3.32.5 \(\int \frac {1}{3} e^{-e^x+\frac {1}{3} e^{-e^x} (12 x^3+100 x^4+(-12 x^2-100 x^3) \log (4))} (36 x^2+400 x^3+(-24 x-300 x^2) \log (4)+e^x (-12 x^3-100 x^4+(12 x^2+100 x^3) \log (4))) \, dx\)

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

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Rubi [F]  time = 6.69, 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 {1}{3} \exp \left (-e^x+\frac {1}{3} e^{-e^x} \left (12 x^3+100 x^4+\left (-12 x^2-100 x^3\right ) \log (4)\right )\right ) \left (36 x^2+400 x^3+\left (-24 x-300 x^2\right ) \log (4)+e^x \left (-12 x^3-100 x^4+\left (12 x^2+100 x^3\right ) \log (4)\right )\right ) \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Int[(E^(-E^x + (12*x^3 + 100*x^4 + (-12*x^2 - 100*x^3)*Log[4])/(3*E^E^x))*(36*x^2 + 400*x^3 + (-24*x - 300*x^2
)*Log[4] + E^x*(-12*x^3 - 100*x^4 + (12*x^2 + 100*x^3)*Log[4])))/3,x]

[Out]

-8*Log[4]*Defer[Int][E^(-E^x + (12*x^3 + 100*x^4 + (-12*x^2 - 100*x^3)*Log[4])/(3*E^E^x))*x, x] + 12*Defer[Int
][E^(-E^x + (12*x^3 + 100*x^4 + (-12*x^2 - 100*x^3)*Log[4])/(3*E^E^x))*x^2, x] - 100*Log[4]*Defer[Int][E^(-E^x
+ (12*x^3 + 100*x^4 + (-12*x^2 - 100*x^3)*Log[4])/(3*E^E^x))*x^2, x] + 4*Log[4]*Defer[Int][E^(-E^x + x + (12*
x^3 + 100*x^4 + (-12*x^2 - 100*x^3)*Log[4])/(3*E^E^x))*x^2, x] + (400*Defer[Int][E^(-E^x + (12*x^3 + 100*x^4 +
(-12*x^2 - 100*x^3)*Log[4])/(3*E^E^x))*x^3, x])/3 - (4*(3 - 25*Log[4])*Defer[Int][E^(-E^x + x + (12*x^3 + 100
*x^4 + (-12*x^2 - 100*x^3)*Log[4])/(3*E^E^x))*x^3, x])/3 - (100*Defer[Int][E^(-E^x + x + (12*x^3 + 100*x^4 + (
-12*x^2 - 100*x^3)*Log[4])/(3*E^E^x))*x^4, x])/3

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\frac {1}{3} \int \exp \left (-e^x+\frac {1}{3} e^{-e^x} \left (12 x^3+100 x^4+\left (-12 x^2-100 x^3\right ) \log (4)\right )\right ) \left (36 x^2+400 x^3+\left (-24 x-300 x^2\right ) \log (4)+e^x \left (-12 x^3-100 x^4+\left (12 x^2+100 x^3\right ) \log (4)\right )\right ) \, dx\\ &=\frac {1}{3} \int \left (36 \exp \left (-e^x+\frac {1}{3} e^{-e^x} \left (12 x^3+100 x^4+\left (-12 x^2-100 x^3\right ) \log (4)\right )\right ) x^2+400 \exp \left (-e^x+\frac {1}{3} e^{-e^x} \left (12 x^3+100 x^4+\left (-12 x^2-100 x^3\right ) \log (4)\right )\right ) x^3-4 \exp \left (-e^x+x+\frac {1}{3} e^{-e^x} \left (12 x^3+100 x^4+\left (-12 x^2-100 x^3\right ) \log (4)\right )\right ) x^2 (3+25 x) (x-\log (4))-12 \exp \left (-e^x+\frac {1}{3} e^{-e^x} \left (12 x^3+100 x^4+\left (-12 x^2-100 x^3\right ) \log (4)\right )\right ) x (2+25 x) \log (4)\right ) \, dx\\ &=-\left (\frac {4}{3} \int \exp \left (-e^x+x+\frac {1}{3} e^{-e^x} \left (12 x^3+100 x^4+\left (-12 x^2-100 x^3\right ) \log (4)\right )\right ) x^2 (3+25 x) (x-\log (4)) \, dx\right )+12 \int \exp \left (-e^x+\frac {1}{3} e^{-e^x} \left (12 x^3+100 x^4+\left (-12 x^2-100 x^3\right ) \log (4)\right )\right ) x^2 \, dx+\frac {400}{3} \int \exp \left (-e^x+\frac {1}{3} e^{-e^x} \left (12 x^3+100 x^4+\left (-12 x^2-100 x^3\right ) \log (4)\right )\right ) x^3 \, dx-(4 \log (4)) \int \exp \left (-e^x+\frac {1}{3} e^{-e^x} \left (12 x^3+100 x^4+\left (-12 x^2-100 x^3\right ) \log (4)\right )\right ) x (2+25 x) \, dx\\ &=-\left (\frac {4}{3} \int \left (25 \exp \left (-e^x+x+\frac {1}{3} e^{-e^x} \left (12 x^3+100 x^4+\left (-12 x^2-100 x^3\right ) \log (4)\right )\right ) x^4-3 \exp \left (-e^x+x+\frac {1}{3} e^{-e^x} \left (12 x^3+100 x^4+\left (-12 x^2-100 x^3\right ) \log (4)\right )\right ) x^2 \log (4)-\exp \left (-e^x+x+\frac {1}{3} e^{-e^x} \left (12 x^3+100 x^4+\left (-12 x^2-100 x^3\right ) \log (4)\right )\right ) x^3 (-3+25 \log (4))\right ) \, dx\right )+12 \int \exp \left (-e^x+\frac {1}{3} e^{-e^x} \left (12 x^3+100 x^4+\left (-12 x^2-100 x^3\right ) \log (4)\right )\right ) x^2 \, dx+\frac {400}{3} \int \exp \left (-e^x+\frac {1}{3} e^{-e^x} \left (12 x^3+100 x^4+\left (-12 x^2-100 x^3\right ) \log (4)\right )\right ) x^3 \, dx-(4 \log (4)) \int \left (2 \exp \left (-e^x+\frac {1}{3} e^{-e^x} \left (12 x^3+100 x^4+\left (-12 x^2-100 x^3\right ) \log (4)\right )\right ) x+25 \exp \left (-e^x+\frac {1}{3} e^{-e^x} \left (12 x^3+100 x^4+\left (-12 x^2-100 x^3\right ) \log (4)\right )\right ) x^2\right ) \, dx\\ &=12 \int \exp \left (-e^x+\frac {1}{3} e^{-e^x} \left (12 x^3+100 x^4+\left (-12 x^2-100 x^3\right ) \log (4)\right )\right ) x^2 \, dx-\frac {100}{3} \int \exp \left (-e^x+x+\frac {1}{3} e^{-e^x} \left (12 x^3+100 x^4+\left (-12 x^2-100 x^3\right ) \log (4)\right )\right ) x^4 \, dx+\frac {400}{3} \int \exp \left (-e^x+\frac {1}{3} e^{-e^x} \left (12 x^3+100 x^4+\left (-12 x^2-100 x^3\right ) \log (4)\right )\right ) x^3 \, dx-\frac {1}{3} (4 (3-25 \log (4))) \int \exp \left (-e^x+x+\frac {1}{3} e^{-e^x} \left (12 x^3+100 x^4+\left (-12 x^2-100 x^3\right ) \log (4)\right )\right ) x^3 \, dx+(4 \log (4)) \int \exp \left (-e^x+x+\frac {1}{3} e^{-e^x} \left (12 x^3+100 x^4+\left (-12 x^2-100 x^3\right ) \log (4)\right )\right ) x^2 \, dx-(8 \log (4)) \int \exp \left (-e^x+\frac {1}{3} e^{-e^x} \left (12 x^3+100 x^4+\left (-12 x^2-100 x^3\right ) \log (4)\right )\right ) x \, dx-(100 \log (4)) \int \exp \left (-e^x+\frac {1}{3} e^{-e^x} \left (12 x^3+100 x^4+\left (-12 x^2-100 x^3\right ) \log (4)\right )\right ) x^2 \, dx\\ \end {aligned} \end {gather*}

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Mathematica [F]  time = 3.17, size = 103, normalized size = 3.81 \begin {gather*} \frac {1}{3} \int e^{-e^x+\frac {1}{3} e^{-e^x} \left (12 x^3+100 x^4+\left (-12 x^2-100 x^3\right ) \log (4)\right )} \left (36 x^2+400 x^3+\left (-24 x-300 x^2\right ) \log (4)+e^x \left (-12 x^3-100 x^4+\left (12 x^2+100 x^3\right ) \log (4)\right )\right ) \, dx \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(E^(-E^x + (12*x^3 + 100*x^4 + (-12*x^2 - 100*x^3)*Log[4])/(3*E^E^x))*(36*x^2 + 400*x^3 + (-24*x - 3
00*x^2)*Log[4] + E^x*(-12*x^3 - 100*x^4 + (12*x^2 + 100*x^3)*Log[4])))/3,x]

[Out]

Integrate[E^(-E^x + (12*x^3 + 100*x^4 + (-12*x^2 - 100*x^3)*Log[4])/(3*E^E^x))*(36*x^2 + 400*x^3 + (-24*x - 30
0*x^2)*Log[4] + E^x*(-12*x^3 - 100*x^4 + (12*x^2 + 100*x^3)*Log[4])), x]/3

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fricas [A]  time = 1.05, size = 34, normalized size = 1.26 \begin {gather*} e^{\left (\frac {4}{3} \, {\left (25 \, x^{4} + 3 \, x^{3} - 2 \, {\left (25 \, x^{3} + 3 \, x^{2}\right )} \log \relax (2)\right )} e^{\left (-e^{x}\right )}\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/3*((2*(100*x^3+12*x^2)*log(2)-100*x^4-12*x^3)*exp(x)+2*(-300*x^2-24*x)*log(2)+400*x^3+36*x^2)*exp(
1/3*(2*(-100*x^3-12*x^2)*log(2)+100*x^4+12*x^3)/exp(exp(x)))/exp(exp(x)),x, algorithm="fricas")

[Out]

e^(4/3*(25*x^4 + 3*x^3 - 2*(25*x^3 + 3*x^2)*log(2))*e^(-e^x))

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giac [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {4}{3} \, {\left (100 \, x^{3} + 9 \, x^{2} - {\left (25 \, x^{4} + 3 \, x^{3} - 2 \, {\left (25 \, x^{3} + 3 \, x^{2}\right )} \log \relax (2)\right )} e^{x} - 6 \, {\left (25 \, x^{2} + 2 \, x\right )} \log \relax (2)\right )} e^{\left (\frac {4}{3} \, {\left (25 \, x^{4} + 3 \, x^{3} - 2 \, {\left (25 \, x^{3} + 3 \, x^{2}\right )} \log \relax (2)\right )} e^{\left (-e^{x}\right )} - e^{x}\right )}\,{d x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/3*((2*(100*x^3+12*x^2)*log(2)-100*x^4-12*x^3)*exp(x)+2*(-300*x^2-24*x)*log(2)+400*x^3+36*x^2)*exp(
1/3*(2*(-100*x^3-12*x^2)*log(2)+100*x^4+12*x^3)/exp(exp(x)))/exp(exp(x)),x, algorithm="giac")

[Out]

integrate(4/3*(100*x^3 + 9*x^2 - (25*x^4 + 3*x^3 - 2*(25*x^3 + 3*x^2)*log(2))*e^x - 6*(25*x^2 + 2*x)*log(2))*e
^(4/3*(25*x^4 + 3*x^3 - 2*(25*x^3 + 3*x^2)*log(2))*e^(-e^x) - e^x), x)

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maple [A]  time = 0.10, size = 25, normalized size = 0.93

method	result	size
risch	\({\mathrm e}^{-\frac {4 x^{2} \left (3+25 x \right ) \left (2 \ln \relax (2)-x \right ) {\mathrm e}^{-{\mathrm e}^{x}}}{3}}\)	\(25\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/3*((2*(100*x^3+12*x^2)*ln(2)-100*x^4-12*x^3)*exp(x)+2*(-300*x^2-24*x)*ln(2)+400*x^3+36*x^2)*exp(1/3*(2*(
-100*x^3-12*x^2)*ln(2)+100*x^4+12*x^3)/exp(exp(x)))/exp(exp(x)),x,method=_RETURNVERBOSE)

[Out]

exp(-4/3*x^2*(3+25*x)*(2*ln(2)-x)*exp(-exp(x)))

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maxima [B]  time = 1.48, size = 46, normalized size = 1.70 \begin {gather*} e^{\left (\frac {100}{3} \, x^{4} e^{\left (-e^{x}\right )} - \frac {200}{3} \, x^{3} e^{\left (-e^{x}\right )} \log \relax (2) + 4 \, x^{3} e^{\left (-e^{x}\right )} - 8 \, x^{2} e^{\left (-e^{x}\right )} \log \relax (2)\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/3*((2*(100*x^3+12*x^2)*log(2)-100*x^4-12*x^3)*exp(x)+2*(-300*x^2-24*x)*log(2)+400*x^3+36*x^2)*exp(
1/3*(2*(-100*x^3-12*x^2)*log(2)+100*x^4+12*x^3)/exp(exp(x)))/exp(exp(x)),x, algorithm="maxima")

[Out]

e^(100/3*x^4*e^(-e^x) - 200/3*x^3*e^(-e^x)*log(2) + 4*x^3*e^(-e^x) - 8*x^2*e^(-e^x)*log(2))

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mupad [F]  time = 0.00, size = -1, normalized size = -0.04 \begin {gather*} \int -\frac {{\mathrm {e}}^{-{\mathrm {e}}^x}\,{\mathrm {e}}^{{\mathrm {e}}^{-{\mathrm {e}}^x}\,\left (4\,x^3-\frac {2\,\ln \relax (2)\,\left (100\,x^3+12\,x^2\right )}{3}+\frac {100\,x^4}{3}\right )}\,\left (2\,\ln \relax (2)\,\left (300\,x^2+24\,x\right )+{\mathrm {e}}^x\,\left (12\,x^3-2\,\ln \relax (2)\,\left (100\,x^3+12\,x^2\right )+100\,x^4\right )-36\,x^2-400\,x^3\right )}{3} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(-(exp(-exp(x))*exp(exp(-exp(x))*(4*x^3 - (2*log(2)*(12*x^2 + 100*x^3))/3 + (100*x^4)/3))*(2*log(2)*(24*x +
300*x^2) + exp(x)*(12*x^3 - 2*log(2)*(12*x^2 + 100*x^3) + 100*x^4) - 36*x^2 - 400*x^3))/3,x)

[Out]

int(-(exp(-exp(x))*exp(exp(-exp(x))*(4*x^3 - (2*log(2)*(12*x^2 + 100*x^3))/3 + (100*x^4)/3))*(2*log(2)*(24*x +
300*x^2) + exp(x)*(12*x^3 - 2*log(2)*(12*x^2 + 100*x^3) + 100*x^4) - 36*x^2 - 400*x^3))/3, x)

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sympy [A]  time = 0.64, size = 34, normalized size = 1.26 \begin {gather*} e^{\left (\frac {100 x^{4}}{3} + 4 x^{3} + \frac {\left (- 200 x^{3} - 24 x^{2}\right ) \log {\relax (2 )}}{3}\right ) e^{- e^{x}}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/3*((2*(100*x**3+12*x**2)*ln(2)-100*x**4-12*x**3)*exp(x)+2*(-300*x**2-24*x)*ln(2)+400*x**3+36*x**2)
*exp(1/3*(2*(-100*x**3-12*x**2)*ln(2)+100*x**4+12*x**3)/exp(exp(x)))/exp(exp(x)),x)

[Out]

exp((100*x**4/3 + 4*x**3 + (-200*x**3 - 24*x**2)*log(2)/3)*exp(-exp(x)))

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