3.97.86 \(\int \frac {e^{\frac {-27 x^4+x^5}{-x^4+(-108+4 x) \log (2)}} (x^8-x^9+(11880 x^4-872 x^5+16 x^6) \log (2)+(11664-864 x+16 x^2) \log ^2(2))}{x^8+(216 x^4-8 x^5) \log (2)+(11664-864 x+16 x^2) \log ^2(2)} \, dx\)

Optimal. Leaf size=25 \[ e^{\frac {x}{\frac {x}{27-x}+\frac {4 \log (2)}{x^3}}} x \]

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Rubi [F]  time = 3.39, 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 {\exp \left (\frac {-27 x^4+x^5}{-x^4+(-108+4 x) \log (2)}\right ) \left (x^8-x^9+\left (11880 x^4-872 x^5+16 x^6\right ) \log (2)+\left (11664-864 x+16 x^2\right ) \log ^2(2)\right )}{x^8+\left (216 x^4-8 x^5\right ) \log (2)+\left (11664-864 x+16 x^2\right ) \log ^2(2)} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Int[(E^((-27*x^4 + x^5)/(-x^4 + (-108 + 4*x)*Log[2]))*(x^8 - x^9 + (11880*x^4 - 872*x^5 + 16*x^6)*Log[2] + (11
664 - 864*x + 16*x^2)*Log[2]^2))/(x^8 + (216*x^4 - 8*x^5)*Log[2] + (11664 - 864*x + 16*x^2)*Log[2]^2),x]

[Out]

Defer[Int][E^(((-27 + x)*x^4)/(-x^4 - 108*Log[2] + 4*x*Log[2])), x] - Defer[Int][E^(((-27 + x)*x^4)/(-x^4 - 10
8*Log[2] + 4*x*Log[2]))*x, x] - 1259712*Log[2]^2*Defer[Int][E^(((-27 + x)*x^4)/(-x^4 - 108*Log[2] + 4*x*Log[2]
))/(x^4 + 108*Log[2] - 4*x*Log[2])^2, x] + 128304*Log[2]^2*Defer[Int][(E^(((-27 + x)*x^4)/(-x^4 - 108*Log[2] +
4*x*Log[2]))*x)/(x^4 + 108*Log[2] - 4*x*Log[2])^2, x] - 4320*Log[2]^2*Defer[Int][(E^(((-27 + x)*x^4)/(-x^4 -
108*Log[2] + 4*x*Log[2]))*x^2)/(x^4 + 108*Log[2] - 4*x*Log[2])^2, x] + 48*Log[2]^2*Defer[Int][(E^(((-27 + x)*x
^4)/(-x^4 - 108*Log[2] + 4*x*Log[2]))*x^3)/(x^4 + 108*Log[2] - 4*x*Log[2])^2, x] + 11664*Log[2]*Defer[Int][E^(
((-27 + x)*x^4)/(-x^4 - 108*Log[2] + 4*x*Log[2]))/(x^4 + 108*Log[2] - 4*x*Log[2]), x] - 648*Log[2]*Defer[Int][
(E^(((-27 + x)*x^4)/(-x^4 - 108*Log[2] + 4*x*Log[2]))*x)/(x^4 + 108*Log[2] - 4*x*Log[2]), x] + 8*Log[2]*Defer[
Int][(E^(((-27 + x)*x^4)/(-x^4 - 108*Log[2] + 4*x*Log[2]))*x^2)/(x^4 + 108*Log[2] - 4*x*Log[2]), x]

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\int \frac {\exp \left (\frac {(-27+x) x^4}{-x^4-108 \log (2)+4 x \log (2)}\right ) \left (x^8-x^9+\left (11880 x^4-872 x^5+16 x^6\right ) \log (2)+\left (11664-864 x+16 x^2\right ) \log ^2(2)\right )}{\left (x^4+108 \log (2)-4 x \log (2)\right )^2} \, dx\\ &=\int \left (\exp \left (\frac {(-27+x) x^4}{-x^4-108 \log (2)+4 x \log (2)}\right )-\exp \left (\frac {(-27+x) x^4}{-x^4-108 \log (2)+4 x \log (2)}\right ) x+\frac {48 \exp \left (\frac {(-27+x) x^4}{-x^4-108 \log (2)+4 x \log (2)}\right ) (-36+x) (-27+x)^2 \log ^2(2)}{\left (x^4+108 \log (2)-4 x \log (2)\right )^2}+\frac {8 \exp \left (\frac {(-27+x) x^4}{-x^4-108 \log (2)+4 x \log (2)}\right ) \left (1458-81 x+x^2\right ) \log (2)}{x^4+108 \log (2)-4 x \log (2)}\right ) \, dx\\ &=(8 \log (2)) \int \frac {\exp \left (\frac {(-27+x) x^4}{-x^4-108 \log (2)+4 x \log (2)}\right ) \left (1458-81 x+x^2\right )}{x^4+108 \log (2)-4 x \log (2)} \, dx+\left (48 \log ^2(2)\right ) \int \frac {\exp \left (\frac {(-27+x) x^4}{-x^4-108 \log (2)+4 x \log (2)}\right ) (-36+x) (-27+x)^2}{\left (x^4+108 \log (2)-4 x \log (2)\right )^2} \, dx+\int \exp \left (\frac {(-27+x) x^4}{-x^4-108 \log (2)+4 x \log (2)}\right ) \, dx-\int \exp \left (\frac {(-27+x) x^4}{-x^4-108 \log (2)+4 x \log (2)}\right ) x \, dx\\ &=(8 \log (2)) \int \left (\frac {1458 \exp \left (\frac {(-27+x) x^4}{-x^4-108 \log (2)+4 x \log (2)}\right )}{x^4+108 \log (2)-4 x \log (2)}-\frac {81 \exp \left (\frac {(-27+x) x^4}{-x^4-108 \log (2)+4 x \log (2)}\right ) x}{x^4+108 \log (2)-4 x \log (2)}+\frac {\exp \left (\frac {(-27+x) x^4}{-x^4-108 \log (2)+4 x \log (2)}\right ) x^2}{x^4+108 \log (2)-4 x \log (2)}\right ) \, dx+\left (48 \log ^2(2)\right ) \int \left (-\frac {26244 \exp \left (\frac {(-27+x) x^4}{-x^4-108 \log (2)+4 x \log (2)}\right )}{\left (x^4+108 \log (2)-4 x \log (2)\right )^2}+\frac {2673 \exp \left (\frac {(-27+x) x^4}{-x^4-108 \log (2)+4 x \log (2)}\right ) x}{\left (x^4+108 \log (2)-4 x \log (2)\right )^2}-\frac {90 \exp \left (\frac {(-27+x) x^4}{-x^4-108 \log (2)+4 x \log (2)}\right ) x^2}{\left (x^4+108 \log (2)-4 x \log (2)\right )^2}+\frac {\exp \left (\frac {(-27+x) x^4}{-x^4-108 \log (2)+4 x \log (2)}\right ) x^3}{\left (x^4+108 \log (2)-4 x \log (2)\right )^2}\right ) \, dx+\int \exp \left (\frac {(-27+x) x^4}{-x^4-108 \log (2)+4 x \log (2)}\right ) \, dx-\int \exp \left (\frac {(-27+x) x^4}{-x^4-108 \log (2)+4 x \log (2)}\right ) x \, dx\\ &=(8 \log (2)) \int \frac {\exp \left (\frac {(-27+x) x^4}{-x^4-108 \log (2)+4 x \log (2)}\right ) x^2}{x^4+108 \log (2)-4 x \log (2)} \, dx-(648 \log (2)) \int \frac {\exp \left (\frac {(-27+x) x^4}{-x^4-108 \log (2)+4 x \log (2)}\right ) x}{x^4+108 \log (2)-4 x \log (2)} \, dx+(11664 \log (2)) \int \frac {\exp \left (\frac {(-27+x) x^4}{-x^4-108 \log (2)+4 x \log (2)}\right )}{x^4+108 \log (2)-4 x \log (2)} \, dx+\left (48 \log ^2(2)\right ) \int \frac {\exp \left (\frac {(-27+x) x^4}{-x^4-108 \log (2)+4 x \log (2)}\right ) x^3}{\left (x^4+108 \log (2)-4 x \log (2)\right )^2} \, dx-\left (4320 \log ^2(2)\right ) \int \frac {\exp \left (\frac {(-27+x) x^4}{-x^4-108 \log (2)+4 x \log (2)}\right ) x^2}{\left (x^4+108 \log (2)-4 x \log (2)\right )^2} \, dx+\left (128304 \log ^2(2)\right ) \int \frac {\exp \left (\frac {(-27+x) x^4}{-x^4-108 \log (2)+4 x \log (2)}\right ) x}{\left (x^4+108 \log (2)-4 x \log (2)\right )^2} \, dx-\left (1259712 \log ^2(2)\right ) \int \frac {\exp \left (\frac {(-27+x) x^4}{-x^4-108 \log (2)+4 x \log (2)}\right )}{\left (x^4+108 \log (2)-4 x \log (2)\right )^2} \, dx+\int \exp \left (\frac {(-27+x) x^4}{-x^4-108 \log (2)+4 x \log (2)}\right ) \, dx-\int \exp \left (\frac {(-27+x) x^4}{-x^4-108 \log (2)+4 x \log (2)}\right ) x \, dx\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.06, size = 33, normalized size = 1.32 \begin {gather*} 2^{-\frac {4 (-27+x)^2}{x^4+108 \log (2)-4 x \log (2)}} e^{27-x} x \end {gather*}

Warning: Unable to verify antiderivative.

[In]

Integrate[(E^((-27*x^4 + x^5)/(-x^4 + (-108 + 4*x)*Log[2]))*(x^8 - x^9 + (11880*x^4 - 872*x^5 + 16*x^6)*Log[2]
+ (11664 - 864*x + 16*x^2)*Log[2]^2))/(x^8 + (216*x^4 - 8*x^5)*Log[2] + (11664 - 864*x + 16*x^2)*Log[2]^2),x]

[Out]

(E^(27 - x)*x)/2^((4*(-27 + x)^2)/(x^4 + 108*Log[2] - 4*x*Log[2]))

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fricas [A]  time = 0.58, size = 27, normalized size = 1.08 \begin {gather*} x e^{\left (-\frac {x^{5} - 27 \, x^{4}}{x^{4} - 4 \, {\left (x - 27\right )} \log \relax (2)}\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((16*x^2-864*x+11664)*log(2)^2+(16*x^6-872*x^5+11880*x^4)*log(2)-x^9+x^8)*exp((x^5-27*x^4)/((4*x-108
)*log(2)-x^4))/((16*x^2-864*x+11664)*log(2)^2+(-8*x^5+216*x^4)*log(2)+x^8),x, algorithm="fricas")

[Out]

x*e^(-(x^5 - 27*x^4)/(x^4 - 4*(x - 27)*log(2)))

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giac [A]  time = 0.30, size = 29, normalized size = 1.16 \begin {gather*} x e^{\left (-\frac {x^{5} - 27 \, x^{4}}{x^{4} - 4 \, x \log \relax (2) + 108 \, \log \relax (2)}\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((16*x^2-864*x+11664)*log(2)^2+(16*x^6-872*x^5+11880*x^4)*log(2)-x^9+x^8)*exp((x^5-27*x^4)/((4*x-108
)*log(2)-x^4))/((16*x^2-864*x+11664)*log(2)^2+(-8*x^5+216*x^4)*log(2)+x^8),x, algorithm="giac")

[Out]

x*e^(-(x^5 - 27*x^4)/(x^4 - 4*x*log(2) + 108*log(2)))

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maple [A]  time = 0.25, size = 28, normalized size = 1.12

method	result	size
gosper	\(x \,{\mathrm e}^{\frac {x^{4} \left (x -27\right )}{-x^{4}+4 x \ln \relax (2)-108 \ln \relax (2)}}\)	\(28\)
risch	\(x \,{\mathrm e}^{\frac {x^{4} \left (x -27\right )}{-x^{4}+4 x \ln \relax (2)-108 \ln \relax (2)}}\)	\(28\)
norman	\(\frac {-x^{5} {\mathrm e}^{\frac {x^{5}-27 x^{4}}{\left (4 x -108\right ) \ln \relax (2)-x^{4}}}-108 x \ln \relax (2) {\mathrm e}^{\frac {x^{5}-27 x^{4}}{\left (4 x -108\right ) \ln \relax (2)-x^{4}}}+4 x^{2} \ln \relax (2) {\mathrm e}^{\frac {x^{5}-27 x^{4}}{\left (4 x -108\right ) \ln \relax (2)-x^{4}}}}{-x^{4}+4 x \ln \relax (2)-108 \ln \relax (2)}\)	\(118\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((16*x^2-864*x+11664)*ln(2)^2+(16*x^6-872*x^5+11880*x^4)*ln(2)-x^9+x^8)*exp((x^5-27*x^4)/((4*x-108)*ln(2)-
x^4))/((16*x^2-864*x+11664)*ln(2)^2+(-8*x^5+216*x^4)*ln(2)+x^8),x,method=_RETURNVERBOSE)

[Out]

x*exp(x^4*(x-27)/(-x^4+4*x*ln(2)-108*ln(2)))

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maxima [B]  time = 0.67, size = 69, normalized size = 2.76 \begin {gather*} x e^{\left (-\frac {4 \, x^{2} \log \relax (2)}{x^{4} - 4 \, x \log \relax (2) + 108 \, \log \relax (2)} - x + \frac {216 \, x \log \relax (2)}{x^{4} - 4 \, x \log \relax (2) + 108 \, \log \relax (2)} - \frac {2916 \, \log \relax (2)}{x^{4} - 4 \, x \log \relax (2) + 108 \, \log \relax (2)} + 27\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((16*x^2-864*x+11664)*log(2)^2+(16*x^6-872*x^5+11880*x^4)*log(2)-x^9+x^8)*exp((x^5-27*x^4)/((4*x-108
)*log(2)-x^4))/((16*x^2-864*x+11664)*log(2)^2+(-8*x^5+216*x^4)*log(2)+x^8),x, algorithm="maxima")

[Out]

x*e^(-4*x^2*log(2)/(x^4 - 4*x*log(2) + 108*log(2)) - x + 216*x*log(2)/(x^4 - 4*x*log(2) + 108*log(2)) - 2916*l
og(2)/(x^4 - 4*x*log(2) + 108*log(2)) + 27)

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mupad [B]  time = 7.39, size = 30, normalized size = 1.20 \begin {gather*} x\,{\mathrm {e}}^{\frac {27\,x^4-x^5}{x^4-4\,\ln \relax (2)\,x+108\,\ln \relax (2)}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((exp(-(27*x^4 - x^5)/(log(2)*(4*x - 108) - x^4))*(log(2)^2*(16*x^2 - 864*x + 11664) + log(2)*(11880*x^4 -
872*x^5 + 16*x^6) + x^8 - x^9))/(log(2)*(216*x^4 - 8*x^5) + log(2)^2*(16*x^2 - 864*x + 11664) + x^8),x)

[Out]

x*exp((27*x^4 - x^5)/(108*log(2) - 4*x*log(2) + x^4))

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sympy [A]  time = 67.52, size = 22, normalized size = 0.88 \begin {gather*} x e^{\frac {x^{5} - 27 x^{4}}{- x^{4} + \left (4 x - 108\right ) \log {\relax (2 )}}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((16*x**2-864*x+11664)*ln(2)**2+(16*x**6-872*x**5+11880*x**4)*ln(2)-x**9+x**8)*exp((x**5-27*x**4)/((
4*x-108)*ln(2)-x**4))/((16*x**2-864*x+11664)*ln(2)**2+(-8*x**5+216*x**4)*ln(2)+x**8),x)

[Out]

x*exp((x**5 - 27*x**4)/(-x**4 + (4*x - 108)*log(2)))

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