\(\int \frac {e^{-5-x} (-1+x-2 x^2+x^3-(-1+x) \log (\frac {7}{4})+e^{5+x} (1-6 x+11 x^2-6 x^3+x^4-(2-6 x+2 x^2) \log (\frac {7}{4})+\log ^2(\frac {7}{4})))}{1-6 x+11 x^2-6 x^3+x^4-(2-6 x+2 x^2) \log (\frac {7}{4})+\log ^2(\frac {7}{4})} \, dx\) [1869]

Optimal result

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

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

Mathematica [A] (verified)

Time = 5.04 (sec) , antiderivative size = 28, normalized size of antiderivative = 1.04 \[ \int \frac {e^{-5-x} \left (-1+x-2 x^2+x^3-(-1+x) \log \left (\frac {7}{4}\right )+e^{5+x} \left (1-6 x+11 x^2-6 x^3+x^4-\left (2-6 x+2 x^2\right ) \log \left (\frac {7}{4}\right )+\log ^2\left (\frac {7}{4}\right )\right )\right )}{1-6 x+11 x^2-6 x^3+x^4-\left (2-6 x+2 x^2\right ) \log \left (\frac {7}{4}\right )+\log ^2\left (\frac {7}{4}\right )} \, dx=x-\frac {e^{-5-x} x}{1-3 x+x^2-\log \left (\frac {7}{4}\right )} \] Input:

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]
 

Output:

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

Rubi [C] (verified)

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

Time = 15.84 (sec) , antiderivative size = 1653, normalized size of antiderivative = 61.22, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.034, Rules used = {2463, 7239, 7293, 2009}

Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.

\(\displaystyle \int \frac {e^{-x-5} \left (x^3-2 x^2+e^{x+5} \left (x^4-6 x^3+11 x^2-\left (2 x^2-6 x+2\right ) \log \left (\frac {7}{4}\right )-6 x+1+\log ^2\left (\frac {7}{4}\right )\right )+x-(x-1) \log \left (\frac {7}{4}\right )-1\right )}{x^4-6 x^3+11 x^2-\left (2 x^2-6 x+2\right ) \log \left (\frac {7}{4}\right )-6 x+1+\log ^2\left (\frac {7}{4}\right )} \, dx\)

\(\Big \downarrow \) 2463

\(\displaystyle \int \left (\frac {4 e^{-x-5} \left (x^3-2 x^2+e^{x+5} \left (x^4-6 x^3+11 x^2-\left (2 x^2-6 x+2\right ) \log \left (\frac {7}{4}\right )-6 x+1+\log ^2\left (\frac {7}{4}\right )\right )+x-(x-1) \log \left (\frac {7}{4}\right )-1\right )}{\left (5+4 \log \left (\frac {7}{4}\right )\right )^{3/2} \left (-2 x+3+\sqrt {5+4 \log \left (\frac {7}{4}\right )}\right )}+\frac {4 e^{-x-5} \left (x^3-2 x^2+e^{x+5} \left (x^4-6 x^3+11 x^2-\left (2 x^2-6 x+2\right ) \log \left (\frac {7}{4}\right )-6 x+1+\log ^2\left (\frac {7}{4}\right )\right )+x-(x-1) \log \left (\frac {7}{4}\right )-1\right )}{\left (5+4 \log \left (\frac {7}{4}\right )\right )^{3/2} \left (2 x-3+\sqrt {5+4 \log \left (\frac {7}{4}\right )}\right )}+\frac {4 e^{-x-5} \left (x^3-2 x^2+e^{x+5} \left (x^4-6 x^3+11 x^2-\left (2 x^2-6 x+2\right ) \log \left (\frac {7}{4}\right )-6 x+1+\log ^2\left (\frac {7}{4}\right )\right )+x-(x-1) \log \left (\frac {7}{4}\right )-1\right )}{\left (5+4 \log \left (\frac {7}{4}\right )\right ) \left (-2 x+3+\sqrt {5+4 \log \left (\frac {7}{4}\right )}\right )^2}+\frac {4 e^{-x-5} \left (x^3-2 x^2+e^{x+5} \left (x^4-6 x^3+11 x^2-\left (2 x^2-6 x+2\right ) \log \left (\frac {7}{4}\right )-6 x+1+\log ^2\left (\frac {7}{4}\right )\right )+x-(x-1) \log \left (\frac {7}{4}\right )-1\right )}{\left (5+4 \log \left (\frac {7}{4}\right )\right ) \left (2 x-3+\sqrt {5+4 \log \left (\frac {7}{4}\right )}\right )^2}\right )dx\)

\(\Big \downarrow \) 7239

\(\displaystyle \int \frac {e^{-x-5} \left (x^3-2 x^2+e^{x+5} \left (x^2-3 x+1-\log \left (\frac {7}{4}\right )\right )^2+x \left (1-\log \left (\frac {7}{4}\right )\right )-1+\log \left (\frac {7}{4}\right )\right )}{\left (x^2-3 x+1-\log \left (\frac {7}{4}\right )\right )^2}dx\)

\(\Big \downarrow \) 7293

\(\displaystyle \int \left (-\frac {2 e^{-x-5} x^2}{\left (x^2-3 x+1-\log \left (\frac {7}{4}\right )\right )^2}-\frac {e^{-x-5} x \left (\log \left (\frac {7}{4}\right )-1\right )}{\left (x^2-3 x+1-\log \left (\frac {7}{4}\right )\right )^2}-\frac {e^{-x-5} \left (1-\log \left (\frac {7}{4}\right )\right )}{\left (x^2-3 x+1-\log \left (\frac {7}{4}\right )\right )^2}+\frac {e^{-x-5} x^3}{\left (x^2-3 x+1-\log \left (\frac {7}{4}\right )\right )^2}+1\right )dx\)

\(\Big \downarrow \) 2009

\(\displaystyle x-\frac {6 e^{-x-5} \left (3-\sqrt {5+4 \log \left (\frac {7}{4}\right )}\right )}{\left (5+4 \log \left (\frac {7}{4}\right )\right ) \left (-2 x-\sqrt {5+4 \log \left (\frac {7}{4}\right )}+3\right )}+\frac {e^{-x-5} \left (8+\log \left (\frac {7}{4}\right )\right ) \left (3-\sqrt {5+4 \log \left (\frac {7}{4}\right )}\right )}{\left (5+4 \log \left (\frac {7}{4}\right )\right ) \left (-2 x-\sqrt {5+4 \log \left (\frac {7}{4}\right )}+3\right )}+\frac {e^{-x-5} \left (1-\log \left (\frac {7}{4}\right )\right ) \left (3-\sqrt {5+4 \log \left (\frac {7}{4}\right )}\right )}{\left (5+4 \log \left (\frac {7}{4}\right )\right ) \left (-2 x-\sqrt {5+4 \log \left (\frac {7}{4}\right )}+3\right )}-\frac {4 e^{-x-5} \left (1-\log \left (\frac {7}{4}\right )\right )}{\left (5+4 \log \left (\frac {7}{4}\right )\right ) \left (-2 x-\sqrt {5+4 \log \left (\frac {7}{4}\right )}+3\right )}-\frac {6 e^{-x-5} \left (3+\sqrt {5+4 \log \left (\frac {7}{4}\right )}\right )}{\left (5+4 \log \left (\frac {7}{4}\right )\right ) \left (-2 x+\sqrt {5+4 \log \left (\frac {7}{4}\right )}+3\right )}+\frac {e^{-x-5} \left (8+\log \left (\frac {7}{4}\right )\right ) \left (3+\sqrt {5+4 \log \left (\frac {7}{4}\right )}\right )}{\left (5+4 \log \left (\frac {7}{4}\right )\right ) \left (-2 x+\sqrt {5+4 \log \left (\frac {7}{4}\right )}+3\right )}+\frac {e^{-x-5} \left (1-\log \left (\frac {7}{4}\right )\right ) \left (3+\sqrt {5+4 \log \left (\frac {7}{4}\right )}\right )}{\left (5+4 \log \left (\frac {7}{4}\right )\right ) \left (-2 x+\sqrt {5+4 \log \left (\frac {7}{4}\right )}+3\right )}-\frac {4 e^{-x-5} \left (1-\log \left (\frac {7}{4}\right )\right )}{\left (5+4 \log \left (\frac {7}{4}\right )\right ) \left (-2 x+\sqrt {5+4 \log \left (\frac {7}{4}\right )}+3\right )}+\frac {3 e^{\frac {1}{2} \left (-13-\sqrt {5+4 \log \left (\frac {7}{4}\right )}\right )} \operatorname {ExpIntegralEi}\left (\frac {1}{2} \left (-2 x+\sqrt {5+4 \log \left (\frac {7}{4}\right )}+3\right )\right ) \left (3+\sqrt {5+4 \log \left (\frac {7}{4}\right )}\right )}{5+4 \log \left (\frac {7}{4}\right )}-\frac {e^{\frac {1}{2} \left (-13-\sqrt {5+4 \log \left (\frac {7}{4}\right )}\right )} \operatorname {ExpIntegralEi}\left (\frac {1}{2} \left (-2 x+\sqrt {5+4 \log \left (\frac {7}{4}\right )}+3\right )\right ) \left (8+\log \left (\frac {7}{4}\right )\right ) \left (3+\sqrt {5+4 \log \left (\frac {7}{4}\right )}\right )}{2 \left (5+4 \log \left (\frac {7}{4}\right )\right )}-\frac {e^{\frac {1}{2} \left (-13-\sqrt {5+4 \log \left (\frac {7}{4}\right )}\right )} \operatorname {ExpIntegralEi}\left (\frac {1}{2} \left (-2 x+\sqrt {5+4 \log \left (\frac {7}{4}\right )}+3\right )\right ) \left (1-\log \left (\frac {7}{4}\right )\right ) \left (3+\sqrt {5+4 \log \left (\frac {7}{4}\right )}\right )}{2 \left (5+4 \log \left (\frac {7}{4}\right )\right )}+\frac {3 e^{\frac {1}{2} \left (-13+\sqrt {5+4 \log \left (\frac {7}{4}\right )}\right )} \operatorname {ExpIntegralEi}\left (\frac {1}{2} \left (-2 x-\sqrt {5+4 \log \left (\frac {7}{4}\right )}+3\right )\right ) \left (3-\sqrt {5+4 \log \left (\frac {7}{4}\right )}\right )}{5+4 \log \left (\frac {7}{4}\right )}-\frac {e^{\frac {1}{2} \left (-13+\sqrt {5+4 \log \left (\frac {7}{4}\right )}\right )} \operatorname {ExpIntegralEi}\left (\frac {1}{2} \left (-2 x-\sqrt {5+4 \log \left (\frac {7}{4}\right )}+3\right )\right ) \left (8+\log \left (\frac {7}{4}\right )\right ) \left (3-\sqrt {5+4 \log \left (\frac {7}{4}\right )}\right )}{2 \left (5+4 \log \left (\frac {7}{4}\right )\right )}-\frac {e^{\frac {1}{2} \left (-13+\sqrt {5+4 \log \left (\frac {7}{4}\right )}\right )} \operatorname {ExpIntegralEi}\left (\frac {1}{2} \left (-2 x-\sqrt {5+4 \log \left (\frac {7}{4}\right )}+3\right )\right ) \left (1-\log \left (\frac {7}{4}\right )\right ) \left (3-\sqrt {5+4 \log \left (\frac {7}{4}\right )}\right )}{2 \left (5+4 \log \left (\frac {7}{4}\right )\right )}+\frac {1}{2} e^{\frac {1}{2} \left (-13-\sqrt {5+4 \log \left (\frac {7}{4}\right )}\right )} \operatorname {ExpIntegralEi}\left (\frac {1}{2} \left (-2 x+\sqrt {5+4 \log \left (\frac {7}{4}\right )}+3\right )\right ) \left (1+\frac {9}{\sqrt {5+4 \log \left (\frac {7}{4}\right )}}\right )+\frac {1}{2} e^{\frac {1}{2} \left (-13+\sqrt {5+4 \log \left (\frac {7}{4}\right )}\right )} \operatorname {ExpIntegralEi}\left (\frac {1}{2} \left (-2 x-\sqrt {5+4 \log \left (\frac {7}{4}\right )}+3\right )\right ) \left (1-\frac {9}{\sqrt {5+4 \log \left (\frac {7}{4}\right )}}\right )+\frac {2 e^{\frac {1}{2} \left (-13+\sqrt {5+4 \log \left (\frac {7}{4}\right )}\right )} \operatorname {ExpIntegralEi}\left (\frac {1}{2} \left (-2 x-\sqrt {5+4 \log \left (\frac {7}{4}\right )}+3\right )\right )}{\sqrt {5+4 \log \left (\frac {7}{4}\right )}}-\frac {2 e^{\frac {1}{2} \left (-13-\sqrt {5+4 \log \left (\frac {7}{4}\right )}\right )} \operatorname {ExpIntegralEi}\left (\frac {1}{2} \left (-2 x+\sqrt {5+4 \log \left (\frac {7}{4}\right )}+3\right )\right )}{\sqrt {5+4 \log \left (\frac {7}{4}\right )}}+\frac {2 e^{\frac {1}{2} \left (-13+\sqrt {5+4 \log \left (\frac {7}{4}\right )}\right )} \operatorname {ExpIntegralEi}\left (\frac {1}{2} \left (-2 x-\sqrt {5+4 \log \left (\frac {7}{4}\right )}+3\right )\right ) \left (1-\log \left (\frac {7}{4}\right )\right )}{5+4 \log \left (\frac {7}{4}\right )}+\frac {2 e^{\frac {1}{2} \left (-13-\sqrt {5+4 \log \left (\frac {7}{4}\right )}\right )} \operatorname {ExpIntegralEi}\left (\frac {1}{2} \left (-2 x+\sqrt {5+4 \log \left (\frac {7}{4}\right )}+3\right )\right ) \left (1-\log \left (\frac {7}{4}\right )\right )}{5+4 \log \left (\frac {7}{4}\right )}-\frac {18 e^{\frac {1}{2} \left (-13+\sqrt {5+4 \log \left (\frac {7}{4}\right )}\right )} \operatorname {ExpIntegralEi}\left (\frac {1}{2} \left (-2 x-\sqrt {5+4 \log \left (\frac {7}{4}\right )}+3\right )\right )}{\left (5+4 \log \left (\frac {7}{4}\right )\right )^{3/2}}+\frac {18 e^{\frac {1}{2} \left (-13-\sqrt {5+4 \log \left (\frac {7}{4}\right )}\right )} \operatorname {ExpIntegralEi}\left (\frac {1}{2} \left (-2 x+\sqrt {5+4 \log \left (\frac {7}{4}\right )}+3\right )\right )}{\left (5+4 \log \left (\frac {7}{4}\right )\right )^{3/2}}+\frac {3 e^{\frac {1}{2} \left (-13+\sqrt {5+4 \log \left (\frac {7}{4}\right )}\right )} \operatorname {ExpIntegralEi}\left (\frac {1}{2} \left (-2 x-\sqrt {5+4 \log \left (\frac {7}{4}\right )}+3\right )\right ) \left (8+\log \left (\frac {7}{4}\right )\right )}{\left (5+4 \log \left (\frac {7}{4}\right )\right )^{3/2}}-\frac {3 e^{\frac {1}{2} \left (-13-\sqrt {5+4 \log \left (\frac {7}{4}\right )}\right )} \operatorname {ExpIntegralEi}\left (\frac {1}{2} \left (-2 x+\sqrt {5+4 \log \left (\frac {7}{4}\right )}+3\right )\right ) \left (8+\log \left (\frac {7}{4}\right )\right )}{\left (5+4 \log \left (\frac {7}{4}\right )\right )^{3/2}}-\frac {e^{\frac {1}{2} \left (-13+\sqrt {5+4 \log \left (\frac {7}{4}\right )}\right )} \operatorname {ExpIntegralEi}\left (\frac {1}{2} \left (-2 x-\sqrt {5+4 \log \left (\frac {7}{4}\right )}+3\right )\right ) \left (1-\log \left (\frac {7}{4}\right )\right )}{\left (5+4 \log \left (\frac {7}{4}\right )\right )^{3/2}}+\frac {e^{\frac {1}{2} \left (-13-\sqrt {5+4 \log \left (\frac {7}{4}\right )}\right )} \operatorname {ExpIntegralEi}\left (\frac {1}{2} \left (-2 x+\sqrt {5+4 \log \left (\frac {7}{4}\right )}+3\right )\right ) \left (1-\log \left (\frac {7}{4}\right )\right )}{\left (5+4 \log \left (\frac {7}{4}\right )\right )^{3/2}}\)

Input:

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]
 

Output:

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 - S 
qrt[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)*ExpIntegral 
Ei[(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)*ExpInt 
egralEi[(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)*ExpInt 
egralEi[(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/Sqrt[5 + 4*Log[7/4]]))/2 + (3*E^((-13 + Sqrt[5 +...
 

Defintions of rubi rules used

rule 2009

Int[u_, x_Symbol] :> Simp[IntSum[u, x], x] /; SumQ[u]
 

rule 2463

Int[(u_.)*(Px_)^(p_), x_Symbol] :> With[{Qx = Factor[Px]}, Int[ExpandIntegr 
and[u, Qx^p, x], x] /;  !SumQ[NonfreeFactors[Qx, x]]] /; PolyQ[Px, x] && Gt 
Q[Expon[Px, x], 2] &&  !BinomialQ[Px, x] &&  !TrinomialQ[Px, x] && ILtQ[p, 
0]
 

rule 7239

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

rule 7293

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

Maple [A] (verified)

Time = 3.40 (sec) , antiderivative size = 30, normalized size of antiderivative = 1.11

method	result	size
risch	\(x -\frac {x \,{\mathrm e}^{-x -5}}{x^{2}+2 \ln \left (2\right )-\ln \left (7\right )-3 x +1}\)	\(30\)
parallelrisch	\(\frac {\left ({\mathrm e}^{5+x} x^{3}+\ln \left (\frac {4}{7}\right ) {\mathrm e}^{5+x} x +3 \ln \left (\frac {4}{7}\right ) {\mathrm e}^{5+x}-8 \,{\mathrm e}^{5+x} x -x +3 \,{\mathrm e}^{5+x}\right ) {\mathrm e}^{-x -5}}{x^{2}+\ln \left (\frac {4}{7}\right )-3 x +1}\)	\(61\)
norman	\(\frac {\left (\left (6 \ln \left (2\right )-3 \ln \left (7\right )+3\right ) {\mathrm e}^{5+x}+{\mathrm e}^{5+x} x^{3}+\left (-8+2 \ln \left (2\right )-\ln \left (7\right )\right ) x \,{\mathrm e}^{5+x}-x \right ) {\mathrm e}^{-x -5}}{x^{2}+\ln \left (\frac {4}{7}\right )-3 x +1}\)	\(63\)
parts	\(x -\frac {2 \ln \left (7\right )^{2} {\mathrm e}^{-x -5}}{\left (8 \ln \left (2\right )-4 \ln \left (7\right )-5\right ) \left (x^{2}+2 \ln \left (2\right )-\ln \left (7\right )-3 x +1\right )}-\frac {8 \ln \left (2\right )^{2} {\mathrm e}^{-x -5}}{\left (8 \ln \left (2\right )-4 \ln \left (7\right )-5\right ) \left (x^{2}+2 \ln \left (2\right )-\ln \left (7\right )-3 x +1\right )}-\frac {181 \,{\mathrm e}^{-x -5} \left (-3+2 x \right )}{\left (8 \ln \left (2\right )-4 \ln \left (7\right )-5\right ) \left (\left (5+x \right )^{2}+2 \ln \left (2\right )-\ln \left (7\right )-24-13 x \right )}-\frac {96 \,{\mathrm e}^{-x -5} \left (17-13 x +4 \ln \left (2\right )-2 \ln \left (7\right )\right )}{\left (8 \ln \left (2\right )-4 \ln \left (7\right )-5\right ) \left (\left (5+x \right )^{2}+2 \ln \left (2\right )-\ln \left (7\right )-24-13 x \right )}+\frac {17 \,{\mathrm e}^{-x -5} \left (4 \ln \left (2\right ) \left (5+x \right )-2 \ln \left (7\right ) \left (5+x \right )+26 \ln \left (2\right )-13 \ln \left (7\right )+98-87 x \right )}{\left (8 \ln \left (2\right )-4 \ln \left (7\right )-5\right ) \left (\left (5+x \right )^{2}+2 \ln \left (2\right )-\ln \left (7\right )-24-13 x \right )}-\frac {\ln \left (7\right ) {\mathrm e}^{-x -5}}{\left (8 \ln \left (2\right )-4 \ln \left (7\right )-5\right ) \left (x^{2}+2 \ln \left (2\right )-\ln \left (7\right )-3 x +1\right )}+\frac {{\mathrm e}^{-x -5} \left (8 \ln \left (2\right )^{2}-8 \ln \left (2\right ) \ln \left (7\right )-78 \ln \left (2\right ) \left (5+x \right )+2 \ln \left (7\right )^{2}+39 \ln \left (7\right ) \left (5+x \right )-10 \ln \left (2\right )+5 \ln \left (7\right )-577+598 x \right )}{\left (8 \ln \left (2\right )-4 \ln \left (7\right )-5\right ) \left (\left (5+x \right )^{2}+2 \ln \left (2\right )-\ln \left (7\right )-24-13 x \right )}+\frac {2 \ln \left (2\right ) {\mathrm e}^{-x -5}}{\left (8 \ln \left (2\right )-4 \ln \left (7\right )-5\right ) \left (x^{2}+2 \ln \left (2\right )-\ln \left (7\right )-3 x +1\right )}+\frac {2 \ln \left (2\right ) {\mathrm e}^{-x -5} x}{\left (8 \ln \left (2\right )-4 \ln \left (7\right )-5\right ) \left (x^{2}+2 \ln \left (2\right )-\ln \left (7\right )-3 x +1\right )}-\frac {\ln \left (7\right ) {\mathrm e}^{-x -5} x}{\left (8 \ln \left (2\right )-4 \ln \left (7\right )-5\right ) \left (x^{2}+2 \ln \left (2\right )-\ln \left (7\right )-3 x +1\right )}+\frac {8 \ln \left (2\right ) \ln \left (7\right ) {\mathrm e}^{-x -5}}{\left (8 \ln \left (2\right )-4 \ln \left (7\right )-5\right ) \left (x^{2}+2 \ln \left (2\right )-\ln \left (7\right )-3 x +1\right )}\)	\(540\)
derivativedivides	\(\text {Expression too large to display}\)	\(5114\)
default	\(\text {Expression too large to display}\)	\(5114\)

Input:

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)
 

Output:

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

Fricas [A] (verification not implemented)

Time = 0.10 (sec) , antiderivative size = 42, normalized size of antiderivative = 1.56 \[ \int \frac {e^{-5-x} \left (-1+x-2 x^2+x^3-(-1+x) \log \left (\frac {7}{4}\right )+e^{5+x} \left (1-6 x+11 x^2-6 x^3+x^4-\left (2-6 x+2 x^2\right ) \log \left (\frac {7}{4}\right )+\log ^2\left (\frac {7}{4}\right )\right )\right )}{1-6 x+11 x^2-6 x^3+x^4-\left (2-6 x+2 x^2\right ) \log \left (\frac {7}{4}\right )+\log ^2\left (\frac {7}{4}\right )} \, dx=\frac {{\left ({\left (x^{3} - 3 \, x^{2} + x \log \left (\frac {4}{7}\right ) + x\right )} e^{\left (x + 5\right )} - x\right )} e^{\left (-x - 5\right )}}{x^{2} - 3 \, x + \log \left (\frac {4}{7}\right ) + 1} \] Input:

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")
 

Output:

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

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)*e 
xp(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)
 

Output:

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 = 0.20 (sec) , antiderivative size = 898, normalized size of antiderivative = 33.26 \[ \int \frac {e^{-5-x} \left (-1+x-2 x^2+x^3-(-1+x) \log \left (\frac {7}{4}\right )+e^{5+x} \left (1-6 x+11 x^2-6 x^3+x^4-\left (2-6 x+2 x^2\right ) \log \left (\frac {7}{4}\right )+\log ^2\left (\frac {7}{4}\right )\right )\right )}{1-6 x+11 x^2-6 x^3+x^4-\left (2-6 x+2 x^2\right ) \log \left (\frac {7}{4}\right )+\log ^2\left (\frac {7}{4}\right )} \, dx=\text {Too large to display} \] Input:

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")
 

Output:

((2*x - 3)/(x^2*(4*log(4/7) - 5) - 3*x*(4*log(4/7) - 5) + 4*log(4/7)^2 - l 
og(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*lo 
g(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*lo 
g(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*lo...
 

Giac [B] (verification not implemented)

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

Time = 0.13 (sec) , antiderivative size = 53, normalized size of antiderivative = 1.96 \[ \int \frac {e^{-5-x} \left (-1+x-2 x^2+x^3-(-1+x) \log \left (\frac {7}{4}\right )+e^{5+x} \left (1-6 x+11 x^2-6 x^3+x^4-\left (2-6 x+2 x^2\right ) \log \left (\frac {7}{4}\right )+\log ^2\left (\frac {7}{4}\right )\right )\right )}{1-6 x+11 x^2-6 x^3+x^4-\left (2-6 x+2 x^2\right ) \log \left (\frac {7}{4}\right )+\log ^2\left (\frac {7}{4}\right )} \, dx=\frac {x^{3} e^{5} - 3 \, x^{2} e^{5} + x e^{5} \log \left (\frac {4}{7}\right ) + x e^{5} - x e^{\left (-x\right )}}{x^{2} e^{5} - 3 \, x e^{5} + e^{5} \log \left (\frac {4}{7}\right ) + e^{5}} \] Input:

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")
 

Output:

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

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)
 

Output:

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)
 

Reduce [B] (verification not implemented)

Time = 0.18 (sec) , antiderivative size = 59, normalized size of antiderivative = 2.19 \[ \int \frac {e^{-5-x} \left (-1+x-2 x^2+x^3-(-1+x) \log \left (\frac {7}{4}\right )+e^{5+x} \left (1-6 x+11 x^2-6 x^3+x^4-\left (2-6 x+2 x^2\right ) \log \left (\frac {7}{4}\right )+\log ^2\left (\frac {7}{4}\right )\right )\right )}{1-6 x+11 x^2-6 x^3+x^4-\left (2-6 x+2 x^2\right ) \log \left (\frac {7}{4}\right )+\log ^2\left (\frac {7}{4}\right )} \, dx=\frac {x \left (e^{x} \mathrm {log}\left (\frac {4}{7}\right ) e^{5}+e^{x} e^{5} x^{2}-3 e^{x} e^{5} x +e^{x} e^{5}-1\right )}{e^{x} e^{5} \left (\mathrm {log}\left (\frac {4}{7}\right )+x^{2}-3 x +1\right )} \] Input:

int(((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)
 

Output:

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