3.98.17 \(\int \frac {1+e^{x+e^x \log ^2(2)} \log ^2(2) (2+i \pi +\log (14))}{2+i \pi +\log (14)} \, dx\)

Optimal. Leaf size=24 \[ e^{e^x \log ^2(2)}+\frac {x}{2+i \pi +\log (14)} \]

________________________________________________________________________________________

Rubi [A]  time = 0.03, antiderivative size = 24, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, integrand size = 40, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.075, Rules used = {12, 2282, 2194} \begin {gather*} e^{e^x \log ^2(2)}+\frac {x}{2+i \pi +\log (14)} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(1 + E^(x + E^x*Log[2]^2)*Log[2]^2*(2 + I*Pi + Log[14]))/(2 + I*Pi + Log[14]),x]

[Out]

E^(E^x*Log[2]^2) + x/(2 + I*Pi + Log[14])

Rule 12

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

Rule 2194

Int[((F_)^((c_.)*((a_.) + (b_.)*(x_))))^(n_.), x_Symbol] :> Simp[(F^(c*(a + b*x)))^n/(b*c*n*Log[F]), x] /; Fre
eQ[{F, a, b, c, n}, x]

Rule 2282

Int[u_, x_Symbol] :> With[{v = FunctionOfExponential[u, x]}, Dist[v/D[v, x], Subst[Int[FunctionOfExponentialFu
nction[u, x]/x, x], x, v], x]] /; FunctionOfExponentialQ[u, x] &&  !MatchQ[u, (w_)*((a_.)*(v_)^(n_))^(m_) /; F
reeQ[{a, m, n}, x] && IntegerQ[m*n]] &&  !MatchQ[u, E^((c_.)*((a_.) + (b_.)*x))*(F_)[v_] /; FreeQ[{a, b, c}, x
] && InverseFunctionQ[F[x]]]

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\frac {\int \left (1+e^{x+e^x \log ^2(2)} \log ^2(2) (2+i \pi +\log (14))\right ) \, dx}{2+i \pi +\log (14)}\\ &=\frac {x}{2+i \pi +\log (14)}+\log ^2(2) \int e^{x+e^x \log ^2(2)} \, dx\\ &=\frac {x}{2+i \pi +\log (14)}+\log ^2(2) \operatorname {Subst}\left (\int e^{x \log ^2(2)} \, dx,x,e^x\right )\\ &=e^{e^x \log ^2(2)}+\frac {x}{2+i \pi +\log (14)}\\ \end {aligned} \end {gather*}

________________________________________________________________________________________

Mathematica [A]  time = 0.04, size = 24, normalized size = 1.00 \begin {gather*} e^{e^x \log ^2(2)}+\frac {x}{2+i \pi +\log (14)} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(1 + E^(x + E^x*Log[2]^2)*Log[2]^2*(2 + I*Pi + Log[14]))/(2 + I*Pi + Log[14]),x]

[Out]

E^(E^x*Log[2]^2) + x/(2 + I*Pi + Log[14])

________________________________________________________________________________________

fricas [B]  time = 0.84, size = 55, normalized size = 2.29 \begin {gather*} \frac {{\left ({\left (i \, \pi + \log \left (14\right ) + 2\right )} e^{\left (x + e^{\left (x + 2 \, \log \left (\log \relax (2)\right )\right )} + 2 \, \log \left (\log \relax (2)\right )\right )} + x e^{\left (x + 2 \, \log \left (\log \relax (2)\right )\right )}\right )} e^{\left (-x - 2 \, \log \left (\log \relax (2)\right )\right )}}{i \, \pi + \log \left (14\right ) + 2} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((log(14)+I*pi+2)*exp(2*log(log(2))+x)*exp(exp(2*log(log(2))+x))+1)/(log(14)+I*pi+2),x, algorithm="f
ricas")

[Out]

((I*pi + log(14) + 2)*e^(x + e^(x + 2*log(log(2))) + 2*log(log(2))) + x*e^(x + 2*log(log(2))))*e^(-x - 2*log(l
og(2)))/(I*pi + log(14) + 2)

________________________________________________________________________________________

giac [A]  time = 0.14, size = 28, normalized size = 1.17 \begin {gather*} \frac {{\left (i \, \pi + \log \left (14\right ) + 2\right )} e^{\left (e^{x} \log \relax (2)^{2}\right )} + x}{i \, \pi + \log \left (14\right ) + 2} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((log(14)+I*pi+2)*exp(2*log(log(2))+x)*exp(exp(2*log(log(2))+x))+1)/(log(14)+I*pi+2),x, algorithm="g
iac")

[Out]

((I*pi + log(14) + 2)*e^(e^x*log(2)^2) + x)/(I*pi + log(14) + 2)

________________________________________________________________________________________

maple [A]  time = 0.07, size = 24, normalized size = 1.00

method	result	size
risch	\(\frac {x}{\ln \relax (2)+\ln \relax (7)+i \pi +2}+{\mathrm e}^{\ln \relax (2)^{2} {\mathrm e}^{x}}\)	\(24\)
norman	\(-\frac {\left (i \pi -\ln \left (14\right )-2\right ) x}{\pi ^{2}+\ln \left (14\right )^{2}+4 \ln \left (14\right )+4}+{\mathrm e}^{{\mathrm e}^{2 \ln \left (\ln \relax (2)\right )+x}}\)	\(39\)
default	\(\frac {x +i \pi \,{\mathrm e}^{{\mathrm e}^{2 \ln \left (\ln \relax (2)\right )+x}}+\ln \left (14\right ) {\mathrm e}^{{\mathrm e}^{2 \ln \left (\ln \relax (2)\right )+x}}+2 \,{\mathrm e}^{{\mathrm e}^{2 \ln \left (\ln \relax (2)\right )+x}}}{\ln \left (14\right )+i \pi +2}\)	\(50\)
derivativedivides	\(\frac {\ln \left ({\mathrm e}^{2 \ln \left (\ln \relax (2)\right )+x}\right )+\ln \left (14\right ) {\mathrm e}^{{\mathrm e}^{2 \ln \left (\ln \relax (2)\right )+x}}+i \pi \,{\mathrm e}^{{\mathrm e}^{2 \ln \left (\ln \relax (2)\right )+x}}+2 \,{\mathrm e}^{{\mathrm e}^{2 \ln \left (\ln \relax (2)\right )+x}}}{\ln \left (14\right )+i \pi +2}\)	\(58\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((ln(14)+I*Pi+2)*exp(2*ln(ln(2))+x)*exp(exp(2*ln(ln(2))+x))+1)/(ln(14)+I*Pi+2),x,method=_RETURNVERBOSE)

[Out]

1/(ln(2)+ln(7)+I*Pi+2)*x+exp(ln(2)^2*exp(x))

________________________________________________________________________________________

maxima [A]  time = 0.44, size = 28, normalized size = 1.17 \begin {gather*} \frac {{\left (i \, \pi + \log \left (14\right ) + 2\right )} e^{\left (e^{x} \log \relax (2)^{2}\right )} + x}{i \, \pi + \log \left (14\right ) + 2} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((log(14)+I*pi+2)*exp(2*log(log(2))+x)*exp(exp(2*log(log(2))+x))+1)/(log(14)+I*pi+2),x, algorithm="m
axima")

[Out]

((I*pi + log(14) + 2)*e^(e^x*log(2)^2) + x)/(I*pi + log(14) + 2)

________________________________________________________________________________________

mupad [B]  time = 0.32, size = 21, normalized size = 0.88 \begin {gather*} {\mathrm {e}}^{{\mathrm {e}}^x\,{\ln \relax (2)}^2}+\frac {x}{\ln \left (14\right )+2+\Pi \,1{}\mathrm {i}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((exp(x + 2*log(log(2)))*exp(exp(x + 2*log(log(2))))*(Pi*1i + log(14) + 2) + 1)/(Pi*1i + log(14) + 2),x)

[Out]

exp(exp(x)*log(2)^2) + x/(Pi*1i + log(14) + 2)

________________________________________________________________________________________

sympy [A]  time = 0.21, size = 19, normalized size = 0.79 \begin {gather*} \frac {x}{2 + \log {\left (14 \right )} + i \pi } + e^{e^{x} \log {\relax (2 )}^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((ln(14)+I*pi+2)*exp(2*ln(ln(2))+x)*exp(exp(2*ln(ln(2))+x))+1)/(ln(14)+I*pi+2),x)

[Out]

x/(2 + log(14) + I*pi) + exp(exp(x)*log(2)**2)

________________________________________________________________________________________