Integrand size = 316, antiderivative size = 31 \[ \int \frac {-4 x^6+2 x^6 \log (2)+e^{4 x^2} \left (-4 x^2+2 x^2 \log (2)\right )+e^{3 x^2} \left (-16 x^3+8 x^3 \log (2)\right )+\left (-4 x^3-3 x^2 \log (2)\right ) \log (4)+e^{2 x^2} \left (-24 x^4+12 x^4 \log (2)+\left (-2 x-4 x^3+\left (-1-4 x^2\right ) \log (2)\right ) \log (4)\right )+e^{x^2} \left (-16 x^5+8 x^5 \log (2)+\left (-6 x^2-4 x^4+\left (-4 x-4 x^3\right ) \log (2)\right ) \log (4)\right )}{16 x^6+16 x^7+4 x^8+e^{4 x^2} \left (16 x^2+16 x^3+4 x^4\right )+e^{3 x^2} \left (64 x^3+64 x^4+16 x^5\right )+\left (8 x^3+4 x^4\right ) \log (4)+\log ^2(4)+e^{2 x^2} \left (96 x^4+96 x^5+24 x^6+\left (8 x+4 x^2\right ) \log (4)\right )+e^{x^2} \left (64 x^5+64 x^6+16 x^7+\left (16 x^2+8 x^3\right ) \log (4)\right )} \, dx=\frac {-x-\log (2)}{4+2 x+\frac {\log (4)}{x \left (e^{x^2}+x\right )^2}} \] Output:
(-x-ln(2))/(2*x+2*ln(2)/(exp(x^2)+x)^2/x+4)
\[ \int \frac {-4 x^6+2 x^6 \log (2)+e^{4 x^2} \left (-4 x^2+2 x^2 \log (2)\right )+e^{3 x^2} \left (-16 x^3+8 x^3 \log (2)\right )+\left (-4 x^3-3 x^2 \log (2)\right ) \log (4)+e^{2 x^2} \left (-24 x^4+12 x^4 \log (2)+\left (-2 x-4 x^3+\left (-1-4 x^2\right ) \log (2)\right ) \log (4)\right )+e^{x^2} \left (-16 x^5+8 x^5 \log (2)+\left (-6 x^2-4 x^4+\left (-4 x-4 x^3\right ) \log (2)\right ) \log (4)\right )}{16 x^6+16 x^7+4 x^8+e^{4 x^2} \left (16 x^2+16 x^3+4 x^4\right )+e^{3 x^2} \left (64 x^3+64 x^4+16 x^5\right )+\left (8 x^3+4 x^4\right ) \log (4)+\log ^2(4)+e^{2 x^2} \left (96 x^4+96 x^5+24 x^6+\left (8 x+4 x^2\right ) \log (4)\right )+e^{x^2} \left (64 x^5+64 x^6+16 x^7+\left (16 x^2+8 x^3\right ) \log (4)\right )} \, dx=\int \frac {-4 x^6+2 x^6 \log (2)+e^{4 x^2} \left (-4 x^2+2 x^2 \log (2)\right )+e^{3 x^2} \left (-16 x^3+8 x^3 \log (2)\right )+\left (-4 x^3-3 x^2 \log (2)\right ) \log (4)+e^{2 x^2} \left (-24 x^4+12 x^4 \log (2)+\left (-2 x-4 x^3+\left (-1-4 x^2\right ) \log (2)\right ) \log (4)\right )+e^{x^2} \left (-16 x^5+8 x^5 \log (2)+\left (-6 x^2-4 x^4+\left (-4 x-4 x^3\right ) \log (2)\right ) \log (4)\right )}{16 x^6+16 x^7+4 x^8+e^{4 x^2} \left (16 x^2+16 x^3+4 x^4\right )+e^{3 x^2} \left (64 x^3+64 x^4+16 x^5\right )+\left (8 x^3+4 x^4\right ) \log (4)+\log ^2(4)+e^{2 x^2} \left (96 x^4+96 x^5+24 x^6+\left (8 x+4 x^2\right ) \log (4)\right )+e^{x^2} \left (64 x^5+64 x^6+16 x^7+\left (16 x^2+8 x^3\right ) \log (4)\right )} \, dx \] Input:
Integrate[(-4*x^6 + 2*x^6*Log[2] + E^(4*x^2)*(-4*x^2 + 2*x^2*Log[2]) + E^( 3*x^2)*(-16*x^3 + 8*x^3*Log[2]) + (-4*x^3 - 3*x^2*Log[2])*Log[4] + E^(2*x^ 2)*(-24*x^4 + 12*x^4*Log[2] + (-2*x - 4*x^3 + (-1 - 4*x^2)*Log[2])*Log[4]) + E^x^2*(-16*x^5 + 8*x^5*Log[2] + (-6*x^2 - 4*x^4 + (-4*x - 4*x^3)*Log[2] )*Log[4]))/(16*x^6 + 16*x^7 + 4*x^8 + E^(4*x^2)*(16*x^2 + 16*x^3 + 4*x^4) + E^(3*x^2)*(64*x^3 + 64*x^4 + 16*x^5) + (8*x^3 + 4*x^4)*Log[4] + Log[4]^2 + E^(2*x^2)*(96*x^4 + 96*x^5 + 24*x^6 + (8*x + 4*x^2)*Log[4]) + E^x^2*(64 *x^5 + 64*x^6 + 16*x^7 + (16*x^2 + 8*x^3)*Log[4])),x]
Output:
Integrate[(-4*x^6 + 2*x^6*Log[2] + E^(4*x^2)*(-4*x^2 + 2*x^2*Log[2]) + E^( 3*x^2)*(-16*x^3 + 8*x^3*Log[2]) + (-4*x^3 - 3*x^2*Log[2])*Log[4] + E^(2*x^ 2)*(-24*x^4 + 12*x^4*Log[2] + (-2*x - 4*x^3 + (-1 - 4*x^2)*Log[2])*Log[4]) + E^x^2*(-16*x^5 + 8*x^5*Log[2] + (-6*x^2 - 4*x^4 + (-4*x - 4*x^3)*Log[2] )*Log[4]))/(16*x^6 + 16*x^7 + 4*x^8 + E^(4*x^2)*(16*x^2 + 16*x^3 + 4*x^4) + E^(3*x^2)*(64*x^3 + 64*x^4 + 16*x^5) + (8*x^3 + 4*x^4)*Log[4] + Log[4]^2 + E^(2*x^2)*(96*x^4 + 96*x^5 + 24*x^6 + (8*x + 4*x^2)*Log[4]) + E^x^2*(64 *x^5 + 64*x^6 + 16*x^7 + (16*x^2 + 8*x^3)*Log[4])), x]
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.
| \(\displaystyle \int \frac {-4 x^6+2 x^6 \log (2)+e^{4 x^2} \left (2 x^2 \log (2)-4 x^2\right )+e^{3 x^2} \left (8 x^3 \log (2)-16 x^3\right )+\log (4) \left (-4 x^3-3 x^2 \log (2)\right )+e^{2 x^2} \left (-24 x^4+12 x^4 \log (2)+\log (4) \left (-4 x^3+\left (-4 x^2-1\right ) \log (2)-2 x\right )\right )+e^{x^2} \left (-16 x^5+8 x^5 \log (2)+\log (4) \left (-4 x^4+\left (-4 x^3-4 x\right ) \log (2)-6 x^2\right )\right )}{4 x^8+16 x^7+16 x^6+\left (4 x^4+8 x^3\right ) \log (4)+e^{4 x^2} \left (4 x^4+16 x^3+16 x^2\right )+e^{2 x^2} \left (24 x^6+96 x^5+96 x^4+\left (4 x^2+8 x\right ) \log (4)\right )+e^{3 x^2} \left (16 x^5+64 x^4+64 x^3\right )+e^{x^2} \left (16 x^7+64 x^6+64 x^5+\left (8 x^3+16 x^2\right ) \log (4)\right )+\log ^2(4)} \, dx\) |
| \(\Big \downarrow \) 6 |
| \(\displaystyle \int \frac {x^6 (2 \log (2)-4)+e^{4 x^2} \left (2 x^2 \log (2)-4 x^2\right )+e^{3 x^2} \left (8 x^3 \log (2)-16 x^3\right )+\log (4) \left (-4 x^3-3 x^2 \log (2)\right )+e^{2 x^2} \left (-24 x^4+12 x^4 \log (2)+\log (4) \left (-4 x^3+\left (-4 x^2-1\right ) \log (2)-2 x\right )\right )+e^{x^2} \left (-16 x^5+8 x^5 \log (2)+\log (4) \left (-4 x^4+\left (-4 x^3-4 x\right ) \log (2)-6 x^2\right )\right )}{4 x^8+16 x^7+16 x^6+\left (4 x^4+8 x^3\right ) \log (4)+e^{4 x^2} \left (4 x^4+16 x^3+16 x^2\right )+e^{2 x^2} \left (24 x^6+96 x^5+96 x^4+\left (4 x^2+8 x\right ) \log (4)\right )+e^{3 x^2} \left (16 x^5+64 x^4+64 x^3\right )+e^{x^2} \left (16 x^7+64 x^6+64 x^5+\left (8 x^3+16 x^2\right ) \log (4)\right )+\log ^2(4)}dx\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle \int \frac {\left (e^{x^2}+x\right ) \left (x \left (x^4 (\log (4)-4)-4 x \log (4)-\log (4) \log (8)\right )+e^{3 x^2} x^2 (\log (4)-4)+e^{2 x^2} x^3 (\log (64)-12)+e^{x^2} \left (x^4 (\log (64)-12)-4 x^3 \log (4)-4 x^2 \log (2) \log (4)-2 x \log (4)-\log (2) \log (4)\right )\right )}{\left (2 x^4+4 x^3+4 e^{x^2} (x+2) x^2+2 e^{2 x^2} (x+2) x+\log (4)\right )^2}dx\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle \int \left (\frac {x^4 (-\log (256))-x^3 \log (4) (8+\log (16))-x^2 \log (4) (2+\log (256))-x \log (4) \log (8)-\log ^2(4)}{2 x (x+2)^2 \left (2 x^4+4 x^3+8 e^{x^2} x^2+2 e^{2 x^2} x^2+4 e^{2 x^2} x+4 e^{x^2} x^3+\log (4)\right )}+\frac {\log (4) \left (16 x^8+64 x^7 \left (1+\frac {\log (2)}{4}\right )+56 x^6 \left (1+\frac {8 \log (2)}{7}\right )-32 x^5 \left (1-\frac {7 \log (2)}{4}\right )-32 x^4 \left (1+\frac {\log (2)}{2}\right )+8 x^3 \log (2) \log (4)-4 e^{x^2} x^2 \log (256)+4 x^2 \log (4) (1+\log (16))+16 e^{x^2} x^7+64 e^{x^2} x^6 \left (1+\frac {\log (2)}{4}\right )+56 e^{x^2} x^5 \left (1+\frac {8 \log (2)}{7}\right )-32 e^{x^2} x^4 \left (1-\frac {7 \log (16)}{16}\right )-32 e^{x^2} x^3 (1+\log (2))+4 x (1+\log (2)) \log (4)+\log (4) \log (16)\right )}{4 x (x+2)^2 \left (2 x^4+4 x^3+8 e^{x^2} x^2+2 e^{2 x^2} x^2+4 e^{2 x^2} x+4 e^{x^2} x^3+\log (4)\right )^2}+\frac {\log (4)-4}{4 (x+2)^2}\right )dx\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle \int \left (\frac {x^4 (-\log (256))-x^3 \log (4) (8+\log (16))-x^2 \log (4) (2+\log (256))-x \log (4) \log (8)-\log ^2(4)}{2 x (x+2)^2 \left (2 x^4+4 x^3+8 e^{x^2} x^2+2 e^{2 x^2} x^2+4 e^{2 x^2} x+4 e^{x^2} x^3+\log (4)\right )}+\frac {\log (4) \left (16 x^8+64 x^7 \left (1+\frac {\log (2)}{4}\right )+56 x^6 \left (1+\frac {8 \log (2)}{7}\right )-32 x^5 \left (1-\frac {\log (128)}{4}\right )-32 x^4 \left (1+\frac {\log (2)}{2}\right )+8 x^3 \log (2) \log (4)-4 e^{x^2} x^2 \log (256)+4 x^2 \log (4) (1+\log (16))+16 e^{x^2} x^7+64 e^{x^2} x^6 \left (1+\frac {\log (2)}{4}\right )+56 e^{x^2} x^5 \left (1+\frac {8 \log (2)}{7}\right )-32 e^{x^2} x^4 \left (1-\frac {7 \log (16)}{16}\right )-32 e^{x^2} x^3 (1+\log (2))+4 x (1+\log (2)) \log (4)+\log (4) \log (16)\right )}{4 x (x+2)^2 \left (2 x^4+4 x^3+8 e^{x^2} x^2+2 e^{2 x^2} x^2+4 e^{2 x^2} x+4 e^{x^2} x^3+\log (4)\right )^2}+\frac {\log (4)-4}{4 (x+2)^2}\right )dx\) |
| \(\Big \downarrow \) 7299 |
| \(\displaystyle \int \left (\frac {x^4 (-\log (256))-x^3 \log (4) (8+\log (16))-x^2 \log (4) (2+\log (256))-x \log (4) \log (8)-\log ^2(4)}{2 x (x+2)^2 \left (2 x^4+4 x^3+8 e^{x^2} x^2+2 e^{2 x^2} x^2+4 e^{2 x^2} x+4 e^{x^2} x^3+\log (4)\right )}+\frac {\log (4) \left (16 x^8+64 x^7 \left (1+\frac {\log (2)}{4}\right )+56 x^6 \left (1+\frac {8 \log (2)}{7}\right )-32 x^5 \left (1-\frac {\log (128)}{4}\right )-32 x^4 \left (1+\frac {\log (2)}{2}\right )+8 x^3 \log (2) \log (4)-4 e^{x^2} x^2 \log (256)+4 x^2 \log (4) (1+\log (16))+16 e^{x^2} x^7+64 e^{x^2} x^6 \left (1+\frac {\log (2)}{4}\right )+56 e^{x^2} x^5 \left (1+\frac {8 \log (2)}{7}\right )-32 e^{x^2} x^4 \left (1-\frac {7 \log (16)}{16}\right )-32 e^{x^2} x^3 (1+\log (2))+4 x (1+\log (2)) \log (4)+\log (4) \log (16)\right )}{4 x (x+2)^2 \left (2 x^4+4 x^3+8 e^{x^2} x^2+2 e^{2 x^2} x^2+4 e^{2 x^2} x+4 e^{x^2} x^3+\log (4)\right )^2}+\frac {\log (4)-4}{4 (x+2)^2}\right )dx\) |
Input:
Int[(-4*x^6 + 2*x^6*Log[2] + E^(4*x^2)*(-4*x^2 + 2*x^2*Log[2]) + E^(3*x^2) *(-16*x^3 + 8*x^3*Log[2]) + (-4*x^3 - 3*x^2*Log[2])*Log[4] + E^(2*x^2)*(-2 4*x^4 + 12*x^4*Log[2] + (-2*x - 4*x^3 + (-1 - 4*x^2)*Log[2])*Log[4]) + E^x ^2*(-16*x^5 + 8*x^5*Log[2] + (-6*x^2 - 4*x^4 + (-4*x - 4*x^3)*Log[2])*Log[ 4]))/(16*x^6 + 16*x^7 + 4*x^8 + E^(4*x^2)*(16*x^2 + 16*x^3 + 4*x^4) + E^(3 *x^2)*(64*x^3 + 64*x^4 + 16*x^5) + (8*x^3 + 4*x^4)*Log[4] + Log[4]^2 + E^( 2*x^2)*(96*x^4 + 96*x^5 + 24*x^6 + (8*x + 4*x^2)*Log[4]) + E^x^2*(64*x^5 + 64*x^6 + 16*x^7 + (16*x^2 + 8*x^3)*Log[4])),x]
Output:
$Aborted
Leaf count of result is larger than twice the leaf count of optimal. \(78\) vs. \(2(31)=62\).
Time = 3.22 (sec) , antiderivative size = 79, normalized size of antiderivative = 2.55
| method | result | size |
| risch | \(\frac {1}{2+x}-\frac {\ln \left (2\right )}{2 \left (2+x \right )}+\frac {\ln \left (2\right ) \left (\ln \left (2\right )+x \right )}{2 \left (2+x \right ) \left ({\mathrm e}^{2 x^{2}} x^{2}+2 x^{3} {\mathrm e}^{x^{2}}+x^{4}+2 x \,{\mathrm e}^{2 x^{2}}+4 x^{2} {\mathrm e}^{x^{2}}+2 x^{3}+\ln \left (2\right )\right )}\) | \(79\) |
| parallelrisch | \(-\frac {2 \,{\mathrm e}^{x^{2}} \ln \left (2\right ) x^{2}+x^{3} \ln \left (2\right )+{\mathrm e}^{2 x^{2}} \ln \left (2\right ) x -4 x^{2} {\mathrm e}^{x^{2}}-2 x^{3}-2 x \,{\mathrm e}^{2 x^{2}}-\ln \left (2\right )}{2 \left ({\mathrm e}^{2 x^{2}} x^{2}+2 x^{3} {\mathrm e}^{x^{2}}+x^{4}+2 x \,{\mathrm e}^{2 x^{2}}+4 x^{2} {\mathrm e}^{x^{2}}+2 x^{3}+\ln \left (2\right )\right )}\) | \(108\) |
Input:
int(((2*x^2*ln(2)-4*x^2)*exp(x^2)^4+(8*x^3*ln(2)-16*x^3)*exp(x^2)^3+(2*((- 4*x^2-1)*ln(2)-4*x^3-2*x)*ln(2)+12*x^4*ln(2)-24*x^4)*exp(x^2)^2+(2*((-4*x^ 3-4*x)*ln(2)-4*x^4-6*x^2)*ln(2)+8*x^5*ln(2)-16*x^5)*exp(x^2)+2*(-3*x^2*ln( 2)-4*x^3)*ln(2)+2*x^6*ln(2)-4*x^6)/((4*x^4+16*x^3+16*x^2)*exp(x^2)^4+(16*x ^5+64*x^4+64*x^3)*exp(x^2)^3+(2*(4*x^2+8*x)*ln(2)+24*x^6+96*x^5+96*x^4)*ex p(x^2)^2+(2*(8*x^3+16*x^2)*ln(2)+16*x^7+64*x^6+64*x^5)*exp(x^2)+4*ln(2)^2+ 2*(4*x^4+8*x^3)*ln(2)+4*x^8+16*x^7+16*x^6),x,method=_RETURNVERBOSE)
Output:
1/(2+x)-1/2*ln(2)/(2+x)+1/2*ln(2)*(ln(2)+x)/(2+x)/(x^2*exp(x^2)^2+2*x^3*ex p(x^2)+x^4+2*x*exp(x^2)^2+4*x^2*exp(x^2)+2*x^3+ln(2))
Leaf count of result is larger than twice the leaf count of optimal. 93 vs. \(2 (25) = 50\).
Time = 0.08 (sec) , antiderivative size = 93, normalized size of antiderivative = 3.00 \[ \int \frac {-4 x^6+2 x^6 \log (2)+e^{4 x^2} \left (-4 x^2+2 x^2 \log (2)\right )+e^{3 x^2} \left (-16 x^3+8 x^3 \log (2)\right )+\left (-4 x^3-3 x^2 \log (2)\right ) \log (4)+e^{2 x^2} \left (-24 x^4+12 x^4 \log (2)+\left (-2 x-4 x^3+\left (-1-4 x^2\right ) \log (2)\right ) \log (4)\right )+e^{x^2} \left (-16 x^5+8 x^5 \log (2)+\left (-6 x^2-4 x^4+\left (-4 x-4 x^3\right ) \log (2)\right ) \log (4)\right )}{16 x^6+16 x^7+4 x^8+e^{4 x^2} \left (16 x^2+16 x^3+4 x^4\right )+e^{3 x^2} \left (64 x^3+64 x^4+16 x^5\right )+\left (8 x^3+4 x^4\right ) \log (4)+\log ^2(4)+e^{2 x^2} \left (96 x^4+96 x^5+24 x^6+\left (8 x+4 x^2\right ) \log (4)\right )+e^{x^2} \left (64 x^5+64 x^6+16 x^7+\left (16 x^2+8 x^3\right ) \log (4)\right )} \, dx=\frac {2 \, x^{3} - {\left (x \log \left (2\right ) - 2 \, x\right )} e^{\left (2 \, x^{2}\right )} - 2 \, {\left (x^{2} \log \left (2\right ) - 2 \, x^{2}\right )} e^{\left (x^{2}\right )} - {\left (x^{3} - 1\right )} \log \left (2\right )}{2 \, {\left (x^{4} + 2 \, x^{3} + {\left (x^{2} + 2 \, x\right )} e^{\left (2 \, x^{2}\right )} + 2 \, {\left (x^{3} + 2 \, x^{2}\right )} e^{\left (x^{2}\right )} + \log \left (2\right )\right )}} \] Input:
integrate(((2*x^2*log(2)-4*x^2)*exp(x^2)^4+(8*x^3*log(2)-16*x^3)*exp(x^2)^ 3+(2*((-4*x^2-1)*log(2)-4*x^3-2*x)*log(2)+12*x^4*log(2)-24*x^4)*exp(x^2)^2 +(2*((-4*x^3-4*x)*log(2)-4*x^4-6*x^2)*log(2)+8*x^5*log(2)-16*x^5)*exp(x^2) +2*(-3*x^2*log(2)-4*x^3)*log(2)+2*x^6*log(2)-4*x^6)/((4*x^4+16*x^3+16*x^2) *exp(x^2)^4+(16*x^5+64*x^4+64*x^3)*exp(x^2)^3+(2*(4*x^2+8*x)*log(2)+24*x^6 +96*x^5+96*x^4)*exp(x^2)^2+(2*(8*x^3+16*x^2)*log(2)+16*x^7+64*x^6+64*x^5)* exp(x^2)+4*log(2)^2+2*(4*x^4+8*x^3)*log(2)+4*x^8+16*x^7+16*x^6),x, algorit hm="fricas")
Output:
1/2*(2*x^3 - (x*log(2) - 2*x)*e^(2*x^2) - 2*(x^2*log(2) - 2*x^2)*e^(x^2) - (x^3 - 1)*log(2))/(x^4 + 2*x^3 + (x^2 + 2*x)*e^(2*x^2) + 2*(x^3 + 2*x^2)* e^(x^2) + log(2))
Leaf count of result is larger than twice the leaf count of optimal. 87 vs. \(2 (26) = 52\).
Time = 0.35 (sec) , antiderivative size = 87, normalized size of antiderivative = 2.81 \[ \int \frac {-4 x^6+2 x^6 \log (2)+e^{4 x^2} \left (-4 x^2+2 x^2 \log (2)\right )+e^{3 x^2} \left (-16 x^3+8 x^3 \log (2)\right )+\left (-4 x^3-3 x^2 \log (2)\right ) \log (4)+e^{2 x^2} \left (-24 x^4+12 x^4 \log (2)+\left (-2 x-4 x^3+\left (-1-4 x^2\right ) \log (2)\right ) \log (4)\right )+e^{x^2} \left (-16 x^5+8 x^5 \log (2)+\left (-6 x^2-4 x^4+\left (-4 x-4 x^3\right ) \log (2)\right ) \log (4)\right )}{16 x^6+16 x^7+4 x^8+e^{4 x^2} \left (16 x^2+16 x^3+4 x^4\right )+e^{3 x^2} \left (64 x^3+64 x^4+16 x^5\right )+\left (8 x^3+4 x^4\right ) \log (4)+\log ^2(4)+e^{2 x^2} \left (96 x^4+96 x^5+24 x^6+\left (8 x+4 x^2\right ) \log (4)\right )+e^{x^2} \left (64 x^5+64 x^6+16 x^7+\left (16 x^2+8 x^3\right ) \log (4)\right )} \, dx=\frac {x \log {\left (2 \right )} + \log {\left (2 \right )}^{2}}{2 x^{5} + 8 x^{4} + 8 x^{3} + 2 x \log {\left (2 \right )} + \left (2 x^{3} + 8 x^{2} + 8 x\right ) e^{2 x^{2}} + \left (4 x^{4} + 16 x^{3} + 16 x^{2}\right ) e^{x^{2}} + 4 \log {\left (2 \right )}} - \frac {-2 + \log {\left (2 \right )}}{2 x + 4} \] Input:
integrate(((2*x**2*ln(2)-4*x**2)*exp(x**2)**4+(8*x**3*ln(2)-16*x**3)*exp(x **2)**3+(2*((-4*x**2-1)*ln(2)-4*x**3-2*x)*ln(2)+12*x**4*ln(2)-24*x**4)*exp (x**2)**2+(2*((-4*x**3-4*x)*ln(2)-4*x**4-6*x**2)*ln(2)+8*x**5*ln(2)-16*x** 5)*exp(x**2)+2*(-3*x**2*ln(2)-4*x**3)*ln(2)+2*x**6*ln(2)-4*x**6)/((4*x**4+ 16*x**3+16*x**2)*exp(x**2)**4+(16*x**5+64*x**4+64*x**3)*exp(x**2)**3+(2*(4 *x**2+8*x)*ln(2)+24*x**6+96*x**5+96*x**4)*exp(x**2)**2+(2*(8*x**3+16*x**2) *ln(2)+16*x**7+64*x**6+64*x**5)*exp(x**2)+4*ln(2)**2+2*(4*x**4+8*x**3)*ln( 2)+4*x**8+16*x**7+16*x**6),x)
Output:
(x*log(2) + log(2)**2)/(2*x**5 + 8*x**4 + 8*x**3 + 2*x*log(2) + (2*x**3 + 8*x**2 + 8*x)*exp(2*x**2) + (4*x**4 + 16*x**3 + 16*x**2)*exp(x**2) + 4*log (2)) - (-2 + log(2))/(2*x + 4)
Leaf count of result is larger than twice the leaf count of optimal. 82 vs. \(2 (25) = 50\).
Time = 0.22 (sec) , antiderivative size = 82, normalized size of antiderivative = 2.65 \[ \int \frac {-4 x^6+2 x^6 \log (2)+e^{4 x^2} \left (-4 x^2+2 x^2 \log (2)\right )+e^{3 x^2} \left (-16 x^3+8 x^3 \log (2)\right )+\left (-4 x^3-3 x^2 \log (2)\right ) \log (4)+e^{2 x^2} \left (-24 x^4+12 x^4 \log (2)+\left (-2 x-4 x^3+\left (-1-4 x^2\right ) \log (2)\right ) \log (4)\right )+e^{x^2} \left (-16 x^5+8 x^5 \log (2)+\left (-6 x^2-4 x^4+\left (-4 x-4 x^3\right ) \log (2)\right ) \log (4)\right )}{16 x^6+16 x^7+4 x^8+e^{4 x^2} \left (16 x^2+16 x^3+4 x^4\right )+e^{3 x^2} \left (64 x^3+64 x^4+16 x^5\right )+\left (8 x^3+4 x^4\right ) \log (4)+\log ^2(4)+e^{2 x^2} \left (96 x^4+96 x^5+24 x^6+\left (8 x+4 x^2\right ) \log (4)\right )+e^{x^2} \left (64 x^5+64 x^6+16 x^7+\left (16 x^2+8 x^3\right ) \log (4)\right )} \, dx=-\frac {x^{3} {\left (\log \left (2\right ) - 2\right )} + 2 \, x^{2} {\left (\log \left (2\right ) - 2\right )} e^{\left (x^{2}\right )} + x {\left (\log \left (2\right ) - 2\right )} e^{\left (2 \, x^{2}\right )} - \log \left (2\right )}{2 \, {\left (x^{4} + 2 \, x^{3} + {\left (x^{2} + 2 \, x\right )} e^{\left (2 \, x^{2}\right )} + 2 \, {\left (x^{3} + 2 \, x^{2}\right )} e^{\left (x^{2}\right )} + \log \left (2\right )\right )}} \] Input:
integrate(((2*x^2*log(2)-4*x^2)*exp(x^2)^4+(8*x^3*log(2)-16*x^3)*exp(x^2)^ 3+(2*((-4*x^2-1)*log(2)-4*x^3-2*x)*log(2)+12*x^4*log(2)-24*x^4)*exp(x^2)^2 +(2*((-4*x^3-4*x)*log(2)-4*x^4-6*x^2)*log(2)+8*x^5*log(2)-16*x^5)*exp(x^2) +2*(-3*x^2*log(2)-4*x^3)*log(2)+2*x^6*log(2)-4*x^6)/((4*x^4+16*x^3+16*x^2) *exp(x^2)^4+(16*x^5+64*x^4+64*x^3)*exp(x^2)^3+(2*(4*x^2+8*x)*log(2)+24*x^6 +96*x^5+96*x^4)*exp(x^2)^2+(2*(8*x^3+16*x^2)*log(2)+16*x^7+64*x^6+64*x^5)* exp(x^2)+4*log(2)^2+2*(4*x^4+8*x^3)*log(2)+4*x^8+16*x^7+16*x^6),x, algorit hm="maxima")
Output:
-1/2*(x^3*(log(2) - 2) + 2*x^2*(log(2) - 2)*e^(x^2) + x*(log(2) - 2)*e^(2* x^2) - log(2))/(x^4 + 2*x^3 + (x^2 + 2*x)*e^(2*x^2) + 2*(x^3 + 2*x^2)*e^(x ^2) + log(2))
Leaf count of result is larger than twice the leaf count of optimal. 205 vs. \(2 (25) = 50\).
Time = 3.69 (sec) , antiderivative size = 205, normalized size of antiderivative = 6.61 \[ \int \frac {-4 x^6+2 x^6 \log (2)+e^{4 x^2} \left (-4 x^2+2 x^2 \log (2)\right )+e^{3 x^2} \left (-16 x^3+8 x^3 \log (2)\right )+\left (-4 x^3-3 x^2 \log (2)\right ) \log (4)+e^{2 x^2} \left (-24 x^4+12 x^4 \log (2)+\left (-2 x-4 x^3+\left (-1-4 x^2\right ) \log (2)\right ) \log (4)\right )+e^{x^2} \left (-16 x^5+8 x^5 \log (2)+\left (-6 x^2-4 x^4+\left (-4 x-4 x^3\right ) \log (2)\right ) \log (4)\right )}{16 x^6+16 x^7+4 x^8+e^{4 x^2} \left (16 x^2+16 x^3+4 x^4\right )+e^{3 x^2} \left (64 x^3+64 x^4+16 x^5\right )+\left (8 x^3+4 x^4\right ) \log (4)+\log ^2(4)+e^{2 x^2} \left (96 x^4+96 x^5+24 x^6+\left (8 x+4 x^2\right ) \log (4)\right )+e^{x^2} \left (64 x^5+64 x^6+16 x^7+\left (16 x^2+8 x^3\right ) \log (4)\right )} \, dx=-\frac {x^{4} \log \left (2\right ) + 2 \, x^{3} e^{\left (x^{2}\right )} \log \left (2\right ) - 2 \, x^{4} - 4 \, x^{3} e^{\left (x^{2}\right )} + 2 \, x^{3} \log \left (2\right ) + x^{2} e^{\left (2 \, x^{2}\right )} \log \left (2\right ) + 4 \, x^{2} e^{\left (x^{2}\right )} \log \left (2\right ) - 4 \, x^{3} - 2 \, x^{2} e^{\left (2 \, x^{2}\right )} - 8 \, x^{2} e^{\left (x^{2}\right )} + 2 \, x e^{\left (2 \, x^{2}\right )} \log \left (2\right ) - 4 \, x e^{\left (2 \, x^{2}\right )} - 2 \, x \log \left (2\right ) - \log \left (2\right )^{2} - 2 \, \log \left (2\right )}{2 \, {\left (x^{5} + 2 \, x^{4} e^{\left (x^{2}\right )} + 4 \, x^{4} + x^{3} e^{\left (2 \, x^{2}\right )} + 8 \, x^{3} e^{\left (x^{2}\right )} + 4 \, x^{3} + 4 \, x^{2} e^{\left (2 \, x^{2}\right )} + 8 \, x^{2} e^{\left (x^{2}\right )} + 4 \, x e^{\left (2 \, x^{2}\right )} + x \log \left (2\right ) + 2 \, \log \left (2\right )\right )}} \] Input:
integrate(((2*x^2*log(2)-4*x^2)*exp(x^2)^4+(8*x^3*log(2)-16*x^3)*exp(x^2)^ 3+(2*((-4*x^2-1)*log(2)-4*x^3-2*x)*log(2)+12*x^4*log(2)-24*x^4)*exp(x^2)^2 +(2*((-4*x^3-4*x)*log(2)-4*x^4-6*x^2)*log(2)+8*x^5*log(2)-16*x^5)*exp(x^2) +2*(-3*x^2*log(2)-4*x^3)*log(2)+2*x^6*log(2)-4*x^6)/((4*x^4+16*x^3+16*x^2) *exp(x^2)^4+(16*x^5+64*x^4+64*x^3)*exp(x^2)^3+(2*(4*x^2+8*x)*log(2)+24*x^6 +96*x^5+96*x^4)*exp(x^2)^2+(2*(8*x^3+16*x^2)*log(2)+16*x^7+64*x^6+64*x^5)* exp(x^2)+4*log(2)^2+2*(4*x^4+8*x^3)*log(2)+4*x^8+16*x^7+16*x^6),x, algorit hm="giac")
Output:
-1/2*(x^4*log(2) + 2*x^3*e^(x^2)*log(2) - 2*x^4 - 4*x^3*e^(x^2) + 2*x^3*lo g(2) + x^2*e^(2*x^2)*log(2) + 4*x^2*e^(x^2)*log(2) - 4*x^3 - 2*x^2*e^(2*x^ 2) - 8*x^2*e^(x^2) + 2*x*e^(2*x^2)*log(2) - 4*x*e^(2*x^2) - 2*x*log(2) - l og(2)^2 - 2*log(2))/(x^5 + 2*x^4*e^(x^2) + 4*x^4 + x^3*e^(2*x^2) + 8*x^3*e ^(x^2) + 4*x^3 + 4*x^2*e^(2*x^2) + 8*x^2*e^(x^2) + 4*x*e^(2*x^2) + x*log(2 ) + 2*log(2))
Timed out. \[ \int \frac {-4 x^6+2 x^6 \log (2)+e^{4 x^2} \left (-4 x^2+2 x^2 \log (2)\right )+e^{3 x^2} \left (-16 x^3+8 x^3 \log (2)\right )+\left (-4 x^3-3 x^2 \log (2)\right ) \log (4)+e^{2 x^2} \left (-24 x^4+12 x^4 \log (2)+\left (-2 x-4 x^3+\left (-1-4 x^2\right ) \log (2)\right ) \log (4)\right )+e^{x^2} \left (-16 x^5+8 x^5 \log (2)+\left (-6 x^2-4 x^4+\left (-4 x-4 x^3\right ) \log (2)\right ) \log (4)\right )}{16 x^6+16 x^7+4 x^8+e^{4 x^2} \left (16 x^2+16 x^3+4 x^4\right )+e^{3 x^2} \left (64 x^3+64 x^4+16 x^5\right )+\left (8 x^3+4 x^4\right ) \log (4)+\log ^2(4)+e^{2 x^2} \left (96 x^4+96 x^5+24 x^6+\left (8 x+4 x^2\right ) \log (4)\right )+e^{x^2} \left (64 x^5+64 x^6+16 x^7+\left (16 x^2+8 x^3\right ) \log (4)\right )} \, dx=\int -\frac {2\,\ln \left (2\right )\,\left (4\,x^3+3\,\ln \left (2\right )\,x^2\right )-{\mathrm {e}}^{4\,x^2}\,\left (2\,x^2\,\ln \left (2\right )-4\,x^2\right )-{\mathrm {e}}^{3\,x^2}\,\left (8\,x^3\,\ln \left (2\right )-16\,x^3\right )-2\,x^6\,\ln \left (2\right )+{\mathrm {e}}^{x^2}\,\left (2\,\ln \left (2\right )\,\left (\ln \left (2\right )\,\left (4\,x^3+4\,x\right )+6\,x^2+4\,x^4\right )-8\,x^5\,\ln \left (2\right )+16\,x^5\right )+{\mathrm {e}}^{2\,x^2}\,\left (2\,\ln \left (2\right )\,\left (2\,x+\ln \left (2\right )\,\left (4\,x^2+1\right )+4\,x^3\right )-12\,x^4\,\ln \left (2\right )+24\,x^4\right )+4\,x^6}{{\mathrm {e}}^{2\,x^2}\,\left (2\,\ln \left (2\right )\,\left (4\,x^2+8\,x\right )+96\,x^4+96\,x^5+24\,x^6\right )+{\mathrm {e}}^{x^2}\,\left (2\,\ln \left (2\right )\,\left (8\,x^3+16\,x^2\right )+64\,x^5+64\,x^6+16\,x^7\right )+2\,\ln \left (2\right )\,\left (4\,x^4+8\,x^3\right )+4\,{\ln \left (2\right )}^2+16\,x^6+16\,x^7+4\,x^8+{\mathrm {e}}^{4\,x^2}\,\left (4\,x^4+16\,x^3+16\,x^2\right )+{\mathrm {e}}^{3\,x^2}\,\left (16\,x^5+64\,x^4+64\,x^3\right )} \,d x \] Input:
int(-(2*log(2)*(3*x^2*log(2) + 4*x^3) - exp(4*x^2)*(2*x^2*log(2) - 4*x^2) - exp(3*x^2)*(8*x^3*log(2) - 16*x^3) - 2*x^6*log(2) + exp(x^2)*(2*log(2)*( log(2)*(4*x + 4*x^3) + 6*x^2 + 4*x^4) - 8*x^5*log(2) + 16*x^5) + exp(2*x^2 )*(2*log(2)*(2*x + log(2)*(4*x^2 + 1) + 4*x^3) - 12*x^4*log(2) + 24*x^4) + 4*x^6)/(exp(2*x^2)*(2*log(2)*(8*x + 4*x^2) + 96*x^4 + 96*x^5 + 24*x^6) + exp(x^2)*(2*log(2)*(16*x^2 + 8*x^3) + 64*x^5 + 64*x^6 + 16*x^7) + 2*log(2) *(8*x^3 + 4*x^4) + 4*log(2)^2 + 16*x^6 + 16*x^7 + 4*x^8 + exp(4*x^2)*(16*x ^2 + 16*x^3 + 4*x^4) + exp(3*x^2)*(64*x^3 + 64*x^4 + 16*x^5)),x)
Output:
int(-(2*log(2)*(3*x^2*log(2) + 4*x^3) - exp(4*x^2)*(2*x^2*log(2) - 4*x^2) - exp(3*x^2)*(8*x^3*log(2) - 16*x^3) - 2*x^6*log(2) + exp(x^2)*(2*log(2)*( log(2)*(4*x + 4*x^3) + 6*x^2 + 4*x^4) - 8*x^5*log(2) + 16*x^5) + exp(2*x^2 )*(2*log(2)*(2*x + log(2)*(4*x^2 + 1) + 4*x^3) - 12*x^4*log(2) + 24*x^4) + 4*x^6)/(exp(2*x^2)*(2*log(2)*(8*x + 4*x^2) + 96*x^4 + 96*x^5 + 24*x^6) + exp(x^2)*(2*log(2)*(16*x^2 + 8*x^3) + 64*x^5 + 64*x^6 + 16*x^7) + 2*log(2) *(8*x^3 + 4*x^4) + 4*log(2)^2 + 16*x^6 + 16*x^7 + 4*x^8 + exp(4*x^2)*(16*x ^2 + 16*x^3 + 4*x^4) + exp(3*x^2)*(64*x^3 + 64*x^4 + 16*x^5)), x)
\[ \int \frac {-4 x^6+2 x^6 \log (2)+e^{4 x^2} \left (-4 x^2+2 x^2 \log (2)\right )+e^{3 x^2} \left (-16 x^3+8 x^3 \log (2)\right )+\left (-4 x^3-3 x^2 \log (2)\right ) \log (4)+e^{2 x^2} \left (-24 x^4+12 x^4 \log (2)+\left (-2 x-4 x^3+\left (-1-4 x^2\right ) \log (2)\right ) \log (4)\right )+e^{x^2} \left (-16 x^5+8 x^5 \log (2)+\left (-6 x^2-4 x^4+\left (-4 x-4 x^3\right ) \log (2)\right ) \log (4)\right )}{16 x^6+16 x^7+4 x^8+e^{4 x^2} \left (16 x^2+16 x^3+4 x^4\right )+e^{3 x^2} \left (64 x^3+64 x^4+16 x^5\right )+\left (8 x^3+4 x^4\right ) \log (4)+\log ^2(4)+e^{2 x^2} \left (96 x^4+96 x^5+24 x^6+\left (8 x+4 x^2\right ) \log (4)\right )+e^{x^2} \left (64 x^5+64 x^6+16 x^7+\left (16 x^2+8 x^3\right ) \log (4)\right )} \, dx=\int \frac {\left (2 \,\mathrm {log}\left (2\right ) x^{2}-4 x^{2}\right ) \left ({\mathrm e}^{x^{2}}\right )^{4}+\left (8 \,\mathrm {log}\left (2\right ) x^{3}-16 x^{3}\right ) \left ({\mathrm e}^{x^{2}}\right )^{3}+\left (2 \left (\left (-4 x^{2}-1\right ) \mathrm {log}\left (2\right )-4 x^{3}-2 x \right ) \mathrm {log}\left (2\right )+12 \,\mathrm {log}\left (2\right ) x^{4}-24 x^{4}\right ) \left ({\mathrm e}^{x^{2}}\right )^{2}+\left (2 \left (\left (-4 x^{3}-4 x \right ) \mathrm {log}\left (2\right )-4 x^{4}-6 x^{2}\right ) \mathrm {log}\left (2\right )+8 x^{5} \mathrm {log}\left (2\right )-16 x^{5}\right ) {\mathrm e}^{x^{2}}+2 \left (-3 \,\mathrm {log}\left (2\right ) x^{2}-4 x^{3}\right ) \mathrm {log}\left (2\right )+2 x^{6} \mathrm {log}\left (2\right )-4 x^{6}}{\left (4 x^{4}+16 x^{3}+16 x^{2}\right ) \left ({\mathrm e}^{x^{2}}\right )^{4}+\left (16 x^{5}+64 x^{4}+64 x^{3}\right ) \left ({\mathrm e}^{x^{2}}\right )^{3}+\left (2 \left (4 x^{2}+8 x \right ) \mathrm {log}\left (2\right )+24 x^{6}+96 x^{5}+96 x^{4}\right ) \left ({\mathrm e}^{x^{2}}\right )^{2}+\left (2 \left (8 x^{3}+16 x^{2}\right ) \mathrm {log}\left (2\right )+16 x^{7}+64 x^{6}+64 x^{5}\right ) {\mathrm e}^{x^{2}}+4 \mathrm {log}\left (2\right )^{2}+2 \left (4 x^{4}+8 x^{3}\right ) \mathrm {log}\left (2\right )+4 x^{8}+16 x^{7}+16 x^{6}}d x \] Input:
int(((2*x^2*log(2)-4*x^2)*exp(x^2)^4+(8*x^3*log(2)-16*x^3)*exp(x^2)^3+(2*( (-4*x^2-1)*log(2)-4*x^3-2*x)*log(2)+12*x^4*log(2)-24*x^4)*exp(x^2)^2+(2*(( -4*x^3-4*x)*log(2)-4*x^4-6*x^2)*log(2)+8*x^5*log(2)-16*x^5)*exp(x^2)+2*(-3 *x^2*log(2)-4*x^3)*log(2)+2*x^6*log(2)-4*x^6)/((4*x^4+16*x^3+16*x^2)*exp(x ^2)^4+(16*x^5+64*x^4+64*x^3)*exp(x^2)^3+(2*(4*x^2+8*x)*log(2)+24*x^6+96*x^ 5+96*x^4)*exp(x^2)^2+(2*(8*x^3+16*x^2)*log(2)+16*x^7+64*x^6+64*x^5)*exp(x^ 2)+4*log(2)^2+2*(4*x^4+8*x^3)*log(2)+4*x^8+16*x^7+16*x^6),x)
Output:
int(((2*x^2*log(2)-4*x^2)*exp(x^2)^4+(8*x^3*log(2)-16*x^3)*exp(x^2)^3+(2*( (-4*x^2-1)*log(2)-4*x^3-2*x)*log(2)+12*x^4*log(2)-24*x^4)*exp(x^2)^2+(2*(( -4*x^3-4*x)*log(2)-4*x^4-6*x^2)*log(2)+8*x^5*log(2)-16*x^5)*exp(x^2)+2*(-3 *x^2*log(2)-4*x^3)*log(2)+2*x^6*log(2)-4*x^6)/((4*x^4+16*x^3+16*x^2)*exp(x ^2)^4+(16*x^5+64*x^4+64*x^3)*exp(x^2)^3+(2*(4*x^2+8*x)*log(2)+24*x^6+96*x^ 5+96*x^4)*exp(x^2)^2+(2*(8*x^3+16*x^2)*log(2)+16*x^7+64*x^6+64*x^5)*exp(x^ 2)+4*log(2)^2+2*(4*x^4+8*x^3)*log(2)+4*x^8+16*x^7+16*x^6),x)