3.49.61 \(\int \frac {1332+e^{20+4 e^3} x^4+e^{10} (5751 x^2+1260 x^3+69 x^4)+e^{20+3 e^3} (-36 x^4-4 x^5)+e^{20} (6561 x^4+2916 x^5+486 x^6+36 x^7+x^8)+e^{2 e^3} (71 e^{10} x^2+e^{20} (486 x^4+108 x^5+6 x^6))+e^{e^3} (e^{10} (-1278 x^2-140 x^3)+e^{20} (-2916 x^4-972 x^5-108 x^6-4 x^7))}{1296+e^{20+4 e^3} x^4+e^{10} (5832 x^2+1296 x^3+72 x^4)+e^{20+3 e^3} (-36 x^4-4 x^5)+e^{20} (6561 x^4+2916 x^5+486 x^6+36 x^7+x^8)+e^{2 e^3} (72 e^{10} x^2+e^{20} (486 x^4+108 x^5+6 x^6))+e^{e^3} (e^{10} (-1296 x^2-144 x^3)+e^{20} (-2916 x^4-972 x^5-108 x^6-4 x^7))} \, dx\)
Optimal. Leaf size=28 \[ -1+x+\frac {x}{36+e^{10} \left (-9+e^{e^3}-x\right )^2 x^2} \]
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Rubi [B] time = 17.49, antiderivative size = 784, normalized size of antiderivative = 28.00,
number of steps used = 27, number of rules used = 13, integrand size = 339, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.038,
Rules used = {2074, 1106, 1094, 634, 618, 206, 628, 1680, 1673, 1178, 12, 1247, 629} \begin {gather*} \frac {2 x-e^{e^3}+9}{2 \left (e^{10} x^4+2 e^{10} \left (9-e^{e^3}\right ) x^3+e^{10} \left (9-e^{e^3}\right )^2 x^2+36\right )}-\frac {9-e^{e^3}}{2 \left (e^{10} x^4+2 e^{10} \left (9-e^{e^3}\right ) x^3+e^{10} \left (9-e^{e^3}\right )^2 x^2+36\right )}+x+\frac {0 \sqrt {-\frac {2}{\left (576+6561 e^{10}-2916 e^{10+e^3}+486 e^{10+2 e^3}-36 e^{10+3 e^3}+e^{10+4 e^3}\right ) \left (81 e^5-18 e^{5+e^3}+e^{5+2 e^3}-\sqrt {576+6561 e^{10}-2916 e^{10+e^3}+486 e^{10+2 e^3}-36 e^{10+3 e^3}+e^{10+4 e^3}}\right )}} \tanh ^{-1}\left (\frac {\frac {1}{\sqrt {\frac {2}{81 e^5-18 e^{5+e^3}+e^{5+2 e^3}+\sqrt {576+6561 e^{10}+486 e^{2 \left (5+e^3\right )}-2916 e^{10+e^3}-36 e^{10+3 e^3}+e^{10+4 e^3}}}}}-e^{5/2} \left (2 x-e^{e^3}+9\right )}{\sqrt {\frac {1}{2} \left (81 e^5-18 e^{5+e^3}+e^{5+2 e^3}-\sqrt {576+6561 e^{10}+486 e^{2 \left (5+e^3\right )}-2916 e^{10+e^3}-36 e^{10+3 e^3}+e^{10+4 e^3}}\right )}}\right )}{e^{5/2}}+\frac {0 \sqrt {-\frac {2}{\left (576+6561 e^{10}-2916 e^{10+e^3}+486 e^{10+2 e^3}-36 e^{10+3 e^3}+e^{10+4 e^3}\right ) \left (81 e^5-18 e^{5+e^3}+e^{5+2 e^3}-\sqrt {576+6561 e^{10}-2916 e^{10+e^3}+486 e^{10+2 e^3}-36 e^{10+3 e^3}+e^{10+4 e^3}}\right )}} \tanh ^{-1}\left (\frac {e^{5/2} \left (2 x-e^{e^3}+9\right )+\frac {1}{\sqrt {\frac {2}{81 e^5-18 e^{5+e^3}+e^{5+2 e^3}+\sqrt {576+6561 e^{10}+486 e^{2 \left (5+e^3\right )}-2916 e^{10+e^3}-36 e^{10+3 e^3}+e^{10+4 e^3}}}}}}{\sqrt {\frac {1}{2} \left (81 e^5-18 e^{5+e^3}+e^{5+2 e^3}-\sqrt {576+6561 e^{10}+486 e^{2 \left (5+e^3\right )}-2916 e^{10+e^3}-36 e^{10+3 e^3}+e^{10+4 e^3}}\right )}}\right )}{e^{5/2}} \end {gather*}
Antiderivative was successfully verified.
[In]
Int[(1332 + E^(20 + 4*E^3)*x^4 + E^10*(5751*x^2 + 1260*x^3 + 69*x^4) + E^(20 + 3*E^3)*(-36*x^4 - 4*x^5) + E^20
*(6561*x^4 + 2916*x^5 + 486*x^6 + 36*x^7 + x^8) + E^(2*E^3)*(71*E^10*x^2 + E^20*(486*x^4 + 108*x^5 + 6*x^6)) +
E^E^3*(E^10*(-1278*x^2 - 140*x^3) + E^20*(-2916*x^4 - 972*x^5 - 108*x^6 - 4*x^7)))/(1296 + E^(20 + 4*E^3)*x^4
+ E^10*(5832*x^2 + 1296*x^3 + 72*x^4) + E^(20 + 3*E^3)*(-36*x^4 - 4*x^5) + E^20*(6561*x^4 + 2916*x^5 + 486*x^
6 + 36*x^7 + x^8) + E^(2*E^3)*(72*E^10*x^2 + E^20*(486*x^4 + 108*x^5 + 6*x^6)) + E^E^3*(E^10*(-1296*x^2 - 144*
x^3) + E^20*(-2916*x^4 - 972*x^5 - 108*x^6 - 4*x^7))),x]
[Out]
x - (9 - E^E^3)/(2*(36 + E^10*(9 - E^E^3)^2*x^2 + 2*E^10*(9 - E^E^3)*x^3 + E^10*x^4)) + (9 - E^E^3 + 2*x)/(2*(
36 + E^10*(9 - E^E^3)^2*x^2 + 2*E^10*(9 - E^E^3)*x^3 + E^10*x^4)) + (0*Sqrt[-2/((576 + 6561*E^10 - 2916*E^(10
+ E^3) + 486*E^(10 + 2*E^3) - 36*E^(10 + 3*E^3) + E^(10 + 4*E^3))*(81*E^5 - 18*E^(5 + E^3) + E^(5 + 2*E^3) - S
qrt[576 + 6561*E^10 - 2916*E^(10 + E^3) + 486*E^(10 + 2*E^3) - 36*E^(10 + 3*E^3) + E^(10 + 4*E^3)]))]*ArcTanh[
(1/Sqrt[2/(81*E^5 - 18*E^(5 + E^3) + E^(5 + 2*E^3) + Sqrt[576 + 6561*E^10 + 486*E^(2*(5 + E^3)) - 2916*E^(10 +
E^3) - 36*E^(10 + 3*E^3) + E^(10 + 4*E^3)])] - E^(5/2)*(9 - E^E^3 + 2*x))/Sqrt[(81*E^5 - 18*E^(5 + E^3) + E^(
5 + 2*E^3) - Sqrt[576 + 6561*E^10 + 486*E^(2*(5 + E^3)) - 2916*E^(10 + E^3) - 36*E^(10 + 3*E^3) + E^(10 + 4*E^
3)])/2]])/E^(5/2) + (0*Sqrt[-2/((576 + 6561*E^10 - 2916*E^(10 + E^3) + 486*E^(10 + 2*E^3) - 36*E^(10 + 3*E^3)
+ E^(10 + 4*E^3))*(81*E^5 - 18*E^(5 + E^3) + E^(5 + 2*E^3) - Sqrt[576 + 6561*E^10 - 2916*E^(10 + E^3) + 486*E^
(10 + 2*E^3) - 36*E^(10 + 3*E^3) + E^(10 + 4*E^3)]))]*ArcTanh[(1/Sqrt[2/(81*E^5 - 18*E^(5 + E^3) + E^(5 + 2*E^
3) + Sqrt[576 + 6561*E^10 + 486*E^(2*(5 + E^3)) - 2916*E^(10 + E^3) - 36*E^(10 + 3*E^3) + E^(10 + 4*E^3)])] +
E^(5/2)*(9 - E^E^3 + 2*x))/Sqrt[(81*E^5 - 18*E^(5 + E^3) + E^(5 + 2*E^3) - Sqrt[576 + 6561*E^10 + 486*E^(2*(5
+ E^3)) - 2916*E^(10 + E^3) - 36*E^(10 + 3*E^3) + E^(10 + 4*E^3)])/2]])/E^(5/2)
Rule 12
Int[(a_)*(u_), x_Symbol] :> Dist[a, Int[u, x], x] /; FreeQ[a, x] && !MatchQ[u, (b_)*(v_) /; FreeQ[b, x]]
Rule 206
Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1*ArcTanh[(Rt[-b, 2]*x)/Rt[a, 2]])/(Rt[a, 2]*Rt[-b, 2]), x]
/; FreeQ[{a, b}, x] && NegQ[a/b] && (GtQ[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 629
Int[((d_) + (e_.)*(x_))*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Simp[(d*(a + b*x + c*x^2)^(p +
1))/(b*(p + 1)), x] /; FreeQ[{a, b, c, d, e, p}, x] && EqQ[2*c*d - b*e, 0] && NeQ[p, -1]
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 1094
Int[((a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4)^(-1), x_Symbol] :> With[{q = Rt[a/c, 2]}, With[{r = Rt[2*q - b/c, 2]}
, Dist[1/(2*c*q*r), Int[(r - x)/(q - r*x + x^2), x], x] + Dist[1/(2*c*q*r), Int[(r + x)/(q + r*x + x^2), x], x
]]] /; FreeQ[{a, b, c}, x] && NeQ[b^2 - 4*a*c, 0] && NegQ[b^2 - 4*a*c]
Rule 1106
Int[(P4_)^(p_), x_Symbol] :> With[{a = Coeff[P4, x, 0], b = Coeff[P4, x, 1], c = Coeff[P4, x, 2], d = Coeff[P4
, x, 3], e = Coeff[P4, x, 4]}, Subst[Int[SimplifyIntegrand[(a + d^4/(256*e^3) - (b*d)/(8*e) + (c - (3*d^2)/(8*
e))*x^2 + e*x^4)^p, x], x], x, d/(4*e) + x] /; EqQ[d^3 - 4*c*d*e + 8*b*e^2, 0] && NeQ[d, 0]] /; FreeQ[p, x] &&
PolyQ[P4, x, 4] && NeQ[p, 2] && NeQ[p, 3]
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 1680
Int[(Pq_)*(Q4_)^(p_), x_Symbol] :> With[{a = Coeff[Q4, x, 0], b = Coeff[Q4, x, 1], c = Coeff[Q4, x, 2], d = Co
eff[Q4, x, 3], e = Coeff[Q4, x, 4]}, Subst[Int[SimplifyIntegrand[(Pq /. x -> -(d/(4*e)) + x)*(a + d^4/(256*e^3
) - (b*d)/(8*e) + (c - (3*d^2)/(8*e))*x^2 + e*x^4)^p, x], x], x, d/(4*e) + x] /; EqQ[d^3 - 4*c*d*e + 8*b*e^2,
0] && NeQ[d, 0]] /; FreeQ[p, x] && PolyQ[Pq, x] && PolyQ[Q4, x, 4] && !IGtQ[p, 0]
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 (1+\frac {3}{-36-e^{10} \left (9-e^{e^3}\right )^2 x^2-2 e^{10} \left (9-e^{e^3}\right ) x^3-e^{10} x^4}+\frac {2 \left (72+e^{10} \left (9-e^{e^3}\right )^2 x^2+e^{10} \left (9-e^{e^3}\right ) x^3\right )}{\left (36+e^{10} \left (9-e^{e^3}\right )^2 x^2+2 e^{10} \left (9-e^{e^3}\right ) x^3+e^{10} x^4\right )^2}\right ) \, dx\\ &=x+2 \int \frac {72+e^{10} \left (9-e^{e^3}\right )^2 x^2+e^{10} \left (9-e^{e^3}\right ) x^3}{\left (36+e^{10} \left (9-e^{e^3}\right )^2 x^2+2 e^{10} \left (9-e^{e^3}\right ) x^3+e^{10} x^4\right )^2} \, dx+3 \int \frac {1}{-36-e^{10} \left (9-e^{e^3}\right )^2 x^2-2 e^{10} \left (9-e^{e^3}\right ) x^3-e^{10} x^4} \, dx\\ &=x+2 \operatorname {Subst}\left (\int \frac {\frac {1}{8} \left (576+6561 e^{10}+486 e^{2 \left (5+e^3\right )}-2916 e^{10+e^3}-36 e^{10+3 e^3}+e^{10+4 e^3}\right )-\frac {1}{4} e^{10} \left (9-e^{e^3}\right )^3 x-\frac {1}{2} e^{10} \left (9-e^{e^3}\right )^2 x^2+e^{10} \left (9-e^{e^3}\right ) x^3}{\left (36+\frac {1}{16} e^{10} \left (-9+e^{e^3}\right )^4-\frac {1}{2} e^{10} \left (-9+e^{e^3}\right )^2 x^2+e^{10} x^4\right )^2} \, dx,x,\frac {1}{2} \left (9-e^{e^3}\right )+x\right )+3 \operatorname {Subst}\left (\int \frac {1}{-36-\frac {1}{16} e^{10} \left (-9+e^{e^3}\right )^4+\frac {1}{2} e^{10} \left (-9+e^{e^3}\right )^2 x^2-e^{10} x^4} \, dx,x,\frac {1}{2} \left (9-e^{e^3}\right )+x\right )\\ &=\text {Rest of rules removed due to large latex content} \end {aligned} \end {gather*}
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Mathematica [B] time = 0.67, size = 88, normalized size = 3.14 \begin {gather*} \frac {x \left (37+e^{2 \left (5+e^3\right )} x^2-2 e^{10+e^3} x^2 (9+x)+e^{10} x^2 (9+x)^2\right )}{36+e^{2 \left (5+e^3\right )} x^2-2 e^{10+e^3} x^2 (9+x)+e^{10} x^2 (9+x)^2} \end {gather*}
Antiderivative was successfully verified.
[In]
Integrate[(1332 + E^(20 + 4*E^3)*x^4 + E^10*(5751*x^2 + 1260*x^3 + 69*x^4) + E^(20 + 3*E^3)*(-36*x^4 - 4*x^5)
+ E^20*(6561*x^4 + 2916*x^5 + 486*x^6 + 36*x^7 + x^8) + E^(2*E^3)*(71*E^10*x^2 + E^20*(486*x^4 + 108*x^5 + 6*x
^6)) + E^E^3*(E^10*(-1278*x^2 - 140*x^3) + E^20*(-2916*x^4 - 972*x^5 - 108*x^6 - 4*x^7)))/(1296 + E^(20 + 4*E^
3)*x^4 + E^10*(5832*x^2 + 1296*x^3 + 72*x^4) + E^(20 + 3*E^3)*(-36*x^4 - 4*x^5) + E^20*(6561*x^4 + 2916*x^5 +
486*x^6 + 36*x^7 + x^8) + E^(2*E^3)*(72*E^10*x^2 + E^20*(486*x^4 + 108*x^5 + 6*x^6)) + E^E^3*(E^10*(-1296*x^2
- 144*x^3) + E^20*(-2916*x^4 - 972*x^5 - 108*x^6 - 4*x^7))),x]
[Out]
(x*(37 + E^(2*(5 + E^3))*x^2 - 2*E^(10 + E^3)*x^2*(9 + x) + E^10*x^2*(9 + x)^2))/(36 + E^(2*(5 + E^3))*x^2 - 2
*E^(10 + E^3)*x^2*(9 + x) + E^10*x^2*(9 + x)^2)
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fricas [B] time = 0.66, size = 97, normalized size = 3.46 \begin {gather*} \frac {x^{3} e^{\left (2 \, e^{3} + 10\right )} + {\left (x^{5} + 18 \, x^{4} + 81 \, x^{3}\right )} e^{10} - 2 \, {\left (x^{4} + 9 \, x^{3}\right )} e^{\left (e^{3} + 10\right )} + 37 \, x}{x^{2} e^{\left (2 \, e^{3} + 10\right )} + {\left (x^{4} + 18 \, x^{3} + 81 \, x^{2}\right )} e^{10} - 2 \, {\left (x^{3} + 9 \, x^{2}\right )} e^{\left (e^{3} + 10\right )} + 36} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate((x^4*exp(5)^4*exp(exp(3))^4+(-4*x^5-36*x^4)*exp(5)^4*exp(exp(3))^3+((6*x^6+108*x^5+486*x^4)*exp(5)^4
+71*x^2*exp(5)^2)*exp(exp(3))^2+((-4*x^7-108*x^6-972*x^5-2916*x^4)*exp(5)^4+(-140*x^3-1278*x^2)*exp(5)^2)*exp(
exp(3))+(x^8+36*x^7+486*x^6+2916*x^5+6561*x^4)*exp(5)^4+(69*x^4+1260*x^3+5751*x^2)*exp(5)^2+1332)/(x^4*exp(5)^
4*exp(exp(3))^4+(-4*x^5-36*x^4)*exp(5)^4*exp(exp(3))^3+((6*x^6+108*x^5+486*x^4)*exp(5)^4+72*x^2*exp(5)^2)*exp(
exp(3))^2+((-4*x^7-108*x^6-972*x^5-2916*x^4)*exp(5)^4+(-144*x^3-1296*x^2)*exp(5)^2)*exp(exp(3))+(x^8+36*x^7+48
6*x^6+2916*x^5+6561*x^4)*exp(5)^4+(72*x^4+1296*x^3+5832*x^2)*exp(5)^2+1296),x, algorithm="fricas")
[Out]
(x^3*e^(2*e^3 + 10) + (x^5 + 18*x^4 + 81*x^3)*e^10 - 2*(x^4 + 9*x^3)*e^(e^3 + 10) + 37*x)/(x^2*e^(2*e^3 + 10)
+ (x^4 + 18*x^3 + 81*x^2)*e^10 - 2*(x^3 + 9*x^2)*e^(e^3 + 10) + 36)
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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((x^4*exp(5)^4*exp(exp(3))^4+(-4*x^5-36*x^4)*exp(5)^4*exp(exp(3))^3+((6*x^6+108*x^5+486*x^4)*exp(5)^4
+71*x^2*exp(5)^2)*exp(exp(3))^2+((-4*x^7-108*x^6-972*x^5-2916*x^4)*exp(5)^4+(-140*x^3-1278*x^2)*exp(5)^2)*exp(
exp(3))+(x^8+36*x^7+486*x^6+2916*x^5+6561*x^4)*exp(5)^4+(69*x^4+1260*x^3+5751*x^2)*exp(5)^2+1332)/(x^4*exp(5)^
4*exp(exp(3))^4+(-4*x^5-36*x^4)*exp(5)^4*exp(exp(3))^3+((6*x^6+108*x^5+486*x^4)*exp(5)^4+72*x^2*exp(5)^2)*exp(
exp(3))^2+((-4*x^7-108*x^6-972*x^5-2916*x^4)*exp(5)^4+(-144*x^3-1296*x^2)*exp(5)^2)*exp(exp(3))+(x^8+36*x^7+48
6*x^6+2916*x^5+6561*x^4)*exp(5)^4+(72*x^4+1296*x^3+5832*x^2)*exp(5)^2+1296),x, algorithm="giac")
[Out]
Timed out
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maple [B] time = 0.48, size = 60, normalized size = 2.14
|
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| method |
result |
size |
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| risch |
\(x +\frac {x}{x^{2} {\mathrm e}^{10+2 \,{\mathrm e}^{3}}-2 x^{3} {\mathrm e}^{10+{\mathrm e}^{3}}+x^{4} {\mathrm e}^{10}-18 x^{2} {\mathrm e}^{10+{\mathrm e}^{3}}+18 x^{3} {\mathrm e}^{10}+81 \,{\mathrm e}^{10} x^{2}+36}\) |
\(60\) |
| norman |
\(\frac {x^{5} {\mathrm e}^{10}+{\mathrm e}^{10} \left ({\mathrm e}^{2 \,{\mathrm e}^{3}}-18 \,{\mathrm e}^{{\mathrm e}^{3}}+81\right ) x^{3}+\left (-2 \,{\mathrm e}^{10} {\mathrm e}^{{\mathrm e}^{3}}+18 \,{\mathrm e}^{10}\right ) x^{4}+37 x}{{\mathrm e}^{10} {\mathrm e}^{2 \,{\mathrm e}^{3}} x^{2}-2 \,{\mathrm e}^{10} {\mathrm e}^{{\mathrm e}^{3}} x^{3}+x^{4} {\mathrm e}^{10}-18 \,{\mathrm e}^{10} {\mathrm e}^{{\mathrm e}^{3}} x^{2}+18 x^{3} {\mathrm e}^{10}+81 \,{\mathrm e}^{10} x^{2}+36}\) |
\(121\) |
| gosper |
\(\frac {x \left ({\mathrm e}^{10} {\mathrm e}^{2 \,{\mathrm e}^{3}} x^{2}-2 \,{\mathrm e}^{10} {\mathrm e}^{{\mathrm e}^{3}} x^{3}+x^{4} {\mathrm e}^{10}-18 \,{\mathrm e}^{10} {\mathrm e}^{{\mathrm e}^{3}} x^{2}+18 x^{3} {\mathrm e}^{10}+81 \,{\mathrm e}^{10} x^{2}+37\right )}{{\mathrm e}^{10} {\mathrm e}^{2 \,{\mathrm e}^{3}} x^{2}-2 \,{\mathrm e}^{10} {\mathrm e}^{{\mathrm e}^{3}} x^{3}+x^{4} {\mathrm e}^{10}-18 \,{\mathrm e}^{10} {\mathrm e}^{{\mathrm e}^{3}} x^{2}+18 x^{3} {\mathrm e}^{10}+81 \,{\mathrm e}^{10} x^{2}+36}\) |
\(135\) |
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Verification of antiderivative is not currently implemented for this CAS.
[In]
int((x^4*exp(5)^4*exp(exp(3))^4+(-4*x^5-36*x^4)*exp(5)^4*exp(exp(3))^3+((6*x^6+108*x^5+486*x^4)*exp(5)^4+71*x^
2*exp(5)^2)*exp(exp(3))^2+((-4*x^7-108*x^6-972*x^5-2916*x^4)*exp(5)^4+(-140*x^3-1278*x^2)*exp(5)^2)*exp(exp(3)
)+(x^8+36*x^7+486*x^6+2916*x^5+6561*x^4)*exp(5)^4+(69*x^4+1260*x^3+5751*x^2)*exp(5)^2+1332)/(x^4*exp(5)^4*exp(
exp(3))^4+(-4*x^5-36*x^4)*exp(5)^4*exp(exp(3))^3+((6*x^6+108*x^5+486*x^4)*exp(5)^4+72*x^2*exp(5)^2)*exp(exp(3)
)^2+((-4*x^7-108*x^6-972*x^5-2916*x^4)*exp(5)^4+(-144*x^3-1296*x^2)*exp(5)^2)*exp(exp(3))+(x^8+36*x^7+486*x^6+
2916*x^5+6561*x^4)*exp(5)^4+(72*x^4+1296*x^3+5832*x^2)*exp(5)^2+1296),x,method=_RETURNVERBOSE)
[Out]
x+x/(x^2*exp(10+2*exp(3))-2*x^3*exp(10+exp(3))+x^4*exp(10)-18*x^2*exp(10+exp(3))+18*x^3*exp(10)+81*exp(10)*x^2
+36)
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maxima [B] time = 0.37, size = 54, normalized size = 1.93 \begin {gather*} x + \frac {x}{x^{4} e^{10} + 2 \, x^{3} {\left (9 \, e^{10} - e^{\left (e^{3} + 10\right )}\right )} + x^{2} {\left (81 \, e^{10} + e^{\left (2 \, e^{3} + 10\right )} - 18 \, e^{\left (e^{3} + 10\right )}\right )} + 36} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate((x^4*exp(5)^4*exp(exp(3))^4+(-4*x^5-36*x^4)*exp(5)^4*exp(exp(3))^3+((6*x^6+108*x^5+486*x^4)*exp(5)^4
+71*x^2*exp(5)^2)*exp(exp(3))^2+((-4*x^7-108*x^6-972*x^5-2916*x^4)*exp(5)^4+(-140*x^3-1278*x^2)*exp(5)^2)*exp(
exp(3))+(x^8+36*x^7+486*x^6+2916*x^5+6561*x^4)*exp(5)^4+(69*x^4+1260*x^3+5751*x^2)*exp(5)^2+1332)/(x^4*exp(5)^
4*exp(exp(3))^4+(-4*x^5-36*x^4)*exp(5)^4*exp(exp(3))^3+((6*x^6+108*x^5+486*x^4)*exp(5)^4+72*x^2*exp(5)^2)*exp(
exp(3))^2+((-4*x^7-108*x^6-972*x^5-2916*x^4)*exp(5)^4+(-144*x^3-1296*x^2)*exp(5)^2)*exp(exp(3))+(x^8+36*x^7+48
6*x^6+2916*x^5+6561*x^4)*exp(5)^4+(72*x^4+1296*x^3+5832*x^2)*exp(5)^2+1296),x, algorithm="maxima")
[Out]
x + x/(x^4*e^10 + 2*x^3*(9*e^10 - e^(e^3 + 10)) + x^2*(81*e^10 + e^(2*e^3 + 10) - 18*e^(e^3 + 10)) + 36)
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mupad [F(-1)] time = 0.00, size = -1, normalized size = -0.04 \begin {gather*} \text {Hanged} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
int((exp(2*exp(3))*(71*x^2*exp(10) + exp(20)*(486*x^4 + 108*x^5 + 6*x^6)) - exp(exp(3))*(exp(10)*(1278*x^2 + 1
40*x^3) + exp(20)*(2916*x^4 + 972*x^5 + 108*x^6 + 4*x^7)) + exp(10)*(5751*x^2 + 1260*x^3 + 69*x^4) + exp(20)*(
6561*x^4 + 2916*x^5 + 486*x^6 + 36*x^7 + x^8) - exp(3*exp(3))*exp(20)*(36*x^4 + 4*x^5) + x^4*exp(4*exp(3))*exp
(20) + 1332)/(exp(2*exp(3))*(72*x^2*exp(10) + exp(20)*(486*x^4 + 108*x^5 + 6*x^6)) - exp(exp(3))*(exp(10)*(129
6*x^2 + 144*x^3) + exp(20)*(2916*x^4 + 972*x^5 + 108*x^6 + 4*x^7)) + exp(10)*(5832*x^2 + 1296*x^3 + 72*x^4) +
exp(20)*(6561*x^4 + 2916*x^5 + 486*x^6 + 36*x^7 + x^8) - exp(3*exp(3))*exp(20)*(36*x^4 + 4*x^5) + x^4*exp(4*ex
p(3))*exp(20) + 1296),x)
[Out]
\text{Hanged}
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sympy [B] time = 6.23, size = 58, normalized size = 2.07 \begin {gather*} x + \frac {x}{x^{4} e^{10} + x^{3} \left (- 2 e^{10} e^{e^{3}} + 18 e^{10}\right ) + x^{2} \left (- 18 e^{10} e^{e^{3}} + 81 e^{10} + e^{10} e^{2 e^{3}}\right ) + 36} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate((x**4*exp(5)**4*exp(exp(3))**4+(-4*x**5-36*x**4)*exp(5)**4*exp(exp(3))**3+((6*x**6+108*x**5+486*x**4
)*exp(5)**4+71*x**2*exp(5)**2)*exp(exp(3))**2+((-4*x**7-108*x**6-972*x**5-2916*x**4)*exp(5)**4+(-140*x**3-1278
*x**2)*exp(5)**2)*exp(exp(3))+(x**8+36*x**7+486*x**6+2916*x**5+6561*x**4)*exp(5)**4+(69*x**4+1260*x**3+5751*x*
*2)*exp(5)**2+1332)/(x**4*exp(5)**4*exp(exp(3))**4+(-4*x**5-36*x**4)*exp(5)**4*exp(exp(3))**3+((6*x**6+108*x**
5+486*x**4)*exp(5)**4+72*x**2*exp(5)**2)*exp(exp(3))**2+((-4*x**7-108*x**6-972*x**5-2916*x**4)*exp(5)**4+(-144
*x**3-1296*x**2)*exp(5)**2)*exp(exp(3))+(x**8+36*x**7+486*x**6+2916*x**5+6561*x**4)*exp(5)**4+(72*x**4+1296*x*
*3+5832*x**2)*exp(5)**2+1296),x)
[Out]
x + x/(x**4*exp(10) + x**3*(-2*exp(10)*exp(exp(3)) + 18*exp(10)) + x**2*(-18*exp(10)*exp(exp(3)) + 81*exp(10)
+ exp(10)*exp(2*exp(3))) + 36)
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