3.29.74 \(\int (e^x (54+x)-9 e^x \log (\log (4))) \, dx\)

Optimal. Leaf size=14 \[ e^x (-1+x-9 (-6+\log (\log (4)))) \]

________________________________________________________________________________________

Rubi [A]  time = 0.01, antiderivative size = 21, normalized size of antiderivative = 1.50, number of steps used = 4, number of rules used = 2, integrand size = 16, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.125, Rules used = {2176, 2194} \begin {gather*} e^x (x+54)-e^x-9 e^x \log (\log (4)) \end {gather*}

Antiderivative was successfully verified.

[In]

Int[E^x*(54 + x) - 9*E^x*Log[Log[4]],x]

[Out]

-E^x + E^x*(54 + x) - 9*E^x*Log[Log[4]]

Rule 2176

Int[((b_.)*(F_)^((g_.)*((e_.) + (f_.)*(x_))))^(n_.)*((c_.) + (d_.)*(x_))^(m_.), x_Symbol] :> Simp[((c + d*x)^m
*(b*F^(g*(e + f*x)))^n)/(f*g*n*Log[F]), x] - Dist[(d*m)/(f*g*n*Log[F]), Int[(c + d*x)^(m - 1)*(b*F^(g*(e + f*x
)))^n, x], x] /; FreeQ[{F, b, c, d, e, f, g, n}, x] && GtQ[m, 0] && IntegerQ[2*m] &&  !$UseGamma === True

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]

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=-\left ((9 \log (\log (4))) \int e^x \, dx\right )+\int e^x (54+x) \, dx\\ &=e^x (54+x)-9 e^x \log (\log (4))-\int e^x \, dx\\ &=-e^x+e^x (54+x)-9 e^x \log (\log (4))\\ \end {aligned} \end {gather*}

________________________________________________________________________________________

Mathematica [A]  time = 0.02, size = 12, normalized size = 0.86 \begin {gather*} e^x (53+x-9 \log (\log (4))) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[E^x*(54 + x) - 9*E^x*Log[Log[4]],x]

[Out]

E^x*(53 + x - 9*Log[Log[4]])

________________________________________________________________________________________

fricas [A]  time = 0.62, size = 16, normalized size = 1.14 \begin {gather*} {\left (x + 53\right )} e^{x} - 9 \, e^{x} \log \left (2 \, \log \relax (2)\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(-9*exp(x)*log(2*log(2))+(x+54)*exp(x),x, algorithm="fricas")

[Out]

(x + 53)*e^x - 9*e^x*log(2*log(2))

________________________________________________________________________________________

giac [A]  time = 0.22, size = 16, normalized size = 1.14 \begin {gather*} {\left (x + 53\right )} e^{x} - 9 \, e^{x} \log \left (2 \, \log \relax (2)\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(-9*exp(x)*log(2*log(2))+(x+54)*exp(x),x, algorithm="giac")

[Out]

(x + 53)*e^x - 9*e^x*log(2*log(2))

________________________________________________________________________________________

maple [A]  time = 0.03, size = 17, normalized size = 1.21

method	result	size
gosper	\(-{\mathrm e}^{x} \left (-x +9 \ln \left (2 \ln \relax (2)\right )-53\right )\)	\(17\)
default	\({\mathrm e}^{x} x +53 \,{\mathrm e}^{x}-9 \,{\mathrm e}^{x} \ln \left (2 \ln \relax (2)\right )\)	\(19\)
norman	\(\left (53-9 \ln \relax (2)-9 \ln \left (\ln \relax (2)\right )\right ) {\mathrm e}^{x}+{\mathrm e}^{x} x\)	\(20\)
risch	\(-9 \,{\mathrm e}^{x} \ln \relax (2)-9 \,{\mathrm e}^{x} \ln \left (\ln \relax (2)\right )+\left (53+x \right ) {\mathrm e}^{x}\)	\(21\)
meijerg	\(1-\frac {\left (-2 x +2\right ) {\mathrm e}^{x}}{2}-\left (-9 \ln \left (2 \ln \relax (2)\right )+54\right ) \left (1-{\mathrm e}^{x}\right )\)	\(29\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(-9*exp(x)*ln(2*ln(2))+(x+54)*exp(x),x,method=_RETURNVERBOSE)

[Out]

-exp(x)*(-x+9*ln(2*ln(2))-53)

________________________________________________________________________________________

maxima [A]  time = 0.36, size = 20, normalized size = 1.43 \begin {gather*} {\left (x - 1\right )} e^{x} - 9 \, e^{x} \log \left (2 \, \log \relax (2)\right ) + 54 \, e^{x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(-9*exp(x)*log(2*log(2))+(x+54)*exp(x),x, algorithm="maxima")

[Out]

(x - 1)*e^x - 9*e^x*log(2*log(2)) + 54*e^x

________________________________________________________________________________________

mupad [B]  time = 1.67, size = 13, normalized size = 0.93 \begin {gather*} {\mathrm {e}}^x\,\left (x-\ln \left ({\ln \relax (4)}^9\right )+53\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(exp(x)*(x + 54) - 9*log(2*log(2))*exp(x),x)

[Out]

exp(x)*(x - log(log(4)^9) + 53)

________________________________________________________________________________________

sympy [A]  time = 0.09, size = 17, normalized size = 1.21 \begin {gather*} \left (x - 9 \log {\relax (2 )} - 9 \log {\left (\log {\relax (2 )} \right )} + 53\right ) e^{x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(-9*exp(x)*ln(2*ln(2))+(x+54)*exp(x),x)

[Out]

(x - 9*log(2) - 9*log(log(2)) + 53)*exp(x)

________________________________________________________________________________________