3.52.96 \(\int \frac {16 x^5+4 x^6+e^{\frac {3 (-1+2 x)}{x}} (-1296 x-756 x^2)+e^{\frac {4 (-1+2 x)}{x}} (1296+324 x-108 x^2)+e^{\frac {2 (-1+2 x)}{x}} (432 x^2+540 x^3+72 x^4)+e^{\frac {-1+2 x}{x}} (-48 x^3-156 x^4-32 x^5)+(e^{\frac {4 (-1+2 x)}{x}} (-1728-432 x+108 x^2)+e^{\frac {3 (-1+2 x)}{x}} (1296 x+756 x^2+36 x^3)+e^{\frac {2 (-1+2 x)}{x}} (-288 x^2-360 x^3-60 x^4)+e^{\frac {-1+2 x}{x}} (16 x^3+52 x^4+12 x^5)) \log (4+x)+(e^{\frac {4 (-1+2 x)}{x}} (864+216 x-36 x^2)+e^{\frac {3 (-1+2 x)}{x}} (-432 x-252 x^2-24 x^3)+e^{\frac {2 (-1+2 x)}{x}} (48 x^2+60 x^3+12 x^4)) \log ^2(4+x)+(e^{\frac {4 (-1+2 x)}{x}} (-192-48 x+4 x^2)+e^{\frac {3 (-1+2 x)}{x}} (48 x+28 x^2+4 x^3)) \log ^3(4+x)+e^{\frac {4 (-1+2 x)}{x}} (16+4 x) \log ^4(4+x)}{4 x^2+x^3} \, dx\)

Optimal. Leaf size=24 \[ \left (-x+e^{2-\frac {1}{x}} (3-\log (4+x))\right )^4 \]

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Rubi [F]  time = 12.21, 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 x^5+4 x^6+e^{\frac {3 (-1+2 x)}{x}} \left (-1296 x-756 x^2\right )+e^{\frac {4 (-1+2 x)}{x}} \left (1296+324 x-108 x^2\right )+e^{\frac {2 (-1+2 x)}{x}} \left (432 x^2+540 x^3+72 x^4\right )+e^{\frac {-1+2 x}{x}} \left (-48 x^3-156 x^4-32 x^5\right )+\left (e^{\frac {4 (-1+2 x)}{x}} \left (-1728-432 x+108 x^2\right )+e^{\frac {3 (-1+2 x)}{x}} \left (1296 x+756 x^2+36 x^3\right )+e^{\frac {2 (-1+2 x)}{x}} \left (-288 x^2-360 x^3-60 x^4\right )+e^{\frac {-1+2 x}{x}} \left (16 x^3+52 x^4+12 x^5\right )\right ) \log (4+x)+\left (e^{\frac {4 (-1+2 x)}{x}} \left (864+216 x-36 x^2\right )+e^{\frac {3 (-1+2 x)}{x}} \left (-432 x-252 x^2-24 x^3\right )+e^{\frac {2 (-1+2 x)}{x}} \left (48 x^2+60 x^3+12 x^4\right )\right ) \log ^2(4+x)+\left (e^{\frac {4 (-1+2 x)}{x}} \left (-192-48 x+4 x^2\right )+e^{\frac {3 (-1+2 x)}{x}} \left (48 x+28 x^2+4 x^3\right )\right ) \log ^3(4+x)+e^{\frac {4 (-1+2 x)}{x}} (16+4 x) \log ^4(4+x)}{4 x^2+x^3} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Int[(16*x^5 + 4*x^6 + E^((3*(-1 + 2*x))/x)*(-1296*x - 756*x^2) + E^((4*(-1 + 2*x))/x)*(1296 + 324*x - 108*x^2)
+ E^((2*(-1 + 2*x))/x)*(432*x^2 + 540*x^3 + 72*x^4) + E^((-1 + 2*x)/x)*(-48*x^3 - 156*x^4 - 32*x^5) + (E^((4*
(-1 + 2*x))/x)*(-1728 - 432*x + 108*x^2) + E^((3*(-1 + 2*x))/x)*(1296*x + 756*x^2 + 36*x^3) + E^((2*(-1 + 2*x)
)/x)*(-288*x^2 - 360*x^3 - 60*x^4) + E^((-1 + 2*x)/x)*(16*x^3 + 52*x^4 + 12*x^5))*Log[4 + x] + (E^((4*(-1 + 2*
x))/x)*(864 + 216*x - 36*x^2) + E^((3*(-1 + 2*x))/x)*(-432*x - 252*x^2 - 24*x^3) + E^((2*(-1 + 2*x))/x)*(48*x^
2 + 60*x^3 + 12*x^4))*Log[4 + x]^2 + (E^((4*(-1 + 2*x))/x)*(-192 - 48*x + 4*x^2) + E^((3*(-1 + 2*x))/x)*(48*x
+ 28*x^2 + 4*x^3))*Log[4 + x]^3 + E^((4*(-1 + 2*x))/x)*(16 + 4*x)*Log[4 + x]^4)/(4*x^2 + x^3),x]

[Out]

-36*E^(6 - 3/x)*x + 90*E^(4 - 2/x)*x + 51*E^(4 - 2/x)*x^2 - 12*E^(2 - x^(-1))*x^3 + x^4 + 504*E^6*ExpIntegralE
i[-3/x] + 516*E^4*ExpIntegralEi[-2/x] - 288*E^(27/4)*ExpIntegralEi[(-3*(4 + x))/(4*x)] - 336*E^(9/2)*ExpIntegr
alEi[-1/2*(4 + x)/x] + 36*E^(6 - 3/x)*x*Log[4 + x] - 60*E^(4 - 2/x)*x*Log[4 + x] - 30*E^(4 - 2/x)*x^2*Log[4 +
x] + 4*E^(2 - x^(-1))*x^3*Log[4 + x] - 504*E^6*ExpIntegralEi[-3/x]*Log[4 + x] - 312*E^4*ExpIntegralEi[-2/x]*Lo
g[4 + x] + 288*E^(27/4)*ExpIntegralEi[(-3*(4 + x))/(4*x)]*Log[4 + x] + 192*E^(9/2)*ExpIntegralEi[-1/2*(4 + x)/
x]*Log[4 + x] + (E^(8 - 4/x)*(3 - Log[4 + x])^3*(12 + 3*x - 4*Log[4 + x] - x*Log[4 + x]))/(4 + x) - 288*E^(27/
4)*Defer[Int][ExpIntegralEi[-3/4 - 3/x]/(4 + x), x] - 192*E^(9/2)*Defer[Int][ExpIntegralEi[-1/2 - 2/x]/(4 + x)
, x] + 504*E^6*Defer[Int][ExpIntegralEi[-3/x]/(4 + x), x] + 312*E^4*Defer[Int][ExpIntegralEi[-2/x]/(4 + x), x]
- 24*Defer[Int][E^(6 - 3/x)*Log[4 + x]^2, x] - 36*Defer[Int][E^(4 - 2/x)*Log[4 + x]^2, x] - 108*Defer[Int][(E
^(6 - 3/x)*Log[4 + x]^2)/x, x] - 48*Defer[Int][(E^(6 - 3/x)*Log[4 + x]^2)/(4 + x), x] + 12*Defer[Int][E^(4 - 2
/x)*(4 + x)*Log[4 + x]^2, x] + 4*Defer[Int][E^(6 - 3/x)*Log[4 + x]^3, x] + 12*Defer[Int][(E^(6 - 3/x)*Log[4 +
x]^3)/x, x]

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\int \frac {16 x^5+4 x^6+e^{\frac {3 (-1+2 x)}{x}} \left (-1296 x-756 x^2\right )+e^{\frac {4 (-1+2 x)}{x}} \left (1296+324 x-108 x^2\right )+e^{\frac {2 (-1+2 x)}{x}} \left (432 x^2+540 x^3+72 x^4\right )+e^{\frac {-1+2 x}{x}} \left (-48 x^3-156 x^4-32 x^5\right )+\left (e^{\frac {4 (-1+2 x)}{x}} \left (-1728-432 x+108 x^2\right )+e^{\frac {3 (-1+2 x)}{x}} \left (1296 x+756 x^2+36 x^3\right )+e^{\frac {2 (-1+2 x)}{x}} \left (-288 x^2-360 x^3-60 x^4\right )+e^{\frac {-1+2 x}{x}} \left (16 x^3+52 x^4+12 x^5\right )\right ) \log (4+x)+\left (e^{\frac {4 (-1+2 x)}{x}} \left (864+216 x-36 x^2\right )+e^{\frac {3 (-1+2 x)}{x}} \left (-432 x-252 x^2-24 x^3\right )+e^{\frac {2 (-1+2 x)}{x}} \left (48 x^2+60 x^3+12 x^4\right )\right ) \log ^2(4+x)+\left (e^{\frac {4 (-1+2 x)}{x}} \left (-192-48 x+4 x^2\right )+e^{\frac {3 (-1+2 x)}{x}} \left (48 x+28 x^2+4 x^3\right )\right ) \log ^3(4+x)+e^{\frac {4 (-1+2 x)}{x}} (16+4 x) \log ^4(4+x)}{x^2 (4+x)} \, dx\\ &=\int \frac {4 e^{-4/x} \left (3 e^2-e^{\frac {1}{x}} x-e^2 \log (4+x)\right )^3 \left (-e^{\frac {1}{x}} x^2 (4+x)-e^2 \left (-12-3 x+x^2\right )-e^2 (4+x) \log (4+x)\right )}{x^2 (4+x)} \, dx\\ &=4 \int \frac {e^{-4/x} \left (3 e^2-e^{\frac {1}{x}} x-e^2 \log (4+x)\right )^3 \left (-e^{\frac {1}{x}} x^2 (4+x)-e^2 \left (-12-3 x+x^2\right )-e^2 (4+x) \log (4+x)\right )}{x^2 (4+x)} \, dx\\ &=4 \int \left (x^3+\frac {e^{8-\frac {4}{x}} (-3+\log (4+x))^3 \left (-12-3 x+x^2+4 \log (4+x)+x \log (4+x)\right )}{x^2 (4+x)}+\frac {3 e^{4-\frac {2}{x}} (-3+\log (4+x)) \left (-12-15 x-2 x^2+4 \log (4+x)+5 x \log (4+x)+x^2 \log (4+x)\right )}{4+x}+\frac {e^{6-\frac {3}{x}} (-3+\log (4+x))^2 \left (-36-21 x+12 \log (4+x)+7 x \log (4+x)+x^2 \log (4+x)\right )}{x (4+x)}+\frac {e^{2-\frac {1}{x}} x \left (-12-39 x-8 x^2+4 \log (4+x)+13 x \log (4+x)+3 x^2 \log (4+x)\right )}{4+x}\right ) \, dx\\ &=x^4+4 \int \frac {e^{8-\frac {4}{x}} (-3+\log (4+x))^3 \left (-12-3 x+x^2+4 \log (4+x)+x \log (4+x)\right )}{x^2 (4+x)} \, dx+4 \int \frac {e^{6-\frac {3}{x}} (-3+\log (4+x))^2 \left (-36-21 x+12 \log (4+x)+7 x \log (4+x)+x^2 \log (4+x)\right )}{x (4+x)} \, dx+4 \int \frac {e^{2-\frac {1}{x}} x \left (-12-39 x-8 x^2+4 \log (4+x)+13 x \log (4+x)+3 x^2 \log (4+x)\right )}{4+x} \, dx+12 \int \frac {e^{4-\frac {2}{x}} (-3+\log (4+x)) \left (-12-15 x-2 x^2+4 \log (4+x)+5 x \log (4+x)+x^2 \log (4+x)\right )}{4+x} \, dx\\ &=x^4+\frac {e^{8-\frac {4}{x}} (3-\log (4+x))^3 (12+3 x-4 \log (4+x)-x \log (4+x))}{4+x}+4 \int \left (-\frac {e^{2-\frac {1}{x}} x \left (12+39 x+8 x^2\right )}{4+x}+e^{2-\frac {1}{x}} x (1+3 x) \log (4+x)\right ) \, dx+4 \int \left (-\frac {27 e^{6-\frac {3}{x}} (12+7 x)}{x (4+x)}+\frac {9 e^{6-\frac {3}{x}} \left (36+21 x+x^2\right ) \log (4+x)}{x (4+x)}-\frac {3 e^{6-\frac {3}{x}} \left (36+21 x+2 x^2\right ) \log ^2(4+x)}{x (4+x)}+\frac {e^{6-\frac {3}{x}} (3+x) \log ^3(4+x)}{x}\right ) \, dx+12 \int \left (\frac {3 e^{4-\frac {2}{x}} \left (12+15 x+2 x^2\right )}{4+x}-\frac {e^{4-\frac {2}{x}} \left (24+30 x+5 x^2\right ) \log (4+x)}{4+x}+e^{4-\frac {2}{x}} (1+x) \log ^2(4+x)\right ) \, dx\\ &=x^4+\frac {e^{8-\frac {4}{x}} (3-\log (4+x))^3 (12+3 x-4 \log (4+x)-x \log (4+x))}{4+x}-4 \int \frac {e^{2-\frac {1}{x}} x \left (12+39 x+8 x^2\right )}{4+x} \, dx+4 \int e^{2-\frac {1}{x}} x (1+3 x) \log (4+x) \, dx+4 \int \frac {e^{6-\frac {3}{x}} (3+x) \log ^3(4+x)}{x} \, dx-12 \int \frac {e^{4-\frac {2}{x}} \left (24+30 x+5 x^2\right ) \log (4+x)}{4+x} \, dx+12 \int e^{4-\frac {2}{x}} (1+x) \log ^2(4+x) \, dx-12 \int \frac {e^{6-\frac {3}{x}} \left (36+21 x+2 x^2\right ) \log ^2(4+x)}{x (4+x)} \, dx+36 \int \frac {e^{4-\frac {2}{x}} \left (12+15 x+2 x^2\right )}{4+x} \, dx+36 \int \frac {e^{6-\frac {3}{x}} \left (36+21 x+x^2\right ) \log (4+x)}{x (4+x)} \, dx-108 \int \frac {e^{6-\frac {3}{x}} (12+7 x)}{x (4+x)} \, dx\\ &=x^4+36 e^{6-\frac {3}{x}} x \log (4+x)-60 e^{4-\frac {2}{x}} x \log (4+x)-30 e^{4-\frac {2}{x}} x^2 \log (4+x)+4 e^{2-\frac {1}{x}} x^3 \log (4+x)-504 e^6 \text {Ei}\left (-\frac {3}{x}\right ) \log (4+x)-312 e^4 \text {Ei}\left (-\frac {2}{x}\right ) \log (4+x)+288 e^{27/4} \text {Ei}\left (-\frac {3 (4+x)}{4 x}\right ) \log (4+x)+192 e^{9/2} \text {Ei}\left (-\frac {4+x}{2 x}\right ) \log (4+x)+\frac {e^{8-\frac {4}{x}} (3-\log (4+x))^3 (12+3 x-4 \log (4+x)-x \log (4+x))}{4+x}-4 \int \frac {e^{2-\frac {1}{x}} x^3}{4+x} \, dx-4 \int \left (-16 e^{2-\frac {1}{x}}+7 e^{2-\frac {1}{x}} x+8 e^{2-\frac {1}{x}} x^2+\frac {64 e^{2-\frac {1}{x}}}{4+x}\right ) \, dx+4 \int \left (e^{6-\frac {3}{x}} \log ^3(4+x)+\frac {3 e^{6-\frac {3}{x}} \log ^3(4+x)}{x}\right ) \, dx+12 \int \frac {e^4 \left (5 e^{-2/x} x (2+x)+52 \text {Ei}\left (-\frac {2}{x}\right )-32 \sqrt {e} \text {Ei}\left (-\frac {4+x}{2 x}\right )\right )}{2 (4+x)} \, dx-12 \int \left (2 e^{6-\frac {3}{x}} \log ^2(4+x)+\frac {9 e^{6-\frac {3}{x}} \log ^2(4+x)}{x}+\frac {4 e^{6-\frac {3}{x}} \log ^2(4+x)}{4+x}\right ) \, dx+12 \int \left (-3 e^{4-\frac {2}{x}} \log ^2(4+x)+e^{4-\frac {2}{x}} (4+x) \log ^2(4+x)\right ) \, dx+36 \int \left (7 e^{4-\frac {2}{x}}+2 e^{4-\frac {2}{x}} x-\frac {16 e^{4-\frac {2}{x}}}{4+x}\right ) \, dx-36 \int \frac {e^6 \left (e^{-3/x} x-14 \text {Ei}\left (-\frac {3}{x}\right )+8 e^{3/4} \text {Ei}\left (-\frac {3 (4+x)}{4 x}\right )\right )}{4+x} \, dx-108 \int \left (\frac {3 e^{6-\frac {3}{x}}}{x}+\frac {4 e^{6-\frac {3}{x}}}{4+x}\right ) \, dx\\ &=x^4+36 e^{6-\frac {3}{x}} x \log (4+x)-60 e^{4-\frac {2}{x}} x \log (4+x)-30 e^{4-\frac {2}{x}} x^2 \log (4+x)+4 e^{2-\frac {1}{x}} x^3 \log (4+x)-504 e^6 \text {Ei}\left (-\frac {3}{x}\right ) \log (4+x)-312 e^4 \text {Ei}\left (-\frac {2}{x}\right ) \log (4+x)+288 e^{27/4} \text {Ei}\left (-\frac {3 (4+x)}{4 x}\right ) \log (4+x)+192 e^{9/2} \text {Ei}\left (-\frac {4+x}{2 x}\right ) \log (4+x)+\frac {e^{8-\frac {4}{x}} (3-\log (4+x))^3 (12+3 x-4 \log (4+x)-x \log (4+x))}{4+x}-4 \int \left (16 e^{2-\frac {1}{x}}-4 e^{2-\frac {1}{x}} x+e^{2-\frac {1}{x}} x^2-\frac {64 e^{2-\frac {1}{x}}}{4+x}\right ) \, dx+4 \int e^{6-\frac {3}{x}} \log ^3(4+x) \, dx+12 \int e^{4-\frac {2}{x}} (4+x) \log ^2(4+x) \, dx+12 \int \frac {e^{6-\frac {3}{x}} \log ^3(4+x)}{x} \, dx-24 \int e^{6-\frac {3}{x}} \log ^2(4+x) \, dx-28 \int e^{2-\frac {1}{x}} x \, dx-32 \int e^{2-\frac {1}{x}} x^2 \, dx-36 \int e^{4-\frac {2}{x}} \log ^2(4+x) \, dx-48 \int \frac {e^{6-\frac {3}{x}} \log ^2(4+x)}{4+x} \, dx+64 \int e^{2-\frac {1}{x}} \, dx+72 \int e^{4-\frac {2}{x}} x \, dx-108 \int \frac {e^{6-\frac {3}{x}} \log ^2(4+x)}{x} \, dx+252 \int e^{4-\frac {2}{x}} \, dx-256 \int \frac {e^{2-\frac {1}{x}}}{4+x} \, dx-324 \int \frac {e^{6-\frac {3}{x}}}{x} \, dx-432 \int \frac {e^{6-\frac {3}{x}}}{4+x} \, dx-576 \int \frac {e^{4-\frac {2}{x}}}{4+x} \, dx+\left (6 e^4\right ) \int \frac {5 e^{-2/x} x (2+x)+52 \text {Ei}\left (-\frac {2}{x}\right )-32 \sqrt {e} \text {Ei}\left (-\frac {4+x}{2 x}\right )}{4+x} \, dx-\left (36 e^6\right ) \int \frac {e^{-3/x} x-14 \text {Ei}\left (-\frac {3}{x}\right )+8 e^{3/4} \text {Ei}\left (-\frac {3 (4+x)}{4 x}\right )}{4+x} \, dx\\ &=252 e^{4-\frac {2}{x}} x+64 e^{2-\frac {1}{x}} x+36 e^{4-\frac {2}{x}} x^2-14 e^{2-\frac {1}{x}} x^2-\frac {32}{3} e^{2-\frac {1}{x}} x^3+x^4+324 e^6 \text {Ei}\left (-\frac {3}{x}\right )+36 e^{6-\frac {3}{x}} x \log (4+x)-60 e^{4-\frac {2}{x}} x \log (4+x)-30 e^{4-\frac {2}{x}} x^2 \log (4+x)+4 e^{2-\frac {1}{x}} x^3 \log (4+x)-504 e^6 \text {Ei}\left (-\frac {3}{x}\right ) \log (4+x)-312 e^4 \text {Ei}\left (-\frac {2}{x}\right ) \log (4+x)+288 e^{27/4} \text {Ei}\left (-\frac {3 (4+x)}{4 x}\right ) \log (4+x)+192 e^{9/2} \text {Ei}\left (-\frac {4+x}{2 x}\right ) \log (4+x)+\frac {e^{8-\frac {4}{x}} (3-\log (4+x))^3 (12+3 x-4 \log (4+x)-x \log (4+x))}{4+x}-4 \int e^{2-\frac {1}{x}} x^2 \, dx+4 \int e^{6-\frac {3}{x}} \log ^3(4+x) \, dx+\frac {32}{3} \int e^{2-\frac {1}{x}} x \, dx+12 \int e^{4-\frac {2}{x}} (4+x) \log ^2(4+x) \, dx+12 \int \frac {e^{6-\frac {3}{x}} \log ^3(4+x)}{x} \, dx+14 \int e^{2-\frac {1}{x}} \, dx+16 \int e^{2-\frac {1}{x}} x \, dx-24 \int e^{6-\frac {3}{x}} \log ^2(4+x) \, dx-36 \int e^{4-\frac {2}{x}} \log ^2(4+x) \, dx-48 \int \frac {e^{6-\frac {3}{x}} \log ^2(4+x)}{4+x} \, dx-64 \int e^{2-\frac {1}{x}} \, dx-64 \int \frac {e^{2-\frac {1}{x}}}{x} \, dx-72 \int e^{4-\frac {2}{x}} \, dx-108 \int \frac {e^{6-\frac {3}{x}} \log ^2(4+x)}{x} \, dx-256 \int \frac {e^{2-\frac {1}{x}}}{x} \, dx+256 \int \frac {e^{2-\frac {1}{x}}}{4+x} \, dx-432 \int \frac {e^{6-\frac {3}{x}}}{x} \, dx-504 \int \frac {e^{4-\frac {2}{x}}}{x} \, dx-576 \int \frac {e^{4-\frac {2}{x}}}{x} \, dx+1024 \int \frac {e^{2-\frac {1}{x}}}{x (4+x)} \, dx+1728 \int \frac {e^{6-\frac {3}{x}}}{x (4+x)} \, dx+2304 \int \frac {e^{4-\frac {2}{x}}}{x (4+x)} \, dx+\left (6 e^4\right ) \int \left (\frac {5 e^{-2/x} x (2+x)}{4+x}-\frac {4 \left (-13 \text {Ei}\left (-\frac {2}{x}\right )+8 \sqrt {e} \text {Ei}\left (-\frac {4+x}{2 x}\right )\right )}{4+x}\right ) \, dx-\left (36 e^6\right ) \int \left (\frac {e^{-3/x} x}{4+x}+\frac {2 \left (-7 \text {Ei}\left (-\frac {3}{x}\right )+4 e^{3/4} \text {Ei}\left (-\frac {3 (4+x)}{4 x}\right )\right )}{4+x}\right ) \, dx\\ &=\text {Rest of rules removed due to large latex content} \end {aligned} \end {gather*}

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Mathematica [A]  time = 1.46, size = 31, normalized size = 1.29 \begin {gather*} e^{-4/x} \left (-3 e^2+e^{\frac {1}{x}} x+e^2 \log (4+x)\right )^4 \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(16*x^5 + 4*x^6 + E^((3*(-1 + 2*x))/x)*(-1296*x - 756*x^2) + E^((4*(-1 + 2*x))/x)*(1296 + 324*x - 10
8*x^2) + E^((2*(-1 + 2*x))/x)*(432*x^2 + 540*x^3 + 72*x^4) + E^((-1 + 2*x)/x)*(-48*x^3 - 156*x^4 - 32*x^5) + (
E^((4*(-1 + 2*x))/x)*(-1728 - 432*x + 108*x^2) + E^((3*(-1 + 2*x))/x)*(1296*x + 756*x^2 + 36*x^3) + E^((2*(-1
+ 2*x))/x)*(-288*x^2 - 360*x^3 - 60*x^4) + E^((-1 + 2*x)/x)*(16*x^3 + 52*x^4 + 12*x^5))*Log[4 + x] + (E^((4*(-
1 + 2*x))/x)*(864 + 216*x - 36*x^2) + E^((3*(-1 + 2*x))/x)*(-432*x - 252*x^2 - 24*x^3) + E^((2*(-1 + 2*x))/x)*
(48*x^2 + 60*x^3 + 12*x^4))*Log[4 + x]^2 + (E^((4*(-1 + 2*x))/x)*(-192 - 48*x + 4*x^2) + E^((3*(-1 + 2*x))/x)*
(48*x + 28*x^2 + 4*x^3))*Log[4 + x]^3 + E^((4*(-1 + 2*x))/x)*(16 + 4*x)*Log[4 + x]^4)/(4*x^2 + x^3),x]

[Out]

(-3*E^2 + E^x^(-1)*x + E^2*Log[4 + x])^4/E^(4/x)

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fricas [B]  time = 0.53, size = 230, normalized size = 9.58 \begin {gather*} e^{\left (\frac {4 \, {\left (2 \, x - 1\right )}}{x}\right )} \log \left (x + 4\right )^{4} + x^{4} - 12 \, x^{3} e^{\left (\frac {2 \, x - 1}{x}\right )} + 4 \, {\left (x e^{\left (\frac {3 \, {\left (2 \, x - 1\right )}}{x}\right )} - 3 \, e^{\left (\frac {4 \, {\left (2 \, x - 1\right )}}{x}\right )}\right )} \log \left (x + 4\right )^{3} + 54 \, x^{2} e^{\left (\frac {2 \, {\left (2 \, x - 1\right )}}{x}\right )} + 6 \, {\left (x^{2} e^{\left (\frac {2 \, {\left (2 \, x - 1\right )}}{x}\right )} - 6 \, x e^{\left (\frac {3 \, {\left (2 \, x - 1\right )}}{x}\right )} + 9 \, e^{\left (\frac {4 \, {\left (2 \, x - 1\right )}}{x}\right )}\right )} \log \left (x + 4\right )^{2} - 108 \, x e^{\left (\frac {3 \, {\left (2 \, x - 1\right )}}{x}\right )} + 4 \, {\left (x^{3} e^{\left (\frac {2 \, x - 1}{x}\right )} - 9 \, x^{2} e^{\left (\frac {2 \, {\left (2 \, x - 1\right )}}{x}\right )} + 27 \, x e^{\left (\frac {3 \, {\left (2 \, x - 1\right )}}{x}\right )} - 27 \, e^{\left (\frac {4 \, {\left (2 \, x - 1\right )}}{x}\right )}\right )} \log \left (x + 4\right ) + 81 \, e^{\left (\frac {4 \, {\left (2 \, x - 1\right )}}{x}\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((4*x+16)*exp((2*x-1)/x)^4*log(4+x)^4+((4*x^2-48*x-192)*exp((2*x-1)/x)^4+(4*x^3+28*x^2+48*x)*exp((2*
x-1)/x)^3)*log(4+x)^3+((-36*x^2+216*x+864)*exp((2*x-1)/x)^4+(-24*x^3-252*x^2-432*x)*exp((2*x-1)/x)^3+(12*x^4+6
0*x^3+48*x^2)*exp((2*x-1)/x)^2)*log(4+x)^2+((108*x^2-432*x-1728)*exp((2*x-1)/x)^4+(36*x^3+756*x^2+1296*x)*exp(
(2*x-1)/x)^3+(-60*x^4-360*x^3-288*x^2)*exp((2*x-1)/x)^2+(12*x^5+52*x^4+16*x^3)*exp((2*x-1)/x))*log(4+x)+(-108*
x^2+324*x+1296)*exp((2*x-1)/x)^4+(-756*x^2-1296*x)*exp((2*x-1)/x)^3+(72*x^4+540*x^3+432*x^2)*exp((2*x-1)/x)^2+
(-32*x^5-156*x^4-48*x^3)*exp((2*x-1)/x)+4*x^6+16*x^5)/(x^3+4*x^2),x, algorithm="fricas")

[Out]

e^(4*(2*x - 1)/x)*log(x + 4)^4 + x^4 - 12*x^3*e^((2*x - 1)/x) + 4*(x*e^(3*(2*x - 1)/x) - 3*e^(4*(2*x - 1)/x))*
log(x + 4)^3 + 54*x^2*e^(2*(2*x - 1)/x) + 6*(x^2*e^(2*(2*x - 1)/x) - 6*x*e^(3*(2*x - 1)/x) + 9*e^(4*(2*x - 1)/
x))*log(x + 4)^2 - 108*x*e^(3*(2*x - 1)/x) + 4*(x^3*e^((2*x - 1)/x) - 9*x^2*e^(2*(2*x - 1)/x) + 27*x*e^(3*(2*x
- 1)/x) - 27*e^(4*(2*x - 1)/x))*log(x + 4) + 81*e^(4*(2*x - 1)/x)

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giac [F(-1)]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((4*x+16)*exp((2*x-1)/x)^4*log(4+x)^4+((4*x^2-48*x-192)*exp((2*x-1)/x)^4+(4*x^3+28*x^2+48*x)*exp((2*
x-1)/x)^3)*log(4+x)^3+((-36*x^2+216*x+864)*exp((2*x-1)/x)^4+(-24*x^3-252*x^2-432*x)*exp((2*x-1)/x)^3+(12*x^4+6
0*x^3+48*x^2)*exp((2*x-1)/x)^2)*log(4+x)^2+((108*x^2-432*x-1728)*exp((2*x-1)/x)^4+(36*x^3+756*x^2+1296*x)*exp(
(2*x-1)/x)^3+(-60*x^4-360*x^3-288*x^2)*exp((2*x-1)/x)^2+(12*x^5+52*x^4+16*x^3)*exp((2*x-1)/x))*log(4+x)+(-108*
x^2+324*x+1296)*exp((2*x-1)/x)^4+(-756*x^2-1296*x)*exp((2*x-1)/x)^3+(72*x^4+540*x^3+432*x^2)*exp((2*x-1)/x)^2+
(-32*x^5-156*x^4-48*x^3)*exp((2*x-1)/x)+4*x^6+16*x^5)/(x^3+4*x^2),x, algorithm="giac")

[Out]

Timed out

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maple [B]  time = 0.20, size = 231, normalized size = 9.62

method	result	size
risch	\({\mathrm e}^{\frac {8 x -4}{x}} \ln \left (4+x \right )^{4}+\left (4 \,{\mathrm e}^{\frac {6 x -3}{x}} x -12 \,{\mathrm e}^{\frac {8 x -4}{x}}\right ) \ln \left (4+x \right )^{3}+\left (6 \,{\mathrm e}^{\frac {4 x -2}{x}} x^{2}-36 \,{\mathrm e}^{\frac {6 x -3}{x}} x +54 \,{\mathrm e}^{\frac {8 x -4}{x}}\right ) \ln \left (4+x \right )^{2}+\left (4 \,{\mathrm e}^{\frac {2 x -1}{x}} x^{3}-36 \,{\mathrm e}^{\frac {4 x -2}{x}} x^{2}+108 \,{\mathrm e}^{\frac {6 x -3}{x}} x -108 \,{\mathrm e}^{\frac {8 x -4}{x}}\right ) \ln \left (4+x \right )+x^{4}-12 \,{\mathrm e}^{\frac {2 x -1}{x}} x^{3}+54 \,{\mathrm e}^{\frac {4 x -2}{x}} x^{2}-108 \,{\mathrm e}^{\frac {6 x -3}{x}} x +81 \,{\mathrm e}^{\frac {8 x -4}{x}}\)	\(231\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((4*x+16)*exp((2*x-1)/x)^4*ln(4+x)^4+((4*x^2-48*x-192)*exp((2*x-1)/x)^4+(4*x^3+28*x^2+48*x)*exp((2*x-1)/x)
^3)*ln(4+x)^3+((-36*x^2+216*x+864)*exp((2*x-1)/x)^4+(-24*x^3-252*x^2-432*x)*exp((2*x-1)/x)^3+(12*x^4+60*x^3+48
*x^2)*exp((2*x-1)/x)^2)*ln(4+x)^2+((108*x^2-432*x-1728)*exp((2*x-1)/x)^4+(36*x^3+756*x^2+1296*x)*exp((2*x-1)/x
)^3+(-60*x^4-360*x^3-288*x^2)*exp((2*x-1)/x)^2+(12*x^5+52*x^4+16*x^3)*exp((2*x-1)/x))*ln(4+x)+(-108*x^2+324*x+
1296)*exp((2*x-1)/x)^4+(-756*x^2-1296*x)*exp((2*x-1)/x)^3+(72*x^4+540*x^3+432*x^2)*exp((2*x-1)/x)^2+(-32*x^5-1
56*x^4-48*x^3)*exp((2*x-1)/x)+4*x^6+16*x^5)/(x^3+4*x^2),x,method=_RETURNVERBOSE)

[Out]

exp(4*(2*x-1)/x)*ln(4+x)^4+(4*exp(3*(2*x-1)/x)*x-12*exp(4*(2*x-1)/x))*ln(4+x)^3+(6*exp(2*(2*x-1)/x)*x^2-36*exp
(3*(2*x-1)/x)*x+54*exp(4*(2*x-1)/x))*ln(4+x)^2+(4*exp((2*x-1)/x)*x^3-36*exp(2*(2*x-1)/x)*x^2+108*exp(3*(2*x-1)
/x)*x-108*exp(4*(2*x-1)/x))*ln(4+x)+x^4-12*exp((2*x-1)/x)*x^3+54*exp(2*(2*x-1)/x)*x^2-108*exp(3*(2*x-1)/x)*x+8
1*exp(4*(2*x-1)/x)

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maxima [B]  time = 0.43, size = 162, normalized size = 6.75 \begin {gather*} x^{4} + 4 \, {\left (x^{3} e^{2} \log \left (x + 4\right ) - 3 \, x^{3} e^{2}\right )} e^{\left (-\frac {1}{x}\right )} + 6 \, {\left (x^{2} e^{4} \log \left (x + 4\right )^{2} - 6 \, x^{2} e^{4} \log \left (x + 4\right ) + 9 \, x^{2} e^{4}\right )} e^{\left (-\frac {2}{x}\right )} + 4 \, {\left (x e^{6} \log \left (x + 4\right )^{3} - 9 \, x e^{6} \log \left (x + 4\right )^{2} + 27 \, x e^{6} \log \left (x + 4\right ) - 27 \, x e^{6}\right )} e^{\left (-\frac {3}{x}\right )} + {\left (e^{8} \log \left (x + 4\right )^{4} - 12 \, e^{8} \log \left (x + 4\right )^{3} + 54 \, e^{8} \log \left (x + 4\right )^{2} - 108 \, e^{8} \log \left (x + 4\right ) + 81 \, e^{8}\right )} e^{\left (-\frac {4}{x}\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((4*x+16)*exp((2*x-1)/x)^4*log(4+x)^4+((4*x^2-48*x-192)*exp((2*x-1)/x)^4+(4*x^3+28*x^2+48*x)*exp((2*
x-1)/x)^3)*log(4+x)^3+((-36*x^2+216*x+864)*exp((2*x-1)/x)^4+(-24*x^3-252*x^2-432*x)*exp((2*x-1)/x)^3+(12*x^4+6
0*x^3+48*x^2)*exp((2*x-1)/x)^2)*log(4+x)^2+((108*x^2-432*x-1728)*exp((2*x-1)/x)^4+(36*x^3+756*x^2+1296*x)*exp(
(2*x-1)/x)^3+(-60*x^4-360*x^3-288*x^2)*exp((2*x-1)/x)^2+(12*x^5+52*x^4+16*x^3)*exp((2*x-1)/x))*log(4+x)+(-108*
x^2+324*x+1296)*exp((2*x-1)/x)^4+(-756*x^2-1296*x)*exp((2*x-1)/x)^3+(72*x^4+540*x^3+432*x^2)*exp((2*x-1)/x)^2+
(-32*x^5-156*x^4-48*x^3)*exp((2*x-1)/x)+4*x^6+16*x^5)/(x^3+4*x^2),x, algorithm="maxima")

[Out]

x^4 + 4*(x^3*e^2*log(x + 4) - 3*x^3*e^2)*e^(-1/x) + 6*(x^2*e^4*log(x + 4)^2 - 6*x^2*e^4*log(x + 4) + 9*x^2*e^4
)*e^(-2/x) + 4*(x*e^6*log(x + 4)^3 - 9*x*e^6*log(x + 4)^2 + 27*x*e^6*log(x + 4) - 27*x*e^6)*e^(-3/x) + (e^8*lo
g(x + 4)^4 - 12*e^8*log(x + 4)^3 + 54*e^8*log(x + 4)^2 - 108*e^8*log(x + 4) + 81*e^8)*e^(-4/x)

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mupad [F]  time = 0.00, size = -1, normalized size = -0.04 \begin {gather*} \int \frac {{\ln \left (x+4\right )}^2\,\left ({\mathrm {e}}^{\frac {2\,\left (2\,x-1\right )}{x}}\,\left (12\,x^4+60\,x^3+48\,x^2\right )+{\mathrm {e}}^{\frac {4\,\left (2\,x-1\right )}{x}}\,\left (-36\,x^2+216\,x+864\right )-{\mathrm {e}}^{\frac {3\,\left (2\,x-1\right )}{x}}\,\left (24\,x^3+252\,x^2+432\,x\right )\right )-{\mathrm {e}}^{\frac {2\,x-1}{x}}\,\left (32\,x^5+156\,x^4+48\,x^3\right )+{\mathrm {e}}^{\frac {2\,\left (2\,x-1\right )}{x}}\,\left (72\,x^4+540\,x^3+432\,x^2\right )+{\mathrm {e}}^{\frac {4\,\left (2\,x-1\right )}{x}}\,\left (-108\,x^2+324\,x+1296\right )-{\mathrm {e}}^{\frac {3\,\left (2\,x-1\right )}{x}}\,\left (756\,x^2+1296\,x\right )+16\,x^5+4\,x^6-{\ln \left (x+4\right )}^3\,\left ({\mathrm {e}}^{\frac {4\,\left (2\,x-1\right )}{x}}\,\left (-4\,x^2+48\,x+192\right )-{\mathrm {e}}^{\frac {3\,\left (2\,x-1\right )}{x}}\,\left (4\,x^3+28\,x^2+48\,x\right )\right )+\ln \left (x+4\right )\,\left ({\mathrm {e}}^{\frac {2\,x-1}{x}}\,\left (12\,x^5+52\,x^4+16\,x^3\right )-{\mathrm {e}}^{\frac {2\,\left (2\,x-1\right )}{x}}\,\left (60\,x^4+360\,x^3+288\,x^2\right )-{\mathrm {e}}^{\frac {4\,\left (2\,x-1\right )}{x}}\,\left (-108\,x^2+432\,x+1728\right )+{\mathrm {e}}^{\frac {3\,\left (2\,x-1\right )}{x}}\,\left (36\,x^3+756\,x^2+1296\,x\right )\right )+{\ln \left (x+4\right )}^4\,{\mathrm {e}}^{\frac {4\,\left (2\,x-1\right )}{x}}\,\left (4\,x+16\right )}{x^3+4\,x^2} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((log(x + 4)^2*(exp((2*(2*x - 1))/x)*(48*x^2 + 60*x^3 + 12*x^4) + exp((4*(2*x - 1))/x)*(216*x - 36*x^2 + 86
4) - exp((3*(2*x - 1))/x)*(432*x + 252*x^2 + 24*x^3)) - exp((2*x - 1)/x)*(48*x^3 + 156*x^4 + 32*x^5) + exp((2*
(2*x - 1))/x)*(432*x^2 + 540*x^3 + 72*x^4) + exp((4*(2*x - 1))/x)*(324*x - 108*x^2 + 1296) - exp((3*(2*x - 1))
/x)*(1296*x + 756*x^2) + 16*x^5 + 4*x^6 - log(x + 4)^3*(exp((4*(2*x - 1))/x)*(48*x - 4*x^2 + 192) - exp((3*(2*
x - 1))/x)*(48*x + 28*x^2 + 4*x^3)) + log(x + 4)*(exp((2*x - 1)/x)*(16*x^3 + 52*x^4 + 12*x^5) - exp((2*(2*x -
1))/x)*(288*x^2 + 360*x^3 + 60*x^4) - exp((4*(2*x - 1))/x)*(432*x - 108*x^2 + 1728) + exp((3*(2*x - 1))/x)*(12
96*x + 756*x^2 + 36*x^3)) + log(x + 4)^4*exp((4*(2*x - 1))/x)*(4*x + 16))/(4*x^2 + x^3),x)

[Out]

int((log(x + 4)^2*(exp((2*(2*x - 1))/x)*(48*x^2 + 60*x^3 + 12*x^4) + exp((4*(2*x - 1))/x)*(216*x - 36*x^2 + 86
4) - exp((3*(2*x - 1))/x)*(432*x + 252*x^2 + 24*x^3)) - exp((2*x - 1)/x)*(48*x^3 + 156*x^4 + 32*x^5) + exp((2*
(2*x - 1))/x)*(432*x^2 + 540*x^3 + 72*x^4) + exp((4*(2*x - 1))/x)*(324*x - 108*x^2 + 1296) - exp((3*(2*x - 1))
/x)*(1296*x + 756*x^2) + 16*x^5 + 4*x^6 - log(x + 4)^3*(exp((4*(2*x - 1))/x)*(48*x - 4*x^2 + 192) - exp((3*(2*
x - 1))/x)*(48*x + 28*x^2 + 4*x^3)) + log(x + 4)*(exp((2*x - 1)/x)*(16*x^3 + 52*x^4 + 12*x^5) - exp((2*(2*x -
1))/x)*(288*x^2 + 360*x^3 + 60*x^4) - exp((4*(2*x - 1))/x)*(432*x - 108*x^2 + 1728) + exp((3*(2*x - 1))/x)*(12
96*x + 756*x^2 + 36*x^3)) + log(x + 4)^4*exp((4*(2*x - 1))/x)*(4*x + 16))/(4*x^2 + x^3), x)

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sympy [B]  time = 80.88, size = 143, normalized size = 5.96 \begin {gather*} x^{4} + \left (4 x^{3} \log {\left (x + 4 \right )} - 12 x^{3}\right ) e^{\frac {2 x - 1}{x}} + \left (6 x^{2} \log {\left (x + 4 \right )}^{2} - 36 x^{2} \log {\left (x + 4 \right )} + 54 x^{2}\right ) e^{\frac {2 \left (2 x - 1\right )}{x}} + \left (4 x \log {\left (x + 4 \right )}^{3} - 36 x \log {\left (x + 4 \right )}^{2} + 108 x \log {\left (x + 4 \right )} - 108 x\right ) e^{\frac {3 \left (2 x - 1\right )}{x}} + \left (\log {\left (x + 4 \right )}^{4} - 12 \log {\left (x + 4 \right )}^{3} + 54 \log {\left (x + 4 \right )}^{2} - 108 \log {\left (x + 4 \right )} + 81\right ) e^{\frac {4 \left (2 x - 1\right )}{x}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((4*x+16)*exp((2*x-1)/x)**4*ln(4+x)**4+((4*x**2-48*x-192)*exp((2*x-1)/x)**4+(4*x**3+28*x**2+48*x)*ex
p((2*x-1)/x)**3)*ln(4+x)**3+((-36*x**2+216*x+864)*exp((2*x-1)/x)**4+(-24*x**3-252*x**2-432*x)*exp((2*x-1)/x)**
3+(12*x**4+60*x**3+48*x**2)*exp((2*x-1)/x)**2)*ln(4+x)**2+((108*x**2-432*x-1728)*exp((2*x-1)/x)**4+(36*x**3+75
6*x**2+1296*x)*exp((2*x-1)/x)**3+(-60*x**4-360*x**3-288*x**2)*exp((2*x-1)/x)**2+(12*x**5+52*x**4+16*x**3)*exp(
(2*x-1)/x))*ln(4+x)+(-108*x**2+324*x+1296)*exp((2*x-1)/x)**4+(-756*x**2-1296*x)*exp((2*x-1)/x)**3+(72*x**4+540
*x**3+432*x**2)*exp((2*x-1)/x)**2+(-32*x**5-156*x**4-48*x**3)*exp((2*x-1)/x)+4*x**6+16*x**5)/(x**3+4*x**2),x)

[Out]

x**4 + (4*x**3*log(x + 4) - 12*x**3)*exp((2*x - 1)/x) + (6*x**2*log(x + 4)**2 - 36*x**2*log(x + 4) + 54*x**2)*
exp(2*(2*x - 1)/x) + (4*x*log(x + 4)**3 - 36*x*log(x + 4)**2 + 108*x*log(x + 4) - 108*x)*exp(3*(2*x - 1)/x) +
(log(x + 4)**4 - 12*log(x + 4)**3 + 54*log(x + 4)**2 - 108*log(x + 4) + 81)*exp(4*(2*x - 1)/x)

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