\(\int e^{-e^4} (3 x^2+e^{e^4} (6 x-2 x \log (2)+2 x \log ^2(2))) \, dx\) [9162]

Optimal result

Integrand size = 36, antiderivative size = 25 \[ \int e^{-e^4} \left (3 x^2+e^{e^4} \left (6 x-2 x \log (2)+2 x \log ^2(2)\right )\right ) \, dx=x \left (x+x \left (2+e^{-e^4} x-\log (2)+\log ^2(2)\right )\right ) \]

[Out]

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

Rubi [A] (verified)

Time = 0.01 (sec) , antiderivative size = 31, normalized size of antiderivative = 1.24, number of steps used = 2, number of rules used = 1, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.028, Rules used = {12} \[ \int e^{-e^4} \left (3 x^2+e^{e^4} \left (6 x-2 x \log (2)+2 x \log ^2(2)\right )\right ) \, dx=e^{-e^4} x^3+\frac {1}{2} x^2 \left (6+2 \log ^2(2)-\log (4)\right ) \]

[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{align*} \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{align*}

Mathematica [A] (verified)

Time = 0.04 (sec) , antiderivative size = 31, normalized size of antiderivative = 1.24 \[ \int e^{-e^4} \left (3 x^2+e^{e^4} \left (6 x-2 x \log (2)+2 x \log ^2(2)\right )\right ) \, dx=e^{-e^4} x^3+\frac {1}{2} x^2 \left (6+2 \log ^2(2)-\log (4)\right ) \]

[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

Maple [A] (verified)

Time = 0.16 (sec) , antiderivative size = 25, normalized size of antiderivative = 1.00

method	result	size
norman	\(\left (\ln \left (2\right )^{2}-\ln \left (2\right )+3\right ) x^{2}+{\mathrm e}^{-{\mathrm e}^{4}} x^{3}\)	\(25\)
risch	\(x^{2} \ln \left (2\right )^{2}-x^{2} \ln \left (2\right )+3 x^{2}+{\mathrm e}^{-{\mathrm e}^{4}} x^{3}\)	\(31\)
gosper	\(\left (\ln \left (2\right )^{2} {\mathrm e}^{{\mathrm e}^{4}}-\ln \left (2\right ) {\mathrm e}^{{\mathrm e}^{4}}+3 \,{\mathrm e}^{{\mathrm e}^{4}}+x \right ) x^{2} {\mathrm e}^{-{\mathrm e}^{4}}\)	\(32\)
default	\({\mathrm e}^{-{\mathrm e}^{4}} \left (x^{3}+\frac {\left (2 \ln \left (2\right )^{2} {\mathrm e}^{{\mathrm e}^{4}}-2 \ln \left (2\right ) {\mathrm e}^{{\mathrm e}^{4}}+6 \,{\mathrm e}^{{\mathrm e}^{4}}\right ) x^{2}}{2}\right )\)	\(38\)
parallelrisch	\({\mathrm e}^{-{\mathrm e}^{4}} \left (\ln \left (2\right )^{2} {\mathrm e}^{{\mathrm e}^{4}} x^{2}-\ln \left (2\right ) {\mathrm e}^{{\mathrm e}^{4}} x^{2}+3 x^{2} {\mathrm e}^{{\mathrm e}^{4}}+x^{3}\right )\)	\(40\)

[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

Fricas [A] (verification not implemented)

none

Time = 0.26 (sec) , antiderivative size = 35, normalized size of antiderivative = 1.40 \[ \int e^{-e^4} \left (3 x^2+e^{e^4} \left (6 x-2 x \log (2)+2 x \log ^2(2)\right )\right ) \, dx={\left (x^{3} + {\left (x^{2} \log \left (2\right )^{2} - x^{2} \log \left (2\right ) + 3 \, x^{2}\right )} e^{\left (e^{4}\right )}\right )} e^{\left (-e^{4}\right )} \]

[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)

Sympy [A] (verification not implemented)

Time = 0.02 (sec) , antiderivative size = 20, normalized size of antiderivative = 0.80 \[ \int e^{-e^4} \left (3 x^2+e^{e^4} \left (6 x-2 x \log (2)+2 x \log ^2(2)\right )\right ) \, dx=\frac {x^{3}}{e^{e^{4}}} + x^{2} \left (- \log {\left (2 \right )} + \log {\left (2 \right )}^{2} + 3\right ) \]

[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)

Maxima [A] (verification not implemented)

none

Time = 0.19 (sec) , antiderivative size = 35, normalized size of antiderivative = 1.40 \[ \int e^{-e^4} \left (3 x^2+e^{e^4} \left (6 x-2 x \log (2)+2 x \log ^2(2)\right )\right ) \, dx={\left (x^{3} + {\left (x^{2} \log \left (2\right )^{2} - x^{2} \log \left (2\right ) + 3 \, x^{2}\right )} e^{\left (e^{4}\right )}\right )} e^{\left (-e^{4}\right )} \]

[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)

Giac [A] (verification not implemented)

none

Time = 0.26 (sec) , antiderivative size = 35, normalized size of antiderivative = 1.40 \[ \int e^{-e^4} \left (3 x^2+e^{e^4} \left (6 x-2 x \log (2)+2 x \log ^2(2)\right )\right ) \, dx={\left (x^{3} + {\left (x^{2} \log \left (2\right )^{2} - x^{2} \log \left (2\right ) + 3 \, x^{2}\right )} e^{\left (e^{4}\right )}\right )} e^{\left (-e^{4}\right )} \]

[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)

Mupad [B] (verification not implemented)

Time = 12.62 (sec) , antiderivative size = 24, normalized size of antiderivative = 0.96 \[ \int e^{-e^4} \left (3 x^2+e^{e^4} \left (6 x-2 x \log (2)+2 x \log ^2(2)\right )\right ) \, dx={\mathrm {e}}^{-{\mathrm {e}}^4}\,x^3+\left ({\ln \left (2\right )}^2-\ln \left (2\right )+3\right )\,x^2 \]

[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))