Integrand size = 374, antiderivative size = 28 \[ \int \frac {e^{3 x} \left (12-80 x+72 x^2-16 x^3+(12-8 x) \log (4)\right )+e^{2 x} \left (24-328 x+440 x^2-160 x^3+16 x^4+\left (52-80 x+16 x^2\right ) \log (4)+4 \log ^2(4)\right )+e^x \left (-288 x+576 x^2-256 x^3+32 x^4+\left (48-128 x+32 x^2\right ) \log (4)+8 \log ^2(4)\right )}{-1728 x^3+1728 x^4-576 x^5+64 x^6+\left (864 x^2-576 x^3+96 x^4\right ) \log (4)+\left (-144 x+48 x^2\right ) \log ^2(4)+8 \log ^3(4)+e^{3 x} \left (-216 x^3+216 x^4-72 x^5+8 x^6+\left (108 x^2-72 x^3+12 x^4\right ) \log (4)+\left (-18 x+6 x^2\right ) \log ^2(4)+\log ^3(4)\right )+e^{2 x} \left (-1296 x^3+1296 x^4-432 x^5+48 x^6+\left (648 x^2-432 x^3+72 x^4\right ) \log (4)+\left (-108 x+36 x^2\right ) \log ^2(4)+6 \log ^3(4)\right )+e^x \left (-2592 x^3+2592 x^4-864 x^5+96 x^6+\left (1296 x^2-864 x^3+144 x^4\right ) \log (4)+\left (-216 x+72 x^2\right ) \log ^2(4)+12 \log ^3(4)\right )} \, dx=\left (1+\frac {e^x}{\left (2+e^x\right ) (-2 (3-x) x+\log (4))}\right )^2 \] Output:
(exp(x)/(2*ln(2)-x*(6-2*x))/(2+exp(x))+1)^2
\[ \int \frac {e^{3 x} \left (12-80 x+72 x^2-16 x^3+(12-8 x) \log (4)\right )+e^{2 x} \left (24-328 x+440 x^2-160 x^3+16 x^4+\left (52-80 x+16 x^2\right ) \log (4)+4 \log ^2(4)\right )+e^x \left (-288 x+576 x^2-256 x^3+32 x^4+\left (48-128 x+32 x^2\right ) \log (4)+8 \log ^2(4)\right )}{-1728 x^3+1728 x^4-576 x^5+64 x^6+\left (864 x^2-576 x^3+96 x^4\right ) \log (4)+\left (-144 x+48 x^2\right ) \log ^2(4)+8 \log ^3(4)+e^{3 x} \left (-216 x^3+216 x^4-72 x^5+8 x^6+\left (108 x^2-72 x^3+12 x^4\right ) \log (4)+\left (-18 x+6 x^2\right ) \log ^2(4)+\log ^3(4)\right )+e^{2 x} \left (-1296 x^3+1296 x^4-432 x^5+48 x^6+\left (648 x^2-432 x^3+72 x^4\right ) \log (4)+\left (-108 x+36 x^2\right ) \log ^2(4)+6 \log ^3(4)\right )+e^x \left (-2592 x^3+2592 x^4-864 x^5+96 x^6+\left (1296 x^2-864 x^3+144 x^4\right ) \log (4)+\left (-216 x+72 x^2\right ) \log ^2(4)+12 \log ^3(4)\right )} \, dx=\int \frac {e^{3 x} \left (12-80 x+72 x^2-16 x^3+(12-8 x) \log (4)\right )+e^{2 x} \left (24-328 x+440 x^2-160 x^3+16 x^4+\left (52-80 x+16 x^2\right ) \log (4)+4 \log ^2(4)\right )+e^x \left (-288 x+576 x^2-256 x^3+32 x^4+\left (48-128 x+32 x^2\right ) \log (4)+8 \log ^2(4)\right )}{-1728 x^3+1728 x^4-576 x^5+64 x^6+\left (864 x^2-576 x^3+96 x^4\right ) \log (4)+\left (-144 x+48 x^2\right ) \log ^2(4)+8 \log ^3(4)+e^{3 x} \left (-216 x^3+216 x^4-72 x^5+8 x^6+\left (108 x^2-72 x^3+12 x^4\right ) \log (4)+\left (-18 x+6 x^2\right ) \log ^2(4)+\log ^3(4)\right )+e^{2 x} \left (-1296 x^3+1296 x^4-432 x^5+48 x^6+\left (648 x^2-432 x^3+72 x^4\right ) \log (4)+\left (-108 x+36 x^2\right ) \log ^2(4)+6 \log ^3(4)\right )+e^x \left (-2592 x^3+2592 x^4-864 x^5+96 x^6+\left (1296 x^2-864 x^3+144 x^4\right ) \log (4)+\left (-216 x+72 x^2\right ) \log ^2(4)+12 \log ^3(4)\right )} \, dx \] Input:
Integrate[(E^(3*x)*(12 - 80*x + 72*x^2 - 16*x^3 + (12 - 8*x)*Log[4]) + E^( 2*x)*(24 - 328*x + 440*x^2 - 160*x^3 + 16*x^4 + (52 - 80*x + 16*x^2)*Log[4 ] + 4*Log[4]^2) + E^x*(-288*x + 576*x^2 - 256*x^3 + 32*x^4 + (48 - 128*x + 32*x^2)*Log[4] + 8*Log[4]^2))/(-1728*x^3 + 1728*x^4 - 576*x^5 + 64*x^6 + (864*x^2 - 576*x^3 + 96*x^4)*Log[4] + (-144*x + 48*x^2)*Log[4]^2 + 8*Log[4 ]^3 + E^(3*x)*(-216*x^3 + 216*x^4 - 72*x^5 + 8*x^6 + (108*x^2 - 72*x^3 + 1 2*x^4)*Log[4] + (-18*x + 6*x^2)*Log[4]^2 + Log[4]^3) + E^(2*x)*(-1296*x^3 + 1296*x^4 - 432*x^5 + 48*x^6 + (648*x^2 - 432*x^3 + 72*x^4)*Log[4] + (-10 8*x + 36*x^2)*Log[4]^2 + 6*Log[4]^3) + E^x*(-2592*x^3 + 2592*x^4 - 864*x^5 + 96*x^6 + (1296*x^2 - 864*x^3 + 144*x^4)*Log[4] + (-216*x + 72*x^2)*Log[ 4]^2 + 12*Log[4]^3)),x]
Output:
Integrate[(E^(3*x)*(12 - 80*x + 72*x^2 - 16*x^3 + (12 - 8*x)*Log[4]) + E^( 2*x)*(24 - 328*x + 440*x^2 - 160*x^3 + 16*x^4 + (52 - 80*x + 16*x^2)*Log[4 ] + 4*Log[4]^2) + E^x*(-288*x + 576*x^2 - 256*x^3 + 32*x^4 + (48 - 128*x + 32*x^2)*Log[4] + 8*Log[4]^2))/(-1728*x^3 + 1728*x^4 - 576*x^5 + 64*x^6 + (864*x^2 - 576*x^3 + 96*x^4)*Log[4] + (-144*x + 48*x^2)*Log[4]^2 + 8*Log[4 ]^3 + E^(3*x)*(-216*x^3 + 216*x^4 - 72*x^5 + 8*x^6 + (108*x^2 - 72*x^3 + 1 2*x^4)*Log[4] + (-18*x + 6*x^2)*Log[4]^2 + Log[4]^3) + E^(2*x)*(-1296*x^3 + 1296*x^4 - 432*x^5 + 48*x^6 + (648*x^2 - 432*x^3 + 72*x^4)*Log[4] + (-10 8*x + 36*x^2)*Log[4]^2 + 6*Log[4]^3) + E^x*(-2592*x^3 + 2592*x^4 - 864*x^5 + 96*x^6 + (1296*x^2 - 864*x^3 + 144*x^4)*Log[4] + (-216*x + 72*x^2)*Log[ 4]^2 + 12*Log[4]^3)), 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.
| \(\displaystyle \int \frac {e^{3 x} \left (-16 x^3+72 x^2-80 x+(12-8 x) \log (4)+12\right )+e^{2 x} \left (16 x^4-160 x^3+440 x^2+\left (16 x^2-80 x+52\right ) \log (4)-328 x+24+4 \log ^2(4)\right )+e^x \left (32 x^4-256 x^3+576 x^2+\left (32 x^2-128 x+48\right ) \log (4)-288 x+8 \log ^2(4)\right )}{64 x^6-576 x^5+1728 x^4-1728 x^3+\left (48 x^2-144 x\right ) \log ^2(4)+\left (96 x^4-576 x^3+864 x^2\right ) \log (4)+e^{3 x} \left (8 x^6-72 x^5+216 x^4-216 x^3+\left (6 x^2-18 x\right ) \log ^2(4)+\left (12 x^4-72 x^3+108 x^2\right ) \log (4)+\log ^3(4)\right )+e^{2 x} \left (48 x^6-432 x^5+1296 x^4-1296 x^3+\left (36 x^2-108 x\right ) \log ^2(4)+\left (72 x^4-432 x^3+648 x^2\right ) \log (4)+6 \log ^3(4)\right )+e^x \left (96 x^6-864 x^5+2592 x^4-2592 x^3+\left (72 x^2-216 x\right ) \log ^2(4)+\left (144 x^4-864 x^3+1296 x^2\right ) \log (4)+12 \log ^3(4)\right )+8 \log ^3(4)} \, dx\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle \int \frac {4 e^x \left (-8 x^4+64 x^3-8 x^2 (18+\log (4))+e^{2 x} (2 x-3) \left (2 x^2-6 x+1+\log (4)\right )-e^x \left (4 x^4-40 x^3+2 x^2 (55+\log (16))-2 x (41+10 \log (4))+6+\log ^2(4)+13 \log (4)\right )+8 x (9+\log (256))-2 \log (4) (6+\log (4))\right )}{\left (e^x+2\right )^3 \left (-2 x^2+6 x-\log (4)\right )^3}dx\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle 4 \int \frac {e^x \left (-8 x^4+64 x^3-8 (18+\log (4)) x^2+8 (9+\log (256)) x-e^{2 x} (3-2 x) \left (2 x^2-6 x+\log (4)+1\right )-e^x \left (4 x^4-40 x^3+2 (55+\log (16)) x^2-2 (41+10 \log (4)) x+\log ^2(4)+13 \log (4)+6\right )-2 \log (4) (6+\log (4))\right )}{\left (2+e^x\right )^3 \left (-2 x^2+6 x-\log (4)\right )^3}dx\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle 4 \int \left (-\frac {e^x (2 x-3) \left (2 x^2-6 x+\log (4)+1\right )}{\left (2+e^x\right ) \left (2 x^2-6 x+\log (4)\right )^3}-\frac {e^x \left (4 x^2-12 x+\log (16)\right )}{\left (2+e^x\right )^3 \left (2 x^2-6 x+\log (4)\right )^3}+\frac {e^x \left (-4 x^4+24 x^3-2 (19+\log (16)) x^2+2 (1+6 \log (4)) x-\log ^2(4)-\log (4)+6\right )}{\left (2+e^x\right )^2 \left (-2 x^2+6 x-\log (4)\right )^3}\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle 4 \left (-6 \int \frac {e^x}{\left (2+e^x\right )^2 \left (2 x^2-6 x+\log (4)\right )^3}dx+3 \int \frac {e^x}{\left (2+e^x\right ) \left (2 x^2-6 x+\log (4)\right )^3}dx+4 \int \frac {e^x x}{\left (2+e^x\right )^2 \left (2 x^2-6 x+\log (4)\right )^3}dx-2 \int \frac {e^x x}{\left (2+e^x\right ) \left (2 x^2-6 x+\log (4)\right )^3}dx+\int \frac {e^x}{\left (2+e^x\right )^2 \left (2 x^2-6 x+\log (4)\right )^2}dx+3 \int \frac {e^x}{\left (2+e^x\right ) \left (2 x^2-6 x+\log (4)\right )^2}dx-2 \int \frac {e^x x}{\left (2+e^x\right ) \left (2 x^2-6 x+\log (4)\right )^2}dx-8 \int \frac {e^x}{\left (2+e^x\right )^3 \left (4 x^2-12 x+\log (16)\right )^2}dx-\frac {8 \int \frac {1}{\left (2+e^x\right ) \left (-4 x+2 \sqrt {9-2 \log (4)}+6\right )^2}dx}{\sqrt {9-2 \log (4)}}+\frac {8 \int \frac {1}{\left (2+e^x\right ) \left (4 x+2 \sqrt {9-2 \log (4)}-6\right )^2}dx}{\sqrt {9-2 \log (4)}}-\frac {1}{\left (e^x+2\right ) \sqrt {9-2 \log (4)} \left (-2 x+3-\sqrt {9-2 \log (4)}\right )}+\frac {1}{\left (e^x+2\right ) \sqrt {9-2 \log (4)} \left (-2 x+3+\sqrt {9-2 \log (4)}\right )}\right )\) |
Input:
Int[(E^(3*x)*(12 - 80*x + 72*x^2 - 16*x^3 + (12 - 8*x)*Log[4]) + E^(2*x)*( 24 - 328*x + 440*x^2 - 160*x^3 + 16*x^4 + (52 - 80*x + 16*x^2)*Log[4] + 4* Log[4]^2) + E^x*(-288*x + 576*x^2 - 256*x^3 + 32*x^4 + (48 - 128*x + 32*x^ 2)*Log[4] + 8*Log[4]^2))/(-1728*x^3 + 1728*x^4 - 576*x^5 + 64*x^6 + (864*x ^2 - 576*x^3 + 96*x^4)*Log[4] + (-144*x + 48*x^2)*Log[4]^2 + 8*Log[4]^3 + E^(3*x)*(-216*x^3 + 216*x^4 - 72*x^5 + 8*x^6 + (108*x^2 - 72*x^3 + 12*x^4) *Log[4] + (-18*x + 6*x^2)*Log[4]^2 + Log[4]^3) + E^(2*x)*(-1296*x^3 + 1296 *x^4 - 432*x^5 + 48*x^6 + (648*x^2 - 432*x^3 + 72*x^4)*Log[4] + (-108*x + 36*x^2)*Log[4]^2 + 6*Log[4]^3) + E^x*(-2592*x^3 + 2592*x^4 - 864*x^5 + 96* x^6 + (1296*x^2 - 864*x^3 + 144*x^4)*Log[4] + (-216*x + 72*x^2)*Log[4]^2 + 12*Log[4]^3)),x]
Output:
$Aborted
Leaf count of result is larger than twice the leaf count of optimal. \(61\) vs. \(2(28)=56\).
Time = 2.19 (sec) , antiderivative size = 62, normalized size of antiderivative = 2.21
| method | result | size |
| norman | \(\frac {2 \,{\mathrm e}^{x} \ln \left (2\right )+\left (\ln \left (2\right )+\frac {1}{4}\right ) {\mathrm e}^{2 x}-6 \,{\mathrm e}^{x} x +{\mathrm e}^{2 x} x^{2}+2 \,{\mathrm e}^{x} x^{2}-3 x \,{\mathrm e}^{2 x}}{\left ({\mathrm e}^{x}+2\right )^{2} \left (x^{2}+\ln \left (2\right )-3 x \right )^{2}}\) | \(62\) |
| parallelrisch | \(\frac {8 \,{\mathrm e}^{x} \ln \left (2\right )+{\mathrm e}^{2 x}+4 \ln \left (2\right ) {\mathrm e}^{2 x}-12 x \,{\mathrm e}^{2 x}+4 \,{\mathrm e}^{2 x} x^{2}-24 \,{\mathrm e}^{x} x +8 \,{\mathrm e}^{x} x^{2}}{4 \left ({\mathrm e}^{2 x}+4 \,{\mathrm e}^{x}+4\right ) \left (x^{4}+2 x^{2} \ln \left (2\right )-6 x^{3}+\ln \left (2\right )^{2}-6 x \ln \left (2\right )+9 x^{2}\right )}\) | \(94\) |
| risch | \(\frac {\frac {1}{4}+x^{2}-3 x +\ln \left (2\right )}{x^{4}+2 x^{2} \ln \left (2\right )-6 x^{3}+\ln \left (2\right )^{2}-6 x \ln \left (2\right )+9 x^{2}}-\frac {2 \,{\mathrm e}^{x} x^{2}+2 \,{\mathrm e}^{x} \ln \left (2\right )+4 x^{2}-6 \,{\mathrm e}^{x} x +4 \ln \left (2\right )-12 x +{\mathrm e}^{x}+1}{\left (x^{4}+2 x^{2} \ln \left (2\right )-6 x^{3}+\ln \left (2\right )^{2}-6 x \ln \left (2\right )+9 x^{2}\right ) \left ({\mathrm e}^{x}+2\right )^{2}}\) | \(119\) |
Input:
int(((2*(-8*x+12)*ln(2)-16*x^3+72*x^2-80*x+12)*exp(x)^3+(16*ln(2)^2+2*(16* x^2-80*x+52)*ln(2)+16*x^4-160*x^3+440*x^2-328*x+24)*exp(x)^2+(32*ln(2)^2+2 *(32*x^2-128*x+48)*ln(2)+32*x^4-256*x^3+576*x^2-288*x)*exp(x))/((8*ln(2)^3 +4*(6*x^2-18*x)*ln(2)^2+2*(12*x^4-72*x^3+108*x^2)*ln(2)+8*x^6-72*x^5+216*x ^4-216*x^3)*exp(x)^3+(48*ln(2)^3+4*(36*x^2-108*x)*ln(2)^2+2*(72*x^4-432*x^ 3+648*x^2)*ln(2)+48*x^6-432*x^5+1296*x^4-1296*x^3)*exp(x)^2+(96*ln(2)^3+4* (72*x^2-216*x)*ln(2)^2+2*(144*x^4-864*x^3+1296*x^2)*ln(2)+96*x^6-864*x^5+2 592*x^4-2592*x^3)*exp(x)+64*ln(2)^3+4*(48*x^2-144*x)*ln(2)^2+2*(96*x^4-576 *x^3+864*x^2)*ln(2)+64*x^6-576*x^5+1728*x^4-1728*x^3),x,method=_RETURNVERB OSE)
Output:
(2*exp(x)*ln(2)+(ln(2)+1/4)*exp(x)^2-6*exp(x)*x+exp(x)^2*x^2+2*exp(x)*x^2- 3*x*exp(x)^2)/(exp(x)+2)^2/(x^2+ln(2)-3*x)^2
Leaf count of result is larger than twice the leaf count of optimal. 137 vs. \(2 (25) = 50\).
Time = 0.08 (sec) , antiderivative size = 137, normalized size of antiderivative = 4.89 \[ \int \frac {e^{3 x} \left (12-80 x+72 x^2-16 x^3+(12-8 x) \log (4)\right )+e^{2 x} \left (24-328 x+440 x^2-160 x^3+16 x^4+\left (52-80 x+16 x^2\right ) \log (4)+4 \log ^2(4)\right )+e^x \left (-288 x+576 x^2-256 x^3+32 x^4+\left (48-128 x+32 x^2\right ) \log (4)+8 \log ^2(4)\right )}{-1728 x^3+1728 x^4-576 x^5+64 x^6+\left (864 x^2-576 x^3+96 x^4\right ) \log (4)+\left (-144 x+48 x^2\right ) \log ^2(4)+8 \log ^3(4)+e^{3 x} \left (-216 x^3+216 x^4-72 x^5+8 x^6+\left (108 x^2-72 x^3+12 x^4\right ) \log (4)+\left (-18 x+6 x^2\right ) \log ^2(4)+\log ^3(4)\right )+e^{2 x} \left (-1296 x^3+1296 x^4-432 x^5+48 x^6+\left (648 x^2-432 x^3+72 x^4\right ) \log (4)+\left (-108 x+36 x^2\right ) \log ^2(4)+6 \log ^3(4)\right )+e^x \left (-2592 x^3+2592 x^4-864 x^5+96 x^6+\left (1296 x^2-864 x^3+144 x^4\right ) \log (4)+\left (-216 x+72 x^2\right ) \log ^2(4)+12 \log ^3(4)\right )} \, dx=\frac {{\left (4 \, x^{2} - 12 \, x + 4 \, \log \left (2\right ) + 1\right )} e^{\left (2 \, x\right )} + 8 \, {\left (x^{2} - 3 \, x + \log \left (2\right )\right )} e^{x}}{4 \, {\left (4 \, x^{4} - 24 \, x^{3} + 36 \, x^{2} + {\left (x^{4} - 6 \, x^{3} + 9 \, x^{2} + 2 \, {\left (x^{2} - 3 \, x\right )} \log \left (2\right ) + \log \left (2\right )^{2}\right )} e^{\left (2 \, x\right )} + 4 \, {\left (x^{4} - 6 \, x^{3} + 9 \, x^{2} + 2 \, {\left (x^{2} - 3 \, x\right )} \log \left (2\right ) + \log \left (2\right )^{2}\right )} e^{x} + 8 \, {\left (x^{2} - 3 \, x\right )} \log \left (2\right ) + 4 \, \log \left (2\right )^{2}\right )}} \] Input:
integrate(((2*(-8*x+12)*log(2)-16*x^3+72*x^2-80*x+12)*exp(x)^3+(16*log(2)^ 2+2*(16*x^2-80*x+52)*log(2)+16*x^4-160*x^3+440*x^2-328*x+24)*exp(x)^2+(32* log(2)^2+2*(32*x^2-128*x+48)*log(2)+32*x^4-256*x^3+576*x^2-288*x)*exp(x))/ ((8*log(2)^3+4*(6*x^2-18*x)*log(2)^2+2*(12*x^4-72*x^3+108*x^2)*log(2)+8*x^ 6-72*x^5+216*x^4-216*x^3)*exp(x)^3+(48*log(2)^3+4*(36*x^2-108*x)*log(2)^2+ 2*(72*x^4-432*x^3+648*x^2)*log(2)+48*x^6-432*x^5+1296*x^4-1296*x^3)*exp(x) ^2+(96*log(2)^3+4*(72*x^2-216*x)*log(2)^2+2*(144*x^4-864*x^3+1296*x^2)*log (2)+96*x^6-864*x^5+2592*x^4-2592*x^3)*exp(x)+64*log(2)^3+4*(48*x^2-144*x)* log(2)^2+2*(96*x^4-576*x^3+864*x^2)*log(2)+64*x^6-576*x^5+1728*x^4-1728*x^ 3),x, algorithm="fricas")
Output:
1/4*((4*x^2 - 12*x + 4*log(2) + 1)*e^(2*x) + 8*(x^2 - 3*x + log(2))*e^x)/( 4*x^4 - 24*x^3 + 36*x^2 + (x^4 - 6*x^3 + 9*x^2 + 2*(x^2 - 3*x)*log(2) + lo g(2)^2)*e^(2*x) + 4*(x^4 - 6*x^3 + 9*x^2 + 2*(x^2 - 3*x)*log(2) + log(2)^2 )*e^x + 8*(x^2 - 3*x)*log(2) + 4*log(2)^2)
Leaf count of result is larger than twice the leaf count of optimal. 199 vs. \(2 (22) = 44\).
Time = 0.65 (sec) , antiderivative size = 199, normalized size of antiderivative = 7.11 \[ \int \frac {e^{3 x} \left (12-80 x+72 x^2-16 x^3+(12-8 x) \log (4)\right )+e^{2 x} \left (24-328 x+440 x^2-160 x^3+16 x^4+\left (52-80 x+16 x^2\right ) \log (4)+4 \log ^2(4)\right )+e^x \left (-288 x+576 x^2-256 x^3+32 x^4+\left (48-128 x+32 x^2\right ) \log (4)+8 \log ^2(4)\right )}{-1728 x^3+1728 x^4-576 x^5+64 x^6+\left (864 x^2-576 x^3+96 x^4\right ) \log (4)+\left (-144 x+48 x^2\right ) \log ^2(4)+8 \log ^3(4)+e^{3 x} \left (-216 x^3+216 x^4-72 x^5+8 x^6+\left (108 x^2-72 x^3+12 x^4\right ) \log (4)+\left (-18 x+6 x^2\right ) \log ^2(4)+\log ^3(4)\right )+e^{2 x} \left (-1296 x^3+1296 x^4-432 x^5+48 x^6+\left (648 x^2-432 x^3+72 x^4\right ) \log (4)+\left (-108 x+36 x^2\right ) \log ^2(4)+6 \log ^3(4)\right )+e^x \left (-2592 x^3+2592 x^4-864 x^5+96 x^6+\left (1296 x^2-864 x^3+144 x^4\right ) \log (4)+\left (-216 x+72 x^2\right ) \log ^2(4)+12 \log ^3(4)\right )} \, dx=- \frac {- 4 x^{2} + 12 x - 4 \log {\left (2 \right )} - 1}{4 x^{4} - 24 x^{3} + x^{2} \cdot \left (8 \log {\left (2 \right )} + 36\right ) - 24 x \log {\left (2 \right )} + 4 \log {\left (2 \right )}^{2}} + \frac {- 4 x^{2} + 12 x + \left (- 2 x^{2} + 6 x - 2 \log {\left (2 \right )} - 1\right ) e^{x} - 4 \log {\left (2 \right )} - 1}{4 x^{4} - 24 x^{3} + 8 x^{2} \log {\left (2 \right )} + 36 x^{2} - 24 x \log {\left (2 \right )} + \left (x^{4} - 6 x^{3} + 2 x^{2} \log {\left (2 \right )} + 9 x^{2} - 6 x \log {\left (2 \right )} + \log {\left (2 \right )}^{2}\right ) e^{2 x} + \left (4 x^{4} - 24 x^{3} + 8 x^{2} \log {\left (2 \right )} + 36 x^{2} - 24 x \log {\left (2 \right )} + 4 \log {\left (2 \right )}^{2}\right ) e^{x} + 4 \log {\left (2 \right )}^{2}} \] Input:
integrate(((2*(-8*x+12)*ln(2)-16*x**3+72*x**2-80*x+12)*exp(x)**3+(16*ln(2) **2+2*(16*x**2-80*x+52)*ln(2)+16*x**4-160*x**3+440*x**2-328*x+24)*exp(x)** 2+(32*ln(2)**2+2*(32*x**2-128*x+48)*ln(2)+32*x**4-256*x**3+576*x**2-288*x) *exp(x))/((8*ln(2)**3+4*(6*x**2-18*x)*ln(2)**2+2*(12*x**4-72*x**3+108*x**2 )*ln(2)+8*x**6-72*x**5+216*x**4-216*x**3)*exp(x)**3+(48*ln(2)**3+4*(36*x** 2-108*x)*ln(2)**2+2*(72*x**4-432*x**3+648*x**2)*ln(2)+48*x**6-432*x**5+129 6*x**4-1296*x**3)*exp(x)**2+(96*ln(2)**3+4*(72*x**2-216*x)*ln(2)**2+2*(144 *x**4-864*x**3+1296*x**2)*ln(2)+96*x**6-864*x**5+2592*x**4-2592*x**3)*exp( x)+64*ln(2)**3+4*(48*x**2-144*x)*ln(2)**2+2*(96*x**4-576*x**3+864*x**2)*ln (2)+64*x**6-576*x**5+1728*x**4-1728*x**3),x)
Output:
-(-4*x**2 + 12*x - 4*log(2) - 1)/(4*x**4 - 24*x**3 + x**2*(8*log(2) + 36) - 24*x*log(2) + 4*log(2)**2) + (-4*x**2 + 12*x + (-2*x**2 + 6*x - 2*log(2) - 1)*exp(x) - 4*log(2) - 1)/(4*x**4 - 24*x**3 + 8*x**2*log(2) + 36*x**2 - 24*x*log(2) + (x**4 - 6*x**3 + 2*x**2*log(2) + 9*x**2 - 6*x*log(2) + log( 2)**2)*exp(2*x) + (4*x**4 - 24*x**3 + 8*x**2*log(2) + 36*x**2 - 24*x*log(2 ) + 4*log(2)**2)*exp(x) + 4*log(2)**2)
Leaf count of result is larger than twice the leaf count of optimal. 135 vs. \(2 (25) = 50\).
Time = 0.23 (sec) , antiderivative size = 135, normalized size of antiderivative = 4.82 \[ \int \frac {e^{3 x} \left (12-80 x+72 x^2-16 x^3+(12-8 x) \log (4)\right )+e^{2 x} \left (24-328 x+440 x^2-160 x^3+16 x^4+\left (52-80 x+16 x^2\right ) \log (4)+4 \log ^2(4)\right )+e^x \left (-288 x+576 x^2-256 x^3+32 x^4+\left (48-128 x+32 x^2\right ) \log (4)+8 \log ^2(4)\right )}{-1728 x^3+1728 x^4-576 x^5+64 x^6+\left (864 x^2-576 x^3+96 x^4\right ) \log (4)+\left (-144 x+48 x^2\right ) \log ^2(4)+8 \log ^3(4)+e^{3 x} \left (-216 x^3+216 x^4-72 x^5+8 x^6+\left (108 x^2-72 x^3+12 x^4\right ) \log (4)+\left (-18 x+6 x^2\right ) \log ^2(4)+\log ^3(4)\right )+e^{2 x} \left (-1296 x^3+1296 x^4-432 x^5+48 x^6+\left (648 x^2-432 x^3+72 x^4\right ) \log (4)+\left (-108 x+36 x^2\right ) \log ^2(4)+6 \log ^3(4)\right )+e^x \left (-2592 x^3+2592 x^4-864 x^5+96 x^6+\left (1296 x^2-864 x^3+144 x^4\right ) \log (4)+\left (-216 x+72 x^2\right ) \log ^2(4)+12 \log ^3(4)\right )} \, dx=\frac {{\left (4 \, x^{2} - 12 \, x + 4 \, \log \left (2\right ) + 1\right )} e^{\left (2 \, x\right )} + 8 \, {\left (x^{2} - 3 \, x + \log \left (2\right )\right )} e^{x}}{4 \, {\left (4 \, x^{4} - 24 \, x^{3} + 4 \, x^{2} {\left (2 \, \log \left (2\right ) + 9\right )} + {\left (x^{4} - 6 \, x^{3} + x^{2} {\left (2 \, \log \left (2\right ) + 9\right )} - 6 \, x \log \left (2\right ) + \log \left (2\right )^{2}\right )} e^{\left (2 \, x\right )} + 4 \, {\left (x^{4} - 6 \, x^{3} + x^{2} {\left (2 \, \log \left (2\right ) + 9\right )} - 6 \, x \log \left (2\right ) + \log \left (2\right )^{2}\right )} e^{x} - 24 \, x \log \left (2\right ) + 4 \, \log \left (2\right )^{2}\right )}} \] Input:
integrate(((2*(-8*x+12)*log(2)-16*x^3+72*x^2-80*x+12)*exp(x)^3+(16*log(2)^ 2+2*(16*x^2-80*x+52)*log(2)+16*x^4-160*x^3+440*x^2-328*x+24)*exp(x)^2+(32* log(2)^2+2*(32*x^2-128*x+48)*log(2)+32*x^4-256*x^3+576*x^2-288*x)*exp(x))/ ((8*log(2)^3+4*(6*x^2-18*x)*log(2)^2+2*(12*x^4-72*x^3+108*x^2)*log(2)+8*x^ 6-72*x^5+216*x^4-216*x^3)*exp(x)^3+(48*log(2)^3+4*(36*x^2-108*x)*log(2)^2+ 2*(72*x^4-432*x^3+648*x^2)*log(2)+48*x^6-432*x^5+1296*x^4-1296*x^3)*exp(x) ^2+(96*log(2)^3+4*(72*x^2-216*x)*log(2)^2+2*(144*x^4-864*x^3+1296*x^2)*log (2)+96*x^6-864*x^5+2592*x^4-2592*x^3)*exp(x)+64*log(2)^3+4*(48*x^2-144*x)* log(2)^2+2*(96*x^4-576*x^3+864*x^2)*log(2)+64*x^6-576*x^5+1728*x^4-1728*x^ 3),x, algorithm="maxima")
Output:
1/4*((4*x^2 - 12*x + 4*log(2) + 1)*e^(2*x) + 8*(x^2 - 3*x + log(2))*e^x)/( 4*x^4 - 24*x^3 + 4*x^2*(2*log(2) + 9) + (x^4 - 6*x^3 + x^2*(2*log(2) + 9) - 6*x*log(2) + log(2)^2)*e^(2*x) + 4*(x^4 - 6*x^3 + x^2*(2*log(2) + 9) - 6 *x*log(2) + log(2)^2)*e^x - 24*x*log(2) + 4*log(2)^2)
Leaf count of result is larger than twice the leaf count of optimal. 185 vs. \(2 (25) = 50\).
Time = 0.17 (sec) , antiderivative size = 185, normalized size of antiderivative = 6.61 \[ \int \frac {e^{3 x} \left (12-80 x+72 x^2-16 x^3+(12-8 x) \log (4)\right )+e^{2 x} \left (24-328 x+440 x^2-160 x^3+16 x^4+\left (52-80 x+16 x^2\right ) \log (4)+4 \log ^2(4)\right )+e^x \left (-288 x+576 x^2-256 x^3+32 x^4+\left (48-128 x+32 x^2\right ) \log (4)+8 \log ^2(4)\right )}{-1728 x^3+1728 x^4-576 x^5+64 x^6+\left (864 x^2-576 x^3+96 x^4\right ) \log (4)+\left (-144 x+48 x^2\right ) \log ^2(4)+8 \log ^3(4)+e^{3 x} \left (-216 x^3+216 x^4-72 x^5+8 x^6+\left (108 x^2-72 x^3+12 x^4\right ) \log (4)+\left (-18 x+6 x^2\right ) \log ^2(4)+\log ^3(4)\right )+e^{2 x} \left (-1296 x^3+1296 x^4-432 x^5+48 x^6+\left (648 x^2-432 x^3+72 x^4\right ) \log (4)+\left (-108 x+36 x^2\right ) \log ^2(4)+6 \log ^3(4)\right )+e^x \left (-2592 x^3+2592 x^4-864 x^5+96 x^6+\left (1296 x^2-864 x^3+144 x^4\right ) \log (4)+\left (-216 x+72 x^2\right ) \log ^2(4)+12 \log ^3(4)\right )} \, dx=\frac {4 \, x^{2} e^{\left (2 \, x\right )} + 8 \, x^{2} e^{x} - 12 \, x e^{\left (2 \, x\right )} - 24 \, x e^{x} + 4 \, e^{\left (2 \, x\right )} \log \left (2\right ) + 8 \, e^{x} \log \left (2\right ) + e^{\left (2 \, x\right )}}{4 \, {\left (x^{4} e^{\left (2 \, x\right )} + 4 \, x^{4} e^{x} + 4 \, x^{4} - 6 \, x^{3} e^{\left (2 \, x\right )} - 24 \, x^{3} e^{x} + 2 \, x^{2} e^{\left (2 \, x\right )} \log \left (2\right ) + 8 \, x^{2} e^{x} \log \left (2\right ) - 24 \, x^{3} + 9 \, x^{2} e^{\left (2 \, x\right )} + 36 \, x^{2} e^{x} + 8 \, x^{2} \log \left (2\right ) - 6 \, x e^{\left (2 \, x\right )} \log \left (2\right ) - 24 \, x e^{x} \log \left (2\right ) + e^{\left (2 \, x\right )} \log \left (2\right )^{2} + 4 \, e^{x} \log \left (2\right )^{2} + 36 \, x^{2} - 24 \, x \log \left (2\right ) + 4 \, \log \left (2\right )^{2}\right )}} \] Input:
integrate(((2*(-8*x+12)*log(2)-16*x^3+72*x^2-80*x+12)*exp(x)^3+(16*log(2)^ 2+2*(16*x^2-80*x+52)*log(2)+16*x^4-160*x^3+440*x^2-328*x+24)*exp(x)^2+(32* log(2)^2+2*(32*x^2-128*x+48)*log(2)+32*x^4-256*x^3+576*x^2-288*x)*exp(x))/ ((8*log(2)^3+4*(6*x^2-18*x)*log(2)^2+2*(12*x^4-72*x^3+108*x^2)*log(2)+8*x^ 6-72*x^5+216*x^4-216*x^3)*exp(x)^3+(48*log(2)^3+4*(36*x^2-108*x)*log(2)^2+ 2*(72*x^4-432*x^3+648*x^2)*log(2)+48*x^6-432*x^5+1296*x^4-1296*x^3)*exp(x) ^2+(96*log(2)^3+4*(72*x^2-216*x)*log(2)^2+2*(144*x^4-864*x^3+1296*x^2)*log (2)+96*x^6-864*x^5+2592*x^4-2592*x^3)*exp(x)+64*log(2)^3+4*(48*x^2-144*x)* log(2)^2+2*(96*x^4-576*x^3+864*x^2)*log(2)+64*x^6-576*x^5+1728*x^4-1728*x^ 3),x, algorithm="giac")
Output:
1/4*(4*x^2*e^(2*x) + 8*x^2*e^x - 12*x*e^(2*x) - 24*x*e^x + 4*e^(2*x)*log(2 ) + 8*e^x*log(2) + e^(2*x))/(x^4*e^(2*x) + 4*x^4*e^x + 4*x^4 - 6*x^3*e^(2* x) - 24*x^3*e^x + 2*x^2*e^(2*x)*log(2) + 8*x^2*e^x*log(2) - 24*x^3 + 9*x^2 *e^(2*x) + 36*x^2*e^x + 8*x^2*log(2) - 6*x*e^(2*x)*log(2) - 24*x*e^x*log(2 ) + e^(2*x)*log(2)^2 + 4*e^x*log(2)^2 + 36*x^2 - 24*x*log(2) + 4*log(2)^2)
Time = 9.11 (sec) , antiderivative size = 33454, normalized size of antiderivative = 1194.79 \[ \int \frac {e^{3 x} \left (12-80 x+72 x^2-16 x^3+(12-8 x) \log (4)\right )+e^{2 x} \left (24-328 x+440 x^2-160 x^3+16 x^4+\left (52-80 x+16 x^2\right ) \log (4)+4 \log ^2(4)\right )+e^x \left (-288 x+576 x^2-256 x^3+32 x^4+\left (48-128 x+32 x^2\right ) \log (4)+8 \log ^2(4)\right )}{-1728 x^3+1728 x^4-576 x^5+64 x^6+\left (864 x^2-576 x^3+96 x^4\right ) \log (4)+\left (-144 x+48 x^2\right ) \log ^2(4)+8 \log ^3(4)+e^{3 x} \left (-216 x^3+216 x^4-72 x^5+8 x^6+\left (108 x^2-72 x^3+12 x^4\right ) \log (4)+\left (-18 x+6 x^2\right ) \log ^2(4)+\log ^3(4)\right )+e^{2 x} \left (-1296 x^3+1296 x^4-432 x^5+48 x^6+\left (648 x^2-432 x^3+72 x^4\right ) \log (4)+\left (-108 x+36 x^2\right ) \log ^2(4)+6 \log ^3(4)\right )+e^x \left (-2592 x^3+2592 x^4-864 x^5+96 x^6+\left (1296 x^2-864 x^3+144 x^4\right ) \log (4)+\left (-216 x+72 x^2\right ) \log ^2(4)+12 \log ^3(4)\right )} \, dx=\text {Too large to display} \] Input:
int((exp(x)*(2*log(2)*(32*x^2 - 128*x + 48) - 288*x + 32*log(2)^2 + 576*x^ 2 - 256*x^3 + 32*x^4) - exp(3*x)*(80*x + 2*log(2)*(8*x - 12) - 72*x^2 + 16 *x^3 - 12) + exp(2*x)*(2*log(2)*(16*x^2 - 80*x + 52) - 328*x + 16*log(2)^2 + 440*x^2 - 160*x^3 + 16*x^4 + 24))/(exp(x)*(96*log(2)^3 - 4*log(2)^2*(21 6*x - 72*x^2) + 2*log(2)*(1296*x^2 - 864*x^3 + 144*x^4) - 2592*x^3 + 2592* x^4 - 864*x^5 + 96*x^6) + exp(3*x)*(8*log(2)^3 - 4*log(2)^2*(18*x - 6*x^2) + 2*log(2)*(108*x^2 - 72*x^3 + 12*x^4) - 216*x^3 + 216*x^4 - 72*x^5 + 8*x ^6) + exp(2*x)*(48*log(2)^3 - 4*log(2)^2*(108*x - 36*x^2) + 2*log(2)*(648* x^2 - 432*x^3 + 72*x^4) - 1296*x^3 + 1296*x^4 - 432*x^5 + 48*x^6) - 4*log( 2)^2*(144*x - 48*x^2) + 64*log(2)^3 + 2*log(2)*(864*x^2 - 576*x^3 + 96*x^4 ) - 1728*x^3 + 1728*x^4 - 576*x^5 + 64*x^6),x)
Output:
(243*x^5)/(378*x^2*log(2)^5 - 1890*x^3*log(2)^4 + 5670*x^4*log(2)^3 - 1020 6*x^5*log(2)^2 + 14*x^2*log(2)^6 - 252*x^3*log(2)^5 + 1890*x^4*log(2)^4 - 7560*x^5*log(2)^3 + 17010*x^6*log(2)^2 + 42*x^4*log(2)^5 - 630*x^5*log(2)^ 4 + 3780*x^6*log(2)^3 - 11340*x^7*log(2)^2 + 70*x^6*log(2)^4 - 840*x^7*log (2)^3 + 3780*x^8*log(2)^2 + 70*x^8*log(2)^3 - 630*x^9*log(2)^2 + 42*x^10*l og(2)^2 + exp(x)*log(2)^7 - 2187*x^7*exp(x) + 5103*x^8*exp(x) - 5103*x^9*e xp(x) + 2835*x^10*exp(x) - 945*x^11*exp(x) + 189*x^12*exp(x) - 21*x^13*exp (x) + x^14*exp(x) - 42*x*log(2)^6 + 10206*x^6*log(2) - 20412*x^7*log(2) + 17010*x^8*log(2) - 7560*x^9*log(2) + 1890*x^10*log(2) - 252*x^11*log(2) + 14*x^12*log(2) + 2*log(2)^7 - 4374*x^7 + 10206*x^8 - 10206*x^9 + 5670*x^10 - 1890*x^11 + 378*x^12 - 42*x^13 + 2*x^14 - 21*x*exp(x)*log(2)^6 + 5103*x ^6*exp(x)*log(2) - 10206*x^7*exp(x)*log(2) + 8505*x^8*exp(x)*log(2) - 3780 *x^9*exp(x)*log(2) + 945*x^10*exp(x)*log(2) - 126*x^11*exp(x)*log(2) + 7*x ^12*exp(x)*log(2) + 189*x^2*exp(x)*log(2)^5 - 945*x^3*exp(x)*log(2)^4 + 28 35*x^4*exp(x)*log(2)^3 - 5103*x^5*exp(x)*log(2)^2 + 7*x^2*exp(x)*log(2)^6 - 126*x^3*exp(x)*log(2)^5 + 945*x^4*exp(x)*log(2)^4 - 3780*x^5*exp(x)*log( 2)^3 + 8505*x^6*exp(x)*log(2)^2 + 21*x^4*exp(x)*log(2)^5 - 315*x^5*exp(x)* log(2)^4 + 1890*x^6*exp(x)*log(2)^3 - 5670*x^7*exp(x)*log(2)^2 + 35*x^6*ex p(x)*log(2)^4 - 420*x^7*exp(x)*log(2)^3 + 1890*x^8*exp(x)*log(2)^2 + 35*x^ 8*exp(x)*log(2)^3 - 315*x^9*exp(x)*log(2)^2 + 21*x^10*exp(x)*log(2)^2) ...
Time = 0.25 (sec) , antiderivative size = 279, normalized size of antiderivative = 9.96 \[ \int \frac {e^{3 x} \left (12-80 x+72 x^2-16 x^3+(12-8 x) \log (4)\right )+e^{2 x} \left (24-328 x+440 x^2-160 x^3+16 x^4+\left (52-80 x+16 x^2\right ) \log (4)+4 \log ^2(4)\right )+e^x \left (-288 x+576 x^2-256 x^3+32 x^4+\left (48-128 x+32 x^2\right ) \log (4)+8 \log ^2(4)\right )}{-1728 x^3+1728 x^4-576 x^5+64 x^6+\left (864 x^2-576 x^3+96 x^4\right ) \log (4)+\left (-144 x+48 x^2\right ) \log ^2(4)+8 \log ^3(4)+e^{3 x} \left (-216 x^3+216 x^4-72 x^5+8 x^6+\left (108 x^2-72 x^3+12 x^4\right ) \log (4)+\left (-18 x+6 x^2\right ) \log ^2(4)+\log ^3(4)\right )+e^{2 x} \left (-1296 x^3+1296 x^4-432 x^5+48 x^6+\left (648 x^2-432 x^3+72 x^4\right ) \log (4)+\left (-108 x+36 x^2\right ) \log ^2(4)+6 \log ^3(4)\right )+e^x \left (-2592 x^3+2592 x^4-864 x^5+96 x^6+\left (1296 x^2-864 x^3+144 x^4\right ) \log (4)+\left (-216 x+72 x^2\right ) \log ^2(4)+12 \log ^3(4)\right )} \, dx=\frac {2 e^{2 x} \mathrm {log}\left (2\right )^{2}+e^{2 x} \mathrm {log}\left (2\right )-2 e^{2 x} x^{4}+12 e^{2 x} x^{3}-18 e^{2 x} x^{2}-8 e^{x} \mathrm {log}\left (2\right ) x^{2}+24 e^{x} \mathrm {log}\left (2\right ) x -8 e^{x} x^{4}+48 e^{x} x^{3}-72 e^{x} x^{2}-8 \mathrm {log}\left (2\right )^{2}-16 \,\mathrm {log}\left (2\right ) x^{2}+48 \,\mathrm {log}\left (2\right ) x -8 x^{4}+48 x^{3}-72 x^{2}}{4 \,\mathrm {log}\left (2\right ) \left (e^{2 x} \mathrm {log}\left (2\right )^{2}+2 e^{2 x} \mathrm {log}\left (2\right ) x^{2}-6 e^{2 x} \mathrm {log}\left (2\right ) x +e^{2 x} x^{4}-6 e^{2 x} x^{3}+9 e^{2 x} x^{2}+4 e^{x} \mathrm {log}\left (2\right )^{2}+8 e^{x} \mathrm {log}\left (2\right ) x^{2}-24 e^{x} \mathrm {log}\left (2\right ) x +4 e^{x} x^{4}-24 e^{x} x^{3}+36 e^{x} x^{2}+4 \mathrm {log}\left (2\right )^{2}+8 \,\mathrm {log}\left (2\right ) x^{2}-24 \,\mathrm {log}\left (2\right ) x +4 x^{4}-24 x^{3}+36 x^{2}\right )} \] Input:
int(((2*(-8*x+12)*log(2)-16*x^3+72*x^2-80*x+12)*exp(x)^3+(16*log(2)^2+2*(1 6*x^2-80*x+52)*log(2)+16*x^4-160*x^3+440*x^2-328*x+24)*exp(x)^2+(32*log(2) ^2+2*(32*x^2-128*x+48)*log(2)+32*x^4-256*x^3+576*x^2-288*x)*exp(x))/((8*lo g(2)^3+4*(6*x^2-18*x)*log(2)^2+2*(12*x^4-72*x^3+108*x^2)*log(2)+8*x^6-72*x ^5+216*x^4-216*x^3)*exp(x)^3+(48*log(2)^3+4*(36*x^2-108*x)*log(2)^2+2*(72* x^4-432*x^3+648*x^2)*log(2)+48*x^6-432*x^5+1296*x^4-1296*x^3)*exp(x)^2+(96 *log(2)^3+4*(72*x^2-216*x)*log(2)^2+2*(144*x^4-864*x^3+1296*x^2)*log(2)+96 *x^6-864*x^5+2592*x^4-2592*x^3)*exp(x)+64*log(2)^3+4*(48*x^2-144*x)*log(2) ^2+2*(96*x^4-576*x^3+864*x^2)*log(2)+64*x^6-576*x^5+1728*x^4-1728*x^3),x)
Output:
(2*e**(2*x)*log(2)**2 + e**(2*x)*log(2) - 2*e**(2*x)*x**4 + 12*e**(2*x)*x* *3 - 18*e**(2*x)*x**2 - 8*e**x*log(2)*x**2 + 24*e**x*log(2)*x - 8*e**x*x** 4 + 48*e**x*x**3 - 72*e**x*x**2 - 8*log(2)**2 - 16*log(2)*x**2 + 48*log(2) *x - 8*x**4 + 48*x**3 - 72*x**2)/(4*log(2)*(e**(2*x)*log(2)**2 + 2*e**(2*x )*log(2)*x**2 - 6*e**(2*x)*log(2)*x + e**(2*x)*x**4 - 6*e**(2*x)*x**3 + 9* e**(2*x)*x**2 + 4*e**x*log(2)**2 + 8*e**x*log(2)*x**2 - 24*e**x*log(2)*x + 4*e**x*x**4 - 24*e**x*x**3 + 36*e**x*x**2 + 4*log(2)**2 + 8*log(2)*x**2 - 24*log(2)*x + 4*x**4 - 24*x**3 + 36*x**2))