3.16.98 \(\int \frac {e^8 (144 x^2-96 x^3+84 x^4-27 x^6)+e^{12} (64-96 x+192 x^2-152 x^3+144 x^4-54 x^5+27 x^6)}{-135 x^6+e^8 (-720 x^2+720 x^3-1260 x^4+540 x^5-405 x^6)+e^{12} (320-480 x+960 x^2-760 x^3+720 x^4-270 x^5+135 x^6)+e^4 (540 x^4-270 x^5+405 x^6)} \, dx\)

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

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Rubi [C]  time = 0.72, antiderivative size = 483, normalized size of antiderivative = 16.66, number of steps used = 11, number of rules used = 5, integrand size = 153, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.033, Rules used = {2074, 638, 614, 618, 204} \begin {gather*} \frac {8 e^4 \left (9-9 e^4+e^8\right ) \left (3 \left (1-e^4\right ) x+e^4\right )}{15 \left (12-11 e^4\right ) \left (1-e^4\right )^4 \left (-3 \left (1-e^4\right ) x^2-2 e^4 x+4 e^4\right )}+\frac {4 e^4 \left (2 e^4 \left (45-27 e^4-86 e^8+66 e^{12}\right )-3 \left (54-24 e^4-149 e^8+163 e^{12}-44 e^{16}\right ) x\right )}{45 \left (12-11 e^4\right ) \left (1-e^4\right )^4 \left (-3 \left (1-e^4\right ) x^2-2 e^4 x+4 e^4\right )}-\frac {16 e^8 \left (2 e^4 \left (72-126 e^4+55 e^8\right )-\left (108-207 e^4+111 e^8-11 e^{12}\right ) x\right )}{45 \left (12-11 e^4\right ) \left (1-e^4\right )^4 \left (-3 \left (1-e^4\right ) x^2-2 e^4 x+4 e^4\right )^2}+\frac {e^8 x}{5 \left (1-e^4\right )^2}-\frac {4 e^2 \left (54+30 e^4-119 e^8+44 e^{12}\right ) \tan ^{-1}\left (\frac {e^4 (1-3 x)+3 x}{e^2 \sqrt {11 e^4-12}}\right )}{15 \left (1-e^4\right )^3 \left (11 e^4-12\right )^{3/2}}+\frac {8 e^2 \left (9-9 e^4+e^8\right ) \tan ^{-1}\left (\frac {e^4 (1-3 x)+3 x}{e^2 \sqrt {11 e^4-12}}\right )}{5 \left (1-e^4\right )^3 \left (11 e^4-12\right )^{3/2}}-\frac {4 e^6 \left (7-4 e^4\right ) \tan ^{-1}\left (\frac {e^4 (1-3 x)+3 x}{e^2 \sqrt {11 e^4-12}}\right )}{15 \left (1-e^4\right )^3 \sqrt {11 e^4-12}} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(E^8*(144*x^2 - 96*x^3 + 84*x^4 - 27*x^6) + E^12*(64 - 96*x + 192*x^2 - 152*x^3 + 144*x^4 - 54*x^5 + 27*x^
6))/(-135*x^6 + E^8*(-720*x^2 + 720*x^3 - 1260*x^4 + 540*x^5 - 405*x^6) + E^12*(320 - 480*x + 960*x^2 - 760*x^
3 + 720*x^4 - 270*x^5 + 135*x^6) + E^4*(540*x^4 - 270*x^5 + 405*x^6)),x]

[Out]

(E^8*x)/(5*(1 - E^4)^2) - (16*E^8*(2*E^4*(72 - 126*E^4 + 55*E^8) - (108 - 207*E^4 + 111*E^8 - 11*E^12)*x))/(45
*(12 - 11*E^4)*(1 - E^4)^4*(4*E^4 - 2*E^4*x - 3*(1 - E^4)*x^2)^2) + (8*E^4*(9 - 9*E^4 + E^8)*(E^4 + 3*(1 - E^4
)*x))/(15*(12 - 11*E^4)*(1 - E^4)^4*(4*E^4 - 2*E^4*x - 3*(1 - E^4)*x^2)) + (4*E^4*(2*E^4*(45 - 27*E^4 - 86*E^8
+ 66*E^12) - 3*(54 - 24*E^4 - 149*E^8 + 163*E^12 - 44*E^16)*x))/(45*(12 - 11*E^4)*(1 - E^4)^4*(4*E^4 - 2*E^4*
x - 3*(1 - E^4)*x^2)) - (4*E^6*(7 - 4*E^4)*ArcTan[(E^4*(1 - 3*x) + 3*x)/(E^2*Sqrt[-12 + 11*E^4])])/(15*(1 - E^
4)^3*Sqrt[-12 + 11*E^4]) + (8*E^2*(9 - 9*E^4 + E^8)*ArcTan[(E^4*(1 - 3*x) + 3*x)/(E^2*Sqrt[-12 + 11*E^4])])/(5
*(1 - E^4)^3*(-12 + 11*E^4)^(3/2)) - (4*E^2*(54 + 30*E^4 - 119*E^8 + 44*E^12)*ArcTan[(E^4*(1 - 3*x) + 3*x)/(E^
2*Sqrt[-12 + 11*E^4])])/(15*(1 - E^4)^3*(-12 + 11*E^4)^(3/2))

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 614

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

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 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 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 {e^8}{5 \left (-1+e^4\right )^2}+\frac {64 e^{12} \left (2 \left (18-24 e^4+7 e^8\right )-\left (45-75 e^4+31 e^8\right ) x\right )}{45 \left (1-e^4\right )^4 \left (4 e^4-2 e^4 x-3 \left (1-e^4\right ) x^2\right )^3}+\frac {8 e^8 \left (-2 \left (27+9 e^4-64 e^8+30 e^{12}\right )+3 \left (12-3 e^4-25 e^8+16 e^{12}\right ) x\right )}{45 \left (1-e^4\right )^4 \left (4 e^4-2 e^4 x-3 \left (1-e^4\right ) x^2\right )^2}+\frac {4 e^8 \left (7-4 e^4\right )}{15 \left (1-e^4\right )^3 \left (4 e^4-2 e^4 x-3 \left (1-e^4\right ) x^2\right )}\right ) \, dx\\ &=\frac {e^8 x}{5 \left (1-e^4\right )^2}+\frac {\left (8 e^8\right ) \int \frac {-2 \left (27+9 e^4-64 e^8+30 e^{12}\right )+3 \left (12-3 e^4-25 e^8+16 e^{12}\right ) x}{\left (4 e^4-2 e^4 x-3 \left (1-e^4\right ) x^2\right )^2} \, dx}{45 \left (1-e^4\right )^4}+\frac {\left (64 e^{12}\right ) \int \frac {2 \left (18-24 e^4+7 e^8\right )-\left (45-75 e^4+31 e^8\right ) x}{\left (4 e^4-2 e^4 x-3 \left (1-e^4\right ) x^2\right )^3} \, dx}{45 \left (1-e^4\right )^4}+\frac {\left (4 e^8 \left (7-4 e^4\right )\right ) \int \frac {1}{4 e^4-2 e^4 x-3 \left (1-e^4\right ) x^2} \, dx}{15 \left (1-e^4\right )^3}\\ &=\frac {e^8 x}{5 \left (1-e^4\right )^2}-\frac {16 e^8 \left (2 e^4 \left (72-126 e^4+55 e^8\right )-\left (108-207 e^4+111 e^8-11 e^{12}\right ) x\right )}{45 \left (12-11 e^4\right ) \left (1-e^4\right )^4 \left (4 e^4-2 e^4 x-3 \left (1-e^4\right ) x^2\right )^2}+\frac {4 e^4 \left (2 e^4 \left (45-27 e^4-86 e^8+66 e^{12}\right )-3 \left (54-24 e^4-149 e^8+163 e^{12}-44 e^{16}\right ) x\right )}{45 \left (12-11 e^4\right ) \left (1-e^4\right )^4 \left (4 e^4-2 e^4 x-3 \left (1-e^4\right ) x^2\right )}-\frac {\left (8 e^8 \left (7-4 e^4\right )\right ) \operatorname {Subst}\left (\int \frac {1}{4 e^4 \left (12-11 e^4\right )-x^2} \, dx,x,-2 e^4-6 \left (1-e^4\right ) x\right )}{15 \left (1-e^4\right )^3}+\frac {\left (16 e^8 \left (9-9 e^4+e^8\right )\right ) \int \frac {1}{\left (4 e^4-2 e^4 x-3 \left (1-e^4\right ) x^2\right )^2} \, dx}{15 \left (1-e^4\right )^4}-\frac {\left (4 e^4 \left (54+30 e^4-119 e^8+44 e^{12}\right )\right ) \int \frac {1}{4 e^4-2 e^4 x-3 \left (1-e^4\right ) x^2} \, dx}{15 \left (12-11 e^4\right ) \left (1-e^4\right )^3}\\ &=\frac {e^8 x}{5 \left (1-e^4\right )^2}-\frac {16 e^8 \left (2 e^4 \left (72-126 e^4+55 e^8\right )-\left (108-207 e^4+111 e^8-11 e^{12}\right ) x\right )}{45 \left (12-11 e^4\right ) \left (1-e^4\right )^4 \left (4 e^4-2 e^4 x-3 \left (1-e^4\right ) x^2\right )^2}+\frac {8 e^4 \left (9-9 e^4+e^8\right ) \left (e^4+3 \left (1-e^4\right ) x\right )}{15 \left (12-11 e^4\right ) \left (1-e^4\right )^4 \left (4 e^4-2 e^4 x-3 \left (1-e^4\right ) x^2\right )}+\frac {4 e^4 \left (2 e^4 \left (45-27 e^4-86 e^8+66 e^{12}\right )-3 \left (54-24 e^4-149 e^8+163 e^{12}-44 e^{16}\right ) x\right )}{45 \left (12-11 e^4\right ) \left (1-e^4\right )^4 \left (4 e^4-2 e^4 x-3 \left (1-e^4\right ) x^2\right )}-\frac {4 e^6 \left (7-4 e^4\right ) \tan ^{-1}\left (\frac {e^4 (1-3 x)+3 x}{e^2 \sqrt {-12+11 e^4}}\right )}{15 \left (1-e^4\right )^3 \sqrt {-12+11 e^4}}+\frac {\left (8 e^4 \left (9-9 e^4+e^8\right )\right ) \int \frac {1}{4 e^4-2 e^4 x-3 \left (1-e^4\right ) x^2} \, dx}{5 \left (12-11 e^4\right ) \left (1-e^4\right )^3}+\frac {\left (8 e^4 \left (54+30 e^4-119 e^8+44 e^{12}\right )\right ) \operatorname {Subst}\left (\int \frac {1}{4 e^4 \left (12-11 e^4\right )-x^2} \, dx,x,-2 e^4-6 \left (1-e^4\right ) x\right )}{15 \left (12-11 e^4\right ) \left (1-e^4\right )^3}\\ &=\frac {e^8 x}{5 \left (1-e^4\right )^2}-\frac {16 e^8 \left (2 e^4 \left (72-126 e^4+55 e^8\right )-\left (108-207 e^4+111 e^8-11 e^{12}\right ) x\right )}{45 \left (12-11 e^4\right ) \left (1-e^4\right )^4 \left (4 e^4-2 e^4 x-3 \left (1-e^4\right ) x^2\right )^2}+\frac {8 e^4 \left (9-9 e^4+e^8\right ) \left (e^4+3 \left (1-e^4\right ) x\right )}{15 \left (12-11 e^4\right ) \left (1-e^4\right )^4 \left (4 e^4-2 e^4 x-3 \left (1-e^4\right ) x^2\right )}+\frac {4 e^4 \left (2 e^4 \left (45-27 e^4-86 e^8+66 e^{12}\right )-3 \left (54-24 e^4-149 e^8+163 e^{12}-44 e^{16}\right ) x\right )}{45 \left (12-11 e^4\right ) \left (1-e^4\right )^4 \left (4 e^4-2 e^4 x-3 \left (1-e^4\right ) x^2\right )}-\frac {4 e^6 \left (7-4 e^4\right ) \tan ^{-1}\left (\frac {e^4 (1-3 x)+3 x}{e^2 \sqrt {-12+11 e^4}}\right )}{15 \left (1-e^4\right )^3 \sqrt {-12+11 e^4}}-\frac {4 e^2 \left (54+30 e^4-119 e^8+44 e^{12}\right ) \tan ^{-1}\left (\frac {e^4 (1-3 x)+3 x}{e^2 \sqrt {-12+11 e^4}}\right )}{15 \left (1-e^4\right )^3 \left (-12+11 e^4\right )^{3/2}}-\frac {\left (16 e^4 \left (9-9 e^4+e^8\right )\right ) \operatorname {Subst}\left (\int \frac {1}{4 e^4 \left (12-11 e^4\right )-x^2} \, dx,x,-2 e^4-6 \left (1-e^4\right ) x\right )}{5 \left (12-11 e^4\right ) \left (1-e^4\right )^3}\\ &=\frac {e^8 x}{5 \left (1-e^4\right )^2}-\frac {16 e^8 \left (2 e^4 \left (72-126 e^4+55 e^8\right )-\left (108-207 e^4+111 e^8-11 e^{12}\right ) x\right )}{45 \left (12-11 e^4\right ) \left (1-e^4\right )^4 \left (4 e^4-2 e^4 x-3 \left (1-e^4\right ) x^2\right )^2}+\frac {8 e^4 \left (9-9 e^4+e^8\right ) \left (e^4+3 \left (1-e^4\right ) x\right )}{15 \left (12-11 e^4\right ) \left (1-e^4\right )^4 \left (4 e^4-2 e^4 x-3 \left (1-e^4\right ) x^2\right )}+\frac {4 e^4 \left (2 e^4 \left (45-27 e^4-86 e^8+66 e^{12}\right )-3 \left (54-24 e^4-149 e^8+163 e^{12}-44 e^{16}\right ) x\right )}{45 \left (12-11 e^4\right ) \left (1-e^4\right )^4 \left (4 e^4-2 e^4 x-3 \left (1-e^4\right ) x^2\right )}-\frac {4 e^6 \left (7-4 e^4\right ) \tan ^{-1}\left (\frac {e^4 (1-3 x)+3 x}{e^2 \sqrt {-12+11 e^4}}\right )}{15 \left (1-e^4\right )^3 \sqrt {-12+11 e^4}}+\frac {8 e^2 \left (9-9 e^4+e^8\right ) \tan ^{-1}\left (\frac {e^4 (1-3 x)+3 x}{e^2 \sqrt {-12+11 e^4}}\right )}{5 \left (1-e^4\right )^3 \left (-12+11 e^4\right )^{3/2}}-\frac {4 e^2 \left (54+30 e^4-119 e^8+44 e^{12}\right ) \tan ^{-1}\left (\frac {e^4 (1-3 x)+3 x}{e^2 \sqrt {-12+11 e^4}}\right )}{15 \left (1-e^4\right )^3 \left (-12+11 e^4\right )^{3/2}}\\ \end {aligned} \end {gather*}

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Mathematica [B]  time = 0.08, size = 121, normalized size = 4.17 \begin {gather*} \frac {e^8 \left (3 e^{12} x \left (4-2 x+3 x^2\right )^2-e^8 (4+9 x) \left (4-2 x+3 x^2\right )^2-3 x \left (16-16 x+28 x^2+9 x^4\right )+3 e^4 x \left (48-16 x+68 x^2-12 x^3+27 x^4\right )\right )}{15 \left (-1+e^4\right )^3 \left (-3 x^2+e^4 \left (4-2 x+3 x^2\right )\right )^2} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(E^8*(144*x^2 - 96*x^3 + 84*x^4 - 27*x^6) + E^12*(64 - 96*x + 192*x^2 - 152*x^3 + 144*x^4 - 54*x^5 +
27*x^6))/(-135*x^6 + E^8*(-720*x^2 + 720*x^3 - 1260*x^4 + 540*x^5 - 405*x^6) + E^12*(320 - 480*x + 960*x^2 -
760*x^3 + 720*x^4 - 270*x^5 + 135*x^6) + E^4*(540*x^4 - 270*x^5 + 405*x^6)),x]

[Out]

(E^8*(3*E^12*x*(4 - 2*x + 3*x^2)^2 - E^8*(4 + 9*x)*(4 - 2*x + 3*x^2)^2 - 3*x*(16 - 16*x + 28*x^2 + 9*x^4) + 3*
E^4*x*(48 - 16*x + 68*x^2 - 12*x^3 + 27*x^4)))/(15*(-1 + E^4)^3*(-3*x^2 + E^4*(4 - 2*x + 3*x^2))^2)

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fricas [B]  time = 0.80, size = 235, normalized size = 8.10 \begin {gather*} -\frac {3 \, {\left (9 \, x^{5} - 12 \, x^{4} + 28 \, x^{3} - 16 \, x^{2} + 16 \, x\right )} e^{20} - {\left (81 \, x^{5} - 72 \, x^{4} + 204 \, x^{3} - 32 \, x^{2} + 80 \, x + 64\right )} e^{16} + 3 \, {\left (27 \, x^{5} - 12 \, x^{4} + 68 \, x^{3} - 16 \, x^{2} + 48 \, x\right )} e^{12} - 3 \, {\left (9 \, x^{5} + 28 \, x^{3} - 16 \, x^{2} + 16 \, x\right )} e^{8}}{15 \, {\left (9 \, x^{4} - {\left (9 \, x^{4} - 12 \, x^{3} + 28 \, x^{2} - 16 \, x + 16\right )} e^{20} + 3 \, {\left (15 \, x^{4} - 16 \, x^{3} + 36 \, x^{2} - 16 \, x + 16\right )} e^{16} - 6 \, {\left (15 \, x^{4} - 12 \, x^{3} + 26 \, x^{2} - 8 \, x + 8\right )} e^{12} + 2 \, {\left (45 \, x^{4} - 24 \, x^{3} + 50 \, x^{2} - 8 \, x + 8\right )} e^{8} - 3 \, {\left (15 \, x^{4} - 4 \, x^{3} + 8 \, x^{2}\right )} e^{4}\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((27*x^6-54*x^5+144*x^4-152*x^3+192*x^2-96*x+64)*exp(4)^3+(-27*x^6+84*x^4-96*x^3+144*x^2)*exp(4)^2)/
((135*x^6-270*x^5+720*x^4-760*x^3+960*x^2-480*x+320)*exp(4)^3+(-405*x^6+540*x^5-1260*x^4+720*x^3-720*x^2)*exp(
4)^2+(405*x^6-270*x^5+540*x^4)*exp(4)-135*x^6),x, algorithm="fricas")

[Out]

-1/15*(3*(9*x^5 - 12*x^4 + 28*x^3 - 16*x^2 + 16*x)*e^20 - (81*x^5 - 72*x^4 + 204*x^3 - 32*x^2 + 80*x + 64)*e^1
6 + 3*(27*x^5 - 12*x^4 + 68*x^3 - 16*x^2 + 48*x)*e^12 - 3*(9*x^5 + 28*x^3 - 16*x^2 + 16*x)*e^8)/(9*x^4 - (9*x^
4 - 12*x^3 + 28*x^2 - 16*x + 16)*e^20 + 3*(15*x^4 - 16*x^3 + 36*x^2 - 16*x + 16)*e^16 - 6*(15*x^4 - 12*x^3 + 2
6*x^2 - 8*x + 8)*e^12 + 2*(45*x^4 - 24*x^3 + 50*x^2 - 8*x + 8)*e^8 - 3*(15*x^4 - 4*x^3 + 8*x^2)*e^4)

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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(((27*x^6-54*x^5+144*x^4-152*x^3+192*x^2-96*x+64)*exp(4)^3+(-27*x^6+84*x^4-96*x^3+144*x^2)*exp(4)^2)/
((135*x^6-270*x^5+720*x^4-760*x^3+960*x^2-480*x+320)*exp(4)^3+(-405*x^6+540*x^5-1260*x^4+720*x^3-720*x^2)*exp(
4)^2+(405*x^6-270*x^5+540*x^4)*exp(4)-135*x^6),x, algorithm="giac")

[Out]

Timed out

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maple [B]  time = 0.42, size = 114, normalized size = 3.93

method	result	size
norman	\(\frac {\frac {9 x^{5} {\mathrm e}^{8}}{5}+\frac {\left (48 \,{\mathrm e}^{12}-96 \,{\mathrm e}^{8}\right ) {\mathrm e}^{-4} x^{2}}{20}+\frac {\left (64 \,{\mathrm e}^{12}+48 \,{\mathrm e}^{8}\right ) {\mathrm e}^{-4} x^{3}}{20}-\frac {3 \left (16 \,{\mathrm e}^{16}+96 \,{\mathrm e}^{12}-48 \,{\mathrm e}^{8}\right ) {\mathrm e}^{-8} x^{4}}{80}+\frac {16 \,{\mathrm e}^{8}}{5}}{\left (3 x^{2} {\mathrm e}^{4}-2 x \,{\mathrm e}^{4}-3 x^{2}+4 \,{\mathrm e}^{4}\right )^{2}}\)	\(114\)
gosper	\(\frac {9 x^{5} {\mathrm e}^{8}-3 x^{4} {\mathrm e}^{8}+16 x^{3} {\mathrm e}^{8}-18 x^{4} {\mathrm e}^{4}+12 x^{2} {\mathrm e}^{8}+12 x^{3} {\mathrm e}^{4}+9 x^{4}-24 x^{2} {\mathrm e}^{4}+16 \,{\mathrm e}^{8}}{45 x^{4} {\mathrm e}^{8}-60 x^{3} {\mathrm e}^{8}-90 x^{4} {\mathrm e}^{4}+140 x^{2} {\mathrm e}^{8}+60 x^{3} {\mathrm e}^{4}+45 x^{4}-80 x \,{\mathrm e}^{8}-120 x^{2} {\mathrm e}^{4}+80 \,{\mathrm e}^{8}}\)	\(141\)
risch	\(\frac {x \,{\mathrm e}^{8}}{5 \,{\mathrm e}^{8}-10 \,{\mathrm e}^{4}+5}+\frac {\left (-\frac {16 \,{\mathrm e}^{12}}{9}+\frac {28 \,{\mathrm e}^{8}}{9}\right ) x^{3}-\frac {16 \,{\mathrm e}^{8} \left ({\mathrm e}^{8}+3 \,{\mathrm e}^{4}-3\right ) x^{2}}{27 \left ({\mathrm e}^{4}-1\right )}-\frac {16 \,{\mathrm e}^{8} \left (2 \,{\mathrm e}^{8}-9 \,{\mathrm e}^{4}+3\right ) x}{27 \left ({\mathrm e}^{4}-1\right )}-\frac {64 \,{\mathrm e}^{16}}{27 \left ({\mathrm e}^{4}-1\right )}}{\left (5 \,{\mathrm e}^{8}-10 \,{\mathrm e}^{4}+5\right ) \left (x^{4} {\mathrm e}^{8}-\frac {4 x^{3} {\mathrm e}^{8}}{3}-2 x^{4} {\mathrm e}^{4}+\frac {28 x^{2} {\mathrm e}^{8}}{9}+\frac {4 x^{3} {\mathrm e}^{4}}{3}+x^{4}-\frac {16 x \,{\mathrm e}^{8}}{9}-\frac {8 x^{2} {\mathrm e}^{4}}{3}+\frac {16 \,{\mathrm e}^{8}}{9}\right )}\)	\(153\)
default	\(\frac {{\mathrm e}^{8} \left (\frac {\left ({\mathrm e}^{4}-1\right ) x}{3 \,{\mathrm e}^{4}+{\mathrm e}^{12}-3 \,{\mathrm e}^{8}-1}+\frac {2 \left (\munderset {\textit {\_R} =\RootOf \left (\left (81 \,{\mathrm e}^{4}+27 \,{\mathrm e}^{12}-81 \,{\mathrm e}^{8}-27\right ) \textit {\_Z}^{6}+\left (-54 \,{\mathrm e}^{4}-54 \,{\mathrm e}^{12}+108 \,{\mathrm e}^{8}\right ) \textit {\_Z}^{5}+\left (108 \,{\mathrm e}^{4}+144 \,{\mathrm e}^{12}-252 \,{\mathrm e}^{8}\right ) \textit {\_Z}^{4}+\left (-152 \,{\mathrm e}^{12}+144 \,{\mathrm e}^{8}\right ) \textit {\_Z}^{3}+\left (192 \,{\mathrm e}^{12}-144 \,{\mathrm e}^{8}\right ) \textit {\_Z}^{2}-96 \textit {\_Z} \,{\mathrm e}^{12}+64 \,{\mathrm e}^{12}\right )}{\sum }\frac {\left (3 \left (-15 \,{\mathrm e}^{8}+4 \,{\mathrm e}^{12}+18 \,{\mathrm e}^{4}-7\right ) \textit {\_R}^{4}+2 \left (12-17 \,{\mathrm e}^{4}+8 \,{\mathrm e}^{12}-3 \,{\mathrm e}^{8}\right ) \textit {\_R}^{3}+12 \left (-2 \,{\mathrm e}^{12}+5 \,{\mathrm e}^{4}-3\right ) \textit {\_R}^{2}+24 \left ({\mathrm e}^{4}+2 \,{\mathrm e}^{12}-3 \,{\mathrm e}^{8}\right ) \textit {\_R} -16 \,{\mathrm e}^{4}-32 \,{\mathrm e}^{12}+48 \,{\mathrm e}^{8}\right ) \ln \left (x -\textit {\_R} \right )}{27 \,{\mathrm e}^{12} \textit {\_R}^{5}-45 \textit {\_R}^{4} {\mathrm e}^{12}-81 \textit {\_R}^{5} {\mathrm e}^{8}+96 \textit {\_R}^{3} {\mathrm e}^{12}+90 \textit {\_R}^{4} {\mathrm e}^{8}+81 \textit {\_R}^{5} {\mathrm e}^{4}-76 \textit {\_R}^{2} {\mathrm e}^{12}-168 \textit {\_R}^{3} {\mathrm e}^{8}-45 \textit {\_R}^{4} {\mathrm e}^{4}-27 \textit {\_R}^{5}+64 \textit {\_R} \,{\mathrm e}^{12}+72 \textit {\_R}^{2} {\mathrm e}^{8}+72 \textit {\_R}^{3} {\mathrm e}^{4}-16 \,{\mathrm e}^{12}-48 \textit {\_R} \,{\mathrm e}^{8}}\right )}{3 \left (3 \,{\mathrm e}^{4}+{\mathrm e}^{12}-3 \,{\mathrm e}^{8}-1\right )}\right )}{5}\)	\(322\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((27*x^6-54*x^5+144*x^4-152*x^3+192*x^2-96*x+64)*exp(4)^3+(-27*x^6+84*x^4-96*x^3+144*x^2)*exp(4)^2)/((135*
x^6-270*x^5+720*x^4-760*x^3+960*x^2-480*x+320)*exp(4)^3+(-405*x^6+540*x^5-1260*x^4+720*x^3-720*x^2)*exp(4)^2+(
405*x^6-270*x^5+540*x^4)*exp(4)-135*x^6),x,method=_RETURNVERBOSE)

[Out]

(9/5*x^5*exp(4)^2+1/20*(48*exp(4)^3-96*exp(4)^2)/exp(4)*x^2+1/20*(64*exp(4)^3+48*exp(4)^2)/exp(4)*x^3-3/80*(16
*exp(4)^4+96*exp(4)^3-48*exp(4)^2)/exp(4)^2*x^4+16/5*exp(4)^2)/(3*x^2*exp(4)-2*x*exp(4)-3*x^2+4*exp(4))^2

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maxima [B]  time = 0.41, size = 183, normalized size = 6.31 \begin {gather*} \frac {x e^{8}}{5 \, {\left (e^{8} - 2 \, e^{4} + 1\right )}} - \frac {4 \, {\left (3 \, x^{3} {\left (4 \, e^{16} - 11 \, e^{12} + 7 \, e^{8}\right )} + 4 \, x^{2} {\left (e^{16} + 3 \, e^{12} - 3 \, e^{8}\right )} + 4 \, x {\left (2 \, e^{16} - 9 \, e^{12} + 3 \, e^{8}\right )} + 16 \, e^{16}\right )}}{15 \, {\left (9 \, x^{4} {\left (e^{20} - 5 \, e^{16} + 10 \, e^{12} - 10 \, e^{8} + 5 \, e^{4} - 1\right )} - 12 \, x^{3} {\left (e^{20} - 4 \, e^{16} + 6 \, e^{12} - 4 \, e^{8} + e^{4}\right )} + 4 \, x^{2} {\left (7 \, e^{20} - 27 \, e^{16} + 39 \, e^{12} - 25 \, e^{8} + 6 \, e^{4}\right )} - 16 \, x {\left (e^{20} - 3 \, e^{16} + 3 \, e^{12} - e^{8}\right )} + 16 \, e^{20} - 48 \, e^{16} + 48 \, e^{12} - 16 \, e^{8}\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((27*x^6-54*x^5+144*x^4-152*x^3+192*x^2-96*x+64)*exp(4)^3+(-27*x^6+84*x^4-96*x^3+144*x^2)*exp(4)^2)/
((135*x^6-270*x^5+720*x^4-760*x^3+960*x^2-480*x+320)*exp(4)^3+(-405*x^6+540*x^5-1260*x^4+720*x^3-720*x^2)*exp(
4)^2+(405*x^6-270*x^5+540*x^4)*exp(4)-135*x^6),x, algorithm="maxima")

[Out]

1/5*x*e^8/(e^8 - 2*e^4 + 1) - 4/15*(3*x^3*(4*e^16 - 11*e^12 + 7*e^8) + 4*x^2*(e^16 + 3*e^12 - 3*e^8) + 4*x*(2*
e^16 - 9*e^12 + 3*e^8) + 16*e^16)/(9*x^4*(e^20 - 5*e^16 + 10*e^12 - 10*e^8 + 5*e^4 - 1) - 12*x^3*(e^20 - 4*e^1
6 + 6*e^12 - 4*e^8 + e^4) + 4*x^2*(7*e^20 - 27*e^16 + 39*e^12 - 25*e^8 + 6*e^4) - 16*x*(e^20 - 3*e^16 + 3*e^12
- e^8) + 16*e^20 - 48*e^16 + 48*e^12 - 16*e^8)

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mupad [B]  time = 1.49, size = 120, normalized size = 4.14 \begin {gather*} \frac {x\,{\mathrm {e}}^8}{5\,{\left ({\mathrm {e}}^4-1\right )}^2}+\frac {16\,{\mathrm {e}}^8\,\left (9\,x-12\,{\mathrm {e}}^4+10\,{\mathrm {e}}^8-9\,x\,{\mathrm {e}}^4+x\,{\mathrm {e}}^8\right )}{45\,{\left ({\mathrm {e}}^4-1\right )}^4\,{\left (\left (3\,{\mathrm {e}}^4-3\right )\,x^2-2\,{\mathrm {e}}^4\,x+4\,{\mathrm {e}}^4\right )}^2}-\frac {4\,{\mathrm {e}}^8\,\left (21\,x-2\,{\mathrm {e}}^4+12\,{\mathrm {e}}^8-33\,x\,{\mathrm {e}}^4+12\,x\,{\mathrm {e}}^8-12\right )}{45\,{\left ({\mathrm {e}}^4-1\right )}^4\,\left (\left (3\,{\mathrm {e}}^4-3\right )\,x^2-2\,{\mathrm {e}}^4\,x+4\,{\mathrm {e}}^4\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((exp(12)*(192*x^2 - 96*x - 152*x^3 + 144*x^4 - 54*x^5 + 27*x^6 + 64) + exp(8)*(144*x^2 - 96*x^3 + 84*x^4 -
27*x^6))/(exp(12)*(960*x^2 - 480*x - 760*x^3 + 720*x^4 - 270*x^5 + 135*x^6 + 320) - exp(8)*(720*x^2 - 720*x^3
+ 1260*x^4 - 540*x^5 + 405*x^6) + exp(4)*(540*x^4 - 270*x^5 + 405*x^6) - 135*x^6),x)

[Out]

(x*exp(8))/(5*(exp(4) - 1)^2) + (16*exp(8)*(9*x - 12*exp(4) + 10*exp(8) - 9*x*exp(4) + x*exp(8)))/(45*(exp(4)
- 1)^4*(4*exp(4) - 2*x*exp(4) + x^2*(3*exp(4) - 3))^2) - (4*exp(8)*(21*x - 2*exp(4) + 12*exp(8) - 33*x*exp(4)
+ 12*x*exp(8) - 12))/(45*(exp(4) - 1)^4*(4*exp(4) - 2*x*exp(4) + x^2*(3*exp(4) - 3)))

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sympy [B]  time = 5.23, size = 206, normalized size = 7.10 \begin {gather*} \frac {x e^{8}}{- 10 e^{4} + 5 + 5 e^{8}} + \frac {x^{3} \left (- 48 e^{16} - 84 e^{8} + 132 e^{12}\right ) + x^{2} \left (- 16 e^{16} - 48 e^{12} + 48 e^{8}\right ) + x \left (- 32 e^{16} - 48 e^{8} + 144 e^{12}\right ) - 64 e^{16}}{x^{4} \left (- 675 e^{16} - 1350 e^{8} - 135 + 675 e^{4} + 1350 e^{12} + 135 e^{20}\right ) + x^{3} \left (- 180 e^{20} - 1080 e^{12} - 180 e^{4} + 720 e^{8} + 720 e^{16}\right ) + x^{2} \left (- 1620 e^{16} - 1500 e^{8} + 360 e^{4} + 2340 e^{12} + 420 e^{20}\right ) + x \left (- 240 e^{20} - 720 e^{12} + 240 e^{8} + 720 e^{16}\right ) - 720 e^{16} - 240 e^{8} + 720 e^{12} + 240 e^{20}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((27*x**6-54*x**5+144*x**4-152*x**3+192*x**2-96*x+64)*exp(4)**3+(-27*x**6+84*x**4-96*x**3+144*x**2)*
exp(4)**2)/((135*x**6-270*x**5+720*x**4-760*x**3+960*x**2-480*x+320)*exp(4)**3+(-405*x**6+540*x**5-1260*x**4+7
20*x**3-720*x**2)*exp(4)**2+(405*x**6-270*x**5+540*x**4)*exp(4)-135*x**6),x)

[Out]

x*exp(8)/(-10*exp(4) + 5 + 5*exp(8)) + (x**3*(-48*exp(16) - 84*exp(8) + 132*exp(12)) + x**2*(-16*exp(16) - 48*
exp(12) + 48*exp(8)) + x*(-32*exp(16) - 48*exp(8) + 144*exp(12)) - 64*exp(16))/(x**4*(-675*exp(16) - 1350*exp(
8) - 135 + 675*exp(4) + 1350*exp(12) + 135*exp(20)) + x**3*(-180*exp(20) - 1080*exp(12) - 180*exp(4) + 720*exp
(8) + 720*exp(16)) + x**2*(-1620*exp(16) - 1500*exp(8) + 360*exp(4) + 2340*exp(12) + 420*exp(20)) + x*(-240*ex
p(20) - 720*exp(12) + 240*exp(8) + 720*exp(16)) - 720*exp(16) - 240*exp(8) + 720*exp(12) + 240*exp(20))

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