\(\int \frac {96+96 x-36 x^2+12 x^2 \log (5)+e^x (-96-96 x+30 x^2-6 x^3-3 x^4-12 x^2 \log (5))+e^{2 x} (24+24 x-6 x^2+3 x^2 \log (5))}{64+48 x^2+9 x^4+(32 x+12 x^3) \log (5)+4 x^2 \log ^2(5)+e^x (-64-40 x^2-6 x^4+(-32 x-10 x^3) \log (5)-4 x^2 \log ^2(5))+e^{2 x} (16+8 x^2+x^4+(8 x+2 x^3) \log (5)+x^2 \log ^2(5))} \, dx\) [5856]

Optimal result

Integrand size = 180, antiderivative size = 31 \[ \int \frac {96+96 x-36 x^2+12 x^2 \log (5)+e^x \left (-96-96 x+30 x^2-6 x^3-3 x^4-12 x^2 \log (5)\right )+e^{2 x} \left (24+24 x-6 x^2+3 x^2 \log (5)\right )}{64+48 x^2+9 x^4+\left (32 x+12 x^3\right ) \log (5)+4 x^2 \log ^2(5)+e^x \left (-64-40 x^2-6 x^4+\left (-32 x-10 x^3\right ) \log (5)-4 x^2 \log ^2(5)\right )+e^{2 x} \left (16+8 x^2+x^4+\left (8 x+2 x^3\right ) \log (5)+x^2 \log ^2(5)\right )} \, dx=9+\frac {3 x (2+x)}{4+x^2+x \left (\frac {x}{2-e^x}+\log (5)\right )} \]

[Out]

3*(2+x)/(x^2+x*(x/(-exp(x)+2)+ln(5))+4)*x+9

Rubi [F]

\[ \int \frac {96+96 x-36 x^2+12 x^2 \log (5)+e^x \left (-96-96 x+30 x^2-6 x^3-3 x^4-12 x^2 \log (5)\right )+e^{2 x} \left (24+24 x-6 x^2+3 x^2 \log (5)\right )}{64+48 x^2+9 x^4+\left (32 x+12 x^3\right ) \log (5)+4 x^2 \log ^2(5)+e^x \left (-64-40 x^2-6 x^4+\left (-32 x-10 x^3\right ) \log (5)-4 x^2 \log ^2(5)\right )+e^{2 x} \left (16+8 x^2+x^4+\left (8 x+2 x^3\right ) \log (5)+x^2 \log ^2(5)\right )} \, dx=\int \frac {96+96 x-36 x^2+12 x^2 \log (5)+e^x \left (-96-96 x+30 x^2-6 x^3-3 x^4-12 x^2 \log (5)\right )+e^{2 x} \left (24+24 x-6 x^2+3 x^2 \log (5)\right )}{64+48 x^2+9 x^4+\left (32 x+12 x^3\right ) \log (5)+4 x^2 \log ^2(5)+e^x \left (-64-40 x^2-6 x^4+\left (-32 x-10 x^3\right ) \log (5)-4 x^2 \log ^2(5)\right )+e^{2 x} \left (16+8 x^2+x^4+\left (8 x+2 x^3\right ) \log (5)+x^2 \log ^2(5)\right )} \, dx \]

[In]

Int[(96 + 96*x - 36*x^2 + 12*x^2*Log[5] + E^x*(-96 - 96*x + 30*x^2 - 6*x^3 - 3*x^4 - 12*x^2*Log[5]) + E^(2*x)*
(24 + 24*x - 6*x^2 + 3*x^2*Log[5]))/(64 + 48*x^2 + 9*x^4 + (32*x + 12*x^3)*Log[5] + 4*x^2*Log[5]^2 + E^x*(-64
- 40*x^2 - 6*x^4 + (-32*x - 10*x^3)*Log[5] - 4*x^2*Log[5]^2) + E^(2*x)*(16 + 8*x^2 + x^4 + (8*x + 2*x^3)*Log[5
] + x^2*Log[5]^2)),x]

[Out]

(3*(x*(2 - Log[5])*(4 - Log[5])*(4 + Log[5]) - 4*(16 - Log[5]^2)))/((4 + x^2 + x*Log[5])*(16 - Log[5]^2)) + 3*
(5*Log[5]^4 - 2*Log[5]^2*(24 - 5*Log[25]) - 32*Log[25] - 3*Log[5]^3*(5 + Log[25]) + 4*Log[5]*(6 + 7*Log[25]))*
Defer[Int][(-8 + 4*E^x - 3*x^2 + E^x*x^2 + E^x*x*Log[5] - x*Log[25])^(-2), x] - 3*(6 - Log[5])*Defer[Int][x^3/
(-8 + 4*E^x - 3*x^2 + E^x*x^2 + E^x*x*Log[5] - x*Log[25])^2, x] - 9*Defer[Int][x^4/(-8 + 4*E^x - 3*x^2 + E^x*x
^2 + E^x*x*Log[5] - x*Log[25])^2, x] - ((6*I)*(64 - 2*Log[5]^6 + 6*Log[5]^4*(12 - Log[25]) - 256*Log[25] + 8*L
og[25]^2 + Log[5]^5*(7 + Log[25]) - Log[5]^3*(184 + 37*Log[25] - Log[25]^2) + 8*Log[5]*(34 - Log[25]^2 - Log[6
25]) + Log[625]*Log[390625]^2)*Defer[Int][1/((-2*x - Log[5] + I*Sqrt[16 - Log[5]^2])*(-8 + 4*E^x - 3*x^2 + E^x
*x^2 + E^x*x*Log[5] - x*Log[25])^2), x])/Sqrt[16 - Log[5]^2] - ((6*I)*(64 - 2*Log[5]^6 + 6*Log[5]^4*(12 - Log[
25]) - 256*Log[25] + 8*Log[25]^2 + Log[5]^5*(7 + Log[25]) - Log[5]^3*(184 + 37*Log[25] - Log[25]^2) + 8*Log[5]
*(34 - Log[25]^2 - Log[625]) + Log[625]*Log[390625]^2)*Defer[Int][1/((2*x + Log[5] + I*Sqrt[16 - Log[5]^2])*(-
8 + 4*E^x - 3*x^2 + E^x*x^2 + E^x*x*Log[5] - x*Log[25])^2), x])/Sqrt[16 - Log[5]^2] + 3*(2 - Log[5]^2 + Log[62
5])*Defer[Int][(-8 + 4*E^x - 3*x^2 + E^x*x^2 + E^x*x*Log[5] - x*Log[25])^(-1), x] - 3*(2 - Log[5])*Defer[Int][
x/(-8 + 4*E^x - 3*x^2 + E^x*x^2 + E^x*x*Log[5] - x*Log[25]), x] - 3*Defer[Int][x^2/(-8 + 4*E^x - 3*x^2 + E^x*x
^2 + E^x*x*Log[5] - x*Log[25]), x] + 3*(1 + (I*Log[5])/Sqrt[16 - Log[5]^2])*(24 - 6*Log[5]^2 + Log[5]^3 - Log[
25])*Defer[Int][1/((2*x + Log[5] - I*Sqrt[16 - Log[5]^2])*(-8 + 4*E^x - 3*x^2 + E^x*x^2 + E^x*x*Log[5] - x*Log
[25])), x] + ((12*I)*(12 + Log[5]^3 - (40*Log[5]^2)/Log[25])*Defer[Int][1/((-2*x - Log[5] + I*Sqrt[16 - Log[5]
^2])*(-8 + 4*E^x - 3*x^2 + E^x*x^2 + E^x*x*Log[5] - x*Log[25])), x])/Sqrt[16 - Log[5]^2] + ((12*I)*(12 + Log[5
]^3 - (40*Log[5]^2)/Log[25])*Defer[Int][1/((2*x + Log[5] + I*Sqrt[16 - Log[5]^2])*(-8 + 4*E^x - 3*x^2 + E^x*x^
2 + E^x*x*Log[5] - x*Log[25])), x])/Sqrt[16 - Log[5]^2] + 3*(1 - (I*Log[5])/Sqrt[16 - Log[5]^2])*(24 - 6*Log[5
]^2 + Log[5]^3 - Log[25])*Defer[Int][1/((2*x + Log[5] + I*Sqrt[16 - Log[5]^2])*(-8 + 4*E^x - 3*x^2 + E^x*x^2 +
E^x*x*Log[5] - x*Log[25])), x] + 3*(16 - 3*Log[5]^3 + Log[5]*(8 - 6*Log[25]) - 7*Log[25] + 2*Log[5]^2*(4 + Lo
g[25]))*Defer[Int][x/(8 - 4*E^x + 3*x^2 - E^x*x^2 - E^x*x*Log[5] + x*Log[25])^2, x] + 3*(4 + Log[5]^2 + Log[5]
*(3 - Log[25]))*Defer[Int][x^2/(8 - 4*E^x + 3*x^2 - E^x*x^2 - E^x*x*Log[5] + x*Log[25])^2, x] - 12*(2*Log[5]^6
+ Log[5]^2*(256 - 80*Log[25]) - Log[5]^4*(52 - 6*Log[25]) - Log[5]^5*(7 + Log[25]) - 4*Log[5]*(44 + 24*Log[25
] - Log[25]^2) + Log[5]^3*(124 + 25*Log[25] - Log[25]^2) - 8*(8 - 16*Log[25] + Log[25]^2))*Defer[Int][1/((4 +
x^2 + x*Log[5])^2*(8 - 4*E^x + 3*x^2 - E^x*x^2 - E^x*x*Log[5] + x*Log[25])^2), x] - 3*(2*Log[5]^7 + 8*Log[5]^3
*(54 - 13*Log[25]) - 16*(8 - 7*Log[25]) - 6*Log[5]^5*(10 - Log[25]) - Log[5]^6*(7 + Log[25]) - 4*Log[5]^2*(140
+ 45*Log[25] - 2*Log[25]^2) + Log[5]^4*(152 + 29*Log[25] - Log[25]^2) - 8*Log[5]*(48 - 44*Log[25] + Log[25]^2
))*Defer[Int][x/((4 + x^2 + x*Log[5])^2*(8 - 4*E^x + 3*x^2 - E^x*x^2 - E^x*x*Log[5] + x*Log[25])^2), x] - 3*(1
+ (I*Log[5])/Sqrt[16 - Log[5]^2])*(96 + 7*Log[5]^5 - 2*Log[5]^3*(52 - 9*Log[25]) + 112*Log[5]*(1 - Log[25]) -
56*Log[25] + Log[5]^2*(152 + 57*Log[25] - 2*Log[25]^2) - 2*Log[5]^4*(11 + Log[625]))*Defer[Int][1/((2*x + Log
[5] - I*Sqrt[16 - Log[5]^2])*(8 - 4*E^x + 3*x^2 - E^x*x^2 - E^x*x*Log[5] + x*Log[25])^2), x] - 3*(1 - (I*Log[5
])/Sqrt[16 - Log[5]^2])*(96 + 7*Log[5]^5 - 2*Log[5]^3*(52 - 9*Log[25]) + 112*Log[5]*(1 - Log[25]) - 56*Log[25]
+ Log[5]^2*(152 + 57*Log[25] - 2*Log[25]^2) - 2*Log[5]^4*(11 + Log[625]))*Defer[Int][1/((2*x + Log[5] + I*Sqr
t[16 - Log[5]^2])*(8 - 4*E^x + 3*x^2 - E^x*x^2 - E^x*x*Log[5] + x*Log[25])^2), x] + 12*(32 - 2*Log[5]^2 + 2*Lo
g[5]^3 - Log[5]*(24 + Log[25]))*Defer[Int][1/((4 + x^2 + x*Log[5])^2*(8 - 4*E^x + 3*x^2 - E^x*x^2 - E^x*x*Log[
5] + x*Log[25])), x] + 3*(40*Log[5] - 2*Log[5]^3 + 2*Log[5]^4 + 4*(16 + Log[25]) - Log[5]^2*(32 + Log[25]))*De
fer[Int][x/((4 + x^2 + x*Log[5])^2*(8 - 4*E^x + 3*x^2 - E^x*x^2 - E^x*x*Log[5] + x*Log[25])), x]

Rubi steps \begin{align*} \text {integral}& = \int \frac {96+96 x+x^2 (-36+12 \log (5))+e^x \left (-96-96 x+30 x^2-6 x^3-3 x^4-12 x^2 \log (5)\right )+e^{2 x} \left (24+24 x-6 x^2+3 x^2 \log (5)\right )}{64+48 x^2+9 x^4+\left (32 x+12 x^3\right ) \log (5)+4 x^2 \log ^2(5)+e^x \left (-64-40 x^2-6 x^4+\left (-32 x-10 x^3\right ) \log (5)-4 x^2 \log ^2(5)\right )+e^{2 x} \left (16+8 x^2+x^4+\left (8 x+2 x^3\right ) \log (5)+x^2 \log ^2(5)\right )} \, dx \\ & = \int \frac {96+96 x+x^2 (-36+12 \log (5))+e^x \left (-96-96 x+30 x^2-6 x^3-3 x^4-12 x^2 \log (5)\right )+e^{2 x} \left (24+24 x-6 x^2+3 x^2 \log (5)\right )}{64+9 x^4+\left (32 x+12 x^3\right ) \log (5)+x^2 \left (48+4 \log ^2(5)\right )+e^x \left (-64-40 x^2-6 x^4+\left (-32 x-10 x^3\right ) \log (5)-4 x^2 \log ^2(5)\right )+e^{2 x} \left (16+8 x^2+x^4+\left (8 x+2 x^3\right ) \log (5)+x^2 \log ^2(5)\right )} \, dx \\ & = \int \frac {3 \left (4 \left (8+8 x+x^2 (-3+\log (5))\right )+e^{2 x} \left (8+8 x+x^2 (-2+\log (5))\right )-e^x \left (32+32 x+2 x^3+x^4+2 x^2 (-5+\log (25))\right )\right )}{\left (8+3 x^2-e^x \left (4+x^2+x \log (5)\right )+x \log (25)\right )^2} \, dx \\ & = 3 \int \frac {4 \left (8+8 x+x^2 (-3+\log (5))\right )+e^{2 x} \left (8+8 x+x^2 (-2+\log (5))\right )-e^x \left (32+32 x+2 x^3+x^4+2 x^2 (-5+\log (25))\right )}{\left (8+3 x^2-e^x \left (4+x^2+x \log (5)\right )+x \log (25)\right )^2} \, dx \\ & = 3 \int \left (\frac {8+8 x-x^2 (2-\log (5))}{\left (4+x^2+x \log (5)\right )^2}+\frac {x^2 \left (-24+6 x^2+x^4+x^3 (2+\log (5))-x (8+\log (25))\right )}{\left (4+x^2+x \log (5)\right )^2 \left (8-4 e^x+3 x^2-e^x x^2-e^x x \log (5)+x \log (25)\right )}+\frac {x^2 \left (-64 x-3 x^6-6 x^5 \left (1+\frac {5 \log (5)}{6}\right )+32 \log ^2(5) \left (1-\frac {\log ^2(25)}{4 \log ^2(5)}\right )-20 x^4 \left (1+\frac {1}{20} \log (5) (9+\log (25))\right )-32 x^3 \left (1+\frac {1}{32} \left (-8 \log ^2(5)+7 \log (25)+6 \log (5) \log (25)\right )\right )-16 x^2 \left (1+\frac {1}{16} \left (12 \log ^2(5)-4 \log ^3(5)+32 \log (25)+\log (5) \left (-32-6 \log (25)+\log ^2(25)\right )\right )\right )\right )}{\left (4+x^2+x \log (5)\right )^2 \left (8-4 e^x+3 x^2-e^x x^2-e^x x \log (5)+x \log (25)\right )^2}\right ) \, dx \\ & = 3 \int \frac {8+8 x-x^2 (2-\log (5))}{\left (4+x^2+x \log (5)\right )^2} \, dx+3 \int \frac {x^2 \left (-24+6 x^2+x^4+x^3 (2+\log (5))-x (8+\log (25))\right )}{\left (4+x^2+x \log (5)\right )^2 \left (8-4 e^x+3 x^2-e^x x^2-e^x x \log (5)+x \log (25)\right )} \, dx+3 \int \frac {x^2 \left (-64 x-3 x^6-6 x^5 \left (1+\frac {5 \log (5)}{6}\right )+32 \log ^2(5) \left (1-\frac {\log ^2(25)}{4 \log ^2(5)}\right )-20 x^4 \left (1+\frac {1}{20} \log (5) (9+\log (25))\right )-32 x^3 \left (1+\frac {1}{32} \left (-8 \log ^2(5)+7 \log (25)+6 \log (5) \log (25)\right )\right )-16 x^2 \left (1+\frac {1}{16} \left (12 \log ^2(5)-4 \log ^3(5)+32 \log (25)+\log (5) \left (-32-6 \log (25)+\log ^2(25)\right )\right )\right )\right )}{\left (4+x^2+x \log (5)\right )^2 \left (8-4 e^x+3 x^2-e^x x^2-e^x x \log (5)+x \log (25)\right )^2} \, dx \\ & = \frac {3 \left (x (2-\log (5)) (4-\log (5)) (4+\log (5))-4 \left (16-\log ^2(5)\right )\right )}{\left (4+x^2+x \log (5)\right ) \left (16-\log ^2(5)\right )}+3 \int \left (-\frac {x^2}{-8+4 e^x-3 x^2+e^x x^2+e^x x \log (5)-x \log (25)}+\frac {x (-2+\log (5))}{-8+4 e^x-3 x^2+e^x x^2+e^x x \log (5)-x \log (25)}+\frac {2 \left (1-\frac {\log ^2(5)}{2}+\log (25)\right )}{-8+4 e^x-3 x^2+e^x x^2+e^x x \log (5)-x \log (25)}+\frac {-24-2 \log ^2(5)-2 \log ^3(5)-x \left (24-6 \log ^2(5)+\log ^3(5)-\log (25)\right )+\log (5) (40+\log (25))}{\left (4+x^2+x \log (5)\right ) \left (8-4 e^x+3 x^2-e^x x^2-e^x x \log (5)+x \log (25)\right )}+\frac {4 \left (32-2 \log ^2(5)+2 \log ^3(5)-\log (5) (24+\log (25))\right )+x \left (40 \log (5)-2 \log ^3(5)+2 \log ^4(5)+4 (16+\log (25))-\log ^2(5) (32+\log (25))\right )}{\left (4+x^2+x \log (5)\right )^2 \left (8-4 e^x+3 x^2-e^x x^2-e^x x \log (5)+x \log (25)\right )}\right ) \, dx+3 \int \left (-\frac {3 x^4}{\left (-8+4 e^x-3 x^2+e^x x^2+e^x x \log (5)-x \log (25)\right )^2}+\frac {x^3 (-6+\log (5))}{\left (-8+4 e^x-3 x^2+e^x x^2+e^x x \log (5)-x \log (25)\right )^2}+\frac {x^2 \left (4+\log ^2(5)+\log (5) (3-\log (25))\right )}{\left (8-4 e^x+3 x^2-e^x x^2-e^x x \log (5)+x \log (25)\right )^2}+\frac {x \left (16-3 \log ^3(5)+\log (5) (8-6 \log (25))-7 \log (25)+2 \log ^2(5) (4+\log (25))\right )}{\left (8-4 e^x+3 x^2-e^x x^2-e^x x \log (5)+x \log (25)\right )^2}+\frac {24 \log (5) \left (1-2 \log (5) \left (1+\frac {5}{16} \log (5) \left (1-\frac {\log (5)}{3}+\frac {\left (32-28 \log (5)-10 \log ^2(5)+3 \log ^3(5)\right ) \log (25)}{15 \log ^3(5)}\right )\right )\right )}{\left (-8+4 e^x-3 x^2+e^x x^2+e^x x \log (5)-x \log (25)\right )^2}+\frac {-x \left (2 \log ^7(5)+8 \log ^3(5) (54-13 \log (25))-16 (8-7 \log (25))-6 \log ^5(5) (10-\log (25))-\log ^6(5) (7+\log (25))-\log ^2(5) \left (560+180 \log (25)-8 \log ^2(25)\right )+\log ^4(5) \left (152+29 \log (25)-\log ^2(25)\right )-8 \log (5) \left (48-44 \log (25)+\log ^2(25)\right )\right )-4 \left (2 \log ^6(5)+\log ^2(5) (256-80 \log (25))-\log ^4(5) (52-6 \log (25))-\log ^5(5) (7+\log (25))-4 \log (5) \left (44+24 \log (25)-\log ^2(25)\right )+\log ^3(5) \left (124+25 \log (25)-\log ^2(25)\right )-8 \left (8-16 \log (25)+\log ^2(25)\right )\right )}{\left (4+x^2+x \log (5)\right )^2 \left (8-4 e^x+3 x^2-e^x x^2-e^x x \log (5)+x \log (25)\right )^2}+\frac {2 \log ^6(5)-6 \log ^4(5) (12-\log (25))-\log ^5(5) (7+\log (25))-8 \log (5) \left (34+26 \log (25)-\log ^2(25)\right )+\log ^3(5) \left (184+37 \log (25)-\log ^2(25)\right )-8 \left (8-32 \log (25)+\log ^2(25)\right )-x \left (96+7 \log ^5(5)-2 \log ^3(5) (52-9 \log (25))+112 \log (5) (1-\log (25))-56 \log (25)-\log ^4(5) (22+4 \log (25))+\log ^2(5) \left (152+57 \log (25)-2 \log ^2(25)\right )\right )+64 \log ^2(5) (7-\log (625))}{\left (4+x^2+x \log (5)\right ) \left (8-4 e^x+3 x^2-e^x x^2-e^x x \log (5)+x \log (25)\right )^2}\right ) \, dx+\frac {3 \int 0 \, dx}{16-\log ^2(5)} \\ & = \frac {3 \left (x (2-\log (5)) (4-\log (5)) (4+\log (5))-4 \left (16-\log ^2(5)\right )\right )}{\left (4+x^2+x \log (5)\right ) \left (16-\log ^2(5)\right )}-3 \int \frac {x^2}{-8+4 e^x-3 x^2+e^x x^2+e^x x \log (5)-x \log (25)} \, dx+3 \int \frac {-24-2 \log ^2(5)-2 \log ^3(5)-x \left (24-6 \log ^2(5)+\log ^3(5)-\log (25)\right )+\log (5) (40+\log (25))}{\left (4+x^2+x \log (5)\right ) \left (8-4 e^x+3 x^2-e^x x^2-e^x x \log (5)+x \log (25)\right )} \, dx+3 \int \frac {4 \left (32-2 \log ^2(5)+2 \log ^3(5)-\log (5) (24+\log (25))\right )+x \left (40 \log (5)-2 \log ^3(5)+2 \log ^4(5)+4 (16+\log (25))-\log ^2(5) (32+\log (25))\right )}{\left (4+x^2+x \log (5)\right )^2 \left (8-4 e^x+3 x^2-e^x x^2-e^x x \log (5)+x \log (25)\right )} \, dx+3 \int \frac {-x \left (2 \log ^7(5)+8 \log ^3(5) (54-13 \log (25))-16 (8-7 \log (25))-6 \log ^5(5) (10-\log (25))-\log ^6(5) (7+\log (25))-\log ^2(5) \left (560+180 \log (25)-8 \log ^2(25)\right )+\log ^4(5) \left (152+29 \log (25)-\log ^2(25)\right )-8 \log (5) \left (48-44 \log (25)+\log ^2(25)\right )\right )-4 \left (2 \log ^6(5)+\log ^2(5) (256-80 \log (25))-\log ^4(5) (52-6 \log (25))-\log ^5(5) (7+\log (25))-4 \log (5) \left (44+24 \log (25)-\log ^2(25)\right )+\log ^3(5) \left (124+25 \log (25)-\log ^2(25)\right )-8 \left (8-16 \log (25)+\log ^2(25)\right )\right )}{\left (4+x^2+x \log (5)\right )^2 \left (8-4 e^x+3 x^2-e^x x^2-e^x x \log (5)+x \log (25)\right )^2} \, dx+3 \int \frac {2 \log ^6(5)-6 \log ^4(5) (12-\log (25))-\log ^5(5) (7+\log (25))-8 \log (5) \left (34+26 \log (25)-\log ^2(25)\right )+\log ^3(5) \left (184+37 \log (25)-\log ^2(25)\right )-8 \left (8-32 \log (25)+\log ^2(25)\right )-x \left (96+7 \log ^5(5)-2 \log ^3(5) (52-9 \log (25))+112 \log (5) (1-\log (25))-56 \log (25)-\log ^4(5) (22+4 \log (25))+\log ^2(5) \left (152+57 \log (25)-2 \log ^2(25)\right )\right )+64 \log ^2(5) (7-\log (625))}{\left (4+x^2+x \log (5)\right ) \left (8-4 e^x+3 x^2-e^x x^2-e^x x \log (5)+x \log (25)\right )^2} \, dx-9 \int \frac {x^4}{\left (-8+4 e^x-3 x^2+e^x x^2+e^x x \log (5)-x \log (25)\right )^2} \, dx-(3 (6-\log (5))) \int \frac {x^3}{\left (-8+4 e^x-3 x^2+e^x x^2+e^x x \log (5)-x \log (25)\right )^2} \, dx+(3 (-2+\log (5))) \int \frac {x}{-8+4 e^x-3 x^2+e^x x^2+e^x x \log (5)-x \log (25)} \, dx+\left (3 \left (4+\log ^2(5)+\log (5) (3-\log (25))\right )\right ) \int \frac {x^2}{\left (8-4 e^x+3 x^2-e^x x^2-e^x x \log (5)+x \log (25)\right )^2} \, dx+\left (3 \left (16-3 \log ^3(5)+\log (5) (8-6 \log (25))-7 \log (25)+2 \log ^2(5) (4+\log (25))\right )\right ) \int \frac {x}{\left (8-4 e^x+3 x^2-e^x x^2-e^x x \log (5)+x \log (25)\right )^2} \, dx+\left (3 \left (5 \log ^4(5)-2 \log ^2(5) (24-5 \log (25))-32 \log (25)-3 \log ^3(5) (5+\log (25))+4 \log (5) (6+7 \log (25))\right )\right ) \int \frac {1}{\left (-8+4 e^x-3 x^2+e^x x^2+e^x x \log (5)-x \log (25)\right )^2} \, dx+\left (3 \left (2-\log ^2(5)+\log (625)\right )\right ) \int \frac {1}{-8+4 e^x-3 x^2+e^x x^2+e^x x \log (5)-x \log (25)} \, dx \\ & = \text {Too large to display} \\ \end{align*}

Mathematica [B] (verified)

Leaf count is larger than twice the leaf count of optimal. \(207\) vs. \(2(31)=62\).

Time = 0.37 (sec) , antiderivative size = 207, normalized size of antiderivative = 6.68 \[ \int \frac {96+96 x-36 x^2+12 x^2 \log (5)+e^x \left (-96-96 x+30 x^2-6 x^3-3 x^4-12 x^2 \log (5)\right )+e^{2 x} \left (24+24 x-6 x^2+3 x^2 \log (5)\right )}{64+48 x^2+9 x^4+\left (32 x+12 x^3\right ) \log (5)+4 x^2 \log ^2(5)+e^x \left (-64-40 x^2-6 x^4+\left (-32 x-10 x^3\right ) \log (5)-4 x^2 \log ^2(5)\right )+e^{2 x} \left (16+8 x^2+x^4+\left (8 x+2 x^3\right ) \log (5)+x^2 \log ^2(5)\right )} \, dx=\frac {3 \left (256+3 x^6+96 x (-2+\log (25))+x^5 (-12+9 \log (5)+\log (25))+4 x^2 \left (56+4 \log (5) (-3+\log (25))-5 \log (25)+\log ^2(25)\right )+x^4 (44-4 \log (25)+\log (5) (-13+6 \log (25)))+x^3 \left (-88-4 \log ^2(5)+20 \log (25)+\log (5) \left (60-3 \log (25)+\log ^2(25)\right )\right )-e^x (4+x (-2+\log (5))) \left (32+3 x^4+x^2 (20+\log (5) (-1+\log (25)))+x^3 \log (3125)+2 x (-4+\log (390625))\right )\right )}{\left (-8-3 x^2+e^x \left (4+x^2+x \log (5)\right )-x \log (25)\right ) \left (32+3 x^4+x^2 (20+\log (5) (-1+\log (25)))+x^3 \log (3125)+2 x (-4+\log (390625))\right )} \]

[In]

Integrate[(96 + 96*x - 36*x^2 + 12*x^2*Log[5] + E^x*(-96 - 96*x + 30*x^2 - 6*x^3 - 3*x^4 - 12*x^2*Log[5]) + E^
(2*x)*(24 + 24*x - 6*x^2 + 3*x^2*Log[5]))/(64 + 48*x^2 + 9*x^4 + (32*x + 12*x^3)*Log[5] + 4*x^2*Log[5]^2 + E^x
*(-64 - 40*x^2 - 6*x^4 + (-32*x - 10*x^3)*Log[5] - 4*x^2*Log[5]^2) + E^(2*x)*(16 + 8*x^2 + x^4 + (8*x + 2*x^3)
*Log[5] + x^2*Log[5]^2)),x]

[Out]

(3*(256 + 3*x^6 + 96*x*(-2 + Log[25]) + x^5*(-12 + 9*Log[5] + Log[25]) + 4*x^2*(56 + 4*Log[5]*(-3 + Log[25]) -
5*Log[25] + Log[25]^2) + x^4*(44 - 4*Log[25] + Log[5]*(-13 + 6*Log[25])) + x^3*(-88 - 4*Log[5]^2 + 20*Log[25]
+ Log[5]*(60 - 3*Log[25] + Log[25]^2)) - E^x*(4 + x*(-2 + Log[5]))*(32 + 3*x^4 + x^2*(20 + Log[5]*(-1 + Log[2
5])) + x^3*Log[3125] + 2*x*(-4 + Log[390625]))))/((-8 - 3*x^2 + E^x*(4 + x^2 + x*Log[5]) - x*Log[25])*(32 + 3*
x^4 + x^2*(20 + Log[5]*(-1 + Log[25])) + x^3*Log[3125] + 2*x*(-4 + Log[390625])))

Maple [A] (verified)

Time = 0.43 (sec) , antiderivative size = 61, normalized size of antiderivative = 1.97

method	result	size
norman	\(\frac {-12 \,{\mathrm e}^{x}+3 x^{2}+\left (-12+6 \ln \left (5\right )\right ) x +\left (-3 \ln \left (5\right )+6\right ) x \,{\mathrm e}^{x}+24}{x \,{\mathrm e}^{x} \ln \left (5\right )+{\mathrm e}^{x} x^{2}-2 x \ln \left (5\right )-3 x^{2}+4 \,{\mathrm e}^{x}-8}\)	\(61\)
parallelrisch	\(\frac {24-3 x \,{\mathrm e}^{x} \ln \left (5\right )+6 x \ln \left (5\right )+3 x^{2}+6 \,{\mathrm e}^{x} x -12 x -12 \,{\mathrm e}^{x}}{x \,{\mathrm e}^{x} \ln \left (5\right )+{\mathrm e}^{x} x^{2}-2 x \ln \left (5\right )-3 x^{2}+4 \,{\mathrm e}^{x}-8}\)	\(63\)
risch	\(\frac {\left (-3 \ln \left (5\right )+6\right ) x -12}{x \ln \left (5\right )+x^{2}+4}+\frac {3 \left (2+x \right ) x^{3}}{\left (x \ln \left (5\right )+x^{2}+4\right ) \left (x \,{\mathrm e}^{x} \ln \left (5\right )+{\mathrm e}^{x} x^{2}-2 x \ln \left (5\right )-3 x^{2}+4 \,{\mathrm e}^{x}-8\right )}\)	\(73\)

[In]

int(((3*x^2*ln(5)-6*x^2+24*x+24)*exp(x)^2+(-12*x^2*ln(5)-3*x^4-6*x^3+30*x^2-96*x-96)*exp(x)+12*x^2*ln(5)-36*x^
2+96*x+96)/((x^2*ln(5)^2+(2*x^3+8*x)*ln(5)+x^4+8*x^2+16)*exp(x)^2+(-4*x^2*ln(5)^2+(-10*x^3-32*x)*ln(5)-6*x^4-4
0*x^2-64)*exp(x)+4*x^2*ln(5)^2+(12*x^3+32*x)*ln(5)+9*x^4+48*x^2+64),x,method=_RETURNVERBOSE)

[Out]

(-12*exp(x)+3*x^2+(-12+6*ln(5))*x+(-3*ln(5)+6)*x*exp(x)+24)/(x*exp(x)*ln(5)+exp(x)*x^2-2*x*ln(5)-3*x^2+4*exp(x
)-8)

Fricas [A] (verification not implemented)

none

Time = 0.26 (sec) , antiderivative size = 55, normalized size of antiderivative = 1.77 \[ \int \frac {96+96 x-36 x^2+12 x^2 \log (5)+e^x \left (-96-96 x+30 x^2-6 x^3-3 x^4-12 x^2 \log (5)\right )+e^{2 x} \left (24+24 x-6 x^2+3 x^2 \log (5)\right )}{64+48 x^2+9 x^4+\left (32 x+12 x^3\right ) \log (5)+4 x^2 \log ^2(5)+e^x \left (-64-40 x^2-6 x^4+\left (-32 x-10 x^3\right ) \log (5)-4 x^2 \log ^2(5)\right )+e^{2 x} \left (16+8 x^2+x^4+\left (8 x+2 x^3\right ) \log (5)+x^2 \log ^2(5)\right )} \, dx=-\frac {3 \, {\left (x^{2} - {\left (x \log \left (5\right ) - 2 \, x + 4\right )} e^{x} + 2 \, x \log \left (5\right ) - 4 \, x + 8\right )}}{3 \, x^{2} - {\left (x^{2} + x \log \left (5\right ) + 4\right )} e^{x} + 2 \, x \log \left (5\right ) + 8} \]

[In]

integrate(((3*x^2*log(5)-6*x^2+24*x+24)*exp(x)^2+(-12*x^2*log(5)-3*x^4-6*x^3+30*x^2-96*x-96)*exp(x)+12*x^2*log
(5)-36*x^2+96*x+96)/((x^2*log(5)^2+(2*x^3+8*x)*log(5)+x^4+8*x^2+16)*exp(x)^2+(-4*x^2*log(5)^2+(-10*x^3-32*x)*l
og(5)-6*x^4-40*x^2-64)*exp(x)+4*x^2*log(5)^2+(12*x^3+32*x)*log(5)+9*x^4+48*x^2+64),x, algorithm="fricas")

[Out]

-3*(x^2 - (x*log(5) - 2*x + 4)*e^x + 2*x*log(5) - 4*x + 8)/(3*x^2 - (x^2 + x*log(5) + 4)*e^x + 2*x*log(5) + 8)

Sympy [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 104 vs. \(2 (24) = 48\).

Time = 0.58 (sec) , antiderivative size = 104, normalized size of antiderivative = 3.35 \[ \int \frac {96+96 x-36 x^2+12 x^2 \log (5)+e^x \left (-96-96 x+30 x^2-6 x^3-3 x^4-12 x^2 \log (5)\right )+e^{2 x} \left (24+24 x-6 x^2+3 x^2 \log (5)\right )}{64+48 x^2+9 x^4+\left (32 x+12 x^3\right ) \log (5)+4 x^2 \log ^2(5)+e^x \left (-64-40 x^2-6 x^4+\left (-32 x-10 x^3\right ) \log (5)-4 x^2 \log ^2(5)\right )+e^{2 x} \left (16+8 x^2+x^4+\left (8 x+2 x^3\right ) \log (5)+x^2 \log ^2(5)\right )} \, dx=\frac {3 x^{4} + 6 x^{3}}{- 3 x^{4} - 5 x^{3} \log {\left (5 \right )} - 20 x^{2} - 2 x^{2} \log {\left (5 \right )}^{2} - 16 x \log {\left (5 \right )} + \left (x^{4} + 2 x^{3} \log {\left (5 \right )} + x^{2} \log {\left (5 \right )}^{2} + 8 x^{2} + 8 x \log {\left (5 \right )} + 16\right ) e^{x} - 32} - \frac {x \left (-6 + 3 \log {\left (5 \right )}\right ) + 12}{x^{2} + x \log {\left (5 \right )} + 4} \]

[In]

integrate(((3*x**2*ln(5)-6*x**2+24*x+24)*exp(x)**2+(-12*x**2*ln(5)-3*x**4-6*x**3+30*x**2-96*x-96)*exp(x)+12*x*
*2*ln(5)-36*x**2+96*x+96)/((x**2*ln(5)**2+(2*x**3+8*x)*ln(5)+x**4+8*x**2+16)*exp(x)**2+(-4*x**2*ln(5)**2+(-10*
x**3-32*x)*ln(5)-6*x**4-40*x**2-64)*exp(x)+4*x**2*ln(5)**2+(12*x**3+32*x)*ln(5)+9*x**4+48*x**2+64),x)

[Out]

(3*x**4 + 6*x**3)/(-3*x**4 - 5*x**3*log(5) - 20*x**2 - 2*x**2*log(5)**2 - 16*x*log(5) + (x**4 + 2*x**3*log(5)
+ x**2*log(5)**2 + 8*x**2 + 8*x*log(5) + 16)*exp(x) - 32) - (x*(-6 + 3*log(5)) + 12)/(x**2 + x*log(5) + 4)

Maxima [A] (verification not implemented)

none

Time = 0.37 (sec) , antiderivative size = 53, normalized size of antiderivative = 1.71 \[ \int \frac {96+96 x-36 x^2+12 x^2 \log (5)+e^x \left (-96-96 x+30 x^2-6 x^3-3 x^4-12 x^2 \log (5)\right )+e^{2 x} \left (24+24 x-6 x^2+3 x^2 \log (5)\right )}{64+48 x^2+9 x^4+\left (32 x+12 x^3\right ) \log (5)+4 x^2 \log ^2(5)+e^x \left (-64-40 x^2-6 x^4+\left (-32 x-10 x^3\right ) \log (5)-4 x^2 \log ^2(5)\right )+e^{2 x} \left (16+8 x^2+x^4+\left (8 x+2 x^3\right ) \log (5)+x^2 \log ^2(5)\right )} \, dx=-\frac {3 \, {\left (x^{2} + 2 \, x {\left (\log \left (5\right ) - 2\right )} - {\left (x {\left (\log \left (5\right ) - 2\right )} + 4\right )} e^{x} + 8\right )}}{3 \, x^{2} - {\left (x^{2} + x \log \left (5\right ) + 4\right )} e^{x} + 2 \, x \log \left (5\right ) + 8} \]

[In]

integrate(((3*x^2*log(5)-6*x^2+24*x+24)*exp(x)^2+(-12*x^2*log(5)-3*x^4-6*x^3+30*x^2-96*x-96)*exp(x)+12*x^2*log
(5)-36*x^2+96*x+96)/((x^2*log(5)^2+(2*x^3+8*x)*log(5)+x^4+8*x^2+16)*exp(x)^2+(-4*x^2*log(5)^2+(-10*x^3-32*x)*l
og(5)-6*x^4-40*x^2-64)*exp(x)+4*x^2*log(5)^2+(12*x^3+32*x)*log(5)+9*x^4+48*x^2+64),x, algorithm="maxima")

[Out]

-3*(x^2 + 2*x*(log(5) - 2) - (x*(log(5) - 2) + 4)*e^x + 8)/(3*x^2 - (x^2 + x*log(5) + 4)*e^x + 2*x*log(5) + 8)

Giac [A] (verification not implemented)

none

Time = 0.38 (sec) , antiderivative size = 62, normalized size of antiderivative = 2.00 \[ \int \frac {96+96 x-36 x^2+12 x^2 \log (5)+e^x \left (-96-96 x+30 x^2-6 x^3-3 x^4-12 x^2 \log (5)\right )+e^{2 x} \left (24+24 x-6 x^2+3 x^2 \log (5)\right )}{64+48 x^2+9 x^4+\left (32 x+12 x^3\right ) \log (5)+4 x^2 \log ^2(5)+e^x \left (-64-40 x^2-6 x^4+\left (-32 x-10 x^3\right ) \log (5)-4 x^2 \log ^2(5)\right )+e^{2 x} \left (16+8 x^2+x^4+\left (8 x+2 x^3\right ) \log (5)+x^2 \log ^2(5)\right )} \, dx=-\frac {3 \, {\left (x e^{x} \log \left (5\right ) - x^{2} - 2 \, x e^{x} - 2 \, x \log \left (5\right ) + 4 \, x + 4 \, e^{x} - 8\right )}}{x^{2} e^{x} + x e^{x} \log \left (5\right ) - 3 \, x^{2} - 2 \, x \log \left (5\right ) + 4 \, e^{x} - 8} \]

[In]

integrate(((3*x^2*log(5)-6*x^2+24*x+24)*exp(x)^2+(-12*x^2*log(5)-3*x^4-6*x^3+30*x^2-96*x-96)*exp(x)+12*x^2*log
(5)-36*x^2+96*x+96)/((x^2*log(5)^2+(2*x^3+8*x)*log(5)+x^4+8*x^2+16)*exp(x)^2+(-4*x^2*log(5)^2+(-10*x^3-32*x)*l
og(5)-6*x^4-40*x^2-64)*exp(x)+4*x^2*log(5)^2+(12*x^3+32*x)*log(5)+9*x^4+48*x^2+64),x, algorithm="giac")

[Out]

-3*(x*e^x*log(5) - x^2 - 2*x*e^x - 2*x*log(5) + 4*x + 4*e^x - 8)/(x^2*e^x + x*e^x*log(5) - 3*x^2 - 2*x*log(5)
+ 4*e^x - 8)

Mupad [F(-1)]

Timed out. \[ \int \frac {96+96 x-36 x^2+12 x^2 \log (5)+e^x \left (-96-96 x+30 x^2-6 x^3-3 x^4-12 x^2 \log (5)\right )+e^{2 x} \left (24+24 x-6 x^2+3 x^2 \log (5)\right )}{64+48 x^2+9 x^4+\left (32 x+12 x^3\right ) \log (5)+4 x^2 \log ^2(5)+e^x \left (-64-40 x^2-6 x^4+\left (-32 x-10 x^3\right ) \log (5)-4 x^2 \log ^2(5)\right )+e^{2 x} \left (16+8 x^2+x^4+\left (8 x+2 x^3\right ) \log (5)+x^2 \log ^2(5)\right )} \, dx=\int \frac {96\,x+12\,x^2\,\ln \left (5\right )+{\mathrm {e}}^{2\,x}\,\left (24\,x+3\,x^2\,\ln \left (5\right )-6\,x^2+24\right )-{\mathrm {e}}^x\,\left (96\,x+12\,x^2\,\ln \left (5\right )-30\,x^2+6\,x^3+3\,x^4+96\right )-36\,x^2+96}{4\,x^2\,{\ln \left (5\right )}^2+\ln \left (5\right )\,\left (12\,x^3+32\,x\right )+{\mathrm {e}}^{2\,x}\,\left (x^2\,{\ln \left (5\right )}^2+\ln \left (5\right )\,\left (2\,x^3+8\,x\right )+8\,x^2+x^4+16\right )-{\mathrm {e}}^x\,\left (4\,x^2\,{\ln \left (5\right )}^2+\ln \left (5\right )\,\left (10\,x^3+32\,x\right )+40\,x^2+6\,x^4+64\right )+48\,x^2+9\,x^4+64} \,d x \]

[In]

int((96*x + 12*x^2*log(5) + exp(2*x)*(24*x + 3*x^2*log(5) - 6*x^2 + 24) - exp(x)*(96*x + 12*x^2*log(5) - 30*x^
2 + 6*x^3 + 3*x^4 + 96) - 36*x^2 + 96)/(4*x^2*log(5)^2 + log(5)*(32*x + 12*x^3) + exp(2*x)*(x^2*log(5)^2 + log
(5)*(8*x + 2*x^3) + 8*x^2 + x^4 + 16) - exp(x)*(4*x^2*log(5)^2 + log(5)*(32*x + 10*x^3) + 40*x^2 + 6*x^4 + 64)
+ 48*x^2 + 9*x^4 + 64),x)

[Out]

int((96*x + 12*x^2*log(5) + exp(2*x)*(24*x + 3*x^2*log(5) - 6*x^2 + 24) - exp(x)*(96*x + 12*x^2*log(5) - 30*x^
2 + 6*x^3 + 3*x^4 + 96) - 36*x^2 + 96)/(4*x^2*log(5)^2 + log(5)*(32*x + 12*x^3) + exp(2*x)*(x^2*log(5)^2 + log
(5)*(8*x + 2*x^3) + 8*x^2 + x^4 + 16) - exp(x)*(4*x^2*log(5)^2 + log(5)*(32*x + 10*x^3) + 40*x^2 + 6*x^4 + 64)
+ 48*x^2 + 9*x^4 + 64), x)