\(\int \frac {-576 x^2+192 x^3+(-144+96 x-144 x^2-624 x^3+240 x^4) \log (1+x^2)+(120 x^2+120 x^4) \log ^2(1+x^2)+(225 x^2+225 x^4) \log ^3(1+x^2)}{(144 x^2-96 x^3+160 x^4-96 x^5+16 x^6) \log ^3(1+x^2)} \, dx\) [2532]

Optimal result

Integrand size = 115, antiderivative size = 32 \[ \int \frac {-576 x^2+192 x^3+\left (-144+96 x-144 x^2-624 x^3+240 x^4\right ) \log \left (1+x^2\right )+\left (120 x^2+120 x^4\right ) \log ^2\left (1+x^2\right )+\left (225 x^2+225 x^4\right ) \log ^3\left (1+x^2\right )}{\left (144 x^2-96 x^3+160 x^4-96 x^5+16 x^6\right ) \log ^3\left (1+x^2\right )} \, dx=\frac {3 \left (x+\frac {1}{4} \left (x+\frac {4}{\log \left (1+x^2\right )}\right )\right )^2}{(3-x) x} \] Output:

3*(1/ln(x^2+1)+5/4*x)^2/(3-x)/x
                                                                                    
                                                                                    
 

Mathematica [A] (verified)

Time = 0.03 (sec) , antiderivative size = 45, normalized size of antiderivative = 1.41 \[ \int \frac {-576 x^2+192 x^3+\left (-144+96 x-144 x^2-624 x^3+240 x^4\right ) \log \left (1+x^2\right )+\left (120 x^2+120 x^4\right ) \log ^2\left (1+x^2\right )+\left (225 x^2+225 x^4\right ) \log ^3\left (1+x^2\right )}{\left (144 x^2-96 x^3+160 x^4-96 x^5+16 x^6\right ) \log ^3\left (1+x^2\right )} \, dx=\frac {3}{16} \left (-\frac {75}{-3+x}-\frac {16}{(-3+x) x \log ^2\left (1+x^2\right )}-\frac {40}{(-3+x) \log \left (1+x^2\right )}\right ) \] Input:

Integrate[(-576*x^2 + 192*x^3 + (-144 + 96*x - 144*x^2 - 624*x^3 + 240*x^4 
)*Log[1 + x^2] + (120*x^2 + 120*x^4)*Log[1 + x^2]^2 + (225*x^2 + 225*x^4)* 
Log[1 + x^2]^3)/((144*x^2 - 96*x^3 + 160*x^4 - 96*x^5 + 16*x^6)*Log[1 + x^ 
2]^3),x]
 

Output:

(3*(-75/(-3 + x) - 16/((-3 + x)*x*Log[1 + x^2]^2) - 40/((-3 + x)*Log[1 + x 
^2])))/16
 

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.

\(\displaystyle \int \frac {192 x^3-576 x^2+\left (225 x^4+225 x^2\right ) \log ^3\left (x^2+1\right )+\left (120 x^4+120 x^2\right ) \log ^2\left (x^2+1\right )+\left (240 x^4-624 x^3-144 x^2+96 x-144\right ) \log \left (x^2+1\right )}{\left (16 x^6-96 x^5+160 x^4-96 x^3+144 x^2\right ) \log ^3\left (x^2+1\right )} \, dx\)

\(\Big \downarrow \) 2026

\(\displaystyle \int \frac {192 x^3-576 x^2+\left (225 x^4+225 x^2\right ) \log ^3\left (x^2+1\right )+\left (120 x^4+120 x^2\right ) \log ^2\left (x^2+1\right )+\left (240 x^4-624 x^3-144 x^2+96 x-144\right ) \log \left (x^2+1\right )}{x^2 \left (16 x^4-96 x^3+160 x^2-96 x+144\right ) \log ^3\left (x^2+1\right )}dx\)

\(\Big \downarrow \) 2463

\(\displaystyle \int \left (\frac {(3 x+4) \left (192 x^3-576 x^2+\left (225 x^4+225 x^2\right ) \log ^3\left (x^2+1\right )+\left (120 x^4+120 x^2\right ) \log ^2\left (x^2+1\right )+\left (240 x^4-624 x^3-144 x^2+96 x-144\right ) \log \left (x^2+1\right )\right )}{800 x^2 \left (x^2+1\right ) \log ^3\left (x^2+1\right )}-\frac {3 \left (192 x^3-576 x^2+\left (225 x^4+225 x^2\right ) \log ^3\left (x^2+1\right )+\left (120 x^4+120 x^2\right ) \log ^2\left (x^2+1\right )+\left (240 x^4-624 x^3-144 x^2+96 x-144\right ) \log \left (x^2+1\right )\right )}{800 (x-3) x^2 \log ^3\left (x^2+1\right )}+\frac {192 x^3-576 x^2+\left (225 x^4+225 x^2\right ) \log ^3\left (x^2+1\right )+\left (120 x^4+120 x^2\right ) \log ^2\left (x^2+1\right )+\left (240 x^4-624 x^3-144 x^2+96 x-144\right ) \log \left (x^2+1\right )}{160 (x-3)^2 x^2 \log ^3\left (x^2+1\right )}\right )dx\)

\(\Big \downarrow \) 2009

\(\displaystyle \frac {6}{5} \int \frac {1}{(x-3) \log ^3\left (x^2+1\right )}dx-\frac {18}{5} \int \frac {1}{\left (x^2+1\right ) \log ^3\left (x^2+1\right )}dx+\int \frac {1}{(x-3)^2 \log ^2\left (x^2+1\right )}dx+\frac {9}{2} \int \frac {1}{(x-3) \log ^2\left (x^2+1\right )}dx-\int \frac {1}{x^2 \log ^2\left (x^2+1\right )}dx+\frac {3}{2} \int \frac {1}{\left (x^2+1\right ) \log ^2\left (x^2+1\right )}dx-\frac {3}{5} \int \frac {1}{\log \left (x^2+1\right )}dx+\frac {15}{2} \int \frac {1}{(x-3)^2 \log \left (x^2+1\right )}dx+\frac {3}{20} \int \frac {3 x+4}{\log \left (x^2+1\right )}dx-\frac {9 \operatorname {LogIntegral}\left (x^2+1\right )}{40}-\frac {27 x^2}{64}+\frac {3}{10 \log ^2\left (x^2+1\right )}+\frac {9}{4 \log \left (x^2+1\right )}-\frac {9 x}{8}+\frac {3}{64} (3 x+4)^2+\frac {225}{16 (3-x)}\)

Input:

Int[(-576*x^2 + 192*x^3 + (-144 + 96*x - 144*x^2 - 624*x^3 + 240*x^4)*Log[ 
1 + x^2] + (120*x^2 + 120*x^4)*Log[1 + x^2]^2 + (225*x^2 + 225*x^4)*Log[1 
+ x^2]^3)/((144*x^2 - 96*x^3 + 160*x^4 - 96*x^5 + 16*x^6)*Log[1 + x^2]^3), 
x]
 

Output:

$Aborted
 

Maple [A] (verified)

Time = 1.76 (sec) , antiderivative size = 38, normalized size of antiderivative = 1.19

method	result	size
risch	\(-\frac {225}{16 \left (-3+x \right )}-\frac {3 \left (5 x \ln \left (x^{2}+1\right )+2\right )}{2 x \left (-3+x \right ) \ln \left (x^{2}+1\right )^{2}}\)	\(38\)
parallelrisch	\(\frac {-48-225 x \ln \left (x^{2}+1\right )^{2}-120 x \ln \left (x^{2}+1\right )}{16 \ln \left (x^{2}+1\right )^{2} x \left (-3+x \right )}\)	\(41\)

Input:

int(((225*x^4+225*x^2)*ln(x^2+1)^3+(120*x^4+120*x^2)*ln(x^2+1)^2+(240*x^4- 
624*x^3-144*x^2+96*x-144)*ln(x^2+1)+192*x^3-576*x^2)/(16*x^6-96*x^5+160*x^ 
4-96*x^3+144*x^2)/ln(x^2+1)^3,x,method=_RETURNVERBOSE)
 

Output:

-225/16/(-3+x)-3/2/x*(5*x*ln(x^2+1)+2)/(-3+x)/ln(x^2+1)^2
 

Fricas [A] (verification not implemented)

Time = 0.09 (sec) , antiderivative size = 41, normalized size of antiderivative = 1.28 \[ \int \frac {-576 x^2+192 x^3+\left (-144+96 x-144 x^2-624 x^3+240 x^4\right ) \log \left (1+x^2\right )+\left (120 x^2+120 x^4\right ) \log ^2\left (1+x^2\right )+\left (225 x^2+225 x^4\right ) \log ^3\left (1+x^2\right )}{\left (144 x^2-96 x^3+160 x^4-96 x^5+16 x^6\right ) \log ^3\left (1+x^2\right )} \, dx=-\frac {3 \, {\left (75 \, x \log \left (x^{2} + 1\right )^{2} + 40 \, x \log \left (x^{2} + 1\right ) + 16\right )}}{16 \, {\left (x^{2} - 3 \, x\right )} \log \left (x^{2} + 1\right )^{2}} \] Input:

integrate(((225*x^4+225*x^2)*log(x^2+1)^3+(120*x^4+120*x^2)*log(x^2+1)^2+( 
240*x^4-624*x^3-144*x^2+96*x-144)*log(x^2+1)+192*x^3-576*x^2)/(16*x^6-96*x 
^5+160*x^4-96*x^3+144*x^2)/log(x^2+1)^3,x, algorithm="fricas")
 

Output:

-3/16*(75*x*log(x^2 + 1)^2 + 40*x*log(x^2 + 1) + 16)/((x^2 - 3*x)*log(x^2 
+ 1)^2)
 

Sympy [A] (verification not implemented)

Time = 0.08 (sec) , antiderivative size = 36, normalized size of antiderivative = 1.12 \[ \int \frac {-576 x^2+192 x^3+\left (-144+96 x-144 x^2-624 x^3+240 x^4\right ) \log \left (1+x^2\right )+\left (120 x^2+120 x^4\right ) \log ^2\left (1+x^2\right )+\left (225 x^2+225 x^4\right ) \log ^3\left (1+x^2\right )}{\left (144 x^2-96 x^3+160 x^4-96 x^5+16 x^6\right ) \log ^3\left (1+x^2\right )} \, dx=\frac {- 15 x \log {\left (x^{2} + 1 \right )} - 6}{\left (2 x^{2} - 6 x\right ) \log {\left (x^{2} + 1 \right )}^{2}} - \frac {225}{16 x - 48} \] Input:

integrate(((225*x**4+225*x**2)*ln(x**2+1)**3+(120*x**4+120*x**2)*ln(x**2+1 
)**2+(240*x**4-624*x**3-144*x**2+96*x-144)*ln(x**2+1)+192*x**3-576*x**2)/( 
16*x**6-96*x**5+160*x**4-96*x**3+144*x**2)/ln(x**2+1)**3,x)
 

Output:

(-15*x*log(x**2 + 1) - 6)/((2*x**2 - 6*x)*log(x**2 + 1)**2) - 225/(16*x - 
48)
 

Maxima [A] (verification not implemented)

Time = 0.15 (sec) , antiderivative size = 41, normalized size of antiderivative = 1.28 \[ \int \frac {-576 x^2+192 x^3+\left (-144+96 x-144 x^2-624 x^3+240 x^4\right ) \log \left (1+x^2\right )+\left (120 x^2+120 x^4\right ) \log ^2\left (1+x^2\right )+\left (225 x^2+225 x^4\right ) \log ^3\left (1+x^2\right )}{\left (144 x^2-96 x^3+160 x^4-96 x^5+16 x^6\right ) \log ^3\left (1+x^2\right )} \, dx=-\frac {3 \, {\left (75 \, x \log \left (x^{2} + 1\right )^{2} + 40 \, x \log \left (x^{2} + 1\right ) + 16\right )}}{16 \, {\left (x^{2} - 3 \, x\right )} \log \left (x^{2} + 1\right )^{2}} \] Input:

integrate(((225*x^4+225*x^2)*log(x^2+1)^3+(120*x^4+120*x^2)*log(x^2+1)^2+( 
240*x^4-624*x^3-144*x^2+96*x-144)*log(x^2+1)+192*x^3-576*x^2)/(16*x^6-96*x 
^5+160*x^4-96*x^3+144*x^2)/log(x^2+1)^3,x, algorithm="maxima")
 

Output:

-3/16*(75*x*log(x^2 + 1)^2 + 40*x*log(x^2 + 1) + 16)/((x^2 - 3*x)*log(x^2 
+ 1)^2)
 

Giac [A] (verification not implemented)

Time = 0.16 (sec) , antiderivative size = 47, normalized size of antiderivative = 1.47 \[ \int \frac {-576 x^2+192 x^3+\left (-144+96 x-144 x^2-624 x^3+240 x^4\right ) \log \left (1+x^2\right )+\left (120 x^2+120 x^4\right ) \log ^2\left (1+x^2\right )+\left (225 x^2+225 x^4\right ) \log ^3\left (1+x^2\right )}{\left (144 x^2-96 x^3+160 x^4-96 x^5+16 x^6\right ) \log ^3\left (1+x^2\right )} \, dx=-\frac {3 \, {\left (5 \, x \log \left (x^{2} + 1\right ) + 2\right )}}{2 \, {\left (x^{2} \log \left (x^{2} + 1\right )^{2} - 3 \, x \log \left (x^{2} + 1\right )^{2}\right )}} - \frac {225}{16 \, {\left (x - 3\right )}} \] Input:

integrate(((225*x^4+225*x^2)*log(x^2+1)^3+(120*x^4+120*x^2)*log(x^2+1)^2+( 
240*x^4-624*x^3-144*x^2+96*x-144)*log(x^2+1)+192*x^3-576*x^2)/(16*x^6-96*x 
^5+160*x^4-96*x^3+144*x^2)/log(x^2+1)^3,x, algorithm="giac")
 

Output:

-3/2*(5*x*log(x^2 + 1) + 2)/(x^2*log(x^2 + 1)^2 - 3*x*log(x^2 + 1)^2) - 22 
5/16/(x - 3)
 

Mupad [B] (verification not implemented)

Time = 3.61 (sec) , antiderivative size = 313, normalized size of antiderivative = 9.78 \[ \int \frac {-576 x^2+192 x^3+\left (-144+96 x-144 x^2-624 x^3+240 x^4\right ) \log \left (1+x^2\right )+\left (120 x^2+120 x^4\right ) \log ^2\left (1+x^2\right )+\left (225 x^2+225 x^4\right ) \log ^3\left (1+x^2\right )}{\left (144 x^2-96 x^3+160 x^4-96 x^5+16 x^6\right ) \log ^3\left (1+x^2\right )} \, dx=\frac {\frac {165\,x^7}{16}-\frac {573\,x^6}{8}+\frac {1911\,x^5}{16}+\frac {153\,x^4}{8}-\frac {27\,x^3}{2}+\frac {33\,x^2}{2}-\frac {81\,x}{8}+\frac {81}{8}}{-x^8+9\,x^7-27\,x^6+27\,x^5}-\frac {\frac {3\,\left (5\,x^4-17\,x^3+3\,x^2-2\,x+3\right )}{4\,x^3\,{\left (x-3\right )}^2}+\frac {3\,\ln \left (x^2+1\right )\,\left (x^2+1\right )\,\left (5\,x^5-19\,x^4+9\,x^3-17\,x^2+27\,x-27\right )}{8\,x^5\,{\left (x-3\right )}^3}-\frac {15\,{\ln \left (x^2+1\right )}^2\,\left (x^2+1\right )\,\left (x^3+3\,x^2+3\,x-3\right )}{16\,x^3\,{\left (x-3\right )}^3}}{\ln \left (x^2+1\right )}-\frac {\frac {3}{x\,\left (x-3\right )}+\frac {15\,{\ln \left (x^2+1\right )}^2\,\left (x^2+1\right )}{8\,x\,{\left (x-3\right )}^2}-\frac {3\,\ln \left (x^2+1\right )\,\left (-5\,x^4+13\,x^3+3\,x^2-2\,x+3\right )}{4\,x^3\,{\left (x-3\right )}^2}}{{\ln \left (x^2+1\right )}^2}+\frac {\ln \left (x^2+1\right )\,\left (\frac {15\,x^5}{16}+\frac {45\,x^4}{16}+\frac {15\,x^3}{4}+\frac {45\,x}{16}-\frac {45}{16}\right )}{-x^6+9\,x^5-27\,x^4+27\,x^3} \] Input:

int((log(x^2 + 1)^2*(120*x^2 + 120*x^4) + log(x^2 + 1)^3*(225*x^2 + 225*x^ 
4) - 576*x^2 + 192*x^3 - log(x^2 + 1)*(144*x^2 - 96*x + 624*x^3 - 240*x^4 
+ 144))/(log(x^2 + 1)^3*(144*x^2 - 96*x^3 + 160*x^4 - 96*x^5 + 16*x^6)),x)
 

Output:

((33*x^2)/2 - (81*x)/8 - (27*x^3)/2 + (153*x^4)/8 + (1911*x^5)/16 - (573*x 
^6)/8 + (165*x^7)/16 + 81/8)/(27*x^5 - 27*x^6 + 9*x^7 - x^8) - ((3*(3*x^2 
- 2*x - 17*x^3 + 5*x^4 + 3))/(4*x^3*(x - 3)^2) + (3*log(x^2 + 1)*(x^2 + 1) 
*(27*x - 17*x^2 + 9*x^3 - 19*x^4 + 5*x^5 - 27))/(8*x^5*(x - 3)^3) - (15*lo 
g(x^2 + 1)^2*(x^2 + 1)*(3*x + 3*x^2 + x^3 - 3))/(16*x^3*(x - 3)^3))/log(x^ 
2 + 1) - (3/(x*(x - 3)) + (15*log(x^2 + 1)^2*(x^2 + 1))/(8*x*(x - 3)^2) - 
(3*log(x^2 + 1)*(3*x^2 - 2*x + 13*x^3 - 5*x^4 + 3))/(4*x^3*(x - 3)^2))/log 
(x^2 + 1)^2 + (log(x^2 + 1)*((45*x)/16 + (15*x^3)/4 + (45*x^4)/16 + (15*x^ 
5)/16 - 45/16))/(27*x^3 - 27*x^4 + 9*x^5 - x^6)
 

Reduce [B] (verification not implemented)

Time = 0.15 (sec) , antiderivative size = 42, normalized size of antiderivative = 1.31 \[ \int \frac {-576 x^2+192 x^3+\left (-144+96 x-144 x^2-624 x^3+240 x^4\right ) \log \left (1+x^2\right )+\left (120 x^2+120 x^4\right ) \log ^2\left (1+x^2\right )+\left (225 x^2+225 x^4\right ) \log ^3\left (1+x^2\right )}{\left (144 x^2-96 x^3+160 x^4-96 x^5+16 x^6\right ) \log ^3\left (1+x^2\right )} \, dx=\frac {-\frac {75 \mathrm {log}\left (x^{2}+1\right )^{2} x^{2}}{16}-\frac {15 \,\mathrm {log}\left (x^{2}+1\right ) x}{2}-3}{\mathrm {log}\left (x^{2}+1\right )^{2} x \left (x -3\right )} \] Input:

int(((225*x^4+225*x^2)*log(x^2+1)^3+(120*x^4+120*x^2)*log(x^2+1)^2+(240*x^ 
4-624*x^3-144*x^2+96*x-144)*log(x^2+1)+192*x^3-576*x^2)/(16*x^6-96*x^5+160 
*x^4-96*x^3+144*x^2)/log(x^2+1)^3,x)
 

Output:

(3*( - 25*log(x**2 + 1)**2*x**2 - 40*log(x**2 + 1)*x - 16))/(16*log(x**2 + 
 1)**2*x*(x - 3))