\(\int \frac {e^{-5-x} (-1+x-2 x^2+x^3-(-1+x) \log (\frac {7}{4})+e^{5+x} (1-6 x+11 x^2-6 x^3+x^4-(2-6 x+2 x^2) \log (\frac {7}{4})+\log ^2(\frac {7}{4})))}{1-6 x+11 x^2-6 x^3+x^4-(2-6 x+2 x^2) \log (\frac {7}{4})+\log ^2(\frac {7}{4})} \, dx\) [4869]

Optimal result

Integrand size = 116, antiderivative size = 27 \[ \int \frac {e^{-5-x} \left (-1+x-2 x^2+x^3-(-1+x) \log \left (\frac {7}{4}\right )+e^{5+x} \left (1-6 x+11 x^2-6 x^3+x^4-\left (2-6 x+2 x^2\right ) \log \left (\frac {7}{4}\right )+\log ^2\left (\frac {7}{4}\right )\right )\right )}{1-6 x+11 x^2-6 x^3+x^4-\left (2-6 x+2 x^2\right ) \log \left (\frac {7}{4}\right )+\log ^2\left (\frac {7}{4}\right )} \, dx=x+\frac {e^{-5-x} x}{-1+3 x-x^2+\log \left (\frac {7}{4}\right )} \]

[Out]

x+x/(-1+3*x-x^2-ln(4/7))/exp(5+x)

Rubi [C] (verified)

Result contains higher order function than in optimal. Order 4 vs. order 3 in optimal.

Time = 5.65 (sec) , antiderivative size = 1653, normalized size of antiderivative = 61.22, number of steps used = 67, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.052, Rules used = {6820, 6874, 2300, 2209, 2208, 6860} \[ \int \frac {e^{-5-x} \left (-1+x-2 x^2+x^3-(-1+x) \log \left (\frac {7}{4}\right )+e^{5+x} \left (1-6 x+11 x^2-6 x^3+x^4-\left (2-6 x+2 x^2\right ) \log \left (\frac {7}{4}\right )+\log ^2\left (\frac {7}{4}\right )\right )\right )}{1-6 x+11 x^2-6 x^3+x^4-\left (2-6 x+2 x^2\right ) \log \left (\frac {7}{4}\right )+\log ^2\left (\frac {7}{4}\right )} \, dx =\text {Too large to display} \]

[In]

Int[(E^(-5 - x)*(-1 + x - 2*x^2 + x^3 - (-1 + x)*Log[7/4] + E^(5 + x)*(1 - 6*x + 11*x^2 - 6*x^3 + x^4 - (2 - 6
*x + 2*x^2)*Log[7/4] + Log[7/4]^2)))/(1 - 6*x + 11*x^2 - 6*x^3 + x^4 - (2 - 6*x + 2*x^2)*Log[7/4] + Log[7/4]^2
),x]

[Out]

x - (18*E^((-13 + Sqrt[5 + 4*Log[7/4]])/2)*ExpIntegralEi[(3 - 2*x - Sqrt[5 + 4*Log[7/4]])/2])/(5 + 4*Log[7/4])
^(3/2) + (18*E^((-13 - Sqrt[5 + 4*Log[7/4]])/2)*ExpIntegralEi[(3 - 2*x + Sqrt[5 + 4*Log[7/4]])/2])/(5 + 4*Log[
7/4])^(3/2) - (E^((-13 + Sqrt[5 + 4*Log[7/4]])/2)*ExpIntegralEi[(3 - 2*x - Sqrt[5 + 4*Log[7/4]])/2]*(1 - Log[7
/4]))/(5 + 4*Log[7/4])^(3/2) + (E^((-13 - Sqrt[5 + 4*Log[7/4]])/2)*ExpIntegralEi[(3 - 2*x + Sqrt[5 + 4*Log[7/4
]])/2]*(1 - Log[7/4]))/(5 + 4*Log[7/4])^(3/2) + (3*E^((-13 + Sqrt[5 + 4*Log[7/4]])/2)*ExpIntegralEi[(3 - 2*x -
Sqrt[5 + 4*Log[7/4]])/2]*(8 + Log[7/4]))/(5 + 4*Log[7/4])^(3/2) - (3*E^((-13 - Sqrt[5 + 4*Log[7/4]])/2)*ExpIn
tegralEi[(3 - 2*x + Sqrt[5 + 4*Log[7/4]])/2]*(8 + Log[7/4]))/(5 + 4*Log[7/4])^(3/2) + (2*E^((-13 + Sqrt[5 + 4*
Log[7/4]])/2)*ExpIntegralEi[(3 - 2*x - Sqrt[5 + 4*Log[7/4]])/2]*(1 - Log[7/4]))/(5 + 4*Log[7/4]) + (2*E^((-13
- Sqrt[5 + 4*Log[7/4]])/2)*ExpIntegralEi[(3 - 2*x + Sqrt[5 + 4*Log[7/4]])/2]*(1 - Log[7/4]))/(5 + 4*Log[7/4])
+ (2*E^((-13 + Sqrt[5 + 4*Log[7/4]])/2)*ExpIntegralEi[(3 - 2*x - Sqrt[5 + 4*Log[7/4]])/2])/Sqrt[5 + 4*Log[7/4]
] - (2*E^((-13 - Sqrt[5 + 4*Log[7/4]])/2)*ExpIntegralEi[(3 - 2*x + Sqrt[5 + 4*Log[7/4]])/2])/Sqrt[5 + 4*Log[7/
4]] + (E^((-13 + Sqrt[5 + 4*Log[7/4]])/2)*ExpIntegralEi[(3 - 2*x - Sqrt[5 + 4*Log[7/4]])/2]*(1 - 9/Sqrt[5 + 4*
Log[7/4]]))/2 + (E^((-13 - Sqrt[5 + 4*Log[7/4]])/2)*ExpIntegralEi[(3 - 2*x + Sqrt[5 + 4*Log[7/4]])/2]*(1 + 9/S
qrt[5 + 4*Log[7/4]]))/2 + (3*E^((-13 + Sqrt[5 + 4*Log[7/4]])/2)*ExpIntegralEi[(3 - 2*x - Sqrt[5 + 4*Log[7/4]])
/2]*(3 - Sqrt[5 + 4*Log[7/4]]))/(5 + 4*Log[7/4]) - (E^((-13 + Sqrt[5 + 4*Log[7/4]])/2)*ExpIntegralEi[(3 - 2*x
- Sqrt[5 + 4*Log[7/4]])/2]*(1 - Log[7/4])*(3 - Sqrt[5 + 4*Log[7/4]]))/(2*(5 + 4*Log[7/4])) - (E^((-13 + Sqrt[5
+ 4*Log[7/4]])/2)*ExpIntegralEi[(3 - 2*x - Sqrt[5 + 4*Log[7/4]])/2]*(8 + Log[7/4])*(3 - Sqrt[5 + 4*Log[7/4]])
)/(2*(5 + 4*Log[7/4])) - (4*E^(-5 - x)*(1 - Log[7/4]))/((5 + 4*Log[7/4])*(3 - 2*x - Sqrt[5 + 4*Log[7/4]])) - (
6*E^(-5 - x)*(3 - Sqrt[5 + 4*Log[7/4]]))/((5 + 4*Log[7/4])*(3 - 2*x - Sqrt[5 + 4*Log[7/4]])) + (E^(-5 - x)*(1
- Log[7/4])*(3 - Sqrt[5 + 4*Log[7/4]]))/((5 + 4*Log[7/4])*(3 - 2*x - Sqrt[5 + 4*Log[7/4]])) + (E^(-5 - x)*(8 +
Log[7/4])*(3 - Sqrt[5 + 4*Log[7/4]]))/((5 + 4*Log[7/4])*(3 - 2*x - Sqrt[5 + 4*Log[7/4]])) + (3*E^((-13 - Sqrt
[5 + 4*Log[7/4]])/2)*ExpIntegralEi[(3 - 2*x + Sqrt[5 + 4*Log[7/4]])/2]*(3 + Sqrt[5 + 4*Log[7/4]]))/(5 + 4*Log[
7/4]) - (E^((-13 - Sqrt[5 + 4*Log[7/4]])/2)*ExpIntegralEi[(3 - 2*x + Sqrt[5 + 4*Log[7/4]])/2]*(1 - Log[7/4])*(
3 + Sqrt[5 + 4*Log[7/4]]))/(2*(5 + 4*Log[7/4])) - (E^((-13 - Sqrt[5 + 4*Log[7/4]])/2)*ExpIntegralEi[(3 - 2*x +
Sqrt[5 + 4*Log[7/4]])/2]*(8 + Log[7/4])*(3 + Sqrt[5 + 4*Log[7/4]]))/(2*(5 + 4*Log[7/4])) - (4*E^(-5 - x)*(1 -
Log[7/4]))/((5 + 4*Log[7/4])*(3 - 2*x + Sqrt[5 + 4*Log[7/4]])) - (6*E^(-5 - x)*(3 + Sqrt[5 + 4*Log[7/4]]))/((
5 + 4*Log[7/4])*(3 - 2*x + Sqrt[5 + 4*Log[7/4]])) + (E^(-5 - x)*(1 - Log[7/4])*(3 + Sqrt[5 + 4*Log[7/4]]))/((5
+ 4*Log[7/4])*(3 - 2*x + Sqrt[5 + 4*Log[7/4]])) + (E^(-5 - x)*(8 + Log[7/4])*(3 + Sqrt[5 + 4*Log[7/4]]))/((5
+ 4*Log[7/4])*(3 - 2*x + Sqrt[5 + 4*Log[7/4]]))

Rule 2208

Int[((b_.)*(F_)^((g_.)*((e_.) + (f_.)*(x_))))^(n_.)*((c_.) + (d_.)*(x_))^(m_), x_Symbol] :> Simp[(c + d*x)^(m
+ 1)*((b*F^(g*(e + f*x)))^n/(d*(m + 1))), x] - Dist[f*g*n*(Log[F]/(d*(m + 1))), 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] && LtQ[m, -1] && IntegerQ[2*m] &&  !TrueQ[$UseGamm
a]

Rule 2209

Int[(F_)^((g_.)*((e_.) + (f_.)*(x_)))/((c_.) + (d_.)*(x_)), x_Symbol] :> Simp[(F^(g*(e - c*(f/d)))/d)*ExpInteg
ralEi[f*g*(c + d*x)*(Log[F]/d)], x] /; FreeQ[{F, c, d, e, f, g}, x] &&  !TrueQ[$UseGamma]

Rule 2300

Int[(F_)^((g_.)*((d_.) + (e_.)*(x_))^(n_.))/((a_.) + (b_.)*(x_) + (c_.)*(x_)^2), x_Symbol] :> Int[ExpandIntegr
and[F^(g*(d + e*x)^n), 1/(a + b*x + c*x^2), x], x] /; FreeQ[{F, a, b, c, d, e, g, n}, x]

Rule 6820

Int[u_, x_Symbol] :> With[{v = SimplifyIntegrand[u, x]}, Int[v, x] /; SimplerIntegrandQ[v, u, x]]

Rule 6860

Int[(u_)/((a_.) + (b_.)*(x_)^(n_.) + (c_.)*(x_)^(n2_.)), x_Symbol] :> With[{v = RationalFunctionExpand[u/(a +
b*x^n + c*x^(2*n)), x]}, Int[v, x] /; SumQ[v]] /; FreeQ[{a, b, c}, x] && EqQ[n2, 2*n] && IGtQ[n, 0]

Rule 6874

Int[u_, x_Symbol] :> With[{v = ExpandIntegrand[u, x]}, Int[v, x] /; SumQ[v]]

Rubi steps \begin{align*} \text {integral}& = \int \frac {e^{-5-x} \left (-1-2 x^2+x^3+x \left (1-\log \left (\frac {7}{4}\right )\right )+e^{5+x} \left (1-3 x+x^2-\log \left (\frac {7}{4}\right )\right )^2+\log \left (\frac {7}{4}\right )\right )}{\left (1-3 x+x^2-\log \left (\frac {7}{4}\right )\right )^2} \, dx \\ & = \int \left (1-\frac {2 e^{-5-x} x^2}{\left (1-3 x+x^2-\log \left (\frac {7}{4}\right )\right )^2}+\frac {e^{-5-x} x^3}{\left (1-3 x+x^2-\log \left (\frac {7}{4}\right )\right )^2}-\frac {e^{-5-x} \left (1-\log \left (\frac {7}{4}\right )\right )}{\left (1-3 x+x^2-\log \left (\frac {7}{4}\right )\right )^2}-\frac {e^{-5-x} x \left (-1+\log \left (\frac {7}{4}\right )\right )}{\left (1-3 x+x^2-\log \left (\frac {7}{4}\right )\right )^2}\right ) \, dx \\ & = x-2 \int \frac {e^{-5-x} x^2}{\left (1-3 x+x^2-\log \left (\frac {7}{4}\right )\right )^2} \, dx+\left (1-\log \left (\frac {7}{4}\right )\right ) \int \frac {e^{-5-x} x}{\left (1-3 x+x^2-\log \left (\frac {7}{4}\right )\right )^2} \, dx+\left (-1+\log \left (\frac {7}{4}\right )\right ) \int \frac {e^{-5-x}}{\left (1-3 x+x^2-\log \left (\frac {7}{4}\right )\right )^2} \, dx+\int \frac {e^{-5-x} x^3}{\left (1-3 x+x^2-\log \left (\frac {7}{4}\right )\right )^2} \, dx \\ & = x-2 \int \left (\frac {e^{-5-x}}{1-3 x+x^2-\log \left (\frac {7}{4}\right )}+\frac {e^{-5-x} \left (-1+3 x+\log \left (\frac {7}{4}\right )\right )}{\left (1-3 x+x^2-\log \left (\frac {7}{4}\right )\right )^2}\right ) \, dx+\left (1-\log \left (\frac {7}{4}\right )\right ) \int \left (\frac {2 e^{-5-x} \left (3+\sqrt {5+4 \log \left (\frac {7}{4}\right )}\right )}{\left (5+4 \log \left (\frac {7}{4}\right )\right ) \left (3-2 x+\sqrt {5+4 \log \left (\frac {7}{4}\right )}\right )^2}+\frac {6 e^{-5-x}}{\left (5+4 \log \left (\frac {7}{4}\right )\right )^{3/2} \left (3-2 x+\sqrt {5+4 \log \left (\frac {7}{4}\right )}\right )}+\frac {2 e^{-5-x} \left (3-\sqrt {5+4 \log \left (\frac {7}{4}\right )}\right )}{\left (5+4 \log \left (\frac {7}{4}\right )\right ) \left (-3+2 x+\sqrt {5+4 \log \left (\frac {7}{4}\right )}\right )^2}+\frac {6 e^{-5-x}}{\left (5+4 \log \left (\frac {7}{4}\right )\right )^{3/2} \left (-3+2 x+\sqrt {5+4 \log \left (\frac {7}{4}\right )}\right )}\right ) \, dx+\left (-1+\log \left (\frac {7}{4}\right )\right ) \int \left (\frac {4 e^{-5-x}}{\left (5+4 \log \left (\frac {7}{4}\right )\right ) \left (3-2 x+\sqrt {5+4 \log \left (\frac {7}{4}\right )}\right )^2}+\frac {4 e^{-5-x}}{\left (5+4 \log \left (\frac {7}{4}\right )\right )^{3/2} \left (3-2 x+\sqrt {5+4 \log \left (\frac {7}{4}\right )}\right )}+\frac {4 e^{-5-x}}{\left (5+4 \log \left (\frac {7}{4}\right )\right ) \left (-3+2 x+\sqrt {5+4 \log \left (\frac {7}{4}\right )}\right )^2}+\frac {4 e^{-5-x}}{\left (5+4 \log \left (\frac {7}{4}\right )\right )^{3/2} \left (-3+2 x+\sqrt {5+4 \log \left (\frac {7}{4}\right )}\right )}\right ) \, dx+\int \left (\frac {e^{-5-x} (3+x)}{1-3 x+x^2-\log \left (\frac {7}{4}\right )}+\frac {e^{-5-x} \left (-3 \left (1-\log \left (\frac {7}{4}\right )\right )+x \left (8+\log \left (\frac {7}{4}\right )\right )\right )}{\left (1-3 x+x^2-\log \left (\frac {7}{4}\right )\right )^2}\right ) \, dx \\ & = x-2 \int \frac {e^{-5-x}}{1-3 x+x^2-\log \left (\frac {7}{4}\right )} \, dx-2 \int \frac {e^{-5-x} \left (-1+3 x+\log \left (\frac {7}{4}\right )\right )}{\left (1-3 x+x^2-\log \left (\frac {7}{4}\right )\right )^2} \, dx-\frac {\left (4 \left (1-\log \left (\frac {7}{4}\right )\right )\right ) \int \frac {e^{-5-x}}{3-2 x+\sqrt {5+4 \log \left (\frac {7}{4}\right )}} \, dx}{\left (5+4 \log \left (\frac {7}{4}\right )\right )^{3/2}}-\frac {\left (4 \left (1-\log \left (\frac {7}{4}\right )\right )\right ) \int \frac {e^{-5-x}}{-3+2 x+\sqrt {5+4 \log \left (\frac {7}{4}\right )}} \, dx}{\left (5+4 \log \left (\frac {7}{4}\right )\right )^{3/2}}+\frac {\left (6 \left (1-\log \left (\frac {7}{4}\right )\right )\right ) \int \frac {e^{-5-x}}{3-2 x+\sqrt {5+4 \log \left (\frac {7}{4}\right )}} \, dx}{\left (5+4 \log \left (\frac {7}{4}\right )\right )^{3/2}}+\frac {\left (6 \left (1-\log \left (\frac {7}{4}\right )\right )\right ) \int \frac {e^{-5-x}}{-3+2 x+\sqrt {5+4 \log \left (\frac {7}{4}\right )}} \, dx}{\left (5+4 \log \left (\frac {7}{4}\right )\right )^{3/2}}-\frac {\left (4 \left (1-\log \left (\frac {7}{4}\right )\right )\right ) \int \frac {e^{-5-x}}{\left (3-2 x+\sqrt {5+4 \log \left (\frac {7}{4}\right )}\right )^2} \, dx}{5+4 \log \left (\frac {7}{4}\right )}-\frac {\left (4 \left (1-\log \left (\frac {7}{4}\right )\right )\right ) \int \frac {e^{-5-x}}{\left (-3+2 x+\sqrt {5+4 \log \left (\frac {7}{4}\right )}\right )^2} \, dx}{5+4 \log \left (\frac {7}{4}\right )}+\frac {\left (2 \left (1-\log \left (\frac {7}{4}\right )\right ) \left (3-\sqrt {5+4 \log \left (\frac {7}{4}\right )}\right )\right ) \int \frac {e^{-5-x}}{\left (-3+2 x+\sqrt {5+4 \log \left (\frac {7}{4}\right )}\right )^2} \, dx}{5+4 \log \left (\frac {7}{4}\right )}+\frac {\left (2 \left (1-\log \left (\frac {7}{4}\right )\right ) \left (3+\sqrt {5+4 \log \left (\frac {7}{4}\right )}\right )\right ) \int \frac {e^{-5-x}}{\left (3-2 x+\sqrt {5+4 \log \left (\frac {7}{4}\right )}\right )^2} \, dx}{5+4 \log \left (\frac {7}{4}\right )}+\int \frac {e^{-5-x} (3+x)}{1-3 x+x^2-\log \left (\frac {7}{4}\right )} \, dx+\int \frac {e^{-5-x} \left (-3 \left (1-\log \left (\frac {7}{4}\right )\right )+x \left (8+\log \left (\frac {7}{4}\right )\right )\right )}{\left (1-3 x+x^2-\log \left (\frac {7}{4}\right )\right )^2} \, dx \\ & = \text {Too large to display} \\ \end{align*}

Mathematica [A] (verified)

Time = 5.07 (sec) , antiderivative size = 28, normalized size of antiderivative = 1.04 \[ \int \frac {e^{-5-x} \left (-1+x-2 x^2+x^3-(-1+x) \log \left (\frac {7}{4}\right )+e^{5+x} \left (1-6 x+11 x^2-6 x^3+x^4-\left (2-6 x+2 x^2\right ) \log \left (\frac {7}{4}\right )+\log ^2\left (\frac {7}{4}\right )\right )\right )}{1-6 x+11 x^2-6 x^3+x^4-\left (2-6 x+2 x^2\right ) \log \left (\frac {7}{4}\right )+\log ^2\left (\frac {7}{4}\right )} \, dx=x-\frac {e^{-5-x} x}{1-3 x+x^2-\log \left (\frac {7}{4}\right )} \]

[In]

Integrate[(E^(-5 - x)*(-1 + x - 2*x^2 + x^3 - (-1 + x)*Log[7/4] + E^(5 + x)*(1 - 6*x + 11*x^2 - 6*x^3 + x^4 -
(2 - 6*x + 2*x^2)*Log[7/4] + Log[7/4]^2)))/(1 - 6*x + 11*x^2 - 6*x^3 + x^4 - (2 - 6*x + 2*x^2)*Log[7/4] + Log[
7/4]^2),x]

[Out]

x - (E^(-5 - x)*x)/(1 - 3*x + x^2 - Log[7/4])

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(60\) vs. \(2(26)=52\).

Time = 0.38 (sec) , antiderivative size = 61, normalized size of antiderivative = 2.26

method	result	size
parallelrisch	\(\frac {\left (x^{3} {\mathrm e}^{5+x}+\ln \left (\frac {4}{7}\right ) {\mathrm e}^{5+x} x +3 \ln \left (\frac {4}{7}\right ) {\mathrm e}^{5+x}-8 x \,{\mathrm e}^{5+x}-x +3 \,{\mathrm e}^{5+x}\right ) {\mathrm e}^{-x -5}}{x^{2}+\ln \left (\frac {4}{7}\right )-3 x +1}\)	\(61\)
norman	\(\frac {\left (x^{3} {\mathrm e}^{5+x}+\left (-3 \ln \left (7\right )+6 \ln \left (2\right )+3\right ) {\mathrm e}^{5+x}+\left (-8+2 \ln \left (2\right )-\ln \left (7\right )\right ) x \,{\mathrm e}^{5+x}-x \right ) {\mathrm e}^{-x -5}}{x^{2}+\ln \left (\frac {4}{7}\right )-3 x +1}\)	\(63\)
parts	\(x +\frac {4 \ln \left (7\right ) {\mathrm e}^{-x -5}}{\left (4 \ln \left (7\right )-8 \ln \left (2\right )+5\right ) \left (-\left (5+x \right )^{2}+\ln \left (7\right )-2 \ln \left (2\right )+24+13 x \right )}+\frac {96 \,{\mathrm e}^{-x -5} \left (-17+13 x +2 \ln \left (7\right )-4 \ln \left (2\right )\right )}{\left (4 \ln \left (7\right )-8 \ln \left (2\right )+5\right ) \left (-\left (5+x \right )^{2}+\ln \left (7\right )-2 \ln \left (2\right )+24+13 x \right )}-\frac {8 \ln \left (2\right ) {\mathrm e}^{-x -5}}{\left (4 \ln \left (7\right )-8 \ln \left (2\right )+5\right ) \left (-\left (5+x \right )^{2}+\ln \left (7\right )-2 \ln \left (2\right )+24+13 x \right )}-\frac {2 \ln \left (7\right )^{2} {\mathrm e}^{-x -5}}{\left (4 \ln \left (7\right )-8 \ln \left (2\right )+5\right ) \left (-\left (5+x \right )^{2}+\ln \left (7\right )-2 \ln \left (2\right )+24+13 x \right )}-\frac {\ln \left (7\right ) {\mathrm e}^{-x -5} \left (5+x \right )}{\left (4 \ln \left (7\right )-8 \ln \left (2\right )+5\right ) \left (-\left (5+x \right )^{2}+\ln \left (7\right )-2 \ln \left (2\right )+24+13 x \right )}+\frac {8 \ln \left (2\right ) \ln \left (7\right ) {\mathrm e}^{-x -5}}{\left (4 \ln \left (7\right )-8 \ln \left (2\right )+5\right ) \left (-\left (5+x \right )^{2}+\ln \left (7\right )-2 \ln \left (2\right )+24+13 x \right )}-\frac {17 \,{\mathrm e}^{-x -5} \left (2 \ln \left (7\right ) \left (5+x \right )-4 \ln \left (2\right ) \left (5+x \right )+13 \ln \left (7\right )-26 \ln \left (2\right )-98+87 x \right )}{\left (4 \ln \left (7\right )-8 \ln \left (2\right )+5\right ) \left (-\left (5+x \right )^{2}+\ln \left (7\right )-2 \ln \left (2\right )+24+13 x \right )}+\frac {{\mathrm e}^{-x -5} \left (2 \ln \left (7\right )^{2}-8 \ln \left (2\right ) \ln \left (7\right )+39 \ln \left (7\right ) \left (5+x \right )+8 \ln \left (2\right )^{2}-78 \ln \left (2\right ) \left (5+x \right )+5 \ln \left (7\right )-10 \ln \left (2\right )-577+598 x \right )}{\left (4 \ln \left (7\right )-8 \ln \left (2\right )+5\right ) \left (-\left (5+x \right )^{2}+\ln \left (7\right )-2 \ln \left (2\right )+24+13 x \right )}-\frac {181 \,{\mathrm e}^{-x -5} \left (-3+2 x \right )}{\left (4 \ln \left (7\right )-8 \ln \left (2\right )+5\right ) \left (-\left (5+x \right )^{2}+\ln \left (7\right )-2 \ln \left (2\right )+24+13 x \right )}+\frac {2 \ln \left (2\right ) {\mathrm e}^{-x -5} \left (5+x \right )}{\left (4 \ln \left (7\right )-8 \ln \left (2\right )+5\right ) \left (-\left (5+x \right )^{2}+\ln \left (7\right )-2 \ln \left (2\right )+24+13 x \right )}-\frac {8 \ln \left (2\right )^{2} {\mathrm e}^{-x -5}}{\left (4 \ln \left (7\right )-8 \ln \left (2\right )+5\right ) \left (-\left (5+x \right )^{2}+\ln \left (7\right )-2 \ln \left (2\right )+24+13 x \right )}\)	\(558\)
derivativedivides	\(\text {Expression too large to display}\)	\(2086\)
default	\(\text {Expression too large to display}\)	\(2086\)

[In]

int(((ln(4/7)^2+(2*x^2-6*x+2)*ln(4/7)+x^4-6*x^3+11*x^2-6*x+1)*exp(5+x)+(-1+x)*ln(4/7)+x^3-2*x^2+x-1)/(ln(4/7)^
2+(2*x^2-6*x+2)*ln(4/7)+x^4-6*x^3+11*x^2-6*x+1)/exp(5+x),x,method=_RETURNVERBOSE)

[Out]

(x^3*exp(5+x)+ln(4/7)*exp(5+x)*x+3*ln(4/7)*exp(5+x)-8*x*exp(5+x)-x+3*exp(5+x))/exp(5+x)/(x^2+ln(4/7)-3*x+1)

Fricas [A] (verification not implemented)

none

Time = 0.25 (sec) , antiderivative size = 42, normalized size of antiderivative = 1.56 \[ \int \frac {e^{-5-x} \left (-1+x-2 x^2+x^3-(-1+x) \log \left (\frac {7}{4}\right )+e^{5+x} \left (1-6 x+11 x^2-6 x^3+x^4-\left (2-6 x+2 x^2\right ) \log \left (\frac {7}{4}\right )+\log ^2\left (\frac {7}{4}\right )\right )\right )}{1-6 x+11 x^2-6 x^3+x^4-\left (2-6 x+2 x^2\right ) \log \left (\frac {7}{4}\right )+\log ^2\left (\frac {7}{4}\right )} \, dx=\frac {{\left ({\left (x^{3} - 3 \, x^{2} + x \log \left (\frac {4}{7}\right ) + x\right )} e^{\left (x + 5\right )} - x\right )} e^{\left (-x - 5\right )}}{x^{2} - 3 \, x + \log \left (\frac {4}{7}\right ) + 1} \]

[In]

integrate(((log(4/7)^2+(2*x^2-6*x+2)*log(4/7)+x^4-6*x^3+11*x^2-6*x+1)*exp(5+x)+(-1+x)*log(4/7)+x^3-2*x^2+x-1)/
(log(4/7)^2+(2*x^2-6*x+2)*log(4/7)+x^4-6*x^3+11*x^2-6*x+1)/exp(5+x),x, algorithm="fricas")

[Out]

((x^3 - 3*x^2 + x*log(4/7) + x)*e^(x + 5) - x)*e^(-x - 5)/(x^2 - 3*x + log(4/7) + 1)

Sympy [A] (verification not implemented)

Time = 0.15 (sec) , antiderivative size = 26, normalized size of antiderivative = 0.96 \[ \int \frac {e^{-5-x} \left (-1+x-2 x^2+x^3-(-1+x) \log \left (\frac {7}{4}\right )+e^{5+x} \left (1-6 x+11 x^2-6 x^3+x^4-\left (2-6 x+2 x^2\right ) \log \left (\frac {7}{4}\right )+\log ^2\left (\frac {7}{4}\right )\right )\right )}{1-6 x+11 x^2-6 x^3+x^4-\left (2-6 x+2 x^2\right ) \log \left (\frac {7}{4}\right )+\log ^2\left (\frac {7}{4}\right )} \, dx=x - \frac {x e^{- x - 5}}{x^{2} - 3 x - \log {\left (7 \right )} + 1 + 2 \log {\left (2 \right )}} \]

[In]

integrate(((ln(4/7)**2+(2*x**2-6*x+2)*ln(4/7)+x**4-6*x**3+11*x**2-6*x+1)*exp(5+x)+(-1+x)*ln(4/7)+x**3-2*x**2+x
-1)/(ln(4/7)**2+(2*x**2-6*x+2)*ln(4/7)+x**4-6*x**3+11*x**2-6*x+1)/exp(5+x),x)

[Out]

x - x*exp(-x - 5)/(x**2 - 3*x - log(7) + 1 + 2*log(2))

Maxima [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 898 vs. \(2 (23) = 46\).

Time = 0.37 (sec) , antiderivative size = 898, normalized size of antiderivative = 33.26 \[ \int \frac {e^{-5-x} \left (-1+x-2 x^2+x^3-(-1+x) \log \left (\frac {7}{4}\right )+e^{5+x} \left (1-6 x+11 x^2-6 x^3+x^4-\left (2-6 x+2 x^2\right ) \log \left (\frac {7}{4}\right )+\log ^2\left (\frac {7}{4}\right )\right )\right )}{1-6 x+11 x^2-6 x^3+x^4-\left (2-6 x+2 x^2\right ) \log \left (\frac {7}{4}\right )+\log ^2\left (\frac {7}{4}\right )} \, dx=\text {Too large to display} \]

[In]

integrate(((log(4/7)^2+(2*x^2-6*x+2)*log(4/7)+x^4-6*x^3+11*x^2-6*x+1)*exp(5+x)+(-1+x)*log(4/7)+x^3-2*x^2+x-1)/
(log(4/7)^2+(2*x^2-6*x+2)*log(4/7)+x^4-6*x^3+11*x^2-6*x+1)/exp(5+x),x, algorithm="maxima")

[Out]

((2*x - 3)/(x^2*(4*log(4/7) - 5) - 3*x*(4*log(4/7) - 5) + 4*log(4/7)^2 - log(4/7) - 5) + 2*log((2*x - sqrt(-4*
log(4/7) + 5) - 3)/(2*x + sqrt(-4*log(4/7) + 5) - 3))/((4*log(4/7) - 5)*sqrt(-4*log(4/7) + 5)))*log(4/7)^2 + 2
*(2*(log(4/7) + 1)*log((2*x - sqrt(-4*log(4/7) + 5) - 3)/(2*x + sqrt(-4*log(4/7) + 5) - 3))/((4*log(4/7) - 5)*
sqrt(-4*log(4/7) + 5)) - (x*(2*log(4/7) - 7) + 3*log(4/7) + 3)/(x^2*(4*log(4/7) - 5) - 3*x*(4*log(4/7) - 5) +
4*log(4/7)^2 - log(4/7) - 5))*log(4/7) - 6*((3*x - 2*log(4/7) - 2)/(x^2*(4*log(4/7) - 5) - 3*x*(4*log(4/7) - 5
) + 4*log(4/7)^2 - log(4/7) - 5) + 3*log((2*x - sqrt(-4*log(4/7) + 5) - 3)/(2*x + sqrt(-4*log(4/7) + 5) - 3))/
((4*log(4/7) - 5)*sqrt(-4*log(4/7) + 5)))*log(4/7) + 2*((2*x - 3)/(x^2*(4*log(4/7) - 5) - 3*x*(4*log(4/7) - 5)
+ 4*log(4/7)^2 - log(4/7) - 5) + 2*log((2*x - sqrt(-4*log(4/7) + 5) - 3)/(2*x + sqrt(-4*log(4/7) + 5) - 3))/(
(4*log(4/7) - 5)*sqrt(-4*log(4/7) + 5)))*log(4/7) + x - x*e^(-x)/(x^2*e^5 - 3*x*e^5 - (log(7) - 2*log(2) - 1)*
e^5) - 3*(2*log(4/7)^2 - 14*log(4/7) + 11)*log((2*x - sqrt(-4*log(4/7) + 5) - 3)/(2*x + sqrt(-4*log(4/7) + 5)
- 3))/((4*log(4/7) - 5)*sqrt(-4*log(4/7) + 5)) - 27*(2*log(4/7) - 1)*log((2*x - sqrt(-4*log(4/7) + 5) - 3)/(2*
x + sqrt(-4*log(4/7) + 5) - 3))/((4*log(4/7) - 5)*sqrt(-4*log(4/7) + 5)) + 22*(log(4/7) + 1)*log((2*x - sqrt(-
4*log(4/7) + 5) - 3)/(2*x + sqrt(-4*log(4/7) + 5) - 3))/((4*log(4/7) - 5)*sqrt(-4*log(4/7) + 5)) + ((2*log(4/7
)^2 - 32*log(4/7) + 47)*x + 9*log(4/7)^2 - 9*log(4/7) - 18)/(x^2*(4*log(4/7) - 5) - 3*x*(4*log(4/7) - 5) + 4*l
og(4/7)^2 - log(4/7) - 5) - 11*(x*(2*log(4/7) - 7) + 3*log(4/7) + 3)/(x^2*(4*log(4/7) - 5) - 3*x*(4*log(4/7) -
5) + 4*log(4/7)^2 - log(4/7) - 5) + 6*(9*x*(log(4/7) - 2) - 2*log(4/7)^2 + 5*log(4/7) + 7)/(x^2*(4*log(4/7) -
5) - 3*x*(4*log(4/7) - 5) + 4*log(4/7)^2 - log(4/7) - 5) - 6*(3*x - 2*log(4/7) - 2)/(x^2*(4*log(4/7) - 5) - 3
*x*(4*log(4/7) - 5) + 4*log(4/7)^2 - log(4/7) - 5) + (2*x - 3)/(x^2*(4*log(4/7) - 5) - 3*x*(4*log(4/7) - 5) +
4*log(4/7)^2 - log(4/7) - 5) - 16*log((2*x - sqrt(-4*log(4/7) + 5) - 3)/(2*x + sqrt(-4*log(4/7) + 5) - 3))/((4
*log(4/7) - 5)*sqrt(-4*log(4/7) + 5))

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 53 vs. \(2 (23) = 46\).

Time = 0.31 (sec) , antiderivative size = 53, normalized size of antiderivative = 1.96 \[ \int \frac {e^{-5-x} \left (-1+x-2 x^2+x^3-(-1+x) \log \left (\frac {7}{4}\right )+e^{5+x} \left (1-6 x+11 x^2-6 x^3+x^4-\left (2-6 x+2 x^2\right ) \log \left (\frac {7}{4}\right )+\log ^2\left (\frac {7}{4}\right )\right )\right )}{1-6 x+11 x^2-6 x^3+x^4-\left (2-6 x+2 x^2\right ) \log \left (\frac {7}{4}\right )+\log ^2\left (\frac {7}{4}\right )} \, dx=\frac {x^{3} e^{5} - 3 \, x^{2} e^{5} + x e^{5} \log \left (\frac {4}{7}\right ) + x e^{5} - x e^{\left (-x\right )}}{x^{2} e^{5} - 3 \, x e^{5} + e^{5} \log \left (\frac {4}{7}\right ) + e^{5}} \]

[In]

integrate(((log(4/7)^2+(2*x^2-6*x+2)*log(4/7)+x^4-6*x^3+11*x^2-6*x+1)*exp(5+x)+(-1+x)*log(4/7)+x^3-2*x^2+x-1)/
(log(4/7)^2+(2*x^2-6*x+2)*log(4/7)+x^4-6*x^3+11*x^2-6*x+1)/exp(5+x),x, algorithm="giac")

[Out]

(x^3*e^5 - 3*x^2*e^5 + x*e^5*log(4/7) + x*e^5 - x*e^(-x))/(x^2*e^5 - 3*x*e^5 + e^5*log(4/7) + e^5)

Mupad [F(-1)]

Timed out. \[ \int \frac {e^{-5-x} \left (-1+x-2 x^2+x^3-(-1+x) \log \left (\frac {7}{4}\right )+e^{5+x} \left (1-6 x+11 x^2-6 x^3+x^4-\left (2-6 x+2 x^2\right ) \log \left (\frac {7}{4}\right )+\log ^2\left (\frac {7}{4}\right )\right )\right )}{1-6 x+11 x^2-6 x^3+x^4-\left (2-6 x+2 x^2\right ) \log \left (\frac {7}{4}\right )+\log ^2\left (\frac {7}{4}\right )} \, dx=\int \frac {{\mathrm {e}}^{-x-5}\,\left (x+{\mathrm {e}}^{x+5}\,\left (\ln \left (\frac {4}{7}\right )\,\left (2\,x^2-6\,x+2\right )-6\,x+{\ln \left (\frac {4}{7}\right )}^2+11\,x^2-6\,x^3+x^4+1\right )+\ln \left (\frac {4}{7}\right )\,\left (x-1\right )-2\,x^2+x^3-1\right )}{\ln \left (\frac {4}{7}\right )\,\left (2\,x^2-6\,x+2\right )-6\,x+{\ln \left (\frac {4}{7}\right )}^2+11\,x^2-6\,x^3+x^4+1} \,d x \]

[In]

int((exp(- x - 5)*(x + exp(x + 5)*(log(4/7)*(2*x^2 - 6*x + 2) - 6*x + log(4/7)^2 + 11*x^2 - 6*x^3 + x^4 + 1) +
log(4/7)*(x - 1) - 2*x^2 + x^3 - 1))/(log(4/7)*(2*x^2 - 6*x + 2) - 6*x + log(4/7)^2 + 11*x^2 - 6*x^3 + x^4 +
1),x)

[Out]

int((exp(- x - 5)*(x + exp(x + 5)*(log(4/7)*(2*x^2 - 6*x + 2) - 6*x + log(4/7)^2 + 11*x^2 - 6*x^3 + x^4 + 1) +
log(4/7)*(x - 1) - 2*x^2 + x^3 - 1))/(log(4/7)*(2*x^2 - 6*x + 2) - 6*x + log(4/7)^2 + 11*x^2 - 6*x^3 + x^4 +
1), x)