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

Optimal result

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

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

Mathematica [A] (verified)

Time = 0.16 (sec) , antiderivative size = 42, normalized size of antiderivative = 1.45 \[ \int \frac {15-21 x-36 x^2+(12+12 x) \log (x)+(2+4 x+(2+4 x) \log (x)) \log \left (x+x^2\right )+\left (-8+16 x+24 x^2+(-7-7 x) \log (x)\right ) \log ^2\left (x+x^2\right )+\left (1-3 x-4 x^2+(1+x) \log (x)\right ) \log ^4\left (x+x^2\right )+\left (3-15 x-18 x^2+(12+12 x) \log (x)+\left (-1+11 x+12 x^2+(-7-7 x) \log (x)\right ) \log ^2\left (x+x^2\right )+\left (-2 x-2 x^2+(1+x) \log (x)\right ) \log ^4\left (x+x^2\right )\right ) \log \left (\frac {-1+6 x-4 \log (x)+(-2 x+\log (x)) \log ^2\left (x+x^2\right )}{-3+\log ^2\left (x+x^2\right )}\right )}{3-15 x-18 x^2+(12+12 x) \log (x)+\left (-1+11 x+12 x^2+(-7-7 x) \log (x)\right ) \log ^2\left (x+x^2\right )+\left (-2 x-2 x^2+(1+x) \log (x)\right ) \log ^4\left (x+x^2\right )} \, dx=x+x \log \left (\frac {-1+6 x-4 \log (x)+(-2 x+\log (x)) \log ^2(x (1+x))}{-3+\log ^2(x (1+x))}\right ) \] Input:

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 - 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 - 1 
5*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]
 

Output:

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

Rubi [F]

Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.

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

\(\Big \downarrow \) 7292

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

\(\Big \downarrow \) 7293

\(\displaystyle \int \left (\frac {36 x^2}{(x+1) \left (\log ^2(x (x+1))-3\right ) \left (-6 x+2 x \log ^2(x (x+1))-\log (x) \log ^2(x (x+1))+4 \log (x)+1\right )}-\frac {(24 x-7 \log (x)-8) \log ^2(x (x+1))}{\left (\log ^2(x (x+1))-3\right ) \left (-6 x+2 x \log ^2(x (x+1))-\log (x) \log ^2(x (x+1))+4 \log (x)+1\right )}-\frac {2 (2 x+1) (\log (x)+1) \log (x (x+1))}{(x+1) \left (\log ^2(x (x+1))-3\right ) \left (-6 x+2 x \log ^2(x (x+1))-\log (x) \log ^2(x (x+1))+4 \log (x)+1\right )}+\log \left (\frac {6 x+(\log (x)-2 x) \log ^2(x (x+1))-4 \log (x)-1}{\log ^2(x (x+1))-3}\right )+\frac {21 x}{(x+1) \left (\log ^2(x (x+1))-3\right ) \left (-6 x+2 x \log ^2(x (x+1))-\log (x) \log ^2(x (x+1))+4 \log (x)+1\right )}-\frac {15}{(x+1) \left (\log ^2(x (x+1))-3\right ) \left (-6 x+2 x \log ^2(x (x+1))-\log (x) \log ^2(x (x+1))+4 \log (x)+1\right )}+\frac {12 \log (x)}{\left (\log ^2(x (x+1))-3\right ) \left (6 x-2 x \log ^2(x (x+1))+\log (x) \log ^2(x (x+1))-4 \log (x)-1\right )}+\frac {(4 x-\log (x)-1) \log ^4(x (x+1))}{\left (\log ^2(x (x+1))-3\right ) \left (-6 x+2 x \log ^2(x (x+1))-\log (x) \log ^2(x (x+1))+4 \log (x)+1\right )}\right )dx\)

\(\Big \downarrow \) 7299

\(\displaystyle \int \left (\frac {36 x^2}{(x+1) \left (\log ^2(x (x+1))-3\right ) \left (-6 x+2 x \log ^2(x (x+1))-\log (x) \log ^2(x (x+1))+4 \log (x)+1\right )}-\frac {(24 x-7 \log (x)-8) \log ^2(x (x+1))}{\left (\log ^2(x (x+1))-3\right ) \left (-6 x+2 x \log ^2(x (x+1))-\log (x) \log ^2(x (x+1))+4 \log (x)+1\right )}-\frac {2 (2 x+1) (\log (x)+1) \log (x (x+1))}{(x+1) \left (\log ^2(x (x+1))-3\right ) \left (-6 x+2 x \log ^2(x (x+1))-\log (x) \log ^2(x (x+1))+4 \log (x)+1\right )}+\log \left (\frac {6 x+(\log (x)-2 x) \log ^2(x (x+1))-4 \log (x)-1}{\log ^2(x (x+1))-3}\right )+\frac {21 x}{(x+1) \left (\log ^2(x (x+1))-3\right ) \left (-6 x+2 x \log ^2(x (x+1))-\log (x) \log ^2(x (x+1))+4 \log (x)+1\right )}-\frac {15}{(x+1) \left (\log ^2(x (x+1))-3\right ) \left (-6 x+2 x \log ^2(x (x+1))-\log (x) \log ^2(x (x+1))+4 \log (x)+1\right )}+\frac {12 \log (x)}{\left (\log ^2(x (x+1))-3\right ) \left (6 x-2 x \log ^2(x (x+1))+\log (x) \log ^2(x (x+1))-4 \log (x)-1\right )}+\frac {(4 x-\log (x)-1) \log ^4(x (x+1))}{\left (\log ^2(x (x+1))-3\right ) \left (-6 x+2 x \log ^2(x (x+1))-\log (x) \log ^2(x (x+1))+4 \log (x)+1\right )}\right )dx\)

Input:

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]
 

Output:

$Aborted
 

Maple [C] (warning: unable to verify)

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

Time = 51.04 (sec) , antiderivative size = 10690, normalized size of antiderivative = 368.62

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

Input:

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)*l 
n(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+1 
2)*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)
 

Output:

result too large to display
 

Fricas [A] (verification not implemented)

Time = 0.08 (sec) , antiderivative size = 45, normalized size of antiderivative = 1.55 \[ \int \frac {15-21 x-36 x^2+(12+12 x) \log (x)+(2+4 x+(2+4 x) \log (x)) \log \left (x+x^2\right )+\left (-8+16 x+24 x^2+(-7-7 x) \log (x)\right ) \log ^2\left (x+x^2\right )+\left (1-3 x-4 x^2+(1+x) \log (x)\right ) \log ^4\left (x+x^2\right )+\left (3-15 x-18 x^2+(12+12 x) \log (x)+\left (-1+11 x+12 x^2+(-7-7 x) \log (x)\right ) \log ^2\left (x+x^2\right )+\left (-2 x-2 x^2+(1+x) \log (x)\right ) \log ^4\left (x+x^2\right )\right ) \log \left (\frac {-1+6 x-4 \log (x)+(-2 x+\log (x)) \log ^2\left (x+x^2\right )}{-3+\log ^2\left (x+x^2\right )}\right )}{3-15 x-18 x^2+(12+12 x) \log (x)+\left (-1+11 x+12 x^2+(-7-7 x) \log (x)\right ) \log ^2\left (x+x^2\right )+\left (-2 x-2 x^2+(1+x) \log (x)\right ) \log ^4\left (x+x^2\right )} \, dx=x \log \left (-\frac {{\left (2 \, x - \log \left (x\right )\right )} \log \left (x^{2} + x\right )^{2} - 6 \, x + 4 \, \log \left (x\right ) + 1}{\log \left (x^{2} + x\right )^{2} - 3}\right ) + x \] Input:

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)*log 
(x^2+x)^4+((-7*x-7)*log(x)+24*x^2+16*x-8)*log(x^2+x)^2+((2+4*x)*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)*log(x) 
-18*x^2-15*x+3),x, algorithm="fricas")
 

Output:

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

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+((2+4*x)*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)
 

Output:

Exception raised: PolynomialError >> 1/(36*_t0**4*x**4 + 72*_t0**4*x**3 + 
36*_t0**4*x**2 - 288*_t0**3*x**5 - 576*_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 = 0.12 (sec) , antiderivative size = 81, normalized size of antiderivative = 2.79 \[ \int \frac {15-21 x-36 x^2+(12+12 x) \log (x)+(2+4 x+(2+4 x) \log (x)) \log \left (x+x^2\right )+\left (-8+16 x+24 x^2+(-7-7 x) \log (x)\right ) \log ^2\left (x+x^2\right )+\left (1-3 x-4 x^2+(1+x) \log (x)\right ) \log ^4\left (x+x^2\right )+\left (3-15 x-18 x^2+(12+12 x) \log (x)+\left (-1+11 x+12 x^2+(-7-7 x) \log (x)\right ) \log ^2\left (x+x^2\right )+\left (-2 x-2 x^2+(1+x) \log (x)\right ) \log ^4\left (x+x^2\right )\right ) \log \left (\frac {-1+6 x-4 \log (x)+(-2 x+\log (x)) \log ^2\left (x+x^2\right )}{-3+\log ^2\left (x+x^2\right )}\right )}{3-15 x-18 x^2+(12+12 x) \log (x)+\left (-1+11 x+12 x^2+(-7-7 x) \log (x)\right ) \log ^2\left (x+x^2\right )+\left (-2 x-2 x^2+(1+x) \log (x)\right ) \log ^4\left (x+x^2\right )} \, dx=x \log \left (-2 \, {\left (x - \log \left (x + 1\right )\right )} \log \left (x\right )^{2} + \log \left (x\right )^{3} - 2 \, {\left (\log \left (x + 1\right )^{2} - 3\right )} x - {\left (4 \, x \log \left (x + 1\right ) - \log \left (x + 1\right )^{2} + 4\right )} \log \left (x\right ) - 1\right ) - x \log \left (\log \left (x + 1\right )^{2} + 2 \, \log \left (x + 1\right ) \log \left (x\right ) + \log \left (x\right )^{2} - 3\right ) + x \] Input:

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)*log 
(x^2+x)^4+((-7*x-7)*log(x)+24*x^2+16*x-8)*log(x^2+x)^2+((2+4*x)*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)*log(x) 
-18*x^2-15*x+3),x, algorithm="maxima")
 

Output:

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)*log(x) - 1) - x*log(log(x + 1)^2 + 2*lo 
g(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 = 2.30 (sec) , antiderivative size = 86, normalized size of antiderivative = 2.97 \[ \int \frac {15-21 x-36 x^2+(12+12 x) \log (x)+(2+4 x+(2+4 x) \log (x)) \log \left (x+x^2\right )+\left (-8+16 x+24 x^2+(-7-7 x) \log (x)\right ) \log ^2\left (x+x^2\right )+\left (1-3 x-4 x^2+(1+x) \log (x)\right ) \log ^4\left (x+x^2\right )+\left (3-15 x-18 x^2+(12+12 x) \log (x)+\left (-1+11 x+12 x^2+(-7-7 x) \log (x)\right ) \log ^2\left (x+x^2\right )+\left (-2 x-2 x^2+(1+x) \log (x)\right ) \log ^4\left (x+x^2\right )\right ) \log \left (\frac {-1+6 x-4 \log (x)+(-2 x+\log (x)) \log ^2\left (x+x^2\right )}{-3+\log ^2\left (x+x^2\right )}\right )}{3-15 x-18 x^2+(12+12 x) \log (x)+\left (-1+11 x+12 x^2+(-7-7 x) \log (x)\right ) \log ^2\left (x+x^2\right )+\left (-2 x-2 x^2+(1+x) \log (x)\right ) \log ^4\left (x+x^2\right )} \, dx=x \log \left (-2 \, x \log \left (x + 1\right )^{2} - 4 \, x \log \left (x + 1\right ) \log \left (x\right ) + \log \left (x + 1\right )^{2} \log \left (x\right ) - 2 \, x \log \left (x\right )^{2} + 2 \, \log \left (x + 1\right ) \log \left (x\right )^{2} + \log \left (x\right )^{3} + 6 \, x - 4 \, \log \left (x\right ) - 1\right ) - x \log \left (\log \left (x + 1\right )^{2} + 2 \, \log \left (x + 1\right ) \log \left (x\right ) + \log \left (x\right )^{2} - 3\right ) + x \] Input:

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)*log 
(x^2+x)^4+((-7*x-7)*log(x)+24*x^2+16*x-8)*log(x^2+x)^2+((2+4*x)*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)*log(x) 
-18*x^2-15*x+3),x, algorithm="giac")
 

Output:

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*lo 
g(log(x + 1)^2 + 2*log(x + 1)*log(x) + log(x)^2 - 3) + x
 

Mupad [B] (verification not implemented)

Time = 2.95 (sec) , antiderivative size = 45, normalized size of antiderivative = 1.55 \[ \int \frac {15-21 x-36 x^2+(12+12 x) \log (x)+(2+4 x+(2+4 x) \log (x)) \log \left (x+x^2\right )+\left (-8+16 x+24 x^2+(-7-7 x) \log (x)\right ) \log ^2\left (x+x^2\right )+\left (1-3 x-4 x^2+(1+x) \log (x)\right ) \log ^4\left (x+x^2\right )+\left (3-15 x-18 x^2+(12+12 x) \log (x)+\left (-1+11 x+12 x^2+(-7-7 x) \log (x)\right ) \log ^2\left (x+x^2\right )+\left (-2 x-2 x^2+(1+x) \log (x)\right ) \log ^4\left (x+x^2\right )\right ) \log \left (\frac {-1+6 x-4 \log (x)+(-2 x+\log (x)) \log ^2\left (x+x^2\right )}{-3+\log ^2\left (x+x^2\right )}\right )}{3-15 x-18 x^2+(12+12 x) \log (x)+\left (-1+11 x+12 x^2+(-7-7 x) \log (x)\right ) \log ^2\left (x+x^2\right )+\left (-2 x-2 x^2+(1+x) \log (x)\right ) \log ^4\left (x+x^2\right )} \, dx=x\,\left (\ln \left (-\frac {\left (2\,x-\ln \left (x\right )\right )\,{\ln \left (x^2+x\right )}^2-6\,x+4\,\ln \left (x\right )+1}{{\ln \left (x^2+x\right )}^2-3}\right )+1\right ) \] Input:

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)
 

Output:

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

Reduce [B] (verification not implemented)

Time = 0.44 (sec) , antiderivative size = 49, normalized size of antiderivative = 1.69 \[ \int \frac {15-21 x-36 x^2+(12+12 x) \log (x)+(2+4 x+(2+4 x) \log (x)) \log \left (x+x^2\right )+\left (-8+16 x+24 x^2+(-7-7 x) \log (x)\right ) \log ^2\left (x+x^2\right )+\left (1-3 x-4 x^2+(1+x) \log (x)\right ) \log ^4\left (x+x^2\right )+\left (3-15 x-18 x^2+(12+12 x) \log (x)+\left (-1+11 x+12 x^2+(-7-7 x) \log (x)\right ) \log ^2\left (x+x^2\right )+\left (-2 x-2 x^2+(1+x) \log (x)\right ) \log ^4\left (x+x^2\right )\right ) \log \left (\frac {-1+6 x-4 \log (x)+(-2 x+\log (x)) \log ^2\left (x+x^2\right )}{-3+\log ^2\left (x+x^2\right )}\right )}{3-15 x-18 x^2+(12+12 x) \log (x)+\left (-1+11 x+12 x^2+(-7-7 x) \log (x)\right ) \log ^2\left (x+x^2\right )+\left (-2 x-2 x^2+(1+x) \log (x)\right ) \log ^4\left (x+x^2\right )} \, dx=x \left (\mathrm {log}\left (\frac {\mathrm {log}\left (x^{2}+x \right )^{2} \mathrm {log}\left (x \right )-2 \mathrm {log}\left (x^{2}+x \right )^{2} x -4 \,\mathrm {log}\left (x \right )+6 x -1}{\mathrm {log}\left (x^{2}+x \right )^{2}-3}\right )+1\right ) \] Input:

int((((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)*log(x^2+x 
)^4+((-7*x-7)*log(x)+24*x^2+16*x-8)*log(x^2+x)^2+((2+4*x)*log(x)+4*x+2)*lo 
g(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)*log(x)-18*x^ 
2-15*x+3),x)
 

Output:

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