\(\int \frac {15-21 x-36 x^2+(12+12 x) \log (x)+(2+4 x+(2+4 x) \log (x)) \log (x+x^2)+(-8+16 x+24 x^2+(-7-7 x) \log (x)) \log ^2(x+x^2)+(1-3 x-4 x^2+(1+x) \log (x)) \log ^4(x+x^2)+(3-15 x-18 x^2+(12+12 x) \log (x)+(-1+11 x+12 x^2+(-7-7 x) \log (x)) \log ^2(x+x^2)+(-2 x-2 x^2+(1+x) \log (x)) \log ^4(x+x^2)) \log (\frac {-1+6 x-4 \log (x)+(-2 x+\log (x)) \log ^2(x+x^2)}{-3+\log ^2(x+x^2)})}{3-15 x-18 x^2+(12+12 x) \log (x)+(-1+11 x+12 x^2+(-7-7 x) \log (x)) \log ^2(x+x^2)+(-2 x-2 x^2+(1+x) \log (x)) \log ^4(x+x^2)} \, dx\) [2875]

Optimal result

Integrand size = 270, antiderivative size = 29 \[ \int \frac {15-21 x-36 x^2+(12+12 x) \log (x)+(2+4 x+(2+4 x) \log (x)) \log \left (x+x^2\right )+\left (-8+16 x+24 x^2+(-7-7 x) \log (x)\right ) \log ^2\left (x+x^2\right )+\left (1-3 x-4 x^2+(1+x) \log (x)\right ) \log ^4\left (x+x^2\right )+\left (3-15 x-18 x^2+(12+12 x) \log (x)+\left (-1+11 x+12 x^2+(-7-7 x) \log (x)\right ) \log ^2\left (x+x^2\right )+\left (-2 x-2 x^2+(1+x) \log (x)\right ) \log ^4\left (x+x^2\right )\right ) \log \left (\frac {-1+6 x-4 \log (x)+(-2 x+\log (x)) \log ^2\left (x+x^2\right )}{-3+\log ^2\left (x+x^2\right )}\right )}{3-15 x-18 x^2+(12+12 x) \log (x)+\left (-1+11 x+12 x^2+(-7-7 x) \log (x)\right ) \log ^2\left (x+x^2\right )+\left (-2 x-2 x^2+(1+x) \log (x)\right ) \log ^4\left (x+x^2\right )} \, dx=x+x \log \left (-2 x+\log (x)-\frac {1+\log (x)}{-3+\log ^2\left (x+x^2\right )}\right ) \]

[Out]

ln(ln(x)-2*x-(ln(x)+1)/(ln(x^2+x)^2-3))*x+x

Rubi [F]

\[ \int \frac {15-21 x-36 x^2+(12+12 x) \log (x)+(2+4 x+(2+4 x) \log (x)) \log \left (x+x^2\right )+\left (-8+16 x+24 x^2+(-7-7 x) \log (x)\right ) \log ^2\left (x+x^2\right )+\left (1-3 x-4 x^2+(1+x) \log (x)\right ) \log ^4\left (x+x^2\right )+\left (3-15 x-18 x^2+(12+12 x) \log (x)+\left (-1+11 x+12 x^2+(-7-7 x) \log (x)\right ) \log ^2\left (x+x^2\right )+\left (-2 x-2 x^2+(1+x) \log (x)\right ) \log ^4\left (x+x^2\right )\right ) \log \left (\frac {-1+6 x-4 \log (x)+(-2 x+\log (x)) \log ^2\left (x+x^2\right )}{-3+\log ^2\left (x+x^2\right )}\right )}{3-15 x-18 x^2+(12+12 x) \log (x)+\left (-1+11 x+12 x^2+(-7-7 x) \log (x)\right ) \log ^2\left (x+x^2\right )+\left (-2 x-2 x^2+(1+x) \log (x)\right ) \log ^4\left (x+x^2\right )} \, dx=\int \frac {15-21 x-36 x^2+(12+12 x) \log (x)+(2+4 x+(2+4 x) \log (x)) \log \left (x+x^2\right )+\left (-8+16 x+24 x^2+(-7-7 x) \log (x)\right ) \log ^2\left (x+x^2\right )+\left (1-3 x-4 x^2+(1+x) \log (x)\right ) \log ^4\left (x+x^2\right )+\left (3-15 x-18 x^2+(12+12 x) \log (x)+\left (-1+11 x+12 x^2+(-7-7 x) \log (x)\right ) \log ^2\left (x+x^2\right )+\left (-2 x-2 x^2+(1+x) \log (x)\right ) \log ^4\left (x+x^2\right )\right ) \log \left (\frac {-1+6 x-4 \log (x)+(-2 x+\log (x)) \log ^2\left (x+x^2\right )}{-3+\log ^2\left (x+x^2\right )}\right )}{3-15 x-18 x^2+(12+12 x) \log (x)+\left (-1+11 x+12 x^2+(-7-7 x) \log (x)\right ) \log ^2\left (x+x^2\right )+\left (-2 x-2 x^2+(1+x) \log (x)\right ) \log ^4\left (x+x^2\right )} \, dx \]

[In]

Int[(15 - 21*x - 36*x^2 + (12 + 12*x)*Log[x] + (2 + 4*x + (2 + 4*x)*Log[x])*Log[x + x^2] + (-8 + 16*x + 24*x^2
+ (-7 - 7*x)*Log[x])*Log[x + x^2]^2 + (1 - 3*x - 4*x^2 + (1 + x)*Log[x])*Log[x + x^2]^4 + (3 - 15*x - 18*x^2
+ (12 + 12*x)*Log[x] + (-1 + 11*x + 12*x^2 + (-7 - 7*x)*Log[x])*Log[x + x^2]^2 + (-2*x - 2*x^2 + (1 + x)*Log[x
])*Log[x + x^2]^4)*Log[(-1 + 6*x - 4*Log[x] + (-2*x + Log[x])*Log[x + x^2]^2)/(-3 + Log[x + x^2]^2)])/(3 - 15*
x - 18*x^2 + (12 + 12*x)*Log[x] + (-1 + 11*x + 12*x^2 + (-7 - 7*x)*Log[x])*Log[x + x^2]^2 + (-2*x - 2*x^2 + (1
+ x)*Log[x])*Log[x + x^2]^4),x]

[Out]

x + x*Log[(1 - 6*x + 4*Log[x] + (2*x - Log[x])*Log[x*(1 + x)]^2)/(3 - Log[x*(1 + x)]^2)] - Defer[Int][1/((2*x
- Log[x])*(1 - 6*x + 4*Log[x] + 2*x*Log[x*(1 + x)]^2 - Log[x]*Log[x*(1 + x)]^2)), x] + 2*Defer[Int][(x*Log[x])
/((2*x - Log[x])*(1 - 6*x + 4*Log[x] + 2*x*Log[x*(1 + x)]^2 - Log[x]*Log[x*(1 + x)]^2)), x] + Defer[Int][1/((2
*x - Log[x])*(1 + Log[x])*(1 - 6*x + 4*Log[x] + 2*x*Log[x*(1 + x)]^2 - Log[x]*Log[x*(1 + x)]^2)), x] - 16*Defe
r[Int][x/((2*x - Log[x])*(1 + Log[x])*(1 - 6*x + 4*Log[x] + 2*x*Log[x*(1 + x)]^2 - Log[x]*Log[x*(1 + x)]^2)),
x] + 84*Defer[Int][x^2/((2*x - Log[x])*(1 + Log[x])*(1 - 6*x + 4*Log[x] + 2*x*Log[x*(1 + x)]^2 - Log[x]*Log[x*
(1 + x)]^2)), x] - 144*Defer[Int][x^3/((2*x - Log[x])*(1 + Log[x])*(1 - 6*x + 4*Log[x] + 2*x*Log[x*(1 + x)]^2
- Log[x]*Log[x*(1 + x)]^2)), x] + 9*Defer[Int][Log[x]/((2*x - Log[x])*(1 + Log[x])*(1 - 6*x + 4*Log[x] + 2*x*L
og[x*(1 + x)]^2 - Log[x]*Log[x*(1 + x)]^2)), x] - 92*Defer[Int][(x*Log[x])/((2*x - Log[x])*(1 + Log[x])*(1 - 6
*x + 4*Log[x] + 2*x*Log[x*(1 + x)]^2 - Log[x]*Log[x*(1 + x)]^2)), x] + 228*Defer[Int][(x^2*Log[x])/((2*x - Log
[x])*(1 + Log[x])*(1 - 6*x + 4*Log[x] + 2*x*Log[x*(1 + x)]^2 - Log[x]*Log[x*(1 + x)]^2)), x] + 24*Defer[Int][L
og[x]^2/((2*x - Log[x])*(1 + Log[x])*(1 - 6*x + 4*Log[x] + 2*x*Log[x*(1 + x)]^2 - Log[x]*Log[x*(1 + x)]^2)), x
] - 112*Defer[Int][(x*Log[x]^2)/((2*x - Log[x])*(1 + Log[x])*(1 - 6*x + 4*Log[x] + 2*x*Log[x*(1 + x)]^2 - Log[
x]*Log[x*(1 + x)]^2)), x] + 16*Defer[Int][Log[x]^3/((2*x - Log[x])*(1 + Log[x])*(1 - 6*x + 4*Log[x] + 2*x*Log[
x*(1 + x)]^2 - Log[x]*Log[x*(1 + x)]^2)), x] - 4*Defer[Int][Log[x*(1 + x)]/(1 - 6*x + 4*Log[x] + 2*x*Log[x*(1
+ x)]^2 - Log[x]*Log[x*(1 + x)]^2), x] + 8*Defer[Int][(x*Log[x*(1 + x)])/(1 - 6*x + 4*Log[x] + 2*x*Log[x*(1 +
x)]^2 - Log[x]*Log[x*(1 + x)]^2), x] + 4*Defer[Int][Log[x*(1 + x)]/((1 + x)*(1 - 6*x + 4*Log[x] + 2*x*Log[x*(1
+ x)]^2 - Log[x]*Log[x*(1 + x)]^2)), x] - 8*Defer[Int][Log[x*(1 + x)]/((2*x - Log[x])*(1 - 6*x + 4*Log[x] + 2
*x*Log[x*(1 + x)]^2 - Log[x]*Log[x*(1 + x)]^2)), x] + 8*Defer[Int][(x*Log[x*(1 + x)])/((2*x - Log[x])*(1 - 6*x
+ 4*Log[x] + 2*x*Log[x*(1 + x)]^2 - Log[x]*Log[x*(1 + x)]^2)), x] - 16*Defer[Int][(x^2*Log[x*(1 + x)])/((2*x
- Log[x])*(1 - 6*x + 4*Log[x] + 2*x*Log[x*(1 + x)]^2 - Log[x]*Log[x*(1 + x)]^2)), x] + 8*Defer[Int][Log[x*(1 +
x)]/((1 + x)*(2*x - Log[x])*(1 - 6*x + 4*Log[x] + 2*x*Log[x*(1 + x)]^2 - Log[x]*Log[x*(1 + x)]^2)), x] + 2*De
fer[Int][(Log[x]*Log[x*(1 + x)])/((1 + x)*(1 - 6*x + 4*Log[x] + 2*x*Log[x*(1 + x)]^2 - Log[x]*Log[x*(1 + x)]^2
)), x] - 8*Defer[Int][(Log[x]*Log[x*(1 + x)])/((2*x - Log[x])*(1 - 6*x + 4*Log[x] + 2*x*Log[x*(1 + x)]^2 - Log
[x]*Log[x*(1 + x)]^2)), x] + 16*Defer[Int][(x*Log[x]*Log[x*(1 + x)])/((2*x - Log[x])*(1 - 6*x + 4*Log[x] + 2*x
*Log[x*(1 + x)]^2 - Log[x]*Log[x*(1 + x)]^2)), x] + 8*Defer[Int][(Log[x]*Log[x*(1 + x)])/((1 + x)*(2*x - Log[x
])*(1 - 6*x + 4*Log[x] + 2*x*Log[x*(1 + x)]^2 - Log[x]*Log[x*(1 + x)]^2)), x] - 4*Defer[Int][(Log[x]^2*Log[x*(
1 + x)])/((2*x - Log[x])*(1 - 6*x + 4*Log[x] + 2*x*Log[x*(1 + x)]^2 - Log[x]*Log[x*(1 + x)]^2)), x] + 2*Defer[
Int][(Log[x]^2*Log[x*(1 + x)])/((1 + x)*(2*x - Log[x])*(1 - 6*x + 4*Log[x] + 2*x*Log[x*(1 + x)]^2 - Log[x]*Log
[x*(1 + x)]^2)), x] - 8*Defer[Int][1/((1 + Log[x])*(-1 + 6*x - 4*Log[x] - 2*x*Log[x*(1 + x)]^2 + Log[x]*Log[x*
(1 + x)]^2)), x] + 42*Defer[Int][x/((1 + Log[x])*(-1 + 6*x - 4*Log[x] - 2*x*Log[x*(1 + x)]^2 + Log[x]*Log[x*(1
+ x)]^2)), x] - 72*Defer[Int][x^2/((1 + Log[x])*(-1 + 6*x - 4*Log[x] - 2*x*Log[x*(1 + x)]^2 + Log[x]*Log[x*(1
+ x)]^2)), x] - 24*Defer[Int][Log[x]/((1 + Log[x])*(-1 + 6*x - 4*Log[x] - 2*x*Log[x*(1 + x)]^2 + Log[x]*Log[x
*(1 + x)]^2)), x] + 78*Defer[Int][(x*Log[x])/((1 + Log[x])*(-1 + 6*x - 4*Log[x] - 2*x*Log[x*(1 + x)]^2 + Log[x
]*Log[x*(1 + x)]^2)), x] - 16*Defer[Int][Log[x]^2/((1 + Log[x])*(-1 + 6*x - 4*Log[x] - 2*x*Log[x*(1 + x)]^2 +
Log[x]*Log[x*(1 + x)]^2)), x] + 4*Defer[Int][(Log[x]*Log[x*(1 + x)])/(-1 + 6*x - 4*Log[x] - 2*x*Log[x*(1 + x)]
^2 + Log[x]*Log[x*(1 + x)]^2), x]

Rubi steps \begin{align*} \text {integral}& = \int \frac {15-21 x-36 x^2+(12+12 x) \log (x)+(2+4 x+(2+4 x) \log (x)) \log \left (x+x^2\right )+\left (-8+16 x+24 x^2+(-7-7 x) \log (x)\right ) \log ^2\left (x+x^2\right )+\left (1-3 x-4 x^2+(1+x) \log (x)\right ) \log ^4\left (x+x^2\right )+\left (3-15 x-18 x^2+(12+12 x) \log (x)+\left (-1+11 x+12 x^2+(-7-7 x) \log (x)\right ) \log ^2\left (x+x^2\right )+\left (-2 x-2 x^2+(1+x) \log (x)\right ) \log ^4\left (x+x^2\right )\right ) \log \left (\frac {-1+6 x-4 \log (x)+(-2 x+\log (x)) \log ^2\left (x+x^2\right )}{-3+\log ^2\left (x+x^2\right )}\right )}{(1+x) \left (3-\log ^2(x (1+x))\right ) \left (1-6 x+4 \log (x)+2 x \log ^2(x (1+x))-\log (x) \log ^2(x (1+x))\right )} \, dx \\ & = \int \left (-\frac {15}{(1+x) \left (-3+\log ^2(x (1+x))\right ) \left (1-6 x+4 \log (x)+2 x \log ^2(x (1+x))-\log (x) \log ^2(x (1+x))\right )}+\frac {21 x}{(1+x) \left (-3+\log ^2(x (1+x))\right ) \left (1-6 x+4 \log (x)+2 x \log ^2(x (1+x))-\log (x) \log ^2(x (1+x))\right )}+\frac {36 x^2}{(1+x) \left (-3+\log ^2(x (1+x))\right ) \left (1-6 x+4 \log (x)+2 x \log ^2(x (1+x))-\log (x) \log ^2(x (1+x))\right )}-\frac {2 (1+2 x) (1+\log (x)) \log (x (1+x))}{(1+x) \left (-3+\log ^2(x (1+x))\right ) \left (1-6 x+4 \log (x)+2 x \log ^2(x (1+x))-\log (x) \log ^2(x (1+x))\right )}-\frac {(-8+24 x-7 \log (x)) \log ^2(x (1+x))}{\left (-3+\log ^2(x (1+x))\right ) \left (1-6 x+4 \log (x)+2 x \log ^2(x (1+x))-\log (x) \log ^2(x (1+x))\right )}+\frac {(-1+4 x-\log (x)) \log ^4(x (1+x))}{\left (-3+\log ^2(x (1+x))\right ) \left (1-6 x+4 \log (x)+2 x \log ^2(x (1+x))-\log (x) \log ^2(x (1+x))\right )}+\frac {12 \log (x)}{\left (-3+\log ^2(x (1+x))\right ) \left (-1+6 x-4 \log (x)-2 x \log ^2(x (1+x))+\log (x) \log ^2(x (1+x))\right )}+\log \left (\frac {-1+6 x-4 \log (x)+(-2 x+\log (x)) \log ^2(x (1+x))}{-3+\log ^2(x (1+x))}\right )\right ) \, dx \\ & = -\left (2 \int \frac {(1+2 x) (1+\log (x)) \log (x (1+x))}{(1+x) \left (-3+\log ^2(x (1+x))\right ) \left (1-6 x+4 \log (x)+2 x \log ^2(x (1+x))-\log (x) \log ^2(x (1+x))\right )} \, dx\right )+12 \int \frac {\log (x)}{\left (-3+\log ^2(x (1+x))\right ) \left (-1+6 x-4 \log (x)-2 x \log ^2(x (1+x))+\log (x) \log ^2(x (1+x))\right )} \, dx-15 \int \frac {1}{(1+x) \left (-3+\log ^2(x (1+x))\right ) \left (1-6 x+4 \log (x)+2 x \log ^2(x (1+x))-\log (x) \log ^2(x (1+x))\right )} \, dx+21 \int \frac {x}{(1+x) \left (-3+\log ^2(x (1+x))\right ) \left (1-6 x+4 \log (x)+2 x \log ^2(x (1+x))-\log (x) \log ^2(x (1+x))\right )} \, dx+36 \int \frac {x^2}{(1+x) \left (-3+\log ^2(x (1+x))\right ) \left (1-6 x+4 \log (x)+2 x \log ^2(x (1+x))-\log (x) \log ^2(x (1+x))\right )} \, dx-\int \frac {(-8+24 x-7 \log (x)) \log ^2(x (1+x))}{\left (-3+\log ^2(x (1+x))\right ) \left (1-6 x+4 \log (x)+2 x \log ^2(x (1+x))-\log (x) \log ^2(x (1+x))\right )} \, dx+\int \frac {(-1+4 x-\log (x)) \log ^4(x (1+x))}{\left (-3+\log ^2(x (1+x))\right ) \left (1-6 x+4 \log (x)+2 x \log ^2(x (1+x))-\log (x) \log ^2(x (1+x))\right )} \, dx+\int \log \left (\frac {-1+6 x-4 \log (x)+(-2 x+\log (x)) \log ^2(x (1+x))}{-3+\log ^2(x (1+x))}\right ) \, dx \\ & = x \log \left (\frac {1-6 x+4 \log (x)+(2 x-\log (x)) \log ^2(x (1+x))}{3-\log ^2(x (1+x))}\right )-2 \int \left (\frac {(1+2 x) \log (x (1+x))}{(1+x) \left (-3+\log ^2(x (1+x))\right )}-\frac {(1+2 x) (2 x-\log (x)) \log (x (1+x))}{(1+x) \left (1-6 x+4 \log (x)+2 x \log ^2(x (1+x))-\log (x) \log ^2(x (1+x))\right )}\right ) \, dx+12 \int \left (-\frac {\log (x)}{(1+\log (x)) \left (-3+\log ^2(x (1+x))\right )}+\frac {\log (x) (-2 x+\log (x))}{(1+\log (x)) \left (-1+6 x-4 \log (x)-2 x \log ^2(x (1+x))+\log (x) \log ^2(x (1+x))\right )}\right ) \, dx-15 \int \left (\frac {1}{(1+x) (1+\log (x)) \left (-3+\log ^2(x (1+x))\right )}+\frac {-2 x+\log (x)}{(1+x) (1+\log (x)) \left (1-6 x+4 \log (x)+2 x \log ^2(x (1+x))-\log (x) \log ^2(x (1+x))\right )}\right ) \, dx+21 \int \left (\frac {x}{(1+x) (1+\log (x)) \left (-3+\log ^2(x (1+x))\right )}-\frac {x (2 x-\log (x))}{(1+x) (1+\log (x)) \left (1-6 x+4 \log (x)+2 x \log ^2(x (1+x))-\log (x) \log ^2(x (1+x))\right )}\right ) \, dx+36 \int \left (\frac {x^2}{(1+x) (1+\log (x)) \left (-3+\log ^2(x (1+x))\right )}-\frac {x^2 (2 x-\log (x))}{(1+x) (1+\log (x)) \left (1-6 x+4 \log (x)+2 x \log ^2(x (1+x))-\log (x) \log ^2(x (1+x))\right )}\right ) \, dx-\int \frac {6 \left (-2+x+3 x^2\right )-2 (1+2 x) (1+\log (x)) \log (x (1+x))+\left (7-5 x-12 x^2\right ) \log ^2(x (1+x))+\left (-1+x+2 x^2\right ) \log ^4(x (1+x))}{(1+x) \left (-3+\log ^2(x (1+x))\right ) \left (1-6 x+2 x \log ^2(x (1+x))-\log (x) \left (-4+\log ^2(x (1+x))\right )\right )} \, dx+\int \left (\frac {-1+4 x-\log (x)}{2 x-\log (x)}+\frac {9 (-1+4 x-\log (x))}{(1+\log (x)) \left (-3+\log ^2(x (1+x))\right )}-\frac {(-1+6 x-4 \log (x))^2 (-1+4 x-\log (x))}{(2 x-\log (x)) (1+\log (x)) \left (1-6 x+4 \log (x)+2 x \log ^2(x (1+x))-\log (x) \log ^2(x (1+x))\right )}\right ) \, dx-\int \left (\frac {3 (-8+24 x-7 \log (x))}{(1+\log (x)) \left (-3+\log ^2(x (1+x))\right )}+\frac {(-8+24 x-7 \log (x)) (-1+6 x-4 \log (x))}{(1+\log (x)) \left (-1+6 x-4 \log (x)-2 x \log ^2(x (1+x))+\log (x) \log ^2(x (1+x))\right )}\right ) \, dx \\ & = x \log \left (\frac {1-6 x+4 \log (x)+(2 x-\log (x)) \log ^2(x (1+x))}{3-\log ^2(x (1+x))}\right )-2 \int \frac {(1+2 x) \log (x (1+x))}{(1+x) \left (-3+\log ^2(x (1+x))\right )} \, dx+2 \int \frac {(1+2 x) (2 x-\log (x)) \log (x (1+x))}{(1+x) \left (1-6 x+4 \log (x)+2 x \log ^2(x (1+x))-\log (x) \log ^2(x (1+x))\right )} \, dx-3 \int \frac {-8+24 x-7 \log (x)}{(1+\log (x)) \left (-3+\log ^2(x (1+x))\right )} \, dx+9 \int \frac {-1+4 x-\log (x)}{(1+\log (x)) \left (-3+\log ^2(x (1+x))\right )} \, dx-12 \int \frac {\log (x)}{(1+\log (x)) \left (-3+\log ^2(x (1+x))\right )} \, dx+12 \int \frac {\log (x) (-2 x+\log (x))}{(1+\log (x)) \left (-1+6 x-4 \log (x)-2 x \log ^2(x (1+x))+\log (x) \log ^2(x (1+x))\right )} \, dx-15 \int \frac {1}{(1+x) (1+\log (x)) \left (-3+\log ^2(x (1+x))\right )} \, dx-15 \int \frac {-2 x+\log (x)}{(1+x) (1+\log (x)) \left (1-6 x+4 \log (x)+2 x \log ^2(x (1+x))-\log (x) \log ^2(x (1+x))\right )} \, dx+21 \int \frac {x}{(1+x) (1+\log (x)) \left (-3+\log ^2(x (1+x))\right )} \, dx-21 \int \frac {x (2 x-\log (x))}{(1+x) (1+\log (x)) \left (1-6 x+4 \log (x)+2 x \log ^2(x (1+x))-\log (x) \log ^2(x (1+x))\right )} \, dx+36 \int \frac {x^2}{(1+x) (1+\log (x)) \left (-3+\log ^2(x (1+x))\right )} \, dx-36 \int \frac {x^2 (2 x-\log (x))}{(1+x) (1+\log (x)) \left (1-6 x+4 \log (x)+2 x \log ^2(x (1+x))-\log (x) \log ^2(x (1+x))\right )} \, dx+\int \frac {-1+4 x-\log (x)}{2 x-\log (x)} \, dx-\int \frac {(-1+6 x-4 \log (x))^2 (-1+4 x-\log (x))}{(2 x-\log (x)) (1+\log (x)) \left (1-6 x+4 \log (x)+2 x \log ^2(x (1+x))-\log (x) \log ^2(x (1+x))\right )} \, dx-\int \frac {(-8+24 x-7 \log (x)) (-1+6 x-4 \log (x))}{(1+\log (x)) \left (-1+6 x-4 \log (x)-2 x \log ^2(x (1+x))+\log (x) \log ^2(x (1+x))\right )} \, dx-\int \left (\frac {-1+2 x}{2 x-\log (x)}-\frac {2 (1+2 x) \log (x (1+x))}{(1+x) \left (-3+\log ^2(x (1+x))\right )}+\frac {1+x-2 x \log (x)-2 x^2 \log (x)+8 x^2 \log (x (1+x))+16 x^3 \log (x (1+x))-8 x \log (x) \log (x (1+x))-16 x^2 \log (x) \log (x (1+x))+2 \log ^2(x) \log (x (1+x))+4 x \log ^2(x) \log (x (1+x))}{(1+x) (2 x-\log (x)) \left (1-6 x+4 \log (x)+2 x \log ^2(x (1+x))-\log (x) \log ^2(x (1+x))\right )}\right ) \, dx \\ & = \text {Too large to display} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.24 (sec) , antiderivative size = 42, normalized size of antiderivative = 1.45 \[ \int \frac {15-21 x-36 x^2+(12+12 x) \log (x)+(2+4 x+(2+4 x) \log (x)) \log \left (x+x^2\right )+\left (-8+16 x+24 x^2+(-7-7 x) \log (x)\right ) \log ^2\left (x+x^2\right )+\left (1-3 x-4 x^2+(1+x) \log (x)\right ) \log ^4\left (x+x^2\right )+\left (3-15 x-18 x^2+(12+12 x) \log (x)+\left (-1+11 x+12 x^2+(-7-7 x) \log (x)\right ) \log ^2\left (x+x^2\right )+\left (-2 x-2 x^2+(1+x) \log (x)\right ) \log ^4\left (x+x^2\right )\right ) \log \left (\frac {-1+6 x-4 \log (x)+(-2 x+\log (x)) \log ^2\left (x+x^2\right )}{-3+\log ^2\left (x+x^2\right )}\right )}{3-15 x-18 x^2+(12+12 x) \log (x)+\left (-1+11 x+12 x^2+(-7-7 x) \log (x)\right ) \log ^2\left (x+x^2\right )+\left (-2 x-2 x^2+(1+x) \log (x)\right ) \log ^4\left (x+x^2\right )} \, dx=x+x \log \left (\frac {-1+6 x-4 \log (x)+(-2 x+\log (x)) \log ^2(x (1+x))}{-3+\log ^2(x (1+x))}\right ) \]

[In]

Integrate[(15 - 21*x - 36*x^2 + (12 + 12*x)*Log[x] + (2 + 4*x + (2 + 4*x)*Log[x])*Log[x + x^2] + (-8 + 16*x +
24*x^2 + (-7 - 7*x)*Log[x])*Log[x + x^2]^2 + (1 - 3*x - 4*x^2 + (1 + x)*Log[x])*Log[x + x^2]^4 + (3 - 15*x - 1
8*x^2 + (12 + 12*x)*Log[x] + (-1 + 11*x + 12*x^2 + (-7 - 7*x)*Log[x])*Log[x + x^2]^2 + (-2*x - 2*x^2 + (1 + x)
*Log[x])*Log[x + x^2]^4)*Log[(-1 + 6*x - 4*Log[x] + (-2*x + Log[x])*Log[x + x^2]^2)/(-3 + Log[x + x^2]^2)])/(3
- 15*x - 18*x^2 + (12 + 12*x)*Log[x] + (-1 + 11*x + 12*x^2 + (-7 - 7*x)*Log[x])*Log[x + x^2]^2 + (-2*x - 2*x^
2 + (1 + x)*Log[x])*Log[x + x^2]^4),x]

[Out]

x + x*Log[(-1 + 6*x - 4*Log[x] + (-2*x + Log[x])*Log[x*(1 + x)]^2)/(-3 + Log[x*(1 + x)]^2)]

Maple [C] (warning: unable to verify)

Result contains higher order function than in optimal. Order 9 vs. order 3.

Time = 27.54 (sec) , antiderivative size = 10690, normalized size of antiderivative = 368.62

\[\text {output too large to display}\]

[In]

int((((ln(x)*(1+x)-2*x^2-2*x)*ln(x^2+x)^4+((-7*x-7)*ln(x)+12*x^2+11*x-1)*ln(x^2+x)^2+(12*x+12)*ln(x)-18*x^2-15
*x+3)*ln(((ln(x)-2*x)*ln(x^2+x)^2-4*ln(x)+6*x-1)/(ln(x^2+x)^2-3))+(ln(x)*(1+x)-4*x^2-3*x+1)*ln(x^2+x)^4+((-7*x
-7)*ln(x)+24*x^2+16*x-8)*ln(x^2+x)^2+((4*x+2)*ln(x)+4*x+2)*ln(x^2+x)+(12*x+12)*ln(x)-36*x^2-21*x+15)/((ln(x)*(
1+x)-2*x^2-2*x)*ln(x^2+x)^4+((-7*x-7)*ln(x)+12*x^2+11*x-1)*ln(x^2+x)^2+(12*x+12)*ln(x)-18*x^2-15*x+3),x)

[Out]

result too large to display

Fricas [A] (verification not implemented)

none

Time = 0.25 (sec) , antiderivative size = 45, normalized size of antiderivative = 1.55 \[ \int \frac {15-21 x-36 x^2+(12+12 x) \log (x)+(2+4 x+(2+4 x) \log (x)) \log \left (x+x^2\right )+\left (-8+16 x+24 x^2+(-7-7 x) \log (x)\right ) \log ^2\left (x+x^2\right )+\left (1-3 x-4 x^2+(1+x) \log (x)\right ) \log ^4\left (x+x^2\right )+\left (3-15 x-18 x^2+(12+12 x) \log (x)+\left (-1+11 x+12 x^2+(-7-7 x) \log (x)\right ) \log ^2\left (x+x^2\right )+\left (-2 x-2 x^2+(1+x) \log (x)\right ) \log ^4\left (x+x^2\right )\right ) \log \left (\frac {-1+6 x-4 \log (x)+(-2 x+\log (x)) \log ^2\left (x+x^2\right )}{-3+\log ^2\left (x+x^2\right )}\right )}{3-15 x-18 x^2+(12+12 x) \log (x)+\left (-1+11 x+12 x^2+(-7-7 x) \log (x)\right ) \log ^2\left (x+x^2\right )+\left (-2 x-2 x^2+(1+x) \log (x)\right ) \log ^4\left (x+x^2\right )} \, dx=x \log \left (-\frac {{\left (2 \, x - \log \left (x\right )\right )} \log \left (x^{2} + x\right )^{2} - 6 \, x + 4 \, \log \left (x\right ) + 1}{\log \left (x^{2} + x\right )^{2} - 3}\right ) + x \]

[In]

integrate((((log(x)*(1+x)-2*x^2-2*x)*log(x^2+x)^4+((-7*x-7)*log(x)+12*x^2+11*x-1)*log(x^2+x)^2+(12*x+12)*log(x
)-18*x^2-15*x+3)*log(((log(x)-2*x)*log(x^2+x)^2-4*log(x)+6*x-1)/(log(x^2+x)^2-3))+(log(x)*(1+x)-4*x^2-3*x+1)*l
og(x^2+x)^4+((-7*x-7)*log(x)+24*x^2+16*x-8)*log(x^2+x)^2+((4*x+2)*log(x)+4*x+2)*log(x^2+x)+(12*x+12)*log(x)-36
*x^2-21*x+15)/((log(x)*(1+x)-2*x^2-2*x)*log(x^2+x)^4+((-7*x-7)*log(x)+12*x^2+11*x-1)*log(x^2+x)^2+(12*x+12)*lo
g(x)-18*x^2-15*x+3),x, algorithm="fricas")

[Out]

x*log(-((2*x - log(x))*log(x^2 + x)^2 - 6*x + 4*log(x) + 1)/(log(x^2 + x)^2 - 3)) + x

Sympy [F(-2)]

Exception generated. \[ \int \frac {15-21 x-36 x^2+(12+12 x) \log (x)+(2+4 x+(2+4 x) \log (x)) \log \left (x+x^2\right )+\left (-8+16 x+24 x^2+(-7-7 x) \log (x)\right ) \log ^2\left (x+x^2\right )+\left (1-3 x-4 x^2+(1+x) \log (x)\right ) \log ^4\left (x+x^2\right )+\left (3-15 x-18 x^2+(12+12 x) \log (x)+\left (-1+11 x+12 x^2+(-7-7 x) \log (x)\right ) \log ^2\left (x+x^2\right )+\left (-2 x-2 x^2+(1+x) \log (x)\right ) \log ^4\left (x+x^2\right )\right ) \log \left (\frac {-1+6 x-4 \log (x)+(-2 x+\log (x)) \log ^2\left (x+x^2\right )}{-3+\log ^2\left (x+x^2\right )}\right )}{3-15 x-18 x^2+(12+12 x) \log (x)+\left (-1+11 x+12 x^2+(-7-7 x) \log (x)\right ) \log ^2\left (x+x^2\right )+\left (-2 x-2 x^2+(1+x) \log (x)\right ) \log ^4\left (x+x^2\right )} \, dx=\text {Exception raised: PolynomialError} \]

[In]

integrate((((ln(x)*(1+x)-2*x**2-2*x)*ln(x**2+x)**4+((-7*x-7)*ln(x)+12*x**2+11*x-1)*ln(x**2+x)**2+(12*x+12)*ln(
x)-18*x**2-15*x+3)*ln(((ln(x)-2*x)*ln(x**2+x)**2-4*ln(x)+6*x-1)/(ln(x**2+x)**2-3))+(ln(x)*(1+x)-4*x**2-3*x+1)*
ln(x**2+x)**4+((-7*x-7)*ln(x)+24*x**2+16*x-8)*ln(x**2+x)**2+((4*x+2)*ln(x)+4*x+2)*ln(x**2+x)+(12*x+12)*ln(x)-3
6*x**2-21*x+15)/((ln(x)*(1+x)-2*x**2-2*x)*ln(x**2+x)**4+((-7*x-7)*ln(x)+12*x**2+11*x-1)*ln(x**2+x)**2+(12*x+12
)*ln(x)-18*x**2-15*x+3),x)

[Out]

Exception raised: PolynomialError >> 1/(36*_t0**4*x**4 + 72*_t0**4*x**3 + 36*_t0**4*x**2 - 288*_t0**3*x**5 - 5
76*_t0**3*x**4 - 288*_t0**3*x**3 + 864*_t0**2*x**6 + 1728*_t0**2*x**5 + 864*_t0**2*x**4 - 1152*_t0*x**7 - 2304
*_t0*x**6 - 1152*

Maxima [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 81 vs. \(2 (29) = 58\).

Time = 0.28 (sec) , antiderivative size = 81, normalized size of antiderivative = 2.79 \[ \int \frac {15-21 x-36 x^2+(12+12 x) \log (x)+(2+4 x+(2+4 x) \log (x)) \log \left (x+x^2\right )+\left (-8+16 x+24 x^2+(-7-7 x) \log (x)\right ) \log ^2\left (x+x^2\right )+\left (1-3 x-4 x^2+(1+x) \log (x)\right ) \log ^4\left (x+x^2\right )+\left (3-15 x-18 x^2+(12+12 x) \log (x)+\left (-1+11 x+12 x^2+(-7-7 x) \log (x)\right ) \log ^2\left (x+x^2\right )+\left (-2 x-2 x^2+(1+x) \log (x)\right ) \log ^4\left (x+x^2\right )\right ) \log \left (\frac {-1+6 x-4 \log (x)+(-2 x+\log (x)) \log ^2\left (x+x^2\right )}{-3+\log ^2\left (x+x^2\right )}\right )}{3-15 x-18 x^2+(12+12 x) \log (x)+\left (-1+11 x+12 x^2+(-7-7 x) \log (x)\right ) \log ^2\left (x+x^2\right )+\left (-2 x-2 x^2+(1+x) \log (x)\right ) \log ^4\left (x+x^2\right )} \, dx=x \log \left (-2 \, {\left (x - \log \left (x + 1\right )\right )} \log \left (x\right )^{2} + \log \left (x\right )^{3} - 2 \, {\left (\log \left (x + 1\right )^{2} - 3\right )} x - {\left (4 \, x \log \left (x + 1\right ) - \log \left (x + 1\right )^{2} + 4\right )} \log \left (x\right ) - 1\right ) - x \log \left (\log \left (x + 1\right )^{2} + 2 \, \log \left (x + 1\right ) \log \left (x\right ) + \log \left (x\right )^{2} - 3\right ) + x \]

[In]

integrate((((log(x)*(1+x)-2*x^2-2*x)*log(x^2+x)^4+((-7*x-7)*log(x)+12*x^2+11*x-1)*log(x^2+x)^2+(12*x+12)*log(x
)-18*x^2-15*x+3)*log(((log(x)-2*x)*log(x^2+x)^2-4*log(x)+6*x-1)/(log(x^2+x)^2-3))+(log(x)*(1+x)-4*x^2-3*x+1)*l
og(x^2+x)^4+((-7*x-7)*log(x)+24*x^2+16*x-8)*log(x^2+x)^2+((4*x+2)*log(x)+4*x+2)*log(x^2+x)+(12*x+12)*log(x)-36
*x^2-21*x+15)/((log(x)*(1+x)-2*x^2-2*x)*log(x^2+x)^4+((-7*x-7)*log(x)+12*x^2+11*x-1)*log(x^2+x)^2+(12*x+12)*lo
g(x)-18*x^2-15*x+3),x, algorithm="maxima")

[Out]

x*log(-2*(x - log(x + 1))*log(x)^2 + log(x)^3 - 2*(log(x + 1)^2 - 3)*x - (4*x*log(x + 1) - log(x + 1)^2 + 4)*l
og(x) - 1) - x*log(log(x + 1)^2 + 2*log(x + 1)*log(x) + log(x)^2 - 3) + x

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 86 vs. \(2 (29) = 58\).

Time = 2.72 (sec) , antiderivative size = 86, normalized size of antiderivative = 2.97 \[ \int \frac {15-21 x-36 x^2+(12+12 x) \log (x)+(2+4 x+(2+4 x) \log (x)) \log \left (x+x^2\right )+\left (-8+16 x+24 x^2+(-7-7 x) \log (x)\right ) \log ^2\left (x+x^2\right )+\left (1-3 x-4 x^2+(1+x) \log (x)\right ) \log ^4\left (x+x^2\right )+\left (3-15 x-18 x^2+(12+12 x) \log (x)+\left (-1+11 x+12 x^2+(-7-7 x) \log (x)\right ) \log ^2\left (x+x^2\right )+\left (-2 x-2 x^2+(1+x) \log (x)\right ) \log ^4\left (x+x^2\right )\right ) \log \left (\frac {-1+6 x-4 \log (x)+(-2 x+\log (x)) \log ^2\left (x+x^2\right )}{-3+\log ^2\left (x+x^2\right )}\right )}{3-15 x-18 x^2+(12+12 x) \log (x)+\left (-1+11 x+12 x^2+(-7-7 x) \log (x)\right ) \log ^2\left (x+x^2\right )+\left (-2 x-2 x^2+(1+x) \log (x)\right ) \log ^4\left (x+x^2\right )} \, dx=x \log \left (-2 \, x \log \left (x + 1\right )^{2} - 4 \, x \log \left (x + 1\right ) \log \left (x\right ) + \log \left (x + 1\right )^{2} \log \left (x\right ) - 2 \, x \log \left (x\right )^{2} + 2 \, \log \left (x + 1\right ) \log \left (x\right )^{2} + \log \left (x\right )^{3} + 6 \, x - 4 \, \log \left (x\right ) - 1\right ) - x \log \left (\log \left (x + 1\right )^{2} + 2 \, \log \left (x + 1\right ) \log \left (x\right ) + \log \left (x\right )^{2} - 3\right ) + x \]

[In]

integrate((((log(x)*(1+x)-2*x^2-2*x)*log(x^2+x)^4+((-7*x-7)*log(x)+12*x^2+11*x-1)*log(x^2+x)^2+(12*x+12)*log(x
)-18*x^2-15*x+3)*log(((log(x)-2*x)*log(x^2+x)^2-4*log(x)+6*x-1)/(log(x^2+x)^2-3))+(log(x)*(1+x)-4*x^2-3*x+1)*l
og(x^2+x)^4+((-7*x-7)*log(x)+24*x^2+16*x-8)*log(x^2+x)^2+((4*x+2)*log(x)+4*x+2)*log(x^2+x)+(12*x+12)*log(x)-36
*x^2-21*x+15)/((log(x)*(1+x)-2*x^2-2*x)*log(x^2+x)^4+((-7*x-7)*log(x)+12*x^2+11*x-1)*log(x^2+x)^2+(12*x+12)*lo
g(x)-18*x^2-15*x+3),x, algorithm="giac")

[Out]

x*log(-2*x*log(x + 1)^2 - 4*x*log(x + 1)*log(x) + log(x + 1)^2*log(x) - 2*x*log(x)^2 + 2*log(x + 1)*log(x)^2 +
log(x)^3 + 6*x - 4*log(x) - 1) - x*log(log(x + 1)^2 + 2*log(x + 1)*log(x) + log(x)^2 - 3) + x

Mupad [B] (verification not implemented)

Time = 9.90 (sec) , antiderivative size = 45, normalized size of antiderivative = 1.55 \[ \int \frac {15-21 x-36 x^2+(12+12 x) \log (x)+(2+4 x+(2+4 x) \log (x)) \log \left (x+x^2\right )+\left (-8+16 x+24 x^2+(-7-7 x) \log (x)\right ) \log ^2\left (x+x^2\right )+\left (1-3 x-4 x^2+(1+x) \log (x)\right ) \log ^4\left (x+x^2\right )+\left (3-15 x-18 x^2+(12+12 x) \log (x)+\left (-1+11 x+12 x^2+(-7-7 x) \log (x)\right ) \log ^2\left (x+x^2\right )+\left (-2 x-2 x^2+(1+x) \log (x)\right ) \log ^4\left (x+x^2\right )\right ) \log \left (\frac {-1+6 x-4 \log (x)+(-2 x+\log (x)) \log ^2\left (x+x^2\right )}{-3+\log ^2\left (x+x^2\right )}\right )}{3-15 x-18 x^2+(12+12 x) \log (x)+\left (-1+11 x+12 x^2+(-7-7 x) \log (x)\right ) \log ^2\left (x+x^2\right )+\left (-2 x-2 x^2+(1+x) \log (x)\right ) \log ^4\left (x+x^2\right )} \, dx=x\,\left (\ln \left (-\frac {\left (2\,x-\ln \left (x\right )\right )\,{\ln \left (x^2+x\right )}^2-6\,x+4\,\ln \left (x\right )+1}{{\ln \left (x^2+x\right )}^2-3}\right )+1\right ) \]

[In]

int((21*x - log(x + x^2)*(4*x + log(x)*(4*x + 2) + 2) + log(-(4*log(x) - 6*x + log(x + x^2)^2*(2*x - log(x)) +
1)/(log(x + x^2)^2 - 3))*(15*x + log(x + x^2)^4*(2*x - log(x)*(x + 1) + 2*x^2) - log(x)*(12*x + 12) - log(x +
x^2)^2*(11*x - log(x)*(7*x + 7) + 12*x^2 - 1) + 18*x^2 - 3) + log(x + x^2)^4*(3*x - log(x)*(x + 1) + 4*x^2 -
1) - log(x)*(12*x + 12) - log(x + x^2)^2*(16*x - log(x)*(7*x + 7) + 24*x^2 - 8) + 36*x^2 - 15)/(15*x + log(x +
x^2)^4*(2*x - log(x)*(x + 1) + 2*x^2) - log(x)*(12*x + 12) - log(x + x^2)^2*(11*x - log(x)*(7*x + 7) + 12*x^2
- 1) + 18*x^2 - 3),x)

[Out]

x*(log(-(4*log(x) - 6*x + log(x + x^2)^2*(2*x - log(x)) + 1)/(log(x + x^2)^2 - 3)) + 1)