3.12.5 \(\int (-1+e^x (14-4 x-8 \log (2)+\log ^2(2))) \, dx\)

Optimal. Leaf size=34 \[ -3+e^4-x+e^x \left (-2+(2-x)^2-x^2+(4-\log (2))^2\right ) \]

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Rubi [A]  time = 0.02, antiderivative size = 26, normalized size of antiderivative = 0.76, number of steps used = 3, number of rules used = 2, integrand size = 19, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.105, Rules used = {2176, 2194} \begin {gather*} -x+4 e^x+e^x \left (-4 x+14+\log ^2(2)-8 \log (2)\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Int[-1 + E^x*(14 - 4*x - 8*Log[2] + Log[2]^2),x]

[Out]

4*E^x - x + E^x*(14 - 4*x - 8*Log[2] + Log[2]^2)

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} &=-x+\int e^x \left (14-4 x-8 \log (2)+\log ^2(2)\right ) \, dx\\ &=-x+e^x \left (14-4 x-8 \log (2)+\log ^2(2)\right )+4 \int e^x \, dx\\ &=4 e^x-x+e^x \left (14-4 x-8 \log (2)+\log ^2(2)\right )\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.03, size = 21, normalized size = 0.62 \begin {gather*} -x+e^x \left (18-4 x-8 \log (2)+\log ^2(2)\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[-1 + E^x*(14 - 4*x - 8*Log[2] + Log[2]^2),x]

[Out]

-x + E^x*(18 - 4*x - 8*Log[2] + Log[2]^2)

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fricas [A]  time = 0.89, size = 20, normalized size = 0.59 \begin {gather*} {\left (\log \relax (2)^{2} - 4 \, x - 8 \, \log \relax (2) + 18\right )} e^{x} - x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((log(2)^2-8*log(2)-4*x+14)*exp(x)-1,x, algorithm="fricas")

[Out]

(log(2)^2 - 4*x - 8*log(2) + 18)*e^x - x

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giac [A]  time = 0.23, size = 20, normalized size = 0.59 \begin {gather*} {\left (\log \relax (2)^{2} - 4 \, x - 8 \, \log \relax (2) + 18\right )} e^{x} - x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((log(2)^2-8*log(2)-4*x+14)*exp(x)-1,x, algorithm="giac")

[Out]

(log(2)^2 - 4*x - 8*log(2) + 18)*e^x - x

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maple [A]  time = 0.02, size = 21, normalized size = 0.62

method	result	size
risch	\(\left (\ln \relax (2)^{2}-8 \ln \relax (2)-4 x +18\right ) {\mathrm e}^{x}-x\)	\(21\)
norman	\(\left (18+\ln \relax (2)^{2}-8 \ln \relax (2)\right ) {\mathrm e}^{x}-x -4 \,{\mathrm e}^{x} x\)	\(23\)
default	\(-x +\ln \relax (2)^{2} {\mathrm e}^{x}-4 \,{\mathrm e}^{x} x +18 \,{\mathrm e}^{x}-8 \,{\mathrm e}^{x} \ln \relax (2)\)	\(27\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((ln(2)^2-8*ln(2)-4*x+14)*exp(x)-1,x,method=_RETURNVERBOSE)

[Out]

(ln(2)^2-8*ln(2)-4*x+18)*exp(x)-x

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maxima [A]  time = 0.61, size = 28, normalized size = 0.82 \begin {gather*} e^{x} \log \relax (2)^{2} - 4 \, {\left (x - 1\right )} e^{x} - 8 \, e^{x} \log \relax (2) - x + 14 \, e^{x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((log(2)^2-8*log(2)-4*x+14)*exp(x)-1,x, algorithm="maxima")

[Out]

e^x*log(2)^2 - 4*(x - 1)*e^x - 8*e^x*log(2) - x + 14*e^x

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mupad [B]  time = 0.72, size = 22, normalized size = 0.65 \begin {gather*} {\mathrm {e}}^x\,\left ({\ln \relax (2)}^2-\ln \left (256\right )+18\right )-x-4\,x\,{\mathrm {e}}^x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(- exp(x)*(4*x + 8*log(2) - log(2)^2 - 14) - 1,x)

[Out]

exp(x)*(log(2)^2 - log(256) + 18) - x - 4*x*exp(x)

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sympy [A]  time = 0.10, size = 19, normalized size = 0.56 \begin {gather*} - x + \left (- 4 x - 8 \log {\relax (2 )} + \log {\relax (2 )}^{2} + 18\right ) e^{x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((ln(2)**2-8*ln(2)-4*x+14)*exp(x)-1,x)

[Out]

-x + (-4*x - 8*log(2) + log(2)**2 + 18)*exp(x)

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