3.59.6 \(\int \frac {-13+e^{1-x} (4-4 x)+3 \log (5)}{3 \log (5)} \, dx\)

Optimal. Leaf size=32 \[ x+\frac {x+4 \left (-x+\frac {1}{3} \left (-x+e^{1-x} x\right )\right )}{\log (5)} \]

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Rubi [A]  time = 0.03, antiderivative size = 51, normalized size of antiderivative = 1.59, number of steps used = 4, number of rules used = 3, integrand size = 27, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.111, Rules used = {12, 2176, 2194} \begin {gather*} -\frac {4 e^{1-x} (1-x)}{3 \log (5)}-\frac {x (13-\log (125))}{3 \log (5)}+\frac {4 e^{1-x}}{3 \log (5)} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(-13 + E^(1 - x)*(4 - 4*x) + 3*Log[5])/(3*Log[5]),x]

[Out]

(4*E^(1 - x))/(3*Log[5]) - (4*E^(1 - x)*(1 - x))/(3*Log[5]) - (x*(13 - Log[125]))/(3*Log[5])

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 {\int \left (-13+e^{1-x} (4-4 x)+3 \log (5)\right ) \, dx}{3 \log (5)}\\ &=-\frac {x (13-\log (125))}{3 \log (5)}+\frac {\int e^{1-x} (4-4 x) \, dx}{3 \log (5)}\\ &=-\frac {4 e^{1-x} (1-x)}{3 \log (5)}-\frac {x (13-\log (125))}{3 \log (5)}-\frac {4 \int e^{1-x} \, dx}{3 \log (5)}\\ &=\frac {4 e^{1-x}}{3 \log (5)}-\frac {4 e^{1-x} (1-x)}{3 \log (5)}-\frac {x (13-\log (125))}{3 \log (5)}\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.02, size = 26, normalized size = 0.81 \begin {gather*} \frac {-13 x+4 e^{1-x} x+x \log (125)}{3 \log (5)} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(-13 + E^(1 - x)*(4 - 4*x) + 3*Log[5])/(3*Log[5]),x]

[Out]

(-13*x + 4*E^(1 - x)*x + x*Log[125])/(3*Log[5])

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fricas [A]  time = 0.83, size = 24, normalized size = 0.75 \begin {gather*} \frac {4 \, x e^{\left (-x + 1\right )} + 3 \, x \log \relax (5) - 13 \, x}{3 \, \log \relax (5)} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/3*((-4*x+4)*exp(-x+1)+3*log(5)-13)/log(5),x, algorithm="fricas")

[Out]

1/3*(4*x*e^(-x + 1) + 3*x*log(5) - 13*x)/log(5)

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giac [A]  time = 0.11, size = 24, normalized size = 0.75 \begin {gather*} \frac {4 \, x e^{\left (-x + 1\right )} + 3 \, x \log \relax (5) - 13 \, x}{3 \, \log \relax (5)} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/3*((-4*x+4)*exp(-x+1)+3*log(5)-13)/log(5),x, algorithm="giac")

[Out]

1/3*(4*x*e^(-x + 1) + 3*x*log(5) - 13*x)/log(5)

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maple [A]  time = 0.06, size = 23, normalized size = 0.72

method	result	size
risch	\(\frac {4 x \,{\mathrm e}^{1-x}}{3 \ln \relax (5)}+x -\frac {13 x}{3 \ln \relax (5)}\)	\(23\)
norman	\(\frac {4 x \,{\mathrm e}^{1-x}}{3 \ln \relax (5)}+\frac {\left (3 \ln \relax (5)-13\right ) x}{3 \ln \relax (5)}\)	\(28\)
default	\(\frac {-13 x -4 \,{\mathrm e}^{1-x} \left (1-x \right )+4 \,{\mathrm e}^{1-x}+3 x \ln \relax (5)}{3 \ln \relax (5)}\)	\(37\)
derivativedivides	\(-\frac {-13+13 x +4 \,{\mathrm e}^{1-x} \left (1-x \right )-4 \,{\mathrm e}^{1-x}+3 \left (1-x \right ) \ln \relax (5)}{3 \ln \relax (5)}\)	\(42\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/3*((-4*x+4)*exp(1-x)+3*ln(5)-13)/ln(5),x,method=_RETURNVERBOSE)

[Out]

4/3*x/ln(5)*exp(1-x)+x-13/3*x/ln(5)

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maxima [A]  time = 0.37, size = 24, normalized size = 0.75 \begin {gather*} \frac {4 \, x e^{\left (-x + 1\right )} + 3 \, x \log \relax (5) - 13 \, x}{3 \, \log \relax (5)} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/3*((-4*x+4)*exp(-x+1)+3*log(5)-13)/log(5),x, algorithm="maxima")

[Out]

1/3*(4*x*e^(-x + 1) + 3*x*log(5) - 13*x)/log(5)

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mupad [B]  time = 4.08, size = 22, normalized size = 0.69 \begin {gather*} \frac {4\,x\,{\mathrm {e}}^{1-x}+x\,\left (\ln \left (125\right )-13\right )}{3\,\ln \relax (5)} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(-((exp(1 - x)*(4*x - 4))/3 - log(5) + 13/3)/log(5),x)

[Out]

(4*x*exp(1 - x) + x*(log(125) - 13))/(3*log(5))

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sympy [A]  time = 0.11, size = 26, normalized size = 0.81 \begin {gather*} \frac {4 x e^{1 - x}}{3 \log {\relax (5 )}} + \frac {x \left (-13 + 3 \log {\relax (5 )}\right )}{3 \log {\relax (5 )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/3*((-4*x+4)*exp(-x+1)+3*ln(5)-13)/ln(5),x)

[Out]

4*x*exp(1 - x)/(3*log(5)) + x*(-13 + 3*log(5))/(3*log(5))

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