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

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

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Rubi [F]  time = 5.88, 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*}

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Mathematica [A]  time = 0.21, 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))))

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fricas [B]  time = 1.09, 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 \relax (2)\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)

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giac [B]  time = 0.39, 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 \relax (2) + 2 \, x^{3} + 2 \, x e^{\left (-x + 3\right )} \log \relax (2) - 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)

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maple [B]  time = 0.44, size = 89, normalized size = 2.70

method	result	size
norman	\(\frac {\left (\left (4 \,{\mathrm e}^{3}-2 \,{\mathrm e}^{6}\right ) {\mathrm e}^{x}-4 \,{\mathrm e}^{x} x +\left (2 \,{\mathrm e}^{6}-8 \,{\mathrm e}^{3}+10\right ) x^{2} {\mathrm e}^{x}+2 x^{2} \ln \relax (2)-2 \,{\mathrm e}^{x} x^{4}-2 \,{\mathrm e}^{3} \ln \relax (2) x \right ) {\mathrm e}^{-x}}{x^{2} {\mathrm e}^{3}-x^{3}-2 x^{2}-{\mathrm e}^{3}+x}\)	\(89\)
default	\(\text {Expression too large to display}\)	\(3946\)

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*exp(3)^2+4*exp(3))*exp(x)-4*exp(x)*x+(2*exp(3)^2-8*exp(3)+10)*x^2*exp(x)+2*x^2*ln(2)-2*exp(x)*x^4-2*exp(3
)*ln(2)*x)/(x^2*exp(3)-x^3-2*x^2-exp(3)+x)/exp(x)

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maxima [A]  time = 0.56, 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 \relax (2) - x e^{3} \log \relax (2)\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)

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

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sympy [A]  time = 0.37, size = 44, normalized size = 1.33 \begin {gather*} 2 x + \frac {\left (- 2 x^{2} \log {\relax (2 )} + 2 x e^{3} \log {\relax (2 )}\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))

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