3.78.80 \(\int \frac {e^{-x} (e^x (2 x^2-8 x^3+4 x^4+8 x^5+2 x^6+e^6 (2-4 x^2+2 x^4)+e^3 (-4 x+8 x^2+8 x^3-8 x^4-4 x^5))+(x^2-x^3+3 x^4+x^5+e^6 (1-x+x^2+x^3)+e^3 (-2 x-4 x^3-2 x^4)) \log (4))}{x^2-4 x^3+2 x^4+4 x^5+x^6+e^6 (1-2 x^2+x^4)+e^3 (-2 x+4 x^2+4 x^3-4 x^4-2 x^5)} \, dx\) [7780]

Optimal. Leaf size=33 \[ 2 x+\frac {e^{-x} \log (4)}{\frac {1}{x}-x+\frac {2 x}{e^3-x}} \]

[Out]

2*x+2*ln(2)/(2*x/(-x+exp(3))+1/x-x)/exp(x)

________________________________________________________________________________________

Rubi [F]
time = 6.37, antiderivative size = 0, normalized size of antiderivative = 0.00, number of steps used = 0, number of rules used = 0, integrand size = 0, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.000, Rules used = {} \begin {gather*} \int \frac {e^{-x} \left (e^x \left (2 x^2-8 x^3+4 x^4+8 x^5+2 x^6+e^6 \left (2-4 x^2+2 x^4\right )+e^3 \left (-4 x+8 x^2+8 x^3-8 x^4-4 x^5\right )\right )+\left (x^2-x^3+3 x^4+x^5+e^6 \left (1-x+x^2+x^3\right )+e^3 \left (-2 x-4 x^3-2 x^4\right )\right ) \log (4)\right )}{x^2-4 x^3+2 x^4+4 x^5+x^6+e^6 \left (1-2 x^2+x^4\right )+e^3 \left (-2 x+4 x^2+4 x^3-4 x^4-2 x^5\right )} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Int[(E^x*(2*x^2 - 8*x^3 + 4*x^4 + 8*x^5 + 2*x^6 + E^6*(2 - 4*x^2 + 2*x^4) + E^3*(-4*x + 8*x^2 + 8*x^3 - 8*x^4
- 4*x^5)) + (x^2 - x^3 + 3*x^4 + x^5 + E^6*(1 - x + x^2 + x^3) + E^3*(-2*x - 4*x^3 - 2*x^4))*Log[4])/(E^x*(x^2
- 4*x^3 + 2*x^4 + 4*x^5 + x^6 + E^6*(1 - 2*x^2 + x^4) + E^3*(-2*x + 4*x^2 + 4*x^3 - 4*x^4 - 2*x^5))),x]

[Out]

2*x + 2*Log[4]*Defer[Int][E^(6 - x)/(-E^3 + x - (2 - E^3)*x^2 - x^3), x] + (2 + E^3 - E^6)*Log[4]*Defer[Int][1
/(E^x*(-E^3 + x - (2 - E^3)*x^2 - x^3)), x] + 2*Log[4]*Defer[Int][(E^(3 - x)*x)/(-E^3 + x - (2 - E^3)*x^2 - x^
3), x] + (2 + E^3 - E^6)*Log[4]*Defer[Int][E^(3 - x)/(E^3 - x + (2 - E^3)*x^2 + x^3)^2, x] + (1 - E^3)*Log[4]*
Defer[Int][E^(6 - x)/(E^3 - x + (2 - E^3)*x^2 + x^3)^2, x] + 2*Log[4]*Defer[Int][E^(9 - x)/(E^3 - x + (2 - E^3
)*x^2 + x^3)^2, x] - 2*Log[4]*Defer[Int][(E^(3 - x)*x)/(E^3 - x + (2 - E^3)*x^2 + x^3)^2, x] - 2*(1 + E^3)*Log
[4]*Defer[Int][x/(E^x*(E^3 - x + (2 - E^3)*x^2 + x^3)^2), x] - 2*(1 - E^3)^2*Log[4]*Defer[Int][(E^(3 - x)*x^2)
/(E^3 - x + (2 - E^3)*x^2 + x^3)^2, x] - (1 - E^3)*Log[4]*Defer[Int][(E^(6 - x)*x^2)/(E^3 - x + (2 - E^3)*x^2
+ x^3)^2, x] + (6 - 3*E^6 + E^9)*Log[4]*Defer[Int][x^2/(E^x*(E^3 - x + (2 - E^3)*x^2 + x^3)^2), x] + Log[4]*De
fer[Int][E^(6 - x)/(E^3 - x + (2 - E^3)*x^2 + x^3), x] + (1 + E^3)*Log[4]*Defer[Int][x/(E^x*(E^3 - x + (2 - E^
3)*x^2 + x^3)), x] + Log[4]*Defer[Int][x^2/(E^x*(E^3 - x + (2 - E^3)*x^2 + x^3)), x]

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\int \frac {e^{-x} \left (2 e^{6+x} \left (-1+x^2\right )^2+2 e^x x^2 \left (-1+2 x+x^2\right )^2-4 e^{3+x} x \left (1-2 x-2 x^2+2 x^3+x^4\right )+e^6 \left (1-x+x^2+x^3\right ) \log (4)-2 e^3 x \left (1+2 x^2+x^3\right ) \log (4)+x^2 \left (1-x+3 x^2+x^3\right ) \log (4)\right )}{\left (e^3-x+\left (2-e^3\right ) x^2+x^3\right )^2} \, dx\\ &=\int \left (2+\frac {2 e^{3-x} x \left (-1-2 x^2-x^3\right ) \log (4)}{\left (e^3-x+\left (2-e^3\right ) x^2+x^3\right )^2}+\frac {e^{6-x} \left (1-x+x^2+x^3\right ) \log (4)}{\left (e^3-x+\left (2-e^3\right ) x^2+x^3\right )^2}+\frac {e^{-x} x^2 \left (1-x+3 x^2+x^3\right ) \log (4)}{\left (e^3-x+\left (2-e^3\right ) x^2+x^3\right )^2}\right ) \, dx\\ &=2 x+\log (4) \int \frac {e^{6-x} \left (1-x+x^2+x^3\right )}{\left (e^3-x+\left (2-e^3\right ) x^2+x^3\right )^2} \, dx+\log (4) \int \frac {e^{-x} x^2 \left (1-x+3 x^2+x^3\right )}{\left (e^3-x+\left (2-e^3\right ) x^2+x^3\right )^2} \, dx+(2 \log (4)) \int \frac {e^{3-x} x \left (-1-2 x^2-x^3\right )}{\left (e^3-x+\left (2-e^3\right ) x^2+x^3\right )^2} \, dx\\ &=2 x+\log (4) \int \left (\frac {e^{6-x} \left (1-e^3\right ) \left (1-x^2\right )}{\left (e^3-x+\left (2-e^3\right ) x^2+x^3\right )^2}+\frac {e^{6-x}}{e^3-x+\left (2-e^3\right ) x^2+x^3}\right ) \, dx+\log (4) \int \left (\frac {e^{-x} \left (e^3 \left (2+e^3-e^6\right )-2 \left (1+e^3\right ) x+\left (6-3 e^6+e^9\right ) x^2\right )}{\left (e^3-x+\left (2-e^3\right ) x^2+x^3\right )^2}+\frac {e^{-x} \left (-2-e^3+e^6+\left (1+e^3\right ) x+x^2\right )}{e^3-x+\left (2-e^3\right ) x^2+x^3}\right ) \, dx+(2 \log (4)) \int \left (\frac {e^{3-x} \left (e^6-x-\left (1-e^3\right )^2 x^2\right )}{\left (e^3-x+\left (2-e^3\right ) x^2+x^3\right )^2}+\frac {e^{3-x} \left (-e^3-x\right )}{e^3-x+\left (2-e^3\right ) x^2+x^3}\right ) \, dx\\ &=2 x+\log (4) \int \frac {e^{-x} \left (e^3 \left (2+e^3-e^6\right )-2 \left (1+e^3\right ) x+\left (6-3 e^6+e^9\right ) x^2\right )}{\left (e^3-x+\left (2-e^3\right ) x^2+x^3\right )^2} \, dx+\log (4) \int \frac {e^{6-x}}{e^3-x+\left (2-e^3\right ) x^2+x^3} \, dx+\log (4) \int \frac {e^{-x} \left (-2-e^3+e^6+\left (1+e^3\right ) x+x^2\right )}{e^3-x+\left (2-e^3\right ) x^2+x^3} \, dx+(2 \log (4)) \int \frac {e^{3-x} \left (e^6-x-\left (1-e^3\right )^2 x^2\right )}{\left (e^3-x+\left (2-e^3\right ) x^2+x^3\right )^2} \, dx+(2 \log (4)) \int \frac {e^{3-x} \left (-e^3-x\right )}{e^3-x+\left (2-e^3\right ) x^2+x^3} \, dx+\left (\left (1-e^3\right ) \log (4)\right ) \int \frac {e^{6-x} \left (1-x^2\right )}{\left (e^3-x+\left (2-e^3\right ) x^2+x^3\right )^2} \, dx\\ &=2 x+\log (4) \int \frac {e^{6-x}}{e^3-x+\left (2-e^3\right ) x^2+x^3} \, dx+\log (4) \int \left (\frac {e^{3-x} \left (2+e^3-e^6\right )}{\left (e^3-x+\left (2-e^3\right ) x^2+x^3\right )^2}+\frac {2 e^{-x} \left (-1-e^3\right ) x}{\left (e^3-x+\left (2-e^3\right ) x^2+x^3\right )^2}+\frac {e^{-x} \left (6-3 e^6+e^9\right ) x^2}{\left (e^3-x+\left (2-e^3\right ) x^2+x^3\right )^2}\right ) \, dx+\log (4) \int \left (\frac {2 e^{-x} \left (1-\frac {1}{2} e^3 \left (-1+e^3\right )\right )}{-e^3+x-\left (2-e^3\right ) x^2-x^3}+\frac {e^{-x} \left (1+e^3\right ) x}{e^3-x+\left (2-e^3\right ) x^2+x^3}+\frac {e^{-x} x^2}{e^3-x+\left (2-e^3\right ) x^2+x^3}\right ) \, dx+(2 \log (4)) \int \left (\frac {e^{6-x}}{-e^3+x-\left (2-e^3\right ) x^2-x^3}+\frac {e^{3-x} x}{-e^3+x-\left (2-e^3\right ) x^2-x^3}\right ) \, dx+(2 \log (4)) \int \left (\frac {e^{9-x}}{\left (e^3-x+\left (2-e^3\right ) x^2+x^3\right )^2}-\frac {e^{3-x} x}{\left (e^3-x+\left (2-e^3\right ) x^2+x^3\right )^2}-\frac {e^{3-x} \left (1-e^3\right )^2 x^2}{\left (e^3-x+\left (2-e^3\right ) x^2+x^3\right )^2}\right ) \, dx+\left (\left (1-e^3\right ) \log (4)\right ) \int \left (\frac {e^{6-x}}{\left (e^3-x+\left (2-e^3\right ) x^2+x^3\right )^2}-\frac {e^{6-x} x^2}{\left (e^3-x+\left (2-e^3\right ) x^2+x^3\right )^2}\right ) \, dx\\ &=2 x+\log (4) \int \frac {e^{6-x}}{e^3-x+\left (2-e^3\right ) x^2+x^3} \, dx+\log (4) \int \frac {e^{-x} x^2}{e^3-x+\left (2-e^3\right ) x^2+x^3} \, dx+(2 \log (4)) \int \frac {e^{6-x}}{-e^3+x-\left (2-e^3\right ) x^2-x^3} \, dx+(2 \log (4)) \int \frac {e^{3-x} x}{-e^3+x-\left (2-e^3\right ) x^2-x^3} \, dx+(2 \log (4)) \int \frac {e^{9-x}}{\left (e^3-x+\left (2-e^3\right ) x^2+x^3\right )^2} \, dx-(2 \log (4)) \int \frac {e^{3-x} x}{\left (e^3-x+\left (2-e^3\right ) x^2+x^3\right )^2} \, dx+\left (\left (1-e^3\right ) \log (4)\right ) \int \frac {e^{6-x}}{\left (e^3-x+\left (2-e^3\right ) x^2+x^3\right )^2} \, dx-\left (\left (1-e^3\right ) \log (4)\right ) \int \frac {e^{6-x} x^2}{\left (e^3-x+\left (2-e^3\right ) x^2+x^3\right )^2} \, dx-\left (2 \left (1-e^3\right )^2 \log (4)\right ) \int \frac {e^{3-x} x^2}{\left (e^3-x+\left (2-e^3\right ) x^2+x^3\right )^2} \, dx+\left (\left (1+e^3\right ) \log (4)\right ) \int \frac {e^{-x} x}{e^3-x+\left (2-e^3\right ) x^2+x^3} \, dx-\left (2 \left (1+e^3\right ) \log (4)\right ) \int \frac {e^{-x} x}{\left (e^3-x+\left (2-e^3\right ) x^2+x^3\right )^2} \, dx+\left (\left (2+e^3-e^6\right ) \log (4)\right ) \int \frac {e^{-x}}{-e^3+x-\left (2-e^3\right ) x^2-x^3} \, dx+\left (\left (2+e^3-e^6\right ) \log (4)\right ) \int \frac {e^{3-x}}{\left (e^3-x+\left (2-e^3\right ) x^2+x^3\right )^2} \, dx+\left (\left (6-3 e^6+e^9\right ) \log (4)\right ) \int \frac {e^{-x} x^2}{\left (e^3-x+\left (2-e^3\right ) x^2+x^3\right )^2} \, dx\\ \end {aligned} \end {gather*}

________________________________________________________________________________________

Mathematica [A]
time = 0.19, size = 43, normalized size = 1.30 \begin {gather*} x \left (2-\frac {e^{-x} \left (e^3-x\right ) \log (4)}{e^3 \left (-1+x^2\right )-x \left (-1+2 x+x^2\right )}\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(E^x*(2*x^2 - 8*x^3 + 4*x^4 + 8*x^5 + 2*x^6 + E^6*(2 - 4*x^2 + 2*x^4) + E^3*(-4*x + 8*x^2 + 8*x^3 -
8*x^4 - 4*x^5)) + (x^2 - x^3 + 3*x^4 + x^5 + E^6*(1 - x + x^2 + x^3) + E^3*(-2*x - 4*x^3 - 2*x^4))*Log[4])/(E^
x*(x^2 - 4*x^3 + 2*x^4 + 4*x^5 + x^6 + E^6*(1 - 2*x^2 + x^4) + E^3*(-2*x + 4*x^2 + 4*x^3 - 4*x^4 - 2*x^5))),x]

[Out]

x*(2 - ((E^3 - x)*Log[4])/(E^x*(E^3*(-1 + x^2) - x*(-1 + 2*x + x^2))))

________________________________________________________________________________________

Maple [C] Result contains higher order function than in optimal. Order 9 vs. order 3.
time = 1.59, size = 3970, normalized size = 120.30

method	result	size
norman	\(\frac {\left (\left (4 \,{\mathrm e}^{3}-2 \,{\mathrm e}^{6}\right ) {\mathrm e}^{x}-4 \,{\mathrm e}^{x} x +\left (2 \,{\mathrm e}^{6}-8 \,{\mathrm e}^{3}+10\right ) x^{2} {\mathrm e}^{x}+2 x^{2} \ln \left (2\right )-2 \,{\mathrm e}^{x} x^{4}-2 \ln \left (2\right ) {\mathrm e}^{3} x \right ) {\mathrm e}^{-x}}{x^{2} {\mathrm e}^{3}-x^{3}-2 x^{2}-{\mathrm e}^{3}+x}\)	\(89\)
default	\(\text {Expression too large to display}\)	\(3970\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((((2*x^4-4*x^2+2)*exp(3)^2+(-4*x^5-8*x^4+8*x^3+8*x^2-4*x)*exp(3)+2*x^6+8*x^5+4*x^4-8*x^3+2*x^2)*exp(x)+2*(
(x^3+x^2-x+1)*exp(3)^2+(-2*x^4-4*x^3-2*x)*exp(3)+x^5+3*x^4-x^3+x^2)*ln(2))/((x^4-2*x^2+1)*exp(3)^2+(-2*x^5-4*x
^4+4*x^3+4*x^2-2*x)*exp(3)+x^6+4*x^5+2*x^4-4*x^3+x^2)/exp(x),x,method=_RETURNVERBOSE)

[Out]

2*x-3*ln(2)*sum((28*_R1^2*exp(3)-2*exp(12)*_R1^2+12*_R1^2*exp(9)-25*_R1^2*exp(6)+16*exp(3)^2+2*exp(9)*exp(3)-1
1*exp(6)*exp(3)-17*_R1*exp(3)+_R1*exp(9)-3*_R1*exp(6)-3*_R1^2-2*exp(3)+2*_R1)/(-exp(12)+6*exp(9)-10*exp(6)+18*
exp(3)-2)/(2*_R1*exp(3)-3*_R1^2-4*_R1+1)*exp(-_R1)*Ei(1,x-_R1),_R1=RootOf(_Z^3+(2-exp(3))*_Z^2-_Z+exp(3)))+2*l
n(2)*exp(-x)/(exp(3)^4-6*exp(3)^3+10*exp(3)^2-18*exp(3)+2)/(x^2*exp(3)-x^3-2*x^2-exp(3)+x)*exp(3)^4*x^2-exp(3)
^2*sum(1/(-2*_R^3*exp(6)+5*_R^4*exp(3)-3*_R^5+8*_R^3*exp(3)-10*_R^4+2*_R*exp(6)-6*_R^2*exp(3)-4*_R^3-4*_R*exp(
3)+6*_R^2+exp(3)-_R)*ln(x-_R),_R=RootOf(_Z^6+(-2*exp(3)+4)*_Z^5+(-4*exp(3)+exp(6)+2)*_Z^4+(4*exp(3)-4)*_Z^3+(4
*exp(3)-2*exp(6)+1)*_Z^2-2*exp(3)*_Z+exp(6)))-4*sum(_R^5/(-2*_R^3*exp(6)+5*_R^4*exp(3)-3*_R^5+8*_R^3*exp(3)-10
*_R^4+2*_R*exp(6)-6*_R^2*exp(3)-4*_R^3-4*_R*exp(3)+6*_R^2+exp(3)-_R)*ln(x-_R),_R=RootOf(_Z^6+(-2*exp(3)+4)*_Z^
5+(-4*exp(3)+exp(6)+2)*_Z^4+(4*exp(3)-4)*_Z^3+(4*exp(3)-2*exp(6)+1)*_Z^2-2*exp(3)*_Z+exp(6)))-36*ln(2)*exp(-x)
/(exp(3)^4-6*exp(3)^3+10*exp(3)^2-18*exp(3)+2)/(x^2*exp(3)-x^3-2*x^2-exp(3)+x)*x^2*exp(3)-4*ln(2)*exp(-x)/(exp
(3)^4-6*exp(3)^3+10*exp(3)^2-18*exp(3)+2)/(x^2*exp(3)-x^3-2*x^2-exp(3)+x)*x*exp(3)+2*exp(3)^2*sum(_R^2/(-2*_R^
3*exp(6)+5*_R^4*exp(3)-3*_R^5+8*_R^3*exp(3)-10*_R^4+2*_R*exp(6)-6*_R^2*exp(3)-4*_R^3-4*_R*exp(3)+6*_R^2+exp(3)
-_R)*ln(x-_R),_R=RootOf(_Z^6+(-2*exp(3)+4)*_Z^5+(-4*exp(3)+exp(6)+2)*_Z^4+(4*exp(3)-4)*_Z^3+(4*exp(3)-2*exp(6)
+1)*_Z^2-2*exp(3)*_Z+exp(6)))-ln(2)*sum((6*_R1^2*exp(3)-3*_R1^2*exp(6)+8*exp(3)^2-exp(6)*exp(3)+9*_R1*exp(3)+_
R1*exp(9)-7*_R1*exp(6)-_R1^2-6*exp(3)-2*_R1)/(-exp(12)+6*exp(9)-10*exp(6)+18*exp(3)-2)/(2*_R1*exp(3)-3*_R1^2-4
*_R1+1)*exp(-_R1)*Ei(1,x-_R1),_R1=RootOf(_Z^3+(2-exp(3))*_Z^2-_Z+exp(3)))-sum((2*(exp(3)-2)*_R^5+(4*exp(3)-exp
(6)-2)*_R^4+4*(-exp(3)+1)*_R^3+(-4*exp(3)+2*exp(6)-1)*_R^2+2*_R*exp(3)-exp(6))/(-2*_R^3*exp(6)+5*_R^4*exp(3)-3
*_R^5+8*_R^3*exp(3)-10*_R^4+2*_R*exp(6)-6*_R^2*exp(3)-4*_R^3-4*_R*exp(3)+6*_R^2+exp(3)-_R)*ln(x-_R),_R=RootOf(
_Z^6+(-2*exp(3)+4)*_Z^5+(-4*exp(3)+exp(6)+2)*_Z^4+(4*exp(3)-4)*_Z^3+(4*exp(3)-2*exp(6)+1)*_Z^2-2*exp(3)*_Z+exp
(6)))-12*ln(2)*exp(-x)/(exp(3)^4-6*exp(3)^3+10*exp(3)^2-18*exp(3)+2)/(x^2*exp(3)-x^3-2*x^2-exp(3)+x)*exp(3)^3*
x^2+2*exp(3)*ln(2)*sum((28*_R1^2*exp(3)-2*exp(12)*_R1^2+12*_R1^2*exp(9)-25*_R1^2*exp(6)+16*exp(3)^2+2*exp(9)*e
xp(3)-11*exp(6)*exp(3)-17*_R1*exp(3)+_R1*exp(9)-3*_R1*exp(6)-3*_R1^2-2*exp(3)+2*_R1)/(-exp(12)+6*exp(9)-10*exp
(6)+18*exp(3)-2)/(2*_R1*exp(3)-3*_R1^2-4*_R1+1)*exp(-_R1)*Ei(1,x-_R1),_R1=RootOf(_Z^3+(2-exp(3))*_Z^2-_Z+exp(3
)))+4*exp(3)*ln(2)*sum((9*_R2*exp(3)^2-2*_R2*exp(6)*exp(3)-10*_R2^2*exp(3)-2*_R2^2*exp(9)+8*_R2^2*exp(6)+3*exp
(6)*exp(3)-3*_R2*exp(3)+_R2^2-10*exp(6))/(-exp(12)+6*exp(9)-10*exp(6)+18*exp(3)-2)/(2*_R2*exp(3)-3*_R2^2-4*_R2
+1)*exp(-_R2)*Ei(1,x-_R2),_R2=RootOf(_Z^3+(2-exp(3))*_Z^2-_Z+exp(3)))-exp(3)^2*ln(2)*sum((4*_R2^2*exp(3)+5*_R2
*exp(3)-_R2*exp(6)+_R2^2-12*exp(3)-exp(9)+6*exp(6)+4*_R2+4)/(-exp(12)+6*exp(9)-10*exp(6)+18*exp(3)-2)/(2*_R2*e
xp(3)-3*_R2^2-4*_R2+1)*exp(-_R2)*Ei(1,x-_R2),_R2=RootOf(_Z^3+(2-exp(3))*_Z^2-_Z+exp(3)))-2*sum(_R^4/(-2*_R^3*e
xp(6)+5*_R^4*exp(3)-3*_R^5+8*_R^3*exp(3)-10*_R^4+2*_R*exp(6)-6*_R^2*exp(3)-4*_R^3-4*_R*exp(3)+6*_R^2+exp(3)-_R
)*ln(x-_R),_R=RootOf(_Z^6+(-2*exp(3)+4)*_Z^5+(-4*exp(3)+exp(6)+2)*_Z^4+(4*exp(3)-4)*_Z^3+(4*exp(3)-2*exp(6)+1)
*_Z^2-2*exp(3)*_Z+exp(6)))+2*exp(3)*sum(_R^5/(-2*_R^3*exp(6)+5*_R^4*exp(3)-3*_R^5+8*_R^3*exp(3)-10*_R^4+2*_R*e
xp(6)-6*_R^2*exp(3)-4*_R^3-4*_R*exp(3)+6*_R^2+exp(3)-_R)*ln(x-_R),_R=RootOf(_Z^6+(-2*exp(3)+4)*_Z^5+(-4*exp(3)
+exp(6)+2)*_Z^4+(4*exp(3)-4)*_Z^3+(4*exp(3)-2*exp(6)+1)*_Z^2-2*exp(3)*_Z+exp(6)))-20*ln(2)*exp(-x)/(exp(3)^4-6
*exp(3)^3+10*exp(3)^2-18*exp(3)+2)/(x^2*exp(3)-x^3-2*x^2-exp(3)+x)*exp(3)^3*x+20*ln(2)*exp(-x)/(exp(3)^4-6*exp
(3)^3+10*exp(3)^2-18*exp(3)+2)/(x^2*exp(3)-x^3-2*x^2-exp(3)+x)*x^2*exp(3)^2+36*ln(2)*exp(-x)/(exp(3)^4-6*exp(3
)^3+10*exp(3)^2-18*exp(3)+2)/(x^2*exp(3)-x^3-2*x^2-exp(3)+x)*x*exp(3)^2-exp(3)^2*sum(_R^4/(-2*_R^3*exp(6)+5*_R
^4*exp(3)-3*_R^5+8*_R^3*exp(3)-10*_R^4+2*_R*exp(6)-6*_R^2*exp(3)-4*_R^3-4*_R*exp(3)+6*_R^2+exp(3)-_R)*ln(x-_R)
,_R=RootOf(_Z^6+(-2*exp(3)+4)*_Z^5+(-4*exp(3)+exp(6)+2)*_Z^4+(4*exp(3)-4)*_Z^3+(4*exp(3)-2*exp(6)+1)*_Z^2-2*ex
p(3)*_Z+exp(6)))-2*exp(3)^5*ln(2)*exp(-x)/(exp(3)^4-6*exp(3)^3+10*exp(3)^2-18*exp(3)+2)/(x^2*exp(3)-x^3-2*x^2-
exp(3)+x)*x+12*exp(3)^4*ln(2)*exp(-x)/(exp(3)^4-6*exp(3)^3+10*exp(3)^2-18*exp(3)+2)/(x^2*exp(3)-x^3-2*x^2-exp(
3)+x)*x+4*sum(_R^3/(-2*_R^3*exp(6)+5*_R^4*exp(3)-3*_R^5+8*_R^3*exp(3)-10*_R^4+2*_R*exp(6)-6*_R^2*exp(3)-4*_R^3
-4*_R*exp(3)+6*_R^2+exp(3)-_R)*ln(x-_R),_R=RootOf(_Z^6+(-2*exp(3)+4)*_Z^5+(-4*exp(3)+exp(6)+2)*_Z^4+(4*exp(3)-
4)*_Z^3+(4*exp(3)-2*exp(6)+1)*_Z^2-2*exp(3)*_Z+exp(6)))-4*exp(3)*sum(_R^2/(-2*_R^3*exp(6)+5*_R^4*exp(3)-3*_R^5
+8*_R^3*exp(3)-10*_R^4+2*_R*exp(6)-6*_R^2*exp(3)-4*_R^3-4*_R*exp(3)+6*_R^2+exp(3)-_R)*ln(x-_R),_R=RootOf(_Z^6+
(-2*exp(3)+4)*_Z^5+(-4*exp(3)+exp(6)+2)*_Z^4+(4*exp(3)-4)*_Z^3+(4*exp(3)-2*exp(6)+1)*_Z^2-2*exp(3)*_Z+exp(6)))
+ln(2)*sum((9*_R2*exp(3)^2-2*_R2*exp(6)*exp(3)-...

________________________________________________________________________________________

Maxima [A]
time = 0.62, size = 64, normalized size = 1.94 \begin {gather*} \frac {2 \, {\left (x^{4} - x^{3} {\left (e^{3} - 2\right )} - x^{2} + x e^{3} - {\left (x^{2} \log \left (2\right ) - x e^{3} \log \left (2\right )\right )} e^{\left (-x\right )}\right )}}{x^{3} - x^{2} {\left (e^{3} - 2\right )} - x + e^{3}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((((2*x^4-4*x^2+2)*exp(3)^2+(-4*x^5-8*x^4+8*x^3+8*x^2-4*x)*exp(3)+2*x^6+8*x^5+4*x^4-8*x^3+2*x^2)*exp(
x)+2*((x^3+x^2-x+1)*exp(3)^2+(-2*x^4-4*x^3-2*x)*exp(3)+x^5+3*x^4-x^3+x^2)*log(2))/((x^4-2*x^2+1)*exp(3)^2+(-2*
x^5-4*x^4+4*x^3+4*x^2-2*x)*exp(3)+x^6+4*x^5+2*x^4-4*x^3+x^2)/exp(x),x, algorithm="maxima")

[Out]

2*(x^4 - x^3*(e^3 - 2) - x^2 + x*e^3 - (x^2*log(2) - x*e^3*log(2))*e^(-x))/(x^3 - x^2*(e^3 - 2) - x + e^3)

________________________________________________________________________________________

Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 71 vs. \(2 (32) = 64\).
time = 0.37, size = 71, normalized size = 2.15 \begin {gather*} \frac {2 \, {\left ({\left (x^{4} + 2 \, x^{3} - x^{2} - {\left (x^{3} - x\right )} e^{3}\right )} e^{x} - {\left (x^{2} - x e^{3}\right )} \log \left (2\right )\right )} e^{\left (-x\right )}}{x^{3} + 2 \, x^{2} - {\left (x^{2} - 1\right )} e^{3} - x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((((2*x^4-4*x^2+2)*exp(3)^2+(-4*x^5-8*x^4+8*x^3+8*x^2-4*x)*exp(3)+2*x^6+8*x^5+4*x^4-8*x^3+2*x^2)*exp(
x)+2*((x^3+x^2-x+1)*exp(3)^2+(-2*x^4-4*x^3-2*x)*exp(3)+x^5+3*x^4-x^3+x^2)*log(2))/((x^4-2*x^2+1)*exp(3)^2+(-2*
x^5-4*x^4+4*x^3+4*x^2-2*x)*exp(3)+x^6+4*x^5+2*x^4-4*x^3+x^2)/exp(x),x, algorithm="fricas")

[Out]

2*((x^4 + 2*x^3 - x^2 - (x^3 - x)*e^3)*e^x - (x^2 - x*e^3)*log(2))*e^(-x)/(x^3 + 2*x^2 - (x^2 - 1)*e^3 - x)

________________________________________________________________________________________

Sympy [A]
time = 0.17, size = 44, normalized size = 1.33 \begin {gather*} 2 x + \frac {\left (- 2 x^{2} \log {\left (2 \right )} + 2 x e^{3} \log {\left (2 \right )}\right ) e^{- x}}{x^{3} - x^{2} e^{3} + 2 x^{2} - x + e^{3}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((((2*x**4-4*x**2+2)*exp(3)**2+(-4*x**5-8*x**4+8*x**3+8*x**2-4*x)*exp(3)+2*x**6+8*x**5+4*x**4-8*x**3+
2*x**2)*exp(x)+2*((x**3+x**2-x+1)*exp(3)**2+(-2*x**4-4*x**3-2*x)*exp(3)+x**5+3*x**4-x**3+x**2)*ln(2))/((x**4-2
*x**2+1)*exp(3)**2+(-2*x**5-4*x**4+4*x**3+4*x**2-2*x)*exp(3)+x**6+4*x**5+2*x**4-4*x**3+x**2)/exp(x),x)

[Out]

2*x + (-2*x**2*log(2) + 2*x*exp(3)*log(2))*exp(-x)/(x**3 - x**2*exp(3) + 2*x**2 - x + exp(3))

________________________________________________________________________________________

Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 72 vs. \(2 (32) = 64\).
time = 0.53, size = 72, normalized size = 2.18 \begin {gather*} \frac {2 \, {\left (x^{4} - x^{3} e^{3} - 2 \, x^{2} e^{\left (-x\right )} \log \left (2\right ) + 2 \, x^{3} + 2 \, x e^{\left (-x + 3\right )} \log \left (2\right ) - x^{2} + x e^{3}\right )}}{x^{3} - x^{2} e^{3} + 2 \, x^{2} - x + e^{3}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((((2*x^4-4*x^2+2)*exp(3)^2+(-4*x^5-8*x^4+8*x^3+8*x^2-4*x)*exp(3)+2*x^6+8*x^5+4*x^4-8*x^3+2*x^2)*exp(
x)+2*((x^3+x^2-x+1)*exp(3)^2+(-2*x^4-4*x^3-2*x)*exp(3)+x^5+3*x^4-x^3+x^2)*log(2))/((x^4-2*x^2+1)*exp(3)^2+(-2*
x^5-4*x^4+4*x^3+4*x^2-2*x)*exp(3)+x^6+4*x^5+2*x^4-4*x^3+x^2)/exp(x),x, algorithm="giac")

[Out]

2*(x^4 - x^3*e^3 - 2*x^2*e^(-x)*log(2) + 2*x^3 + 2*x*e^(-x + 3)*log(2) - x^2 + x*e^3)/(x^3 - x^2*e^3 + 2*x^2 -
x + e^3)

________________________________________________________________________________________

Mupad [B]
time = 8.40, size = 45, normalized size = 1.36 \begin {gather*} 2\,x+\frac {{\mathrm {e}}^{-x}\,\left (2\,x^2\,\ln \left (2\right )-2\,x\,{\mathrm {e}}^3\,\ln \left (2\right )\right )}{-x^3+\left ({\mathrm {e}}^3-2\right )\,x^2+x-{\mathrm {e}}^3} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((exp(-x)*(exp(x)*(exp(6)*(2*x^4 - 4*x^2 + 2) - exp(3)*(4*x - 8*x^2 - 8*x^3 + 8*x^4 + 4*x^5) + 2*x^2 - 8*x^
3 + 4*x^4 + 8*x^5 + 2*x^6) + 2*log(2)*(exp(6)*(x^2 - x + x^3 + 1) - exp(3)*(2*x + 4*x^3 + 2*x^4) + x^2 - x^3 +
3*x^4 + x^5)))/(exp(6)*(x^4 - 2*x^2 + 1) - exp(3)*(2*x - 4*x^2 - 4*x^3 + 4*x^4 + 2*x^5) + x^2 - 4*x^3 + 2*x^4
+ 4*x^5 + x^6),x)

[Out]

2*x + (exp(-x)*(2*x^2*log(2) - 2*x*exp(3)*log(2)))/(x - exp(3) - x^3 + x^2*(exp(3) - 2))

________________________________________________________________________________________