3.92.62 \(\int e^{-e^4} (3 x^2+e^{e^4} (6 x-2 x \log (2)+2 x \log ^2(2))) \, dx\)

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

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Rubi [A]  time = 0.02, antiderivative size = 31, normalized size of antiderivative = 1.24, number of steps used = 2, number of rules used = 1, integrand size = 36, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.028, Rules used = {12} \begin {gather*} e^{-e^4} x^3+\frac {1}{2} x^2 \left (6+2 \log ^2(2)-\log (4)\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

x^3/E^E^4 + (x^2*(6 + 2*Log[2]^2 - Log[4]))/2

Rule 12

Int[(a_)*(u_), x_Symbol] :> Dist[a, Int[u, x], x] /; FreeQ[a, x] &&  !MatchQ[u, (b_)*(v_) /; FreeQ[b, x]]

Rubi steps

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

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Mathematica [A]  time = 0.03, size = 31, normalized size = 1.24 \begin {gather*} e^{-e^4} x^3+\frac {1}{2} x^2 \left (6+2 \log ^2(2)-\log (4)\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

x^3/E^E^4 + (x^2*(6 + 2*Log[2]^2 - Log[4]))/2

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fricas [A]  time = 0.88, size = 35, normalized size = 1.40 \begin {gather*} {\left (x^{3} + {\left (x^{2} \log \relax (2)^{2} - x^{2} \log \relax (2) + 3 \, x^{2}\right )} e^{\left (e^{4}\right )}\right )} e^{\left (-e^{4}\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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giac [A]  time = 0.19, size = 35, normalized size = 1.40 \begin {gather*} {\left (x^{3} + {\left (x^{2} \log \relax (2)^{2} - x^{2} \log \relax (2) + 3 \, x^{2}\right )} e^{\left (e^{4}\right )}\right )} e^{\left (-e^{4}\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maple [A]  time = 0.04, size = 25, normalized size = 1.00

method	result	size
norman	\(\left (\ln \relax (2)^{2}-\ln \relax (2)+3\right ) x^{2}+{\mathrm e}^{-{\mathrm e}^{4}} x^{3}\)	\(25\)
gosper	\(\left (\ln \relax (2)^{2} {\mathrm e}^{{\mathrm e}^{4}}-\ln \relax (2) {\mathrm e}^{{\mathrm e}^{4}}+3 \,{\mathrm e}^{{\mathrm e}^{4}}+x \right ) x^{2} {\mathrm e}^{-{\mathrm e}^{4}}\)	\(32\)
default	\({\mathrm e}^{-{\mathrm e}^{4}} \left ({\mathrm e}^{{\mathrm e}^{4}} \left (x^{2} \ln \relax (2)^{2}-x^{2} \ln \relax (2)+3 x^{2}\right )+x^{3}\right )\)	\(36\)
risch	\({\mathrm e}^{-{\mathrm e}^{4}} x^{3}+\ln \relax (2)^{2} {\mathrm e}^{-{\mathrm e}^{4}} {\mathrm e}^{{\mathrm e}^{4}} x^{2}-\ln \relax (2) {\mathrm e}^{-{\mathrm e}^{4}} {\mathrm e}^{{\mathrm e}^{4}} x^{2}+3 \,{\mathrm e}^{-{\mathrm e}^{4}} {\mathrm e}^{{\mathrm e}^{4}} x^{2}\)	\(55\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(ln(2)^2-ln(2)+3)*x^2+1/exp(exp(4))*x^3

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maxima [A]  time = 0.35, size = 35, normalized size = 1.40 \begin {gather*} {\left (x^{3} + {\left (x^{2} \log \relax (2)^{2} - x^{2} \log \relax (2) + 3 \, x^{2}\right )} e^{\left (e^{4}\right )}\right )} e^{\left (-e^{4}\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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mupad [B]  time = 6.55, size = 24, normalized size = 0.96 \begin {gather*} {\mathrm {e}}^{-{\mathrm {e}}^4}\,x^3+\left ({\ln \relax (2)}^2-\ln \relax (2)+3\right )\,x^2 \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(exp(-exp(4))*(exp(exp(4))*(6*x - 2*x*log(2) + 2*x*log(2)^2) + 3*x^2),x)

[Out]

x^2*(log(2)^2 - log(2) + 3) + x^3*exp(-exp(4))

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sympy [A]  time = 0.06, size = 20, normalized size = 0.80 \begin {gather*} \frac {x^{3}}{e^{e^{4}}} + x^{2} \left (- \log {\relax (2 )} + \log {\relax (2 )}^{2} + 3\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

x**3*exp(-exp(4)) + x**2*(-log(2) + log(2)**2 + 3)

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