3.7.15 \(\int \frac {144-8 x+8 x \log (3)+(-108-48 x+16 x^2+(-36+8 x) \log (3)) \log (\frac {1}{9} (45-10 x))+(9 x^2-2 x^3) \log ^2(\frac {1}{9} (45-10 x))}{-144+32 x+(72 x-16 x^2) \log (\frac {1}{9} (45-10 x))+(-9 x^2+2 x^3) \log ^2(\frac {1}{9} (45-10 x))} \, dx\) [615]

3.7.15.1 Optimal result

Integrand size = 115, antiderivative size = 29 \[ \int \frac {144-8 x+8 x \log (3)+\left (-108-48 x+16 x^2+(-36+8 x) \log (3)\right ) \log \left (\frac {1}{9} (45-10 x)\right )+\left (9 x^2-2 x^3\right ) \log ^2\left (\frac {1}{9} (45-10 x)\right )}{-144+32 x+\left (72 x-16 x^2\right ) \log \left (\frac {1}{9} (45-10 x)\right )+\left (-9 x^2+2 x^3\right ) \log ^2\left (\frac {1}{9} (45-10 x)\right )} \, dx=-25-x+\frac {x (3+\log (3))}{-x+\frac {4}{\log \left (5-\frac {10 x}{9}\right )}} \]

output

(3+ln(3))*x/(4/ln(-10/9*x+5)-x)-x-25
 

3.7.15.2 Mathematica [A] (verified)

Time = 0.52 (sec) , antiderivative size = 24, normalized size of antiderivative = 0.83 \[ \int \frac {144-8 x+8 x \log (3)+\left (-108-48 x+16 x^2+(-36+8 x) \log (3)\right ) \log \left (\frac {1}{9} (45-10 x)\right )+\left (9 x^2-2 x^3\right ) \log ^2\left (\frac {1}{9} (45-10 x)\right )}{-144+32 x+\left (72 x-16 x^2\right ) \log \left (\frac {1}{9} (45-10 x)\right )+\left (-9 x^2+2 x^3\right ) \log ^2\left (\frac {1}{9} (45-10 x)\right )} \, dx=-x-\frac {4 (3+\log (3))}{-4+x \log \left (5-\frac {10 x}{9}\right )} \]

input

Integrate[(144 - 8*x + 8*x*Log[3] + (-108 - 48*x + 16*x^2 + (-36 + 8*x)*Lo 
g[3])*Log[(45 - 10*x)/9] + (9*x^2 - 2*x^3)*Log[(45 - 10*x)/9]^2)/(-144 + 3 
2*x + (72*x - 16*x^2)*Log[(45 - 10*x)/9] + (-9*x^2 + 2*x^3)*Log[(45 - 10*x 
)/9]^2),x]
 

output

-x - (4*(3 + Log[3]))/(-4 + x*Log[5 - (10*x)/9])
 

3.7.15.3 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.

\(\displaystyle \int \frac {\left (16 x^2-48 x+(8 x-36) \log (3)-108\right ) \log \left (\frac {1}{9} (45-10 x)\right )+\left (9 x^2-2 x^3\right ) \log ^2\left (\frac {1}{9} (45-10 x)\right )-8 x+8 x \log (3)+144}{\left (72 x-16 x^2\right ) \log \left (\frac {1}{9} (45-10 x)\right )+\left (2 x^3-9 x^2\right ) \log ^2\left (\frac {1}{9} (45-10 x)\right )+32 x-144} \, dx\)

\(\Big \downarrow \) 6

\(\displaystyle \int \frac {\left (16 x^2-48 x+(8 x-36) \log (3)-108\right ) \log \left (\frac {1}{9} (45-10 x)\right )+\left (9 x^2-2 x^3\right ) \log ^2\left (\frac {1}{9} (45-10 x)\right )+x (8 \log (3)-8)+144}{\left (72 x-16 x^2\right ) \log \left (\frac {1}{9} (45-10 x)\right )+\left (2 x^3-9 x^2\right ) \log ^2\left (\frac {1}{9} (45-10 x)\right )+32 x-144}dx\)

\(\Big \downarrow \) 7292

\(\displaystyle \int \frac {-\left (16 x^2-48 x+(8 x-36) \log (3)-108\right ) \log \left (\frac {1}{9} (45-10 x)\right )-\left (\left (9 x^2-2 x^3\right ) \log ^2\left (\frac {1}{9} (45-10 x)\right )\right )-x (8 \log (3)-8)-144}{(9-2 x) \left (4-x \log \left (5-\frac {10 x}{9}\right )\right )^2}dx\)

\(\Big \downarrow \) 7293

\(\displaystyle \int \left (-\frac {2 x^3 \log ^2\left (5-\frac {10 x}{9}\right )}{(2 x-9) \left (x \log \left (5-\frac {10 x}{9}\right )-4\right )^2}+\frac {9 x^2 \log ^2\left (5-\frac {10 x}{9}\right )}{(2 x-9) \left (x \log \left (5-\frac {10 x}{9}\right )-4\right )^2}+\frac {16 x^2 \log \left (5-\frac {10 x}{9}\right )}{(2 x-9) \left (x \log \left (5-\frac {10 x}{9}\right )-4\right )^2}-\frac {48 x \left (1-\frac {\log (3)}{6}\right ) \log \left (5-\frac {10 x}{9}\right )}{(2 x-9) \left (x \log \left (5-\frac {10 x}{9}\right )-4\right )^2}+\frac {8 x (\log (3)-1)}{(2 x-9) \left (x \log \left (5-\frac {10 x}{9}\right )-4\right )^2}-\frac {108 \left (1+\frac {\log (3)}{3}\right ) \log \left (5-\frac {10 x}{9}\right )}{(2 x-9) \left (x \log \left (5-\frac {10 x}{9}\right )-4\right )^2}+\frac {144}{(2 x-9) \left (x \log \left (5-\frac {10 x}{9}\right )-4\right )^2}\right )dx\)

\(\Big \downarrow \) 2009

\(\displaystyle -4 (1-\log (3)) \int \frac {1}{\left (x \log \left (5-\frac {10 x}{9}\right )-4\right )^2}dx+16 \int \frac {1}{\left (x \log \left (5-\frac {10 x}{9}\right )-4\right )^2}dx+16 (3+\log (3)) \int \frac {1}{x \left (x \log \left (5-\frac {10 x}{9}\right )-4\right )^2}dx-32 (3+\log (3)) \int \frac {1}{(2 x-9) \left (x \log \left (5-\frac {10 x}{9}\right )-4\right )^2}dx-32 (6-\log (3)) \int \frac {1}{(2 x-9) \left (x \log \left (5-\frac {10 x}{9}\right )-4\right )^2}dx-36 (1-\log (3)) \int \frac {1}{(2 x-9) \left (x \log \left (5-\frac {10 x}{9}\right )-4\right )^2}dx+432 \int \frac {1}{(2 x-9) \left (x \log \left (5-\frac {10 x}{9}\right )-4\right )^2}dx+4 (3+\log (3)) \int \frac {1}{x \left (x \log \left (5-\frac {10 x}{9}\right )-4\right )}dx-8 (3+\log (3)) \int \frac {1}{(2 x-9) \left (x \log \left (5-\frac {10 x}{9}\right )-4\right )}dx-8 (6-\log (3)) \int \frac {1}{(2 x-9) \left (x \log \left (5-\frac {10 x}{9}\right )-4\right )}dx+72 \int \frac {1}{(2 x-9) \left (x \log \left (5-\frac {10 x}{9}\right )-4\right )}dx-x\)

input

Int[(144 - 8*x + 8*x*Log[3] + (-108 - 48*x + 16*x^2 + (-36 + 8*x)*Log[3])* 
Log[(45 - 10*x)/9] + (9*x^2 - 2*x^3)*Log[(45 - 10*x)/9]^2)/(-144 + 32*x + 
(72*x - 16*x^2)*Log[(45 - 10*x)/9] + (-9*x^2 + 2*x^3)*Log[(45 - 10*x)/9]^2 
),x]
 

output

$Aborted
 

3.7.15.3.1 Defintions of rubi rules used

rule 6

Int[(u_.)*((v_.) + (a_.)*(Fx_) + (b_.)*(Fx_))^(p_.), x_Symbol] :> Int[u*(v 
+ (a + b)*Fx)^p, x] /; FreeQ[{a, b}, x] &&  !FreeQ[Fx, x]
 

rule 2009

Int[u_, x_Symbol] :> Simp[IntSum[u, x], x] /; SumQ[u]
 

rule 7292

Int[u_, x_Symbol] :> With[{v = NormalizeIntegrand[u, x]}, Int[v, x] /; v =! 
= u]
 

rule 7293

Int[u_, x_Symbol] :> With[{v = ExpandIntegrand[u, x]}, Int[v, x] /; SumQ[v] 
]
 

3.7.15.4 Maple [A] (verified)

Time = 0.86 (sec) , antiderivative size = 34, normalized size of antiderivative = 1.17

method	result	size
norman	\(\frac {4 x -x^{2} \ln \left (-\frac {10 x}{9}+5\right )-12-4 \ln \left (3\right )}{\ln \left (-\frac {10 x}{9}+5\right ) x -4}\)	\(34\)
risch	\(-x -\frac {12}{\ln \left (-\frac {10 x}{9}+5\right ) x -4}-\frac {4 \ln \left (3\right )}{\ln \left (-\frac {10 x}{9}+5\right ) x -4}\)	\(35\)
parallelrisch	\(-\frac {-96+4 x^{2} \ln \left (-\frac {10 x}{9}+5\right )+36 \ln \left (-\frac {10 x}{9}+5\right ) x +16 \ln \left (3\right )-16 x}{4 \left (\ln \left (-\frac {10 x}{9}+5\right ) x -4\right )}\)	\(44\)
derivativedivides	\(\frac {-\frac {405 \ln \left (-\frac {10 x}{9}+5\right )}{2}+\frac {81 \ln \left (-\frac {10 x}{9}+5\right ) \left (-\frac {10 x}{9}+5\right )^{2}}{10}-40 x +480}{9 \ln \left (-\frac {10 x}{9}+5\right ) \left (-\frac {10 x}{9}+5\right )-45 \ln \left (-\frac {10 x}{9}+5\right )+40}+\frac {40 \ln \left (3\right )}{9 \ln \left (-\frac {10 x}{9}+5\right ) \left (-\frac {10 x}{9}+5\right )-45 \ln \left (-\frac {10 x}{9}+5\right )+40}\)	\(86\)
default	\(\frac {-\frac {405 \ln \left (-\frac {10 x}{9}+5\right )}{2}+\frac {81 \ln \left (-\frac {10 x}{9}+5\right ) \left (-\frac {10 x}{9}+5\right )^{2}}{10}-40 x +480}{9 \ln \left (-\frac {10 x}{9}+5\right ) \left (-\frac {10 x}{9}+5\right )-45 \ln \left (-\frac {10 x}{9}+5\right )+40}+\frac {40 \ln \left (3\right )}{9 \ln \left (-\frac {10 x}{9}+5\right ) \left (-\frac {10 x}{9}+5\right )-45 \ln \left (-\frac {10 x}{9}+5\right )+40}\)	\(86\)

input

int(((-2*x^3+9*x^2)*ln(-10/9*x+5)^2+((8*x-36)*ln(3)+16*x^2-48*x-108)*ln(-1 
0/9*x+5)+8*x*ln(3)-8*x+144)/((2*x^3-9*x^2)*ln(-10/9*x+5)^2+(-16*x^2+72*x)* 
ln(-10/9*x+5)+32*x-144),x,method=_RETURNVERBOSE)
 

output

(4*x-x^2*ln(-10/9*x+5)-12-4*ln(3))/(ln(-10/9*x+5)*x-4)
 

3.7.15.5 Fricas [A] (verification not implemented)

Time = 0.29 (sec) , antiderivative size = 33, normalized size of antiderivative = 1.14 \[ \int \frac {144-8 x+8 x \log (3)+\left (-108-48 x+16 x^2+(-36+8 x) \log (3)\right ) \log \left (\frac {1}{9} (45-10 x)\right )+\left (9 x^2-2 x^3\right ) \log ^2\left (\frac {1}{9} (45-10 x)\right )}{-144+32 x+\left (72 x-16 x^2\right ) \log \left (\frac {1}{9} (45-10 x)\right )+\left (-9 x^2+2 x^3\right ) \log ^2\left (\frac {1}{9} (45-10 x)\right )} \, dx=-\frac {x^{2} \log \left (-\frac {10}{9} \, x + 5\right ) - 4 \, x + 4 \, \log \left (3\right ) + 12}{x \log \left (-\frac {10}{9} \, x + 5\right ) - 4} \]

input

integrate(((-2*x^3+9*x^2)*log(-10/9*x+5)^2+((8*x-36)*log(3)+16*x^2-48*x-10 
8)*log(-10/9*x+5)+8*x*log(3)-8*x+144)/((2*x^3-9*x^2)*log(-10/9*x+5)^2+(-16 
*x^2+72*x)*log(-10/9*x+5)+32*x-144),x, algorithm=\
 

output

-(x^2*log(-10/9*x + 5) - 4*x + 4*log(3) + 12)/(x*log(-10/9*x + 5) - 4)
 

3.7.15.6 Sympy [A] (verification not implemented)

Time = 0.09 (sec) , antiderivative size = 20, normalized size of antiderivative = 0.69 \[ \int \frac {144-8 x+8 x \log (3)+\left (-108-48 x+16 x^2+(-36+8 x) \log (3)\right ) \log \left (\frac {1}{9} (45-10 x)\right )+\left (9 x^2-2 x^3\right ) \log ^2\left (\frac {1}{9} (45-10 x)\right )}{-144+32 x+\left (72 x-16 x^2\right ) \log \left (\frac {1}{9} (45-10 x)\right )+\left (-9 x^2+2 x^3\right ) \log ^2\left (\frac {1}{9} (45-10 x)\right )} \, dx=- x + \frac {-12 - 4 \log {\left (3 \right )}}{x \log {\left (5 - \frac {10 x}{9} \right )} - 4} \]

input

integrate(((-2*x**3+9*x**2)*ln(-10/9*x+5)**2+((8*x-36)*ln(3)+16*x**2-48*x- 
108)*ln(-10/9*x+5)+8*x*ln(3)-8*x+144)/((2*x**3-9*x**2)*ln(-10/9*x+5)**2+(- 
16*x**2+72*x)*ln(-10/9*x+5)+32*x-144),x)
 

output

-x + (-12 - 4*log(3))/(x*log(5 - 10*x/9) - 4)
 

3.7.15.7 Maxima [C] (verification not implemented)

Result contains complex when optimal does not.

Time = 0.34 (sec) , antiderivative size = 59, normalized size of antiderivative = 2.03 \[ \int \frac {144-8 x+8 x \log (3)+\left (-108-48 x+16 x^2+(-36+8 x) \log (3)\right ) \log \left (\frac {1}{9} (45-10 x)\right )+\left (9 x^2-2 x^3\right ) \log ^2\left (\frac {1}{9} (45-10 x)\right )}{-144+32 x+\left (72 x-16 x^2\right ) \log \left (\frac {1}{9} (45-10 x)\right )+\left (-9 x^2+2 x^3\right ) \log ^2\left (\frac {1}{9} (45-10 x)\right )} \, dx=-\frac {{\left (i \, \pi + \log \left (5\right ) - 2 \, \log \left (3\right )\right )} x^{2} + x^{2} \log \left (2 \, x - 9\right ) - 4 \, x + 4 \, \log \left (3\right ) + 12}{{\left (i \, \pi + \log \left (5\right ) - 2 \, \log \left (3\right )\right )} x + x \log \left (2 \, x - 9\right ) - 4} \]

input

integrate(((-2*x^3+9*x^2)*log(-10/9*x+5)^2+((8*x-36)*log(3)+16*x^2-48*x-10 
8)*log(-10/9*x+5)+8*x*log(3)-8*x+144)/((2*x^3-9*x^2)*log(-10/9*x+5)^2+(-16 
*x^2+72*x)*log(-10/9*x+5)+32*x-144),x, algorithm=\
 

output

-((I*pi + log(5) - 2*log(3))*x^2 + x^2*log(2*x - 9) - 4*x + 4*log(3) + 12) 
/((I*pi + log(5) - 2*log(3))*x + x*log(2*x - 9) - 4)
 

3.7.15.8 Giac [A] (verification not implemented)

Time = 0.28 (sec) , antiderivative size = 22, normalized size of antiderivative = 0.76 \[ \int \frac {144-8 x+8 x \log (3)+\left (-108-48 x+16 x^2+(-36+8 x) \log (3)\right ) \log \left (\frac {1}{9} (45-10 x)\right )+\left (9 x^2-2 x^3\right ) \log ^2\left (\frac {1}{9} (45-10 x)\right )}{-144+32 x+\left (72 x-16 x^2\right ) \log \left (\frac {1}{9} (45-10 x)\right )+\left (-9 x^2+2 x^3\right ) \log ^2\left (\frac {1}{9} (45-10 x)\right )} \, dx=-x - \frac {4 \, {\left (\log \left (3\right ) + 3\right )}}{x \log \left (-\frac {10}{9} \, x + 5\right ) - 4} \]

input

integrate(((-2*x^3+9*x^2)*log(-10/9*x+5)^2+((8*x-36)*log(3)+16*x^2-48*x-10 
8)*log(-10/9*x+5)+8*x*log(3)-8*x+144)/((2*x^3-9*x^2)*log(-10/9*x+5)^2+(-16 
*x^2+72*x)*log(-10/9*x+5)+32*x-144),x, algorithm=\
 

output

-x - 4*(log(3) + 3)/(x*log(-10/9*x + 5) - 4)
 

3.7.15.9 Mupad [B] (verification not implemented)

Time = 11.94 (sec) , antiderivative size = 22, normalized size of antiderivative = 0.76 \[ \int \frac {144-8 x+8 x \log (3)+\left (-108-48 x+16 x^2+(-36+8 x) \log (3)\right ) \log \left (\frac {1}{9} (45-10 x)\right )+\left (9 x^2-2 x^3\right ) \log ^2\left (\frac {1}{9} (45-10 x)\right )}{-144+32 x+\left (72 x-16 x^2\right ) \log \left (\frac {1}{9} (45-10 x)\right )+\left (-9 x^2+2 x^3\right ) \log ^2\left (\frac {1}{9} (45-10 x)\right )} \, dx=-x-\frac {\ln \left (81\right )+12}{x\,\ln \left (5-\frac {10\,x}{9}\right )-4} \]

input

int((log(5 - (10*x)/9)^2*(9*x^2 - 2*x^3) - 8*x + 8*x*log(3) - log(5 - (10* 
x)/9)*(48*x - log(3)*(8*x - 36) - 16*x^2 + 108) + 144)/(32*x - log(5 - (10 
*x)/9)^2*(9*x^2 - 2*x^3) + log(5 - (10*x)/9)*(72*x - 16*x^2) - 144),x)
 

output

- x - (log(81) + 12)/(x*log(5 - (10*x)/9) - 4)