3.9.73 \(\int \frac {28+e^4 (-27+x)-x}{e^4 (-27+x)} \, dx\)

Optimal. Leaf size=23 \[ 4+\frac {1}{2} (-3+2 x)+\frac {-x+\log (-27+x)}{e^4} \]

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Rubi [A]  time = 0.02, antiderivative size = 20, normalized size of antiderivative = 0.87, number of steps used = 4, number of rules used = 3, integrand size = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.143, Rules used = {12, 186, 43} \begin {gather*} \left (1-\frac {1}{e^4}\right ) x+\frac {\log (27-x)}{e^4} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(28 + E^4*(-27 + x) - x)/(E^4*(-27 + x)),x]

[Out]

(1 - E^(-4))*x + Log[27 - x]/E^4

Rule 12

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

Rule 43

Int[((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_.), x_Symbol] :> Int[ExpandIntegrand[(a + b*x)^m*(c + d
*x)^n, x], x] /; FreeQ[{a, b, c, d, n}, x] && NeQ[b*c - a*d, 0] && IGtQ[m, 0] && ( !IntegerQ[n] || (EqQ[c, 0]
&& LeQ[7*m + 4*n + 4, 0]) || LtQ[9*m + 5*(n + 1), 0] || GtQ[m + n + 2, 0])

Rule 186

Int[(u_)^(m_.)*(v_)^(n_.), x_Symbol] :> Int[ExpandToSum[u, x]^m*ExpandToSum[v, x]^n, x] /; FreeQ[{m, n}, x] &&
LinearQ[{u, v}, x] &&  !LinearMatchQ[{u, v}, x]

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\frac {\int \frac {28+e^4 (-27+x)-x}{-27+x} \, dx}{e^4}\\ &=\frac {\int \frac {28-27 e^4-\left (1-e^4\right ) x}{-27+x} \, dx}{e^4}\\ &=\frac {\int \left (-1+e^4+\frac {1}{-27+x}\right ) \, dx}{e^4}\\ &=\left (1-\frac {1}{e^4}\right ) x+\frac {\log (27-x)}{e^4}\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.00, size = 18, normalized size = 0.78 \begin {gather*} \frac {\left (-1+e^4\right ) (-27+x)+\log (-27+x)}{e^4} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(28 + E^4*(-27 + x) - x)/(E^4*(-27 + x)),x]

[Out]

((-1 + E^4)*(-27 + x) + Log[-27 + x])/E^4

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fricas [A]  time = 0.59, size = 15, normalized size = 0.65 \begin {gather*} {\left (x e^{4} - x + \log \left (x - 27\right )\right )} e^{\left (-4\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((x-27)*exp(4)-x+28)/(x-27)/exp(4),x, algorithm="fricas")

[Out]

(x*e^4 - x + log(x - 27))*e^(-4)

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giac [A]  time = 0.34, size = 16, normalized size = 0.70 \begin {gather*} {\left (x e^{4} - x + \log \left ({\left | x - 27 \right |}\right )\right )} e^{\left (-4\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((x-27)*exp(4)-x+28)/(x-27)/exp(4),x, algorithm="giac")

[Out]

(x*e^4 - x + log(abs(x - 27)))*e^(-4)

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maple [A]  time = 0.21, size = 18, normalized size = 0.78

method	result	size
default	\({\mathrm e}^{-4} \left (x \,{\mathrm e}^{4}-x +\ln \left (x -27\right )\right )\)	\(18\)
risch	\({\mathrm e}^{-4} x \,{\mathrm e}^{4}-{\mathrm e}^{-4} x +{\mathrm e}^{-4} \ln \left (x -27\right )\)	\(20\)
norman	\(\left ({\mathrm e}^{4}-1\right ) {\mathrm e}^{-4} x +{\mathrm e}^{-4} \ln \left (x -27\right )\)	\(21\)
meijerg	\(28 \,{\mathrm e}^{-4} \ln \left (1-\frac {x}{27}\right )+729 \left (-\frac {{\mathrm e}^{4}}{27}+\frac {1}{27}\right ) {\mathrm e}^{-4} \left (-\frac {x}{27}-\ln \left (1-\frac {x}{27}\right )\right )-27 \ln \left (1-\frac {x}{27}\right )\)	\(42\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((x-27)*exp(4)-x+28)/(x-27)/exp(4),x,method=_RETURNVERBOSE)

[Out]

1/exp(4)*(x*exp(4)-x+ln(x-27))

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maxima [A]  time = 0.73, size = 14, normalized size = 0.61 \begin {gather*} {\left (x {\left (e^{4} - 1\right )} + \log \left (x - 27\right )\right )} e^{\left (-4\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((x-27)*exp(4)-x+28)/(x-27)/exp(4),x, algorithm="maxima")

[Out]

(x*(e^4 - 1) + log(x - 27))*e^(-4)

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mupad [B]  time = 0.63, size = 16, normalized size = 0.70 \begin {gather*} \ln \left (x-27\right )\,{\mathrm {e}}^{-4}+x\,{\mathrm {e}}^{-4}\,\left ({\mathrm {e}}^4-1\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((exp(-4)*(exp(4)*(x - 27) - x + 28))/(x - 27),x)

[Out]

log(x - 27)*exp(-4) + x*exp(-4)*(exp(4) - 1)

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sympy [A]  time = 0.11, size = 15, normalized size = 0.65 \begin {gather*} - x \left (-1 + e^{-4}\right ) + \frac {\log {\left (x - 27 \right )}}{e^{4}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((x-27)*exp(4)-x+28)/(x-27)/exp(4),x)

[Out]

-x*(-1 + exp(-4)) + exp(-4)*log(x - 27)

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