Integrand size = 235, antiderivative size = 34 \[ \int \frac {50-75 x-20 x^2+35 x^3+12 x^4+x^5+e^{20} \left (2 x^2+x^3\right )+e^{10-x} \left (-45 x+25 x^2+x^3-14 x^4-2 x^5\right )+e^x \left (e^{10-x} \left (20 x-10 x^2-14 x^3-2 x^4\right )+e^{20-2 x} \left (-5 x^2+2 x^3+2 x^4\right )\right )+\left (e^{20-x} \left (2 x^2+x^3\right )+e^{10-x} \left (-10+5 x-7 x^2-8 x^3-x^4\right )\right ) \log (2+x)}{50-75 x-20 x^2+35 x^3+12 x^4+x^5+e^{20} \left (2 x^2+x^3\right )+e^{10} \left (20 x-10 x^2-14 x^3-2 x^4\right )} \, dx=x-\frac {2 x+\log (2+x)}{e^x+e^{-10+x} \left (-5+\frac {5}{x}-x\right )} \] Output:
x-(ln(2+x)+2*x)/(exp(x)+(5/x-5-x)/exp(10-x))
Time = 0.13 (sec) , antiderivative size = 57, normalized size of antiderivative = 1.68 \[ \int \frac {50-75 x-20 x^2+35 x^3+12 x^4+x^5+e^{20} \left (2 x^2+x^3\right )+e^{10-x} \left (-45 x+25 x^2+x^3-14 x^4-2 x^5\right )+e^x \left (e^{10-x} \left (20 x-10 x^2-14 x^3-2 x^4\right )+e^{20-2 x} \left (-5 x^2+2 x^3+2 x^4\right )\right )+\left (e^{20-x} \left (2 x^2+x^3\right )+e^{10-x} \left (-10+5 x-7 x^2-8 x^3-x^4\right )\right ) \log (2+x)}{50-75 x-20 x^2+35 x^3+12 x^4+x^5+e^{20} \left (2 x^2+x^3\right )+e^{10} \left (20 x-10 x^2-14 x^3-2 x^4\right )} \, dx=\frac {e^{-x} x \left (2 e^{10} x-e^{10+x} x+e^x \left (-5+5 x+x^2\right )+e^{10} \log (2+x)\right )}{-5-\left (-5+e^{10}\right ) x+x^2} \] Input:
Integrate[(50 - 75*x - 20*x^2 + 35*x^3 + 12*x^4 + x^5 + E^20*(2*x^2 + x^3) + E^(10 - x)*(-45*x + 25*x^2 + x^3 - 14*x^4 - 2*x^5) + E^x*(E^(10 - x)*(2 0*x - 10*x^2 - 14*x^3 - 2*x^4) + E^(20 - 2*x)*(-5*x^2 + 2*x^3 + 2*x^4)) + (E^(20 - x)*(2*x^2 + x^3) + E^(10 - x)*(-10 + 5*x - 7*x^2 - 8*x^3 - x^4))* Log[2 + x])/(50 - 75*x - 20*x^2 + 35*x^3 + 12*x^4 + x^5 + E^20*(2*x^2 + x^ 3) + E^10*(20*x - 10*x^2 - 14*x^3 - 2*x^4)),x]
Output:
(x*(2*E^10*x - E^(10 + x)*x + E^x*(-5 + 5*x + x^2) + E^10*Log[2 + x]))/(E^ x*(-5 - (-5 + E^10)*x + x^2))
Result contains higher order function than in optimal. Order 4 vs. order 3 in optimal.
Time = 22.59 (sec) , antiderivative size = 4186, normalized size of antiderivative = 123.12, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.017, Rules used = {2463, 7239, 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.
| \(\displaystyle \int \frac {x^5+12 x^4+35 x^3-20 x^2+e^{20} \left (x^3+2 x^2\right )+e^x \left (e^{10-x} \left (-2 x^4-14 x^3-10 x^2+20 x\right )+e^{20-2 x} \left (2 x^4+2 x^3-5 x^2\right )\right )+\left (e^{20-x} \left (x^3+2 x^2\right )+e^{10-x} \left (-x^4-8 x^3-7 x^2+5 x-10\right )\right ) \log (x+2)+e^{10-x} \left (-2 x^5-14 x^4+x^3+25 x^2-45 x\right )-75 x+50}{x^5+12 x^4+35 x^3-20 x^2+e^{20} \left (x^3+2 x^2\right )+e^{10} \left (-2 x^4-14 x^3-10 x^2+20 x\right )-75 x+50} \, dx\) |
| \(\Big \downarrow \) 2463 |
| \(\displaystyle \int \left (\frac {x^5+12 x^4+35 x^3-20 x^2+e^{20} \left (x^3+2 x^2\right )+e^x \left (e^{10-x} \left (-2 x^4-14 x^3-10 x^2+20 x\right )+e^{20-2 x} \left (2 x^4+2 x^3-5 x^2\right )\right )+\left (e^{20-x} \left (x^3+2 x^2\right )+e^{10-x} \left (-x^4-8 x^3-7 x^2+5 x-10\right )\right ) \log (x+2)+e^{10-x} \left (-2 x^5-14 x^4+x^3+25 x^2-45 x\right )-75 x+50}{\left (2 e^{10}-11\right )^2 (x+2)}+\frac {\left (x-e^{10}+3\right ) \left (x^5+12 x^4+35 x^3-20 x^2+e^{20} \left (x^3+2 x^2\right )+e^x \left (e^{10-x} \left (-2 x^4-14 x^3-10 x^2+20 x\right )+e^{20-2 x} \left (2 x^4+2 x^3-5 x^2\right )\right )+\left (e^{20-x} \left (x^3+2 x^2\right )+e^{10-x} \left (-x^4-8 x^3-7 x^2+5 x-10\right )\right ) \log (x+2)+e^{10-x} \left (-2 x^5-14 x^4+x^3+25 x^2-45 x\right )-75 x+50\right )}{\left (11-2 e^{10}\right )^2 \left (-x^2-\left (5-e^{10}\right ) x+5\right )}+\frac {\left (x-e^{10}+3\right ) \left (x^5+12 x^4+35 x^3-20 x^2+e^{20} \left (x^3+2 x^2\right )+e^x \left (e^{10-x} \left (-2 x^4-14 x^3-10 x^2+20 x\right )+e^{20-2 x} \left (2 x^4+2 x^3-5 x^2\right )\right )+\left (e^{20-x} \left (x^3+2 x^2\right )+e^{10-x} \left (-x^4-8 x^3-7 x^2+5 x-10\right )\right ) \log (x+2)+e^{10-x} \left (-2 x^5-14 x^4+x^3+25 x^2-45 x\right )-75 x+50\right )}{\left (11-2 e^{10}\right ) \left (-x^2-\left (5-e^{10}\right ) x+5\right )^2}\right )dx\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle \int \frac {e^{-x} \left (e^{x+20} (x+2) x^2+e^{20} \left (2 x^2+2 x-5\right ) x^2+e^x (x+2) \left (x^2+5 x-5\right )^2-2 e^{x+10} \left (x^3+7 x^2+5 x-10\right ) x-e^{10} (x+2) \left (x^3-\left (e^{10}-6\right ) x^2-5 x+5\right ) \log (x+2)+e^{10} \left (-2 x^4-14 x^3+x^2+25 x-45\right ) x\right )}{(x+2) \left (-x^2-\left (5-e^{10}\right ) x+5\right )^2}dx\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle \int \left (\frac {e^{20-x} \left (2 x^2+2 x-5\right ) x^2}{(x+2) \left (-x^2-\left (5-e^{10}\right ) x+5\right )^2}+\frac {e^{10-x} \left (-x^3-\left (6-e^{10}\right ) x^2+5 x-5\right ) \log (x+2)}{\left (-x^2-\left (5-e^{10}\right ) x+5\right )^2}+\frac {e^{10-x} \left (-2 x^4-14 x^3+x^2+25 x-45\right ) x}{(x+2) \left (-x^2-\left (5-e^{10}\right ) x+5\right )^2}+1\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle x+2 e^{10-x}+\frac {e^{\frac {1}{2} \left (25-e^{10}+\sqrt {45-10 e^{10}+e^{20}}\right )} \left (594-462 e^{10}+108 e^{20}-8 e^{30}+\frac {2915-4092 e^{10}+1680 e^{20}-276 e^{30}+16 e^{40}}{\sqrt {45-10 e^{10}+e^{20}}}\right ) \operatorname {ExpIntegralEi}\left (\frac {1}{2} \left (-2 x-\sqrt {45-10 e^{10}+e^{20}}+e^{10}-5\right )\right )}{\left (11-2 e^{10}\right )^2}+\frac {e^{\frac {1}{2} \left (45-e^{10}+\sqrt {45-10 e^{10}+e^{20}}\right )} \left (123-44 e^{10}+4 e^{20}+\frac {2055-1109 e^{10}+200 e^{20}-12 e^{30}}{\sqrt {45-10 e^{10}+e^{20}}}\right ) \operatorname {ExpIntegralEi}\left (\frac {1}{2} \left (-2 x-\sqrt {45-10 e^{10}+e^{20}}+e^{10}-5\right )\right )}{\left (11-2 e^{10}\right )^2}-\frac {e^{\frac {1}{2} \left (25-e^{10}+\sqrt {45-10 e^{10}+e^{20}}\right )} \left (770+395 e^{10}-318 e^{20}+62 e^{30}-4 e^{40}\right ) \left (5-e^{10}+\sqrt {45-10 e^{10}+e^{20}}\right ) \operatorname {ExpIntegralEi}\left (\frac {1}{2} \left (-2 x-\sqrt {45-10 e^{10}+e^{20}}+e^{10}-5\right )\right )}{2 \left (11-2 e^{10}\right ) \left (45-10 e^{10}+e^{20}\right )}+\frac {e^{\frac {1}{2} \left (45-e^{10}+\sqrt {45-10 e^{10}+e^{20}}\right )} \left (755-380 e^{10}+66 e^{20}-4 e^{30}\right ) \left (5-e^{10}+\sqrt {45-10 e^{10}+e^{20}}\right ) \operatorname {ExpIntegralEi}\left (\frac {1}{2} \left (-2 x-\sqrt {45-10 e^{10}+e^{20}}+e^{10}-5\right )\right )}{2 \left (11-2 e^{10}\right ) \left (45-10 e^{10}+e^{20}\right )}+\frac {e^{\frac {1}{2} \left (25-e^{10}+\sqrt {45-10 e^{10}+e^{20}}\right )} \left (1+\frac {10-e^{10}}{\sqrt {45-10 e^{10}+e^{20}}}\right ) \operatorname {ExpIntegralEi}\left (\frac {1}{2} \left (-2 x-\sqrt {45-10 e^{10}+e^{20}}+e^{10}-5\right )\right )}{11-2 e^{10}}+\frac {e^{\frac {1}{2} \left (25-e^{10}+\sqrt {45-10 e^{10}+e^{20}}\right )} \left (5-e^{10}\right ) \left (770+395 e^{10}-318 e^{20}+62 e^{30}-4 e^{40}\right ) \operatorname {ExpIntegralEi}\left (\frac {1}{2} \left (-2 x-\sqrt {45-10 e^{10}+e^{20}}+e^{10}-5\right )\right )}{\left (11-2 e^{10}\right ) \left (45-10 e^{10}+e^{20}\right )^{3/2}}-\frac {10 e^{\frac {1}{2} \left (25-e^{10}+\sqrt {45-10 e^{10}+e^{20}}\right )} \left (55+44 e^{10}-21 e^{20}+2 e^{30}\right ) \operatorname {ExpIntegralEi}\left (\frac {1}{2} \left (-2 x-\sqrt {45-10 e^{10}+e^{20}}+e^{10}-5\right )\right )}{\left (11-2 e^{10}\right ) \left (45-10 e^{10}+e^{20}\right )}+\frac {20 e^{\frac {1}{2} \left (25-e^{10}+\sqrt {45-10 e^{10}+e^{20}}\right )} \left (55+44 e^{10}-21 e^{20}+2 e^{30}\right ) \operatorname {ExpIntegralEi}\left (\frac {1}{2} \left (-2 x-\sqrt {45-10 e^{10}+e^{20}}+e^{10}-5\right )\right )}{\left (11-2 e^{10}\right ) \left (45-10 e^{10}+e^{20}\right )^{3/2}}-\frac {e^{\frac {1}{2} \left (45-e^{10}+\sqrt {45-10 e^{10}+e^{20}}\right )} \left (5-e^{10}\right ) \left (755-380 e^{10}+66 e^{20}-4 e^{30}\right ) \operatorname {ExpIntegralEi}\left (\frac {1}{2} \left (-2 x-\sqrt {45-10 e^{10}+e^{20}}+e^{10}-5\right )\right )}{\left (11-2 e^{10}\right ) \left (45-10 e^{10}+e^{20}\right )^{3/2}}+\frac {10 e^{\frac {1}{2} \left (45-e^{10}+\sqrt {45-10 e^{10}+e^{20}}\right )} \left (65-23 e^{10}+2 e^{20}\right ) \operatorname {ExpIntegralEi}\left (\frac {1}{2} \left (-2 x-\sqrt {45-10 e^{10}+e^{20}}+e^{10}-5\right )\right )}{\left (11-2 e^{10}\right ) \left (45-10 e^{10}+e^{20}\right )}-\frac {20 e^{\frac {1}{2} \left (45-e^{10}+\sqrt {45-10 e^{10}+e^{20}}\right )} \left (65-23 e^{10}+2 e^{20}\right ) \operatorname {ExpIntegralEi}\left (\frac {1}{2} \left (-2 x-\sqrt {45-10 e^{10}+e^{20}}+e^{10}-5\right )\right )}{\left (11-2 e^{10}\right ) \left (45-10 e^{10}+e^{20}\right )^{3/2}}+\frac {e^{\frac {1}{2} \left (25-e^{10}-\sqrt {45-10 e^{10}+e^{20}}\right )} \left (594-462 e^{10}+108 e^{20}-8 e^{30}-\frac {2915-4092 e^{10}+1680 e^{20}-276 e^{30}+16 e^{40}}{\sqrt {45-10 e^{10}+e^{20}}}\right ) \operatorname {ExpIntegralEi}\left (\frac {1}{2} \left (-2 x+\sqrt {45-10 e^{10}+e^{20}}+e^{10}-5\right )\right )}{\left (11-2 e^{10}\right )^2}+\frac {e^{\frac {1}{2} \left (45-e^{10}-\sqrt {45-10 e^{10}+e^{20}}\right )} \left (123-44 e^{10}+4 e^{20}-\frac {2055-1109 e^{10}+200 e^{20}-12 e^{30}}{\sqrt {45-10 e^{10}+e^{20}}}\right ) \operatorname {ExpIntegralEi}\left (\frac {1}{2} \left (-2 x+\sqrt {45-10 e^{10}+e^{20}}+e^{10}-5\right )\right )}{\left (11-2 e^{10}\right )^2}-\frac {e^{\frac {1}{2} \left (25-e^{10}-\sqrt {45-10 e^{10}+e^{20}}\right )} \left (770+395 e^{10}-318 e^{20}+62 e^{30}-4 e^{40}\right ) \left (5-e^{10}-\sqrt {45-10 e^{10}+e^{20}}\right ) \operatorname {ExpIntegralEi}\left (\frac {1}{2} \left (-2 x+\sqrt {45-10 e^{10}+e^{20}}+e^{10}-5\right )\right )}{2 \left (11-2 e^{10}\right ) \left (45-10 e^{10}+e^{20}\right )}+\frac {e^{\frac {1}{2} \left (45-e^{10}-\sqrt {45-10 e^{10}+e^{20}}\right )} \left (755-380 e^{10}+66 e^{20}-4 e^{30}\right ) \left (5-e^{10}-\sqrt {45-10 e^{10}+e^{20}}\right ) \operatorname {ExpIntegralEi}\left (\frac {1}{2} \left (-2 x+\sqrt {45-10 e^{10}+e^{20}}+e^{10}-5\right )\right )}{2 \left (11-2 e^{10}\right ) \left (45-10 e^{10}+e^{20}\right )}+\frac {e^{\frac {1}{2} \left (25-e^{10}-\sqrt {45-10 e^{10}+e^{20}}\right )} \left (1-\frac {10-e^{10}}{\sqrt {45-10 e^{10}+e^{20}}}\right ) \operatorname {ExpIntegralEi}\left (\frac {1}{2} \left (-2 x+\sqrt {45-10 e^{10}+e^{20}}+e^{10}-5\right )\right )}{11-2 e^{10}}-\frac {e^{\frac {1}{2} \left (25-e^{10}-\sqrt {45-10 e^{10}+e^{20}}\right )} \left (5-e^{10}\right ) \left (770+395 e^{10}-318 e^{20}+62 e^{30}-4 e^{40}\right ) \operatorname {ExpIntegralEi}\left (\frac {1}{2} \left (-2 x+\sqrt {45-10 e^{10}+e^{20}}+e^{10}-5\right )\right )}{\left (11-2 e^{10}\right ) \left (45-10 e^{10}+e^{20}\right )^{3/2}}-\frac {10 e^{\frac {1}{2} \left (25-e^{10}-\sqrt {45-10 e^{10}+e^{20}}\right )} \left (55+44 e^{10}-21 e^{20}+2 e^{30}\right ) \operatorname {ExpIntegralEi}\left (\frac {1}{2} \left (-2 x+\sqrt {45-10 e^{10}+e^{20}}+e^{10}-5\right )\right )}{\left (11-2 e^{10}\right ) \left (45-10 e^{10}+e^{20}\right )}-\frac {20 e^{\frac {1}{2} \left (25-e^{10}-\sqrt {45-10 e^{10}+e^{20}}\right )} \left (55+44 e^{10}-21 e^{20}+2 e^{30}\right ) \operatorname {ExpIntegralEi}\left (\frac {1}{2} \left (-2 x+\sqrt {45-10 e^{10}+e^{20}}+e^{10}-5\right )\right )}{\left (11-2 e^{10}\right ) \left (45-10 e^{10}+e^{20}\right )^{3/2}}+\frac {e^{\frac {1}{2} \left (45-e^{10}-\sqrt {45-10 e^{10}+e^{20}}\right )} \left (5-e^{10}\right ) \left (755-380 e^{10}+66 e^{20}-4 e^{30}\right ) \operatorname {ExpIntegralEi}\left (\frac {1}{2} \left (-2 x+\sqrt {45-10 e^{10}+e^{20}}+e^{10}-5\right )\right )}{\left (11-2 e^{10}\right ) \left (45-10 e^{10}+e^{20}\right )^{3/2}}+\frac {10 e^{\frac {1}{2} \left (45-e^{10}-\sqrt {45-10 e^{10}+e^{20}}\right )} \left (65-23 e^{10}+2 e^{20}\right ) \operatorname {ExpIntegralEi}\left (\frac {1}{2} \left (-2 x+\sqrt {45-10 e^{10}+e^{20}}+e^{10}-5\right )\right )}{\left (11-2 e^{10}\right ) \left (45-10 e^{10}+e^{20}\right )}+\frac {20 e^{\frac {1}{2} \left (45-e^{10}-\sqrt {45-10 e^{10}+e^{20}}\right )} \left (65-23 e^{10}+2 e^{20}\right ) \operatorname {ExpIntegralEi}\left (\frac {1}{2} \left (-2 x+\sqrt {45-10 e^{10}+e^{20}}+e^{10}-5\right )\right )}{\left (11-2 e^{10}\right ) \left (45-10 e^{10}+e^{20}\right )^{3/2}}-\frac {2 e^{12} \operatorname {ExpIntegralEi}(-x-2)}{11-2 e^{10}}-\frac {4 e^{22} \operatorname {ExpIntegralEi}(-x-2)}{\left (11-2 e^{10}\right )^2}+\frac {22 e^{12} \operatorname {ExpIntegralEi}(-x-2)}{\left (11-2 e^{10}\right )^2}+\frac {e^{\frac {1}{2} \left (25-e^{10}+\sqrt {45-10 e^{10}+e^{20}}\right )} \left (5-e^{10}\right ) \left (5-e^{10}+\sqrt {45-10 e^{10}+e^{20}}\right ) \operatorname {ExpIntegralEi}\left (\frac {1}{2} \left (-2 x-\sqrt {45-10 e^{10}+e^{20}}+e^{10}-5\right )\right ) \log (x+2)}{2 \left (45-10 e^{10}+e^{20}\right )}-\frac {1}{2} e^{\frac {1}{2} \left (25-e^{10}+\sqrt {45-10 e^{10}+e^{20}}\right )} \left (1+\frac {3-e^{10}}{\sqrt {45-10 e^{10}+e^{20}}}\right ) \operatorname {ExpIntegralEi}\left (\frac {1}{2} \left (-2 x-\sqrt {45-10 e^{10}+e^{20}}+e^{10}-5\right )\right ) \log (x+2)+\frac {10 e^{\frac {1}{2} \left (25-e^{10}+\sqrt {45-10 e^{10}+e^{20}}\right )} \operatorname {ExpIntegralEi}\left (\frac {1}{2} \left (-2 x-\sqrt {45-10 e^{10}+e^{20}}+e^{10}-5\right )\right ) \log (x+2)}{45-10 e^{10}+e^{20}}-\frac {e^{\frac {1}{2} \left (25-e^{10}+\sqrt {45-10 e^{10}+e^{20}}\right )} \left (5-e^{10}\right )^2 \operatorname {ExpIntegralEi}\left (\frac {1}{2} \left (-2 x-\sqrt {45-10 e^{10}+e^{20}}+e^{10}-5\right )\right ) \log (x+2)}{\left (45-10 e^{10}+e^{20}\right )^{3/2}}-\frac {20 e^{\frac {1}{2} \left (25-e^{10}+\sqrt {45-10 e^{10}+e^{20}}\right )} \operatorname {ExpIntegralEi}\left (\frac {1}{2} \left (-2 x-\sqrt {45-10 e^{10}+e^{20}}+e^{10}-5\right )\right ) \log (x+2)}{\left (45-10 e^{10}+e^{20}\right )^{3/2}}+\frac {e^{\frac {1}{2} \left (25-e^{10}-\sqrt {45-10 e^{10}+e^{20}}\right )} \left (5-e^{10}\right ) \left (5-e^{10}-\sqrt {45-10 e^{10}+e^{20}}\right ) \operatorname {ExpIntegralEi}\left (\frac {1}{2} \left (-2 x+\sqrt {45-10 e^{10}+e^{20}}+e^{10}-5\right )\right ) \log (x+2)}{2 \left (45-10 e^{10}+e^{20}\right )}-\frac {1}{2} e^{\frac {1}{2} \left (25-e^{10}-\sqrt {45-10 e^{10}+e^{20}}\right )} \left (1-\frac {3-e^{10}}{\sqrt {45-10 e^{10}+e^{20}}}\right ) \operatorname {ExpIntegralEi}\left (\frac {1}{2} \left (-2 x+\sqrt {45-10 e^{10}+e^{20}}+e^{10}-5\right )\right ) \log (x+2)+\frac {10 e^{\frac {1}{2} \left (25-e^{10}-\sqrt {45-10 e^{10}+e^{20}}\right )} \operatorname {ExpIntegralEi}\left (\frac {1}{2} \left (-2 x+\sqrt {45-10 e^{10}+e^{20}}+e^{10}-5\right )\right ) \log (x+2)}{45-10 e^{10}+e^{20}}+\frac {e^{\frac {1}{2} \left (25-e^{10}-\sqrt {45-10 e^{10}+e^{20}}\right )} \left (5-e^{10}\right )^2 \operatorname {ExpIntegralEi}\left (\frac {1}{2} \left (-2 x+\sqrt {45-10 e^{10}+e^{20}}+e^{10}-5\right )\right ) \log (x+2)}{\left (45-10 e^{10}+e^{20}\right )^{3/2}}+\frac {20 e^{\frac {1}{2} \left (25-e^{10}-\sqrt {45-10 e^{10}+e^{20}}\right )} \operatorname {ExpIntegralEi}\left (\frac {1}{2} \left (-2 x+\sqrt {45-10 e^{10}+e^{20}}+e^{10}-5\right )\right ) \log (x+2)}{\left (45-10 e^{10}+e^{20}\right )^{3/2}}+\frac {e^{10-x} \left (5-e^{10}\right ) \left (5-e^{10}-\sqrt {45-10 e^{10}+e^{20}}\right ) \log (x+2)}{\left (45-10 e^{10}+e^{20}\right ) \left (2 x-\sqrt {45-10 e^{10}+e^{20}}-e^{10}+5\right )}+\frac {20 e^{10-x} \log (x+2)}{\left (45-10 e^{10}+e^{20}\right ) \left (2 x-\sqrt {45-10 e^{10}+e^{20}}-e^{10}+5\right )}+\frac {e^{10-x} \left (5-e^{10}\right ) \left (5-e^{10}+\sqrt {45-10 e^{10}+e^{20}}\right ) \log (x+2)}{\left (45-10 e^{10}+e^{20}\right ) \left (2 x+\sqrt {45-10 e^{10}+e^{20}}-e^{10}+5\right )}+\frac {20 e^{10-x} \log (x+2)}{\left (45-10 e^{10}+e^{20}\right ) \left (2 x+\sqrt {45-10 e^{10}+e^{20}}-e^{10}+5\right )}-\frac {e^{10-x} \left (770+395 e^{10}-318 e^{20}+62 e^{30}-4 e^{40}\right ) \left (5-e^{10}-\sqrt {45-10 e^{10}+e^{20}}\right )}{\left (11-2 e^{10}\right ) \left (45-10 e^{10}+e^{20}\right ) \left (2 x-\sqrt {45-10 e^{10}+e^{20}}-e^{10}+5\right )}+\frac {e^{20-x} \left (755-380 e^{10}+66 e^{20}-4 e^{30}\right ) \left (5-e^{10}-\sqrt {45-10 e^{10}+e^{20}}\right )}{\left (11-2 e^{10}\right ) \left (45-10 e^{10}+e^{20}\right ) \left (2 x-\sqrt {45-10 e^{10}+e^{20}}-e^{10}+5\right )}-\frac {20 e^{10-x} \left (55+44 e^{10}-21 e^{20}+2 e^{30}\right )}{\left (11-2 e^{10}\right ) \left (45-10 e^{10}+e^{20}\right ) \left (2 x-\sqrt {45-10 e^{10}+e^{20}}-e^{10}+5\right )}+\frac {20 e^{20-x} \left (65-23 e^{10}+2 e^{20}\right )}{\left (11-2 e^{10}\right ) \left (45-10 e^{10}+e^{20}\right ) \left (2 x-\sqrt {45-10 e^{10}+e^{20}}-e^{10}+5\right )}-\frac {e^{10-x} \left (770+395 e^{10}-318 e^{20}+62 e^{30}-4 e^{40}\right ) \left (5-e^{10}+\sqrt {45-10 e^{10}+e^{20}}\right )}{\left (11-2 e^{10}\right ) \left (45-10 e^{10}+e^{20}\right ) \left (2 x+\sqrt {45-10 e^{10}+e^{20}}-e^{10}+5\right )}+\frac {e^{20-x} \left (755-380 e^{10}+66 e^{20}-4 e^{30}\right ) \left (5-e^{10}+\sqrt {45-10 e^{10}+e^{20}}\right )}{\left (11-2 e^{10}\right ) \left (45-10 e^{10}+e^{20}\right ) \left (2 x+\sqrt {45-10 e^{10}+e^{20}}-e^{10}+5\right )}-\frac {20 e^{10-x} \left (55+44 e^{10}-21 e^{20}+2 e^{30}\right )}{\left (11-2 e^{10}\right ) \left (45-10 e^{10}+e^{20}\right ) \left (2 x+\sqrt {45-10 e^{10}+e^{20}}-e^{10}+5\right )}+\frac {20 e^{20-x} \left (65-23 e^{10}+2 e^{20}\right )}{\left (11-2 e^{10}\right ) \left (45-10 e^{10}+e^{20}\right ) \left (2 x+\sqrt {45-10 e^{10}+e^{20}}-e^{10}+5\right )}\) |
Input:
Int[(50 - 75*x - 20*x^2 + 35*x^3 + 12*x^4 + x^5 + E^20*(2*x^2 + x^3) + E^( 10 - x)*(-45*x + 25*x^2 + x^3 - 14*x^4 - 2*x^5) + E^x*(E^(10 - x)*(20*x - 10*x^2 - 14*x^3 - 2*x^4) + E^(20 - 2*x)*(-5*x^2 + 2*x^3 + 2*x^4)) + (E^(20 - x)*(2*x^2 + x^3) + E^(10 - x)*(-10 + 5*x - 7*x^2 - 8*x^3 - x^4))*Log[2 + x])/(50 - 75*x - 20*x^2 + 35*x^3 + 12*x^4 + x^5 + E^20*(2*x^2 + x^3) + E ^10*(20*x - 10*x^2 - 14*x^3 - 2*x^4)),x]
Output:
2*E^(10 - x) + x + (20*E^(20 - x)*(65 - 23*E^10 + 2*E^20))/((11 - 2*E^10)* (45 - 10*E^10 + E^20)*(5 - E^10 - Sqrt[45 - 10*E^10 + E^20] + 2*x)) - (20* E^(10 - x)*(55 + 44*E^10 - 21*E^20 + 2*E^30))/((11 - 2*E^10)*(45 - 10*E^10 + E^20)*(5 - E^10 - Sqrt[45 - 10*E^10 + E^20] + 2*x)) + (E^(20 - x)*(755 - 380*E^10 + 66*E^20 - 4*E^30)*(5 - E^10 - Sqrt[45 - 10*E^10 + E^20]))/((1 1 - 2*E^10)*(45 - 10*E^10 + E^20)*(5 - E^10 - Sqrt[45 - 10*E^10 + E^20] + 2*x)) - (E^(10 - x)*(770 + 395*E^10 - 318*E^20 + 62*E^30 - 4*E^40)*(5 - E^ 10 - Sqrt[45 - 10*E^10 + E^20]))/((11 - 2*E^10)*(45 - 10*E^10 + E^20)*(5 - E^10 - Sqrt[45 - 10*E^10 + E^20] + 2*x)) + (20*E^(20 - x)*(65 - 23*E^10 + 2*E^20))/((11 - 2*E^10)*(45 - 10*E^10 + E^20)*(5 - E^10 + Sqrt[45 - 10*E^ 10 + E^20] + 2*x)) - (20*E^(10 - x)*(55 + 44*E^10 - 21*E^20 + 2*E^30))/((1 1 - 2*E^10)*(45 - 10*E^10 + E^20)*(5 - E^10 + Sqrt[45 - 10*E^10 + E^20] + 2*x)) + (E^(20 - x)*(755 - 380*E^10 + 66*E^20 - 4*E^30)*(5 - E^10 + Sqrt[4 5 - 10*E^10 + E^20]))/((11 - 2*E^10)*(45 - 10*E^10 + E^20)*(5 - E^10 + Sqr t[45 - 10*E^10 + E^20] + 2*x)) - (E^(10 - x)*(770 + 395*E^10 - 318*E^20 + 62*E^30 - 4*E^40)*(5 - E^10 + Sqrt[45 - 10*E^10 + E^20]))/((11 - 2*E^10)*( 45 - 10*E^10 + E^20)*(5 - E^10 + Sqrt[45 - 10*E^10 + E^20] + 2*x)) - (20*E ^((45 - E^10 + Sqrt[45 - 10*E^10 + E^20])/2)*(65 - 23*E^10 + 2*E^20)*ExpIn tegralEi[(-5 + E^10 - Sqrt[45 - 10*E^10 + E^20] - 2*x)/2])/((11 - 2*E^10)* (45 - 10*E^10 + E^20)^(3/2)) + (10*E^((45 - E^10 + Sqrt[45 - 10*E^10 + ...
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]
Int[u_, x_Symbol] :> With[{v = SimplifyIntegrand[u, x]}, Int[v, x] /; Simpl erIntegrandQ[v, u, x]]
Leaf count of result is larger than twice the leaf count of optimal. \(157\) vs. \(2(36)=72\).
Time = 40.94 (sec) , antiderivative size = 158, normalized size of antiderivative = 4.65
| method | result | size |
| parallelrisch | \(\frac {30780-34200 x +400 \,{\mathrm e}^{20-2 x} {\mathrm e}^{2 x}+1900 \,{\mathrm e}^{10-x} {\mathrm e}^{x}+1368 x^{2} {\mathrm e}^{10-x}-2736 x^{2}+684 x^{3}-20 x \,{\mathrm e}^{20-2 x} {\mathrm e}^{2 x}+684 \,{\mathrm e}^{10-x} \ln \left (2+x \right ) x -80 \,{\mathrm e}^{20-2 x} {\mathrm e}^{2 x} x^{2}+4256 \,{\mathrm e}^{10-x} x \,{\mathrm e}^{x}-1064 \,{\mathrm e}^{10-x} {\mathrm e}^{x} x^{2}+80 \,{\mathrm e}^{30-3 x} x \,{\mathrm e}^{3 x}}{-684 \,{\mathrm e}^{10-x} x \,{\mathrm e}^{x}+684 x^{2}+3420 x -3420}\) | \(158\) |
Input:
int((((x^3+2*x^2)*exp(10-x)^2*exp(x)+(-x^4-8*x^3-7*x^2+5*x-10)*exp(10-x))* ln(2+x)+(x^3+2*x^2)*exp(10-x)^2*exp(x)^2+((2*x^4+2*x^3-5*x^2)*exp(10-x)^2+ (-2*x^4-14*x^3-10*x^2+20*x)*exp(10-x))*exp(x)+(-2*x^5-14*x^4+x^3+25*x^2-45 *x)*exp(10-x)+x^5+12*x^4+35*x^3-20*x^2-75*x+50)/((x^3+2*x^2)*exp(10-x)^2*e xp(x)^2+(-2*x^4-14*x^3-10*x^2+20*x)*exp(10-x)*exp(x)+x^5+12*x^4+35*x^3-20* x^2-75*x+50),x,method=_RETURNVERBOSE)
Output:
1/684*(30780-34200*x+400*exp(10-x)^2*exp(x)^2+1900*exp(10-x)*exp(x)+1368*x ^2*exp(10-x)-2736*x^2+684*x^3-20*x*exp(10-x)^2*exp(x)^2+684*exp(10-x)*ln(2 +x)*x-80*exp(10-x)^2*exp(x)^2*x^2+4256*exp(10-x)*x*exp(x)-1064*exp(10-x)*e xp(x)*x^2+80*exp(10-x)^3*x*exp(x)^3)/(-exp(10-x)*x*exp(x)+x^2+5*x-5)
Time = 0.10 (sec) , antiderivative size = 58, normalized size of antiderivative = 1.71 \[ \int \frac {50-75 x-20 x^2+35 x^3+12 x^4+x^5+e^{20} \left (2 x^2+x^3\right )+e^{10-x} \left (-45 x+25 x^2+x^3-14 x^4-2 x^5\right )+e^x \left (e^{10-x} \left (20 x-10 x^2-14 x^3-2 x^4\right )+e^{20-2 x} \left (-5 x^2+2 x^3+2 x^4\right )\right )+\left (e^{20-x} \left (2 x^2+x^3\right )+e^{10-x} \left (-10+5 x-7 x^2-8 x^3-x^4\right )\right ) \log (2+x)}{50-75 x-20 x^2+35 x^3+12 x^4+x^5+e^{20} \left (2 x^2+x^3\right )+e^{10} \left (20 x-10 x^2-14 x^3-2 x^4\right )} \, dx=\frac {{\left (2 \, x^{2} e^{10} + x e^{10} \log \left (x + 2\right ) + {\left (x^{3} - x^{2} e^{10} + 5 \, x^{2} - 5 \, x\right )} e^{x}\right )} e^{\left (-x\right )}}{x^{2} - x e^{10} + 5 \, x - 5} \] Input:
integrate((((x^3+2*x^2)*exp(10-x)^2*exp(x)+(-x^4-8*x^3-7*x^2+5*x-10)*exp(1 0-x))*log(2+x)+(x^3+2*x^2)*exp(10-x)^2*exp(x)^2+((2*x^4+2*x^3-5*x^2)*exp(1 0-x)^2+(-2*x^4-14*x^3-10*x^2+20*x)*exp(10-x))*exp(x)+(-2*x^5-14*x^4+x^3+25 *x^2-45*x)*exp(10-x)+x^5+12*x^4+35*x^3-20*x^2-75*x+50)/((x^3+2*x^2)*exp(10 -x)^2*exp(x)^2+(-2*x^4-14*x^3-10*x^2+20*x)*exp(10-x)*exp(x)+x^5+12*x^4+35* x^3-20*x^2-75*x+50),x, algorithm="fricas")
Output:
(2*x^2*e^10 + x*e^10*log(x + 2) + (x^3 - x^2*e^10 + 5*x^2 - 5*x)*e^x)*e^(- x)/(x^2 - x*e^10 + 5*x - 5)
Time = 0.31 (sec) , antiderivative size = 36, normalized size of antiderivative = 1.06 \[ \int \frac {50-75 x-20 x^2+35 x^3+12 x^4+x^5+e^{20} \left (2 x^2+x^3\right )+e^{10-x} \left (-45 x+25 x^2+x^3-14 x^4-2 x^5\right )+e^x \left (e^{10-x} \left (20 x-10 x^2-14 x^3-2 x^4\right )+e^{20-2 x} \left (-5 x^2+2 x^3+2 x^4\right )\right )+\left (e^{20-x} \left (2 x^2+x^3\right )+e^{10-x} \left (-10+5 x-7 x^2-8 x^3-x^4\right )\right ) \log (2+x)}{50-75 x-20 x^2+35 x^3+12 x^4+x^5+e^{20} \left (2 x^2+x^3\right )+e^{10} \left (20 x-10 x^2-14 x^3-2 x^4\right )} \, dx=x + \frac {\left (2 x^{2} e^{10} + x e^{10} \log {\left (x + 2 \right )}\right ) e^{- x}}{x^{2} - x e^{10} + 5 x - 5} \] Input:
integrate((((x**3+2*x**2)*exp(10-x)**2*exp(x)+(-x**4-8*x**3-7*x**2+5*x-10) *exp(10-x))*ln(2+x)+(x**3+2*x**2)*exp(10-x)**2*exp(x)**2+((2*x**4+2*x**3-5 *x**2)*exp(10-x)**2+(-2*x**4-14*x**3-10*x**2+20*x)*exp(10-x))*exp(x)+(-2*x **5-14*x**4+x**3+25*x**2-45*x)*exp(10-x)+x**5+12*x**4+35*x**3-20*x**2-75*x +50)/((x**3+2*x**2)*exp(10-x)**2*exp(x)**2+(-2*x**4-14*x**3-10*x**2+20*x)* exp(10-x)*exp(x)+x**5+12*x**4+35*x**3-20*x**2-75*x+50),x)
Output:
x + (2*x**2*exp(10) + x*exp(10)*log(x + 2))*exp(-x)/(x**2 - x*exp(10) + 5* x - 5)
Leaf count of result is larger than twice the leaf count of optimal. 2515 vs. \(2 (31) = 62\).
Time = 0.20 (sec) , antiderivative size = 2515, normalized size of antiderivative = 73.97 \[ \int \frac {50-75 x-20 x^2+35 x^3+12 x^4+x^5+e^{20} \left (2 x^2+x^3\right )+e^{10-x} \left (-45 x+25 x^2+x^3-14 x^4-2 x^5\right )+e^x \left (e^{10-x} \left (20 x-10 x^2-14 x^3-2 x^4\right )+e^{20-2 x} \left (-5 x^2+2 x^3+2 x^4\right )\right )+\left (e^{20-x} \left (2 x^2+x^3\right )+e^{10-x} \left (-10+5 x-7 x^2-8 x^3-x^4\right )\right ) \log (2+x)}{50-75 x-20 x^2+35 x^3+12 x^4+x^5+e^{20} \left (2 x^2+x^3\right )+e^{10} \left (20 x-10 x^2-14 x^3-2 x^4\right )} \, dx=\text {Too large to display} \] Input:
integrate((((x^3+2*x^2)*exp(10-x)^2*exp(x)+(-x^4-8*x^3-7*x^2+5*x-10)*exp(1 0-x))*log(2+x)+(x^3+2*x^2)*exp(10-x)^2*exp(x)^2+((2*x^4+2*x^3-5*x^2)*exp(1 0-x)^2+(-2*x^4-14*x^3-10*x^2+20*x)*exp(10-x))*exp(x)+(-2*x^5-14*x^4+x^3+25 *x^2-45*x)*exp(10-x)+x^5+12*x^4+35*x^3-20*x^2-75*x+50)/((x^3+2*x^2)*exp(10 -x)^2*exp(x)^2+(-2*x^4-14*x^3-10*x^2+20*x)*exp(10-x)*exp(x)+x^5+12*x^4+35* x^3-20*x^2-75*x+50),x, algorithm="maxima")
Output:
(2*(2*e^30 - 30*e^20 + 135*e^10 - 350)*log((2*x - sqrt(e^20 - 10*e^10 + 45 ) - e^10 + 5)/(2*x + sqrt(e^20 - 10*e^10 + 45) - e^10 + 5))/((4*e^40 - 84* e^30 + 741*e^20 - 3190*e^10 + 5445)*sqrt(e^20 - 10*e^10 + 45)) - (x*(2*e^3 0 - 35*e^20 + 230*e^10 - 575) + 10*e^20 - 125*e^10 + 475)/(x^2*(2*e^30 - 3 1*e^20 + 200*e^10 - 495) - x*(2*e^40 - 41*e^30 + 355*e^20 - 1495*e^10 + 24 75) - 10*e^30 + 155*e^20 - 1000*e^10 + 2475) + 4*log(x^2 - x*(e^10 - 5) - 5)/(4*e^20 - 44*e^10 + 121) - 8*log(x + 2)/(4*e^20 - 44*e^10 + 121))*e^20 - 2*((2*e^30 - 50*e^20 + 355*e^10 - 955)*log((2*x - sqrt(e^20 - 10*e^10 + 45) - e^10 + 5)/(2*x + sqrt(e^20 - 10*e^10 + 45) - e^10 + 5))/((4*e^40 - 8 4*e^30 + 741*e^20 - 3190*e^10 + 5445)*sqrt(e^20 - 10*e^10 + 45)) + (x*(2*e ^20 - 25*e^10 + 95) + 10*e^10 - 100)/(x^2*(2*e^30 - 31*e^20 + 200*e^10 - 4 95) - x*(2*e^40 - 41*e^30 + 355*e^20 - 1495*e^10 + 2475) - 10*e^30 + 155*e ^20 - 1000*e^10 + 2475) + 2*log(x^2 - x*(e^10 - 5) - 5)/(4*e^20 - 44*e^10 + 121) - 4*log(x + 2)/(4*e^20 - 44*e^10 + 121))*e^20 - ((4*e^20 - 44*e^10 + 105)*log(x^2 - x*(e^10 - 5) - 5)/(4*e^20 - 44*e^10 + 121) + (4*e^50 - 10 4*e^40 + 1185*e^30 - 7295*e^20 + 23725*e^10 - 30475)*log((2*x - sqrt(e^20 - 10*e^10 + 45) - e^10 + 5)/(2*x + sqrt(e^20 - 10*e^10 + 45) - e^10 + 5))/ ((4*e^40 - 84*e^30 + 741*e^20 - 3190*e^10 + 5445)*sqrt(e^20 - 10*e^10 + 45 )) - 2*(x*(2*e^40 - 45*e^30 + 415*e^20 - 1850*e^10 + 3350) + 10*e^30 - 175 *e^20 + 1150*e^10 - 2875)/(x^2*(2*e^30 - 31*e^20 + 200*e^10 - 495) - x*...
Leaf count of result is larger than twice the leaf count of optimal. 115 vs. \(2 (31) = 62\).
Time = 0.18 (sec) , antiderivative size = 115, normalized size of antiderivative = 3.38 \[ \int \frac {50-75 x-20 x^2+35 x^3+12 x^4+x^5+e^{20} \left (2 x^2+x^3\right )+e^{10-x} \left (-45 x+25 x^2+x^3-14 x^4-2 x^5\right )+e^x \left (e^{10-x} \left (20 x-10 x^2-14 x^3-2 x^4\right )+e^{20-2 x} \left (-5 x^2+2 x^3+2 x^4\right )\right )+\left (e^{20-x} \left (2 x^2+x^3\right )+e^{10-x} \left (-10+5 x-7 x^2-8 x^3-x^4\right )\right ) \log (2+x)}{50-75 x-20 x^2+35 x^3+12 x^4+x^5+e^{20} \left (2 x^2+x^3\right )+e^{10} \left (20 x-10 x^2-14 x^3-2 x^4\right )} \, dx=\frac {{\left (x - 10\right )}^{3} - {\left (x - 10\right )}^{2} e^{10} + 2 \, {\left (x - 10\right )}^{2} e^{\left (-x + 10\right )} + {\left (x - 10\right )} e^{\left (-x + 10\right )} \log \left (x + 2\right ) + 25 \, {\left (x - 10\right )}^{2} - 10 \, {\left (x - 10\right )} e^{10} + 40 \, {\left (x - 10\right )} e^{\left (-x + 10\right )} + 10 \, e^{\left (-x + 10\right )} \log \left (x + 2\right ) + 145 \, x + 200 \, e^{\left (-x + 10\right )} - 1450}{{\left (x - 10\right )}^{2} - {\left (x - 10\right )} e^{10} + 25 \, x - 10 \, e^{10} - 105} \] Input:
integrate((((x^3+2*x^2)*exp(10-x)^2*exp(x)+(-x^4-8*x^3-7*x^2+5*x-10)*exp(1 0-x))*log(2+x)+(x^3+2*x^2)*exp(10-x)^2*exp(x)^2+((2*x^4+2*x^3-5*x^2)*exp(1 0-x)^2+(-2*x^4-14*x^3-10*x^2+20*x)*exp(10-x))*exp(x)+(-2*x^5-14*x^4+x^3+25 *x^2-45*x)*exp(10-x)+x^5+12*x^4+35*x^3-20*x^2-75*x+50)/((x^3+2*x^2)*exp(10 -x)^2*exp(x)^2+(-2*x^4-14*x^3-10*x^2+20*x)*exp(10-x)*exp(x)+x^5+12*x^4+35* x^3-20*x^2-75*x+50),x, algorithm="giac")
Output:
((x - 10)^3 - (x - 10)^2*e^10 + 2*(x - 10)^2*e^(-x + 10) + (x - 10)*e^(-x + 10)*log(x + 2) + 25*(x - 10)^2 - 10*(x - 10)*e^10 + 40*(x - 10)*e^(-x + 10) + 10*e^(-x + 10)*log(x + 2) + 145*x + 200*e^(-x + 10) - 1450)/((x - 10 )^2 - (x - 10)*e^10 + 25*x - 10*e^10 - 105)
Timed out. \[ \int \frac {50-75 x-20 x^2+35 x^3+12 x^4+x^5+e^{20} \left (2 x^2+x^3\right )+e^{10-x} \left (-45 x+25 x^2+x^3-14 x^4-2 x^5\right )+e^x \left (e^{10-x} \left (20 x-10 x^2-14 x^3-2 x^4\right )+e^{20-2 x} \left (-5 x^2+2 x^3+2 x^4\right )\right )+\left (e^{20-x} \left (2 x^2+x^3\right )+e^{10-x} \left (-10+5 x-7 x^2-8 x^3-x^4\right )\right ) \log (2+x)}{50-75 x-20 x^2+35 x^3+12 x^4+x^5+e^{20} \left (2 x^2+x^3\right )+e^{10} \left (20 x-10 x^2-14 x^3-2 x^4\right )} \, dx=-\int -\frac {{\mathrm {e}}^{20}\,\left (x^3+2\,x^2\right )-{\mathrm {e}}^{10-x}\,\left (2\,x^5+14\,x^4-x^3-25\,x^2+45\,x\right )-75\,x-{\mathrm {e}}^x\,\left ({\mathrm {e}}^{10-x}\,\left (2\,x^4+14\,x^3+10\,x^2-20\,x\right )-{\mathrm {e}}^{20-2\,x}\,\left (2\,x^4+2\,x^3-5\,x^2\right )\right )-20\,x^2+35\,x^3+12\,x^4+x^5+\ln \left (x+2\right )\,\left ({\mathrm {e}}^{20-x}\,\left (x^3+2\,x^2\right )-{\mathrm {e}}^{10-x}\,\left (x^4+8\,x^3+7\,x^2-5\,x+10\right )\right )+50}{{\mathrm {e}}^{20}\,\left (x^3+2\,x^2\right )-75\,x-{\mathrm {e}}^{10}\,\left (2\,x^4+14\,x^3+10\,x^2-20\,x\right )-20\,x^2+35\,x^3+12\,x^4+x^5+50} \,d x \] Input:
int(-(75*x + exp(10 - x)*(45*x - 25*x^2 - x^3 + 14*x^4 + 2*x^5) + exp(x)*( exp(10 - x)*(10*x^2 - 20*x + 14*x^3 + 2*x^4) - exp(20 - 2*x)*(2*x^3 - 5*x^ 2 + 2*x^4)) + log(x + 2)*(exp(10 - x)*(7*x^2 - 5*x + 8*x^3 + x^4 + 10) - e xp(20 - 2*x)*exp(x)*(2*x^2 + x^3)) + 20*x^2 - 35*x^3 - 12*x^4 - x^5 - exp( 2*x)*exp(20 - 2*x)*(2*x^2 + x^3) - 50)/(35*x^3 - 20*x^2 - 75*x + 12*x^4 + x^5 + exp(2*x)*exp(20 - 2*x)*(2*x^2 + x^3) - exp(10 - x)*exp(x)*(10*x^2 - 20*x + 14*x^3 + 2*x^4) + 50),x)
Output:
-int(-(exp(20)*(2*x^2 + x^3) - exp(10 - x)*(45*x - 25*x^2 - x^3 + 14*x^4 + 2*x^5) - 75*x - exp(x)*(exp(10 - x)*(10*x^2 - 20*x + 14*x^3 + 2*x^4) - ex p(20 - 2*x)*(2*x^3 - 5*x^2 + 2*x^4)) - 20*x^2 + 35*x^3 + 12*x^4 + x^5 + lo g(x + 2)*(exp(20 - x)*(2*x^2 + x^3) - exp(10 - x)*(7*x^2 - 5*x + 8*x^3 + x ^4 + 10)) + 50)/(exp(20)*(2*x^2 + x^3) - 75*x - exp(10)*(10*x^2 - 20*x + 1 4*x^3 + 2*x^4) - 20*x^2 + 35*x^3 + 12*x^4 + x^5 + 50), x)
Time = 0.19 (sec) , antiderivative size = 67, normalized size of antiderivative = 1.97 \[ \int \frac {50-75 x-20 x^2+35 x^3+12 x^4+x^5+e^{20} \left (2 x^2+x^3\right )+e^{10-x} \left (-45 x+25 x^2+x^3-14 x^4-2 x^5\right )+e^x \left (e^{10-x} \left (20 x-10 x^2-14 x^3-2 x^4\right )+e^{20-2 x} \left (-5 x^2+2 x^3+2 x^4\right )\right )+\left (e^{20-x} \left (2 x^2+x^3\right )+e^{10-x} \left (-10+5 x-7 x^2-8 x^3-x^4\right )\right ) \log (2+x)}{50-75 x-20 x^2+35 x^3+12 x^4+x^5+e^{20} \left (2 x^2+x^3\right )+e^{10} \left (20 x-10 x^2-14 x^3-2 x^4\right )} \, dx=\frac {x \left (e^{x} e^{10} x -e^{x} x^{2}-5 e^{x} x +5 e^{x}-\mathrm {log}\left (x +2\right ) e^{10}-2 e^{10} x \right )}{e^{x} \left (e^{10} x -x^{2}-5 x +5\right )} \] Input:
int((((x^3+2*x^2)*exp(10-x)^2*exp(x)+(-x^4-8*x^3-7*x^2+5*x-10)*exp(10-x))* log(2+x)+(x^3+2*x^2)*exp(10-x)^2*exp(x)^2+((2*x^4+2*x^3-5*x^2)*exp(10-x)^2 +(-2*x^4-14*x^3-10*x^2+20*x)*exp(10-x))*exp(x)+(-2*x^5-14*x^4+x^3+25*x^2-4 5*x)*exp(10-x)+x^5+12*x^4+35*x^3-20*x^2-75*x+50)/((x^3+2*x^2)*exp(10-x)^2* exp(x)^2+(-2*x^4-14*x^3-10*x^2+20*x)*exp(10-x)*exp(x)+x^5+12*x^4+35*x^3-20 *x^2-75*x+50),x)
Output:
(x*(e**x*e**10*x - e**x*x**2 - 5*e**x*x + 5*e**x - log(x + 2)*e**10 - 2*e* *10*x))/(e**x*(e**10*x - x**2 - 5*x + 5))