x^3\right )\right ) \log ^2(-4+x)} \, dx \end {gather*}
Verification is not applicable to the result.
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Rubi steps
\begin {gather*} \begin {aligned} \text {integral} &=\int \frac {3 \left (-2048+64 (23+32 e) x-2 \left (-77+248 e+256 e^2\right ) x^2-\left (227+389 e+128 e^2\right ) x^3+32 e (1+e) x^4+8 (1+e)^2 x^5-(-4+x) \left (256-4 (47+64 e) x+\left (-1+63 e+64 e^2\right ) x^2+16 (1+e)^2 x^3\right ) \log (-4+x)+8 (-4+x) (-2+x+e x)^2 \log ^2(-4+x)\right )}{2 (4-x) (2-(1+e) x)^2 (4+x-\log (-4+x))^2} \, dx\\ &=\frac {3}{2} \int \frac {-2048+64 (23+32 e) x-2 \left (-77+248 e+256 e^2\right ) x^2-\left (227+389 e+128 e^2\right ) x^3+32 e (1+e) x^4+8 (1+e)^2 x^5-(-4+x) \left (256-4 (47+64 e) x+\left (-1+63 e+64 e^2\right ) x^2+16 (1+e)^2 x^3\right ) \log (-4+x)+8 (-4+x) (-2+x+e x)^2 \log ^2(-4+x)}{(4-x) (2-(1+e) x)^2 (4+x-\log (-4+x))^2} \, dx\\ &=\frac {3}{2} \int \left (-8+\frac {(5-x) x^2}{(4-x) (2-(1+e) x) (4+x-\log (-4+x))^2}+\frac {x (-4+(1+e) x)}{(2-(1+e) x)^2 (4+x-\log (-4+x))}\right ) \, dx\\ &=-12 x+\frac {3}{2} \int \frac {(5-x) x^2}{(4-x) (2-(1+e) x) (4+x-\log (-4+x))^2} \, dx+\frac {3}{2} \int \frac {x (-4+(1+e) x)}{(2-(1+e) x)^2 (4+x-\log (-4+x))} \, dx\\ &=-12 x+\frac {3}{2} \int \left (\frac {-1+e}{(1+e)^2 (4+x-\log (-4+x))^2}+\frac {8}{(1+2 e) (-4+x) (4+x-\log (-4+x))^2}-\frac {x}{(1+e) (4+x-\log (-4+x))^2}+\frac {2 (3+5 e)}{(1+e)^2 (1+2 e) (2-(1+e) x) (4+x-\log (-4+x))^2}\right ) \, dx+\frac {3}{2} \int \left (\frac {1}{(1+e) (4+x-\log (-4+x))}+\frac {4}{(-1-e) (2-(1+e) x)^2 (4+x-\log (-4+x))}\right ) \, dx\\ &=-12 x-\frac {(3 (1-e)) \int \frac {1}{(4+x-\log (-4+x))^2} \, dx}{2 (1+e)^2}-\frac {3 \int \frac {x}{(4+x-\log (-4+x))^2} \, dx}{2 (1+e)}+\frac {3 \int \frac {1}{4+x-\log (-4+x)} \, dx}{2 (1+e)}-\frac {6 \int \frac {1}{(2-(1+e) x)^2 (4+x-\log (-4+x))} \, dx}{1+e}+\frac {12 \int \frac {1}{(-4+x) (4+x-\log (-4+x))^2} \, dx}{1+2 e}+\frac {(3 (3+5 e)) \int \frac {1}{(2-(1+e) x) (4+x-\log (-4+x))^2} \, dx}{(1+e)^2 (1+2 e)}\\ \end {aligned} \end {gather*}
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Mathematica [A]
time = 0.14, size = 33, normalized size = 1.14 \begin {gather*} -\frac {3}{2} \left (-32+8 x-\frac {x^2}{(-2+x+e x) (4+x-\log (-4+x))}\right ) \end {gather*}
Antiderivative was successfully verified.
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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(136\) vs.
\(2(30)=60\).
time = 10.65, size = 137, normalized size = 4.72
method | result | size |
risch | \(-12 x +\frac {3 x^{2}}{2 \left (x \,{\mathrm e}+x -2\right ) \left (4+x -\ln \left (x -4\right )\right )}\) | \(30\) |
norman | \(\frac {\left (-12 \,{\mathrm e}-12\right ) x^{3}+96 x +\left (-\frac {45}{2}-48 \,{\mathrm e}\right ) x^{2}+\left (12 \,{\mathrm e}+12\right ) x^{2} \ln \left (x -4\right )-24 x \ln \left (x -4\right )}{\left (4+x -\ln \left (x -4\right )\right ) \left (x \,{\mathrm e}+x -2\right )}\) | \(67\) |
derivativedivides | \(-\frac {3 \left (-\frac {32 \left (4 \,{\mathrm e}+1+4 \,{\mathrm e}^{2}\right ) \ln \left (x -4\right )}{1+{\mathrm e}}+\frac {8 \left (16 \,{\mathrm e}^{2}+17 \,{\mathrm e}+5\right ) \left (x -4\right )}{1+{\mathrm e}}+\left (-8 \,{\mathrm e}-8\right ) \left (x -4\right )^{3}+\left (-64 \,{\mathrm e}-63\right ) \left (x -4\right )^{2}+\left (8 \,{\mathrm e}+8\right ) \ln \left (x -4\right ) \left (x -4\right )^{2}+\frac {1024 \,{\mathrm e}^{2}+1040 \,{\mathrm e}+272}{1+{\mathrm e}}\right )}{2 \left (\ln \left (x -4\right )-x -4\right ) \left ({\mathrm e} \left (x -4\right )+4 \,{\mathrm e}+x -2\right )}\) | \(137\) |
default | \(-\frac {3 \left (-\frac {32 \left (4 \,{\mathrm e}+1+4 \,{\mathrm e}^{2}\right ) \ln \left (x -4\right )}{1+{\mathrm e}}+\frac {8 \left (16 \,{\mathrm e}^{2}+17 \,{\mathrm e}+5\right ) \left (x -4\right )}{1+{\mathrm e}}+\left (-8 \,{\mathrm e}-8\right ) \left (x -4\right )^{3}+\left (-64 \,{\mathrm e}-63\right ) \left (x -4\right )^{2}+\left (8 \,{\mathrm e}+8\right ) \ln \left (x -4\right ) \left (x -4\right )^{2}+\frac {1024 \,{\mathrm e}^{2}+1040 \,{\mathrm e}+272}{1+{\mathrm e}}\right )}{2 \left (\ln \left (x -4\right )-x -4\right ) \left ({\mathrm e} \left (x -4\right )+4 \,{\mathrm e}+x -2\right )}\) | \(137\) |
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. 78 vs.
\(2 (27) = 54\).
time = 0.47, size = 78, normalized size = 2.69 \begin {gather*} -\frac {3 \, {\left (8 \, x^{3} {\left (e + 1\right )} + x^{2} {\left (32 \, e + 15\right )} - 8 \, {\left (x^{2} {\left (e + 1\right )} - 2 \, x\right )} \log \left (x - 4\right ) - 64 \, x\right )}}{2 \, {\left (x^{2} {\left (e + 1\right )} + 2 \, x {\left (2 \, e + 1\right )} - {\left (x {\left (e + 1\right )} - 2\right )} \log \left (x - 4\right ) - 8\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. 81 vs.
\(2 (27) = 54\).
time = 0.37, size = 81, normalized size = 2.79 \begin {gather*} -\frac {3 \, {\left (8 \, x^{3} + 15 \, x^{2} + 8 \, {\left (x^{3} + 4 \, x^{2}\right )} e - 8 \, {\left (x^{2} e + x^{2} - 2 \, x\right )} \log \left (x - 4\right ) - 64 \, x\right )}}{2 \, {\left (x^{2} + {\left (x^{2} + 4 \, x\right )} e - {\left (x e + x - 2\right )} \log \left (x - 4\right ) + 2 \, x - 8\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 0.16, size = 51, normalized size = 1.76 \begin {gather*} - \frac {3 x^{2}}{- 2 e x^{2} - 2 x^{2} - 8 e x - 4 x + \left (2 x + 2 e x - 4\right ) \log {\left (x - 4 \right )} + 16} - 12 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. 100 vs.
\(2 (27) = 54\).
time = 0.46, size = 100, normalized size = 3.45 \begin {gather*} -\frac {3 \, {\left (8 \, x^{3} e - 8 \, x^{2} e \log \left (x - 4\right ) + 8 \, x^{3} + 32 \, x^{2} e - 8 \, x^{2} \log \left (x - 4\right ) + 15 \, x^{2} + 16 \, x \log \left (x - 4\right ) - 64 \, x\right )}}{2 \, {\left (x^{2} e - x e \log \left (x - 4\right ) + x^{2} + 4 \, x e - x \log \left (x - 4\right ) + 2 \, x + 2 \, \log \left (x - 4\right ) - 8\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 4.40, size = 129, normalized size = 4.45 \begin {gather*} -\frac {3\,\left (2\,\ln \left (x-4\right )-62\,x+15\,x\,\ln \left (x-4\right )-60\,x\,\mathrm {e}-8\,x^2\,\ln \left (x-4\right )+48\,x^2\,\mathrm {e}+32\,x^2\,{\mathrm {e}}^2+16\,x^3\,\mathrm {e}+8\,x^3\,{\mathrm {e}}^2+16\,x^2+8\,x^3+15\,x\,\ln \left (x-4\right )\,\mathrm {e}-16\,x^2\,\ln \left (x-4\right )\,\mathrm {e}-8\,x^2\,\ln \left (x-4\right )\,{\mathrm {e}}^2-8\right )}{2\,\left (\mathrm {e}+1\right )\,\left (x+x\,\mathrm {e}-2\right )\,\left (x-\ln \left (x-4\right )+4\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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