3.75.75 \(\int \frac {e^{-\frac {e^x}{\log (2)}} (e^x (-33 x^2-8 x^3+x^4)+(88 x+17 x^2-x^3) \log (2)+(e^x (-33 x-8 x^2+x^3)+(33+11 x) \log (2)) \log (9+6 x+x^2))}{(1452+220 x-76 x^2+4 x^3) \log (2)} \, dx\)

Optimal. Leaf size=31 \[ \frac {e^{-\frac {e^x}{\log (2)}} x \left (x+\log \left ((3+x)^2\right )\right )}{4 (11-x)} \]

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Rubi [F]  time = 4.79, 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^{-\frac {e^x}{\log (2)}} \left (e^x \left (-33 x^2-8 x^3+x^4\right )+\left (88 x+17 x^2-x^3\right ) \log (2)+\left (e^x \left (-33 x-8 x^2+x^3\right )+(33+11 x) \log (2)\right ) \log \left (9+6 x+x^2\right )\right )}{\left (1452+220 x-76 x^2+4 x^3\right ) \log (2)} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Int[(E^x*(-33*x^2 - 8*x^3 + x^4) + (88*x + 17*x^2 - x^3)*Log[2] + (E^x*(-33*x - 8*x^2 + x^3) + (33 + 11*x)*Log
[2])*Log[9 + 6*x + x^2])/(E^(E^x/Log[2])*(1452 + 220*x - 76*x^2 + 4*x^3)*Log[2]),x]

[Out]

-11/(4*E^(E^x/Log[2])) - ExpIntegralEi[-(E^x/Log[2])]/4 - Log[(3 + x)^2]/(4*E^(E^x/Log[2])) + (121*Defer[Int][
1/(E^(E^x/Log[2])*(-11 + x)^2), x])/4 + (11*Log[(3 + x)^2]*Defer[Int][1/(E^(E^x/Log[2])*(-11 + x)^2), x])/4 +
(121*Defer[Int][E^(x - E^x/Log[2])/(-11 + x), x])/(4*Log[2]) + (11*Log[(3 + x)^2]*Defer[Int][E^(x - E^x/Log[2]
)/(-11 + x), x])/(4*Log[2]) - (11*Defer[Int][1/(E^(E^x/Log[2])*(-11 + x)), x])/28 + Defer[Int][E^(x - E^x/Log[
2])*x, x]/(4*Log[2]) + (11*Defer[Int][1/(E^(E^x/Log[2])*(3 + x)), x])/28 - (11*Defer[Int][Defer[Int][1/(E^(E^x
/Log[2])*(-11 + x)^2), x]/(3 + x), x])/2 - (11*Defer[Int][Defer[Int][E^(x - E^x/Log[2])/(-11 + x), x]/(3 + x),
x])/(2*Log[2])

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\frac {\int \frac {e^{-\frac {e^x}{\log (2)}} \left (e^x \left (-33 x^2-8 x^3+x^4\right )+\left (88 x+17 x^2-x^3\right ) \log (2)+\left (e^x \left (-33 x-8 x^2+x^3\right )+(33+11 x) \log (2)\right ) \log \left (9+6 x+x^2\right )\right )}{1452+220 x-76 x^2+4 x^3} \, dx}{\log (2)}\\ &=\frac {\int \left (\frac {e^{x-\frac {e^x}{\log (2)}} x \left (x+\log \left ((3+x)^2\right )\right )}{4 (-11+x)}-\frac {e^{-\frac {e^x}{\log (2)}} \log (2) \left (-88 x-17 x^2+x^3-33 \log \left ((3+x)^2\right )-11 x \log \left ((3+x)^2\right )\right )}{4 (-11+x)^2 (3+x)}\right ) \, dx}{\log (2)}\\ &=-\left (\frac {1}{4} \int \frac {e^{-\frac {e^x}{\log (2)}} \left (-88 x-17 x^2+x^3-33 \log \left ((3+x)^2\right )-11 x \log \left ((3+x)^2\right )\right )}{(-11+x)^2 (3+x)} \, dx\right )+\frac {\int \frac {e^{x-\frac {e^x}{\log (2)}} x \left (x+\log \left ((3+x)^2\right )\right )}{-11+x} \, dx}{4 \log (2)}\\ &=-\left (\frac {1}{4} \int \left (-\frac {88 e^{-\frac {e^x}{\log (2)}} x}{(-11+x)^2 (3+x)}-\frac {17 e^{-\frac {e^x}{\log (2)}} x^2}{(-11+x)^2 (3+x)}+\frac {e^{-\frac {e^x}{\log (2)}} x^3}{(-11+x)^2 (3+x)}-\frac {11 e^{-\frac {e^x}{\log (2)}} \log \left ((3+x)^2\right )}{(-11+x)^2}\right ) \, dx\right )+\frac {\int \left (\frac {e^{x-\frac {e^x}{\log (2)}} x^2}{-11+x}+\frac {e^{x-\frac {e^x}{\log (2)}} x \log \left ((3+x)^2\right )}{-11+x}\right ) \, dx}{4 \log (2)}\\ &=-\left (\frac {1}{4} \int \frac {e^{-\frac {e^x}{\log (2)}} x^3}{(-11+x)^2 (3+x)} \, dx\right )+\frac {11}{4} \int \frac {e^{-\frac {e^x}{\log (2)}} \log \left ((3+x)^2\right )}{(-11+x)^2} \, dx+\frac {17}{4} \int \frac {e^{-\frac {e^x}{\log (2)}} x^2}{(-11+x)^2 (3+x)} \, dx+22 \int \frac {e^{-\frac {e^x}{\log (2)}} x}{(-11+x)^2 (3+x)} \, dx+\frac {\int \frac {e^{x-\frac {e^x}{\log (2)}} x^2}{-11+x} \, dx}{4 \log (2)}+\frac {\int \frac {e^{x-\frac {e^x}{\log (2)}} x \log \left ((3+x)^2\right )}{-11+x} \, dx}{4 \log (2)}\\ &=-\frac {1}{4} e^{-\frac {e^x}{\log (2)}} \log \left ((3+x)^2\right )-\frac {1}{4} \int \left (e^{-\frac {e^x}{\log (2)}}+\frac {1331 e^{-\frac {e^x}{\log (2)}}}{14 (-11+x)^2}+\frac {3751 e^{-\frac {e^x}{\log (2)}}}{196 (-11+x)}-\frac {27 e^{-\frac {e^x}{\log (2)}}}{196 (3+x)}\right ) \, dx-\frac {11}{4} \int \frac {2 \int \frac {e^{-\frac {e^x}{\log (2)}}}{(-11+x)^2} \, dx}{3+x} \, dx+\frac {17}{4} \int \left (\frac {121 e^{-\frac {e^x}{\log (2)}}}{14 (-11+x)^2}+\frac {187 e^{-\frac {e^x}{\log (2)}}}{196 (-11+x)}+\frac {9 e^{-\frac {e^x}{\log (2)}}}{196 (3+x)}\right ) \, dx+22 \int \left (\frac {11 e^{-\frac {e^x}{\log (2)}}}{14 (-11+x)^2}+\frac {3 e^{-\frac {e^x}{\log (2)}}}{196 (-11+x)}-\frac {3 e^{-\frac {e^x}{\log (2)}}}{196 (3+x)}\right ) \, dx+\frac {\int \left (11 e^{x-\frac {e^x}{\log (2)}}+\frac {121 e^{x-\frac {e^x}{\log (2)}}}{-11+x}+e^{x-\frac {e^x}{\log (2)}} x\right ) \, dx}{4 \log (2)}-\frac {\int \frac {2 \left (-e^{-\frac {e^x}{\log (2)}} \log (2)+11 \int \frac {e^{x-\frac {e^x}{\log (2)}}}{-11+x} \, dx\right )}{3+x} \, dx}{4 \log (2)}+\frac {1}{4} \left (11 \log \left ((3+x)^2\right )\right ) \int \frac {e^{-\frac {e^x}{\log (2)}}}{(-11+x)^2} \, dx+\frac {\left (11 \log \left ((3+x)^2\right )\right ) \int \frac {e^{x-\frac {e^x}{\log (2)}}}{-11+x} \, dx}{4 \log (2)}\\ &=-\frac {1}{4} e^{-\frac {e^x}{\log (2)}} \log \left ((3+x)^2\right )+\frac {27}{784} \int \frac {e^{-\frac {e^x}{\log (2)}}}{3+x} \, dx+\frac {153}{784} \int \frac {e^{-\frac {e^x}{\log (2)}}}{3+x} \, dx-\frac {1}{4} \int e^{-\frac {e^x}{\log (2)}} \, dx+\frac {33}{98} \int \frac {e^{-\frac {e^x}{\log (2)}}}{-11+x} \, dx-\frac {33}{98} \int \frac {e^{-\frac {e^x}{\log (2)}}}{3+x} \, dx+\frac {3179}{784} \int \frac {e^{-\frac {e^x}{\log (2)}}}{-11+x} \, dx-\frac {3751}{784} \int \frac {e^{-\frac {e^x}{\log (2)}}}{-11+x} \, dx-\frac {11}{2} \int \frac {\int \frac {e^{-\frac {e^x}{\log (2)}}}{(-11+x)^2} \, dx}{3+x} \, dx+\frac {121}{7} \int \frac {e^{-\frac {e^x}{\log (2)}}}{(-11+x)^2} \, dx-\frac {1331}{56} \int \frac {e^{-\frac {e^x}{\log (2)}}}{(-11+x)^2} \, dx+\frac {2057}{56} \int \frac {e^{-\frac {e^x}{\log (2)}}}{(-11+x)^2} \, dx+\frac {\int e^{x-\frac {e^x}{\log (2)}} x \, dx}{4 \log (2)}-\frac {\int \frac {-e^{-\frac {e^x}{\log (2)}} \log (2)+11 \int \frac {e^{x-\frac {e^x}{\log (2)}}}{-11+x} \, dx}{3+x} \, dx}{2 \log (2)}+\frac {11 \int e^{x-\frac {e^x}{\log (2)}} \, dx}{4 \log (2)}+\frac {121 \int \frac {e^{x-\frac {e^x}{\log (2)}}}{-11+x} \, dx}{4 \log (2)}+\frac {1}{4} \left (11 \log \left ((3+x)^2\right )\right ) \int \frac {e^{-\frac {e^x}{\log (2)}}}{(-11+x)^2} \, dx+\frac {\left (11 \log \left ((3+x)^2\right )\right ) \int \frac {e^{x-\frac {e^x}{\log (2)}}}{-11+x} \, dx}{4 \log (2)}\\ &=-\frac {1}{4} e^{-\frac {e^x}{\log (2)}} \log \left ((3+x)^2\right )+\frac {27}{784} \int \frac {e^{-\frac {e^x}{\log (2)}}}{3+x} \, dx+\frac {153}{784} \int \frac {e^{-\frac {e^x}{\log (2)}}}{3+x} \, dx-\frac {1}{4} \operatorname {Subst}\left (\int \frac {e^{-\frac {x}{\log (2)}}}{x} \, dx,x,e^x\right )+\frac {33}{98} \int \frac {e^{-\frac {e^x}{\log (2)}}}{-11+x} \, dx-\frac {33}{98} \int \frac {e^{-\frac {e^x}{\log (2)}}}{3+x} \, dx+\frac {3179}{784} \int \frac {e^{-\frac {e^x}{\log (2)}}}{-11+x} \, dx-\frac {3751}{784} \int \frac {e^{-\frac {e^x}{\log (2)}}}{-11+x} \, dx-\frac {11}{2} \int \frac {\int \frac {e^{-\frac {e^x}{\log (2)}}}{(-11+x)^2} \, dx}{3+x} \, dx+\frac {121}{7} \int \frac {e^{-\frac {e^x}{\log (2)}}}{(-11+x)^2} \, dx-\frac {1331}{56} \int \frac {e^{-\frac {e^x}{\log (2)}}}{(-11+x)^2} \, dx+\frac {2057}{56} \int \frac {e^{-\frac {e^x}{\log (2)}}}{(-11+x)^2} \, dx+\frac {\int e^{x-\frac {e^x}{\log (2)}} x \, dx}{4 \log (2)}-\frac {\int \left (-\frac {e^{-\frac {e^x}{\log (2)}} \log (2)}{3+x}+\frac {11 \int \frac {e^{x-\frac {e^x}{\log (2)}}}{-11+x} \, dx}{3+x}\right ) \, dx}{2 \log (2)}+\frac {11 \operatorname {Subst}\left (\int e^{-\frac {x}{\log (2)}} \, dx,x,e^x\right )}{4 \log (2)}+\frac {121 \int \frac {e^{x-\frac {e^x}{\log (2)}}}{-11+x} \, dx}{4 \log (2)}+\frac {1}{4} \left (11 \log \left ((3+x)^2\right )\right ) \int \frac {e^{-\frac {e^x}{\log (2)}}}{(-11+x)^2} \, dx+\frac {\left (11 \log \left ((3+x)^2\right )\right ) \int \frac {e^{x-\frac {e^x}{\log (2)}}}{-11+x} \, dx}{4 \log (2)}\\ &=-\frac {11}{4} e^{-\frac {e^x}{\log (2)}}-\frac {1}{4} \text {Ei}\left (-\frac {e^x}{\log (2)}\right )-\frac {1}{4} e^{-\frac {e^x}{\log (2)}} \log \left ((3+x)^2\right )+\frac {27}{784} \int \frac {e^{-\frac {e^x}{\log (2)}}}{3+x} \, dx+\frac {153}{784} \int \frac {e^{-\frac {e^x}{\log (2)}}}{3+x} \, dx+\frac {33}{98} \int \frac {e^{-\frac {e^x}{\log (2)}}}{-11+x} \, dx-\frac {33}{98} \int \frac {e^{-\frac {e^x}{\log (2)}}}{3+x} \, dx+\frac {1}{2} \int \frac {e^{-\frac {e^x}{\log (2)}}}{3+x} \, dx+\frac {3179}{784} \int \frac {e^{-\frac {e^x}{\log (2)}}}{-11+x} \, dx-\frac {3751}{784} \int \frac {e^{-\frac {e^x}{\log (2)}}}{-11+x} \, dx-\frac {11}{2} \int \frac {\int \frac {e^{-\frac {e^x}{\log (2)}}}{(-11+x)^2} \, dx}{3+x} \, dx+\frac {121}{7} \int \frac {e^{-\frac {e^x}{\log (2)}}}{(-11+x)^2} \, dx-\frac {1331}{56} \int \frac {e^{-\frac {e^x}{\log (2)}}}{(-11+x)^2} \, dx+\frac {2057}{56} \int \frac {e^{-\frac {e^x}{\log (2)}}}{(-11+x)^2} \, dx+\frac {\int e^{x-\frac {e^x}{\log (2)}} x \, dx}{4 \log (2)}-\frac {11 \int \frac {\int \frac {e^{x-\frac {e^x}{\log (2)}}}{-11+x} \, dx}{3+x} \, dx}{2 \log (2)}+\frac {121 \int \frac {e^{x-\frac {e^x}{\log (2)}}}{-11+x} \, dx}{4 \log (2)}+\frac {1}{4} \left (11 \log \left ((3+x)^2\right )\right ) \int \frac {e^{-\frac {e^x}{\log (2)}}}{(-11+x)^2} \, dx+\frac {\left (11 \log \left ((3+x)^2\right )\right ) \int \frac {e^{x-\frac {e^x}{\log (2)}}}{-11+x} \, dx}{4 \log (2)}\\ \end {aligned} \end {gather*}

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Mathematica [F]  time = 2.33, size = 107, normalized size = 3.45 \begin {gather*} \frac {\int \frac {e^{-\frac {e^x}{\log (2)}} \left (e^x \left (-33 x^2-8 x^3+x^4\right )+\left (88 x+17 x^2-x^3\right ) \log (2)+\left (e^x \left (-33 x-8 x^2+x^3\right )+(33+11 x) \log (2)\right ) \log \left (9+6 x+x^2\right )\right )}{1452+220 x-76 x^2+4 x^3} \, dx}{\log (2)} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(E^x*(-33*x^2 - 8*x^3 + x^4) + (88*x + 17*x^2 - x^3)*Log[2] + (E^x*(-33*x - 8*x^2 + x^3) + (33 + 11*
x)*Log[2])*Log[9 + 6*x + x^2])/(E^(E^x/Log[2])*(1452 + 220*x - 76*x^2 + 4*x^3)*Log[2]),x]

[Out]

Integrate[(E^x*(-33*x^2 - 8*x^3 + x^4) + (88*x + 17*x^2 - x^3)*Log[2] + (E^x*(-33*x - 8*x^2 + x^3) + (33 + 11*
x)*Log[2])*Log[9 + 6*x + x^2])/(E^(E^x/Log[2])*(1452 + 220*x - 76*x^2 + 4*x^3)), x]/Log[2]

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fricas [A]  time = 0.59, size = 31, normalized size = 1.00 \begin {gather*} -\frac {{\left (x^{2} + x \log \left (x^{2} + 6 \, x + 9\right )\right )} e^{\left (-\frac {e^{x}}{\log \relax (2)}\right )}}{4 \, {\left (x - 11\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((((x^3-8*x^2-33*x)*exp(x)+(11*x+33)*log(2))*log(x^2+6*x+9)+(x^4-8*x^3-33*x^2)*exp(x)+(-x^3+17*x^2+88
*x)*log(2))/(4*x^3-76*x^2+220*x+1452)/log(2)/exp(exp(x)/log(2)),x, algorithm="fricas")

[Out]

-1/4*(x^2 + x*log(x^2 + 6*x + 9))*e^(-e^x/log(2))/(x - 11)

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giac [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {{\left ({\left (x^{4} - 8 \, x^{3} - 33 \, x^{2}\right )} e^{x} - {\left (x^{3} - 17 \, x^{2} - 88 \, x\right )} \log \relax (2) + {\left ({\left (x^{3} - 8 \, x^{2} - 33 \, x\right )} e^{x} + 11 \, {\left (x + 3\right )} \log \relax (2)\right )} \log \left (x^{2} + 6 \, x + 9\right )\right )} e^{\left (-\frac {e^{x}}{\log \relax (2)}\right )}}{4 \, {\left (x^{3} - 19 \, x^{2} + 55 \, x + 363\right )} \log \relax (2)}\,{d x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((((x^3-8*x^2-33*x)*exp(x)+(11*x+33)*log(2))*log(x^2+6*x+9)+(x^4-8*x^3-33*x^2)*exp(x)+(-x^3+17*x^2+88
*x)*log(2))/(4*x^3-76*x^2+220*x+1452)/log(2)/exp(exp(x)/log(2)),x, algorithm="giac")

[Out]

integrate(1/4*((x^4 - 8*x^3 - 33*x^2)*e^x - (x^3 - 17*x^2 - 88*x)*log(2) + ((x^3 - 8*x^2 - 33*x)*e^x + 11*(x +
3)*log(2))*log(x^2 + 6*x + 9))*e^(-e^x/log(2))/((x^3 - 19*x^2 + 55*x + 363)*log(2)), x)

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maple [C]  time = 0.25, size = 87, normalized size = 2.81

method	result	size
risch	\(-\frac {x \left (-i \pi \mathrm {csgn}\left (i \left (3+x \right )\right )^{2} \mathrm {csgn}\left (i \left (3+x \right )^{2}\right )+2 i \pi \,\mathrm {csgn}\left (i \left (3+x \right )\right ) \mathrm {csgn}\left (i \left (3+x \right )^{2}\right )^{2}-i \pi \mathrm {csgn}\left (i \left (3+x \right )^{2}\right )^{3}+2 x +4 \ln \left (3+x \right )\right ) {\mathrm e}^{-\frac {{\mathrm e}^{x}}{\ln \relax (2)}}}{8 \left (x -11\right )}\)	\(87\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((((x^3-8*x^2-33*x)*exp(x)+(11*x+33)*ln(2))*ln(x^2+6*x+9)+(x^4-8*x^3-33*x^2)*exp(x)+(-x^3+17*x^2+88*x)*ln(2
))/(4*x^3-76*x^2+220*x+1452)/ln(2)/exp(exp(x)/ln(2)),x,method=_RETURNVERBOSE)

[Out]

-1/8*x*(-I*Pi*csgn(I*(3+x))^2*csgn(I*(3+x)^2)+2*I*Pi*csgn(I*(3+x))*csgn(I*(3+x)^2)^2-I*Pi*csgn(I*(3+x)^2)^3+2*
x+4*ln(3+x))/(x-11)*exp(-exp(x)/ln(2))

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maxima [A]  time = 0.53, size = 36, normalized size = 1.16 \begin {gather*} -\frac {{\left (x^{2} \log \relax (2) + 2 \, x \log \relax (2) \log \left (x + 3\right )\right )} e^{\left (-\frac {e^{x}}{\log \relax (2)}\right )}}{4 \, {\left (x - 11\right )} \log \relax (2)} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((((x^3-8*x^2-33*x)*exp(x)+(11*x+33)*log(2))*log(x^2+6*x+9)+(x^4-8*x^3-33*x^2)*exp(x)+(-x^3+17*x^2+88
*x)*log(2))/(4*x^3-76*x^2+220*x+1452)/log(2)/exp(exp(x)/log(2)),x, algorithm="maxima")

[Out]

-1/4*(x^2*log(2) + 2*x*log(2)*log(x + 3))*e^(-e^x/log(2))/((x - 11)*log(2))

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mupad [F]  time = 0.00, size = -1, normalized size = -0.03 \begin {gather*} -\int -\frac {{\mathrm {e}}^{-\frac {{\mathrm {e}}^x}{\ln \relax (2)}}\,\left (\ln \left (x^2+6\,x+9\right )\,\left (\ln \relax (2)\,\left (11\,x+33\right )-{\mathrm {e}}^x\,\left (-x^3+8\,x^2+33\,x\right )\right )-{\mathrm {e}}^x\,\left (-x^4+8\,x^3+33\,x^2\right )+\ln \relax (2)\,\left (-x^3+17\,x^2+88\,x\right )\right )}{\ln \relax (2)\,\left (4\,x^3-76\,x^2+220\,x+1452\right )} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((exp(-exp(x)/log(2))*(log(6*x + x^2 + 9)*(log(2)*(11*x + 33) - exp(x)*(33*x + 8*x^2 - x^3)) - exp(x)*(33*x
^2 + 8*x^3 - x^4) + log(2)*(88*x + 17*x^2 - x^3)))/(log(2)*(220*x - 76*x^2 + 4*x^3 + 1452)),x)

[Out]

-int(-(exp(-exp(x)/log(2))*(log(6*x + x^2 + 9)*(log(2)*(11*x + 33) - exp(x)*(33*x + 8*x^2 - x^3)) - exp(x)*(33
*x^2 + 8*x^3 - x^4) + log(2)*(88*x + 17*x^2 - x^3)))/(log(2)*(220*x - 76*x^2 + 4*x^3 + 1452)), x)

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sympy [A]  time = 0.52, size = 29, normalized size = 0.94 \begin {gather*} \frac {\left (- x^{2} - x \log {\left (x^{2} + 6 x + 9 \right )}\right ) e^{- \frac {e^{x}}{\log {\relax (2 )}}}}{4 x - 44} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((((x**3-8*x**2-33*x)*exp(x)+(11*x+33)*ln(2))*ln(x**2+6*x+9)+(x**4-8*x**3-33*x**2)*exp(x)+(-x**3+17*x
**2+88*x)*ln(2))/(4*x**3-76*x**2+220*x+1452)/ln(2)/exp(exp(x)/ln(2)),x)

[Out]

(-x**2 - x*log(x**2 + 6*x + 9))*exp(-exp(x)/log(2))/(4*x - 44)

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