3.45.99 \(\int \frac {-839808 x-2239488 x^2-2620512 x^3-1670400 x^4-595968 x^5-110592 x^6-8192 x^7+e^{8 x} (-13436928 x-53747712 x^2-10077696 x^3+13436928 x^4)+e^{6 x} (-26873856 x-98537472 x^2-55987200 x^3+12690432 x^4+8957952 x^5)+e^{4 x} (-20155392 x-67184640 x^2-58910976 x^3-9082368 x^4+6967296 x^5+1990656 x^6)+e^{2 x} (-6718464 x-20155392 x^2-21959424 x^3-9374976 x^4-340992 x^5+774144 x^6+147456 x^7)}{-6718464+8398080 x^2-4199040 x^4+1049760 x^6-131220 x^8+6561 x^{10}} \, dx\)

Optimal. Leaf size=27 \[ \frac {\left (2+4 e^{2 x}+\frac {8 x}{9}\right )^4 x^2}{\left (-4+x^2\right )^4} \]

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Rubi [B]  time = 3.94, antiderivative size = 532, normalized size of antiderivative = 19.70, number of steps used = 130, number of rules used = 14, integrand size = 194, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.072, Rules used = {6688, 12, 6742, 261, 288, 199, 207, 266, 43, 264, 2288, 2177, 2178, 2269} \begin {gather*} \frac {1280 x}{729 \left (4-x^2\right )^2}-\frac {12896 x}{729 \left (4-x^2\right )^3}+\frac {128 x}{3 \left (4-x^2\right )^4}+\frac {49664}{2187 \left (4-x^2\right )^2}-\frac {1231376}{6561 \left (4-x^2\right )^3}+\frac {869056}{2187 \left (4-x^2\right )^4}+\frac {256 x^8}{6561 \left (4-x^2\right )^4}+\frac {512 x^5}{243 \left (4-x^2\right )^4}-\frac {1280 x^3}{729 \left (4-x^2\right )^3}+\frac {23200 x^3}{729 \left (4-x^2\right )^4}+\frac {256 e^{8 x} \left (4 x-x^3\right ) x}{\left (4-x^2\right )^5}+\frac {37 e^{2 x}}{216 (2-x)}-\frac {17 e^{4 x}}{216 (2-x)}-\frac {e^{6 x}}{2 (2-x)}+\frac {37 e^{2 x}}{216 (x+2)}-\frac {17 e^{4 x}}{216 (x+2)}-\frac {e^{6 x}}{2 (x+2)}+\frac {1615 e^{2 x}}{2916 (2-x)^2}-\frac {161 e^{4 x}}{108 (2-x)^2}-\frac {17 e^{6 x}}{9 (2-x)^2}+\frac {383 e^{2 x}}{2916 (x+2)^2}+\frac {127 e^{4 x}}{108 (x+2)^2}-\frac {e^{6 x}}{9 (x+2)^2}-\frac {2312 e^{2 x}}{243 (2-x)^3}-\frac {272 e^{4 x}}{27 (2-x)^3}-\frac {32 e^{6 x}}{9 (2-x)^3}+\frac {8 e^{2 x}}{243 (x+2)^3}+\frac {16 e^{4 x}}{27 (x+2)^3}+\frac {32 e^{6 x}}{9 (x+2)^3}+\frac {9826 e^{2 x}}{729 (2-x)^4}+\frac {578 e^{4 x}}{27 (2-x)^4}+\frac {136 e^{6 x}}{9 (2-x)^4}+\frac {2 e^{2 x}}{729 (x+2)^4}+\frac {2 e^{4 x}}{27 (x+2)^4}+\frac {8 e^{6 x}}{9 (x+2)^4} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(-839808*x - 2239488*x^2 - 2620512*x^3 - 1670400*x^4 - 595968*x^5 - 110592*x^6 - 8192*x^7 + E^(8*x)*(-1343
6928*x - 53747712*x^2 - 10077696*x^3 + 13436928*x^4) + E^(6*x)*(-26873856*x - 98537472*x^2 - 55987200*x^3 + 12
690432*x^4 + 8957952*x^5) + E^(4*x)*(-20155392*x - 67184640*x^2 - 58910976*x^3 - 9082368*x^4 + 6967296*x^5 + 1
990656*x^6) + E^(2*x)*(-6718464*x - 20155392*x^2 - 21959424*x^3 - 9374976*x^4 - 340992*x^5 + 774144*x^6 + 1474
56*x^7))/(-6718464 + 8398080*x^2 - 4199040*x^4 + 1049760*x^6 - 131220*x^8 + 6561*x^10),x]

[Out]

(9826*E^(2*x))/(729*(2 - x)^4) + (578*E^(4*x))/(27*(2 - x)^4) + (136*E^(6*x))/(9*(2 - x)^4) - (2312*E^(2*x))/(
243*(2 - x)^3) - (272*E^(4*x))/(27*(2 - x)^3) - (32*E^(6*x))/(9*(2 - x)^3) + (1615*E^(2*x))/(2916*(2 - x)^2) -
(161*E^(4*x))/(108*(2 - x)^2) - (17*E^(6*x))/(9*(2 - x)^2) + (37*E^(2*x))/(216*(2 - x)) - (17*E^(4*x))/(216*(
2 - x)) - E^(6*x)/(2*(2 - x)) + (2*E^(2*x))/(729*(2 + x)^4) + (2*E^(4*x))/(27*(2 + x)^4) + (8*E^(6*x))/(9*(2 +
x)^4) + (8*E^(2*x))/(243*(2 + x)^3) + (16*E^(4*x))/(27*(2 + x)^3) + (32*E^(6*x))/(9*(2 + x)^3) + (383*E^(2*x)
)/(2916*(2 + x)^2) + (127*E^(4*x))/(108*(2 + x)^2) - E^(6*x)/(9*(2 + x)^2) + (37*E^(2*x))/(216*(2 + x)) - (17*
E^(4*x))/(216*(2 + x)) - E^(6*x)/(2*(2 + x)) + 869056/(2187*(4 - x^2)^4) + (128*x)/(3*(4 - x^2)^4) + (23200*x^
3)/(729*(4 - x^2)^4) + (512*x^5)/(243*(4 - x^2)^4) + (256*x^8)/(6561*(4 - x^2)^4) - 1231376/(6561*(4 - x^2)^3)
- (12896*x)/(729*(4 - x^2)^3) - (1280*x^3)/(729*(4 - x^2)^3) + 49664/(2187*(4 - x^2)^2) + (1280*x)/(729*(4 -
x^2)^2) + (256*E^(8*x)*x*(4*x - x^3))/(4 - x^2)^5

Rule 12

Int[(a_)*(u_), x_Symbol] :> Dist[a, Int[u, x], x] /; FreeQ[a, x] &&  !MatchQ[u, (b_)*(v_) /; FreeQ[b, x]]

Rule 43

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, n}, x] && NeQ[b*c - a*d, 0] && IGtQ[m, 0] && ( !IntegerQ[n] || (EqQ[c, 0]
&& LeQ[7*m + 4*n + 4, 0]) || LtQ[9*m + 5*(n + 1), 0] || GtQ[m + n + 2, 0])

Rule 199

Int[((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> -Simp[(x*(a + b*x^n)^(p + 1))/(a*n*(p + 1)), x] + Dist[(n*(p +
1) + 1)/(a*n*(p + 1)), Int[(a + b*x^n)^(p + 1), x], x] /; FreeQ[{a, b}, x] && IGtQ[n, 0] && LtQ[p, -1] && (In
tegerQ[2*p] || (n == 2 && IntegerQ[4*p]) || (n == 2 && IntegerQ[3*p]) || Denominator[p + 1/n] < Denominator[p]
)

Rule 207

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

Rule 261

Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[(a + b*x^n)^(p + 1)/(b*n*(p + 1)), x] /; FreeQ
[{a, b, m, n, p}, x] && EqQ[m, n - 1] && NeQ[p, -1]

Rule 264

Int[((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[((c*x)^(m + 1)*(a + b*x^n)^(p + 1))/(a
*c*(m + 1)), x] /; FreeQ[{a, b, c, m, n, p}, x] && EqQ[(m + 1)/n + p + 1, 0] && NeQ[m, -1]

Rule 266

Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Dist[1/n, Subst[Int[x^(Simplify[(m + 1)/n] - 1)*(a
+ b*x)^p, x], x, x^n], x] /; FreeQ[{a, b, m, n, p}, x] && IntegerQ[Simplify[(m + 1)/n]]

Rule 288

Int[((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[(c^(n - 1)*(c*x)^(m - n + 1)*(a + b*x^
n)^(p + 1))/(b*n*(p + 1)), x] - Dist[(c^n*(m - n + 1))/(b*n*(p + 1)), Int[(c*x)^(m - n)*(a + b*x^n)^(p + 1), x
], x] /; FreeQ[{a, b, c}, x] && IGtQ[n, 0] && LtQ[p, -1] && GtQ[m + 1, n] &&  !ILtQ[(m + n*(p + 1) + 1)/n, 0]
&& IntBinomialQ[a, b, c, n, m, p, x]

Rule 2177

Int[((b_.)*(F_)^((g_.)*((e_.) + (f_.)*(x_))))^(n_.)*((c_.) + (d_.)*(x_))^(m_), x_Symbol] :> Simp[((c + d*x)^(m
+ 1)*(b*F^(g*(e + f*x)))^n)/(d*(m + 1)), x] - Dist[(f*g*n*Log[F])/(d*(m + 1)), Int[(c + d*x)^(m + 1)*(b*F^(g*
(e + f*x)))^n, x], x] /; FreeQ[{F, b, c, d, e, f, g, n}, x] && LtQ[m, -1] && IntegerQ[2*m] &&  !$UseGamma ===
True

Rule 2178

Int[(F_)^((g_.)*((e_.) + (f_.)*(x_)))/((c_.) + (d_.)*(x_)), x_Symbol] :> Simp[(F^(g*(e - (c*f)/d))*ExpIntegral
Ei[(f*g*(c + d*x)*Log[F])/d])/d, x] /; FreeQ[{F, c, d, e, f, g}, x] &&  !$UseGamma === True

Rule 2269

Int[(F_)^((g_.)*((d_.) + (e_.)*(x_))^(n_.))/((a_) + (c_.)*(x_)^2), x_Symbol] :> Int[ExpandIntegrand[F^(g*(d +
e*x)^n), 1/(a + c*x^2), x], x] /; FreeQ[{F, a, c, d, e, g, n}, x]

Rule 2288

Int[(y_.)*(F_)^(u_)*((v_) + (w_)), x_Symbol] :> With[{z = (v*y)/(Log[F]*D[u, x])}, Simp[F^u*z, x] /; EqQ[D[z,
x], w*y]] /; FreeQ[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 {32 x \left (9+18 e^{2 x}+4 x\right )^3 \left (36+48 x+27 x^2+4 x^3-18 e^{2 x} \left (-4-16 x-3 x^2+4 x^3\right )\right )}{6561 \left (4-x^2\right )^5} \, dx\\ &=\frac {32 \int \frac {x \left (9+18 e^{2 x}+4 x\right )^3 \left (36+48 x+27 x^2+4 x^3-18 e^{2 x} \left (-4-16 x-3 x^2+4 x^3\right )\right )}{\left (4-x^2\right )^5} \, dx}{6561}\\ &=\frac {32 \int \left (-\frac {26244 x}{\left (-4+x^2\right )^5}-\frac {69984 x^2}{\left (-4+x^2\right )^5}-\frac {81891 x^3}{\left (-4+x^2\right )^5}-\frac {52200 x^4}{\left (-4+x^2\right )^5}-\frac {18624 x^5}{\left (-4+x^2\right )^5}-\frac {3456 x^6}{\left (-4+x^2\right )^5}-\frac {256 x^7}{\left (-4+x^2\right )^5}+\frac {104976 e^{8 x} x \left (-4-16 x-3 x^2+4 x^3\right )}{\left (-4+x^2\right )^5}+\frac {72 e^{2 x} x (9+4 x)^2 \left (-36-76 x-43 x^2+3 x^3+4 x^4\right )}{\left (-4+x^2\right )^5}+\frac {23328 e^{6 x} x \left (-36-132 x-75 x^2+17 x^3+12 x^4\right )}{\left (-4+x^2\right )^5}+\frac {1944 e^{4 x} x \left (-324-1080 x-947 x^2-146 x^3+112 x^4+32 x^5\right )}{\left (-4+x^2\right )^5}\right ) \, dx}{6561}\\ &=\frac {256}{729} \int \frac {e^{2 x} x (9+4 x)^2 \left (-36-76 x-43 x^2+3 x^3+4 x^4\right )}{\left (-4+x^2\right )^5} \, dx-\frac {8192 \int \frac {x^7}{\left (-4+x^2\right )^5} \, dx}{6561}+\frac {256}{27} \int \frac {e^{4 x} x \left (-324-1080 x-947 x^2-146 x^3+112 x^4+32 x^5\right )}{\left (-4+x^2\right )^5} \, dx-\frac {4096}{243} \int \frac {x^6}{\left (-4+x^2\right )^5} \, dx-\frac {198656 \int \frac {x^5}{\left (-4+x^2\right )^5} \, dx}{2187}+\frac {1024}{9} \int \frac {e^{6 x} x \left (-36-132 x-75 x^2+17 x^3+12 x^4\right )}{\left (-4+x^2\right )^5} \, dx-128 \int \frac {x}{\left (-4+x^2\right )^5} \, dx-\frac {185600}{729} \int \frac {x^4}{\left (-4+x^2\right )^5} \, dx-\frac {1024}{3} \int \frac {x^2}{\left (-4+x^2\right )^5} \, dx-\frac {10784}{27} \int \frac {x^3}{\left (-4+x^2\right )^5} \, dx+512 \int \frac {e^{8 x} x \left (-4-16 x-3 x^2+4 x^3\right )}{\left (-4+x^2\right )^5} \, dx\\ &=\frac {16}{\left (4-x^2\right )^4}+\frac {128 x}{3 \left (4-x^2\right )^4}+\frac {23200 x^3}{729 \left (4-x^2\right )^4}+\frac {512 x^5}{243 \left (4-x^2\right )^4}+\frac {256 x^8}{6561 \left (4-x^2\right )^4}+\frac {256 e^{8 x} x \left (4 x-x^3\right )}{\left (4-x^2\right )^5}+\frac {256}{729} \int \left (-\frac {4913 e^{2 x}}{32 (-2+x)^5}-\frac {289 e^{2 x}}{64 (-2+x)^4}+\frac {26129 e^{2 x}}{512 (-2+x)^3}+\frac {7459 e^{2 x}}{2048 (-2+x)^2}-\frac {e^{2 x}}{32 (2+x)^5}-\frac {17 e^{2 x}}{64 (2+x)^4}-\frac {287 e^{2 x}}{512 (2+x)^3}+\frac {533 e^{2 x}}{2048 (2+x)^2}-\frac {999 e^{2 x}}{256 \left (-4+x^2\right )}\right ) \, dx+\frac {256}{27} \int \left (-\frac {289 e^{4 x}}{32 (-2+x)^5}+\frac {187 e^{4 x}}{32 (-2+x)^4}+\frac {2337 e^{4 x}}{512 (-2+x)^3}-\frac {1305 e^{4 x}}{2048 (-2+x)^2}-\frac {e^{4 x}}{32 (2+x)^5}-\frac {5 e^{4 x}}{32 (2+x)^4}+\frac {e^{4 x}}{512 (2+x)^3}+\frac {1033 e^{4 x}}{2048 (2+x)^2}+\frac {17 e^{4 x}}{128 \left (-4+x^2\right )}\right ) \, dx-\frac {2560}{243} \int \frac {x^4}{\left (-4+x^2\right )^4} \, dx-\frac {128}{3} \int \frac {1}{\left (-4+x^2\right )^4} \, dx-\frac {99328 \operatorname {Subst}\left (\int \frac {x^2}{(-4+x)^5} \, dx,x,x^2\right )}{2187}-\frac {23200}{243} \int \frac {x^2}{\left (-4+x^2\right )^4} \, dx+\frac {1024}{9} \int \left (-\frac {17 e^{6 x}}{32 (-2+x)^5}+\frac {45 e^{6 x}}{64 (-2+x)^4}+\frac {113 e^{6 x}}{512 (-2+x)^3}-\frac {213 e^{6 x}}{2048 (-2+x)^2}-\frac {e^{6 x}}{32 (2+x)^5}-\frac {3 e^{6 x}}{64 (2+x)^4}+\frac {97 e^{6 x}}{512 (2+x)^3}-\frac {3 e^{6 x}}{2048 (2+x)^2}+\frac {27 e^{6 x}}{256 \left (-4+x^2\right )}\right ) \, dx-\frac {5392}{27} \operatorname {Subst}\left (\int \frac {x}{(-4+x)^5} \, dx,x,x^2\right )\\ &=\text {Rest of rules removed due to large latex content} \end {aligned} \end {gather*}

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Mathematica [A]  time = 1.26, size = 28, normalized size = 1.04 \begin {gather*} \frac {16 x^2 \left (9+18 e^{2 x}+4 x\right )^4}{6561 \left (-4+x^2\right )^4} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(-839808*x - 2239488*x^2 - 2620512*x^3 - 1670400*x^4 - 595968*x^5 - 110592*x^6 - 8192*x^7 + E^(8*x)*
(-13436928*x - 53747712*x^2 - 10077696*x^3 + 13436928*x^4) + E^(6*x)*(-26873856*x - 98537472*x^2 - 55987200*x^
3 + 12690432*x^4 + 8957952*x^5) + E^(4*x)*(-20155392*x - 67184640*x^2 - 58910976*x^3 - 9082368*x^4 + 6967296*x
^5 + 1990656*x^6) + E^(2*x)*(-6718464*x - 20155392*x^2 - 21959424*x^3 - 9374976*x^4 - 340992*x^5 + 774144*x^6
+ 147456*x^7))/(-6718464 + 8398080*x^2 - 4199040*x^4 + 1049760*x^6 - 131220*x^8 + 6561*x^10),x]

[Out]

(16*x^2*(9 + 18*E^(2*x) + 4*x)^4)/(6561*(-4 + x^2)^4)

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fricas [B]  time = 0.57, size = 125, normalized size = 4.63 \begin {gather*} \frac {16 \, {\left (256 \, x^{6} + 2304 \, x^{5} + 7776 \, x^{4} + 11664 \, x^{3} + 104976 \, x^{2} e^{\left (8 \, x\right )} + 6561 \, x^{2} + 23328 \, {\left (4 \, x^{3} + 9 \, x^{2}\right )} e^{\left (6 \, x\right )} + 1944 \, {\left (16 \, x^{4} + 72 \, x^{3} + 81 \, x^{2}\right )} e^{\left (4 \, x\right )} + 72 \, {\left (64 \, x^{5} + 432 \, x^{4} + 972 \, x^{3} + 729 \, x^{2}\right )} e^{\left (2 \, x\right )}\right )}}{6561 \, {\left (x^{8} - 16 \, x^{6} + 96 \, x^{4} - 256 \, x^{2} + 256\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((13436928*x^4-10077696*x^3-53747712*x^2-13436928*x)*exp(x)^8+(8957952*x^5+12690432*x^4-55987200*x^3
-98537472*x^2-26873856*x)*exp(x)^6+(1990656*x^6+6967296*x^5-9082368*x^4-58910976*x^3-67184640*x^2-20155392*x)*
exp(x)^4+(147456*x^7+774144*x^6-340992*x^5-9374976*x^4-21959424*x^3-20155392*x^2-6718464*x)*exp(x)^2-8192*x^7-
110592*x^6-595968*x^5-1670400*x^4-2620512*x^3-2239488*x^2-839808*x)/(6561*x^10-131220*x^8+1049760*x^6-4199040*
x^4+8398080*x^2-6718464),x, algorithm="fricas")

[Out]

16/6561*(256*x^6 + 2304*x^5 + 7776*x^4 + 11664*x^3 + 104976*x^2*e^(8*x) + 6561*x^2 + 23328*(4*x^3 + 9*x^2)*e^(
6*x) + 1944*(16*x^4 + 72*x^3 + 81*x^2)*e^(4*x) + 72*(64*x^5 + 432*x^4 + 972*x^3 + 729*x^2)*e^(2*x))/(x^8 - 16*
x^6 + 96*x^4 - 256*x^2 + 256)

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giac [B]  time = 0.15, size = 140, normalized size = 5.19 \begin {gather*} \frac {16 \, {\left (256 \, x^{6} + 4608 \, x^{5} e^{\left (2 \, x\right )} + 2304 \, x^{5} + 31104 \, x^{4} e^{\left (4 \, x\right )} + 31104 \, x^{4} e^{\left (2 \, x\right )} + 7776 \, x^{4} + 93312 \, x^{3} e^{\left (6 \, x\right )} + 139968 \, x^{3} e^{\left (4 \, x\right )} + 69984 \, x^{3} e^{\left (2 \, x\right )} + 11664 \, x^{3} + 104976 \, x^{2} e^{\left (8 \, x\right )} + 209952 \, x^{2} e^{\left (6 \, x\right )} + 157464 \, x^{2} e^{\left (4 \, x\right )} + 52488 \, x^{2} e^{\left (2 \, x\right )} + 6561 \, x^{2}\right )}}{6561 \, {\left (x^{8} - 16 \, x^{6} + 96 \, x^{4} - 256 \, x^{2} + 256\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((13436928*x^4-10077696*x^3-53747712*x^2-13436928*x)*exp(x)^8+(8957952*x^5+12690432*x^4-55987200*x^3
-98537472*x^2-26873856*x)*exp(x)^6+(1990656*x^6+6967296*x^5-9082368*x^4-58910976*x^3-67184640*x^2-20155392*x)*
exp(x)^4+(147456*x^7+774144*x^6-340992*x^5-9374976*x^4-21959424*x^3-20155392*x^2-6718464*x)*exp(x)^2-8192*x^7-
110592*x^6-595968*x^5-1670400*x^4-2620512*x^3-2239488*x^2-839808*x)/(6561*x^10-131220*x^8+1049760*x^6-4199040*
x^4+8398080*x^2-6718464),x, algorithm="giac")

[Out]

16/6561*(256*x^6 + 4608*x^5*e^(2*x) + 2304*x^5 + 31104*x^4*e^(4*x) + 31104*x^4*e^(2*x) + 7776*x^4 + 93312*x^3*
e^(6*x) + 139968*x^3*e^(4*x) + 69984*x^3*e^(2*x) + 11664*x^3 + 104976*x^2*e^(8*x) + 209952*x^2*e^(6*x) + 15746
4*x^2*e^(4*x) + 52488*x^2*e^(2*x) + 6561*x^2)/(x^8 - 16*x^6 + 96*x^4 - 256*x^2 + 256)

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maple [B]  time = 0.21, size = 145, normalized size = 5.37

method	result	size
risch	\(\frac {\frac {4096}{6561} x^{6}+\frac {4096}{729} x^{5}+\frac {512}{27} x^{4}+\frac {256}{9} x^{3}+16 x^{2}}{x^{8}-16 x^{6}+96 x^{4}-256 x^{2}+256}+\frac {256 x^{2} {\mathrm e}^{8 x}}{\left (x^{2}-4\right )^{4}}+\frac {512 x^{2} \left (4 x +9\right ) {\mathrm e}^{6 x}}{9 \left (x^{2}-4\right )^{4}}+\frac {128 x^{2} \left (16 x^{2}+72 x +81\right ) {\mathrm e}^{4 x}}{27 \left (x^{2}-4\right )^{4}}+\frac {128 x^{2} \left (64 x^{3}+432 x^{2}+972 x +729\right ) {\mathrm e}^{2 x}}{729 \left (x^{2}-4\right )^{4}}\)	\(145\)
default	\(\frac {364}{243 \left (2+x \right )^{3}}-\frac {1160}{729 \left (2+x \right )^{4}}+\frac {1160}{729 \left (x -2\right )^{4}}+\frac {4096}{6561 \left (x^{2}-4\right )}-\frac {239}{729 \left (2+x \right )^{2}}+\frac {239}{729 \left (x -2\right )^{2}}+\frac {364}{243 \left (x -2\right )^{3}}+\frac {2672704}{6561 \left (x^{2}-4\right )^{4}}+\frac {57856}{2187 \left (x^{2}-4\right )^{2}}+\frac {432304}{2187 \left (x^{2}-4\right )^{3}}+\frac {{\mathrm e}^{8 x}}{4 x -8}+\frac {{\mathrm e}^{6 x}}{2 x -4}-\frac {37 \,{\mathrm e}^{2 x}}{216 \left (x -2\right )}+\frac {1615 \,{\mathrm e}^{2 x}}{2916 \left (x -2\right )^{2}}+\frac {2312 \,{\mathrm e}^{2 x}}{243 \left (x -2\right )^{3}}+\frac {37 \,{\mathrm e}^{2 x}}{216 \left (2+x \right )}+\frac {383 \,{\mathrm e}^{2 x}}{2916 \left (2+x \right )^{2}}+\frac {8 \,{\mathrm e}^{2 x}}{243 \left (2+x \right )^{3}}+\frac {9826 \,{\mathrm e}^{2 x}}{729 \left (x -2\right )^{4}}+\frac {2 \,{\mathrm e}^{2 x}}{729 \left (2+x \right )^{4}}+\frac {136 \,{\mathrm e}^{6 x}}{9 \left (x -2\right )^{4}}+\frac {17 \,{\mathrm e}^{4 x}}{216 \left (x -2\right )}-\frac {161 \,{\mathrm e}^{4 x}}{108 \left (x -2\right )^{2}}+\frac {272 \,{\mathrm e}^{4 x}}{27 \left (x -2\right )^{3}}+\frac {578 \,{\mathrm e}^{4 x}}{27 \left (x -2\right )^{4}}+\frac {2 \,{\mathrm e}^{4 x}}{27 \left (2+x \right )^{4}}-\frac {17 \,{\mathrm e}^{4 x}}{216 \left (2+x \right )}+\frac {127 \,{\mathrm e}^{4 x}}{108 \left (2+x \right )^{2}}+\frac {16 \,{\mathrm e}^{4 x}}{27 \left (2+x \right )^{3}}+\frac {4 \,{\mathrm e}^{8 x}}{\left (x -2\right )^{4}}-\frac {{\mathrm e}^{8 x}}{4 \left (2+x \right )}-\frac {{\mathrm e}^{8 x}}{2 \left (2+x \right )^{2}}-\frac {{\mathrm e}^{8 x}}{2 \left (x -2\right )^{2}}+\frac {4 \,{\mathrm e}^{8 x}}{\left (2+x \right )^{4}}+\frac {8 \,{\mathrm e}^{6 x}}{9 \left (2+x \right )^{4}}-\frac {{\mathrm e}^{6 x}}{2 \left (2+x \right )}-\frac {{\mathrm e}^{6 x}}{9 \left (2+x \right )^{2}}+\frac {32 \,{\mathrm e}^{6 x}}{9 \left (2+x \right )^{3}}-\frac {17 \,{\mathrm e}^{6 x}}{9 \left (x -2\right )^{2}}+\frac {32 \,{\mathrm e}^{6 x}}{9 \left (x -2\right )^{3}}\)	\(410\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((13436928*x^4-10077696*x^3-53747712*x^2-13436928*x)*exp(x)^8+(8957952*x^5+12690432*x^4-55987200*x^3-98537
472*x^2-26873856*x)*exp(x)^6+(1990656*x^6+6967296*x^5-9082368*x^4-58910976*x^3-67184640*x^2-20155392*x)*exp(x)
^4+(147456*x^7+774144*x^6-340992*x^5-9374976*x^4-21959424*x^3-20155392*x^2-6718464*x)*exp(x)^2-8192*x^7-110592
*x^6-595968*x^5-1670400*x^4-2620512*x^3-2239488*x^2-839808*x)/(6561*x^10-131220*x^8+1049760*x^6-4199040*x^4+83
98080*x^2-6718464),x,method=_RETURNVERBOSE)

[Out]

(4096/6561*x^6+4096/729*x^5+512/27*x^4+256/9*x^3+16*x^2)/(x^8-16*x^6+96*x^4-256*x^2+256)+256*x^2/(x^2-4)^4*exp
(8*x)+512/9*x^2*(4*x+9)/(x^2-4)^4*exp(6*x)+128/27*x^2*(16*x^2+72*x+81)/(x^2-4)^4*exp(4*x)+128/729*x^2*(64*x^3+
432*x^2+972*x+729)/(x^2-4)^4*exp(2*x)

________________________________________________________________________________________

maxima [B]  time = 0.44, size = 357, normalized size = 13.22 \begin {gather*} \frac {8 \, {\left (15 \, x^{7} + 292 \, x^{5} - 880 \, x^{3} + 960 \, x\right )}}{729 \, {\left (x^{8} - 16 \, x^{6} + 96 \, x^{4} - 256 \, x^{2} + 256\right )}} + \frac {15 \, x^{7} - 220 \, x^{5} + 1168 \, x^{3} + 960 \, x}{72 \, {\left (x^{8} - 16 \, x^{6} + 96 \, x^{4} - 256 \, x^{2} + 256\right )}} - \frac {725 \, {\left (3 \, x^{7} - 44 \, x^{5} - 176 \, x^{3} + 192 \, x\right )}}{5832 \, {\left (x^{8} - 16 \, x^{6} + 96 \, x^{4} - 256 \, x^{2} + 256\right )}} + \frac {4096 \, {\left (x^{6} - 6 \, x^{4} + 16 \, x^{2} - 16\right )}}{6561 \, {\left (x^{8} - 16 \, x^{6} + 96 \, x^{4} - 256 \, x^{2} + 256\right )}} + \frac {49664 \, {\left (3 \, x^{4} - 8 \, x^{2} + 8\right )}}{6561 \, {\left (x^{8} - 16 \, x^{6} + 96 \, x^{4} - 256 \, x^{2} + 256\right )}} + \frac {128 \, {\left (1458 \, x^{2} e^{\left (8 \, x\right )} + 324 \, {\left (4 \, x^{3} + 9 \, x^{2}\right )} e^{\left (6 \, x\right )} + 27 \, {\left (16 \, x^{4} + 72 \, x^{3} + 81 \, x^{2}\right )} e^{\left (4 \, x\right )} + {\left (64 \, x^{5} + 432 \, x^{4} + 972 \, x^{3} + 729 \, x^{2}\right )} e^{\left (2 \, x\right )}\right )}}{729 \, {\left (x^{8} - 16 \, x^{6} + 96 \, x^{4} - 256 \, x^{2} + 256\right )}} + \frac {5392 \, {\left (x^{2} - 1\right )}}{81 \, {\left (x^{8} - 16 \, x^{6} + 96 \, x^{4} - 256 \, x^{2} + 256\right )}} + \frac {16}{x^{8} - 16 \, x^{6} + 96 \, x^{4} - 256 \, x^{2} + 256} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((13436928*x^4-10077696*x^3-53747712*x^2-13436928*x)*exp(x)^8+(8957952*x^5+12690432*x^4-55987200*x^3
-98537472*x^2-26873856*x)*exp(x)^6+(1990656*x^6+6967296*x^5-9082368*x^4-58910976*x^3-67184640*x^2-20155392*x)*
exp(x)^4+(147456*x^7+774144*x^6-340992*x^5-9374976*x^4-21959424*x^3-20155392*x^2-6718464*x)*exp(x)^2-8192*x^7-
110592*x^6-595968*x^5-1670400*x^4-2620512*x^3-2239488*x^2-839808*x)/(6561*x^10-131220*x^8+1049760*x^6-4199040*
x^4+8398080*x^2-6718464),x, algorithm="maxima")

[Out]

8/729*(15*x^7 + 292*x^5 - 880*x^3 + 960*x)/(x^8 - 16*x^6 + 96*x^4 - 256*x^2 + 256) + 1/72*(15*x^7 - 220*x^5 +
1168*x^3 + 960*x)/(x^8 - 16*x^6 + 96*x^4 - 256*x^2 + 256) - 725/5832*(3*x^7 - 44*x^5 - 176*x^3 + 192*x)/(x^8 -
16*x^6 + 96*x^4 - 256*x^2 + 256) + 4096/6561*(x^6 - 6*x^4 + 16*x^2 - 16)/(x^8 - 16*x^6 + 96*x^4 - 256*x^2 + 2
56) + 49664/6561*(3*x^4 - 8*x^2 + 8)/(x^8 - 16*x^6 + 96*x^4 - 256*x^2 + 256) + 128/729*(1458*x^2*e^(8*x) + 324
*(4*x^3 + 9*x^2)*e^(6*x) + 27*(16*x^4 + 72*x^3 + 81*x^2)*e^(4*x) + (64*x^5 + 432*x^4 + 972*x^3 + 729*x^2)*e^(2
*x))/(x^8 - 16*x^6 + 96*x^4 - 256*x^2 + 256) + 5392/81*(x^2 - 1)/(x^8 - 16*x^6 + 96*x^4 - 256*x^2 + 256) + 16/
(x^8 - 16*x^6 + 96*x^4 - 256*x^2 + 256)

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mupad [B]  time = 0.46, size = 210, normalized size = 7.78 \begin {gather*} \frac {\frac {4096\,x^6}{6561}+\frac {4096\,x^5}{729}+\frac {512\,x^4}{27}+\frac {256\,x^3}{9}+16\,x^2}{x^8-16\,x^6+96\,x^4-256\,x^2+256}+\frac {{\mathrm {e}}^{2\,x}\,\left (\frac {8192\,x^5}{729}+\frac {2048\,x^4}{27}+\frac {512\,x^3}{3}+128\,x^2\right )}{x^8-16\,x^6+96\,x^4-256\,x^2+256}+\frac {{\mathrm {e}}^{6\,x}\,\left (\frac {2048\,x^3}{9}+512\,x^2\right )}{x^8-16\,x^6+96\,x^4-256\,x^2+256}+\frac {256\,x^2\,{\mathrm {e}}^{8\,x}}{x^8-16\,x^6+96\,x^4-256\,x^2+256}+\frac {{\mathrm {e}}^{4\,x}\,\left (\frac {2048\,x^4}{27}+\frac {1024\,x^3}{3}+384\,x^2\right )}{x^8-16\,x^6+96\,x^4-256\,x^2+256} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(-(839808*x + exp(2*x)*(6718464*x + 20155392*x^2 + 21959424*x^3 + 9374976*x^4 + 340992*x^5 - 774144*x^6 - 1
47456*x^7) + exp(8*x)*(13436928*x + 53747712*x^2 + 10077696*x^3 - 13436928*x^4) + exp(6*x)*(26873856*x + 98537
472*x^2 + 55987200*x^3 - 12690432*x^4 - 8957952*x^5) + 2239488*x^2 + 2620512*x^3 + 1670400*x^4 + 595968*x^5 +
110592*x^6 + 8192*x^7 + exp(4*x)*(20155392*x + 67184640*x^2 + 58910976*x^3 + 9082368*x^4 - 6967296*x^5 - 19906
56*x^6))/(8398080*x^2 - 4199040*x^4 + 1049760*x^6 - 131220*x^8 + 6561*x^10 - 6718464),x)

[Out]

(16*x^2 + (256*x^3)/9 + (512*x^4)/27 + (4096*x^5)/729 + (4096*x^6)/6561)/(96*x^4 - 256*x^2 - 16*x^6 + x^8 + 25
6) + (exp(2*x)*(128*x^2 + (512*x^3)/3 + (2048*x^4)/27 + (8192*x^5)/729))/(96*x^4 - 256*x^2 - 16*x^6 + x^8 + 25
6) + (exp(6*x)*(512*x^2 + (2048*x^3)/9))/(96*x^4 - 256*x^2 - 16*x^6 + x^8 + 256) + (256*x^2*exp(8*x))/(96*x^4
- 256*x^2 - 16*x^6 + x^8 + 256) + (exp(4*x)*(384*x^2 + (1024*x^3)/3 + (2048*x^4)/27))/(96*x^4 - 256*x^2 - 16*x
^6 + x^8 + 256)

________________________________________________________________________________________

sympy [B]  time = 0.47, size = 631, normalized size = 23.37 \begin {gather*} \frac {\left (45349632 x^{26} - 2176782336 x^{24} + 47889211392 x^{22} - 638522818560 x^{20} + 5746705367040 x^{18} - 36778914349056 x^{16} + 171634933628928 x^{14} - 588462629584896 x^{12} + 1471156573962240 x^{10} - 2615389464821760 x^{8} + 3138467357786112 x^{6} - 2282521714753536 x^{4} + 760840571584512 x^{2}\right ) e^{8 x} + \left (40310784 x^{27} + 90699264 x^{26} - 1934917632 x^{25} - 4353564672 x^{24} + 42568187904 x^{23} + 95778422784 x^{22} - 567575838720 x^{21} - 1277045637120 x^{20} + 5108182548480 x^{19} + 11493410734080 x^{18} - 32692368310272 x^{17} - 73557828698112 x^{16} + 152564385447936 x^{15} + 343269867257856 x^{14} - 523077892964352 x^{13} - 1176925259169792 x^{12} + 1307694732410880 x^{11} + 2942313147924480 x^{10} - 2324790635397120 x^{9} - 5230778929643520 x^{8} + 2789748762476544 x^{7} + 6276934715572224 x^{6} - 2028908190892032 x^{5} - 4565043429507072 x^{4} + 676302730297344 x^{3} + 1521681143169024 x^{2}\right ) e^{6 x} + \left (13436928 x^{28} + 60466176 x^{27} - 576948096 x^{26} - 2902376448 x^{25} + 10924222464 x^{24} + 63852281856 x^{23} - 117358129152 x^{22} - 851363758080 x^{21} + 744943288320 x^{20} + 7662273822720 x^{19} - 2277398052864 x^{18} - 49038552465408 x^{17} - 4313576374272 x^{16} + 228846578171904 x^{15} + 83093102788608 x^{14} - 784616839446528 x^{13} - 446795700240384 x^{12} + 1961542098616320 x^{11} + 1431804649144320 x^{10} - 3487185953095680 x^{9} - 2993167943073792 x^{8} + 4184623143714816 x^{7} + 4031398306381824 x^{6} - 3043362286338048 x^{5} - 3198348328697856 x^{4} + 1014454095446016 x^{3} + 1141260857376768 x^{2}\right ) e^{4 x} + \left (1990656 x^{29} + 13436928 x^{28} - 65318400 x^{27} - 622297728 x^{26} + 650944512 x^{25} + 13101004800 x^{24} + 3897704448 x^{23} - 165247340544 x^{22} - 173425950720 x^{21} + 1383466106880 x^{20} + 2216698970112 x^{19} - 8024103419904 x^{18} - 16985232506880 x^{17} + 32465337974784 x^{16} + 88592282025984 x^{15} - 88541830840320 x^{14} - 327730902073344 x^{13} + 141666929344512 x^{12} + 865966573486080 x^{11} - 39351924817920 x^{10} - 1605827605561344 x^{9} - 377778478252032 x^{8} + 1992118574776320 x^{7} + 892930948595712 x^{6} - 1488283477475328 x^{5} - 915826613944320 x^{4} + 507227047723008 x^{3} + 380420285792256 x^{2}\right ) e^{2 x}}{177147 x^{32} - 11337408 x^{30} + 340122240 x^{28} - 6348948480 x^{26} + 82536330240 x^{24} - 792348770304 x^{22} + 5810557648896 x^{20} - 33203186565120 x^{18} + 149414339543040 x^{16} - 531250985041920 x^{14} + 1487502758117376 x^{12} - 3245460563165184 x^{10} + 5409100938608640 x^{8} - 6657355001364480 x^{6} + 5706304286883840 x^{4} - 3043362286338048 x^{2} + 760840571584512} - \frac {- 4096 x^{6} - 36864 x^{5} - 124416 x^{4} - 186624 x^{3} - 104976 x^{2}}{6561 x^{8} - 104976 x^{6} + 629856 x^{4} - 1679616 x^{2} + 1679616} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((13436928*x**4-10077696*x**3-53747712*x**2-13436928*x)*exp(x)**8+(8957952*x**5+12690432*x**4-559872
00*x**3-98537472*x**2-26873856*x)*exp(x)**6+(1990656*x**6+6967296*x**5-9082368*x**4-58910976*x**3-67184640*x**
2-20155392*x)*exp(x)**4+(147456*x**7+774144*x**6-340992*x**5-9374976*x**4-21959424*x**3-20155392*x**2-6718464*
x)*exp(x)**2-8192*x**7-110592*x**6-595968*x**5-1670400*x**4-2620512*x**3-2239488*x**2-839808*x)/(6561*x**10-13
1220*x**8+1049760*x**6-4199040*x**4+8398080*x**2-6718464),x)

[Out]

((45349632*x**26 - 2176782336*x**24 + 47889211392*x**22 - 638522818560*x**20 + 5746705367040*x**18 - 367789143
49056*x**16 + 171634933628928*x**14 - 588462629584896*x**12 + 1471156573962240*x**10 - 2615389464821760*x**8 +
3138467357786112*x**6 - 2282521714753536*x**4 + 760840571584512*x**2)*exp(8*x) + (40310784*x**27 + 90699264*x
**26 - 1934917632*x**25 - 4353564672*x**24 + 42568187904*x**23 + 95778422784*x**22 - 567575838720*x**21 - 1277
045637120*x**20 + 5108182548480*x**19 + 11493410734080*x**18 - 32692368310272*x**17 - 73557828698112*x**16 + 1
52564385447936*x**15 + 343269867257856*x**14 - 523077892964352*x**13 - 1176925259169792*x**12 + 13076947324108
80*x**11 + 2942313147924480*x**10 - 2324790635397120*x**9 - 5230778929643520*x**8 + 2789748762476544*x**7 + 62
76934715572224*x**6 - 2028908190892032*x**5 - 4565043429507072*x**4 + 676302730297344*x**3 + 1521681143169024*
x**2)*exp(6*x) + (13436928*x**28 + 60466176*x**27 - 576948096*x**26 - 2902376448*x**25 + 10924222464*x**24 + 6
3852281856*x**23 - 117358129152*x**22 - 851363758080*x**21 + 744943288320*x**20 + 7662273822720*x**19 - 227739
8052864*x**18 - 49038552465408*x**17 - 4313576374272*x**16 + 228846578171904*x**15 + 83093102788608*x**14 - 78
4616839446528*x**13 - 446795700240384*x**12 + 1961542098616320*x**11 + 1431804649144320*x**10 - 34871859530956
80*x**9 - 2993167943073792*x**8 + 4184623143714816*x**7 + 4031398306381824*x**6 - 3043362286338048*x**5 - 3198
348328697856*x**4 + 1014454095446016*x**3 + 1141260857376768*x**2)*exp(4*x) + (1990656*x**29 + 13436928*x**28
- 65318400*x**27 - 622297728*x**26 + 650944512*x**25 + 13101004800*x**24 + 3897704448*x**23 - 165247340544*x**
22 - 173425950720*x**21 + 1383466106880*x**20 + 2216698970112*x**19 - 8024103419904*x**18 - 16985232506880*x**
17 + 32465337974784*x**16 + 88592282025984*x**15 - 88541830840320*x**14 - 327730902073344*x**13 + 141666929344
512*x**12 + 865966573486080*x**11 - 39351924817920*x**10 - 1605827605561344*x**9 - 377778478252032*x**8 + 1992
118574776320*x**7 + 892930948595712*x**6 - 1488283477475328*x**5 - 915826613944320*x**4 + 507227047723008*x**3
+ 380420285792256*x**2)*exp(2*x))/(177147*x**32 - 11337408*x**30 + 340122240*x**28 - 6348948480*x**26 + 82536
330240*x**24 - 792348770304*x**22 + 5810557648896*x**20 - 33203186565120*x**18 + 149414339543040*x**16 - 53125
0985041920*x**14 + 1487502758117376*x**12 - 3245460563165184*x**10 + 5409100938608640*x**8 - 6657355001364480*
x**6 + 5706304286883840*x**4 - 3043362286338048*x**2 + 760840571584512) - (-4096*x**6 - 36864*x**5 - 124416*x*
*4 - 186624*x**3 - 104976*x**2)/(6561*x**8 - 104976*x**6 + 629856*x**4 - 1679616*x**2 + 1679616)

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