3.93.35 \(\int \frac {1600+4 e^4-2240 x+480 x^2-112 x^3+20 x^4}{7840000-2240000 x+1728000 x^2-448000 x^3+149600 x^4-33600 x^5+6320 x^6-1120 x^7+129 x^8-14 x^9+x^{10}+e^8 (49-14 x+x^2)+e^4 (39200-11200 x+4720 x^2-1120 x^3+178 x^4-28 x^5+2 x^6)} \, dx\)

Optimal. Leaf size=22 \[ 2-\frac {4}{(-7+x) \left (e^4+\left (20+x^2\right )^2\right )} \]

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Rubi [C]  time = 1.25, antiderivative size = 259, normalized size of antiderivative = 11.77, number of steps used = 31, number of rules used = 12, integrand size = 122, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.098, Rules used = {2074, 1673, 1178, 1169, 634, 618, 204, 628, 1247, 638, 12, 1107} \begin {gather*} \frac {4 x \left (x^2+89\right )}{\left (4761+e^4\right ) \left (x^4+40 x^2+e^4+400\right )}+\frac {28 \left (x^2+89\right )}{\left (4761+e^4\right ) \left (x^4+40 x^2+e^4+400\right )}+\frac {4}{\left (4761+e^4\right ) (7-x)}-\frac {\sqrt {2} \left (\sqrt {\sqrt {400+e^4}-20}+187 \sqrt {\frac {\sqrt {400+e^4}-20}{400+e^4}}\right ) \tan ^{-1}\left (\frac {2 x+\sqrt {2 \left (\sqrt {400+e^4}-20\right )}}{\sqrt {2 \left (20+\sqrt {400+e^4}\right )}}\right )}{e^2 \left (4761+e^4\right )}+\frac {\sqrt {\frac {2}{\left (400+e^4\right ) \left (20+\sqrt {400+e^4}\right )}} \left (187+\sqrt {400+e^4}\right ) \tan ^{-1}\left (\frac {2 x+\sqrt {2 \left (\sqrt {400+e^4}-20\right )}}{\sqrt {2 \left (20+\sqrt {400+e^4}\right )}}\right )}{4761+e^4} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(1600 + 4*E^4 - 2240*x + 480*x^2 - 112*x^3 + 20*x^4)/(7840000 - 2240000*x + 1728000*x^2 - 448000*x^3 + 149
600*x^4 - 33600*x^5 + 6320*x^6 - 1120*x^7 + 129*x^8 - 14*x^9 + x^10 + E^8*(49 - 14*x + x^2) + E^4*(39200 - 112
00*x + 4720*x^2 - 1120*x^3 + 178*x^4 - 28*x^5 + 2*x^6)),x]

[Out]

4/((4761 + E^4)*(7 - x)) + (28*(89 + x^2))/((4761 + E^4)*(400 + E^4 + 40*x^2 + x^4)) + (4*x*(89 + x^2))/((4761
+ E^4)*(400 + E^4 + 40*x^2 + x^4)) + (Sqrt[2/((400 + E^4)*(20 + Sqrt[400 + E^4]))]*(187 + Sqrt[400 + E^4])*Ar
cTan[(Sqrt[2*(-20 + Sqrt[400 + E^4])] + 2*x)/Sqrt[2*(20 + Sqrt[400 + E^4])]])/(4761 + E^4) - (Sqrt[2]*(Sqrt[-2
0 + Sqrt[400 + E^4]] + 187*Sqrt[(-20 + Sqrt[400 + E^4])/(400 + E^4)])*ArcTan[(Sqrt[2*(-20 + Sqrt[400 + E^4])]
+ 2*x)/Sqrt[2*(20 + Sqrt[400 + E^4])]])/(E^2*(4761 + 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 204

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> -Simp[ArcTan[(Rt[-b, 2]*x)/Rt[-a, 2]]/(Rt[-a, 2]*Rt[-b, 2]), x] /
; FreeQ[{a, b}, x] && PosQ[a/b] && (LtQ[a, 0] || LtQ[b, 0])

Rule 618

Int[((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(-1), x_Symbol] :> Dist[-2, Subst[Int[1/Simp[b^2 - 4*a*c - x^2, x], x]
, x, b + 2*c*x], x] /; FreeQ[{a, b, c}, x] && NeQ[b^2 - 4*a*c, 0]

Rule 628

Int[((d_) + (e_.)*(x_))/((a_.) + (b_.)*(x_) + (c_.)*(x_)^2), x_Symbol] :> Simp[(d*Log[RemoveContent[a + b*x +
c*x^2, x]])/b, x] /; FreeQ[{a, b, c, d, e}, x] && EqQ[2*c*d - b*e, 0]

Rule 634

Int[((d_.) + (e_.)*(x_))/((a_) + (b_.)*(x_) + (c_.)*(x_)^2), x_Symbol] :> Dist[(2*c*d - b*e)/(2*c), Int[1/(a +
b*x + c*x^2), x], x] + Dist[e/(2*c), Int[(b + 2*c*x)/(a + b*x + c*x^2), x], x] /; FreeQ[{a, b, c, d, e}, x] &
& NeQ[2*c*d - b*e, 0] && NeQ[b^2 - 4*a*c, 0] &&  !NiceSqrtQ[b^2 - 4*a*c]

Rule 638

Int[((d_.) + (e_.)*(x_))*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> Simp[((b*d - 2*a*e + (2*c*d -
b*e)*x)*(a + b*x + c*x^2)^(p + 1))/((p + 1)*(b^2 - 4*a*c)), x] - Dist[((2*p + 3)*(2*c*d - b*e))/((p + 1)*(b^2
- 4*a*c)), Int[(a + b*x + c*x^2)^(p + 1), x], x] /; FreeQ[{a, b, c, d, e}, x] && NeQ[2*c*d - b*e, 0] && NeQ[b^
2 - 4*a*c, 0] && LtQ[p, -1] && NeQ[p, -3/2]

Rule 1107

Int[(x_)*((a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4)^(p_.), x_Symbol] :> Dist[1/2, Subst[Int[(a + b*x + c*x^2)^p, x],
x, x^2], x] /; FreeQ[{a, b, c, p}, x]

Rule 1169

Int[((d_) + (e_.)*(x_)^2)/((a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4), x_Symbol] :> With[{q = Rt[a/c, 2]}, With[{r =
Rt[2*q - b/c, 2]}, Dist[1/(2*c*q*r), Int[(d*r - (d - e*q)*x)/(q - r*x + x^2), x], x] + Dist[1/(2*c*q*r), Int[(
d*r + (d - e*q)*x)/(q + r*x + x^2), x], x]]] /; FreeQ[{a, b, c, d, e}, x] && NeQ[b^2 - 4*a*c, 0] && NeQ[c*d^2
- b*d*e + a*e^2, 0] && NegQ[b^2 - 4*a*c]

Rule 1178

Int[((d_) + (e_.)*(x_)^2)*((a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4)^(p_), x_Symbol] :> Simp[(x*(a*b*e - d*(b^2 - 2*
a*c) - c*(b*d - 2*a*e)*x^2)*(a + b*x^2 + c*x^4)^(p + 1))/(2*a*(p + 1)*(b^2 - 4*a*c)), x] + Dist[1/(2*a*(p + 1)
*(b^2 - 4*a*c)), Int[Simp[(2*p + 3)*d*b^2 - a*b*e - 2*a*c*d*(4*p + 5) + (4*p + 7)*(d*b - 2*a*e)*c*x^2, x]*(a +
b*x^2 + c*x^4)^(p + 1), x], x] /; FreeQ[{a, b, c, d, e}, x] && NeQ[b^2 - 4*a*c, 0] && NeQ[c*d^2 - b*d*e + a*e
^2, 0] && LtQ[p, -1] && IntegerQ[2*p]

Rule 1247

Int[(x_)*((d_) + (e_.)*(x_)^2)^(q_.)*((a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4)^(p_.), x_Symbol] :> Dist[1/2, Subst[
Int[(d + e*x)^q*(a + b*x + c*x^2)^p, x], x, x^2], x] /; FreeQ[{a, b, c, d, e, p, q}, x]

Rule 1673

Int[(Pq_)*((a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4)^(p_), x_Symbol] :> Module[{q = Expon[Pq, x], k}, Int[Sum[Coeff[
Pq, x, 2*k]*x^(2*k), {k, 0, q/2}]*(a + b*x^2 + c*x^4)^p, x] + Int[x*Sum[Coeff[Pq, x, 2*k + 1]*x^(2*k), {k, 0,
(q - 1)/2}]*(a + b*x^2 + c*x^4)^p, x]] /; FreeQ[{a, b, c, p}, x] && PolyQ[Pq, x] &&  !PolyQ[Pq, x^2]

Rule 2074

Int[(P_)^(p_)*(Q_)^(q_.), x_Symbol] :> With[{PP = Factor[P]}, Int[ExpandIntegrand[PP^p*Q^q, x], x] /;  !SumQ[N
onfreeFactors[PP, x]]] /; FreeQ[q, x] && PolyQ[P, x] && PolyQ[Q, x] && IntegerQ[p] && NeQ[P, x]

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\int \left (\frac {4}{\left (4761+e^4\right ) (-7+x)^2}+\frac {16 \left (69 \left (400+e^4\right )-7 \left (1380-e^4\right ) x+\left (1380+e^4\right ) x^2-483 x^3\right )}{\left (4761+e^4\right ) \left (400+e^4+40 x^2+x^4\right )^2}-\frac {4 \left (187+14 x+x^2\right )}{\left (4761+e^4\right ) \left (400+e^4+40 x^2+x^4\right )}\right ) \, dx\\ &=\frac {4}{\left (4761+e^4\right ) (7-x)}-\frac {4 \int \frac {187+14 x+x^2}{400+e^4+40 x^2+x^4} \, dx}{4761+e^4}+\frac {16 \int \frac {69 \left (400+e^4\right )-7 \left (1380-e^4\right ) x+\left (1380+e^4\right ) x^2-483 x^3}{\left (400+e^4+40 x^2+x^4\right )^2} \, dx}{4761+e^4}\\ &=\frac {4}{\left (4761+e^4\right ) (7-x)}-\frac {4 \int \frac {14 x}{400+e^4+40 x^2+x^4} \, dx}{4761+e^4}-\frac {4 \int \frac {187+x^2}{400+e^4+40 x^2+x^4} \, dx}{4761+e^4}+\frac {16 \int \frac {x \left (-7 \left (1380-e^4\right )-483 x^2\right )}{\left (400+e^4+40 x^2+x^4\right )^2} \, dx}{4761+e^4}+\frac {16 \int \frac {69 \left (400+e^4\right )+\left (1380+e^4\right ) x^2}{\left (400+e^4+40 x^2+x^4\right )^2} \, dx}{4761+e^4}\\ &=\frac {4}{\left (4761+e^4\right ) (7-x)}+\frac {4 x \left (89+x^2\right )}{\left (4761+e^4\right ) \left (400+e^4+40 x^2+x^4\right )}+\frac {8 \operatorname {Subst}\left (\int \frac {-7 \left (1380-e^4\right )-483 x}{\left (400+e^4+40 x+x^2\right )^2} \, dx,x,x^2\right )}{4761+e^4}-\frac {56 \int \frac {x}{400+e^4+40 x^2+x^4} \, dx}{4761+e^4}+\frac {2 \int \frac {374 e^4 \left (400+e^4\right )+2 e^4 \left (400+e^4\right ) x^2}{400+e^4+40 x^2+x^4} \, dx}{e^4 \left (400+e^4\right ) \left (4761+e^4\right )}-\frac {\sqrt {\frac {2}{\left (400+e^4\right ) \left (-20+\sqrt {400+e^4}\right )}} \int \frac {187 \sqrt {2 \left (-20+\sqrt {400+e^4}\right )}-\left (187-\sqrt {400+e^4}\right ) x}{\sqrt {400+e^4}-\sqrt {2 \left (-20+\sqrt {400+e^4}\right )} x+x^2} \, dx}{4761+e^4}-\frac {\sqrt {\frac {2}{\left (400+e^4\right ) \left (-20+\sqrt {400+e^4}\right )}} \int \frac {187 \sqrt {2 \left (-20+\sqrt {400+e^4}\right )}+\left (187-\sqrt {400+e^4}\right ) x}{\sqrt {400+e^4}+\sqrt {2 \left (-20+\sqrt {400+e^4}\right )} x+x^2} \, dx}{4761+e^4}\\ &=\frac {4}{\left (4761+e^4\right ) (7-x)}+\frac {28 \left (89+x^2\right )}{\left (4761+e^4\right ) \left (400+e^4+40 x^2+x^4\right )}+\frac {4 x \left (89+x^2\right )}{\left (4761+e^4\right ) \left (400+e^4+40 x^2+x^4\right )}-\frac {\left (1+\frac {187}{\sqrt {400+e^4}}\right ) \int \frac {1}{\sqrt {400+e^4}-\sqrt {2 \left (-20+\sqrt {400+e^4}\right )} x+x^2} \, dx}{4761+e^4}-\frac {\left (1+\frac {187}{\sqrt {400+e^4}}\right ) \int \frac {1}{\sqrt {400+e^4}+\sqrt {2 \left (-20+\sqrt {400+e^4}\right )} x+x^2} \, dx}{4761+e^4}+\frac {\int \frac {374 e^4 \left (400+e^4\right ) \sqrt {2 \left (-20+\sqrt {400+e^4}\right )}-\left (374 e^4 \left (400+e^4\right )-2 e^4 \left (400+e^4\right )^{3/2}\right ) x}{\sqrt {400+e^4}-\sqrt {2 \left (-20+\sqrt {400+e^4}\right )} x+x^2} \, dx}{e^4 \left (400+e^4\right )^{3/2} \left (4761+e^4\right ) \sqrt {2 \left (-20+\sqrt {400+e^4}\right )}}+\frac {\int \frac {374 e^4 \left (400+e^4\right ) \sqrt {2 \left (-20+\sqrt {400+e^4}\right )}+\left (374 e^4 \left (400+e^4\right )-2 e^4 \left (400+e^4\right )^{3/2}\right ) x}{\sqrt {400+e^4}+\sqrt {2 \left (-20+\sqrt {400+e^4}\right )} x+x^2} \, dx}{e^4 \left (400+e^4\right )^{3/2} \left (4761+e^4\right ) \sqrt {2 \left (-20+\sqrt {400+e^4}\right )}}+\frac {\left (187-\sqrt {400+e^4}\right ) \int \frac {-\sqrt {2 \left (-20+\sqrt {400+e^4}\right )}+2 x}{\sqrt {400+e^4}-\sqrt {2 \left (-20+\sqrt {400+e^4}\right )} x+x^2} \, dx}{\left (4761+e^4\right ) \sqrt {2 \left (400+e^4\right ) \left (-20+\sqrt {400+e^4}\right )}}-\frac {\left (187-\sqrt {400+e^4}\right ) \int \frac {\sqrt {2 \left (-20+\sqrt {400+e^4}\right )}+2 x}{\sqrt {400+e^4}+\sqrt {2 \left (-20+\sqrt {400+e^4}\right )} x+x^2} \, dx}{\left (4761+e^4\right ) \sqrt {2 \left (400+e^4\right ) \left (-20+\sqrt {400+e^4}\right )}}\\ &=\frac {4}{\left (4761+e^4\right ) (7-x)}+\frac {28 \left (89+x^2\right )}{\left (4761+e^4\right ) \left (400+e^4+40 x^2+x^4\right )}+\frac {4 x \left (89+x^2\right )}{\left (4761+e^4\right ) \left (400+e^4+40 x^2+x^4\right )}+\frac {\left (187-\sqrt {400+e^4}\right ) \log \left (\sqrt {400+e^4}-\sqrt {2 \left (-20+\sqrt {400+e^4}\right )} x+x^2\right )}{\left (4761+e^4\right ) \sqrt {2 \left (400+e^4\right ) \left (-20+\sqrt {400+e^4}\right )}}-\frac {\left (187-\sqrt {400+e^4}\right ) \log \left (\sqrt {400+e^4}+\sqrt {2 \left (-20+\sqrt {400+e^4}\right )} x+x^2\right )}{\left (4761+e^4\right ) \sqrt {2 \left (400+e^4\right ) \left (-20+\sqrt {400+e^4}\right )}}+\frac {\left (1+\frac {187}{\sqrt {400+e^4}}\right ) \int \frac {1}{\sqrt {400+e^4}-\sqrt {2 \left (-20+\sqrt {400+e^4}\right )} x+x^2} \, dx}{4761+e^4}+\frac {\left (1+\frac {187}{\sqrt {400+e^4}}\right ) \int \frac {1}{\sqrt {400+e^4}+\sqrt {2 \left (-20+\sqrt {400+e^4}\right )} x+x^2} \, dx}{4761+e^4}+\frac {\left (2 \left (1+\frac {187}{\sqrt {400+e^4}}\right )\right ) \operatorname {Subst}\left (\int \frac {1}{-2 \left (20+\sqrt {400+e^4}\right )-x^2} \, dx,x,-\sqrt {2 \left (-20+\sqrt {400+e^4}\right )}+2 x\right )}{4761+e^4}+\frac {\left (2 \left (1+\frac {187}{\sqrt {400+e^4}}\right )\right ) \operatorname {Subst}\left (\int \frac {1}{-2 \left (20+\sqrt {400+e^4}\right )-x^2} \, dx,x,\sqrt {2 \left (-20+\sqrt {400+e^4}\right )}+2 x\right )}{4761+e^4}-\frac {\left (187-\sqrt {400+e^4}\right ) \int \frac {-\sqrt {2 \left (-20+\sqrt {400+e^4}\right )}+2 x}{\sqrt {400+e^4}-\sqrt {2 \left (-20+\sqrt {400+e^4}\right )} x+x^2} \, dx}{\left (4761+e^4\right ) \sqrt {2 \left (400+e^4\right ) \left (-20+\sqrt {400+e^4}\right )}}+\frac {\left (187-\sqrt {400+e^4}\right ) \int \frac {\sqrt {2 \left (-20+\sqrt {400+e^4}\right )}+2 x}{\sqrt {400+e^4}+\sqrt {2 \left (-20+\sqrt {400+e^4}\right )} x+x^2} \, dx}{\left (4761+e^4\right ) \sqrt {2 \left (400+e^4\right ) \left (-20+\sqrt {400+e^4}\right )}}\\ &=\frac {4}{\left (4761+e^4\right ) (7-x)}+\frac {28 \left (89+x^2\right )}{\left (4761+e^4\right ) \left (400+e^4+40 x^2+x^4\right )}+\frac {4 x \left (89+x^2\right )}{\left (4761+e^4\right ) \left (400+e^4+40 x^2+x^4\right )}+\frac {\left (1+\frac {187}{\sqrt {400+e^4}}\right ) \sqrt {\frac {2}{20+\sqrt {400+e^4}}} \tan ^{-1}\left (\frac {\sqrt {2 \left (-20+\sqrt {400+e^4}\right )}-2 x}{\sqrt {2 \left (20+\sqrt {400+e^4}\right )}}\right )}{4761+e^4}-\frac {\left (1+\frac {187}{\sqrt {400+e^4}}\right ) \sqrt {\frac {2}{20+\sqrt {400+e^4}}} \tan ^{-1}\left (\frac {\sqrt {2 \left (-20+\sqrt {400+e^4}\right )}+2 x}{\sqrt {2 \left (20+\sqrt {400+e^4}\right )}}\right )}{4761+e^4}-\frac {\left (2 \left (1+\frac {187}{\sqrt {400+e^4}}\right )\right ) \operatorname {Subst}\left (\int \frac {1}{-2 \left (20+\sqrt {400+e^4}\right )-x^2} \, dx,x,-\sqrt {2 \left (-20+\sqrt {400+e^4}\right )}+2 x\right )}{4761+e^4}-\frac {\left (2 \left (1+\frac {187}{\sqrt {400+e^4}}\right )\right ) \operatorname {Subst}\left (\int \frac {1}{-2 \left (20+\sqrt {400+e^4}\right )-x^2} \, dx,x,\sqrt {2 \left (-20+\sqrt {400+e^4}\right )}+2 x\right )}{4761+e^4}\\ &=\frac {4}{\left (4761+e^4\right ) (7-x)}+\frac {28 \left (89+x^2\right )}{\left (4761+e^4\right ) \left (400+e^4+40 x^2+x^4\right )}+\frac {4 x \left (89+x^2\right )}{\left (4761+e^4\right ) \left (400+e^4+40 x^2+x^4\right )}\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.06, size = 20, normalized size = 0.91 \begin {gather*} -\frac {4}{(-7+x) \left (e^4+\left (20+x^2\right )^2\right )} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(1600 + 4*E^4 - 2240*x + 480*x^2 - 112*x^3 + 20*x^4)/(7840000 - 2240000*x + 1728000*x^2 - 448000*x^3
+ 149600*x^4 - 33600*x^5 + 6320*x^6 - 1120*x^7 + 129*x^8 - 14*x^9 + x^10 + E^8*(49 - 14*x + x^2) + E^4*(39200
- 11200*x + 4720*x^2 - 1120*x^3 + 178*x^4 - 28*x^5 + 2*x^6)),x]

[Out]

-4/((-7 + x)*(E^4 + (20 + x^2)^2))

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fricas [A]  time = 0.55, size = 33, normalized size = 1.50 \begin {gather*} -\frac {4}{x^{5} - 7 \, x^{4} + 40 \, x^{3} - 280 \, x^{2} + {\left (x - 7\right )} e^{4} + 400 \, x - 2800} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((4*exp(4)+20*x^4-112*x^3+480*x^2-2240*x+1600)/((x^2-14*x+49)*exp(4)^2+(2*x^6-28*x^5+178*x^4-1120*x^3
+4720*x^2-11200*x+39200)*exp(4)+x^10-14*x^9+129*x^8-1120*x^7+6320*x^6-33600*x^5+149600*x^4-448000*x^3+1728000*
x^2-2240000*x+7840000),x, algorithm="fricas")

[Out]

-4/(x^5 - 7*x^4 + 40*x^3 - 280*x^2 + (x - 7)*e^4 + 400*x - 2800)

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giac [F(-1)]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((4*exp(4)+20*x^4-112*x^3+480*x^2-2240*x+1600)/((x^2-14*x+49)*exp(4)^2+(2*x^6-28*x^5+178*x^4-1120*x^3
+4720*x^2-11200*x+39200)*exp(4)+x^10-14*x^9+129*x^8-1120*x^7+6320*x^6-33600*x^5+149600*x^4-448000*x^3+1728000*
x^2-2240000*x+7840000),x, algorithm="giac")

[Out]

Timed out

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maple [A]  time = 0.22, size = 22, normalized size = 1.00

method	result	size
norman	\(-\frac {4}{\left (x -7\right ) \left (x^{4}+40 x^{2}+{\mathrm e}^{4}+400\right )}\)	\(22\)
gosper	\(-\frac {4}{x^{5}-7 x^{4}+40 x^{3}+x \,{\mathrm e}^{4}-280 x^{2}-7 \,{\mathrm e}^{4}+400 x -2800}\)	\(36\)
risch	\(-\frac {4}{x^{5}-7 x^{4}+40 x^{3}+x \,{\mathrm e}^{4}-280 x^{2}-7 \,{\mathrm e}^{4}+400 x -2800}\)	\(36\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((4*exp(4)+20*x^4-112*x^3+480*x^2-2240*x+1600)/((x^2-14*x+49)*exp(4)^2+(2*x^6-28*x^5+178*x^4-1120*x^3+4720*
x^2-11200*x+39200)*exp(4)+x^10-14*x^9+129*x^8-1120*x^7+6320*x^6-33600*x^5+149600*x^4-448000*x^3+1728000*x^2-22
40000*x+7840000),x,method=_RETURNVERBOSE)

[Out]

-4/(x-7)/(x^4+40*x^2+exp(4)+400)

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maxima [A]  time = 0.36, size = 34, normalized size = 1.55 \begin {gather*} -\frac {4}{x^{5} - 7 \, x^{4} + 40 \, x^{3} - 280 \, x^{2} + x {\left (e^{4} + 400\right )} - 7 \, e^{4} - 2800} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((4*exp(4)+20*x^4-112*x^3+480*x^2-2240*x+1600)/((x^2-14*x+49)*exp(4)^2+(2*x^6-28*x^5+178*x^4-1120*x^3
+4720*x^2-11200*x+39200)*exp(4)+x^10-14*x^9+129*x^8-1120*x^7+6320*x^6-33600*x^5+149600*x^4-448000*x^3+1728000*
x^2-2240000*x+7840000),x, algorithm="maxima")

[Out]

-4/(x^5 - 7*x^4 + 40*x^3 - 280*x^2 + x*(e^4 + 400) - 7*e^4 - 2800)

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mupad [B]  time = 0.29, size = 21, normalized size = 0.95 \begin {gather*} -\frac {4}{\left (x-7\right )\,\left (x^4+40\,x^2+{\mathrm {e}}^4+400\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((4*exp(4) - 2240*x + 480*x^2 - 112*x^3 + 20*x^4 + 1600)/(exp(4)*(4720*x^2 - 11200*x - 1120*x^3 + 178*x^4 -
28*x^5 + 2*x^6 + 39200) - 2240000*x + exp(8)*(x^2 - 14*x + 49) + 1728000*x^2 - 448000*x^3 + 149600*x^4 - 3360
0*x^5 + 6320*x^6 - 1120*x^7 + 129*x^8 - 14*x^9 + x^10 + 7840000),x)

[Out]

-4/((x - 7)*(exp(4) + 40*x^2 + x^4 + 400))

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sympy [B]  time = 3.28, size = 34, normalized size = 1.55 \begin {gather*} - \frac {4}{x^{5} - 7 x^{4} + 40 x^{3} - 280 x^{2} + x \left (e^{4} + 400\right ) - 2800 - 7 e^{4}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((4*exp(4)+20*x**4-112*x**3+480*x**2-2240*x+1600)/((x**2-14*x+49)*exp(4)**2+(2*x**6-28*x**5+178*x**4-
1120*x**3+4720*x**2-11200*x+39200)*exp(4)+x**10-14*x**9+129*x**8-1120*x**7+6320*x**6-33600*x**5+149600*x**4-44
8000*x**3+1728000*x**2-2240000*x+7840000),x)

[Out]

-4/(x**5 - 7*x**4 + 40*x**3 - 280*x**2 + x*(exp(4) + 400) - 2800 - 7*exp(4))

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