Integrand size = 83, antiderivative size = 30 \[ \int \frac {e^x \left (36 x-60 x^2+42 x^3-24 x^4+6 x^5+\left (-12+48 x^2-48 x^3+12 x^4\right ) \log (2)\right )}{x^4+\left (-4 x^2+4 x^3\right ) \log (2)+\left (4-8 x+4 x^2\right ) \log ^2(2)} \, dx=\frac {3 e^x x \left (-4+\frac {3}{x}+x\right )}{\frac {x^2}{-2+2 x}+\log (2)} \] Output:
3*x*(x+3/x-4)*exp(x)/(ln(2)+x^2/(-2+2*x))
Result contains higher order function than in optimal. Order 4 vs. order 3 in optimal.
Time = 7.50 (sec) , antiderivative size = 602, normalized size of antiderivative = 20.07 \[ \int \frac {e^x \left (36 x-60 x^2+42 x^3-24 x^4+6 x^5+\left (-12+48 x^2-48 x^3+12 x^4\right ) \log (2)\right )}{x^4+\left (-4 x^2+4 x^3\right ) \log (2)+\left (4-8 x+4 x^2\right ) \log ^2(2)} \, dx=\frac {3 e^{-\frac {1}{2} \sqrt {\log (4) (4+\log (4))}} \left (-3 e^{\sqrt {\log (4) (4+\log (4))}} \operatorname {ExpIntegralEi}\left (x+\log (2)-\frac {1}{2} \sqrt {\log (4) (4+\log (4))}\right ) \left (-5 \log ^3(4)+3 \log ^{\frac {5}{2}}(4) \sqrt {4+\log (4)}+\log (4) \log (16)-\log ^{\frac {3}{2}}(4) \sqrt {4+\log (4)} (2+\log (64))+\log (16) \sqrt {\log ^2(4)+\log (256)}+\log ^2(4) (-2+\log (1024))\right )+\frac {4 e^{x+\frac {1}{2} \sqrt {\log (4) (4+\log (4))}} \left (-5 x^2 \left (\log ^{\frac {3}{2}}(4) \sqrt {4+\log (4)}+4 \sqrt {\log (4) (4+\log (4))}\right )+x^3 \left (\log ^{\frac {3}{2}}(4) \sqrt {4+\log (4)}+4 \sqrt {\log (4) (4+\log (4))}\right )-3 \left ((1-6 \log (2)) \log ^{\frac {3}{2}}(4) \sqrt {4+\log (4)}+3 \log ^{\frac {5}{2}}(4) \sqrt {4+\log (4)}+4 \sqrt {\log (4) (4+\log (4))}\right )-x \left (-13 \log ^{\frac {3}{2}}(4) \sqrt {4+\log (4)}-3 \log ^{\frac {7}{2}}(4) \sqrt {4+\log (4)}+\log ^{\frac {5}{2}}(4) \sqrt {4+\log (4)} (-12+\log (64))+2 \sqrt {\log (4) (4+\log (4))} (-14+\log (64)+\log (8) \log (256))\right )\right )-3 \operatorname {ExpIntegralEi}\left (x+\frac {1}{2} \left (\log (4)+\sqrt {\log (4) (4+\log (4))}\right )\right ) \left (x^2 \left (5 \log ^3(4)+3 \log ^{\frac {5}{2}}(4) \sqrt {4+\log (4)}-\log (4) \log (16)-\log ^{\frac {3}{2}}(4) \sqrt {4+\log (4)} (2+\log (64))+\log (16) \sqrt {\log ^2(4)+\log (256)}-\log ^2(4) (-2+\log (1024))\right )+x \log (4) \left (5 \log ^3(4)+3 \log ^{\frac {5}{2}}(4) \sqrt {4+\log (4)}-\log (4) \log (16)-\log ^{\frac {3}{2}}(4) \sqrt {4+\log (4)} (2+\log (64))+\log (16) \sqrt {\log ^2(4)+\log (256)}-\log ^2(4) (-2+\log (1024))\right )+\log ^2(4) \left (-5 \log ^2(4)-3 \log ^{\frac {3}{2}}(4) \sqrt {4+\log (4)}+\log (64) \sqrt {\log ^2(4)+\log (256)}+\log (4) \log (1024)\right )\right )}{x^2-\log (4)+x \log (4)}\right )}{2 \sqrt {\log (4)} (4+\log (4))^{3/2}} \] Input:
Integrate[(E^x*(36*x - 60*x^2 + 42*x^3 - 24*x^4 + 6*x^5 + (-12 + 48*x^2 - 48*x^3 + 12*x^4)*Log[2]))/(x^4 + (-4*x^2 + 4*x^3)*Log[2] + (4 - 8*x + 4*x^ 2)*Log[2]^2),x]
Output:
(3*(-3*E^Sqrt[Log[4]*(4 + Log[4])]*ExpIntegralEi[x + Log[2] - Sqrt[Log[4]* (4 + Log[4])]/2]*(-5*Log[4]^3 + 3*Log[4]^(5/2)*Sqrt[4 + Log[4]] + Log[4]*L og[16] - Log[4]^(3/2)*Sqrt[4 + Log[4]]*(2 + Log[64]) + Log[16]*Sqrt[Log[4] ^2 + Log[256]] + Log[4]^2*(-2 + Log[1024])) + (4*E^(x + Sqrt[Log[4]*(4 + L og[4])]/2)*(-5*x^2*(Log[4]^(3/2)*Sqrt[4 + Log[4]] + 4*Sqrt[Log[4]*(4 + Log [4])]) + x^3*(Log[4]^(3/2)*Sqrt[4 + Log[4]] + 4*Sqrt[Log[4]*(4 + Log[4])]) - 3*((1 - 6*Log[2])*Log[4]^(3/2)*Sqrt[4 + Log[4]] + 3*Log[4]^(5/2)*Sqrt[4 + Log[4]] + 4*Sqrt[Log[4]*(4 + Log[4])]) - x*(-13*Log[4]^(3/2)*Sqrt[4 + L og[4]] - 3*Log[4]^(7/2)*Sqrt[4 + Log[4]] + Log[4]^(5/2)*Sqrt[4 + Log[4]]*( -12 + Log[64]) + 2*Sqrt[Log[4]*(4 + Log[4])]*(-14 + Log[64] + Log[8]*Log[2 56]))) - 3*ExpIntegralEi[x + (Log[4] + Sqrt[Log[4]*(4 + Log[4])])/2]*(x^2* (5*Log[4]^3 + 3*Log[4]^(5/2)*Sqrt[4 + Log[4]] - Log[4]*Log[16] - Log[4]^(3 /2)*Sqrt[4 + Log[4]]*(2 + Log[64]) + Log[16]*Sqrt[Log[4]^2 + Log[256]] - L og[4]^2*(-2 + Log[1024])) + x*Log[4]*(5*Log[4]^3 + 3*Log[4]^(5/2)*Sqrt[4 + Log[4]] - Log[4]*Log[16] - Log[4]^(3/2)*Sqrt[4 + Log[4]]*(2 + Log[64]) + Log[16]*Sqrt[Log[4]^2 + Log[256]] - Log[4]^2*(-2 + Log[1024])) + Log[4]^2* (-5*Log[4]^2 - 3*Log[4]^(3/2)*Sqrt[4 + Log[4]] + Log[64]*Sqrt[Log[4]^2 + L og[256]] + Log[4]*Log[1024])))/(x^2 - Log[4] + x*Log[4])))/(2*E^(Sqrt[Log[ 4]*(4 + Log[4])]/2)*Sqrt[Log[4]]*(4 + Log[4])^(3/2))
Result contains higher order function than in optimal. Order 4 vs. order 3 in optimal.
Time = 9.03 (sec) , antiderivative size = 4123, normalized size of antiderivative = 137.43, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.024, Rules used = {2463, 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.
| \(\displaystyle \int \frac {e^x \left (6 x^5-24 x^4+42 x^3-60 x^2+\left (12 x^4-48 x^3+48 x^2-12\right ) \log (2)+36 x\right )}{x^4+\left (4 x^2-8 x+4\right ) \log ^2(2)+\left (4 x^3-4 x^2\right ) \log (2)} \, dx\) |
| \(\Big \downarrow \) 2463 |
| \(\displaystyle \int \left (\frac {e^x \left (6 x^5-24 x^4+42 x^3-60 x^2+\left (12 x^4-48 x^3+48 x^2-12\right ) \log (2)+36 x\right )}{2 \left (\log ^2(2)+\log (4)\right )^{3/2} \left (-2 x+2 \sqrt {\log ^2(2)+\log (4)}-2 \log (2)\right )}+\frac {e^x \left (6 x^5-24 x^4+42 x^3-60 x^2+\left (12 x^4-48 x^3+48 x^2-12\right ) \log (2)+36 x\right )}{2 \left (\log ^2(2)+\log (4)\right )^{3/2} \left (2 x+2 \sqrt {\log ^2(2)+\log (4)}+\log (4)\right )}+\frac {e^x \left (6 x^5-24 x^4+42 x^3-60 x^2+\left (12 x^4-48 x^3+48 x^2-12\right ) \log (2)+36 x\right )}{\left (\log ^2(2)+\log (4)\right ) \left (-2 x+2 \sqrt {\log ^2(2)+\log (4)}-2 \log (2)\right )^2}+\frac {e^x \left (6 x^5-24 x^4+42 x^3-60 x^2+\left (12 x^4-48 x^3+48 x^2-12\right ) \log (2)+36 x\right )}{\left (\log ^2(2)+\log (4)\right ) \left (2 x+2 \sqrt {\log ^2(2)+\log (4)}+\log (4)\right )^2}\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle -\frac {3 e^x \left (8-\log (4)+2 \sqrt {\log ^2(2)+\log (4)}\right ) x^3}{4 \left (\log ^2(2)+\log (4)\right )^{3/2}}+\frac {3 e^x \left (4-\log (2)-\sqrt {\log ^2(2)+\log (4)}\right ) x^3}{2 \left (\log ^2(2)+\log (4)\right )^{3/2}}+\frac {3 e^x x^3}{\log ^2(2)+\log (4)}+\frac {3 e^x \left (28+4 \log ^2(2)+\log ^2(4)+16 \sqrt {\log ^2(2)+\log (4)}-\log (4) \log (16)-\log (256)\right ) x^2}{8 \left (\log ^2(2)+\log (4)\right )^{3/2}}+\frac {9 e^x \left (8-\log (4)+2 \sqrt {\log ^2(2)+\log (4)}\right ) x^2}{4 \left (\log ^2(2)+\log (4)\right )^{3/2}}-\frac {3 e^x \left (2+\sqrt {\log ^2(2)+\log (4)}\right ) x^2}{\log ^2(2)+\log (4)}-\frac {9 e^x \left (4-\log (2)-\sqrt {\log ^2(2)+\log (4)}\right ) x^2}{2 \left (\log ^2(2)+\log (4)\right )^{3/2}}-\frac {3 e^x \left (2-\sqrt {\log ^2(2)+\log (4)}\right ) x^2}{\log ^2(2)+\log (4)}-\frac {3 e^x \left (7-\log (4)-4 \sqrt {\log ^2(2)+\log (4)}\right ) x^2}{2 \left (\log ^2(2)+\log (4)\right )^{3/2}}-\frac {9 e^x x^2}{\log ^2(2)+\log (4)}-\frac {3 e^x \left (28+4 \log ^2(2)+\log ^2(4)+16 \sqrt {\log ^2(2)+\log (4)}-\log (4) \log (16)-\log (256)\right ) x}{4 \left (\log ^2(2)+\log (4)\right )^{3/2}}-\frac {3 e^x \left (80-16 \log ^3(2)+\log ^3(4)+56 \sqrt {\log ^2(2)+\log (4)}+20 \log (4) \left (3+2 \sqrt {\log ^2(2)+\log (4)}\right )+\log ^2(4) \left (20+6 \sqrt {\log ^2(2)+\log (4)}\right )-4 \log (2) \left (\log ^2(4)+16 \left (1+\sqrt {\log ^2(2)+\log (4)}\right )+4 \log (4) \left (3+\sqrt {\log ^2(2)+\log (4)}\right )\right )+4 \log ^2(2) \left (8+2 \sqrt {\log ^2(2)+\log (4)}+\log (64)\right )\right ) x}{16 \left (\log ^2(2)+\log (4)\right )^{3/2}}+\frac {3 e^x \left (7+2 \log ^2(2)-(8+\log (4)) \sqrt {\log ^2(2)+\log (4)}+\log (64)\right ) x}{2 \left (\log ^2(2)+\log (4)\right )}+\frac {3 e^x \left (12 \log ^2(2)+3 \log ^2(4)-8 \log (2) \left (4+\log (4)+2 \sqrt {\log ^2(2)+\log (4)}\right )+4 \log (4) \left (7+3 \sqrt {\log ^2(2)+\log (4)}\right )+4 \left (7+8 \sqrt {\log ^2(2)+\log (4)}\right )\right ) x}{8 \left (\log ^2(2)+\log (4)\right )}+\frac {3 e^x \left (10-\log (2) (1-\log (4))-7 \sqrt {\log ^2(2)+\log (4)}+\log (4) \left (4-\sqrt {\log ^2(2)+\log (4)}\right )\right ) x}{2 \left (\log ^2(2)+\log (4)\right )^{3/2}}-\frac {9 e^x \left (8-\log (4)+2 \sqrt {\log ^2(2)+\log (4)}\right ) x}{2 \left (\log ^2(2)+\log (4)\right )^{3/2}}+\frac {6 e^x \left (2+\sqrt {\log ^2(2)+\log (4)}\right ) x}{\log ^2(2)+\log (4)}+\frac {9 e^x \left (4-\log (2)-\sqrt {\log ^2(2)+\log (4)}\right ) x}{\left (\log ^2(2)+\log (4)\right )^{3/2}}+\frac {6 e^x \left (2-\sqrt {\log ^2(2)+\log (4)}\right ) x}{\log ^2(2)+\log (4)}+\frac {3 e^x \left (7-\log (4)-4 \sqrt {\log ^2(2)+\log (4)}\right ) x}{\left (\log ^2(2)+\log (4)\right )^{3/2}}+\frac {18 e^x x}{\log ^2(2)+\log (4)}-\frac {3 e^x \left (5+2 \log ^3(2)+2 \log ^2(2) \left (2-\sqrt {\log ^2(2)+\log (4)}\right )+\log (4) \log (8)-\sqrt {\log ^2(2)+\log (4)} (7+\log (256))+\log (32768)\right )}{\log ^2(2)+\log (4)}+\frac {3 e^x \left (4 \log ^3(2) (3+\log (4))-6 \sqrt {\log ^2(2)+\log (4)}+\log (4) \left (10-7 \sqrt {\log ^2(2)+\log (4)}\right )-\log (2) \left (20 \sqrt {\log ^2(2)+\log (4)}-\log (4) \left (13-8 \sqrt {\log ^2(2)+\log (4)}\right )\right )+4 \log ^2(2) \left (5-3 \sqrt {\log ^2(2)+\log (4)}+\log (4) \left (2-\sqrt {\log ^2(2)+\log (4)}\right )\right )+\log ^2(4) \left (4-\sqrt {\log ^2(2)+\log (4)}+\log (8)\right )+\log (256)\right )}{2 \left (\log ^2(2)+\log (4)\right ) \left (x-\sqrt {\log ^2(2)+\log (4)}+\log (2)\right )}-\frac {3 e^{\sqrt {\log ^2(2)+\log (4)}} \operatorname {ExpIntegralEi}\left (x-\sqrt {\log ^2(2)+\log (4)}+\log (2)\right ) \left (4 \log ^3(2) (3+\log (4))-6 \sqrt {\log ^2(2)+\log (4)}+\log (4) \left (10-7 \sqrt {\log ^2(2)+\log (4)}\right )-\log (2) \left (20 \sqrt {\log ^2(2)+\log (4)}-\log (4) \left (13-8 \sqrt {\log ^2(2)+\log (4)}\right )\right )+4 \log ^2(2) \left (5-3 \sqrt {\log ^2(2)+\log (4)}+\log (4) \left (2-\sqrt {\log ^2(2)+\log (4)}\right )\right )+\log ^2(4) \left (4-\sqrt {\log ^2(2)+\log (4)}+\log (8)\right )+\log (256)\right )}{4 \left (\log ^2(2)+\log (4)\right )}+\frac {3 e^x \left (28+4 \log ^2(2)+\log ^2(4)+16 \sqrt {\log ^2(2)+\log (4)}-\log (4) \log (16)-\log (256)\right )}{4 \left (\log ^2(2)+\log (4)\right )^{3/2}}+\frac {3 e^x \left (16 \log ^4(2)+\log ^4(4)+8 \log ^3(4) \left (4+\sqrt {\log ^2(2)+\log (4)}\right )+20 \log ^2(4) \left (7+4 \sqrt {\log ^2(2)+\log (4)}\right )+32 \left (3+5 \sqrt {\log ^2(2)+\log (4)}\right )+16 \log (4) \left (12+11 \sqrt {\log ^2(2)+\log (4)}\right )+8 \log ^2(2) \left (14+3 \log ^2(4)+8 \sqrt {\log ^2(2)+\log (4)}+4 \log (4) \left (4+\sqrt {\log ^2(2)+\log (4)}\right )\right )-4 \log (2) \left (\log ^3(4)+32 \sqrt {\log ^2(2)+\log (4)}+8 \log (4) \left (6+5 \sqrt {\log ^2(2)+\log (4)}\right )+\log ^2(4) \left (20+6 \sqrt {\log ^2(2)+\log (4)}\right )\right )-16 \log ^3(2) \left (8+2 \sqrt {\log ^2(2)+\log (4)}+\log (64)\right )\right )}{32 \left (\log ^2(2)+\log (4)\right )^{3/2}}+\frac {3 e^x \left (80-16 \log ^3(2)+\log ^3(4)+56 \sqrt {\log ^2(2)+\log (4)}+20 \log (4) \left (3+2 \sqrt {\log ^2(2)+\log (4)}\right )+\log ^2(4) \left (20+6 \sqrt {\log ^2(2)+\log (4)}\right )-4 \log (2) \left (\log ^2(4)+16 \left (1+\sqrt {\log ^2(2)+\log (4)}\right )+4 \log (4) \left (3+\sqrt {\log ^2(2)+\log (4)}\right )\right )+4 \log ^2(2) \left (8+2 \sqrt {\log ^2(2)+\log (4)}+\log (64)\right )\right )}{16 \left (\log ^2(2)+\log (4)\right )^{3/2}}+\frac {3 e^{-\sqrt {\log ^2(2)+\log (4)}} \operatorname {ExpIntegralEi}\left (\frac {1}{2} \left (2 x+2 \sqrt {\log ^2(2)+\log (4)}+\log (4)\right )\right ) \left (96+80 \log ^4(2)+5 \log ^4(4)+320 \sqrt {\log ^2(2)+\log (4)}+8 \log ^3(4) \left (19+5 \sqrt {\log ^2(2)+\log (4)}\right )+16 \log (4) \left (31+37 \sqrt {\log ^2(2)+\log (4)}\right )+4 \log ^2(4) \left (137+88 \sqrt {\log ^2(2)+\log (4)}\right )-16 \log (2) \left (\log ^3(4)+16 \sqrt {\log ^2(2)+\log (4)}+32 \log (4) \left (1+\sqrt {\log ^2(2)+\log (4)}\right )+6 \log ^2(4) \left (3+\sqrt {\log ^2(2)+\log (4)}\right )\right )+8 \log ^2(2) \left (42+15 \log ^2(4)+32 \sqrt {\log ^2(2)+\log (4)}+4 \log (4) \left (17+5 \sqrt {\log ^2(2)+\log (4)}\right )\right )-64 \log ^3(2) \left (6+2 \sqrt {\log ^2(2)+\log (4)}+\log (64)\right )\right )}{64 \left (\log ^2(2)+\log (4)\right )}+\frac {3 e^x \left (12 \log ^3(2)-\log ^3(4)-6 \log ^2(4) \left (3+\sqrt {\log ^2(2)+\log (4)}\right )-4 \left (5+7 \sqrt {\log ^2(2)+\log (4)}\right )-\log (4) \left (38+32 \sqrt {\log ^2(2)+\log (4)}\right )+\log (2) \left (16+3 \log ^2(4)+32 \sqrt {\log ^2(2)+\log (4)}+4 \log (4) \left (7+3 \sqrt {\log ^2(2)+\log (4)}\right )\right )-4 \log ^2(2) \left (6+2 \sqrt {\log ^2(2)+\log (4)}+\log (64)\right )\right )}{4 \left (\log ^2(2)+\log (4)\right )}-\frac {3 e^x \left (6+2 \log ^2(2) (3+\log (4))+\log (4) \left (7-4 \sqrt {\log ^2(2)+\log (4)}\right )+\log ^2(4) \left (1-\sqrt {\log ^2(2)+\log (4)}\right )+2 \log (2) (5+\log (16))-\sqrt {\log ^2(2)+\log (4)} (10+\log (64))\right )}{2 \left (\log ^2(2)+\log (4)\right )^{3/2}}-\frac {3 e^x \left (7+2 \log ^2(2)-(8+\log (4)) \sqrt {\log ^2(2)+\log (4)}+\log (64)\right )}{2 \left (\log ^2(2)+\log (4)\right )}-\frac {3 e^x \left (64 \log ^5(2)-\log ^5(4)-192 \sqrt {\log ^2(2)+\log (4)}-16 \log ^4(2) \left (8+5 \log (4)+2 \sqrt {\log ^2(2)+\log (4)}\right )-2 \log ^4(4) \left (24+5 \sqrt {\log ^2(2)+\log (4)}\right )-12 \log ^3(4) \left (25+12 \sqrt {\log ^2(2)+\log (4)}\right )-32 \log (4) \left (13+17 \sqrt {\log ^2(2)+\log (4)}\right )-8 \log ^2(4) \left (68+57 \sqrt {\log ^2(2)+\log (4)}\right )+32 \log ^3(2) \left (3 \log ^2(4)+8 \left (1+\sqrt {\log ^2(2)+\log (4)}\right )+4 \log (4) \left (4+\sqrt {\log ^2(2)+\log (4)}\right )\right )-4 \log (2) \left (16-\log ^4(4)-8 \log ^3(4) \left (4+\sqrt {\log ^2(2)+\log (4)}\right )-64 \log (4) \left (1+2 \sqrt {\log ^2(2)+\log (4)}\right )-16 \log ^2(4) \left (8+5 \sqrt {\log ^2(2)+\log (4)}\right )\right )-8 \log ^2(2) \left (40+5 \log ^3(4)+28 \sqrt {\log ^2(2)+\log (4)}+2 \log ^2(4) \left (22+5 \sqrt {\log ^2(2)+\log (4)}\right )+\log (4) \left (74+40 \sqrt {\log ^2(2)+\log (4)}\right )\right )\right )}{32 \left (\log ^2(2)+\log (4)\right ) \left (2 x+2 \sqrt {\log ^2(2)+\log (4)}+\log (4)\right )}+\frac {3 e^{-\sqrt {\log ^2(2)+\log (4)}} \operatorname {ExpIntegralEi}\left (\frac {1}{2} \left (2 x+2 \sqrt {\log ^2(2)+\log (4)}+\log (4)\right )\right ) \left (64 \log ^5(2)-\log ^5(4)-192 \sqrt {\log ^2(2)+\log (4)}-16 \log ^4(2) \left (8+5 \log (4)+2 \sqrt {\log ^2(2)+\log (4)}\right )-2 \log ^4(4) \left (24+5 \sqrt {\log ^2(2)+\log (4)}\right )-12 \log ^3(4) \left (25+12 \sqrt {\log ^2(2)+\log (4)}\right )-32 \log (4) \left (13+17 \sqrt {\log ^2(2)+\log (4)}\right )-8 \log ^2(4) \left (68+57 \sqrt {\log ^2(2)+\log (4)}\right )+32 \log ^3(2) \left (3 \log ^2(4)+8 \left (1+\sqrt {\log ^2(2)+\log (4)}\right )+4 \log (4) \left (4+\sqrt {\log ^2(2)+\log (4)}\right )\right )-4 \log (2) \left (16-\log ^4(4)-8 \log ^3(4) \left (4+\sqrt {\log ^2(2)+\log (4)}\right )-64 \log (4) \left (1+2 \sqrt {\log ^2(2)+\log (4)}\right )-16 \log ^2(4) \left (8+5 \sqrt {\log ^2(2)+\log (4)}\right )\right )-8 \log ^2(2) \left (40+5 \log ^3(4)+28 \sqrt {\log ^2(2)+\log (4)}+2 \log ^2(4) \left (22+5 \sqrt {\log ^2(2)+\log (4)}\right )+\log (4) \left (74+40 \sqrt {\log ^2(2)+\log (4)}\right )\right )\right )}{128 \left (\log ^2(2)+\log (4)\right )}+\frac {3 e^{-\sqrt {\log ^2(2)+\log (4)}} \operatorname {ExpIntegralEi}\left (\frac {1}{2} \left (2 x+2 \sqrt {\log ^2(2)+\log (4)}+\log (4)\right )\right ) \left (64 \log ^5(2)-\log ^5(4)-192 \sqrt {\log ^2(2)+\log (4)}-16 \log ^4(2) \left (8+5 \log (4)+2 \sqrt {\log ^2(2)+\log (4)}\right )-2 \log ^4(4) \left (24+5 \sqrt {\log ^2(2)+\log (4)}\right )-12 \log ^3(4) \left (25+12 \sqrt {\log ^2(2)+\log (4)}\right )-32 \log (4) \left (13+17 \sqrt {\log ^2(2)+\log (4)}\right )-8 \log ^2(4) \left (68+57 \sqrt {\log ^2(2)+\log (4)}\right )+32 \log ^3(2) \left (3 \log ^2(4)+8 \left (1+\sqrt {\log ^2(2)+\log (4)}\right )+4 \log (4) \left (4+\sqrt {\log ^2(2)+\log (4)}\right )\right )-4 \log (2) \left (16-\log ^4(4)-8 \log ^3(4) \left (4+\sqrt {\log ^2(2)+\log (4)}\right )-64 \log (4) \left (1+2 \sqrt {\log ^2(2)+\log (4)}\right )-16 \log ^2(4) \left (8+5 \sqrt {\log ^2(2)+\log (4)}\right )\right )-8 \log ^2(2) \left (40+5 \log ^3(4)+28 \sqrt {\log ^2(2)+\log (4)}+2 \log ^2(4) \left (22+5 \sqrt {\log ^2(2)+\log (4)}\right )+\log (4) \left (74+40 \sqrt {\log ^2(2)+\log (4)}\right )\right )\right )}{128 \left (\log ^2(2)+\log (4)\right )^{3/2}}+\frac {3 e^{\sqrt {\log ^2(2)+\log (4)}} \operatorname {ExpIntegralEi}\left (\frac {1}{2} \left (2 x-2 \sqrt {\log ^2(2)+\log (4)}+\log (4)\right )\right ) \left (6+8 \log ^4(2)+5 \log ^2(4)-20 \sqrt {\log ^2(2)+\log (4)}+\log (4) \left (21-16 \sqrt {\log ^2(2)+\log (4)}\right )+\log ^2(2) \left (26+16 \log (4)-16 \sqrt {\log ^2(2)+\log (4)}\right )+8 \log ^3(2) \left (2-\sqrt {\log ^2(2)+\log (4)}\right )+\log (2) \left (20-26 \sqrt {\log ^2(2)+\log (4)}+12 \log (4) \left (2-\sqrt {\log ^2(2)+\log (4)}\right )\right )\right )}{4 \left (\log ^2(2)+\log (4)\right )}+\frac {3 e^{\sqrt {\log ^2(2)+\log (4)}} \operatorname {ExpIntegralEi}\left (x-\sqrt {\log ^2(2)+\log (4)}+\log (2)\right ) \left (4 \log ^3(2) (3+\log (4))-6 \sqrt {\log ^2(2)+\log (4)}+\log (4) \left (10-7 \sqrt {\log ^2(2)+\log (4)}\right )+\log ^2(4) \left (4-\sqrt {\log ^2(2)+\log (4)}\right )+\log (2) \left (8+3 \log ^2(4)-20 \sqrt {\log ^2(2)+\log (4)}+\log (4) \left (13-8 \sqrt {\log ^2(2)+\log (4)}\right )\right )+4 \log ^2(2) \left (5-3 \sqrt {\log ^2(2)+\log (4)}+\log (4) \left (2-\sqrt {\log ^2(2)+\log (4)}\right )\right )\right )}{4 \left (\log ^2(2)+\log (4)\right )^{3/2}}-\frac {3 e^x \left (12 \log ^2(2)+3 \log ^2(4)-8 \log (2) \left (4+\log (4)+2 \sqrt {\log ^2(2)+\log (4)}\right )+4 \log (4) \left (7+3 \sqrt {\log ^2(2)+\log (4)}\right )+4 \left (7+8 \sqrt {\log ^2(2)+\log (4)}\right )\right )}{8 \left (\log ^2(2)+\log (4)\right )}-\frac {3 e^x \left (10-\log (2) (1-\log (4))-7 \sqrt {\log ^2(2)+\log (4)}+\log (4) \left (4-\sqrt {\log ^2(2)+\log (4)}\right )\right )}{2 \left (\log ^2(2)+\log (4)\right )^{3/2}}+\frac {9 e^x \left (8-\log (4)+2 \sqrt {\log ^2(2)+\log (4)}\right )}{2 \left (\log ^2(2)+\log (4)\right )^{3/2}}-\frac {6 e^x \left (2+\sqrt {\log ^2(2)+\log (4)}\right )}{\log ^2(2)+\log (4)}-\frac {9 e^x \left (4-\log (2)-\sqrt {\log ^2(2)+\log (4)}\right )}{\left (\log ^2(2)+\log (4)\right )^{3/2}}-\frac {6 e^x \left (2-\sqrt {\log ^2(2)+\log (4)}\right )}{\log ^2(2)+\log (4)}-\frac {3 e^x \left (7-\log (4)-4 \sqrt {\log ^2(2)+\log (4)}\right )}{\left (\log ^2(2)+\log (4)\right )^{3/2}}-\frac {18 e^x}{\log ^2(2)+\log (4)}\) |
Input:
Int[(E^x*(36*x - 60*x^2 + 42*x^3 - 24*x^4 + 6*x^5 + (-12 + 48*x^2 - 48*x^3 + 12*x^4)*Log[2]))/(x^4 + (-4*x^2 + 4*x^3)*Log[2] + (4 - 8*x + 4*x^2)*Log [2]^2),x]
Output:
(-18*E^x)/(Log[2]^2 + Log[4]) + (18*E^x*x)/(Log[2]^2 + Log[4]) - (9*E^x*x^ 2)/(Log[2]^2 + Log[4]) + (3*E^x*x^3)/(Log[2]^2 + Log[4]) - (3*E^x*(7 - Log [4] - 4*Sqrt[Log[2]^2 + Log[4]]))/(Log[2]^2 + Log[4])^(3/2) + (3*E^x*x*(7 - Log[4] - 4*Sqrt[Log[2]^2 + Log[4]]))/(Log[2]^2 + Log[4])^(3/2) - (3*E^x* x^2*(7 - Log[4] - 4*Sqrt[Log[2]^2 + Log[4]]))/(2*(Log[2]^2 + Log[4])^(3/2) ) - (6*E^x*(2 - Sqrt[Log[2]^2 + Log[4]]))/(Log[2]^2 + Log[4]) + (6*E^x*x*( 2 - Sqrt[Log[2]^2 + Log[4]]))/(Log[2]^2 + Log[4]) - (3*E^x*x^2*(2 - Sqrt[L og[2]^2 + Log[4]]))/(Log[2]^2 + Log[4]) - (9*E^x*(4 - Log[2] - Sqrt[Log[2] ^2 + Log[4]]))/(Log[2]^2 + Log[4])^(3/2) + (9*E^x*x*(4 - Log[2] - Sqrt[Log [2]^2 + Log[4]]))/(Log[2]^2 + Log[4])^(3/2) - (9*E^x*x^2*(4 - Log[2] - Sqr t[Log[2]^2 + Log[4]]))/(2*(Log[2]^2 + Log[4])^(3/2)) + (3*E^x*x^3*(4 - Log [2] - Sqrt[Log[2]^2 + Log[4]]))/(2*(Log[2]^2 + Log[4])^(3/2)) - (6*E^x*(2 + Sqrt[Log[2]^2 + Log[4]]))/(Log[2]^2 + Log[4]) + (6*E^x*x*(2 + Sqrt[Log[2 ]^2 + Log[4]]))/(Log[2]^2 + Log[4]) - (3*E^x*x^2*(2 + Sqrt[Log[2]^2 + Log[ 4]]))/(Log[2]^2 + Log[4]) + (9*E^x*(8 - Log[4] + 2*Sqrt[Log[2]^2 + Log[4]] ))/(2*(Log[2]^2 + Log[4])^(3/2)) - (9*E^x*x*(8 - Log[4] + 2*Sqrt[Log[2]^2 + Log[4]]))/(2*(Log[2]^2 + Log[4])^(3/2)) + (9*E^x*x^2*(8 - Log[4] + 2*Sqr t[Log[2]^2 + Log[4]]))/(4*(Log[2]^2 + Log[4])^(3/2)) - (3*E^x*x^3*(8 - Log [4] + 2*Sqrt[Log[2]^2 + Log[4]]))/(4*(Log[2]^2 + Log[4])^(3/2)) - (3*E^x*( 10 - Log[2]*(1 - Log[4]) - 7*Sqrt[Log[2]^2 + Log[4]] + Log[4]*(4 - Sqrt...
Int[(u_.)*(Px_)^(p_), x_Symbol] :> With[{Qx = Factor[Px]}, Int[ExpandIntegr and[u, Qx^p, x], x] /; !SumQ[NonfreeFactors[Qx, x]]] /; PolyQ[Px, x] && Gt Q[Expon[Px, x], 2] && !BinomialQ[Px, x] && !TrinomialQ[Px, x] && ILtQ[p, 0]
Time = 0.33 (sec) , antiderivative size = 33, normalized size of antiderivative = 1.10
| method | result | size |
| gosper | \(\frac {6 \left (x^{3}-5 x^{2}+7 x -3\right ) {\mathrm e}^{x}}{2 x \ln \left (2\right )+x^{2}-2 \ln \left (2\right )}\) | \(33\) |
| risch | \(\frac {6 \left (x^{3}-5 x^{2}+7 x -3\right ) {\mathrm e}^{x}}{2 x \ln \left (2\right )+x^{2}-2 \ln \left (2\right )}\) | \(33\) |
| norman | \(\frac {42 \,{\mathrm e}^{x} x -30 \,{\mathrm e}^{x} x^{2}+6 \,{\mathrm e}^{x} x^{3}-18 \,{\mathrm e}^{x}}{2 x \ln \left (2\right )+x^{2}-2 \ln \left (2\right )}\) | \(41\) |
| parallelrisch | \(\frac {42 \,{\mathrm e}^{x} x -30 \,{\mathrm e}^{x} x^{2}+6 \,{\mathrm e}^{x} x^{3}-18 \,{\mathrm e}^{x}}{2 x \ln \left (2\right )+x^{2}-2 \ln \left (2\right )}\) | \(41\) |
| default | \(\text {Expression too large to display}\) | \(5402\) |
Input:
int(((12*x^4-48*x^3+48*x^2-12)*ln(2)+6*x^5-24*x^4+42*x^3-60*x^2+36*x)*exp( x)/((4*x^2-8*x+4)*ln(2)^2+(4*x^3-4*x^2)*ln(2)+x^4),x,method=_RETURNVERBOSE )
Output:
6*(x^3-5*x^2+7*x-3)*exp(x)/(2*x*ln(2)+x^2-2*ln(2))
Time = 0.09 (sec) , antiderivative size = 30, normalized size of antiderivative = 1.00 \[ \int \frac {e^x \left (36 x-60 x^2+42 x^3-24 x^4+6 x^5+\left (-12+48 x^2-48 x^3+12 x^4\right ) \log (2)\right )}{x^4+\left (-4 x^2+4 x^3\right ) \log (2)+\left (4-8 x+4 x^2\right ) \log ^2(2)} \, dx=\frac {6 \, {\left (x^{3} - 5 \, x^{2} + 7 \, x - 3\right )} e^{x}}{x^{2} + 2 \, {\left (x - 1\right )} \log \left (2\right )} \] Input:
integrate(((12*x^4-48*x^3+48*x^2-12)*log(2)+6*x^5-24*x^4+42*x^3-60*x^2+36* x)*exp(x)/((4*x^2-8*x+4)*log(2)^2+(4*x^3-4*x^2)*log(2)+x^4),x, algorithm=" fricas")
Output:
6*(x^3 - 5*x^2 + 7*x - 3)*e^x/(x^2 + 2*(x - 1)*log(2))
Time = 0.11 (sec) , antiderivative size = 32, normalized size of antiderivative = 1.07 \[ \int \frac {e^x \left (36 x-60 x^2+42 x^3-24 x^4+6 x^5+\left (-12+48 x^2-48 x^3+12 x^4\right ) \log (2)\right )}{x^4+\left (-4 x^2+4 x^3\right ) \log (2)+\left (4-8 x+4 x^2\right ) \log ^2(2)} \, dx=\frac {\left (6 x^{3} - 30 x^{2} + 42 x - 18\right ) e^{x}}{x^{2} + 2 x \log {\left (2 \right )} - 2 \log {\left (2 \right )}} \] Input:
integrate(((12*x**4-48*x**3+48*x**2-12)*ln(2)+6*x**5-24*x**4+42*x**3-60*x* *2+36*x)*exp(x)/((4*x**2-8*x+4)*ln(2)**2+(4*x**3-4*x**2)*ln(2)+x**4),x)
Output:
(6*x**3 - 30*x**2 + 42*x - 18)*exp(x)/(x**2 + 2*x*log(2) - 2*log(2))
Time = 0.16 (sec) , antiderivative size = 32, normalized size of antiderivative = 1.07 \[ \int \frac {e^x \left (36 x-60 x^2+42 x^3-24 x^4+6 x^5+\left (-12+48 x^2-48 x^3+12 x^4\right ) \log (2)\right )}{x^4+\left (-4 x^2+4 x^3\right ) \log (2)+\left (4-8 x+4 x^2\right ) \log ^2(2)} \, dx=\frac {6 \, {\left (x^{3} - 5 \, x^{2} + 7 \, x - 3\right )} e^{x}}{x^{2} + 2 \, x \log \left (2\right ) - 2 \, \log \left (2\right )} \] Input:
integrate(((12*x^4-48*x^3+48*x^2-12)*log(2)+6*x^5-24*x^4+42*x^3-60*x^2+36* x)*exp(x)/((4*x^2-8*x+4)*log(2)^2+(4*x^3-4*x^2)*log(2)+x^4),x, algorithm=" maxima")
Output:
6*(x^3 - 5*x^2 + 7*x - 3)*e^x/(x^2 + 2*x*log(2) - 2*log(2))
Time = 0.12 (sec) , antiderivative size = 40, normalized size of antiderivative = 1.33 \[ \int \frac {e^x \left (36 x-60 x^2+42 x^3-24 x^4+6 x^5+\left (-12+48 x^2-48 x^3+12 x^4\right ) \log (2)\right )}{x^4+\left (-4 x^2+4 x^3\right ) \log (2)+\left (4-8 x+4 x^2\right ) \log ^2(2)} \, dx=\frac {6 \, {\left (x^{3} e^{x} - 5 \, x^{2} e^{x} + 7 \, x e^{x} - 3 \, e^{x}\right )}}{x^{2} + 2 \, x \log \left (2\right ) - 2 \, \log \left (2\right )} \] Input:
integrate(((12*x^4-48*x^3+48*x^2-12)*log(2)+6*x^5-24*x^4+42*x^3-60*x^2+36* x)*exp(x)/((4*x^2-8*x+4)*log(2)^2+(4*x^3-4*x^2)*log(2)+x^4),x, algorithm=" giac")
Output:
6*(x^3*e^x - 5*x^2*e^x + 7*x*e^x - 3*e^x)/(x^2 + 2*x*log(2) - 2*log(2))
Timed out. \[ \int \frac {e^x \left (36 x-60 x^2+42 x^3-24 x^4+6 x^5+\left (-12+48 x^2-48 x^3+12 x^4\right ) \log (2)\right )}{x^4+\left (-4 x^2+4 x^3\right ) \log (2)+\left (4-8 x+4 x^2\right ) \log ^2(2)} \, dx=\int \frac {{\mathrm {e}}^x\,\left (36\,x+\ln \left (2\right )\,\left (12\,x^4-48\,x^3+48\,x^2-12\right )-60\,x^2+42\,x^3-24\,x^4+6\,x^5\right )}{{\ln \left (2\right )}^2\,\left (4\,x^2-8\,x+4\right )-\ln \left (2\right )\,\left (4\,x^2-4\,x^3\right )+x^4} \,d x \] Input:
int((exp(x)*(36*x + log(2)*(48*x^2 - 48*x^3 + 12*x^4 - 12) - 60*x^2 + 42*x ^3 - 24*x^4 + 6*x^5))/(log(2)^2*(4*x^2 - 8*x + 4) - log(2)*(4*x^2 - 4*x^3) + x^4),x)
Output:
int((exp(x)*(36*x + log(2)*(48*x^2 - 48*x^3 + 12*x^4 - 12) - 60*x^2 + 42*x ^3 - 24*x^4 + 6*x^5))/(log(2)^2*(4*x^2 - 8*x + 4) - log(2)*(4*x^2 - 4*x^3) + x^4), x)
Time = 0.20 (sec) , antiderivative size = 33, normalized size of antiderivative = 1.10 \[ \int \frac {e^x \left (36 x-60 x^2+42 x^3-24 x^4+6 x^5+\left (-12+48 x^2-48 x^3+12 x^4\right ) \log (2)\right )}{x^4+\left (-4 x^2+4 x^3\right ) \log (2)+\left (4-8 x+4 x^2\right ) \log ^2(2)} \, dx=\frac {6 e^{x} \left (x^{3}-5 x^{2}+7 x -3\right )}{2 \,\mathrm {log}\left (2\right ) x -2 \,\mathrm {log}\left (2\right )+x^{2}} \] Input:
int(((12*x^4-48*x^3+48*x^2-12)*log(2)+6*x^5-24*x^4+42*x^3-60*x^2+36*x)*exp (x)/((4*x^2-8*x+4)*log(2)^2+(4*x^3-4*x^2)*log(2)+x^4),x)
Output:
(6*e**x*(x**3 - 5*x**2 + 7*x - 3))/(2*log(2)*x - 2*log(2) + x**2)