Integrand size = 177, antiderivative size = 27 \[ \int \frac {e^{-\frac {2 x+3 x^2}{\log ^2(4)}} \left (-32 x^4-96 x^5-e^{\frac {2 x+3 x^2}{\log ^2(4)}} \log ^2(4)+64 x^3 \log ^2(4)+e^{\frac {3 \left (2 x+3 x^2\right )}{4 \log ^2(4)}} \left (-256 x-768 x^2+512 \log ^2(4)\right )+e^{\frac {2 x+3 x^2}{2 \log ^2(4)}} \left (-384 x^2-1152 x^3+768 x \log ^2(4)\right )+e^{\frac {2 x+3 x^2}{4 \log ^2(4)}} \left (-192 x^3-576 x^4+384 x^2 \log ^2(4)\right )\right )}{\log ^2(4)} \, dx=-x+\left (4+2 e^{-\frac {x (2+3 x)}{4 \log ^2(4)}} x\right )^4 \] Output:
(2*x/exp(1/16*x*(2+3*x)/ln(2)^2)+4)^4-x
Leaf count is larger than twice the leaf count of optimal. \(94\) vs. \(2(27)=54\).
Time = 0.19 (sec) , antiderivative size = 94, normalized size of antiderivative = 3.48 \[ \int \frac {e^{-\frac {2 x+3 x^2}{\log ^2(4)}} \left (-32 x^4-96 x^5-e^{\frac {2 x+3 x^2}{\log ^2(4)}} \log ^2(4)+64 x^3 \log ^2(4)+e^{\frac {3 \left (2 x+3 x^2\right )}{4 \log ^2(4)}} \left (-256 x-768 x^2+512 \log ^2(4)\right )+e^{\frac {2 x+3 x^2}{2 \log ^2(4)}} \left (-384 x^2-1152 x^3+768 x \log ^2(4)\right )+e^{\frac {2 x+3 x^2}{4 \log ^2(4)}} \left (-192 x^3-576 x^4+384 x^2 \log ^2(4)\right )\right )}{\log ^2(4)} \, dx=-e^{-\frac {x (2+3 x)}{\log ^2(4)}} x \left (-512 e^{\frac {3 x (2+3 x)}{4 \log ^2(4)}}+e^{\frac {x (2+3 x)}{\log ^2(4)}}-384 e^{\frac {x (2+3 x)}{2 \log ^2(4)}} x-128 e^{\frac {x (2+3 x)}{4 \log ^2(4)}} x^2-16 x^3\right ) \] Input:
Integrate[(-32*x^4 - 96*x^5 - E^((2*x + 3*x^2)/Log[4]^2)*Log[4]^2 + 64*x^3 *Log[4]^2 + E^((3*(2*x + 3*x^2))/(4*Log[4]^2))*(-256*x - 768*x^2 + 512*Log [4]^2) + E^((2*x + 3*x^2)/(2*Log[4]^2))*(-384*x^2 - 1152*x^3 + 768*x*Log[4 ]^2) + E^((2*x + 3*x^2)/(4*Log[4]^2))*(-192*x^3 - 576*x^4 + 384*x^2*Log[4] ^2))/(E^((2*x + 3*x^2)/Log[4]^2)*Log[4]^2),x]
Output:
-((x*(-512*E^((3*x*(2 + 3*x))/(4*Log[4]^2)) + E^((x*(2 + 3*x))/Log[4]^2) - 384*E^((x*(2 + 3*x))/(2*Log[4]^2))*x - 128*E^((x*(2 + 3*x))/(4*Log[4]^2)) *x^2 - 16*x^3))/E^((x*(2 + 3*x))/Log[4]^2))
Leaf count is larger than twice the leaf count of optimal. \(180\) vs. \(2(27)=54\).
Time = 9.15 (sec) , antiderivative size = 180, normalized size of antiderivative = 6.67, number of steps used = 5, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.028, Rules used = {27, 25, 7292, 7293, 2009}
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.
| \(\displaystyle \int \frac {e^{-\frac {3 x^2+2 x}{\log ^2(4)}} \left (-96 x^5-32 x^4+64 x^3 \log ^2(4)+e^{\frac {3 \left (3 x^2+2 x\right )}{4 \log ^2(4)}} \left (-768 x^2-256 x+512 \log ^2(4)\right )-\log ^2(4) e^{\frac {3 x^2+2 x}{\log ^2(4)}}+e^{\frac {3 x^2+2 x}{2 \log ^2(4)}} \left (-1152 x^3-384 x^2+768 x \log ^2(4)\right )+e^{\frac {3 x^2+2 x}{4 \log ^2(4)}} \left (-576 x^4-192 x^3+384 x^2 \log ^2(4)\right )\right )}{\log ^2(4)} \, dx\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\int -e^{-\frac {3 x^2+2 x}{\log ^2(4)}} \left (96 x^5+32 x^4-64 \log ^2(4) x^3+256 e^{\frac {3 \left (3 x^2+2 x\right )}{4 \log ^2(4)}} \left (3 x^2+x-2 \log ^2(4)\right )+384 e^{\frac {3 x^2+2 x}{2 \log ^2(4)}} \left (3 x^3+x^2-2 \log ^2(4) x\right )+192 e^{\frac {3 x^2+2 x}{4 \log ^2(4)}} \left (3 x^4+x^3-2 \log ^2(4) x^2\right )+e^{\frac {3 x^2+2 x}{\log ^2(4)}} \log ^2(4)\right )dx}{\log ^2(4)}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle -\frac {\int e^{-\frac {3 x^2+2 x}{\log ^2(4)}} \left (96 x^5+32 x^4-64 \log ^2(4) x^3+256 e^{\frac {3 \left (3 x^2+2 x\right )}{4 \log ^2(4)}} \left (3 x^2+x-2 \log ^2(4)\right )+384 e^{\frac {3 x^2+2 x}{2 \log ^2(4)}} \left (3 x^3+x^2-2 \log ^2(4) x\right )+192 e^{\frac {3 x^2+2 x}{4 \log ^2(4)}} \left (3 x^4+x^3-2 \log ^2(4) x^2\right )+e^{\frac {3 x^2+2 x}{\log ^2(4)}} \log ^2(4)\right )dx}{\log ^2(4)}\) |
| \(\Big \downarrow \) 7292 |
| \(\displaystyle -\frac {\int e^{-\frac {x (3 x+2)}{\log ^2(4)}} \left (96 x^5+32 x^4-64 \log ^2(4) x^3+256 e^{\frac {3 \left (3 x^2+2 x\right )}{4 \log ^2(4)}} \left (3 x^2+x-2 \log ^2(4)\right )+384 e^{\frac {3 x^2+2 x}{2 \log ^2(4)}} \left (3 x^3+x^2-2 \log ^2(4) x\right )+192 e^{\frac {3 x^2+2 x}{4 \log ^2(4)}} \left (3 x^4+x^3-2 \log ^2(4) x^2\right )+e^{\frac {3 x^2+2 x}{\log ^2(4)}} \log ^2(4)\right )dx}{\log ^2(4)}\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle -\frac {\int \left (96 e^{-\frac {x (3 x+2)}{\log ^2(4)}} x^5+32 e^{-\frac {x (3 x+2)}{\log ^2(4)}} x^4-64 e^{-\frac {x (3 x+2)}{\log ^2(4)}} \log ^2(4) x^3+192 e^{-\frac {3 x (3 x+2)}{4 \log ^2(4)}} \left (3 x^2+x-2 \log ^2(4)\right ) x^2+384 e^{-\frac {x (3 x+2)}{2 \log ^2(4)}} \left (3 x^2+x-2 \log ^2(4)\right ) x+256 e^{-\frac {x (3 x+2)}{4 \log ^2(4)}} \left (3 x^2+x-2 \log ^2(4)\right )+\log ^2(4)\right )dx}{\log ^2(4)}\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle -\frac {-\frac {256 \left (3 x^2+x\right ) x^2 e^{-\frac {3 x (3 x+2)}{4 \log ^2(4)}}}{\frac {3 x}{\log ^2(4)}+\frac {3 x+2}{\log ^2(4)}}-\frac {768 \left (3 x^2+x\right ) x e^{-\frac {x (3 x+2)}{2 \log ^2(4)}}}{\frac {3 x}{\log ^2(4)}+\frac {3 x+2}{\log ^2(4)}}-\frac {1024 \left (3 x^2+x\right ) e^{-\frac {x (3 x+2)}{4 \log ^2(4)}}}{\frac {3 x}{\log ^2(4)}+\frac {3 x+2}{\log ^2(4)}}-16 x^4 \log ^2(4) e^{-\frac {3 x^2}{\log ^2(4)}-\frac {2 x}{\log ^2(4)}}+x \log ^2(4)}{\log ^2(4)}\) |
Input:
Int[(-32*x^4 - 96*x^5 - E^((2*x + 3*x^2)/Log[4]^2)*Log[4]^2 + 64*x^3*Log[4 ]^2 + E^((3*(2*x + 3*x^2))/(4*Log[4]^2))*(-256*x - 768*x^2 + 512*Log[4]^2) + E^((2*x + 3*x^2)/(2*Log[4]^2))*(-384*x^2 - 1152*x^3 + 768*x*Log[4]^2) + E^((2*x + 3*x^2)/(4*Log[4]^2))*(-192*x^3 - 576*x^4 + 384*x^2*Log[4]^2))/( E^((2*x + 3*x^2)/Log[4]^2)*Log[4]^2),x]
Output:
-(((-1024*(x + 3*x^2))/(E^((x*(2 + 3*x))/(4*Log[4]^2))*((3*x)/Log[4]^2 + ( 2 + 3*x)/Log[4]^2)) - (768*x*(x + 3*x^2))/(E^((x*(2 + 3*x))/(2*Log[4]^2))* ((3*x)/Log[4]^2 + (2 + 3*x)/Log[4]^2)) - (256*x^2*(x + 3*x^2))/(E^((3*x*(2 + 3*x))/(4*Log[4]^2))*((3*x)/Log[4]^2 + (2 + 3*x)/Log[4]^2)) + x*Log[4]^2 - 16*E^((-2*x)/Log[4]^2 - (3*x^2)/Log[4]^2)*x^4*Log[4]^2)/Log[4]^2)
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
Leaf count of result is larger than twice the leaf count of optimal. \(74\) vs. \(2(26)=52\).
Time = 13.47 (sec) , antiderivative size = 75, normalized size of antiderivative = 2.78
| method | result | size |
| risch | \(-x +512 x \,{\mathrm e}^{-\frac {x \left (2+3 x \right )}{16 \ln \left (2\right )^{2}}}+384 x^{2} {\mathrm e}^{-\frac {x \left (2+3 x \right )}{8 \ln \left (2\right )^{2}}}+128 x^{3} {\mathrm e}^{-\frac {3 x \left (2+3 x \right )}{16 \ln \left (2\right )^{2}}}+16 x^{4} {\mathrm e}^{-\frac {x \left (2+3 x \right )}{4 \ln \left (2\right )^{2}}}\) | \(75\) |
| parts | \(-x +512 x \,{\mathrm e}^{-\frac {3 x^{2}}{16 \ln \left (2\right )^{2}}-\frac {x}{8 \ln \left (2\right )^{2}}}+384 x^{2} {\mathrm e}^{-\frac {3 x^{2}}{8 \ln \left (2\right )^{2}}-\frac {x}{4 \ln \left (2\right )^{2}}}+128 x^{3} {\mathrm e}^{-\frac {9 x^{2}}{16 \ln \left (2\right )^{2}}-\frac {3 x}{8 \ln \left (2\right )^{2}}}+16 x^{4} {\mathrm e}^{-\frac {3 x^{2}}{4 \ln \left (2\right )^{2}}-\frac {x}{2 \ln \left (2\right )^{2}}}\) | \(95\) |
| default | \(\frac {-4 x \ln \left (2\right )^{2}+64 \ln \left (2\right )^{2} x^{4} {\mathrm e}^{-\frac {3 x^{2}}{4 \ln \left (2\right )^{2}}-\frac {x}{2 \ln \left (2\right )^{2}}}+512 \ln \left (2\right )^{2} x^{3} {\mathrm e}^{-\frac {9 x^{2}}{16 \ln \left (2\right )^{2}}-\frac {3 x}{8 \ln \left (2\right )^{2}}}+1536 \ln \left (2\right )^{2} x^{2} {\mathrm e}^{-\frac {3 x^{2}}{8 \ln \left (2\right )^{2}}-\frac {x}{4 \ln \left (2\right )^{2}}}+2048 \ln \left (2\right )^{2} x \,{\mathrm e}^{-\frac {3 x^{2}}{16 \ln \left (2\right )^{2}}-\frac {x}{8 \ln \left (2\right )^{2}}}}{4 \ln \left (2\right )^{2}}\) | \(121\) |
| parallelrisch | \(-\frac {\left (12 x \,{\mathrm e}^{\frac {3 x^{2}+2 x}{4 \ln \left (2\right )^{2}}} \ln \left (2\right )^{2}-192 x^{4} \ln \left (2\right )^{2}-1536 \ln \left (2\right )^{2} x^{3} {\mathrm e}^{\frac {3 x^{2}+2 x}{16 \ln \left (2\right )^{2}}}-4608 \ln \left (2\right )^{2} x^{2} {\mathrm e}^{\frac {3 x^{2}+2 x}{8 \ln \left (2\right )^{2}}}-6144 \ln \left (2\right )^{2} x \,{\mathrm e}^{\frac {\frac {9}{16} x^{2}+\frac {3}{8} x}{\ln \left (2\right )^{2}}}\right ) {\mathrm e}^{-\frac {x \left (2+3 x \right )}{4 \ln \left (2\right )^{2}}}}{12 \ln \left (2\right )^{2}}\) | \(134\) |
Input:
int(1/4*(-4*ln(2)^2*exp(1/16*(3*x^2+2*x)/ln(2)^2)^4+(2048*ln(2)^2-768*x^2- 256*x)*exp(1/16*(3*x^2+2*x)/ln(2)^2)^3+(3072*x*ln(2)^2-1152*x^3-384*x^2)*e xp(1/16*(3*x^2+2*x)/ln(2)^2)^2+(1536*x^2*ln(2)^2-576*x^4-192*x^3)*exp(1/16 *(3*x^2+2*x)/ln(2)^2)+256*x^3*ln(2)^2-96*x^5-32*x^4)/ln(2)^2/exp(1/16*(3*x ^2+2*x)/ln(2)^2)^4,x,method=_RETURNVERBOSE)
Output:
-x+512*x*exp(-1/16*x*(2+3*x)/ln(2)^2)+384*x^2*exp(-1/8*x*(2+3*x)/ln(2)^2)+ 128*x^3*exp(-3/16*x*(2+3*x)/ln(2)^2)+16*x^4*exp(-1/4*x*(2+3*x)/ln(2)^2)
Leaf count of result is larger than twice the leaf count of optimal. 103 vs. \(2 (25) = 50\).
Time = 0.09 (sec) , antiderivative size = 103, normalized size of antiderivative = 3.81 \[ \int \frac {e^{-\frac {2 x+3 x^2}{\log ^2(4)}} \left (-32 x^4-96 x^5-e^{\frac {2 x+3 x^2}{\log ^2(4)}} \log ^2(4)+64 x^3 \log ^2(4)+e^{\frac {3 \left (2 x+3 x^2\right )}{4 \log ^2(4)}} \left (-256 x-768 x^2+512 \log ^2(4)\right )+e^{\frac {2 x+3 x^2}{2 \log ^2(4)}} \left (-384 x^2-1152 x^3+768 x \log ^2(4)\right )+e^{\frac {2 x+3 x^2}{4 \log ^2(4)}} \left (-192 x^3-576 x^4+384 x^2 \log ^2(4)\right )\right )}{\log ^2(4)} \, dx={\left (16 \, x^{4} + 128 \, x^{3} e^{\left (\frac {3 \, x^{2} + 2 \, x}{16 \, \log \left (2\right )^{2}}\right )} + 384 \, x^{2} e^{\left (\frac {3 \, x^{2} + 2 \, x}{8 \, \log \left (2\right )^{2}}\right )} - x e^{\left (\frac {3 \, x^{2} + 2 \, x}{4 \, \log \left (2\right )^{2}}\right )} + 512 \, x e^{\left (\frac {3 \, {\left (3 \, x^{2} + 2 \, x\right )}}{16 \, \log \left (2\right )^{2}}\right )}\right )} e^{\left (-\frac {3 \, x^{2} + 2 \, x}{4 \, \log \left (2\right )^{2}}\right )} \] Input:
integrate(1/4*(-4*log(2)^2*exp(1/16*(3*x^2+2*x)/log(2)^2)^4+(2048*log(2)^2 -768*x^2-256*x)*exp(1/16*(3*x^2+2*x)/log(2)^2)^3+(3072*x*log(2)^2-1152*x^3 -384*x^2)*exp(1/16*(3*x^2+2*x)/log(2)^2)^2+(1536*x^2*log(2)^2-576*x^4-192* x^3)*exp(1/16*(3*x^2+2*x)/log(2)^2)+256*x^3*log(2)^2-96*x^5-32*x^4)/log(2) ^2/exp(1/16*(3*x^2+2*x)/log(2)^2)^4,x, algorithm="fricas")
Output:
(16*x^4 + 128*x^3*e^(1/16*(3*x^2 + 2*x)/log(2)^2) + 384*x^2*e^(1/8*(3*x^2 + 2*x)/log(2)^2) - x*e^(1/4*(3*x^2 + 2*x)/log(2)^2) + 512*x*e^(3/16*(3*x^2 + 2*x)/log(2)^2))*e^(-1/4*(3*x^2 + 2*x)/log(2)^2)
Leaf count of result is larger than twice the leaf count of optimal. 88 vs. \(2 (22) = 44\).
Time = 0.18 (sec) , antiderivative size = 88, normalized size of antiderivative = 3.26 \[ \int \frac {e^{-\frac {2 x+3 x^2}{\log ^2(4)}} \left (-32 x^4-96 x^5-e^{\frac {2 x+3 x^2}{\log ^2(4)}} \log ^2(4)+64 x^3 \log ^2(4)+e^{\frac {3 \left (2 x+3 x^2\right )}{4 \log ^2(4)}} \left (-256 x-768 x^2+512 \log ^2(4)\right )+e^{\frac {2 x+3 x^2}{2 \log ^2(4)}} \left (-384 x^2-1152 x^3+768 x \log ^2(4)\right )+e^{\frac {2 x+3 x^2}{4 \log ^2(4)}} \left (-192 x^3-576 x^4+384 x^2 \log ^2(4)\right )\right )}{\log ^2(4)} \, dx=16 x^{4} e^{- \frac {4 \cdot \left (\frac {3 x^{2}}{16} + \frac {x}{8}\right )}{\log {\left (2 \right )}^{2}}} + 128 x^{3} e^{- \frac {3 \cdot \left (\frac {3 x^{2}}{16} + \frac {x}{8}\right )}{\log {\left (2 \right )}^{2}}} + 384 x^{2} e^{- \frac {2 \cdot \left (\frac {3 x^{2}}{16} + \frac {x}{8}\right )}{\log {\left (2 \right )}^{2}}} - x + 512 x e^{- \frac {\frac {3 x^{2}}{16} + \frac {x}{8}}{\log {\left (2 \right )}^{2}}} \] Input:
integrate(1/4*(-4*ln(2)**2*exp(1/16*(3*x**2+2*x)/ln(2)**2)**4+(2048*ln(2)* *2-768*x**2-256*x)*exp(1/16*(3*x**2+2*x)/ln(2)**2)**3+(3072*x*ln(2)**2-115 2*x**3-384*x**2)*exp(1/16*(3*x**2+2*x)/ln(2)**2)**2+(1536*x**2*ln(2)**2-57 6*x**4-192*x**3)*exp(1/16*(3*x**2+2*x)/ln(2)**2)+256*x**3*ln(2)**2-96*x**5 -32*x**4)/ln(2)**2/exp(1/16*(3*x**2+2*x)/ln(2)**2)**4,x)
Output:
16*x**4*exp(-4*(3*x**2/16 + x/8)/log(2)**2) + 128*x**3*exp(-3*(3*x**2/16 + x/8)/log(2)**2) + 384*x**2*exp(-2*(3*x**2/16 + x/8)/log(2)**2) - x + 512* x*exp(-(3*x**2/16 + x/8)/log(2)**2)
Result contains higher order function than in optimal. Order 4 vs. order 3.
Time = 0.37 (sec) , antiderivative size = 2622, normalized size of antiderivative = 97.11 \[ \int \frac {e^{-\frac {2 x+3 x^2}{\log ^2(4)}} \left (-32 x^4-96 x^5-e^{\frac {2 x+3 x^2}{\log ^2(4)}} \log ^2(4)+64 x^3 \log ^2(4)+e^{\frac {3 \left (2 x+3 x^2\right )}{4 \log ^2(4)}} \left (-256 x-768 x^2+512 \log ^2(4)\right )+e^{\frac {2 x+3 x^2}{2 \log ^2(4)}} \left (-384 x^2-1152 x^3+768 x \log ^2(4)\right )+e^{\frac {2 x+3 x^2}{4 \log ^2(4)}} \left (-192 x^3-576 x^4+384 x^2 \log ^2(4)\right )\right )}{\log ^2(4)} \, dx=\text {Too large to display} \] Input:
integrate(1/4*(-4*log(2)^2*exp(1/16*(3*x^2+2*x)/log(2)^2)^4+(2048*log(2)^2 -768*x^2-256*x)*exp(1/16*(3*x^2+2*x)/log(2)^2)^3+(3072*x*log(2)^2-1152*x^3 -384*x^2)*exp(1/16*(3*x^2+2*x)/log(2)^2)^2+(1536*x^2*log(2)^2-576*x^4-192* x^3)*exp(1/16*(3*x^2+2*x)/log(2)^2)+256*x^3*log(2)^2-96*x^5-32*x^4)/log(2) ^2/exp(1/16*(3*x^2+2*x)/log(2)^2)^4,x, algorithm="maxima")
Output:
1/243*(82944*sqrt(3)*sqrt(pi)*erf(1/4*sqrt(3)*x/log(2) + 1/12*sqrt(3)/log( 2))*e^(1/48/log(2)^2)*log(2)^3 + 192*sqrt(3)*(36*sqrt(3)*sqrt(1/3)*(3*x/lo g(2)^2 + 1/log(2)^2)^3*gamma(3/2, 1/12*(3*x/log(2)^2 + 1/log(2)^2)^2*log(2 )^2)/(((3*x/log(2)^2 + 1/log(2)^2)^2)^(3/2)*(-1/log(2)^2)^(7/2)*log(2)^5) - 24*sqrt(3)*gamma(2, 1/12*(3*x/log(2)^2 + 1/log(2)^2)^2*log(2)^2)/((-1/lo g(2)^2)^(7/2)*log(2)^4) - sqrt(3)*sqrt(1/3)*sqrt(pi)*(3*x/log(2)^2 + 1/log (2)^2)*(erf(1/2*sqrt(1/3)*sqrt((3*x/log(2)^2 + 1/log(2)^2)^2)*log(2)) - 1) /(sqrt((3*x/log(2)^2 + 1/log(2)^2)^2)*(-1/log(2)^2)^(7/2)*log(2)^7) - 6*sq rt(3)*e^(-1/12*(3*x/log(2)^2 + 1/log(2)^2)^2*log(2)^2)/((-1/log(2)^2)^(7/2 )*log(2)^6))*e^(1/12/log(2)^2)*log(2)^2/sqrt(-1/log(2)^2) - 82944*sqrt(3/2 )*(sqrt(3/2)*sqrt(1/6)*sqrt(pi)*(3*x/log(2)^2 + 1/log(2)^2)*(erf(1/2*sqrt( 1/6)*sqrt((3*x/log(2)^2 + 1/log(2)^2)^2)*log(2)) - 1)/(sqrt((3*x/log(2)^2 + 1/log(2)^2)^2)*(-1/log(2)^2)^(3/2)*log(2)^3) + 2*sqrt(3/2)*e^(-1/24*(3*x /log(2)^2 + 1/log(2)^2)^2*log(2)^2)/((-1/log(2)^2)^(3/2)*log(2)^2))*e^(1/2 4/log(2)^2)*log(2)^2/sqrt(-1/log(2)^2) + 6912*(16*(3*x/log(2)^2 + 1/log(2) ^2)^3*gamma(3/2, 1/16*(3*x/log(2)^2 + 1/log(2)^2)^2*log(2)^2)/(((3*x/log(2 )^2 + 1/log(2)^2)^2)^(3/2)*(-1/log(2)^2)^(5/2)*log(2)^3) - sqrt(pi)*(3*x/l og(2)^2 + 1/log(2)^2)*(erf(1/4*sqrt((3*x/log(2)^2 + 1/log(2)^2)^2)*log(2)) - 1)/(sqrt((3*x/log(2)^2 + 1/log(2)^2)^2)*(-1/log(2)^2)^(5/2)*log(2)^5) - 8*e^(-1/16*(3*x/log(2)^2 + 1/log(2)^2)^2*log(2)^2)/((-1/log(2)^2)^(5/2...
Leaf count of result is larger than twice the leaf count of optimal. 228 vs. \(2 (25) = 50\).
Time = 0.16 (sec) , antiderivative size = 228, normalized size of antiderivative = 8.44 \[ \int \frac {e^{-\frac {2 x+3 x^2}{\log ^2(4)}} \left (-32 x^4-96 x^5-e^{\frac {2 x+3 x^2}{\log ^2(4)}} \log ^2(4)+64 x^3 \log ^2(4)+e^{\frac {3 \left (2 x+3 x^2\right )}{4 \log ^2(4)}} \left (-256 x-768 x^2+512 \log ^2(4)\right )+e^{\frac {2 x+3 x^2}{2 \log ^2(4)}} \left (-384 x^2-1152 x^3+768 x \log ^2(4)\right )+e^{\frac {2 x+3 x^2}{4 \log ^2(4)}} \left (-192 x^3-576 x^4+384 x^2 \log ^2(4)\right )\right )}{\log ^2(4)} \, dx=-\frac {81 \, x \log \left (2\right )^{2} - 13824 \, {\left ({\left (3 \, x + 1\right )} \log \left (2\right )^{2} - \log \left (2\right )^{2}\right )} e^{\left (-\frac {3 \, x^{2} + 2 \, x}{16 \, \log \left (2\right )^{2}}\right )} - 3456 \, {\left ({\left (3 \, x + 1\right )}^{2} \log \left (2\right )^{2} - 2 \, {\left (3 \, x + 1\right )} \log \left (2\right )^{2} + \log \left (2\right )^{2}\right )} e^{\left (-\frac {3 \, x^{2} + 2 \, x}{8 \, \log \left (2\right )^{2}}\right )} - 384 \, {\left ({\left (3 \, x + 1\right )}^{3} \log \left (2\right )^{2} - 3 \, {\left (3 \, x + 1\right )}^{2} \log \left (2\right )^{2} + 3 \, {\left (3 \, x + 1\right )} \log \left (2\right )^{2} - \log \left (2\right )^{2}\right )} e^{\left (-\frac {3 \, {\left (3 \, x^{2} + 2 \, x\right )}}{16 \, \log \left (2\right )^{2}}\right )} - 16 \, {\left ({\left (3 \, x + 1\right )}^{4} \log \left (2\right )^{2} - 4 \, {\left (3 \, x + 1\right )}^{3} \log \left (2\right )^{2} + 6 \, {\left (3 \, x + 1\right )}^{2} \log \left (2\right )^{2} - 4 \, {\left (3 \, x + 1\right )} \log \left (2\right )^{2} + \log \left (2\right )^{2}\right )} e^{\left (-\frac {3 \, x^{2} + 2 \, x}{4 \, \log \left (2\right )^{2}}\right )}}{81 \, \log \left (2\right )^{2}} \] Input:
integrate(1/4*(-4*log(2)^2*exp(1/16*(3*x^2+2*x)/log(2)^2)^4+(2048*log(2)^2 -768*x^2-256*x)*exp(1/16*(3*x^2+2*x)/log(2)^2)^3+(3072*x*log(2)^2-1152*x^3 -384*x^2)*exp(1/16*(3*x^2+2*x)/log(2)^2)^2+(1536*x^2*log(2)^2-576*x^4-192* x^3)*exp(1/16*(3*x^2+2*x)/log(2)^2)+256*x^3*log(2)^2-96*x^5-32*x^4)/log(2) ^2/exp(1/16*(3*x^2+2*x)/log(2)^2)^4,x, algorithm="giac")
Output:
-1/81*(81*x*log(2)^2 - 13824*((3*x + 1)*log(2)^2 - log(2)^2)*e^(-1/16*(3*x ^2 + 2*x)/log(2)^2) - 3456*((3*x + 1)^2*log(2)^2 - 2*(3*x + 1)*log(2)^2 + log(2)^2)*e^(-1/8*(3*x^2 + 2*x)/log(2)^2) - 384*((3*x + 1)^3*log(2)^2 - 3* (3*x + 1)^2*log(2)^2 + 3*(3*x + 1)*log(2)^2 - log(2)^2)*e^(-3/16*(3*x^2 + 2*x)/log(2)^2) - 16*((3*x + 1)^4*log(2)^2 - 4*(3*x + 1)^3*log(2)^2 + 6*(3* x + 1)^2*log(2)^2 - 4*(3*x + 1)*log(2)^2 + log(2)^2)*e^(-1/4*(3*x^2 + 2*x) /log(2)^2))/log(2)^2
Time = 0.43 (sec) , antiderivative size = 94, normalized size of antiderivative = 3.48 \[ \int \frac {e^{-\frac {2 x+3 x^2}{\log ^2(4)}} \left (-32 x^4-96 x^5-e^{\frac {2 x+3 x^2}{\log ^2(4)}} \log ^2(4)+64 x^3 \log ^2(4)+e^{\frac {3 \left (2 x+3 x^2\right )}{4 \log ^2(4)}} \left (-256 x-768 x^2+512 \log ^2(4)\right )+e^{\frac {2 x+3 x^2}{2 \log ^2(4)}} \left (-384 x^2-1152 x^3+768 x \log ^2(4)\right )+e^{\frac {2 x+3 x^2}{4 \log ^2(4)}} \left (-192 x^3-576 x^4+384 x^2 \log ^2(4)\right )\right )}{\log ^2(4)} \, dx=512\,x\,{\mathrm {e}}^{-\frac {3\,x^2}{16\,{\ln \left (2\right )}^2}-\frac {x}{8\,{\ln \left (2\right )}^2}}-x+16\,x^4\,{\mathrm {e}}^{-\frac {3\,x^2}{4\,{\ln \left (2\right )}^2}-\frac {x}{2\,{\ln \left (2\right )}^2}}+384\,x^2\,{\mathrm {e}}^{-\frac {3\,x^2}{8\,{\ln \left (2\right )}^2}-\frac {x}{4\,{\ln \left (2\right )}^2}}+128\,x^3\,{\mathrm {e}}^{-\frac {9\,x^2}{16\,{\ln \left (2\right )}^2}-\frac {3\,x}{8\,{\ln \left (2\right )}^2}} \] Input:
int(-(exp(-(4*(x/8 + (3*x^2)/16))/log(2)^2)*(exp((4*(x/8 + (3*x^2)/16))/lo g(2)^2)*log(2)^2 - 64*x^3*log(2)^2 + (exp((x/8 + (3*x^2)/16)/log(2)^2)*(19 2*x^3 - 1536*x^2*log(2)^2 + 576*x^4))/4 + (exp((2*(x/8 + (3*x^2)/16))/log( 2)^2)*(384*x^2 - 3072*x*log(2)^2 + 1152*x^3))/4 + 8*x^4 + 24*x^5 + (exp((3 *(x/8 + (3*x^2)/16))/log(2)^2)*(256*x - 2048*log(2)^2 + 768*x^2))/4))/log( 2)^2,x)
Output:
512*x*exp(- (3*x^2)/(16*log(2)^2) - x/(8*log(2)^2)) - x + 16*x^4*exp(- (3* x^2)/(4*log(2)^2) - x/(2*log(2)^2)) + 384*x^2*exp(- (3*x^2)/(8*log(2)^2) - x/(4*log(2)^2)) + 128*x^3*exp(- (9*x^2)/(16*log(2)^2) - (3*x)/(8*log(2)^2 ))
Time = 0.16 (sec) , antiderivative size = 107, normalized size of antiderivative = 3.96 \[ \int \frac {e^{-\frac {2 x+3 x^2}{\log ^2(4)}} \left (-32 x^4-96 x^5-e^{\frac {2 x+3 x^2}{\log ^2(4)}} \log ^2(4)+64 x^3 \log ^2(4)+e^{\frac {3 \left (2 x+3 x^2\right )}{4 \log ^2(4)}} \left (-256 x-768 x^2+512 \log ^2(4)\right )+e^{\frac {2 x+3 x^2}{2 \log ^2(4)}} \left (-384 x^2-1152 x^3+768 x \log ^2(4)\right )+e^{\frac {2 x+3 x^2}{4 \log ^2(4)}} \left (-192 x^3-576 x^4+384 x^2 \log ^2(4)\right )\right )}{\log ^2(4)} \, dx=\frac {x \left (512 e^{\frac {9 x^{2}+6 x}{16 \mathrm {log}\left (2\right )^{2}}}+128 e^{\frac {3 x^{2}+2 x}{16 \mathrm {log}\left (2\right )^{2}}} x^{2}+384 e^{\frac {3 x^{2}+2 x}{8 \mathrm {log}\left (2\right )^{2}}} x -e^{\frac {3 x^{2}+2 x}{4 \mathrm {log}\left (2\right )^{2}}}+16 x^{3}\right )}{e^{\frac {3 x^{2}+2 x}{4 \mathrm {log}\left (2\right )^{2}}}} \] Input:
int(1/4*(-4*log(2)^2*exp(1/16*(3*x^2+2*x)/log(2)^2)^4+(2048*log(2)^2-768*x ^2-256*x)*exp(1/16*(3*x^2+2*x)/log(2)^2)^3+(3072*x*log(2)^2-1152*x^3-384*x ^2)*exp(1/16*(3*x^2+2*x)/log(2)^2)^2+(1536*x^2*log(2)^2-576*x^4-192*x^3)*e xp(1/16*(3*x^2+2*x)/log(2)^2)+256*x^3*log(2)^2-96*x^5-32*x^4)/log(2)^2/exp (1/16*(3*x^2+2*x)/log(2)^2)^4,x)
Output:
(x*(512*e**((9*x**2 + 6*x)/(16*log(2)**2)) + 128*e**((3*x**2 + 2*x)/(16*lo g(2)**2))*x**2 + 384*e**((3*x**2 + 2*x)/(8*log(2)**2))*x - e**((3*x**2 + 2 *x)/(4*log(2)**2)) + 16*x**3))/e**((3*x**2 + 2*x)/(4*log(2)**2))