Integrand size = 150, antiderivative size = 17 \[ \int \frac {(4-12 x) \log (4)+x^{x^2} \left (\left (2-8 x^2\right ) \log (4)-16 x^2 \log (4) \log (x)\right )}{32+160 x+320 x^2+320 x^3+160 x^4+32 x^5+x^{5 x^2}+x^{4 x^2} (10+10 x)+x^{3 x^2} \left (40+80 x+40 x^2\right )+x^{2 x^2} \left (80+240 x+240 x^2+80 x^3\right )+x^{x^2} \left (80+320 x+480 x^2+320 x^3+80 x^4\right )} \, dx=\frac {2 x \log (4)}{\left (2+2 x+x^{x^2}\right )^4} \] Output:
4*x*ln(2)/(2*x+2+exp(x^2*ln(x)))^4
Time = 0.11 (sec) , antiderivative size = 17, normalized size of antiderivative = 1.00 \[ \int \frac {(4-12 x) \log (4)+x^{x^2} \left (\left (2-8 x^2\right ) \log (4)-16 x^2 \log (4) \log (x)\right )}{32+160 x+320 x^2+320 x^3+160 x^4+32 x^5+x^{5 x^2}+x^{4 x^2} (10+10 x)+x^{3 x^2} \left (40+80 x+40 x^2\right )+x^{2 x^2} \left (80+240 x+240 x^2+80 x^3\right )+x^{x^2} \left (80+320 x+480 x^2+320 x^3+80 x^4\right )} \, dx=\frac {2 x \log (4)}{\left (2+2 x+x^{x^2}\right )^4} \] Input:
Integrate[((4 - 12*x)*Log[4] + x^x^2*((2 - 8*x^2)*Log[4] - 16*x^2*Log[4]*L og[x]))/(32 + 160*x + 320*x^2 + 320*x^3 + 160*x^4 + 32*x^5 + x^(5*x^2) + x ^(4*x^2)*(10 + 10*x) + x^(3*x^2)*(40 + 80*x + 40*x^2) + x^(2*x^2)*(80 + 24 0*x + 240*x^2 + 80*x^3) + x^x^2*(80 + 320*x + 480*x^2 + 320*x^3 + 80*x^4)) ,x]
Output:
(2*x*Log[4])/(2 + 2*x + x^x^2)^4
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.
| \(\displaystyle \int \frac {x^{x^2} \left (\left (2-8 x^2\right ) \log (4)-16 x^2 \log (4) \log (x)\right )+(4-12 x) \log (4)}{32 x^5+160 x^4+320 x^3+320 x^2+\left (40 x^2+80 x+40\right ) x^{3 x^2}+(10 x+10) x^{4 x^2}+x^{5 x^2}+\left (80 x^3+240 x^2+240 x+80\right ) x^{2 x^2}+\left (80 x^4+320 x^3+480 x^2+320 x+80\right ) x^{x^2}+160 x+32} \, dx\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle \int \frac {2 \log (4) \left (-4 x^{x^2+2}-8 x^{x^2+2} \log (x)+x^{x^2}-6 x+2\right )}{\left (x^{x^2}+2 x+2\right )^5}dx\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle 2 \log (4) \int \frac {x^{x^2}-8 \log (x) x^{x^2+2}-4 x^{x^2+2}-6 x+2}{\left (x^{x^2}+2 x+2\right )^5}dx\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle 2 \log (4) \int \left (\frac {8 x \left (2 \log (x) x^2+x^2+2 \log (x) x+x-1\right )}{\left (x^{x^2}+2 x+2\right )^5}-\frac {8 \log (x) x^2+4 x^2-1}{\left (x^{x^2}+2 x+2\right )^4}\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle 2 \log (4) \left (-8 \int \frac {x}{\left (x^{x^2}+2 x+2\right )^5}dx+8 \int \frac {x^2}{\left (x^{x^2}+2 x+2\right )^5}dx+\int \frac {1}{\left (x^{x^2}+2 x+2\right )^4}dx-4 \int \frac {x^2}{\left (x^{x^2}+2 x+2\right )^4}dx-16 \int \frac {\int \frac {x^2}{\left (x^{x^2}+2 x+2\right )^5}dx}{x}dx+8 \int \frac {\int \frac {x^2}{\left (x^{x^2}+2 x+2\right )^4}dx}{x}dx+16 \log (x) \int \frac {x^2}{\left (x^{x^2}+2 x+2\right )^5}dx-8 \log (x) \int \frac {x^2}{\left (x^{x^2}+2 x+2\right )^4}dx+8 \int \frac {x^3}{\left (x^{x^2}+2 x+2\right )^5}dx-16 \int \frac {\int \frac {x^3}{\left (x^{x^2}+2 x+2\right )^5}dx}{x}dx+16 \log (x) \int \frac {x^3}{\left (x^{x^2}+2 x+2\right )^5}dx\right )\) |
Input:
Int[((4 - 12*x)*Log[4] + x^x^2*((2 - 8*x^2)*Log[4] - 16*x^2*Log[4]*Log[x]) )/(32 + 160*x + 320*x^2 + 320*x^3 + 160*x^4 + 32*x^5 + x^(5*x^2) + x^(4*x^ 2)*(10 + 10*x) + x^(3*x^2)*(40 + 80*x + 40*x^2) + x^(2*x^2)*(80 + 240*x + 240*x^2 + 80*x^3) + x^x^2*(80 + 320*x + 480*x^2 + 320*x^3 + 80*x^4)),x]
Output:
$Aborted
Time = 1.24 (sec) , antiderivative size = 18, normalized size of antiderivative = 1.06
| method | result | size |
| risch | \(\frac {4 x \ln \left (2\right )}{\left (2 x +2+x^{x^{2}}\right )^{4}}\) | \(18\) |
| parallelrisch | \(\frac {4 x \ln \left (2\right )}{16 x^{4}+32 \,{\mathrm e}^{x^{2} \ln \left (x \right )} x^{3}+24 \,{\mathrm e}^{2 x^{2} \ln \left (x \right )} x^{2}+8 \,{\mathrm e}^{3 x^{2} \ln \left (x \right )} x +{\mathrm e}^{4 x^{2} \ln \left (x \right )}+64 x^{3}+96 \,{\mathrm e}^{x^{2} \ln \left (x \right )} x^{2}+48 \,{\mathrm e}^{2 x^{2} \ln \left (x \right )} x +8 \,{\mathrm e}^{3 x^{2} \ln \left (x \right )}+96 x^{2}+96 \,{\mathrm e}^{x^{2} \ln \left (x \right )} x +24 \,{\mathrm e}^{2 x^{2} \ln \left (x \right )}+64 x +32 \,{\mathrm e}^{x^{2} \ln \left (x \right )}+16}\) | \(140\) |
Input:
int(((-32*x^2*ln(2)*ln(x)+2*(-8*x^2+2)*ln(2))*exp(x^2*ln(x))+2*(-12*x+4)*l n(2))/(exp(x^2*ln(x))^5+(10*x+10)*exp(x^2*ln(x))^4+(40*x^2+80*x+40)*exp(x^ 2*ln(x))^3+(80*x^3+240*x^2+240*x+80)*exp(x^2*ln(x))^2+(80*x^4+320*x^3+480* x^2+320*x+80)*exp(x^2*ln(x))+32*x^5+160*x^4+320*x^3+320*x^2+160*x+32),x,me thod=_RETURNVERBOSE)
Output:
4*x*ln(2)/(2*x+2+x^(x^2))^4
Leaf count of result is larger than twice the leaf count of optimal. 83 vs. \(2 (17) = 34\).
Time = 0.07 (sec) , antiderivative size = 83, normalized size of antiderivative = 4.88 \[ \int \frac {(4-12 x) \log (4)+x^{x^2} \left (\left (2-8 x^2\right ) \log (4)-16 x^2 \log (4) \log (x)\right )}{32+160 x+320 x^2+320 x^3+160 x^4+32 x^5+x^{5 x^2}+x^{4 x^2} (10+10 x)+x^{3 x^2} \left (40+80 x+40 x^2\right )+x^{2 x^2} \left (80+240 x+240 x^2+80 x^3\right )+x^{x^2} \left (80+320 x+480 x^2+320 x^3+80 x^4\right )} \, dx=\frac {4 \, x \log \left (2\right )}{16 \, x^{4} + 64 \, x^{3} + 8 \, {\left (x + 1\right )} x^{3 \, x^{2}} + 24 \, {\left (x^{2} + 2 \, x + 1\right )} x^{2 \, x^{2}} + 32 \, {\left (x^{3} + 3 \, x^{2} + 3 \, x + 1\right )} x^{\left (x^{2}\right )} + 96 \, x^{2} + 64 \, x + x^{4 \, x^{2}} + 16} \] Input:
integrate(((-32*x^2*log(2)*log(x)+2*(-8*x^2+2)*log(2))*exp(x^2*log(x))+2*( -12*x+4)*log(2))/(exp(x^2*log(x))^5+(10*x+10)*exp(x^2*log(x))^4+(40*x^2+80 *x+40)*exp(x^2*log(x))^3+(80*x^3+240*x^2+240*x+80)*exp(x^2*log(x))^2+(80*x ^4+320*x^3+480*x^2+320*x+80)*exp(x^2*log(x))+32*x^5+160*x^4+320*x^3+320*x^ 2+160*x+32),x, algorithm="fricas")
Output:
4*x*log(2)/(16*x^4 + 64*x^3 + 8*(x + 1)*x^(3*x^2) + 24*(x^2 + 2*x + 1)*x^( 2*x^2) + 32*(x^3 + 3*x^2 + 3*x + 1)*x^(x^2) + 96*x^2 + 64*x + x^(4*x^2) + 16)
Leaf count of result is larger than twice the leaf count of optimal. 95 vs. \(2 (20) = 40\).
Time = 0.22 (sec) , antiderivative size = 95, normalized size of antiderivative = 5.59 \[ \int \frac {(4-12 x) \log (4)+x^{x^2} \left (\left (2-8 x^2\right ) \log (4)-16 x^2 \log (4) \log (x)\right )}{32+160 x+320 x^2+320 x^3+160 x^4+32 x^5+x^{5 x^2}+x^{4 x^2} (10+10 x)+x^{3 x^2} \left (40+80 x+40 x^2\right )+x^{2 x^2} \left (80+240 x+240 x^2+80 x^3\right )+x^{x^2} \left (80+320 x+480 x^2+320 x^3+80 x^4\right )} \, dx=\frac {4 x \log {\left (2 \right )}}{16 x^{4} + 64 x^{3} + 96 x^{2} + 64 x + \left (8 x + 8\right ) e^{3 x^{2} \log {\left (x \right )}} + \left (24 x^{2} + 48 x + 24\right ) e^{2 x^{2} \log {\left (x \right )}} + \left (32 x^{3} + 96 x^{2} + 96 x + 32\right ) e^{x^{2} \log {\left (x \right )}} + e^{4 x^{2} \log {\left (x \right )}} + 16} \] Input:
integrate(((-32*x**2*ln(2)*ln(x)+2*(-8*x**2+2)*ln(2))*exp(x**2*ln(x))+2*(- 12*x+4)*ln(2))/(exp(x**2*ln(x))**5+(10*x+10)*exp(x**2*ln(x))**4+(40*x**2+8 0*x+40)*exp(x**2*ln(x))**3+(80*x**3+240*x**2+240*x+80)*exp(x**2*ln(x))**2+ (80*x**4+320*x**3+480*x**2+320*x+80)*exp(x**2*ln(x))+32*x**5+160*x**4+320* x**3+320*x**2+160*x+32),x)
Output:
4*x*log(2)/(16*x**4 + 64*x**3 + 96*x**2 + 64*x + (8*x + 8)*exp(3*x**2*log( x)) + (24*x**2 + 48*x + 24)*exp(2*x**2*log(x)) + (32*x**3 + 96*x**2 + 96*x + 32)*exp(x**2*log(x)) + exp(4*x**2*log(x)) + 16)
Leaf count of result is larger than twice the leaf count of optimal. 83 vs. \(2 (17) = 34\).
Time = 0.52 (sec) , antiderivative size = 83, normalized size of antiderivative = 4.88 \[ \int \frac {(4-12 x) \log (4)+x^{x^2} \left (\left (2-8 x^2\right ) \log (4)-16 x^2 \log (4) \log (x)\right )}{32+160 x+320 x^2+320 x^3+160 x^4+32 x^5+x^{5 x^2}+x^{4 x^2} (10+10 x)+x^{3 x^2} \left (40+80 x+40 x^2\right )+x^{2 x^2} \left (80+240 x+240 x^2+80 x^3\right )+x^{x^2} \left (80+320 x+480 x^2+320 x^3+80 x^4\right )} \, dx=\frac {4 \, x \log \left (2\right )}{16 \, x^{4} + 64 \, x^{3} + 8 \, {\left (x + 1\right )} x^{3 \, x^{2}} + 24 \, {\left (x^{2} + 2 \, x + 1\right )} x^{2 \, x^{2}} + 32 \, {\left (x^{3} + 3 \, x^{2} + 3 \, x + 1\right )} x^{\left (x^{2}\right )} + 96 \, x^{2} + 64 \, x + x^{4 \, x^{2}} + 16} \] Input:
integrate(((-32*x^2*log(2)*log(x)+2*(-8*x^2+2)*log(2))*exp(x^2*log(x))+2*( -12*x+4)*log(2))/(exp(x^2*log(x))^5+(10*x+10)*exp(x^2*log(x))^4+(40*x^2+80 *x+40)*exp(x^2*log(x))^3+(80*x^3+240*x^2+240*x+80)*exp(x^2*log(x))^2+(80*x ^4+320*x^3+480*x^2+320*x+80)*exp(x^2*log(x))+32*x^5+160*x^4+320*x^3+320*x^ 2+160*x+32),x, algorithm="maxima")
Output:
4*x*log(2)/(16*x^4 + 64*x^3 + 8*(x + 1)*x^(3*x^2) + 24*(x^2 + 2*x + 1)*x^( 2*x^2) + 32*(x^3 + 3*x^2 + 3*x + 1)*x^(x^2) + 96*x^2 + 64*x + x^(4*x^2) + 16)
Leaf count of result is larger than twice the leaf count of optimal. 146 vs. \(2 (17) = 34\).
Time = 2.34 (sec) , antiderivative size = 146, normalized size of antiderivative = 8.59 \[ \int \frac {(4-12 x) \log (4)+x^{x^2} \left (\left (2-8 x^2\right ) \log (4)-16 x^2 \log (4) \log (x)\right )}{32+160 x+320 x^2+320 x^3+160 x^4+32 x^5+x^{5 x^2}+x^{4 x^2} (10+10 x)+x^{3 x^2} \left (40+80 x+40 x^2\right )+x^{2 x^2} \left (80+240 x+240 x^2+80 x^3\right )+x^{x^2} \left (80+320 x+480 x^2+320 x^3+80 x^4\right )} \, dx=\frac {4 \, x^{13} \log \left (2\right )}{32 \, x^{15} x^{\left (x^{2}\right )} + 16 \, x^{16} + 24 \, x^{14} x^{2 \, x^{2}} + 96 \, x^{14} x^{\left (x^{2}\right )} + 64 \, x^{15} + 8 \, x^{13} x^{3 \, x^{2}} + 48 \, x^{13} x^{2 \, x^{2}} + 96 \, x^{13} x^{\left (x^{2}\right )} + 96 \, x^{14} + x^{12} x^{4 \, x^{2}} + 8 \, x^{12} x^{3 \, x^{2}} + 24 \, x^{12} x^{2 \, x^{2}} + 32 \, x^{12} x^{\left (x^{2}\right )} + 64 \, x^{13} + 16 \, x^{12}} \] Input:
integrate(((-32*x^2*log(2)*log(x)+2*(-8*x^2+2)*log(2))*exp(x^2*log(x))+2*( -12*x+4)*log(2))/(exp(x^2*log(x))^5+(10*x+10)*exp(x^2*log(x))^4+(40*x^2+80 *x+40)*exp(x^2*log(x))^3+(80*x^3+240*x^2+240*x+80)*exp(x^2*log(x))^2+(80*x ^4+320*x^3+480*x^2+320*x+80)*exp(x^2*log(x))+32*x^5+160*x^4+320*x^3+320*x^ 2+160*x+32),x, algorithm="giac")
Output:
4*x^13*log(2)/(32*x^15*x^(x^2) + 16*x^16 + 24*x^14*x^(2*x^2) + 96*x^14*x^( x^2) + 64*x^15 + 8*x^13*x^(3*x^2) + 48*x^13*x^(2*x^2) + 96*x^13*x^(x^2) + 96*x^14 + x^12*x^(4*x^2) + 8*x^12*x^(3*x^2) + 24*x^12*x^(2*x^2) + 32*x^12* x^(x^2) + 64*x^13 + 16*x^12)
Timed out. \[ \int \frac {(4-12 x) \log (4)+x^{x^2} \left (\left (2-8 x^2\right ) \log (4)-16 x^2 \log (4) \log (x)\right )}{32+160 x+320 x^2+320 x^3+160 x^4+32 x^5+x^{5 x^2}+x^{4 x^2} (10+10 x)+x^{3 x^2} \left (40+80 x+40 x^2\right )+x^{2 x^2} \left (80+240 x+240 x^2+80 x^3\right )+x^{x^2} \left (80+320 x+480 x^2+320 x^3+80 x^4\right )} \, dx=\int -\frac {2\,\ln \left (2\right )\,\left (12\,x-4\right )+{\mathrm {e}}^{x^2\,\ln \left (x\right )}\,\left (2\,\ln \left (2\right )\,\left (8\,x^2-2\right )+32\,x^2\,\ln \left (2\right )\,\ln \left (x\right )\right )}{160\,x+{\mathrm {e}}^{5\,x^2\,\ln \left (x\right )}+{\mathrm {e}}^{4\,x^2\,\ln \left (x\right )}\,\left (10\,x+10\right )+{\mathrm {e}}^{3\,x^2\,\ln \left (x\right )}\,\left (40\,x^2+80\,x+40\right )+{\mathrm {e}}^{2\,x^2\,\ln \left (x\right )}\,\left (80\,x^3+240\,x^2+240\,x+80\right )+{\mathrm {e}}^{x^2\,\ln \left (x\right )}\,\left (80\,x^4+320\,x^3+480\,x^2+320\,x+80\right )+320\,x^2+320\,x^3+160\,x^4+32\,x^5+32} \,d x \] Input:
int(-(2*log(2)*(12*x - 4) + exp(x^2*log(x))*(2*log(2)*(8*x^2 - 2) + 32*x^2 *log(2)*log(x)))/(160*x + exp(5*x^2*log(x)) + exp(4*x^2*log(x))*(10*x + 10 ) + exp(3*x^2*log(x))*(80*x + 40*x^2 + 40) + exp(2*x^2*log(x))*(240*x + 24 0*x^2 + 80*x^3 + 80) + exp(x^2*log(x))*(320*x + 480*x^2 + 320*x^3 + 80*x^4 + 80) + 320*x^2 + 320*x^3 + 160*x^4 + 32*x^5 + 32),x)
Output:
int(-(2*log(2)*(12*x - 4) + exp(x^2*log(x))*(2*log(2)*(8*x^2 - 2) + 32*x^2 *log(2)*log(x)))/(160*x + exp(5*x^2*log(x)) + exp(4*x^2*log(x))*(10*x + 10 ) + exp(3*x^2*log(x))*(80*x + 40*x^2 + 40) + exp(2*x^2*log(x))*(240*x + 24 0*x^2 + 80*x^3 + 80) + exp(x^2*log(x))*(320*x + 480*x^2 + 320*x^3 + 80*x^4 + 80) + 320*x^2 + 320*x^3 + 160*x^4 + 32*x^5 + 32), x)
Time = 0.24 (sec) , antiderivative size = 119, normalized size of antiderivative = 7.00 \[ \int \frac {(4-12 x) \log (4)+x^{x^2} \left (\left (2-8 x^2\right ) \log (4)-16 x^2 \log (4) \log (x)\right )}{32+160 x+320 x^2+320 x^3+160 x^4+32 x^5+x^{5 x^2}+x^{4 x^2} (10+10 x)+x^{3 x^2} \left (40+80 x+40 x^2\right )+x^{2 x^2} \left (80+240 x+240 x^2+80 x^3\right )+x^{x^2} \left (80+320 x+480 x^2+320 x^3+80 x^4\right )} \, dx=\frac {4 \,\mathrm {log}\left (2\right ) x}{x^{4 x^{2}}+8 x^{3 x^{2}} x +8 x^{3 x^{2}}+24 x^{2 x^{2}} x^{2}+48 x^{2 x^{2}} x +24 x^{2 x^{2}}+32 x^{x^{2}} x^{3}+96 x^{x^{2}} x^{2}+96 x^{x^{2}} x +32 x^{x^{2}}+16 x^{4}+64 x^{3}+96 x^{2}+64 x +16} \] Input:
int(((-32*x^2*log(2)*log(x)+2*(-8*x^2+2)*log(2))*exp(x^2*log(x))+2*(-12*x+ 4)*log(2))/(exp(x^2*log(x))^5+(10*x+10)*exp(x^2*log(x))^4+(40*x^2+80*x+40) *exp(x^2*log(x))^3+(80*x^3+240*x^2+240*x+80)*exp(x^2*log(x))^2+(80*x^4+320 *x^3+480*x^2+320*x+80)*exp(x^2*log(x))+32*x^5+160*x^4+320*x^3+320*x^2+160* x+32),x)
Output:
(4*log(2)*x)/(x**(4*x**2) + 8*x**(3*x**2)*x + 8*x**(3*x**2) + 24*x**(2*x** 2)*x**2 + 48*x**(2*x**2)*x + 24*x**(2*x**2) + 32*x**(x**2)*x**3 + 96*x**(x **2)*x**2 + 96*x**(x**2)*x + 32*x**(x**2) + 16*x**4 + 64*x**3 + 96*x**2 + 64*x + 16)