\(\int \frac {-56 x^2+62 x^4+28 x^5+10 x^6+4 x^7+e^5 (-14-28 x-2 x^2-4 x^3)+(-112 x^2-168 x^3-72 x^4-24 x^5-8 x^6) \log (5)+(56+112 x+106 x^2+44 x^3+14 x^4+4 x^5) \log ^2(5)+(56 x^2+48 x^3+22 x^4+18 x^5+6 x^6+e^5 (-14-2 x-6 x^2)+(-112 x-112 x^2-60 x^3-40 x^4-12 x^5) \log (5)+(56+64 x+38 x^2+22 x^3+6 x^4) \log ^2(5)) \log (-\frac {5 x}{-e^5+4 x^2+4 x^3+x^4+(-8 x-8 x^2-2 x^3) \log (5)+(4+4 x+x^2) \log ^2(5)})+(-2 e^5 x+8 x^3+8 x^4+2 x^5+(-16 x^2-16 x^3-4 x^4) \log (5)+(8 x+8 x^2+2 x^3) \log ^2(5)) \log ^2(-\frac {5 x}{-e^5+4 x^2+4 x^3+x^4+(-8 x-8 x^2-2 x^3) \log (5)+(4+4 x+x^2) \log ^2(5)})}{-e^5+4 x^2+4 x^3+x^4+(-8 x-8 x^2-2 x^3) \log (5)+(4+4 x+x^2) \log ^2(5)} \, dx\) [1370]

Optimal result

Integrand size = 422, antiderivative size = 33 \[ \int \frac {-56 x^2+62 x^4+28 x^5+10 x^6+4 x^7+e^5 \left (-14-28 x-2 x^2-4 x^3\right )+\left (-112 x^2-168 x^3-72 x^4-24 x^5-8 x^6\right ) \log (5)+\left (56+112 x+106 x^2+44 x^3+14 x^4+4 x^5\right ) \log ^2(5)+\left (56 x^2+48 x^3+22 x^4+18 x^5+6 x^6+e^5 \left (-14-2 x-6 x^2\right )+\left (-112 x-112 x^2-60 x^3-40 x^4-12 x^5\right ) \log (5)+\left (56+64 x+38 x^2+22 x^3+6 x^4\right ) \log ^2(5)\right ) \log \left (-\frac {5 x}{-e^5+4 x^2+4 x^3+x^4+\left (-8 x-8 x^2-2 x^3\right ) \log (5)+\left (4+4 x+x^2\right ) \log ^2(5)}\right )+\left (-2 e^5 x+8 x^3+8 x^4+2 x^5+\left (-16 x^2-16 x^3-4 x^4\right ) \log (5)+\left (8 x+8 x^2+2 x^3\right ) \log ^2(5)\right ) \log ^2\left (-\frac {5 x}{-e^5+4 x^2+4 x^3+x^4+\left (-8 x-8 x^2-2 x^3\right ) \log (5)+\left (4+4 x+x^2\right ) \log ^2(5)}\right )}{-e^5+4 x^2+4 x^3+x^4+\left (-8 x-8 x^2-2 x^3\right ) \log (5)+\left (4+4 x+x^2\right ) \log ^2(5)} \, dx=\left (7+x \left (x+\log \left (\frac {5 x}{e^5-(2+x)^2 (x-\log (5))^2}\right )\right )\right )^2 \]

[Out]

(7+x*(x+ln(5*x/(exp(5)-(-ln(5)+x)^2*(2+x)^2))))^2

Rubi [F]

\[ \int \frac {-56 x^2+62 x^4+28 x^5+10 x^6+4 x^7+e^5 \left (-14-28 x-2 x^2-4 x^3\right )+\left (-112 x^2-168 x^3-72 x^4-24 x^5-8 x^6\right ) \log (5)+\left (56+112 x+106 x^2+44 x^3+14 x^4+4 x^5\right ) \log ^2(5)+\left (56 x^2+48 x^3+22 x^4+18 x^5+6 x^6+e^5 \left (-14-2 x-6 x^2\right )+\left (-112 x-112 x^2-60 x^3-40 x^4-12 x^5\right ) \log (5)+\left (56+64 x+38 x^2+22 x^3+6 x^4\right ) \log ^2(5)\right ) \log \left (-\frac {5 x}{-e^5+4 x^2+4 x^3+x^4+\left (-8 x-8 x^2-2 x^3\right ) \log (5)+\left (4+4 x+x^2\right ) \log ^2(5)}\right )+\left (-2 e^5 x+8 x^3+8 x^4+2 x^5+\left (-16 x^2-16 x^3-4 x^4\right ) \log (5)+\left (8 x+8 x^2+2 x^3\right ) \log ^2(5)\right ) \log ^2\left (-\frac {5 x}{-e^5+4 x^2+4 x^3+x^4+\left (-8 x-8 x^2-2 x^3\right ) \log (5)+\left (4+4 x+x^2\right ) \log ^2(5)}\right )}{-e^5+4 x^2+4 x^3+x^4+\left (-8 x-8 x^2-2 x^3\right ) \log (5)+\left (4+4 x+x^2\right ) \log ^2(5)} \, dx=\int \frac {-56 x^2+62 x^4+28 x^5+10 x^6+4 x^7+e^5 \left (-14-28 x-2 x^2-4 x^3\right )+\left (-112 x^2-168 x^3-72 x^4-24 x^5-8 x^6\right ) \log (5)+\left (56+112 x+106 x^2+44 x^3+14 x^4+4 x^5\right ) \log ^2(5)+\left (56 x^2+48 x^3+22 x^4+18 x^5+6 x^6+e^5 \left (-14-2 x-6 x^2\right )+\left (-112 x-112 x^2-60 x^3-40 x^4-12 x^5\right ) \log (5)+\left (56+64 x+38 x^2+22 x^3+6 x^4\right ) \log ^2(5)\right ) \log \left (-\frac {5 x}{-e^5+4 x^2+4 x^3+x^4+\left (-8 x-8 x^2-2 x^3\right ) \log (5)+\left (4+4 x+x^2\right ) \log ^2(5)}\right )+\left (-2 e^5 x+8 x^3+8 x^4+2 x^5+\left (-16 x^2-16 x^3-4 x^4\right ) \log (5)+\left (8 x+8 x^2+2 x^3\right ) \log ^2(5)\right ) \log ^2\left (-\frac {5 x}{-e^5+4 x^2+4 x^3+x^4+\left (-8 x-8 x^2-2 x^3\right ) \log (5)+\left (4+4 x+x^2\right ) \log ^2(5)}\right )}{-e^5+4 x^2+4 x^3+x^4+\left (-8 x-8 x^2-2 x^3\right ) \log (5)+\left (4+4 x+x^2\right ) \log ^2(5)} \, dx \]

[In]

Int[(-56*x^2 + 62*x^4 + 28*x^5 + 10*x^6 + 4*x^7 + E^5*(-14 - 28*x - 2*x^2 - 4*x^3) + (-112*x^2 - 168*x^3 - 72*
x^4 - 24*x^5 - 8*x^6)*Log[5] + (56 + 112*x + 106*x^2 + 44*x^3 + 14*x^4 + 4*x^5)*Log[5]^2 + (56*x^2 + 48*x^3 +
22*x^4 + 18*x^5 + 6*x^6 + E^5*(-14 - 2*x - 6*x^2) + (-112*x - 112*x^2 - 60*x^3 - 40*x^4 - 12*x^5)*Log[5] + (56
+ 64*x + 38*x^2 + 22*x^3 + 6*x^4)*Log[5]^2)*Log[(-5*x)/(-E^5 + 4*x^2 + 4*x^3 + x^4 + (-8*x - 8*x^2 - 2*x^3)*L
og[5] + (4 + 4*x + x^2)*Log[5]^2)] + (-2*E^5*x + 8*x^3 + 8*x^4 + 2*x^5 + (-16*x^2 - 16*x^3 - 4*x^4)*Log[5] + (
8*x + 8*x^2 + 2*x^3)*Log[5]^2)*Log[(-5*x)/(-E^5 + 4*x^2 + 4*x^3 + x^4 + (-8*x - 8*x^2 - 2*x^3)*Log[5] + (4 + 4
*x + x^2)*Log[5]^2)]^2)/(-E^5 + 4*x^2 + 4*x^3 + x^4 + (-8*x - 8*x^2 - 2*x^3)*Log[5] + (4 + 4*x + x^2)*Log[5]^2
),x]

[Out]

62*x + (x + (2 - Log[5])/2)^4 - 82*x*(2 - Log[5]) - (28*ArcTanh[(x^2 + x*(2 - Log[5]) - 2*Log[5])/E^(5/2)]*(2
- Log[5]))/E^(5/2) + (7*(2 + 2*x - Log[5])^2)/2 - 5*(2 - Log[5])*(2 + 2*x - Log[5])^2 + (2*(2 + 2*x - Log[5])^
3)/3 - (7*(2 - Log[5])*(2 + 2*x - Log[5])^3)/12 + 3*(1 - Log[5])*(2 + 2*x - Log[5])^2*Log[5] - ((2 + 2*x - Log
[5])^3*Log[5])/3 + ((2 + 2*x - Log[5])^2*Log[5]^2)/2 + E^(5/2)*ArcTanh[(x^2 + x*(2 - Log[5]) - 2*Log[5])/E^(5/
2)]*(20 - 3*Log[5] + 2*Log[5]^2) - (7*x*(2 - Log[5])*(28 - 12*Log[5] + 7*Log[5]^2))/2 + (x*(60 - 28*Log[5] + 1
5*Log[5]^2))/2 + (5*x*(68 - 52*Log[5] + 17*Log[5]^2))/2 - 2*x*Log[5]*(44 - 22*Log[5] + 17*Log[5]^2) + ((2 + 2*
x - Log[5])^2*(92 - 76*Log[5] + 23*Log[5]^2))/8 + (5*ArcTanh[(x^2 + x*(2 - Log[5]) - 2*Log[5])/E^(5/2)]*(2 - L
og[5])*(16 + 3*E^5 + 4*Log[5]^2 + Log[5]^4))/E^(5/2) - (14*ArcTanh[(x^2 + x*(2 - Log[5]) - 2*Log[5])/E^(5/2)]*
(16 + E^5 - 8*Log[5] + 4*Log[5]^2 - 2*Log[5]^3 + Log[5]^4))/E^(5/2) + (28*ArcTan[(2 + 2*x - Log[5])/Sqrt[4*E^(
5/2) - (2 + Log[5])^2]]*(4 - 2*E^(5/2) + Log[5]^2))/(E^(5/2)*Sqrt[4*E^(5/2) - (2 + Log[5])^2]) + 4*ArcTan[(2 +
2*x - Log[5])/Sqrt[4*E^(5/2) - (2 + Log[5])^2]]*(2 - Log[5])*Sqrt[4*E^(5/2) - (2 + Log[5])^2] + (28*ArcTanh[(
2 + 2*x - Log[5])/Sqrt[4*E^(5/2) + (2 + Log[5])^2]]*(4 + 2*E^(5/2) + Log[5]^2))/(E^(5/2)*Sqrt[4*E^(5/2) + (2 +
Log[5])^2]) + 4*ArcTanh[(2 + 2*x - Log[5])/Sqrt[4*E^(5/2) + (2 + Log[5])^2]]*(2 - Log[5])*Sqrt[4*E^(5/2) + (2
+ Log[5])^2] - 2*ArcTanh[(2 + 2*x - Log[5])/Sqrt[4*E^(5/2) + (2 + Log[5])^2]]*(4 + E^(5/2) - 2*Log[5] + Log[5
]^2)*Sqrt[4*E^(5/2) + (2 + Log[5])^2] - (5*ArcTan[(2 + 2*x - Log[5])/Sqrt[4*E^(5/2) - (2 + Log[5])^2]]*(4 - 2*
E^(5/2) + Log[5]^2)*(16 + E^5 - 4*Log[5]^2 + Log[5]^4 - 4*E^(5/2)*(4 - 3*Log[5] + Log[5]^2)))/(E^(5/2)*Sqrt[4*
E^(5/2) - (2 + Log[5])^2]) - (5*ArcTanh[(2 + 2*x - Log[5])/Sqrt[4*E^(5/2) + (2 + Log[5])^2]]*(4 + 2*E^(5/2) +
Log[5]^2)*(16 + E^5 - 4*Log[5]^2 + Log[5]^4 + 4*E^(5/2)*(4 - 3*Log[5] + Log[5]^2)))/(E^(5/2)*Sqrt[4*E^(5/2) +
(2 + Log[5])^2]) - (2*ArcTanh[(x^2 + x*(2 - Log[5]) - 2*Log[5])/E^(5/2)]*(64 - 32*Log[5] + 16*Log[5]^2 - 8*Log
[5]^3 + 4*Log[5]^4 - 2*Log[5]^5 + Log[5]^6 + 6*E^5*(4 - 3*Log[5] + Log[5]^2)))/E^(5/2) - (31*ArcTan[(2 + 2*x -
Log[5])/Sqrt[4*E^(5/2) - (2 + Log[5])^2]]*(16 + 2*E^5 + Log[5]^4 - 4*E^(5/2)*(4 - 2*Log[5] + Log[5]^2)))/(E^(
5/2)*Sqrt[4*E^(5/2) - (2 + Log[5])^2]) - (31*ArcTanh[(2 + 2*x - Log[5])/Sqrt[4*E^(5/2) + (2 + Log[5])^2]]*(16
+ 2*E^5 + Log[5]^4 + 4*E^(5/2)*(4 - 2*Log[5] + Log[5]^2)))/(E^(5/2)*Sqrt[4*E^(5/2) + (2 + Log[5])^2]) - (4*Arc
Tanh[(x^2 + x*(2 - Log[5]) - 2*Log[5])/E^(5/2)]*Log[5]*(3*E^5*(1 - Log[5]) - Log[5]^2*(7 + 7*Log[5] + Log[5]^2
+ Log[5]^3)))/E^(5/2) + (2*ArcTan[(2 + 2*x - Log[5])/Sqrt[4*E^(5/2) - (2 + Log[5])^2]]*(2 - Log[5])*(64 - 7*E
^(15/2) + 32*Log[5] + 16*Log[5]^2 + 8*Log[5]^3 + 4*Log[5]^4 + 2*Log[5]^5 + Log[5]^6 + 14*E^5*(4 - Log[5] + Log
[5]^2) - 7*E^(5/2)*(16 + 4*Log[5]^2 + Log[5]^4)))/(E^(5/2)*Sqrt[4*E^(5/2) - (2 + Log[5])^2]) + (2*ArcTanh[(2 +
2*x - Log[5])/Sqrt[4*E^(5/2) + (2 + Log[5])^2]]*(2 - Log[5])*(64 + 7*E^(15/2) + 32*Log[5] + 16*Log[5]^2 + 8*L
og[5]^3 + 4*Log[5]^4 + 2*Log[5]^5 + Log[5]^6 + 14*E^5*(4 - Log[5] + Log[5]^2) + 7*E^(5/2)*(16 + 4*Log[5]^2 + L
og[5]^4)))/(E^(5/2)*Sqrt[4*E^(5/2) + (2 + Log[5])^2]) + 6*x*(11 - 3*Log[5]^2 - Log[5]*(2 - Log[125])) - 2*ArcT
an[(2 + 2*x - Log[5])/Sqrt[4*E^(5/2) - (2 + Log[5])^2]]*Sqrt[4*E^(5/2) - (2 + Log[5])^2]*(11 - 3*Log[5]^2 - Lo
g[5]*(2 - Log[125])) - 2*ArcTanh[(2 + 2*x - Log[5])/Sqrt[4*E^(5/2) + (2 + Log[5])^2]]*Sqrt[4*E^(5/2) + (2 + Lo
g[5])^2]*(11 - 3*Log[5]^2 - Log[5]*(2 - Log[125])) - (E^(5/2)*ArcTan[(2 + 2*x - Log[5])/Sqrt[4*E^(5/2) - (2 +
Log[5])^2]]*(26 - 14*Log[5] - Log[5]^2 - 2*Log[5]^3 - 2*E^(5/2)*(5 - Log[125])))/Sqrt[4*E^(5/2) - (2 + Log[5])
^2] - (E^(5/2)*ArcTanh[(2 + 2*x - Log[5])/Sqrt[4*E^(5/2) + (2 + Log[5])^2]]*(26 - 14*Log[5] - Log[5]^2 - 2*Log
[5]^3 + 2*E^(5/2)*(5 - Log[125])))/Sqrt[4*E^(5/2) + (2 + Log[5])^2] - (ArcTan[(2 + 2*x - Log[5])/Sqrt[4*E^(5/2
) - (2 + Log[5])^2]]*Log[5]*(8*E^(15/2) - 12*E^5*(8 - 3*Log[5] + 3*Log[5]^2) - Log[5]*(2 + Log[5])*(25*Log[5]
+ 28*Log[5]^2 + 4*Log[5]^3 + 4*Log[5]^4 + Log[125]) + 4*E^(5/2)*(22 + 30*Log[5]^2 + 3*Log[5]^3 + 6*Log[5]^4 +
Log[125])))/(E^(5/2)*Sqrt[4*E^(5/2) - (2 + Log[5])^2]) + (ArcTanh[(2 + 2*x - Log[5])/Sqrt[4*E^(5/2) + (2 + Log
[5])^2]]*Log[5]*(8*E^(15/2) + 12*E^5*(8 - 3*Log[5] + 3*Log[5]^2) + Log[5]*(2 + Log[5])*(25*Log[5] + 28*Log[5]^
2 + 4*Log[5]^3 + 4*Log[5]^4 + Log[125]) + 4*E^(5/2)*(22 + 30*Log[5]^2 + 3*Log[5]^3 + 6*Log[5]^4 + Log[125])))/
(E^(5/2)*Sqrt[4*E^(5/2) + (2 + Log[5])^2]) + (2*ArcTan[(2 + 2*x - Log[5])/Sqrt[4*E^(5/2) - (2 + Log[5])^2]]*(1
6 + 4*E^5 + 8*Log[5] + 2*Log[5]^3 + Log[5]^4 - E^(5/2)*(20 + 5*Log[5]^2 - Log[625])))/Sqrt[4*E^(5/2) - (2 + Lo
g[5])^2] + (7*ArcTan[(2 + 2*x - Log[5])/Sqrt[4*E^(5/2) - (2 + Log[5])^2]]*(20*E^5*(2 - Log[5]) + 20*Log[5]^3 -
4*Log[5]^5 + 4*(32 - 5*Log[625]) + 20*Log[5]*(4 - Log[625]) + 5*Log[5]^2*(16 - Log[625]) - 20*E^(5/2)*(8 + 2*
Log[5]^2 - Log[5]^3 - Log[625])))/(2*E^(5/2)*Sqrt[4*E^(5/2) - (2 + Log[5])^2]) + (7*ArcTanh[(2 + 2*x - Log[5])
/Sqrt[4*E^(5/2) + (2 + Log[5])^2]]*(20*E^5*(2 - Log[5]) + 20*Log[5]^3 - 4*Log[5]^5 + 4*(32 - 5*Log[625]) + 20*
Log[5]*(4 - Log[625]) + 5*Log[5]^2*(16 - Log[625]) + 20*E^(5/2)*(8 + 2*Log[5]^2 - Log[5]^3 - Log[625])))/(2*E^
(5/2)*Sqrt[4*E^(5/2) + (2 + Log[5])^2]) + (31*ArcTanh[(x^2 + x*(2 - Log[5]) - 2*Log[5])/E^(5/2)]*(8 + 2*Log[5]
^2 - Log[5]^3 - Log[625]))/E^(5/2) + (ArcTanh[(x^2 + x*(2 - Log[5]) - 2*Log[5])/E^(5/2)]*(2 - Log[5])*(12*E^5
+ 2*Log[5]^3 + Log[5]^2*(8 - Log[25]) - Log[25]*Log[625]))/(2*E^(5/2)) - (ArcTanh[(x^2 + x*(2 - Log[5]) - 2*Lo
g[5])/E^(5/2)]*Log[5]^2*(8*E^5 + (7 + Log[5]^2)*(8 + 4*Log[5]^2 + 3*Log[625] + Log[5]*Log[625])))/(4*E^(5/2))
- (ArcTan[(2 + 2*x - Log[5])/Sqrt[4*E^(5/2) - (2 + Log[5])^2]]*Log[5]^2*(448 + 16*Log[5]^5 - 16*E^5*(6 - 6*Log
[5] - Log[625]) + 112*Log[625] + 4*Log[625]^2 + Log[625]^3 + 4*Log[5]^4*(12 + Log[625]) + 32*Log[5]*(14 + 3*Lo
g[625]) + 4*Log[5]^3*(44 + 3*Log[625]) + Log[5]^2*(400 + 28*Log[625] + Log[625]^2) - 4*E^(5/2)*(24 + 72*Log[5]
+ 20*Log[5]^3 + 32*Log[625] + Log[625]^2 + Log[5]^2*(16 + 5*Log[625]))))/(16*E^(5/2)*Sqrt[4*E^(5/2) - (2 + Lo
g[5])^2]) - (ArcTanh[(2 + 2*x - Log[5])/Sqrt[4*E^(5/2) + (2 + Log[5])^2]]*Log[5]^2*(448 + 16*Log[5]^5 - 16*E^5
*(6 - 6*Log[5] - Log[625]) + 112*Log[625] + 4*Log[625]^2 + Log[625]^3 + 4*Log[5]^4*(12 + Log[625]) + 32*Log[5]
*(14 + 3*Log[625]) + 4*Log[5]^3*(44 + 3*Log[625]) + Log[5]^2*(400 + 28*Log[625] + Log[625]^2) + 4*E^(5/2)*(24
+ 72*Log[5] + 20*Log[5]^3 + 32*Log[625] + Log[625]^2 + Log[5]^2*(16 + 5*Log[625]))))/(16*E^(5/2)*Sqrt[4*E^(5/2
) + (2 + Log[5])^2]) - 2*x*Log[5]^2*(3 - Log[3125]) - E^5*Log[E^5 - 4*x^2 - 4*x^3 - x^4 + 8*x*Log[5] + 8*x^2*L
og[5] + 2*x^3*Log[5] - 4*Log[5]^2 - 4*x*Log[5]^2 - x^2*Log[5]^2] - 31*(2 - Log[5])*Log[E^5 - 4*x^2 - 4*x^3 - x
^4 + 8*x*Log[5] + 8*x^2*Log[5] + 2*x^3*Log[5] - 4*Log[5]^2 - 4*x*Log[5]^2 - x^2*Log[5]^2] + 2*(2 - Log[5])^2*L
og[E^5 - 4*x^2 - 4*x^3 - x^4 + 8*x*Log[5] + 8*x^2*Log[5] + 2*x^3*Log[5] - 4*Log[5]^2 - 4*x*Log[5]^2 - x^2*Log[
5]^2] - 10*(2 - Log[5])*(4 - Log[5] + Log[5]^2)*Log[E^5 - 4*x^2 - 4*x^3 - x^4 + 8*x*Log[5] + 8*x^2*Log[5] + 2*
x^3*Log[5] - 4*Log[5]^2 - 4*x*Log[5]^2 - x^2*Log[5]^2] + 7*(12 - 8*Log[5] + 3*Log[5]^2)*Log[E^5 - 4*x^2 - 4*x^
3 - x^4 + 8*x*Log[5] + 8*x^2*Log[5] + 2*x^3*Log[5] - 4*Log[5]^2 - 4*x*Log[5]^2 - x^2*Log[5]^2] + 2*Log[5]*(11
- 18*Log[5] + 3*Log[5]^2 - 4*Log[5]^3)*Log[E^5 - 4*x^2 - 4*x^3 - x^4 + 8*x*Log[5] + 8*x^2*Log[5] + 2*x^3*Log[5
] - 4*Log[5]^2 - 4*x*Log[5]^2 - x^2*Log[5]^2] - (8 - Log[5]^3)*Log[E^5 - 4*x^2 - 4*x^3 - x^4 + 8*x*Log[5] + 8*
x^2*Log[5] + 2*x^3*Log[5] - 4*Log[5]^2 - 4*x*Log[5]^2 - x^2*Log[5]^2] + (80 + E^5 - 64*Log[5] + 36*Log[5]^2 -
16*Log[5]^3 + 5*Log[5]^4)*Log[E^5 - 4*x^2 - 4*x^3 - x^4 + 8*x*Log[5] + 8*x^2*Log[5] + 2*x^3*Log[5] - 4*Log[5]^
2 - 4*x*Log[5]^2 - x^2*Log[5]^2] - (2 - Log[5])*(11 - 3*Log[5]^2 - Log[25] + Log[5]*Log[125])*Log[E^5 - 4*x^2
- 4*x^3 - x^4 + 8*x*Log[5] + 8*x^2*Log[5] + 2*x^3*Log[5] - 4*Log[5]^2 - 4*x*Log[5]^2 - x^2*Log[5]^2] + (Log[5]
^2*(72 + 12*Log[5]^2 - 2*Log[625] + 3*Log[5]*Log[625])*Log[E^5 - 4*x^2 - 4*x^3 - x^4 + 8*x*Log[5] + 8*x^2*Log[
5] + 2*x^3*Log[5] - 4*Log[5]^2 - 4*x*Log[5]^2 - x^2*Log[5]^2])/8 + 2*x^3*Log[(5*x)/(E^5 - x^4 - 2*x^3*(2 - Log
[5]) + 4*x*(2 - Log[5])*Log[5] - 4*Log[5]^2 - x^2*(4 - 8*Log[5] + Log[5]^2))] - 4*x*(2 - Log[5])*Log[(5*x)/(E^
5 - x^4 - 2*x^3*(2 - Log[5]) + 4*x*(2 - Log[5])*Log[5] - 4*Log[5]^2 - x^2*(4 - 8*Log[5] + Log[5]^2))] + 2*x*(1
1 - 3*Log[5]^2 - Log[5]*(2 - Log[125]))*Log[(5*x)/(E^5 - x^4 - 2*x^3*(2 - Log[5]) + 4*x*(2 - Log[5])*Log[5] -
4*Log[5]^2 - x^2*(4 - 8*Log[5] + Log[5]^2))] + x^2*Log[(5*x)/(E^5 - x^4 - 2*x^3*(2 - Log[5]) + 4*x*(2 - Log[5]
)*Log[5] - 4*Log[5]^2 - x^2*(4 - 8*Log[5] + Log[5]^2))]^2 + 4*(2 - Log[5])*(E^5 - 4*Log[5]^2)*Defer[Int][Log[(
-5*x)/(-E^5 + x^4 + 2*x^3*(2 - Log[5]) - 4*x*(2 - Log[5])*Log[5] + 4*Log[5]^2 + x^2*(4 - 8*Log[5] + Log[5]^2))
]/(E^5 - x^4 - 2*x^3*(2 - Log[5]) + 4*x*(2 - Log[5])*Log[5] - 4*Log[5]^2 - x^2*(4 - 8*Log[5] + Log[5]^2)), x]
- 2*(E^5 - 4*Log[5]^2)*(4 - 3*Log[5]^2 - Log[25] + Log[5]*Log[125])*Defer[Int][Log[(-5*x)/(-E^5 + x^4 + 2*x^3*
(2 - Log[5]) - 4*x*(2 - Log[5])*Log[5] + 4*Log[5]^2 + x^2*(4 - 8*Log[5] + Log[5]^2))]/(E^5 - x^4 - 2*x^3*(2 -
Log[5]) + 4*x*(2 - Log[5])*Log[5] - 4*Log[5]^2 - x^2*(4 - 8*Log[5] + Log[5]^2)), x] - 8*(E^5 - 2*Log[5]*(4 - 2
*Log[5] + Log[5]^2))*Defer[Int][(x*Log[(-5*x)/(-E^5 + x^4 + 2*x^3*(2 - Log[5]) - 4*x*(2 - Log[5])*Log[5] + 4*L
og[5]^2 + x^2*(4 - 8*Log[5] + Log[5]^2))])/(E^5 - x^4 - 2*x^3*(2 - Log[5]) + 4*x*(2 - Log[5])*Log[5] - 4*Log[5
]^2 - x^2*(4 - 8*Log[5] + Log[5]^2)), x] + 8*(E^5 - Log[5]*(8 + 3*Log[5]^3 + Log[25]*Log[125] - Log[5]^2*(4 +
Log[125]) - Log[625]))*Defer[Int][(x*Log[(-5*x)/(-E^5 + x^4 + 2*x^3*(2 - Log[5]) - 4*x*(2 - Log[5])*Log[5] + 4
*Log[5]^2 + x^2*(4 - 8*Log[5] + Log[5]^2))])/(E^5 - x^4 - 2*x^3*(2 - Log[5]) + 4*x*(2 - Log[5])*Log[5] - 4*Log
[5]^2 - x^2*(4 - 8*Log[5] + Log[5]^2)), x] - 4*(2 - Log[5])*(4 - 2*Log[5] + Log[5]^2)*Defer[Int][(x^2*Log[(-5*
x)/(-E^5 + x^4 + 2*x^3*(2 - Log[5]) - 4*x*(2 - Log[5])*Log[5] + 4*Log[5]^2 + x^2*(4 - 8*Log[5] + Log[5]^2))])/
(E^5 - x^4 - 2*x^3*(2 - Log[5]) + 4*x*(2 - Log[5])*Log[5] - 4*Log[5]^2 - x^2*(4 - 8*Log[5] + Log[5]^2)), x] +
2*(16 - 3*Log[5]^4 - 4*Log[5]*(4 - Log[125]) + Log[5]^3*(22 + Log[125]) - 4*Log[5]^2*(1 + Log[15625]))*Defer[I
nt][(x^2*Log[(-5*x)/(-E^5 + x^4 + 2*x^3*(2 - Log[5]) - 4*x*(2 - Log[5])*Log[5] + 4*Log[5]^2 + x^2*(4 - 8*Log[5
] + Log[5]^2))])/(E^5 - x^4 - 2*x^3*(2 - Log[5]) + 4*x*(2 - Log[5])*Log[5] - 4*Log[5]^2 - x^2*(4 - 8*Log[5] +
Log[5]^2)), x] - 4*(4 + Log[5]^2)*Defer[Int][(x^3*Log[(-5*x)/(-E^5 + x^4 + 2*x^3*(2 - Log[5]) - 4*x*(2 - Log[5
])*Log[5] + 4*Log[5]^2 + x^2*(4 - 8*Log[5] + Log[5]^2))])/(E^5 - x^4 - 2*x^3*(2 - Log[5]) + 4*x*(2 - Log[5])*L
og[5] - 4*Log[5]^2 - x^2*(4 - 8*Log[5] + Log[5]^2)), x] + 4*(4 + 3*Log[5]^3 + Log[25]*Log[125] - Log[5]^2*(5 +
Log[125]))*Defer[Int][(x^3*Log[(-5*x)/(-E^5 + x^4 + 2*x^3*(2 - Log[5]) - 4*x*(2 - Log[5])*Log[5] + 4*Log[5]^2
+ x^2*(4 - 8*Log[5] + Log[5]^2))])/(E^5 - x^4 - 2*x^3*(2 - Log[5]) + 4*x*(2 - Log[5])*Log[5] - 4*Log[5]^2 - x
^2*(4 - 8*Log[5] + Log[5]^2)), x]

Rubi steps \begin{align*} \text {integral}& = \int \frac {56 x^2-62 x^4-28 x^5-10 x^6-4 x^7-e^5 \left (-14-28 x-2 x^2-4 x^3\right )-\left (-112 x^2-168 x^3-72 x^4-24 x^5-8 x^6\right ) \log (5)-\left (56+112 x+106 x^2+44 x^3+14 x^4+4 x^5\right ) \log ^2(5)-\left (56 x^2+48 x^3+22 x^4+18 x^5+6 x^6+e^5 \left (-14-2 x-6 x^2\right )+\left (-112 x-112 x^2-60 x^3-40 x^4-12 x^5\right ) \log (5)+\left (56+64 x+38 x^2+22 x^3+6 x^4\right ) \log ^2(5)\right ) \log \left (-\frac {5 x}{-e^5+4 x^2+4 x^3+x^4+\left (-8 x-8 x^2-2 x^3\right ) \log (5)+\left (4+4 x+x^2\right ) \log ^2(5)}\right )-\left (-2 e^5 x+8 x^3+8 x^4+2 x^5+\left (-16 x^2-16 x^3-4 x^4\right ) \log (5)+\left (8 x+8 x^2+2 x^3\right ) \log ^2(5)\right ) \log ^2\left (-\frac {5 x}{-e^5+4 x^2+4 x^3+x^4+\left (-8 x-8 x^2-2 x^3\right ) \log (5)+\left (4+4 x+x^2\right ) \log ^2(5)}\right )}{e^5-x^4-2 x^3 (2-\log (5))+4 x (2-\log (5)) \log (5)-4 \log ^2(5)-x^2 \left (4-8 \log (5)+\log ^2(5)\right )} \, dx \\ & = \int \frac {56 x^2-62 x^4-28 x^5-10 x^6-4 x^7+2 e^5 \left (7+14 x+x^2+2 x^3\right )+8 x^2 \left (14+21 x+9 x^2+3 x^3+x^4\right ) \log (5)-2 \left (28+56 x+53 x^2+22 x^3+7 x^4+2 x^5\right ) \log ^2(5)-\left (-2 e^5 \left (7+x+3 x^2\right )+2 (2+x) \left (3 x^5+x^4 (3-6 \log (5))+14 \log ^2(5)+x \log (5) (-28+9 \log (5))+x^3 \left (5-8 \log (5)+3 \log ^2(5)\right )+x^2 \left (14-14 \log (5)+5 \log ^2(5)\right )\right )\right ) \log \left (-\frac {5 x}{-e^5+(2+x)^2 (x-\log (5))^2}\right )-2 x \left (-e^5+(2+x)^2 (x-\log (5))^2\right ) \log ^2\left (-\frac {5 x}{-e^5+(2+x)^2 (x-\log (5))^2}\right )}{e^5-x^4-2 x^3 (2-\log (5))+4 x (2-\log (5)) \log (5)-4 \log ^2(5)-x^2 \left (4-8 \log (5)+\log ^2(5)\right )} \, dx \\ & = \int \left (\frac {56 x^2}{e^5-x^4-2 x^3 (2-\log (5))+4 x (2-\log (5)) \log (5)-4 \log ^2(5)-x^2 \left (4-8 \log (5)+\log ^2(5)\right )}+\frac {2 e^5 (1+2 x) \left (7+x^2\right )}{e^5-x^4-2 x^3 (2-\log (5))+4 x (2-\log (5)) \log (5)-4 \log ^2(5)-x^2 \left (4-8 \log (5)+\log ^2(5)\right )}+\frac {8 x^2 (1+x) (2+x) \left (7+x^2\right ) \log (5)}{e^5-x^4-2 x^3 (2-\log (5))+4 x (2-\log (5)) \log (5)-4 \log ^2(5)-x^2 \left (4-8 \log (5)+\log ^2(5)\right )}+\frac {2 (2+x) \left (-2-3 x-2 x^2\right ) \left (7+x^2\right ) \log ^2(5)}{e^5-x^4-2 x^3 (2-\log (5))+4 x (2-\log (5)) \log (5)-4 \log ^2(5)-x^2 \left (4-8 \log (5)+\log ^2(5)\right )}+\frac {62 x^4}{-e^5+x^4+2 x^3 (2-\log (5))-4 x (2-\log (5)) \log (5)+4 \log ^2(5)+x^2 \left (4-8 \log (5)+\log ^2(5)\right )}+\frac {28 x^5}{-e^5+x^4+2 x^3 (2-\log (5))-4 x (2-\log (5)) \log (5)+4 \log ^2(5)+x^2 \left (4-8 \log (5)+\log ^2(5)\right )}+\frac {10 x^6}{-e^5+x^4+2 x^3 (2-\log (5))-4 x (2-\log (5)) \log (5)+4 \log ^2(5)+x^2 \left (4-8 \log (5)+\log ^2(5)\right )}+\frac {4 x^7}{-e^5+x^4+2 x^3 (2-\log (5))-4 x (2-\log (5)) \log (5)+4 \log ^2(5)+x^2 \left (4-8 \log (5)+\log ^2(5)\right )}+\frac {2 \left (-3 x^6-9 x^5 \left (1-\frac {2 \log (5)}{3}\right )+7 e^5 \left (1-\frac {4 \log ^2(5)}{e^5}\right )-24 x^3 \left (1+\frac {1}{24} \log (5) (-30+11 \log (5))\right )-28 x^2 \left (1+\frac {1}{28} \left (-3 e^5+\log (5) (-56+19 \log (5))\right )\right )-11 x^4 \left (1+\frac {1}{11} \log (5) (-20+\log (125))\right )+e^5 x \left (1-\frac {8 \log (5) (-7+\log (625))}{e^5}\right )\right ) \log \left (-\frac {5 x}{-e^5+x^4+2 x^3 (2-\log (5))-4 x (2-\log (5)) \log (5)+4 \log ^2(5)+x^2 \left (4-8 \log (5)+\log ^2(5)\right )}\right )}{e^5-x^4-2 x^3 (2-\log (5))+4 x (2-\log (5)) \log (5)-4 \log ^2(5)-x^2 \left (4-8 \log (5)+\log ^2(5)\right )}+2 x \log ^2\left (-\frac {5 x}{-e^5+x^4+2 x^3 (2-\log (5))-4 x (2-\log (5)) \log (5)+4 \log ^2(5)+x^2 \left (4-8 \log (5)+\log ^2(5)\right )}\right )\right ) \, dx \\ & = 2 \int \frac {\left (-3 x^6-9 x^5 \left (1-\frac {2 \log (5)}{3}\right )+7 e^5 \left (1-\frac {4 \log ^2(5)}{e^5}\right )-24 x^3 \left (1+\frac {1}{24} \log (5) (-30+11 \log (5))\right )-28 x^2 \left (1+\frac {1}{28} \left (-3 e^5+\log (5) (-56+19 \log (5))\right )\right )-11 x^4 \left (1+\frac {1}{11} \log (5) (-20+\log (125))\right )+e^5 x \left (1-\frac {8 \log (5) (-7+\log (625))}{e^5}\right )\right ) \log \left (-\frac {5 x}{-e^5+x^4+2 x^3 (2-\log (5))-4 x (2-\log (5)) \log (5)+4 \log ^2(5)+x^2 \left (4-8 \log (5)+\log ^2(5)\right )}\right )}{e^5-x^4-2 x^3 (2-\log (5))+4 x (2-\log (5)) \log (5)-4 \log ^2(5)-x^2 \left (4-8 \log (5)+\log ^2(5)\right )} \, dx+2 \int x \log ^2\left (-\frac {5 x}{-e^5+x^4+2 x^3 (2-\log (5))-4 x (2-\log (5)) \log (5)+4 \log ^2(5)+x^2 \left (4-8 \log (5)+\log ^2(5)\right )}\right ) \, dx+4 \int \frac {x^7}{-e^5+x^4+2 x^3 (2-\log (5))-4 x (2-\log (5)) \log (5)+4 \log ^2(5)+x^2 \left (4-8 \log (5)+\log ^2(5)\right )} \, dx+10 \int \frac {x^6}{-e^5+x^4+2 x^3 (2-\log (5))-4 x (2-\log (5)) \log (5)+4 \log ^2(5)+x^2 \left (4-8 \log (5)+\log ^2(5)\right )} \, dx+28 \int \frac {x^5}{-e^5+x^4+2 x^3 (2-\log (5))-4 x (2-\log (5)) \log (5)+4 \log ^2(5)+x^2 \left (4-8 \log (5)+\log ^2(5)\right )} \, dx+56 \int \frac {x^2}{e^5-x^4-2 x^3 (2-\log (5))+4 x (2-\log (5)) \log (5)-4 \log ^2(5)-x^2 \left (4-8 \log (5)+\log ^2(5)\right )} \, dx+62 \int \frac {x^4}{-e^5+x^4+2 x^3 (2-\log (5))-4 x (2-\log (5)) \log (5)+4 \log ^2(5)+x^2 \left (4-8 \log (5)+\log ^2(5)\right )} \, dx+\left (2 e^5\right ) \int \frac {(1+2 x) \left (7+x^2\right )}{e^5-x^4-2 x^3 (2-\log (5))+4 x (2-\log (5)) \log (5)-4 \log ^2(5)-x^2 \left (4-8 \log (5)+\log ^2(5)\right )} \, dx+(8 \log (5)) \int \frac {x^2 (1+x) (2+x) \left (7+x^2\right )}{e^5-x^4-2 x^3 (2-\log (5))+4 x (2-\log (5)) \log (5)-4 \log ^2(5)-x^2 \left (4-8 \log (5)+\log ^2(5)\right )} \, dx+\left (2 \log ^2(5)\right ) \int \frac {(2+x) \left (-2-3 x-2 x^2\right ) \left (7+x^2\right )}{e^5-x^4-2 x^3 (2-\log (5))+4 x (2-\log (5)) \log (5)-4 \log ^2(5)-x^2 \left (4-8 \log (5)+\log ^2(5)\right )} \, dx \\ & = \text {Too large to display} \\ \end{align*}

Mathematica [B] (verified)

Leaf count is larger than twice the leaf count of optimal. \(72\) vs. \(2(33)=66\).

Time = 0.27 (sec) , antiderivative size = 72, normalized size of antiderivative = 2.18 \[ \int \frac {-56 x^2+62 x^4+28 x^5+10 x^6+4 x^7+e^5 \left (-14-28 x-2 x^2-4 x^3\right )+\left (-112 x^2-168 x^3-72 x^4-24 x^5-8 x^6\right ) \log (5)+\left (56+112 x+106 x^2+44 x^3+14 x^4+4 x^5\right ) \log ^2(5)+\left (56 x^2+48 x^3+22 x^4+18 x^5+6 x^6+e^5 \left (-14-2 x-6 x^2\right )+\left (-112 x-112 x^2-60 x^3-40 x^4-12 x^5\right ) \log (5)+\left (56+64 x+38 x^2+22 x^3+6 x^4\right ) \log ^2(5)\right ) \log \left (-\frac {5 x}{-e^5+4 x^2+4 x^3+x^4+\left (-8 x-8 x^2-2 x^3\right ) \log (5)+\left (4+4 x+x^2\right ) \log ^2(5)}\right )+\left (-2 e^5 x+8 x^3+8 x^4+2 x^5+\left (-16 x^2-16 x^3-4 x^4\right ) \log (5)+\left (8 x+8 x^2+2 x^3\right ) \log ^2(5)\right ) \log ^2\left (-\frac {5 x}{-e^5+4 x^2+4 x^3+x^4+\left (-8 x-8 x^2-2 x^3\right ) \log (5)+\left (4+4 x+x^2\right ) \log ^2(5)}\right )}{-e^5+4 x^2+4 x^3+x^4+\left (-8 x-8 x^2-2 x^3\right ) \log (5)+\left (4+4 x+x^2\right ) \log ^2(5)} \, dx=x \left (14 x+x^3+2 \left (7+x^2\right ) \log \left (-\frac {5 x}{-e^5+(2+x)^2 (x-\log (5))^2}\right )+x \log ^2\left (-\frac {5 x}{-e^5+(2+x)^2 (x-\log (5))^2}\right )\right ) \]

[In]

Integrate[(-56*x^2 + 62*x^4 + 28*x^5 + 10*x^6 + 4*x^7 + E^5*(-14 - 28*x - 2*x^2 - 4*x^3) + (-112*x^2 - 168*x^3
- 72*x^4 - 24*x^5 - 8*x^6)*Log[5] + (56 + 112*x + 106*x^2 + 44*x^3 + 14*x^4 + 4*x^5)*Log[5]^2 + (56*x^2 + 48*
x^3 + 22*x^4 + 18*x^5 + 6*x^6 + E^5*(-14 - 2*x - 6*x^2) + (-112*x - 112*x^2 - 60*x^3 - 40*x^4 - 12*x^5)*Log[5]
+ (56 + 64*x + 38*x^2 + 22*x^3 + 6*x^4)*Log[5]^2)*Log[(-5*x)/(-E^5 + 4*x^2 + 4*x^3 + x^4 + (-8*x - 8*x^2 - 2*
x^3)*Log[5] + (4 + 4*x + x^2)*Log[5]^2)] + (-2*E^5*x + 8*x^3 + 8*x^4 + 2*x^5 + (-16*x^2 - 16*x^3 - 4*x^4)*Log[
5] + (8*x + 8*x^2 + 2*x^3)*Log[5]^2)*Log[(-5*x)/(-E^5 + 4*x^2 + 4*x^3 + x^4 + (-8*x - 8*x^2 - 2*x^3)*Log[5] +
(4 + 4*x + x^2)*Log[5]^2)]^2)/(-E^5 + 4*x^2 + 4*x^3 + x^4 + (-8*x - 8*x^2 - 2*x^3)*Log[5] + (4 + 4*x + x^2)*Lo
g[5]^2),x]

[Out]

x*(14*x + x^3 + 2*(7 + x^2)*Log[(-5*x)/(-E^5 + (2 + x)^2*(x - Log[5])^2)] + x*Log[(-5*x)/(-E^5 + (2 + x)^2*(x
- Log[5])^2)]^2)

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(132\) vs. \(2(32)=64\).

Time = 1.52 (sec) , antiderivative size = 133, normalized size of antiderivative = 4.03

method	result	size
risch	\(x^{2} {\ln \left (-\frac {5 x}{\left (x^{2}+4 x +4\right ) \ln \left (5\right )^{2}+\left (-2 x^{3}-8 x^{2}-8 x \right ) \ln \left (5\right )-{\mathrm e}^{5}+x^{4}+4 x^{3}+4 x^{2}}\right )}^{2}+\left (2 x^{3}+14 x \right ) \ln \left (-\frac {5 x}{\left (x^{2}+4 x +4\right ) \ln \left (5\right )^{2}+\left (-2 x^{3}-8 x^{2}-8 x \right ) \ln \left (5\right )-{\mathrm e}^{5}+x^{4}+4 x^{3}+4 x^{2}}\right )+\left (x^{2}+7\right )^{2}\)	\(133\)
norman	\(x^{4}+x^{2} {\ln \left (-\frac {5 x}{\left (x^{2}+4 x +4\right ) \ln \left (5\right )^{2}+\left (-2 x^{3}-8 x^{2}-8 x \right ) \ln \left (5\right )-{\mathrm e}^{5}+x^{4}+4 x^{3}+4 x^{2}}\right )}^{2}+14 x^{2}+14 x \ln \left (-\frac {5 x}{\left (x^{2}+4 x +4\right ) \ln \left (5\right )^{2}+\left (-2 x^{3}-8 x^{2}-8 x \right ) \ln \left (5\right )-{\mathrm e}^{5}+x^{4}+4 x^{3}+4 x^{2}}\right )+2 x^{3} \ln \left (-\frac {5 x}{\left (x^{2}+4 x +4\right ) \ln \left (5\right )^{2}+\left (-2 x^{3}-8 x^{2}-8 x \right ) \ln \left (5\right )-{\mathrm e}^{5}+x^{4}+4 x^{3}+4 x^{2}}\right )\)	\(186\)
parallelrisch	\(\text {Expression too large to display}\)	\(995\)

[In]

int((((2*x^3+8*x^2+8*x)*ln(5)^2+(-4*x^4-16*x^3-16*x^2)*ln(5)-2*x*exp(5)+2*x^5+8*x^4+8*x^3)*ln(-5*x/((x^2+4*x+4
)*ln(5)^2+(-2*x^3-8*x^2-8*x)*ln(5)-exp(5)+x^4+4*x^3+4*x^2))^2+((6*x^4+22*x^3+38*x^2+64*x+56)*ln(5)^2+(-12*x^5-
40*x^4-60*x^3-112*x^2-112*x)*ln(5)+(-6*x^2-2*x-14)*exp(5)+6*x^6+18*x^5+22*x^4+48*x^3+56*x^2)*ln(-5*x/((x^2+4*x
+4)*ln(5)^2+(-2*x^3-8*x^2-8*x)*ln(5)-exp(5)+x^4+4*x^3+4*x^2))+(4*x^5+14*x^4+44*x^3+106*x^2+112*x+56)*ln(5)^2+(
-8*x^6-24*x^5-72*x^4-168*x^3-112*x^2)*ln(5)+(-4*x^3-2*x^2-28*x-14)*exp(5)+4*x^7+10*x^6+28*x^5+62*x^4-56*x^2)/(
(x^2+4*x+4)*ln(5)^2+(-2*x^3-8*x^2-8*x)*ln(5)-exp(5)+x^4+4*x^3+4*x^2),x,method=_RETURNVERBOSE)

[Out]

x^2*ln(-5*x/((x^2+4*x+4)*ln(5)^2+(-2*x^3-8*x^2-8*x)*ln(5)-exp(5)+x^4+4*x^3+4*x^2))^2+(2*x^3+14*x)*ln(-5*x/((x^
2+4*x+4)*ln(5)^2+(-2*x^3-8*x^2-8*x)*ln(5)-exp(5)+x^4+4*x^3+4*x^2))+(x^2+7)^2

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 130 vs. \(2 (33) = 66\).

Time = 0.27 (sec) , antiderivative size = 130, normalized size of antiderivative = 3.94 \[ \int \frac {-56 x^2+62 x^4+28 x^5+10 x^6+4 x^7+e^5 \left (-14-28 x-2 x^2-4 x^3\right )+\left (-112 x^2-168 x^3-72 x^4-24 x^5-8 x^6\right ) \log (5)+\left (56+112 x+106 x^2+44 x^3+14 x^4+4 x^5\right ) \log ^2(5)+\left (56 x^2+48 x^3+22 x^4+18 x^5+6 x^6+e^5 \left (-14-2 x-6 x^2\right )+\left (-112 x-112 x^2-60 x^3-40 x^4-12 x^5\right ) \log (5)+\left (56+64 x+38 x^2+22 x^3+6 x^4\right ) \log ^2(5)\right ) \log \left (-\frac {5 x}{-e^5+4 x^2+4 x^3+x^4+\left (-8 x-8 x^2-2 x^3\right ) \log (5)+\left (4+4 x+x^2\right ) \log ^2(5)}\right )+\left (-2 e^5 x+8 x^3+8 x^4+2 x^5+\left (-16 x^2-16 x^3-4 x^4\right ) \log (5)+\left (8 x+8 x^2+2 x^3\right ) \log ^2(5)\right ) \log ^2\left (-\frac {5 x}{-e^5+4 x^2+4 x^3+x^4+\left (-8 x-8 x^2-2 x^3\right ) \log (5)+\left (4+4 x+x^2\right ) \log ^2(5)}\right )}{-e^5+4 x^2+4 x^3+x^4+\left (-8 x-8 x^2-2 x^3\right ) \log (5)+\left (4+4 x+x^2\right ) \log ^2(5)} \, dx=x^{4} + x^{2} \log \left (-\frac {5 \, x}{x^{4} + 4 \, x^{3} + {\left (x^{2} + 4 \, x + 4\right )} \log \left (5\right )^{2} + 4 \, x^{2} - 2 \, {\left (x^{3} + 4 \, x^{2} + 4 \, x\right )} \log \left (5\right ) - e^{5}}\right )^{2} + 14 \, x^{2} + 2 \, {\left (x^{3} + 7 \, x\right )} \log \left (-\frac {5 \, x}{x^{4} + 4 \, x^{3} + {\left (x^{2} + 4 \, x + 4\right )} \log \left (5\right )^{2} + 4 \, x^{2} - 2 \, {\left (x^{3} + 4 \, x^{2} + 4 \, x\right )} \log \left (5\right ) - e^{5}}\right ) \]

[In]

integrate((((2*x^3+8*x^2+8*x)*log(5)^2+(-4*x^4-16*x^3-16*x^2)*log(5)-2*x*exp(5)+2*x^5+8*x^4+8*x^3)*log(-5*x/((
x^2+4*x+4)*log(5)^2+(-2*x^3-8*x^2-8*x)*log(5)-exp(5)+x^4+4*x^3+4*x^2))^2+((6*x^4+22*x^3+38*x^2+64*x+56)*log(5)
^2+(-12*x^5-40*x^4-60*x^3-112*x^2-112*x)*log(5)+(-6*x^2-2*x-14)*exp(5)+6*x^6+18*x^5+22*x^4+48*x^3+56*x^2)*log(
-5*x/((x^2+4*x+4)*log(5)^2+(-2*x^3-8*x^2-8*x)*log(5)-exp(5)+x^4+4*x^3+4*x^2))+(4*x^5+14*x^4+44*x^3+106*x^2+112
*x+56)*log(5)^2+(-8*x^6-24*x^5-72*x^4-168*x^3-112*x^2)*log(5)+(-4*x^3-2*x^2-28*x-14)*exp(5)+4*x^7+10*x^6+28*x^
5+62*x^4-56*x^2)/((x^2+4*x+4)*log(5)^2+(-2*x^3-8*x^2-8*x)*log(5)-exp(5)+x^4+4*x^3+4*x^2),x, algorithm="fricas"
)

[Out]

x^4 + x^2*log(-5*x/(x^4 + 4*x^3 + (x^2 + 4*x + 4)*log(5)^2 + 4*x^2 - 2*(x^3 + 4*x^2 + 4*x)*log(5) - e^5))^2 +
14*x^2 + 2*(x^3 + 7*x)*log(-5*x/(x^4 + 4*x^3 + (x^2 + 4*x + 4)*log(5)^2 + 4*x^2 - 2*(x^3 + 4*x^2 + 4*x)*log(5)
- e^5))

Sympy [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 133 vs. \(2 (26) = 52\).

Time = 0.39 (sec) , antiderivative size = 133, normalized size of antiderivative = 4.03 \[ \int \frac {-56 x^2+62 x^4+28 x^5+10 x^6+4 x^7+e^5 \left (-14-28 x-2 x^2-4 x^3\right )+\left (-112 x^2-168 x^3-72 x^4-24 x^5-8 x^6\right ) \log (5)+\left (56+112 x+106 x^2+44 x^3+14 x^4+4 x^5\right ) \log ^2(5)+\left (56 x^2+48 x^3+22 x^4+18 x^5+6 x^6+e^5 \left (-14-2 x-6 x^2\right )+\left (-112 x-112 x^2-60 x^3-40 x^4-12 x^5\right ) \log (5)+\left (56+64 x+38 x^2+22 x^3+6 x^4\right ) \log ^2(5)\right ) \log \left (-\frac {5 x}{-e^5+4 x^2+4 x^3+x^4+\left (-8 x-8 x^2-2 x^3\right ) \log (5)+\left (4+4 x+x^2\right ) \log ^2(5)}\right )+\left (-2 e^5 x+8 x^3+8 x^4+2 x^5+\left (-16 x^2-16 x^3-4 x^4\right ) \log (5)+\left (8 x+8 x^2+2 x^3\right ) \log ^2(5)\right ) \log ^2\left (-\frac {5 x}{-e^5+4 x^2+4 x^3+x^4+\left (-8 x-8 x^2-2 x^3\right ) \log (5)+\left (4+4 x+x^2\right ) \log ^2(5)}\right )}{-e^5+4 x^2+4 x^3+x^4+\left (-8 x-8 x^2-2 x^3\right ) \log (5)+\left (4+4 x+x^2\right ) \log ^2(5)} \, dx=x^{4} + x^{2} \log {\left (- \frac {5 x}{x^{4} + 4 x^{3} + 4 x^{2} + \left (x^{2} + 4 x + 4\right ) \log {\left (5 \right )}^{2} + \left (- 2 x^{3} - 8 x^{2} - 8 x\right ) \log {\left (5 \right )} - e^{5}} \right )}^{2} + 14 x^{2} + \left (2 x^{3} + 14 x\right ) \log {\left (- \frac {5 x}{x^{4} + 4 x^{3} + 4 x^{2} + \left (x^{2} + 4 x + 4\right ) \log {\left (5 \right )}^{2} + \left (- 2 x^{3} - 8 x^{2} - 8 x\right ) \log {\left (5 \right )} - e^{5}} \right )} \]

[In]

integrate((((2*x**3+8*x**2+8*x)*ln(5)**2+(-4*x**4-16*x**3-16*x**2)*ln(5)-2*x*exp(5)+2*x**5+8*x**4+8*x**3)*ln(-
5*x/((x**2+4*x+4)*ln(5)**2+(-2*x**3-8*x**2-8*x)*ln(5)-exp(5)+x**4+4*x**3+4*x**2))**2+((6*x**4+22*x**3+38*x**2+
64*x+56)*ln(5)**2+(-12*x**5-40*x**4-60*x**3-112*x**2-112*x)*ln(5)+(-6*x**2-2*x-14)*exp(5)+6*x**6+18*x**5+22*x*
*4+48*x**3+56*x**2)*ln(-5*x/((x**2+4*x+4)*ln(5)**2+(-2*x**3-8*x**2-8*x)*ln(5)-exp(5)+x**4+4*x**3+4*x**2))+(4*x
**5+14*x**4+44*x**3+106*x**2+112*x+56)*ln(5)**2+(-8*x**6-24*x**5-72*x**4-168*x**3-112*x**2)*ln(5)+(-4*x**3-2*x
**2-28*x-14)*exp(5)+4*x**7+10*x**6+28*x**5+62*x**4-56*x**2)/((x**2+4*x+4)*ln(5)**2+(-2*x**3-8*x**2-8*x)*ln(5)-
exp(5)+x**4+4*x**3+4*x**2),x)

[Out]

x**4 + x**2*log(-5*x/(x**4 + 4*x**3 + 4*x**2 + (x**2 + 4*x + 4)*log(5)**2 + (-2*x**3 - 8*x**2 - 8*x)*log(5) -
exp(5)))**2 + 14*x**2 + (2*x**3 + 14*x)*log(-5*x/(x**4 + 4*x**3 + 4*x**2 + (x**2 + 4*x + 4)*log(5)**2 + (-2*x*
*3 - 8*x**2 - 8*x)*log(5) - exp(5)))

Maxima [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 180 vs. \(2 (33) = 66\).

Time = 0.29 (sec) , antiderivative size = 180, normalized size of antiderivative = 5.45 \[ \int \frac {-56 x^2+62 x^4+28 x^5+10 x^6+4 x^7+e^5 \left (-14-28 x-2 x^2-4 x^3\right )+\left (-112 x^2-168 x^3-72 x^4-24 x^5-8 x^6\right ) \log (5)+\left (56+112 x+106 x^2+44 x^3+14 x^4+4 x^5\right ) \log ^2(5)+\left (56 x^2+48 x^3+22 x^4+18 x^5+6 x^6+e^5 \left (-14-2 x-6 x^2\right )+\left (-112 x-112 x^2-60 x^3-40 x^4-12 x^5\right ) \log (5)+\left (56+64 x+38 x^2+22 x^3+6 x^4\right ) \log ^2(5)\right ) \log \left (-\frac {5 x}{-e^5+4 x^2+4 x^3+x^4+\left (-8 x-8 x^2-2 x^3\right ) \log (5)+\left (4+4 x+x^2\right ) \log ^2(5)}\right )+\left (-2 e^5 x+8 x^3+8 x^4+2 x^5+\left (-16 x^2-16 x^3-4 x^4\right ) \log (5)+\left (8 x+8 x^2+2 x^3\right ) \log ^2(5)\right ) \log ^2\left (-\frac {5 x}{-e^5+4 x^2+4 x^3+x^4+\left (-8 x-8 x^2-2 x^3\right ) \log (5)+\left (4+4 x+x^2\right ) \log ^2(5)}\right )}{-e^5+4 x^2+4 x^3+x^4+\left (-8 x-8 x^2-2 x^3\right ) \log (5)+\left (4+4 x+x^2\right ) \log ^2(5)} \, dx=x^{4} + 2 \, x^{3} \log \left (5\right ) + x^{2} \log \left (-x^{4} + 2 \, x^{3} {\left (\log \left (5\right ) - 2\right )} - {\left (\log \left (5\right )^{2} - 8 \, \log \left (5\right ) + 4\right )} x^{2} - 4 \, {\left (\log \left (5\right )^{2} - 2 \, \log \left (5\right )\right )} x - 4 \, \log \left (5\right )^{2} + e^{5}\right )^{2} + x^{2} \log \left (x\right )^{2} + {\left (\log \left (5\right )^{2} + 14\right )} x^{2} + 14 \, x \log \left (5\right ) - 2 \, {\left (x^{3} + x^{2} \log \left (5\right ) + x^{2} \log \left (x\right ) + 7 \, x\right )} \log \left (-x^{4} + 2 \, x^{3} {\left (\log \left (5\right ) - 2\right )} - {\left (\log \left (5\right )^{2} - 8 \, \log \left (5\right ) + 4\right )} x^{2} - 4 \, {\left (\log \left (5\right )^{2} - 2 \, \log \left (5\right )\right )} x - 4 \, \log \left (5\right )^{2} + e^{5}\right ) + 2 \, {\left (x^{3} + x^{2} \log \left (5\right ) + 7 \, x\right )} \log \left (x\right ) \]

[In]

integrate((((2*x^3+8*x^2+8*x)*log(5)^2+(-4*x^4-16*x^3-16*x^2)*log(5)-2*x*exp(5)+2*x^5+8*x^4+8*x^3)*log(-5*x/((
x^2+4*x+4)*log(5)^2+(-2*x^3-8*x^2-8*x)*log(5)-exp(5)+x^4+4*x^3+4*x^2))^2+((6*x^4+22*x^3+38*x^2+64*x+56)*log(5)
^2+(-12*x^5-40*x^4-60*x^3-112*x^2-112*x)*log(5)+(-6*x^2-2*x-14)*exp(5)+6*x^6+18*x^5+22*x^4+48*x^3+56*x^2)*log(
-5*x/((x^2+4*x+4)*log(5)^2+(-2*x^3-8*x^2-8*x)*log(5)-exp(5)+x^4+4*x^3+4*x^2))+(4*x^5+14*x^4+44*x^3+106*x^2+112
*x+56)*log(5)^2+(-8*x^6-24*x^5-72*x^4-168*x^3-112*x^2)*log(5)+(-4*x^3-2*x^2-28*x-14)*exp(5)+4*x^7+10*x^6+28*x^
5+62*x^4-56*x^2)/((x^2+4*x+4)*log(5)^2+(-2*x^3-8*x^2-8*x)*log(5)-exp(5)+x^4+4*x^3+4*x^2),x, algorithm="maxima"
)

[Out]

x^4 + 2*x^3*log(5) + x^2*log(-x^4 + 2*x^3*(log(5) - 2) - (log(5)^2 - 8*log(5) + 4)*x^2 - 4*(log(5)^2 - 2*log(5
))*x - 4*log(5)^2 + e^5)^2 + x^2*log(x)^2 + (log(5)^2 + 14)*x^2 + 14*x*log(5) - 2*(x^3 + x^2*log(5) + x^2*log(
x) + 7*x)*log(-x^4 + 2*x^3*(log(5) - 2) - (log(5)^2 - 8*log(5) + 4)*x^2 - 4*(log(5)^2 - 2*log(5))*x - 4*log(5)
^2 + e^5) + 2*(x^3 + x^2*log(5) + 7*x)*log(x)

Giac [F(-1)]

Timed out. \[ \int \frac {-56 x^2+62 x^4+28 x^5+10 x^6+4 x^7+e^5 \left (-14-28 x-2 x^2-4 x^3\right )+\left (-112 x^2-168 x^3-72 x^4-24 x^5-8 x^6\right ) \log (5)+\left (56+112 x+106 x^2+44 x^3+14 x^4+4 x^5\right ) \log ^2(5)+\left (56 x^2+48 x^3+22 x^4+18 x^5+6 x^6+e^5 \left (-14-2 x-6 x^2\right )+\left (-112 x-112 x^2-60 x^3-40 x^4-12 x^5\right ) \log (5)+\left (56+64 x+38 x^2+22 x^3+6 x^4\right ) \log ^2(5)\right ) \log \left (-\frac {5 x}{-e^5+4 x^2+4 x^3+x^4+\left (-8 x-8 x^2-2 x^3\right ) \log (5)+\left (4+4 x+x^2\right ) \log ^2(5)}\right )+\left (-2 e^5 x+8 x^3+8 x^4+2 x^5+\left (-16 x^2-16 x^3-4 x^4\right ) \log (5)+\left (8 x+8 x^2+2 x^3\right ) \log ^2(5)\right ) \log ^2\left (-\frac {5 x}{-e^5+4 x^2+4 x^3+x^4+\left (-8 x-8 x^2-2 x^3\right ) \log (5)+\left (4+4 x+x^2\right ) \log ^2(5)}\right )}{-e^5+4 x^2+4 x^3+x^4+\left (-8 x-8 x^2-2 x^3\right ) \log (5)+\left (4+4 x+x^2\right ) \log ^2(5)} \, dx=\text {Timed out} \]

[In]

integrate((((2*x^3+8*x^2+8*x)*log(5)^2+(-4*x^4-16*x^3-16*x^2)*log(5)-2*x*exp(5)+2*x^5+8*x^4+8*x^3)*log(-5*x/((
x^2+4*x+4)*log(5)^2+(-2*x^3-8*x^2-8*x)*log(5)-exp(5)+x^4+4*x^3+4*x^2))^2+((6*x^4+22*x^3+38*x^2+64*x+56)*log(5)
^2+(-12*x^5-40*x^4-60*x^3-112*x^2-112*x)*log(5)+(-6*x^2-2*x-14)*exp(5)+6*x^6+18*x^5+22*x^4+48*x^3+56*x^2)*log(
-5*x/((x^2+4*x+4)*log(5)^2+(-2*x^3-8*x^2-8*x)*log(5)-exp(5)+x^4+4*x^3+4*x^2))+(4*x^5+14*x^4+44*x^3+106*x^2+112
*x+56)*log(5)^2+(-8*x^6-24*x^5-72*x^4-168*x^3-112*x^2)*log(5)+(-4*x^3-2*x^2-28*x-14)*exp(5)+4*x^7+10*x^6+28*x^
5+62*x^4-56*x^2)/((x^2+4*x+4)*log(5)^2+(-2*x^3-8*x^2-8*x)*log(5)-exp(5)+x^4+4*x^3+4*x^2),x, algorithm="giac")

[Out]

Timed out

Mupad [B] (verification not implemented)

Time = 12.01 (sec) , antiderivative size = 135, normalized size of antiderivative = 4.09 \[ \int \frac {-56 x^2+62 x^4+28 x^5+10 x^6+4 x^7+e^5 \left (-14-28 x-2 x^2-4 x^3\right )+\left (-112 x^2-168 x^3-72 x^4-24 x^5-8 x^6\right ) \log (5)+\left (56+112 x+106 x^2+44 x^3+14 x^4+4 x^5\right ) \log ^2(5)+\left (56 x^2+48 x^3+22 x^4+18 x^5+6 x^6+e^5 \left (-14-2 x-6 x^2\right )+\left (-112 x-112 x^2-60 x^3-40 x^4-12 x^5\right ) \log (5)+\left (56+64 x+38 x^2+22 x^3+6 x^4\right ) \log ^2(5)\right ) \log \left (-\frac {5 x}{-e^5+4 x^2+4 x^3+x^4+\left (-8 x-8 x^2-2 x^3\right ) \log (5)+\left (4+4 x+x^2\right ) \log ^2(5)}\right )+\left (-2 e^5 x+8 x^3+8 x^4+2 x^5+\left (-16 x^2-16 x^3-4 x^4\right ) \log (5)+\left (8 x+8 x^2+2 x^3\right ) \log ^2(5)\right ) \log ^2\left (-\frac {5 x}{-e^5+4 x^2+4 x^3+x^4+\left (-8 x-8 x^2-2 x^3\right ) \log (5)+\left (4+4 x+x^2\right ) \log ^2(5)}\right )}{-e^5+4 x^2+4 x^3+x^4+\left (-8 x-8 x^2-2 x^3\right ) \log (5)+\left (4+4 x+x^2\right ) \log ^2(5)} \, dx=x^2\,{\ln \left (-\frac {5\,x}{{\ln \left (5\right )}^2\,\left (x^2+4\,x+4\right )-{\mathrm {e}}^5-\ln \left (5\right )\,\left (2\,x^3+8\,x^2+8\,x\right )+4\,x^2+4\,x^3+x^4}\right )}^2+\ln \left (-\frac {5\,x}{{\ln \left (5\right )}^2\,\left (x^2+4\,x+4\right )-{\mathrm {e}}^5-\ln \left (5\right )\,\left (2\,x^3+8\,x^2+8\,x\right )+4\,x^2+4\,x^3+x^4}\right )\,\left (2\,x^3+14\,x\right )+14\,x^2+x^4 \]

[In]

int((log(5)^2*(112*x + 106*x^2 + 44*x^3 + 14*x^4 + 4*x^5 + 56) - log(5)*(112*x^2 + 168*x^3 + 72*x^4 + 24*x^5 +
8*x^6) + log(-(5*x)/(log(5)^2*(4*x + x^2 + 4) - exp(5) - log(5)*(8*x + 8*x^2 + 2*x^3) + 4*x^2 + 4*x^3 + x^4))
^2*(log(5)^2*(8*x + 8*x^2 + 2*x^3) - 2*x*exp(5) - log(5)*(16*x^2 + 16*x^3 + 4*x^4) + 8*x^3 + 8*x^4 + 2*x^5) -
exp(5)*(28*x + 2*x^2 + 4*x^3 + 14) + log(-(5*x)/(log(5)^2*(4*x + x^2 + 4) - exp(5) - log(5)*(8*x + 8*x^2 + 2*x
^3) + 4*x^2 + 4*x^3 + x^4))*(log(5)^2*(64*x + 38*x^2 + 22*x^3 + 6*x^4 + 56) - exp(5)*(2*x + 6*x^2 + 14) + 56*x
^2 + 48*x^3 + 22*x^4 + 18*x^5 + 6*x^6 - log(5)*(112*x + 112*x^2 + 60*x^3 + 40*x^4 + 12*x^5)) - 56*x^2 + 62*x^4
+ 28*x^5 + 10*x^6 + 4*x^7)/(log(5)^2*(4*x + x^2 + 4) - exp(5) - log(5)*(8*x + 8*x^2 + 2*x^3) + 4*x^2 + 4*x^3
+ x^4),x)

[Out]

x^2*log(-(5*x)/(log(5)^2*(4*x + x^2 + 4) - exp(5) - log(5)*(8*x + 8*x^2 + 2*x^3) + 4*x^2 + 4*x^3 + x^4))^2 + l
og(-(5*x)/(log(5)^2*(4*x + x^2 + 4) - exp(5) - log(5)*(8*x + 8*x^2 + 2*x^3) + 4*x^2 + 4*x^3 + x^4))*(14*x + 2*
x^3) + 14*x^2 + x^4