\(\int \frac {-x+4 e^{60+4 x} x+16 x^2+64 x^4+e^{45+3 x} (8 x+24 x^2)+(4+16 x^2) \log (2)+e^{30+2 x} (4+48 x^2+48 x^3+4 x \log (2))+e^{15+x} (16 x+96 x^3+32 x^4+(8 x+8 x^2) \log (2))+(4+4 e^{30+2 x} x+16 x^2+e^{15+x} (8 x+8 x^2)) \log (3 x^2)}{e^{60+4 x} x-x^2+8 e^{45+3 x} x^2+16 x^5+8 x^3 \log (2)+x \log ^2(2)+e^{30+2 x} (24 x^3+2 x \log (2))+e^{15+x} (32 x^4+8 x^2 \log (2))+(2 e^{30+2 x} x+8 e^{15+x} x^2+8 x^3+2 x \log (2)) \log (3 x^2)+x \log ^2(3 x^2)} \, dx\) [138]

Optimal result

Integrand size = 281, antiderivative size = 27 \[ \int \frac {-x+4 e^{60+4 x} x+16 x^2+64 x^4+e^{45+3 x} \left (8 x+24 x^2\right )+\left (4+16 x^2\right ) \log (2)+e^{30+2 x} \left (4+48 x^2+48 x^3+4 x \log (2)\right )+e^{15+x} \left (16 x+96 x^3+32 x^4+\left (8 x+8 x^2\right ) \log (2)\right )+\left (4+4 e^{30+2 x} x+16 x^2+e^{15+x} \left (8 x+8 x^2\right )\right ) \log \left (3 x^2\right )}{e^{60+4 x} x-x^2+8 e^{45+3 x} x^2+16 x^5+8 x^3 \log (2)+x \log ^2(2)+e^{30+2 x} \left (24 x^3+2 x \log (2)\right )+e^{15+x} \left (32 x^4+8 x^2 \log (2)\right )+\left (2 e^{30+2 x} x+8 e^{15+x} x^2+8 x^3+2 x \log (2)\right ) \log \left (3 x^2\right )+x \log ^2\left (3 x^2\right )} \, dx=\log \left (-x+\left (\left (e^{15+x}+2 x\right )^2+\log (2)+\log \left (3 x^2\right )\right )^2\right ) \] Output:

ln((ln(3*x^2)+(2*x+exp(x+15))^2+ln(2))^2-x)
                                                                                    
                                                                                    
 

Mathematica [F]

\[ \int \frac {-x+4 e^{60+4 x} x+16 x^2+64 x^4+e^{45+3 x} \left (8 x+24 x^2\right )+\left (4+16 x^2\right ) \log (2)+e^{30+2 x} \left (4+48 x^2+48 x^3+4 x \log (2)\right )+e^{15+x} \left (16 x+96 x^3+32 x^4+\left (8 x+8 x^2\right ) \log (2)\right )+\left (4+4 e^{30+2 x} x+16 x^2+e^{15+x} \left (8 x+8 x^2\right )\right ) \log \left (3 x^2\right )}{e^{60+4 x} x-x^2+8 e^{45+3 x} x^2+16 x^5+8 x^3 \log (2)+x \log ^2(2)+e^{30+2 x} \left (24 x^3+2 x \log (2)\right )+e^{15+x} \left (32 x^4+8 x^2 \log (2)\right )+\left (2 e^{30+2 x} x+8 e^{15+x} x^2+8 x^3+2 x \log (2)\right ) \log \left (3 x^2\right )+x \log ^2\left (3 x^2\right )} \, dx=\int \frac {-x+4 e^{60+4 x} x+16 x^2+64 x^4+e^{45+3 x} \left (8 x+24 x^2\right )+\left (4+16 x^2\right ) \log (2)+e^{30+2 x} \left (4+48 x^2+48 x^3+4 x \log (2)\right )+e^{15+x} \left (16 x+96 x^3+32 x^4+\left (8 x+8 x^2\right ) \log (2)\right )+\left (4+4 e^{30+2 x} x+16 x^2+e^{15+x} \left (8 x+8 x^2\right )\right ) \log \left (3 x^2\right )}{e^{60+4 x} x-x^2+8 e^{45+3 x} x^2+16 x^5+8 x^3 \log (2)+x \log ^2(2)+e^{30+2 x} \left (24 x^3+2 x \log (2)\right )+e^{15+x} \left (32 x^4+8 x^2 \log (2)\right )+\left (2 e^{30+2 x} x+8 e^{15+x} x^2+8 x^3+2 x \log (2)\right ) \log \left (3 x^2\right )+x \log ^2\left (3 x^2\right )} \, dx \] Input:

Integrate[(-x + 4*E^(60 + 4*x)*x + 16*x^2 + 64*x^4 + E^(45 + 3*x)*(8*x + 2 
4*x^2) + (4 + 16*x^2)*Log[2] + E^(30 + 2*x)*(4 + 48*x^2 + 48*x^3 + 4*x*Log 
[2]) + E^(15 + x)*(16*x + 96*x^3 + 32*x^4 + (8*x + 8*x^2)*Log[2]) + (4 + 4 
*E^(30 + 2*x)*x + 16*x^2 + E^(15 + x)*(8*x + 8*x^2))*Log[3*x^2])/(E^(60 + 
4*x)*x - x^2 + 8*E^(45 + 3*x)*x^2 + 16*x^5 + 8*x^3*Log[2] + x*Log[2]^2 + E 
^(30 + 2*x)*(24*x^3 + 2*x*Log[2]) + E^(15 + x)*(32*x^4 + 8*x^2*Log[2]) + ( 
2*E^(30 + 2*x)*x + 8*E^(15 + x)*x^2 + 8*x^3 + 2*x*Log[2])*Log[3*x^2] + x*L 
og[3*x^2]^2),x]
 

Output:

Integrate[(-x + 4*E^(60 + 4*x)*x + 16*x^2 + 64*x^4 + E^(45 + 3*x)*(8*x + 2 
4*x^2) + (4 + 16*x^2)*Log[2] + E^(30 + 2*x)*(4 + 48*x^2 + 48*x^3 + 4*x*Log 
[2]) + E^(15 + x)*(16*x + 96*x^3 + 32*x^4 + (8*x + 8*x^2)*Log[2]) + (4 + 4 
*E^(30 + 2*x)*x + 16*x^2 + E^(15 + x)*(8*x + 8*x^2))*Log[3*x^2])/(E^(60 + 
4*x)*x - x^2 + 8*E^(45 + 3*x)*x^2 + 16*x^5 + 8*x^3*Log[2] + x*Log[2]^2 + E 
^(30 + 2*x)*(24*x^3 + 2*x*Log[2]) + E^(15 + x)*(32*x^4 + 8*x^2*Log[2]) + ( 
2*E^(30 + 2*x)*x + 8*E^(15 + x)*x^2 + 8*x^3 + 2*x*Log[2])*Log[3*x^2] + x*L 
og[3*x^2]^2), x]
 

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.

\(\displaystyle \int \frac {64 x^4+16 x^2+e^{3 x+45} \left (24 x^2+8 x\right )+\left (16 x^2+e^{x+15} \left (8 x^2+8 x\right )+4 e^{2 x+30} x+4\right ) \log \left (3 x^2\right )+\left (16 x^2+4\right ) \log (2)+e^{2 x+30} \left (48 x^3+48 x^2+4 x \log (2)+4\right )+e^{x+15} \left (32 x^4+96 x^3+\left (8 x^2+8 x\right ) \log (2)+16 x\right )+4 e^{4 x+60} x-x}{16 x^5+8 x^3 \log (2)+e^{2 x+30} \left (24 x^3+2 x \log (2)\right )+8 e^{3 x+45} x^2-x^2+x \log ^2\left (3 x^2\right )+e^{x+15} \left (32 x^4+8 x^2 \log (2)\right )+\left (8 x^3+8 e^{x+15} x^2+2 e^{2 x+30} x+2 x \log (2)\right ) \log \left (3 x^2\right )+e^{4 x+60} x+x \log ^2(2)} \, dx\)

\(\Big \downarrow \) 7293

\(\displaystyle \int \left (\frac {-64 x^5-96 e^{x+15} x^4+64 x^4+96 e^{x+15} x^3-48 e^{2 x+30} x^3-32 x^3 \log (2)+48 e^{2 x+30} x^2-8 e^{3 x+45} x^2-4 x \log ^2\left (3 x^2\right )-24 e^{x+15} x^2 \log \left (3 x^2\right )+16 x^2 \log \left (3 x^2\right )-8 e^{x+15} x^2 \log (8)+20 x^2 \left (1+\frac {4 \log (2)}{5}\right )+8 e^{x+15} x \log \left (3 x^2\right )-4 e^{2 x+30} x \log \left (3 x^2\right )-4 x \log (4) \log \left (3 x^2\right )+4 \log \left (3 x^2\right )-32 x^3 \log \left (3 x^2\right )+8 e^{3 x+45} x+4 e^{2 x+30}-x \left (1+4 \log ^2(2)\right )-4 e^{2 x+30} x \log (2)+16 e^{x+15} x \left (1+\frac {\log (2)}{2}\right )+\log (16)}{x \left (16 x^4+32 e^{x+15} x^3+24 e^{2 x+30} x^2+\log ^2\left (3 x^2\right )+8 x^2 \log \left (3 x^2\right )+8 x^2 \log (2)+8 e^{x+15} x \log \left (3 x^2\right )+2 e^{2 x+30} \log \left (3 x^2\right )+2 \log (2) \log \left (3 x^2\right )+8 e^{3 x+45} x-x+e^{4 x+60}+8 e^{x+15} x \log (2)+e^{2 x+30} \log (4)+\log ^2(2)\right )}+4\right )dx\)

\(\Big \downarrow \) 7299

\(\displaystyle \int \left (\frac {-64 x^5-96 e^{x+15} x^4+64 x^4+96 e^{x+15} x^3-48 e^{2 x+30} x^3-32 x^3 \log (2)+48 e^{2 x+30} x^2-8 e^{3 x+45} x^2-4 x \log ^2\left (3 x^2\right )-24 e^{x+15} x^2 \log \left (3 x^2\right )+16 x^2 \log \left (3 x^2\right )-8 e^{x+15} x^2 \log (8)+20 x^2 \left (1+\frac {4 \log (2)}{5}\right )+8 e^{x+15} x \log \left (3 x^2\right )-4 e^{2 x+30} x \log \left (3 x^2\right )-4 x \log (4) \log \left (3 x^2\right )+4 \log \left (3 x^2\right )-32 x^3 \log \left (3 x^2\right )+8 e^{3 x+45} x+4 e^{2 x+30}-x \left (1+4 \log ^2(2)\right )-4 e^{2 x+30} x \log (2)+16 e^{x+15} x \left (1+\frac {\log (2)}{2}\right )+\log (16)}{x \left (16 x^4+32 e^{x+15} x^3+24 e^{2 x+30} x^2+\log ^2\left (3 x^2\right )+8 x^2 \log \left (3 x^2\right )+8 x^2 \log (2)+8 e^{x+15} x \log \left (3 x^2\right )+2 e^{2 x+30} \log \left (3 x^2\right )+2 \log (2) \log \left (3 x^2\right )+8 e^{3 x+45} x-x+e^{4 x+60}+8 e^{x+15} x \log (2)+e^{2 x+30} \log (4)+\log ^2(2)\right )}+4\right )dx\)

Input:

Int[(-x + 4*E^(60 + 4*x)*x + 16*x^2 + 64*x^4 + E^(45 + 3*x)*(8*x + 24*x^2) 
 + (4 + 16*x^2)*Log[2] + E^(30 + 2*x)*(4 + 48*x^2 + 48*x^3 + 4*x*Log[2]) + 
 E^(15 + x)*(16*x + 96*x^3 + 32*x^4 + (8*x + 8*x^2)*Log[2]) + (4 + 4*E^(30 
 + 2*x)*x + 16*x^2 + E^(15 + x)*(8*x + 8*x^2))*Log[3*x^2])/(E^(60 + 4*x)*x 
 - x^2 + 8*E^(45 + 3*x)*x^2 + 16*x^5 + 8*x^3*Log[2] + x*Log[2]^2 + E^(30 + 
 2*x)*(24*x^3 + 2*x*Log[2]) + E^(15 + x)*(32*x^4 + 8*x^2*Log[2]) + (2*E^(3 
0 + 2*x)*x + 8*E^(15 + x)*x^2 + 8*x^3 + 2*x*Log[2])*Log[3*x^2] + x*Log[3*x 
^2]^2),x]
 

Output:

$Aborted
 

Maple [C] (warning: unable to verify)

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

Time = 0.20 (sec) , antiderivative size = 592, normalized size of antiderivative = 21.93

\[\ln \left (-\frac {x}{4}+\frac {\ln \left (2\right ) \ln \left (3\right )}{2}+2 x^{2} \ln \left (3\right )+8 \,{\mathrm e}^{x +15} x^{3}+2 x^{2} \ln \left (2\right )+\frac {\ln \left (3\right )^{2}}{4}+\frac {\ln \left (2\right )^{2}}{4}+\ln \left (x \right )^{2}+4 x^{4}-\frac {i \pi \ln \left (2\right ) \operatorname {csgn}\left (i x \right )^{2} \operatorname {csgn}\left (i x^{2}\right )}{4}-i \pi \,x^{2} \operatorname {csgn}\left (i x \right )^{2} \operatorname {csgn}\left (i x^{2}\right )+2 i \pi \,x^{2} \operatorname {csgn}\left (i x \right ) \operatorname {csgn}\left (i x^{2}\right )^{2}+\frac {i \pi \,\operatorname {csgn}\left (i x \right ) \operatorname {csgn}\left (i x^{2}\right )^{2} {\mathrm e}^{2 x +30}}{2}+\frac {i \pi \ln \left (2\right ) \operatorname {csgn}\left (i x \right ) \operatorname {csgn}\left (i x^{2}\right )^{2}}{2}-\frac {i \pi \operatorname {csgn}\left (i x \right )^{2} \operatorname {csgn}\left (i x^{2}\right ) {\mathrm e}^{2 x +30}}{4}+\frac {\ln \left (3\right ) {\mathrm e}^{2 x +30}}{2}+\frac {{\mathrm e}^{2 x +30} \ln \left (2\right )}{2}+6 \,{\mathrm e}^{2 x +30} x^{2}+2 \,{\mathrm e}^{3 x +45} x +2 \,{\mathrm e}^{x +15} \ln \left (2\right ) x +\frac {{\mathrm e}^{4 x +60}}{4}-i \pi \,x^{2} \operatorname {csgn}\left (i x^{2}\right )^{3}-\frac {i \pi \operatorname {csgn}\left (i x^{2}\right )^{3} {\mathrm e}^{2 x +30}}{4}-\frac {i \ln \left (3\right ) \pi \operatorname {csgn}\left (i x^{2}\right )^{3}}{4}-\frac {i \pi \ln \left (2\right ) \operatorname {csgn}\left (i x^{2}\right )^{3}}{4}+2 \ln \left (3\right ) {\mathrm e}^{x +15} x -i \pi x \operatorname {csgn}\left (i x^{2}\right )^{3} {\mathrm e}^{x +15}-\frac {i \ln \left (3\right ) \pi \operatorname {csgn}\left (i x \right )^{2} \operatorname {csgn}\left (i x^{2}\right )}{4}+\frac {i \ln \left (3\right ) \pi \,\operatorname {csgn}\left (i x \right ) \operatorname {csgn}\left (i x^{2}\right )^{2}}{2}+\frac {\pi ^{2} \operatorname {csgn}\left (i x^{2}\right )^{5} \operatorname {csgn}\left (i x \right )}{4}-\frac {3 \pi ^{2} \operatorname {csgn}\left (i x^{2}\right )^{4} \operatorname {csgn}\left (i x \right )^{2}}{8}+\frac {\pi ^{2} \operatorname {csgn}\left (i x^{2}\right )^{3} \operatorname {csgn}\left (i x \right )^{3}}{4}-\frac {\pi ^{2} \operatorname {csgn}\left (i x^{2}\right )^{2} \operatorname {csgn}\left (i x \right )^{4}}{16}-\frac {\pi ^{2} \operatorname {csgn}\left (i x^{2}\right )^{6}}{16}+\left (\ln \left (3\right )-\frac {i \pi \operatorname {csgn}\left (i x^{2}\right )^{3}}{2}-\frac {i \pi \operatorname {csgn}\left (i x \right )^{2} \operatorname {csgn}\left (i x^{2}\right )}{2}+i \pi \,\operatorname {csgn}\left (i x \right ) \operatorname {csgn}\left (i x^{2}\right )^{2}+{\mathrm e}^{2 x +30}+\ln \left (2\right )+4 x \,{\mathrm e}^{x +15}+4 x^{2}\right ) \ln \left (x \right )-i \pi x \operatorname {csgn}\left (i x \right )^{2} \operatorname {csgn}\left (i x^{2}\right ) {\mathrm e}^{x +15}+2 i \pi x \,\operatorname {csgn}\left (i x \right ) \operatorname {csgn}\left (i x^{2}\right )^{2} {\mathrm e}^{x +15}\right )\]

Input:

int(((4*x*exp(x+15)^2+(8*x^2+8*x)*exp(x+15)+16*x^2+4)*ln(3*x^2)+4*x*exp(x+ 
15)^4+(24*x^2+8*x)*exp(x+15)^3+(4*x*ln(2)+48*x^3+48*x^2+4)*exp(x+15)^2+((8 
*x^2+8*x)*ln(2)+32*x^4+96*x^3+16*x)*exp(x+15)+(16*x^2+4)*ln(2)+64*x^4+16*x 
^2-x)/(x*ln(3*x^2)^2+(2*x*exp(x+15)^2+8*x^2*exp(x+15)+2*x*ln(2)+8*x^3)*ln( 
3*x^2)+x*exp(x+15)^4+8*x^2*exp(x+15)^3+(2*x*ln(2)+24*x^3)*exp(x+15)^2+(8*x 
^2*ln(2)+32*x^4)*exp(x+15)+x*ln(2)^2+8*x^3*ln(2)+16*x^5-x^2),x)
 

Output:

ln(-1/4*x-I*Pi*x*csgn(I*x)^2*csgn(I*x^2)*exp(x+15)+2*I*Pi*x*csgn(I*x)*csgn 
(I*x^2)^2*exp(x+15)+1/2*ln(2)*ln(3)+2*x^2*ln(3)+8*exp(x+15)*x^3-I*Pi*x^2*c 
sgn(I*x^2)^3+2*x^2*ln(2)+1/4*ln(3)^2+1/4*ln(2)^2+ln(x)^2+4*x^4+1/4*Pi^2*cs 
gn(I*x^2)^5*csgn(I*x)+1/2*ln(3)*exp(2*x+30)+1/2*exp(2*x+30)*ln(2)+6*exp(2* 
x+30)*x^2+2*exp(3*x+45)*x-3/8*Pi^2*csgn(I*x^2)^4*csgn(I*x)^2-I*Pi*x*csgn(I 
*x^2)^3*exp(x+15)-1/4*I*ln(3)*Pi*csgn(I*x)^2*csgn(I*x^2)+1/2*I*ln(3)*Pi*cs 
gn(I*x)*csgn(I*x^2)^2-1/4*I*Pi*ln(2)*csgn(I*x)^2*csgn(I*x^2)+2*exp(x+15)*l 
n(2)*x-1/16*Pi^2*csgn(I*x^2)^6+1/4*exp(4*x+60)-I*Pi*x^2*csgn(I*x)^2*csgn(I 
*x^2)+2*I*Pi*x^2*csgn(I*x)*csgn(I*x^2)^2+2*ln(3)*exp(x+15)*x+1/4*Pi^2*csgn 
(I*x^2)^3*csgn(I*x)^3+1/2*I*Pi*csgn(I*x)*csgn(I*x^2)^2*exp(2*x+30)-1/16*Pi 
^2*csgn(I*x^2)^2*csgn(I*x)^4+(ln(3)-1/2*I*Pi*csgn(I*x^2)^3-1/2*I*Pi*csgn(I 
*x)^2*csgn(I*x^2)+I*Pi*csgn(I*x)*csgn(I*x^2)^2+exp(2*x+30)+ln(2)+4*x*exp(x 
+15)+4*x^2)*ln(x)+1/2*I*Pi*ln(2)*csgn(I*x)*csgn(I*x^2)^2-1/4*I*Pi*csgn(I*x 
^2)^3*exp(2*x+30)-1/4*I*Pi*csgn(I*x)^2*csgn(I*x^2)*exp(2*x+30)-1/4*I*ln(3) 
*Pi*csgn(I*x^2)^3-1/4*I*Pi*ln(2)*csgn(I*x^2)^3)
 

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 105 vs. \(2 (26) = 52\).

Time = 0.11 (sec) , antiderivative size = 105, normalized size of antiderivative = 3.89 \[ \int \frac {-x+4 e^{60+4 x} x+16 x^2+64 x^4+e^{45+3 x} \left (8 x+24 x^2\right )+\left (4+16 x^2\right ) \log (2)+e^{30+2 x} \left (4+48 x^2+48 x^3+4 x \log (2)\right )+e^{15+x} \left (16 x+96 x^3+32 x^4+\left (8 x+8 x^2\right ) \log (2)\right )+\left (4+4 e^{30+2 x} x+16 x^2+e^{15+x} \left (8 x+8 x^2\right )\right ) \log \left (3 x^2\right )}{e^{60+4 x} x-x^2+8 e^{45+3 x} x^2+16 x^5+8 x^3 \log (2)+x \log ^2(2)+e^{30+2 x} \left (24 x^3+2 x \log (2)\right )+e^{15+x} \left (32 x^4+8 x^2 \log (2)\right )+\left (2 e^{30+2 x} x+8 e^{15+x} x^2+8 x^3+2 x \log (2)\right ) \log \left (3 x^2\right )+x \log ^2\left (3 x^2\right )} \, dx=\log \left (16 \, x^{4} + 8 \, x^{2} \log \left (2\right ) + 8 \, x e^{\left (3 \, x + 45\right )} + 2 \, {\left (12 \, x^{2} + \log \left (2\right )\right )} e^{\left (2 \, x + 30\right )} + 8 \, {\left (4 \, x^{3} + x \log \left (2\right )\right )} e^{\left (x + 15\right )} + \log \left (2\right )^{2} + 2 \, {\left (4 \, x^{2} + 4 \, x e^{\left (x + 15\right )} + e^{\left (2 \, x + 30\right )} + \log \left (2\right )\right )} \log \left (3 \, x^{2}\right ) + \log \left (3 \, x^{2}\right )^{2} - x + e^{\left (4 \, x + 60\right )}\right ) \] Input:

integrate(((4*x*exp(x+15)^2+(8*x^2+8*x)*exp(x+15)+16*x^2+4)*log(3*x^2)+4*x 
*exp(x+15)^4+(24*x^2+8*x)*exp(x+15)^3+(4*x*log(2)+48*x^3+48*x^2+4)*exp(x+1 
5)^2+((8*x^2+8*x)*log(2)+32*x^4+96*x^3+16*x)*exp(x+15)+(16*x^2+4)*log(2)+6 
4*x^4+16*x^2-x)/(x*log(3*x^2)^2+(2*x*exp(x+15)^2+8*x^2*exp(x+15)+2*x*log(2 
)+8*x^3)*log(3*x^2)+x*exp(x+15)^4+8*x^2*exp(x+15)^3+(2*x*log(2)+24*x^3)*ex 
p(x+15)^2+(8*x^2*log(2)+32*x^4)*exp(x+15)+x*log(2)^2+8*x^3*log(2)+16*x^5-x 
^2),x, algorithm="fricas")
 

Output:

log(16*x^4 + 8*x^2*log(2) + 8*x*e^(3*x + 45) + 2*(12*x^2 + log(2))*e^(2*x 
+ 30) + 8*(4*x^3 + x*log(2))*e^(x + 15) + log(2)^2 + 2*(4*x^2 + 4*x*e^(x + 
 15) + e^(2*x + 30) + log(2))*log(3*x^2) + log(3*x^2)^2 - x + e^(4*x + 60) 
)
 

Sympy [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 122 vs. \(2 (24) = 48\).

Time = 1.18 (sec) , antiderivative size = 122, normalized size of antiderivative = 4.52 \[ \int \frac {-x+4 e^{60+4 x} x+16 x^2+64 x^4+e^{45+3 x} \left (8 x+24 x^2\right )+\left (4+16 x^2\right ) \log (2)+e^{30+2 x} \left (4+48 x^2+48 x^3+4 x \log (2)\right )+e^{15+x} \left (16 x+96 x^3+32 x^4+\left (8 x+8 x^2\right ) \log (2)\right )+\left (4+4 e^{30+2 x} x+16 x^2+e^{15+x} \left (8 x+8 x^2\right )\right ) \log \left (3 x^2\right )}{e^{60+4 x} x-x^2+8 e^{45+3 x} x^2+16 x^5+8 x^3 \log (2)+x \log ^2(2)+e^{30+2 x} \left (24 x^3+2 x \log (2)\right )+e^{15+x} \left (32 x^4+8 x^2 \log (2)\right )+\left (2 e^{30+2 x} x+8 e^{15+x} x^2+8 x^3+2 x \log (2)\right ) \log \left (3 x^2\right )+x \log ^2\left (3 x^2\right )} \, dx=\log {\left (16 x^{4} + 8 x^{2} \log {\left (3 x^{2} \right )} + 8 x^{2} \log {\left (2 \right )} + 8 x e^{3 x + 45} - x + \left (24 x^{2} + 2 \log {\left (3 x^{2} \right )} + 2 \log {\left (2 \right )}\right ) e^{2 x + 30} + \left (32 x^{3} + 8 x \log {\left (3 x^{2} \right )} + 8 x \log {\left (2 \right )}\right ) e^{x + 15} + e^{4 x + 60} + \log {\left (3 x^{2} \right )}^{2} + 2 \log {\left (2 \right )} \log {\left (3 x^{2} \right )} + \log {\left (2 \right )}^{2} \right )} \] Input:

integrate(((4*x*exp(x+15)**2+(8*x**2+8*x)*exp(x+15)+16*x**2+4)*ln(3*x**2)+ 
4*x*exp(x+15)**4+(24*x**2+8*x)*exp(x+15)**3+(4*x*ln(2)+48*x**3+48*x**2+4)* 
exp(x+15)**2+((8*x**2+8*x)*ln(2)+32*x**4+96*x**3+16*x)*exp(x+15)+(16*x**2+ 
4)*ln(2)+64*x**4+16*x**2-x)/(x*ln(3*x**2)**2+(2*x*exp(x+15)**2+8*x**2*exp( 
x+15)+2*x*ln(2)+8*x**3)*ln(3*x**2)+x*exp(x+15)**4+8*x**2*exp(x+15)**3+(2*x 
*ln(2)+24*x**3)*exp(x+15)**2+(8*x**2*ln(2)+32*x**4)*exp(x+15)+x*ln(2)**2+8 
*x**3*ln(2)+16*x**5-x**2),x)
 

Output:

log(16*x**4 + 8*x**2*log(3*x**2) + 8*x**2*log(2) + 8*x*exp(3*x + 45) - x + 
 (24*x**2 + 2*log(3*x**2) + 2*log(2))*exp(2*x + 30) + (32*x**3 + 8*x*log(3 
*x**2) + 8*x*log(2))*exp(x + 15) + exp(4*x + 60) + log(3*x**2)**2 + 2*log( 
2)*log(3*x**2) + log(2)**2)
 

Maxima [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 128 vs. \(2 (26) = 52\).

Time = 0.22 (sec) , antiderivative size = 128, normalized size of antiderivative = 4.74 \[ \int \frac {-x+4 e^{60+4 x} x+16 x^2+64 x^4+e^{45+3 x} \left (8 x+24 x^2\right )+\left (4+16 x^2\right ) \log (2)+e^{30+2 x} \left (4+48 x^2+48 x^3+4 x \log (2)\right )+e^{15+x} \left (16 x+96 x^3+32 x^4+\left (8 x+8 x^2\right ) \log (2)\right )+\left (4+4 e^{30+2 x} x+16 x^2+e^{15+x} \left (8 x+8 x^2\right )\right ) \log \left (3 x^2\right )}{e^{60+4 x} x-x^2+8 e^{45+3 x} x^2+16 x^5+8 x^3 \log (2)+x \log ^2(2)+e^{30+2 x} \left (24 x^3+2 x \log (2)\right )+e^{15+x} \left (32 x^4+8 x^2 \log (2)\right )+\left (2 e^{30+2 x} x+8 e^{15+x} x^2+8 x^3+2 x \log (2)\right ) \log \left (3 x^2\right )+x \log ^2\left (3 x^2\right )} \, dx=\log \left ({\left (16 \, x^{4} + 8 \, x^{2} {\left (\log \left (3\right ) + \log \left (2\right )\right )} + 2 \, {\left (12 \, x^{2} e^{30} + {\left (\log \left (3\right ) + \log \left (2\right )\right )} e^{30} + 2 \, e^{30} \log \left (x\right )\right )} e^{\left (2 \, x\right )} + 8 \, x e^{\left (3 \, x + 45\right )} + 8 \, {\left (4 \, x^{3} e^{15} + x {\left (\log \left (3\right ) + \log \left (2\right )\right )} e^{15} + 2 \, x e^{15} \log \left (x\right )\right )} e^{x} + \log \left (3\right )^{2} + 2 \, \log \left (3\right ) \log \left (2\right ) + \log \left (2\right )^{2} + 4 \, {\left (4 \, x^{2} + \log \left (3\right ) + \log \left (2\right )\right )} \log \left (x\right ) + 4 \, \log \left (x\right )^{2} - x + e^{\left (4 \, x + 60\right )}\right )} e^{\left (-60\right )}\right ) \] Input:

integrate(((4*x*exp(x+15)^2+(8*x^2+8*x)*exp(x+15)+16*x^2+4)*log(3*x^2)+4*x 
*exp(x+15)^4+(24*x^2+8*x)*exp(x+15)^3+(4*x*log(2)+48*x^3+48*x^2+4)*exp(x+1 
5)^2+((8*x^2+8*x)*log(2)+32*x^4+96*x^3+16*x)*exp(x+15)+(16*x^2+4)*log(2)+6 
4*x^4+16*x^2-x)/(x*log(3*x^2)^2+(2*x*exp(x+15)^2+8*x^2*exp(x+15)+2*x*log(2 
)+8*x^3)*log(3*x^2)+x*exp(x+15)^4+8*x^2*exp(x+15)^3+(2*x*log(2)+24*x^3)*ex 
p(x+15)^2+(8*x^2*log(2)+32*x^4)*exp(x+15)+x*log(2)^2+8*x^3*log(2)+16*x^5-x 
^2),x, algorithm="maxima")
 

Output:

log((16*x^4 + 8*x^2*(log(3) + log(2)) + 2*(12*x^2*e^30 + (log(3) + log(2)) 
*e^30 + 2*e^30*log(x))*e^(2*x) + 8*x*e^(3*x + 45) + 8*(4*x^3*e^15 + x*(log 
(3) + log(2))*e^15 + 2*x*e^15*log(x))*e^x + log(3)^2 + 2*log(3)*log(2) + l 
og(2)^2 + 4*(4*x^2 + log(3) + log(2))*log(x) + 4*log(x)^2 - x + e^(4*x + 6 
0))*e^(-60))
 

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 131 vs. \(2 (26) = 52\).

Time = 0.28 (sec) , antiderivative size = 131, normalized size of antiderivative = 4.85 \[ \int \frac {-x+4 e^{60+4 x} x+16 x^2+64 x^4+e^{45+3 x} \left (8 x+24 x^2\right )+\left (4+16 x^2\right ) \log (2)+e^{30+2 x} \left (4+48 x^2+48 x^3+4 x \log (2)\right )+e^{15+x} \left (16 x+96 x^3+32 x^4+\left (8 x+8 x^2\right ) \log (2)\right )+\left (4+4 e^{30+2 x} x+16 x^2+e^{15+x} \left (8 x+8 x^2\right )\right ) \log \left (3 x^2\right )}{e^{60+4 x} x-x^2+8 e^{45+3 x} x^2+16 x^5+8 x^3 \log (2)+x \log ^2(2)+e^{30+2 x} \left (24 x^3+2 x \log (2)\right )+e^{15+x} \left (32 x^4+8 x^2 \log (2)\right )+\left (2 e^{30+2 x} x+8 e^{15+x} x^2+8 x^3+2 x \log (2)\right ) \log \left (3 x^2\right )+x \log ^2\left (3 x^2\right )} \, dx=\log \left (16 \, x^{4} + 32 \, x^{3} e^{\left (x + 15\right )} + 24 \, x^{2} e^{\left (2 \, x + 30\right )} + 8 \, x^{2} \log \left (2\right ) + 8 \, x e^{\left (x + 15\right )} \log \left (2\right ) + 8 \, x^{2} \log \left (3 \, x^{2}\right ) + 8 \, x e^{\left (x + 15\right )} \log \left (3 \, x^{2}\right ) + 8 \, x e^{\left (3 \, x + 45\right )} + 2 \, e^{\left (2 \, x + 30\right )} \log \left (2\right ) + \log \left (2\right )^{2} + 2 \, e^{\left (2 \, x + 30\right )} \log \left (3 \, x^{2}\right ) + 2 \, \log \left (2\right ) \log \left (3 \, x^{2}\right ) + \log \left (3 \, x^{2}\right )^{2} - x + e^{\left (4 \, x + 60\right )}\right ) \] Input:

integrate(((4*x*exp(x+15)^2+(8*x^2+8*x)*exp(x+15)+16*x^2+4)*log(3*x^2)+4*x 
*exp(x+15)^4+(24*x^2+8*x)*exp(x+15)^3+(4*x*log(2)+48*x^3+48*x^2+4)*exp(x+1 
5)^2+((8*x^2+8*x)*log(2)+32*x^4+96*x^3+16*x)*exp(x+15)+(16*x^2+4)*log(2)+6 
4*x^4+16*x^2-x)/(x*log(3*x^2)^2+(2*x*exp(x+15)^2+8*x^2*exp(x+15)+2*x*log(2 
)+8*x^3)*log(3*x^2)+x*exp(x+15)^4+8*x^2*exp(x+15)^3+(2*x*log(2)+24*x^3)*ex 
p(x+15)^2+(8*x^2*log(2)+32*x^4)*exp(x+15)+x*log(2)^2+8*x^3*log(2)+16*x^5-x 
^2),x, algorithm="giac")
 

Output:

log(16*x^4 + 32*x^3*e^(x + 15) + 24*x^2*e^(2*x + 30) + 8*x^2*log(2) + 8*x* 
e^(x + 15)*log(2) + 8*x^2*log(3*x^2) + 8*x*e^(x + 15)*log(3*x^2) + 8*x*e^( 
3*x + 45) + 2*e^(2*x + 30)*log(2) + log(2)^2 + 2*e^(2*x + 30)*log(3*x^2) + 
 2*log(2)*log(3*x^2) + log(3*x^2)^2 - x + e^(4*x + 60))
 

Mupad [F(-1)]

Timed out. \[ \int \frac {-x+4 e^{60+4 x} x+16 x^2+64 x^4+e^{45+3 x} \left (8 x+24 x^2\right )+\left (4+16 x^2\right ) \log (2)+e^{30+2 x} \left (4+48 x^2+48 x^3+4 x \log (2)\right )+e^{15+x} \left (16 x+96 x^3+32 x^4+\left (8 x+8 x^2\right ) \log (2)\right )+\left (4+4 e^{30+2 x} x+16 x^2+e^{15+x} \left (8 x+8 x^2\right )\right ) \log \left (3 x^2\right )}{e^{60+4 x} x-x^2+8 e^{45+3 x} x^2+16 x^5+8 x^3 \log (2)+x \log ^2(2)+e^{30+2 x} \left (24 x^3+2 x \log (2)\right )+e^{15+x} \left (32 x^4+8 x^2 \log (2)\right )+\left (2 e^{30+2 x} x+8 e^{15+x} x^2+8 x^3+2 x \log (2)\right ) \log \left (3 x^2\right )+x \log ^2\left (3 x^2\right )} \, dx=\int \frac {{\mathrm {e}}^{2\,x+30}\,\left (48\,x^3+48\,x^2+4\,\ln \left (2\right )\,x+4\right )-x+\ln \left (3\,x^2\right )\,\left ({\mathrm {e}}^{x+15}\,\left (8\,x^2+8\,x\right )+4\,x\,{\mathrm {e}}^{2\,x+30}+16\,x^2+4\right )+\ln \left (2\right )\,\left (16\,x^2+4\right )+{\mathrm {e}}^{3\,x+45}\,\left (24\,x^2+8\,x\right )+4\,x\,{\mathrm {e}}^{4\,x+60}+{\mathrm {e}}^{x+15}\,\left (16\,x+\ln \left (2\right )\,\left (8\,x^2+8\,x\right )+96\,x^3+32\,x^4\right )+16\,x^2+64\,x^4}{x\,{\mathrm {e}}^{4\,x+60}+x\,{\ln \left (2\right )}^2+8\,x^3\,\ln \left (2\right )+\ln \left (3\,x^2\right )\,\left (2\,x\,\ln \left (2\right )+2\,x\,{\mathrm {e}}^{2\,x+30}+8\,x^2\,{\mathrm {e}}^{x+15}+8\,x^3\right )+8\,x^2\,{\mathrm {e}}^{3\,x+45}+{\mathrm {e}}^{2\,x+30}\,\left (24\,x^3+2\,\ln \left (2\right )\,x\right )+{\mathrm {e}}^{x+15}\,\left (32\,x^4+8\,\ln \left (2\right )\,x^2\right )-x^2+16\,x^5+x\,{\ln \left (3\,x^2\right )}^2} \,d x \] Input:

int((exp(2*x + 30)*(4*x*log(2) + 48*x^2 + 48*x^3 + 4) - x + log(3*x^2)*(ex 
p(x + 15)*(8*x + 8*x^2) + 4*x*exp(2*x + 30) + 16*x^2 + 4) + log(2)*(16*x^2 
 + 4) + exp(3*x + 45)*(8*x + 24*x^2) + 4*x*exp(4*x + 60) + exp(x + 15)*(16 
*x + log(2)*(8*x + 8*x^2) + 96*x^3 + 32*x^4) + 16*x^2 + 64*x^4)/(x*exp(4*x 
 + 60) + x*log(2)^2 + 8*x^3*log(2) + log(3*x^2)*(2*x*log(2) + 2*x*exp(2*x 
+ 30) + 8*x^2*exp(x + 15) + 8*x^3) + 8*x^2*exp(3*x + 45) + exp(2*x + 30)*( 
2*x*log(2) + 24*x^3) + exp(x + 15)*(8*x^2*log(2) + 32*x^4) - x^2 + 16*x^5 
+ x*log(3*x^2)^2),x)
 

Output:

int((exp(2*x + 30)*(4*x*log(2) + 48*x^2 + 48*x^3 + 4) - x + log(3*x^2)*(ex 
p(x + 15)*(8*x + 8*x^2) + 4*x*exp(2*x + 30) + 16*x^2 + 4) + log(2)*(16*x^2 
 + 4) + exp(3*x + 45)*(8*x + 24*x^2) + 4*x*exp(4*x + 60) + exp(x + 15)*(16 
*x + log(2)*(8*x + 8*x^2) + 96*x^3 + 32*x^4) + 16*x^2 + 64*x^4)/(x*exp(4*x 
 + 60) + x*log(2)^2 + 8*x^3*log(2) + log(3*x^2)*(2*x*log(2) + 2*x*exp(2*x 
+ 30) + 8*x^2*exp(x + 15) + 8*x^3) + 8*x^2*exp(3*x + 45) + exp(2*x + 30)*( 
2*x*log(2) + 24*x^3) + exp(x + 15)*(8*x^2*log(2) + 32*x^4) - x^2 + 16*x^5 
+ x*log(3*x^2)^2), x)
 

Reduce [B] (verification not implemented)

Time = 0.18 (sec) , antiderivative size = 73, normalized size of antiderivative = 2.70 \[ \int \frac {-x+4 e^{60+4 x} x+16 x^2+64 x^4+e^{45+3 x} \left (8 x+24 x^2\right )+\left (4+16 x^2\right ) \log (2)+e^{30+2 x} \left (4+48 x^2+48 x^3+4 x \log (2)\right )+e^{15+x} \left (16 x+96 x^3+32 x^4+\left (8 x+8 x^2\right ) \log (2)\right )+\left (4+4 e^{30+2 x} x+16 x^2+e^{15+x} \left (8 x+8 x^2\right )\right ) \log \left (3 x^2\right )}{e^{60+4 x} x-x^2+8 e^{45+3 x} x^2+16 x^5+8 x^3 \log (2)+x \log ^2(2)+e^{30+2 x} \left (24 x^3+2 x \log (2)\right )+e^{15+x} \left (32 x^4+8 x^2 \log (2)\right )+\left (2 e^{30+2 x} x+8 e^{15+x} x^2+8 x^3+2 x \log (2)\right ) \log \left (3 x^2\right )+x \log ^2\left (3 x^2\right )} \, dx=\mathrm {log}\left (e^{2 x} e^{30}+4 e^{x} e^{15} x -\sqrt {x}+\mathrm {log}\left (3 x^{2}\right )+\mathrm {log}\left (2\right )+4 x^{2}\right )+\mathrm {log}\left (e^{2 x} e^{30}+4 e^{x} e^{15} x +\sqrt {x}+\mathrm {log}\left (3 x^{2}\right )+\mathrm {log}\left (2\right )+4 x^{2}\right ) \] Input:

int(((4*x*exp(x+15)^2+(8*x^2+8*x)*exp(x+15)+16*x^2+4)*log(3*x^2)+4*x*exp(x 
+15)^4+(24*x^2+8*x)*exp(x+15)^3+(4*x*log(2)+48*x^3+48*x^2+4)*exp(x+15)^2+( 
(8*x^2+8*x)*log(2)+32*x^4+96*x^3+16*x)*exp(x+15)+(16*x^2+4)*log(2)+64*x^4+ 
16*x^2-x)/(x*log(3*x^2)^2+(2*x*exp(x+15)^2+8*x^2*exp(x+15)+2*x*log(2)+8*x^ 
3)*log(3*x^2)+x*exp(x+15)^4+8*x^2*exp(x+15)^3+(2*x*log(2)+24*x^3)*exp(x+15 
)^2+(8*x^2*log(2)+32*x^4)*exp(x+15)+x*log(2)^2+8*x^3*log(2)+16*x^5-x^2),x)
 

Output:

log(e**(2*x)*e**30 + 4*e**x*e**15*x - sqrt(x) + log(3*x**2) + log(2) + 4*x 
**2) + log(e**(2*x)*e**30 + 4*e**x*e**15*x + sqrt(x) + log(3*x**2) + log(2 
) + 4*x**2)