3.44.91 \(\int \frac {-180-72 x+9 x^2+e^2 (120+48 x-6 x^2)+e^4 (-20-8 x+x^2)+e^{-8+2 x} (-20-8 x+x^2)+e^{-4+x} (120+48 x-131 x^2-50 x^3-5 x^4+e^2 (-40-16 x+2 x^2))}{225 x^2+90 x^3+9 x^4+e^2 (-150 x^2-60 x^3-6 x^4)+e^4 (25 x^2+10 x^3+x^4)+e^{-8+2 x} (25 x^2+10 x^3+x^4)+e^{-4+x} (-150 x^2-60 x^3-6 x^4+e^2 (50 x^2+20 x^3+2 x^4))} \, dx\)

Optimal. Leaf size=29 \[ 1+\frac {5}{-3+e^2+e^{-4+x}}-\frac {-4+x}{x (5+x)} \]

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Rubi [A]  time = 1.79, antiderivative size = 40, normalized size of antiderivative = 1.38, number of steps used = 12, number of rules used = 8, integrand size = 213, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.038, Rules used = {6688, 6742, 2282, 44, 36, 31, 29, 893} \begin {gather*} \frac {4}{5 x}-\frac {9}{5 (x+5)}+\frac {5 e^4}{e^x-e^4 \left (3-e^2\right )} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(-180 - 72*x + 9*x^2 + E^2*(120 + 48*x - 6*x^2) + E^4*(-20 - 8*x + x^2) + E^(-8 + 2*x)*(-20 - 8*x + x^2) +
E^(-4 + x)*(120 + 48*x - 131*x^2 - 50*x^3 - 5*x^4 + E^2*(-40 - 16*x + 2*x^2)))/(225*x^2 + 90*x^3 + 9*x^4 + E^
2*(-150*x^2 - 60*x^3 - 6*x^4) + E^4*(25*x^2 + 10*x^3 + x^4) + E^(-8 + 2*x)*(25*x^2 + 10*x^3 + x^4) + E^(-4 + x
)*(-150*x^2 - 60*x^3 - 6*x^4 + E^2*(50*x^2 + 20*x^3 + 2*x^4))),x]

[Out]

(5*E^4)/(E^x - E^4*(3 - E^2)) + 4/(5*x) - 9/(5*(5 + x))

Rule 29

Int[(x_)^(-1), x_Symbol] :> Simp[Log[x], x]

Rule 31

Int[((a_) + (b_.)*(x_))^(-1), x_Symbol] :> Simp[Log[RemoveContent[a + b*x, x]]/b, x] /; FreeQ[{a, b}, x]

Rule 36

Int[1/(((a_.) + (b_.)*(x_))*((c_.) + (d_.)*(x_))), x_Symbol] :> Dist[b/(b*c - a*d), Int[1/(a + b*x), x], x] -
Dist[d/(b*c - a*d), Int[1/(c + d*x), x], x] /; FreeQ[{a, b, c, d}, x] && NeQ[b*c - a*d, 0]

Rule 44

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}, x] && NeQ[b*c - a*d, 0] && ILtQ[m, 0] && IntegerQ[n] &&  !(IGtQ[n, 0] && L
tQ[m + n + 2, 0])

Rule 893

Int[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))^(n_)*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :
> Int[ExpandIntegrand[(d + e*x)^m*(f + g*x)^n*(a + b*x + c*x^2)^p, x], x] /; FreeQ[{a, b, c, d, e, f, g}, x] &
& NeQ[e*f - d*g, 0] && NeQ[b^2 - 4*a*c, 0] && NeQ[c*d^2 - b*d*e + a*e^2, 0] && IntegerQ[p] && ((EqQ[p, 1] && I
ntegersQ[m, n]) || (ILtQ[m, 0] && ILtQ[n, 0]))

Rule 2282

Int[u_, x_Symbol] :> With[{v = FunctionOfExponential[u, x]}, Dist[v/D[v, x], Subst[Int[FunctionOfExponentialFu
nction[u, x]/x, x], x, v], x]] /; FunctionOfExponentialQ[u, x] &&  !MatchQ[u, (w_)*((a_.)*(v_)^(n_))^(m_) /; F
reeQ[{a, m, n}, x] && IntegerQ[m*n]] &&  !MatchQ[u, E^((c_.)*((a_.) + (b_.)*x))*(F_)[v_] /; FreeQ[{a, b, c}, x
] && InverseFunctionQ[F[x]]]

Rule 6688

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

Rule 6742

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

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\int \frac {e^{2 x} \left (-20-8 x+x^2\right )+2 e^{6+x} \left (-20-8 x+x^2\right )+9 e^8 \left (1+\frac {1}{9} e^2 \left (-6+e^2\right )\right ) \left (-20-8 x+x^2\right )+e^{4+x} \left (120+48 x-131 x^2-50 x^3-5 x^4\right )}{\left (e^x-3 e^4 \left (1-\frac {e^2}{3}\right )\right )^2 x^2 (5+x)^2} \, dx\\ &=\int \left (\frac {5 e^8 \left (-3+e^2\right )}{\left (e^x-3 e^4 \left (1-\frac {e^2}{3}\right )\right )^2}+\frac {5 e^4}{-e^x+3 e^4 \left (1-\frac {e^2}{3}\right )}+\frac {-20-8 x+x^2}{x^2 (5+x)^2}\right ) \, dx\\ &=\left (5 e^4\right ) \int \frac {1}{-e^x+3 e^4 \left (1-\frac {e^2}{3}\right )} \, dx-\left (5 e^8 \left (3-e^2\right )\right ) \int \frac {1}{\left (e^x-3 e^4 \left (1-\frac {e^2}{3}\right )\right )^2} \, dx+\int \frac {-20-8 x+x^2}{x^2 (5+x)^2} \, dx\\ &=\left (5 e^4\right ) \operatorname {Subst}\left (\int \frac {1}{\left (3 e^4-e^6-x\right ) x} \, dx,x,e^x\right )-\left (5 e^8 \left (3-e^2\right )\right ) \operatorname {Subst}\left (\int \frac {1}{\left (3 e^4-e^6-x\right )^2 x} \, dx,x,e^x\right )+\int \left (-\frac {4}{5 x^2}+\frac {9}{5 (5+x)^2}\right ) \, dx\\ &=\frac {4}{5 x}-\frac {9}{5 (5+x)}+\frac {5 \operatorname {Subst}\left (\int \frac {1}{3 e^4-e^6-x} \, dx,x,e^x\right )}{3-e^2}+\frac {5 \operatorname {Subst}\left (\int \frac {1}{x} \, dx,x,e^x\right )}{3-e^2}-\left (5 e^8 \left (3-e^2\right )\right ) \operatorname {Subst}\left (\int \left (\frac {1}{e^8 \left (-3+e^2\right )^2 x}-\frac {1}{e^4 \left (-3+e^2\right ) \left (-3 e^4+e^6+x\right )^2}-\frac {1}{e^8 \left (-3+e^2\right )^2 \left (-3 e^4+e^6+x\right )}\right ) \, dx,x,e^x\right )\\ &=\frac {5 e^4}{e^x-e^4 \left (3-e^2\right )}+\frac {4}{5 x}-\frac {9}{5 (5+x)}\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.08, size = 36, normalized size = 1.24 \begin {gather*} \frac {5 e^4}{-3 e^4+e^6+e^x}+\frac {4}{5 x}-\frac {9}{5 (5+x)} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(-180 - 72*x + 9*x^2 + E^2*(120 + 48*x - 6*x^2) + E^4*(-20 - 8*x + x^2) + E^(-8 + 2*x)*(-20 - 8*x +
x^2) + E^(-4 + x)*(120 + 48*x - 131*x^2 - 50*x^3 - 5*x^4 + E^2*(-40 - 16*x + 2*x^2)))/(225*x^2 + 90*x^3 + 9*x^
4 + E^2*(-150*x^2 - 60*x^3 - 6*x^4) + E^4*(25*x^2 + 10*x^3 + x^4) + E^(-8 + 2*x)*(25*x^2 + 10*x^3 + x^4) + E^(
-4 + x)*(-150*x^2 - 60*x^3 - 6*x^4 + E^2*(50*x^2 + 20*x^3 + 2*x^4))),x]

[Out]

(5*E^4)/(-3*E^4 + E^6 + E^x) + 4/(5*x) - 9/(5*(5 + x))

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fricas [B]  time = 0.64, size = 63, normalized size = 2.17 \begin {gather*} -\frac {5 \, x^{2} - {\left (x - 4\right )} e^{2} - {\left (x - 4\right )} e^{\left (x - 4\right )} + 28 \, x - 12}{3 \, x^{2} - {\left (x^{2} + 5 \, x\right )} e^{2} - {\left (x^{2} + 5 \, x\right )} e^{\left (x - 4\right )} + 15 \, x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((x^2-8*x-20)*exp(x-4)^2+((2*x^2-16*x-40)*exp(2)-5*x^4-50*x^3-131*x^2+48*x+120)*exp(x-4)+(x^2-8*x-20
)*exp(2)^2+(-6*x^2+48*x+120)*exp(2)+9*x^2-72*x-180)/((x^4+10*x^3+25*x^2)*exp(x-4)^2+((2*x^4+20*x^3+50*x^2)*exp
(2)-6*x^4-60*x^3-150*x^2)*exp(x-4)+(x^4+10*x^3+25*x^2)*exp(2)^2+(-6*x^4-60*x^3-150*x^2)*exp(2)+9*x^4+90*x^3+22
5*x^2),x, algorithm="fricas")

[Out]

-(5*x^2 - (x - 4)*e^2 - (x - 4)*e^(x - 4) + 28*x - 12)/(3*x^2 - (x^2 + 5*x)*e^2 - (x^2 + 5*x)*e^(x - 4) + 15*x
)

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giac [B]  time = 0.25, size = 73, normalized size = 2.52 \begin {gather*} \frac {5 \, x^{2} e^{4} - x e^{6} + 28 \, x e^{4} - x e^{x} + 4 \, e^{6} - 12 \, e^{4} + 4 \, e^{x}}{x^{2} e^{6} - 3 \, x^{2} e^{4} + x^{2} e^{x} + 5 \, x e^{6} - 15 \, x e^{4} + 5 \, x e^{x}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((x^2-8*x-20)*exp(x-4)^2+((2*x^2-16*x-40)*exp(2)-5*x^4-50*x^3-131*x^2+48*x+120)*exp(x-4)+(x^2-8*x-20
)*exp(2)^2+(-6*x^2+48*x+120)*exp(2)+9*x^2-72*x-180)/((x^4+10*x^3+25*x^2)*exp(x-4)^2+((2*x^4+20*x^3+50*x^2)*exp
(2)-6*x^4-60*x^3-150*x^2)*exp(x-4)+(x^4+10*x^3+25*x^2)*exp(2)^2+(-6*x^4-60*x^3-150*x^2)*exp(2)+9*x^4+90*x^3+22
5*x^2),x, algorithm="giac")

[Out]

(5*x^2*e^4 - x*e^6 + 28*x*e^4 - x*e^x + 4*e^6 - 12*e^4 + 4*e^x)/(x^2*e^6 - 3*x^2*e^4 + x^2*e^x + 5*x*e^6 - 15*
x*e^4 + 5*x*e^x)

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maple [A]  time = 0.77, size = 28, normalized size = 0.97

method	result	size
risch	\(\frac {-x +4}{\left (5+x \right ) x}+\frac {5}{{\mathrm e}^{x -4}+{\mathrm e}^{2}-3}\)	\(28\)
norman	\(\frac {5 x^{2}+\left (-{\mathrm e}^{2}+28\right ) x -x \,{\mathrm e}^{x -4}+4 \,{\mathrm e}^{x -4}+4 \,{\mathrm e}^{2}-12}{x \left (5+x \right ) \left ({\mathrm e}^{x -4}+{\mathrm e}^{2}-3\right )}\)	\(52\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((x^2-8*x-20)*exp(x-4)^2+((2*x^2-16*x-40)*exp(2)-5*x^4-50*x^3-131*x^2+48*x+120)*exp(x-4)+(x^2-8*x-20)*exp(
2)^2+(-6*x^2+48*x+120)*exp(2)+9*x^2-72*x-180)/((x^4+10*x^3+25*x^2)*exp(x-4)^2+((2*x^4+20*x^3+50*x^2)*exp(2)-6*
x^4-60*x^3-150*x^2)*exp(x-4)+(x^4+10*x^3+25*x^2)*exp(2)^2+(-6*x^4-60*x^3-150*x^2)*exp(2)+9*x^4+90*x^3+225*x^2)
,x,method=_RETURNVERBOSE)

[Out]

(-x+4)/(5+x)/x+5/(exp(x-4)+exp(2)-3)

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maxima [B]  time = 0.48, size = 68, normalized size = 2.34 \begin {gather*} \frac {5 \, x^{2} e^{4} - x {\left (e^{6} - 28 \, e^{4}\right )} - {\left (x - 4\right )} e^{x} + 4 \, e^{6} - 12 \, e^{4}}{x^{2} {\left (e^{6} - 3 \, e^{4}\right )} + 5 \, x {\left (e^{6} - 3 \, e^{4}\right )} + {\left (x^{2} + 5 \, x\right )} e^{x}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((x^2-8*x-20)*exp(x-4)^2+((2*x^2-16*x-40)*exp(2)-5*x^4-50*x^3-131*x^2+48*x+120)*exp(x-4)+(x^2-8*x-20
)*exp(2)^2+(-6*x^2+48*x+120)*exp(2)+9*x^2-72*x-180)/((x^4+10*x^3+25*x^2)*exp(x-4)^2+((2*x^4+20*x^3+50*x^2)*exp
(2)-6*x^4-60*x^3-150*x^2)*exp(x-4)+(x^4+10*x^3+25*x^2)*exp(2)^2+(-6*x^4-60*x^3-150*x^2)*exp(2)+9*x^4+90*x^3+22
5*x^2),x, algorithm="maxima")

[Out]

(5*x^2*e^4 - x*(e^6 - 28*e^4) - (x - 4)*e^x + 4*e^6 - 12*e^4)/(x^2*(e^6 - 3*e^4) + 5*x*(e^6 - 3*e^4) + (x^2 +
5*x)*e^x)

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mupad [B]  time = 0.37, size = 27, normalized size = 0.93 \begin {gather*} \frac {5}{{\mathrm {e}}^{x-4}+{\mathrm {e}}^2-3}-\frac {x-4}{x^2+5\,x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(-(72*x + exp(4)*(8*x - x^2 + 20) - exp(2)*(48*x - 6*x^2 + 120) + exp(2*x - 8)*(8*x - x^2 + 20) + exp(x - 4
)*(exp(2)*(16*x - 2*x^2 + 40) - 48*x + 131*x^2 + 50*x^3 + 5*x^4 - 120) - 9*x^2 + 180)/(exp(4)*(25*x^2 + 10*x^3
+ x^4) - exp(x - 4)*(150*x^2 - exp(2)*(50*x^2 + 20*x^3 + 2*x^4) + 60*x^3 + 6*x^4) + exp(2*x - 8)*(25*x^2 + 10
*x^3 + x^4) - exp(2)*(150*x^2 + 60*x^3 + 6*x^4) + 225*x^2 + 90*x^3 + 9*x^4),x)

[Out]

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

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sympy [A]  time = 0.29, size = 20, normalized size = 0.69 \begin {gather*} \frac {4 - x}{x^{2} + 5 x} + \frac {5}{e^{x - 4} - 3 + e^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((x**2-8*x-20)*exp(x-4)**2+((2*x**2-16*x-40)*exp(2)-5*x**4-50*x**3-131*x**2+48*x+120)*exp(x-4)+(x**2
-8*x-20)*exp(2)**2+(-6*x**2+48*x+120)*exp(2)+9*x**2-72*x-180)/((x**4+10*x**3+25*x**2)*exp(x-4)**2+((2*x**4+20*
x**3+50*x**2)*exp(2)-6*x**4-60*x**3-150*x**2)*exp(x-4)+(x**4+10*x**3+25*x**2)*exp(2)**2+(-6*x**4-60*x**3-150*x
**2)*exp(2)+9*x**4+90*x**3+225*x**2),x)

[Out]

(4 - x)/(x**2 + 5*x) + 5/(exp(x - 4) - 3 + exp(2))

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