Integrand size = 74, antiderivative size = 25 \[ \int \frac {4 x^2+e^{625 x^6} \left (2-7500 x^6\right )}{3 e^{1250 x^6}+12 x^4+12 x^3 \log (4)+3 x^2 \log ^2(4)+e^{625 x^6} \left (-12 x^2-6 x \log (4)\right )} \, dx=\frac {2}{3 \left (\frac {e^{625 x^6}}{x}-2 x-\log (4)\right )} \] Output:
2/(3*exp(625*x^6)/x-6*ln(2)-6*x)
Time = 0.19 (sec) , antiderivative size = 25, normalized size of antiderivative = 1.00 \[ \int \frac {4 x^2+e^{625 x^6} \left (2-7500 x^6\right )}{3 e^{1250 x^6}+12 x^4+12 x^3 \log (4)+3 x^2 \log ^2(4)+e^{625 x^6} \left (-12 x^2-6 x \log (4)\right )} \, dx=\frac {2 x}{3 \left (e^{625 x^6}-2 x^2-x \log (4)\right )} \] Input:
Integrate[(4*x^2 + E^(625*x^6)*(2 - 7500*x^6))/(3*E^(1250*x^6) + 12*x^4 + 12*x^3*Log[4] + 3*x^2*Log[4]^2 + E^(625*x^6)*(-12*x^2 - 6*x*Log[4])),x]
Output:
(2*x)/(3*(E^(625*x^6) - 2*x^2 - x*Log[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.
\(\displaystyle \int \frac {e^{625 x^6} \left (2-7500 x^6\right )+4 x^2}{3 e^{1250 x^6}+12 x^4+12 x^3 \log (4)+3 x^2 \log ^2(4)+e^{625 x^6} \left (-12 x^2-6 x \log (4)\right )} \, dx\) |
\(\Big \downarrow \) 7239 |
\(\displaystyle \int \frac {e^{625 x^6} \left (2-7500 x^6\right )+4 x^2}{3 \left (e^{625 x^6}-x (2 x+\log (4))\right )^2}dx\) |
\(\Big \downarrow \) 27 |
\(\displaystyle \frac {1}{3} \int \frac {2 \left (2 x^2+e^{625 x^6} \left (1-3750 x^6\right )\right )}{\left (e^{625 x^6}-x (2 x+\log (4))\right )^2}dx\) |
\(\Big \downarrow \) 27 |
\(\displaystyle \frac {2}{3} \int \frac {2 x^2+e^{625 x^6} \left (1-3750 x^6\right )}{\left (e^{625 x^6}-x (2 x+\log (4))\right )^2}dx\) |
\(\Big \downarrow \) 7293 |
\(\displaystyle \frac {2}{3} \int \left (\frac {3750 x^6-1}{2 x^2+\log (4) x-e^{625 x^6}}-\frac {x \left (7500 x^7+3750 \log (4) x^6-4 x-\log (4)\right )}{\left (2 x^2+\log (4) x-e^{625 x^6}\right )^2}\right )dx\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle \frac {2}{3} \left (\int \frac {1}{-2 x^2-\log (4) x+e^{625 x^6}}dx+\log (4) \int \frac {x}{\left (2 x^2+\log (4) x-e^{625 x^6}\right )^2}dx+4 \int \frac {x^2}{\left (2 x^2+\log (4) x-e^{625 x^6}\right )^2}dx+3750 \int \frac {x^6}{2 x^2+\log (4) x-e^{625 x^6}}dx-7500 \int \frac {x^8}{\left (2 x^2+\log (4) x-e^{625 x^6}\right )^2}dx-3750 \log (4) \int \frac {x^7}{\left (2 x^2+\log (4) x-e^{625 x^6}\right )^2}dx\right )\) |
Input:
Int[(4*x^2 + E^(625*x^6)*(2 - 7500*x^6))/(3*E^(1250*x^6) + 12*x^4 + 12*x^3 *Log[4] + 3*x^2*Log[4]^2 + E^(625*x^6)*(-12*x^2 - 6*x*Log[4])),x]
Output:
$Aborted
Time = 0.27 (sec) , antiderivative size = 25, normalized size of antiderivative = 1.00
method | result | size |
norman | \(-\frac {2 x}{3 \left (2 x \ln \left (2\right )+2 x^{2}-{\mathrm e}^{625 x^{6}}\right )}\) | \(25\) |
risch | \(-\frac {2 x}{3 \left (2 x \ln \left (2\right )+2 x^{2}-{\mathrm e}^{625 x^{6}}\right )}\) | \(25\) |
parallelrisch | \(-\frac {2 x}{3 \left (2 x \ln \left (2\right )+2 x^{2}-{\mathrm e}^{625 x^{6}}\right )}\) | \(25\) |
Input:
int(((-7500*x^6+2)*exp(625*x^6)+4*x^2)/(3*exp(625*x^6)^2+(-12*x*ln(2)-12*x ^2)*exp(625*x^6)+12*x^2*ln(2)^2+24*x^3*ln(2)+12*x^4),x,method=_RETURNVERBO SE)
Output:
-2/3*x/(2*x*ln(2)+2*x^2-exp(625*x^6))
Time = 0.09 (sec) , antiderivative size = 24, normalized size of antiderivative = 0.96 \[ \int \frac {4 x^2+e^{625 x^6} \left (2-7500 x^6\right )}{3 e^{1250 x^6}+12 x^4+12 x^3 \log (4)+3 x^2 \log ^2(4)+e^{625 x^6} \left (-12 x^2-6 x \log (4)\right )} \, dx=-\frac {2 \, x}{3 \, {\left (2 \, x^{2} + 2 \, x \log \left (2\right ) - e^{\left (625 \, x^{6}\right )}\right )}} \] Input:
integrate(((-7500*x^6+2)*exp(625*x^6)+4*x^2)/(3*exp(625*x^6)^2+(-12*x*log( 2)-12*x^2)*exp(625*x^6)+12*x^2*log(2)^2+24*x^3*log(2)+12*x^4),x, algorithm ="fricas")
Output:
-2/3*x/(2*x^2 + 2*x*log(2) - e^(625*x^6))
Time = 0.07 (sec) , antiderivative size = 22, normalized size of antiderivative = 0.88 \[ \int \frac {4 x^2+e^{625 x^6} \left (2-7500 x^6\right )}{3 e^{1250 x^6}+12 x^4+12 x^3 \log (4)+3 x^2 \log ^2(4)+e^{625 x^6} \left (-12 x^2-6 x \log (4)\right )} \, dx=\frac {2 x}{- 6 x^{2} - 6 x \log {\left (2 \right )} + 3 e^{625 x^{6}}} \] Input:
integrate(((-7500*x**6+2)*exp(625*x**6)+4*x**2)/(3*exp(625*x**6)**2+(-12*x *ln(2)-12*x**2)*exp(625*x**6)+12*x**2*ln(2)**2+24*x**3*ln(2)+12*x**4),x)
Output:
2*x/(-6*x**2 - 6*x*log(2) + 3*exp(625*x**6))
Time = 0.17 (sec) , antiderivative size = 24, normalized size of antiderivative = 0.96 \[ \int \frac {4 x^2+e^{625 x^6} \left (2-7500 x^6\right )}{3 e^{1250 x^6}+12 x^4+12 x^3 \log (4)+3 x^2 \log ^2(4)+e^{625 x^6} \left (-12 x^2-6 x \log (4)\right )} \, dx=-\frac {2 \, x}{3 \, {\left (2 \, x^{2} + 2 \, x \log \left (2\right ) - e^{\left (625 \, x^{6}\right )}\right )}} \] Input:
integrate(((-7500*x^6+2)*exp(625*x^6)+4*x^2)/(3*exp(625*x^6)^2+(-12*x*log( 2)-12*x^2)*exp(625*x^6)+12*x^2*log(2)^2+24*x^3*log(2)+12*x^4),x, algorithm ="maxima")
Output:
-2/3*x/(2*x^2 + 2*x*log(2) - e^(625*x^6))
Time = 0.17 (sec) , antiderivative size = 24, normalized size of antiderivative = 0.96 \[ \int \frac {4 x^2+e^{625 x^6} \left (2-7500 x^6\right )}{3 e^{1250 x^6}+12 x^4+12 x^3 \log (4)+3 x^2 \log ^2(4)+e^{625 x^6} \left (-12 x^2-6 x \log (4)\right )} \, dx=-\frac {2 \, x}{3 \, {\left (2 \, x^{2} + 2 \, x \log \left (2\right ) - e^{\left (625 \, x^{6}\right )}\right )}} \] Input:
integrate(((-7500*x^6+2)*exp(625*x^6)+4*x^2)/(3*exp(625*x^6)^2+(-12*x*log( 2)-12*x^2)*exp(625*x^6)+12*x^2*log(2)^2+24*x^3*log(2)+12*x^4),x, algorithm ="giac")
Output:
-2/3*x/(2*x^2 + 2*x*log(2) - e^(625*x^6))
Timed out. \[ \int \frac {4 x^2+e^{625 x^6} \left (2-7500 x^6\right )}{3 e^{1250 x^6}+12 x^4+12 x^3 \log (4)+3 x^2 \log ^2(4)+e^{625 x^6} \left (-12 x^2-6 x \log (4)\right )} \, dx=-\int \frac {{\mathrm {e}}^{625\,x^6}\,\left (7500\,x^6-2\right )-4\,x^2}{3\,{\mathrm {e}}^{1250\,x^6}+12\,x^2\,{\ln \left (2\right )}^2+24\,x^3\,\ln \left (2\right )-{\mathrm {e}}^{625\,x^6}\,\left (12\,x^2+12\,\ln \left (2\right )\,x\right )+12\,x^4} \,d x \] Input:
int(-(exp(625*x^6)*(7500*x^6 - 2) - 4*x^2)/(3*exp(1250*x^6) + 12*x^2*log(2 )^2 + 24*x^3*log(2) - exp(625*x^6)*(12*x*log(2) + 12*x^2) + 12*x^4),x)
Output:
-int((exp(625*x^6)*(7500*x^6 - 2) - 4*x^2)/(3*exp(1250*x^6) + 12*x^2*log(2 )^2 + 24*x^3*log(2) - exp(625*x^6)*(12*x*log(2) + 12*x^2) + 12*x^4), x)
\[ \int \frac {4 x^2+e^{625 x^6} \left (2-7500 x^6\right )}{3 e^{1250 x^6}+12 x^4+12 x^3 \log (4)+3 x^2 \log ^2(4)+e^{625 x^6} \left (-12 x^2-6 x \log (4)\right )} \, dx=\frac {2 \left (\int \frac {e^{625 x^{6}}}{e^{1250 x^{6}}-4 e^{625 x^{6}} \mathrm {log}\left (2\right ) x -4 e^{625 x^{6}} x^{2}+4 \mathrm {log}\left (2\right )^{2} x^{2}+8 \,\mathrm {log}\left (2\right ) x^{3}+4 x^{4}}d x \right )}{3}+\frac {4 \left (\int \frac {x^{2}}{e^{1250 x^{6}}-4 e^{625 x^{6}} \mathrm {log}\left (2\right ) x -4 e^{625 x^{6}} x^{2}+4 \mathrm {log}\left (2\right )^{2} x^{2}+8 \,\mathrm {log}\left (2\right ) x^{3}+4 x^{4}}d x \right )}{3}-2500 \left (\int \frac {e^{625 x^{6}} x^{6}}{e^{1250 x^{6}}-4 e^{625 x^{6}} \mathrm {log}\left (2\right ) x -4 e^{625 x^{6}} x^{2}+4 \mathrm {log}\left (2\right )^{2} x^{2}+8 \,\mathrm {log}\left (2\right ) x^{3}+4 x^{4}}d x \right ) \] Input:
int(((-7500*x^6+2)*exp(625*x^6)+4*x^2)/(3*exp(625*x^6)^2+(-12*x*log(2)-12* x^2)*exp(625*x^6)+12*x^2*log(2)^2+24*x^3*log(2)+12*x^4),x)
Output:
(2*(int(e**(625*x**6)/(e**(1250*x**6) - 4*e**(625*x**6)*log(2)*x - 4*e**(6 25*x**6)*x**2 + 4*log(2)**2*x**2 + 8*log(2)*x**3 + 4*x**4),x) + 2*int(x**2 /(e**(1250*x**6) - 4*e**(625*x**6)*log(2)*x - 4*e**(625*x**6)*x**2 + 4*log (2)**2*x**2 + 8*log(2)*x**3 + 4*x**4),x) - 3750*int((e**(625*x**6)*x**6)/( e**(1250*x**6) - 4*e**(625*x**6)*log(2)*x - 4*e**(625*x**6)*x**2 + 4*log(2 )**2*x**2 + 8*log(2)*x**3 + 4*x**4),x)))/3