3.46.95 \(\int \frac {1}{8} (33+e^{6-2 x}-x+e^{3-x} (-26+x-\log (3))+\log (3)) \, dx\)

Optimal. Leaf size=28 \[ x-\left (7+\frac {1}{4} \left (-3-e^{3-x}-x+\log (3)\right )\right )^2 \]

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Rubi [B]  time = 0.03, antiderivative size = 57, normalized size of antiderivative = 2.04, number of steps used = 5, number of rules used = 3, integrand size = 33, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.091, Rules used = {12, 2194, 2176} \begin {gather*} -\frac {x^2}{16}-\frac {1}{16} e^{6-2 x}-\frac {e^{3-x}}{8}+\frac {1}{8} x (33+\log (3))+\frac {1}{8} e^{3-x} (-x+26+\log (3)) \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(33 + E^(6 - 2*x) - x + E^(3 - x)*(-26 + x - Log[3]) + Log[3])/8,x]

[Out]

-1/16*E^(6 - 2*x) - E^(3 - x)/8 - x^2/16 + (x*(33 + Log[3]))/8 + (E^(3 - x)*(26 - x + Log[3]))/8

Rule 12

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

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} &=\frac {1}{8} \int \left (33+e^{6-2 x}-x+e^{3-x} (-26+x-\log (3))+\log (3)\right ) \, dx\\ &=-\frac {x^2}{16}+\frac {1}{8} x (33+\log (3))+\frac {1}{8} \int e^{6-2 x} \, dx+\frac {1}{8} \int e^{3-x} (-26+x-\log (3)) \, dx\\ &=-\frac {1}{16} e^{6-2 x}-\frac {x^2}{16}+\frac {1}{8} x (33+\log (3))+\frac {1}{8} e^{3-x} (26-x+\log (3))+\frac {1}{8} \int e^{3-x} \, dx\\ &=-\frac {1}{16} e^{6-2 x}-\frac {e^{3-x}}{8}-\frac {x^2}{16}+\frac {1}{8} x (33+\log (3))+\frac {1}{8} e^{3-x} (26-x+\log (3))\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.07, size = 51, normalized size = 1.82 \begin {gather*} \frac {1}{8} \left (-\frac {1}{2} e^{6-2 x}+33 x-\frac {x^2}{2}+x \log (3)+e^{-x} \left (-e^3 x+e^3 (25+\log (3))\right )\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(33 + E^(6 - 2*x) - x + E^(3 - x)*(-26 + x - Log[3]) + Log[3])/8,x]

[Out]

(-1/2*E^(6 - 2*x) + 33*x - x^2/2 + x*Log[3] + (-(E^3*x) + E^3*(25 + Log[3]))/E^x)/8

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fricas [A]  time = 0.57, size = 37, normalized size = 1.32 \begin {gather*} -\frac {1}{16} \, x^{2} - \frac {1}{8} \, {\left (x - \log \relax (3) - 25\right )} e^{\left (-x + 3\right )} + \frac {1}{8} \, x \log \relax (3) + \frac {33}{8} \, x - \frac {1}{16} \, e^{\left (-2 \, x + 6\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/8*exp(3-x)^2+1/8*(-log(3)+x-26)*exp(3-x)+1/8*log(3)-1/8*x+33/8,x, algorithm="fricas")

[Out]

-1/16*x^2 - 1/8*(x - log(3) - 25)*e^(-x + 3) + 1/8*x*log(3) + 33/8*x - 1/16*e^(-2*x + 6)

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giac [A]  time = 0.24, size = 37, normalized size = 1.32 \begin {gather*} -\frac {1}{16} \, x^{2} - \frac {1}{8} \, {\left (x - \log \relax (3) - 25\right )} e^{\left (-x + 3\right )} + \frac {1}{8} \, x \log \relax (3) + \frac {33}{8} \, x - \frac {1}{16} \, e^{\left (-2 \, x + 6\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/8*exp(3-x)^2+1/8*(-log(3)+x-26)*exp(3-x)+1/8*log(3)-1/8*x+33/8,x, algorithm="giac")

[Out]

-1/16*x^2 - 1/8*(x - log(3) - 25)*e^(-x + 3) + 1/8*x*log(3) + 33/8*x - 1/16*e^(-2*x + 6)

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maple [A]  time = 0.05, size = 38, normalized size = 1.36

method	result	size
risch	\(-\frac {{\mathrm e}^{6-2 x}}{16}+\frac {\left (\ln \relax (3)+25-x \right ) {\mathrm e}^{3-x}}{8}+\frac {x \ln \relax (3)}{8}-\frac {x^{2}}{16}+\frac {33 x}{8}\)	\(38\)
norman	\(\left (\frac {25}{8}+\frac {\ln \relax (3)}{8}\right ) {\mathrm e}^{3-x}+\left (\frac {\ln \relax (3)}{8}+\frac {33}{8}\right ) x -\frac {x^{2}}{16}-\frac {{\mathrm e}^{6-2 x}}{16}-\frac {x \,{\mathrm e}^{3-x}}{8}\)	\(47\)
default	\(\frac {33 x}{8}-\frac {x^{2}}{16}-\frac {{\mathrm e}^{6-2 x}}{16}+\frac {{\mathrm e}^{3-x} \left (3-x \right )}{8}+\frac {11 \,{\mathrm e}^{3-x}}{4}+\frac {{\mathrm e}^{3-x} \ln \relax (3)}{8}+\frac {x \ln \relax (3)}{8}\)	\(56\)
derivativedivides	\(-\frac {45}{4}+\frac {15 x}{4}-\frac {\left (3-x \right )^{2}}{16}-\frac {{\mathrm e}^{6-2 x}}{16}+\frac {{\mathrm e}^{3-x} \left (3-x \right )}{8}+\frac {11 \,{\mathrm e}^{3-x}}{4}+\frac {{\mathrm e}^{3-x} \ln \relax (3)}{8}-\frac {\ln \relax (3) \left (3-x \right )}{8}\)	\(65\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/8*exp(3-x)^2+1/8*(-ln(3)+x-26)*exp(3-x)+1/8*ln(3)-1/8*x+33/8,x,method=_RETURNVERBOSE)

[Out]

-1/16*exp(6-2*x)+1/8*(ln(3)+25-x)*exp(3-x)+1/8*x*ln(3)-1/16*x^2+33/8*x

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maxima [B]  time = 0.46, size = 41, normalized size = 1.46 \begin {gather*} -\frac {1}{16} \, x^{2} - \frac {1}{8} \, {\left (x e^{3} - {\left (\log \relax (3) + 25\right )} e^{3}\right )} e^{\left (-x\right )} + \frac {1}{8} \, x \log \relax (3) + \frac {33}{8} \, x - \frac {1}{16} \, e^{\left (-2 \, x + 6\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/8*exp(3-x)^2+1/8*(-log(3)+x-26)*exp(3-x)+1/8*log(3)-1/8*x+33/8,x, algorithm="maxima")

[Out]

-1/16*x^2 - 1/8*(x*e^3 - (log(3) + 25)*e^3)*e^(-x) + 1/8*x*log(3) + 33/8*x - 1/16*e^(-2*x + 6)

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mupad [B]  time = 3.24, size = 44, normalized size = 1.57 \begin {gather*} x\,\left (\frac {\ln \relax (3)}{8}+\frac {33}{8}\right )-\frac {{\mathrm {e}}^{6-2\,x}}{16}+{\mathrm {e}}^{3-x}\,\left (\frac {\ln \relax (3)}{8}+\frac {25}{8}\right )-\frac {x\,{\mathrm {e}}^{3-x}}{8}-\frac {x^2}{16} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(log(3)/8 - x/8 + exp(6 - 2*x)/8 - (exp(3 - x)*(log(3) - x + 26))/8 + 33/8,x)

[Out]

x*(log(3)/8 + 33/8) - exp(6 - 2*x)/16 + exp(3 - x)*(log(3)/8 + 25/8) - (x*exp(3 - x))/8 - x^2/16

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sympy [A]  time = 0.14, size = 39, normalized size = 1.39 \begin {gather*} - \frac {x^{2}}{16} + x \left (\frac {\log {\relax (3 )}}{8} + \frac {33}{8}\right ) + \frac {\left (- 16 x + 16 \log {\relax (3 )} + 400\right ) e^{3 - x}}{128} - \frac {e^{6 - 2 x}}{16} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/8*exp(3-x)**2+1/8*(-ln(3)+x-26)*exp(3-x)+1/8*ln(3)-1/8*x+33/8,x)

[Out]

-x**2/16 + x*(log(3)/8 + 33/8) + (-16*x + 16*log(3) + 400)*exp(3 - x)/128 - exp(6 - 2*x)/16

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