3.61.14 \(\int \frac {e^{-16-16 x} (-32 e^{2+2 x} x^{21}+3 x^{24}-2 x^{25}+(7168 e^{4+4 x} x^{18}+e^{2+2 x} (-672 x^{21}+448 x^{22})) \log (x)+(-688128 e^{6+6 x} x^{15}+e^{4+4 x} (64512 x^{18}-43008 x^{19})) \log ^2(x)+(36700160 e^{8+8 x} x^{12}+e^{6+6 x} (-3440640 x^{15}+2293760 x^{16})) \log ^3(x)+(-1174405120 e^{10+10 x} x^9+e^{8+8 x} (110100480 x^{12}-73400320 x^{13})) \log ^4(x)+(22548578304 e^{12+12 x} x^6+e^{10+10 x} (-2113929216 x^9+1409286144 x^{10})) \log ^5(x)+(-240518168576 e^{14+14 x} x^3+e^{12+12 x} (22548578304 x^6-15032385536 x^7)) \log ^6(x)+(1099511627776 e^{16+16 x}+e^{14+14 x} (-103079215104 x^3+68719476736 x^4)) \log ^7(x))}{2097152 x} \, dx\)

Optimal. Leaf size=23 \[ 256 \left (-\frac {1}{16} e^{-2-2 x} x^3+2 \log (x)\right )^8 \]

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Rubi [B]  time = 25.42, antiderivative size = 139, normalized size of antiderivative = 6.04, number of steps used = 64, number of rules used = 9, integrand size = 291, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.031, Rules used = {12, 6688, 6742, 2196, 2176, 2194, 2302, 30, 2288} \begin {gather*} \frac {e^{-16 x-16} x^{24}}{16777216}-\frac {e^{-14 x-14} x^{21} \log (x)}{65536}+\frac {7 e^{-12 x-12} x^{18} \log ^2(x)}{4096}-\frac {7}{64} e^{-10 x-10} x^{15} \log ^3(x)+\frac {35}{8} e^{-8 x-8} x^{12} \log ^4(x)-112 e^{-6 x-6} x^9 \log ^5(x)+1792 e^{-4 x-4} x^6 \log ^6(x)-16384 e^{-2 x-2} x^3 \log ^7(x)+65536 \log ^8(x) \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(E^(-16 - 16*x)*(-32*E^(2 + 2*x)*x^21 + 3*x^24 - 2*x^25 + (7168*E^(4 + 4*x)*x^18 + E^(2 + 2*x)*(-672*x^21
+ 448*x^22))*Log[x] + (-688128*E^(6 + 6*x)*x^15 + E^(4 + 4*x)*(64512*x^18 - 43008*x^19))*Log[x]^2 + (36700160*
E^(8 + 8*x)*x^12 + E^(6 + 6*x)*(-3440640*x^15 + 2293760*x^16))*Log[x]^3 + (-1174405120*E^(10 + 10*x)*x^9 + E^(
8 + 8*x)*(110100480*x^12 - 73400320*x^13))*Log[x]^4 + (22548578304*E^(12 + 12*x)*x^6 + E^(10 + 10*x)*(-2113929
216*x^9 + 1409286144*x^10))*Log[x]^5 + (-240518168576*E^(14 + 14*x)*x^3 + E^(12 + 12*x)*(22548578304*x^6 - 150
32385536*x^7))*Log[x]^6 + (1099511627776*E^(16 + 16*x) + E^(14 + 14*x)*(-103079215104*x^3 + 68719476736*x^4))*
Log[x]^7))/(2097152*x),x]

[Out]

(E^(-16 - 16*x)*x^24)/16777216 - (E^(-14 - 14*x)*x^21*Log[x])/65536 + (7*E^(-12 - 12*x)*x^18*Log[x]^2)/4096 -
(7*E^(-10 - 10*x)*x^15*Log[x]^3)/64 + (35*E^(-8 - 8*x)*x^12*Log[x]^4)/8 - 112*E^(-6 - 6*x)*x^9*Log[x]^5 + 1792
*E^(-4 - 4*x)*x^6*Log[x]^6 - 16384*E^(-2 - 2*x)*x^3*Log[x]^7 + 65536*Log[x]^8

Rule 12

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

Rule 30

Int[(x_)^(m_.), x_Symbol] :> Simp[x^(m + 1)/(m + 1), x] /; FreeQ[m, x] && NeQ[m, -1]

Rule 2176

Int[((b_.)*(F_)^((g_.)*((e_.) + (f_.)*(x_))))^(n_.)*((c_.) + (d_.)*(x_))^(m_.), x_Symbol] :> Simp[((c + d*x)^m
*(b*F^(g*(e + f*x)))^n)/(f*g*n*Log[F]), x] - Dist[(d*m)/(f*g*n*Log[F]), 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] && GtQ[m, 0] && IntegerQ[2*m] &&  !$UseGamma === True

Rule 2194

Int[((F_)^((c_.)*((a_.) + (b_.)*(x_))))^(n_.), x_Symbol] :> Simp[(F^(c*(a + b*x)))^n/(b*c*n*Log[F]), x] /; Fre
eQ[{F, a, b, c, n}, x]

Rule 2196

Int[(F_)^((c_.)*(v_))*(u_), x_Symbol] :> Int[ExpandIntegrand[F^(c*ExpandToSum[v, x]), u, x], x] /; FreeQ[{F, c
}, x] && PolynomialQ[u, x] && LinearQ[v, x] &&  !$UseGamma === True

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 2302

Int[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))^(p_.)/(x_), x_Symbol] :> Dist[1/(b*n), Subst[Int[x^p, x], x, a + b*L
og[c*x^n]], x] /; FreeQ[{a, b, c, n, p}, 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} &=\frac {\int \frac {e^{-16-16 x} \left (-32 e^{2+2 x} x^{21}+3 x^{24}-2 x^{25}+\left (7168 e^{4+4 x} x^{18}+e^{2+2 x} \left (-672 x^{21}+448 x^{22}\right )\right ) \log (x)+\left (-688128 e^{6+6 x} x^{15}+e^{4+4 x} \left (64512 x^{18}-43008 x^{19}\right )\right ) \log ^2(x)+\left (36700160 e^{8+8 x} x^{12}+e^{6+6 x} \left (-3440640 x^{15}+2293760 x^{16}\right )\right ) \log ^3(x)+\left (-1174405120 e^{10+10 x} x^9+e^{8+8 x} \left (110100480 x^{12}-73400320 x^{13}\right )\right ) \log ^4(x)+\left (22548578304 e^{12+12 x} x^6+e^{10+10 x} \left (-2113929216 x^9+1409286144 x^{10}\right )\right ) \log ^5(x)+\left (-240518168576 e^{14+14 x} x^3+e^{12+12 x} \left (22548578304 x^6-15032385536 x^7\right )\right ) \log ^6(x)+\left (1099511627776 e^{16+16 x}+e^{14+14 x} \left (-103079215104 x^3+68719476736 x^4\right )\right ) \log ^7(x)\right )}{x} \, dx}{2097152}\\ &=\frac {\int \frac {e^{-16-16 x} \left (-32 e^{2+2 x}-x^3 (-3+2 x)\right ) \left (x^3-32 e^{2+2 x} \log (x)\right )^7}{x} \, dx}{2097152}\\ &=\frac {\int \left (-e^{-16-16 x} x^{23} (-3+2 x)+\frac {1099511627776 \log ^7(x)}{x}[phone] e^{-2-2 x} x^2 \log ^6(x) (-7-3 \log (x)+2 x \log (x))-7516192768 e^{-4-4 x} x^5 \log ^5(x) (-3-3 \log (x)+2 x \log (x))-36700160 e^{-8-8 x} x^{11} \log ^3(x) (-1-3 \log (x)+2 x \log (x))[phone] e^{-6-6 x} x^8 \log ^4(x) (-5-9 \log (x)+6 x \log (x))-7168 e^{-12-12 x} x^{17} \log (x) (-1-9 \log (x)+6 x \log (x))+229376 e^{-10-10 x} x^{14} \log ^2(x) (-3-15 \log (x)+10 x \log (x))+32 e^{-14-14 x} x^{20} (-1-21 \log (x)+14 x \log (x))\right ) \, dx}{2097152}\\ &=-\frac {\int e^{-16-16 x} x^{23} (-3+2 x) \, dx}{2097152}+\frac {\int e^{-14-14 x} x^{20} (-1-21 \log (x)+14 x \log (x)) \, dx}{65536}-\frac {7 \int e^{-12-12 x} x^{17} \log (x) (-1-9 \log (x)+6 x \log (x)) \, dx}{2048}+\frac {7}{64} \int e^{-10-10 x} x^{14} \log ^2(x) (-3-15 \log (x)+10 x \log (x)) \, dx-\frac {35}{2} \int e^{-8-8 x} x^{11} \log ^3(x) (-1-3 \log (x)+2 x \log (x)) \, dx+112 \int e^{-6-6 x} x^8 \log ^4(x) (-5-9 \log (x)+6 x \log (x)) \, dx-3584 \int e^{-4-4 x} x^5 \log ^5(x) (-3-3 \log (x)+2 x \log (x)) \, dx+16384 \int e^{-2-2 x} x^2 \log ^6(x) (-7-3 \log (x)+2 x \log (x)) \, dx+524288 \int \frac {\log ^7(x)}{x} \, dx\\ &=-\frac {e^{-14-14 x} x^{21} \log (x)}{65536}+\frac {7 e^{-12-12 x} x^{18} \log ^2(x)}{4096}-\frac {7}{64} e^{-10-10 x} x^{15} \log ^3(x)+\frac {35}{8} e^{-8-8 x} x^{12} \log ^4(x)-112 e^{-6-6 x} x^9 \log ^5(x)+1792 e^{-4-4 x} x^6 \log ^6(x)-16384 e^{-2-2 x} x^3 \log ^7(x)-\frac {\int \left (-3 e^{-16-16 x} x^{23}+2 e^{-16-16 x} x^{24}\right ) \, dx}{2097152}+524288 \operatorname {Subst}\left (\int x^7 \, dx,x,\log (x)\right )\\ &=-\frac {e^{-14-14 x} x^{21} \log (x)}{65536}+\frac {7 e^{-12-12 x} x^{18} \log ^2(x)}{4096}-\frac {7}{64} e^{-10-10 x} x^{15} \log ^3(x)+\frac {35}{8} e^{-8-8 x} x^{12} \log ^4(x)-112 e^{-6-6 x} x^9 \log ^5(x)+1792 e^{-4-4 x} x^6 \log ^6(x)-16384 e^{-2-2 x} x^3 \log ^7(x)+65536 \log ^8(x)-\frac {\int e^{-16-16 x} x^{24} \, dx}{1048576}+\frac {3 \int e^{-16-16 x} x^{23} \, dx}{2097152}\\ &=-\frac {3 e^{-16-16 x} x^{23}}{33554432}+\frac {e^{-16-16 x} x^{24}}{16777216}-\frac {e^{-14-14 x} x^{21} \log (x)}{65536}+\frac {7 e^{-12-12 x} x^{18} \log ^2(x)}{4096}-\frac {7}{64} e^{-10-10 x} x^{15} \log ^3(x)+\frac {35}{8} e^{-8-8 x} x^{12} \log ^4(x)-112 e^{-6-6 x} x^9 \log ^5(x)+1792 e^{-4-4 x} x^6 \log ^6(x)-16384 e^{-2-2 x} x^3 \log ^7(x)+65536 \log ^8(x)-\frac {3 \int e^{-16-16 x} x^{23} \, dx}{2097152}+\frac {69 \int e^{-16-16 x} x^{22} \, dx}{33554432}\\ &=-\frac {69 e^{-16-16 x} x^{22}}{536870912}+\frac {e^{-16-16 x} x^{24}}{16777216}-\frac {e^{-14-14 x} x^{21} \log (x)}{65536}+\frac {7 e^{-12-12 x} x^{18} \log ^2(x)}{4096}-\frac {7}{64} e^{-10-10 x} x^{15} \log ^3(x)+\frac {35}{8} e^{-8-8 x} x^{12} \log ^4(x)-112 e^{-6-6 x} x^9 \log ^5(x)+1792 e^{-4-4 x} x^6 \log ^6(x)-16384 e^{-2-2 x} x^3 \log ^7(x)+65536 \log ^8(x)-\frac {69 \int e^{-16-16 x} x^{22} \, dx}{33554432}+\frac {759 \int e^{-16-16 x} x^{21} \, dx}{268435456}\\ &=-\frac {759 e^{-16-16 x} x^{21}}{4294967296}+\frac {e^{-16-16 x} x^{24}}{16777216}-\frac {e^{-14-14 x} x^{21} \log (x)}{65536}+\frac {7 e^{-12-12 x} x^{18} \log ^2(x)}{4096}-\frac {7}{64} e^{-10-10 x} x^{15} \log ^3(x)+\frac {35}{8} e^{-8-8 x} x^{12} \log ^4(x)-112 e^{-6-6 x} x^9 \log ^5(x)+1792 e^{-4-4 x} x^6 \log ^6(x)-16384 e^{-2-2 x} x^3 \log ^7(x)+65536 \log ^8(x)-\frac {759 \int e^{-16-16 x} x^{21} \, dx}{268435456}+\frac {15939 \int e^{-16-16 x} x^{20} \, dx}{4294967296}\\ &=-\frac {15939 e^{-16-16 x} x^{20}}{68719476736}+\frac {e^{-16-16 x} x^{24}}{16777216}-\frac {e^{-14-14 x} x^{21} \log (x)}{65536}+\frac {7 e^{-12-12 x} x^{18} \log ^2(x)}{4096}-\frac {7}{64} e^{-10-10 x} x^{15} \log ^3(x)+\frac {35}{8} e^{-8-8 x} x^{12} \log ^4(x)-112 e^{-6-6 x} x^9 \log ^5(x)+1792 e^{-4-4 x} x^6 \log ^6(x)-16384 e^{-2-2 x} x^3 \log ^7(x)+65536 \log ^8(x)-\frac {15939 \int e^{-16-16 x} x^{20} \, dx}{4294967296}+\frac {79695 \int e^{-16-16 x} x^{19} \, dx}{17179869184}\\ &=-\frac {79695 e^{-16-16 x} x^{19}}{274877906944}+\frac {e^{-16-16 x} x^{24}}{16777216}-\frac {e^{-14-14 x} x^{21} \log (x)}{65536}+\frac {7 e^{-12-12 x} x^{18} \log ^2(x)}{4096}-\frac {7}{64} e^{-10-10 x} x^{15} \log ^3(x)+\frac {35}{8} e^{-8-8 x} x^{12} \log ^4(x)-112 e^{-6-6 x} x^9 \log ^5(x)+1792 e^{-4-4 x} x^6 \log ^6(x)-16384 e^{-2-2 x} x^3 \log ^7(x)+65536 \log ^8(x)-\frac {79695 \int e^{-16-16 x} x^{19} \, dx}{17179869184}+\frac {1514205 \int e^{-16-16 x} x^{18} \, dx}{274877906944}\\ &=-\frac {1514205 e^{-16-16 x} x^{18}}{4398046511104}+\frac {e^{-16-16 x} x^{24}}{16777216}-\frac {e^{-14-14 x} x^{21} \log (x)}{65536}+\frac {7 e^{-12-12 x} x^{18} \log ^2(x)}{4096}-\frac {7}{64} e^{-10-10 x} x^{15} \log ^3(x)+\frac {35}{8} e^{-8-8 x} x^{12} \log ^4(x)-112 e^{-6-6 x} x^9 \log ^5(x)+1792 e^{-4-4 x} x^6 \log ^6(x)-16384 e^{-2-2 x} x^3 \log ^7(x)+65536 \log ^8(x)-\frac {1514205 \int e^{-16-16 x} x^{18} \, dx}{274877906944}+\frac {13627845 \int e^{-16-16 x} x^{17} \, dx}{2199023255552}\\ &=-\frac {13627845 e^{-16-16 x} x^{17}}{35184372088832}+\frac {e^{-16-16 x} x^{24}}{16777216}-\frac {e^{-14-14 x} x^{21} \log (x)}{65536}+\frac {7 e^{-12-12 x} x^{18} \log ^2(x)}{4096}-\frac {7}{64} e^{-10-10 x} x^{15} \log ^3(x)+\frac {35}{8} e^{-8-8 x} x^{12} \log ^4(x)-112 e^{-6-6 x} x^9 \log ^5(x)+1792 e^{-4-4 x} x^6 \log ^6(x)-16384 e^{-2-2 x} x^3 \log ^7(x)+65536 \log ^8(x)-\frac {13627845 \int e^{-16-16 x} x^{17} \, dx}{2199023255552}+\frac {231673365 \int e^{-16-16 x} x^{16} \, dx}{35184372088832}\\ &=-\frac {231673365 e^{-16-16 x} x^{16}}{562949953421312}+\frac {e^{-16-16 x} x^{24}}{16777216}-\frac {e^{-14-14 x} x^{21} \log (x)}{65536}+\frac {7 e^{-12-12 x} x^{18} \log ^2(x)}{4096}-\frac {7}{64} e^{-10-10 x} x^{15} \log ^3(x)+\frac {35}{8} e^{-8-8 x} x^{12} \log ^4(x)-112 e^{-6-6 x} x^9 \log ^5(x)+1792 e^{-4-4 x} x^6 \log ^6(x)-16384 e^{-2-2 x} x^3 \log ^7(x)+65536 \log ^8(x)+\frac {231673365 \int e^{-16-16 x} x^{15} \, dx}{35184372088832}-\frac {231673365 \int e^{-16-16 x} x^{16} \, dx}{35184372088832}\\ &=-\frac {231673365 e^{-16-16 x} x^{15}}{562949953421312}+\frac {e^{-16-16 x} x^{24}}{16777216}-\frac {e^{-14-14 x} x^{21} \log (x)}{65536}+\frac {7 e^{-12-12 x} x^{18} \log ^2(x)}{4096}-\frac {7}{64} e^{-10-10 x} x^{15} \log ^3(x)+\frac {35}{8} e^{-8-8 x} x^{12} \log ^4(x)-112 e^{-6-6 x} x^9 \log ^5(x)+1792 e^{-4-4 x} x^6 \log ^6(x)-16384 e^{-2-2 x} x^3 \log ^7(x)+65536 \log ^8(x)+\frac {3475100475 \int e^{-16-16 x} x^{14} \, dx}{562949953421312}-\frac {231673365 \int e^{-16-16 x} x^{15} \, dx}{35184372088832}\\ &=-\frac {3475100475 e^{-16-16 x} x^{14}}{9007199254740992}+\frac {e^{-16-16 x} x^{24}}{16777216}-\frac {e^{-14-14 x} x^{21} \log (x)}{65536}+\frac {7 e^{-12-12 x} x^{18} \log ^2(x)}{4096}-\frac {7}{64} e^{-10-10 x} x^{15} \log ^3(x)+\frac {35}{8} e^{-8-8 x} x^{12} \log ^4(x)-112 e^{-6-6 x} x^9 \log ^5(x)+1792 e^{-4-4 x} x^6 \log ^6(x)-16384 e^{-2-2 x} x^3 \log ^7(x)+65536 \log ^8(x)+\frac {24325703325 \int e^{-16-16 x} x^{13} \, dx}{4503599627370496}-\frac {3475100475 \int e^{-16-16 x} x^{14} \, dx}{562949953421312}\\ &=-\frac {24325703325 e^{-16-16 x} x^{13}}{72057594037927936}+\frac {e^{-16-16 x} x^{24}}{16777216}-\frac {e^{-14-14 x} x^{21} \log (x)}{65536}+\frac {7 e^{-12-12 x} x^{18} \log ^2(x)}{4096}-\frac {7}{64} e^{-10-10 x} x^{15} \log ^3(x)+\frac {35}{8} e^{-8-8 x} x^{12} \log ^4(x)-112 e^{-6-6 x} x^9 \log ^5(x)+1792 e^{-4-4 x} x^6 \log ^6(x)-16384 e^{-2-2 x} x^3 \log ^7(x)+65536 \log ^8(x)+\frac {316234143225 \int e^{-16-16 x} x^{12} \, dx}{72057594037927936}-\frac {24325703325 \int e^{-16-16 x} x^{13} \, dx}{4503599627370496}\\ &=-\frac {316234143225 e^{-16-16 x} x^{12}}{1152921504606846976}+\frac {e^{-16-16 x} x^{24}}{16777216}-\frac {e^{-14-14 x} x^{21} \log (x)}{65536}+\frac {7 e^{-12-12 x} x^{18} \log ^2(x)}{4096}-\frac {7}{64} e^{-10-10 x} x^{15} \log ^3(x)+\frac {35}{8} e^{-8-8 x} x^{12} \log ^4(x)-112 e^{-6-6 x} x^9 \log ^5(x)+1792 e^{-4-4 x} x^6 \log ^6(x)-16384 e^{-2-2 x} x^3 \log ^7(x)+65536 \log ^8(x)+\frac {948702429675 \int e^{-16-16 x} x^{11} \, dx}{288230376151711744}-\frac {316234143225 \int e^{-16-16 x} x^{12} \, dx}{72057594037927936}\\ &=-\frac {948702429675 e^{-16-16 x} x^{11}}{4611686018427387904}+\frac {e^{-16-16 x} x^{24}}{16777216}-\frac {e^{-14-14 x} x^{21} \log (x)}{65536}+\frac {7 e^{-12-12 x} x^{18} \log ^2(x)}{4096}-\frac {7}{64} e^{-10-10 x} x^{15} \log ^3(x)+\frac {35}{8} e^{-8-8 x} x^{12} \log ^4(x)-112 e^{-6-6 x} x^9 \log ^5(x)+1792 e^{-4-4 x} x^6 \log ^6(x)-16384 e^{-2-2 x} x^3 \log ^7(x)+65536 \log ^8(x)+\frac {10435726726425 \int e^{-16-16 x} x^{10} \, dx}{4611686018427387904}-\frac {948702429675 \int e^{-16-16 x} x^{11} \, dx}{288230376151711744}\\ &=-\frac {10435726726425 e^{-16-16 x} x^{10}}{73786976294838206464}+\frac {e^{-16-16 x} x^{24}}{16777216}-\frac {e^{-14-14 x} x^{21} \log (x)}{65536}+\frac {7 e^{-12-12 x} x^{18} \log ^2(x)}{4096}-\frac {7}{64} e^{-10-10 x} x^{15} \log ^3(x)+\frac {35}{8} e^{-8-8 x} x^{12} \log ^4(x)-112 e^{-6-6 x} x^9 \log ^5(x)+1792 e^{-4-4 x} x^6 \log ^6(x)-16384 e^{-2-2 x} x^3 \log ^7(x)+65536 \log ^8(x)+\frac {52178633632125 \int e^{-16-16 x} x^9 \, dx}{36893488147419103232}-\frac {10435726726425 \int e^{-16-16 x} x^{10} \, dx}{4611686018427387904}\\ &=-\frac {52178633632125 e^{-16-16 x} x^9}{590295810358705651712}+\frac {e^{-16-16 x} x^{24}}{16777216}-\frac {e^{-14-14 x} x^{21} \log (x)}{65536}+\frac {7 e^{-12-12 x} x^{18} \log ^2(x)}{4096}-\frac {7}{64} e^{-10-10 x} x^{15} \log ^3(x)+\frac {35}{8} e^{-8-8 x} x^{12} \log ^4(x)-112 e^{-6-6 x} x^9 \log ^5(x)+1792 e^{-4-4 x} x^6 \log ^6(x)-16384 e^{-2-2 x} x^3 \log ^7(x)+65536 \log ^8(x)+\frac {469607702689125 \int e^{-16-16 x} x^8 \, dx}{590295810358705651712}-\frac {52178633632125 \int e^{-16-16 x} x^9 \, dx}{36893488147419103232}\\ &=-\frac {469607702689125 e^{-16-16 x} x^8}{9444732965739290427392}+\frac {e^{-16-16 x} x^{24}}{16777216}-\frac {e^{-14-14 x} x^{21} \log (x)}{65536}+\frac {7 e^{-12-12 x} x^{18} \log ^2(x)}{4096}-\frac {7}{64} e^{-10-10 x} x^{15} \log ^3(x)+\frac {35}{8} e^{-8-8 x} x^{12} \log ^4(x)-112 e^{-6-6 x} x^9 \log ^5(x)+1792 e^{-4-4 x} x^6 \log ^6(x)-16384 e^{-2-2 x} x^3 \log ^7(x)+65536 \log ^8(x)+\frac {469607702689125 \int e^{-16-16 x} x^7 \, dx}{1180591620717411303424}-\frac {469607702689125 \int e^{-16-16 x} x^8 \, dx}{590295810358705651712}\\ &=-\frac {469607702689125 e^{-16-16 x} x^7}{18889465931478580854784}+\frac {e^{-16-16 x} x^{24}}{16777216}-\frac {e^{-14-14 x} x^{21} \log (x)}{65536}+\frac {7 e^{-12-12 x} x^{18} \log ^2(x)}{4096}-\frac {7}{64} e^{-10-10 x} x^{15} \log ^3(x)+\frac {35}{8} e^{-8-8 x} x^{12} \log ^4(x)-112 e^{-6-6 x} x^9 \log ^5(x)+1792 e^{-4-4 x} x^6 \log ^6(x)-16384 e^{-2-2 x} x^3 \log ^7(x)+65536 \log ^8(x)+\frac {3287253918823875 \int e^{-16-16 x} x^6 \, dx}{18889465931478580854784}-\frac {469607702689125 \int e^{-16-16 x} x^7 \, dx}{1180591620717411303424}\\ &=-\frac {3287253918823875 e^{-16-16 x} x^6}{302231454903657293676544}+\frac {e^{-16-16 x} x^{24}}{16777216}-\frac {e^{-14-14 x} x^{21} \log (x)}{65536}+\frac {7 e^{-12-12 x} x^{18} \log ^2(x)}{4096}-\frac {7}{64} e^{-10-10 x} x^{15} \log ^3(x)+\frac {35}{8} e^{-8-8 x} x^{12} \log ^4(x)-112 e^{-6-6 x} x^9 \log ^5(x)+1792 e^{-4-4 x} x^6 \log ^6(x)-16384 e^{-2-2 x} x^3 \log ^7(x)+65536 \log ^8(x)+\frac {9861761756471625 \int e^{-16-16 x} x^5 \, dx}{151115727451828646838272}-\frac {3287253918823875 \int e^{-16-16 x} x^6 \, dx}{18889465931478580854784}\\ &=-\frac {9861761756471625 e^{-16-16 x} x^5}{2417851639229258349412352}+\frac {e^{-16-16 x} x^{24}}{16777216}-\frac {e^{-14-14 x} x^{21} \log (x)}{65536}+\frac {7 e^{-12-12 x} x^{18} \log ^2(x)}{4096}-\frac {7}{64} e^{-10-10 x} x^{15} \log ^3(x)+\frac {35}{8} e^{-8-8 x} x^{12} \log ^4(x)-112 e^{-6-6 x} x^9 \log ^5(x)+1792 e^{-4-4 x} x^6 \log ^6(x)-16384 e^{-2-2 x} x^3 \log ^7(x)+65536 \log ^8(x)+\frac {49308808782358125 \int e^{-16-16 x} x^4 \, dx}{2417851639229258349412352}-\frac {9861761756471625 \int e^{-16-16 x} x^5 \, dx}{151115727451828646838272}\\ &=-\frac {49308808782358125 e^{-16-16 x} x^4}{38685626227668133590597632}+\frac {e^{-16-16 x} x^{24}}{16777216}-\frac {e^{-14-14 x} x^{21} \log (x)}{65536}+\frac {7 e^{-12-12 x} x^{18} \log ^2(x)}{4096}-\frac {7}{64} e^{-10-10 x} x^{15} \log ^3(x)+\frac {35}{8} e^{-8-8 x} x^{12} \log ^4(x)-112 e^{-6-6 x} x^9 \log ^5(x)+1792 e^{-4-4 x} x^6 \log ^6(x)-16384 e^{-2-2 x} x^3 \log ^7(x)+65536 \log ^8(x)+\frac {49308808782358125 \int e^{-16-16 x} x^3 \, dx}{9671406556917033397649408}-\frac {49308808782358125 \int e^{-16-16 x} x^4 \, dx}{2417851639229258349412352}\\ &=-\frac {49308808782358125 e^{-16-16 x} x^3}{154742504910672534362390528}+\frac {e^{-16-16 x} x^{24}}{16777216}-\frac {e^{-14-14 x} x^{21} \log (x)}{65536}+\frac {7 e^{-12-12 x} x^{18} \log ^2(x)}{4096}-\frac {7}{64} e^{-10-10 x} x^{15} \log ^3(x)+\frac {35}{8} e^{-8-8 x} x^{12} \log ^4(x)-112 e^{-6-6 x} x^9 \log ^5(x)+1792 e^{-4-4 x} x^6 \log ^6(x)-16384 e^{-2-2 x} x^3 \log ^7(x)+65536 \log ^8(x)+\frac {147926426347074375 \int e^{-16-16 x} x^2 \, dx}{154742504910672534362390528}-\frac {49308808782358125 \int e^{-16-16 x} x^3 \, dx}{9671406556917033397649408}\\ &=-\frac {147926426347074375 e^{-16-16 x} x^2}{2475880078570760549798248448}+\frac {e^{-16-16 x} x^{24}}{16777216}-\frac {e^{-14-14 x} x^{21} \log (x)}{65536}+\frac {7 e^{-12-12 x} x^{18} \log ^2(x)}{4096}-\frac {7}{64} e^{-10-10 x} x^{15} \log ^3(x)+\frac {35}{8} e^{-8-8 x} x^{12} \log ^4(x)-112 e^{-6-6 x} x^9 \log ^5(x)+1792 e^{-4-4 x} x^6 \log ^6(x)-16384 e^{-2-2 x} x^3 \log ^7(x)+65536 \log ^8(x)+\frac {147926426347074375 \int e^{-16-16 x} x \, dx}{1237940039285380274899124224}-\frac {147926426347074375 \int e^{-16-16 x} x^2 \, dx}{154742504910672534362390528}\\ &=-\frac {147926426347074375 e^{-16-16 x} x}{19807040628566084398385987584}+\frac {e^{-16-16 x} x^{24}}{16777216}-\frac {e^{-14-14 x} x^{21} \log (x)}{65536}+\frac {7 e^{-12-12 x} x^{18} \log ^2(x)}{4096}-\frac {7}{64} e^{-10-10 x} x^{15} \log ^3(x)+\frac {35}{8} e^{-8-8 x} x^{12} \log ^4(x)-112 e^{-6-6 x} x^9 \log ^5(x)+1792 e^{-4-4 x} x^6 \log ^6(x)-16384 e^{-2-2 x} x^3 \log ^7(x)+65536 \log ^8(x)+\frac {147926426347074375 \int e^{-16-16 x} \, dx}{19807040628566084398385987584}-\frac {147926426347074375 \int e^{-16-16 x} x \, dx}{1237940039285380274899124224}\\ &=-\frac {147926426347074375 e^{-16-16 x}}{316912650057057350374175801344}+\frac {e^{-16-16 x} x^{24}}{16777216}-\frac {e^{-14-14 x} x^{21} \log (x)}{65536}+\frac {7 e^{-12-12 x} x^{18} \log ^2(x)}{4096}-\frac {7}{64} e^{-10-10 x} x^{15} \log ^3(x)+\frac {35}{8} e^{-8-8 x} x^{12} \log ^4(x)-112 e^{-6-6 x} x^9 \log ^5(x)+1792 e^{-4-4 x} x^6 \log ^6(x)-16384 e^{-2-2 x} x^3 \log ^7(x)+65536 \log ^8(x)-\frac {147926426347074375 \int e^{-16-16 x} \, dx}{19807040628566084398385987584}\\ &=\frac {e^{-16-16 x} x^{24}}{16777216}-\frac {e^{-14-14 x} x^{21} \log (x)}{65536}+\frac {7 e^{-12-12 x} x^{18} \log ^2(x)}{4096}-\frac {7}{64} e^{-10-10 x} x^{15} \log ^3(x)+\frac {35}{8} e^{-8-8 x} x^{12} \log ^4(x)-112 e^{-6-6 x} x^9 \log ^5(x)+1792 e^{-4-4 x} x^6 \log ^6(x)-16384 e^{-2-2 x} x^3 \log ^7(x)+65536 \log ^8(x)\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.68, size = 28, normalized size = 1.22 \begin {gather*} \frac {e^{-16 (1+x)} \left (x^3-32 e^{2+2 x} \log (x)\right )^8}{16777216} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(E^(-16 - 16*x)*(-32*E^(2 + 2*x)*x^21 + 3*x^24 - 2*x^25 + (7168*E^(4 + 4*x)*x^18 + E^(2 + 2*x)*(-672
*x^21 + 448*x^22))*Log[x] + (-688128*E^(6 + 6*x)*x^15 + E^(4 + 4*x)*(64512*x^18 - 43008*x^19))*Log[x]^2 + (367
00160*E^(8 + 8*x)*x^12 + E^(6 + 6*x)*(-3440640*x^15 + 2293760*x^16))*Log[x]^3 + (-1174405120*E^(10 + 10*x)*x^9
+ E^(8 + 8*x)*(110100480*x^12 - 73400320*x^13))*Log[x]^4 + (22548578304*E^(12 + 12*x)*x^6 + E^(10 + 10*x)*(-2
113929216*x^9 + 1409286144*x^10))*Log[x]^5 + (-240518168576*E^(14 + 14*x)*x^3 + E^(12 + 12*x)*(22548578304*x^6
- 15032385536*x^7))*Log[x]^6 + (1099511627776*E^(16 + 16*x) + E^(14 + 14*x)*(-103079215104*x^3 + 68719476736*
x^4))*Log[x]^7))/(2097152*x),x]

[Out]

(x^3 - 32*E^(2 + 2*x)*Log[x])^8/(16777216*E^(16*(1 + x)))

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fricas [B]  time = 0.66, size = 127, normalized size = 5.52 \begin {gather*} \frac {1}{16777216} \, {\left (x^{24} - 256 \, x^{21} e^{\left (2 \, x + 2\right )} \log \relax (x) + 28672 \, x^{18} e^{\left (4 \, x + 4\right )} \log \relax (x)^{2} - 1835008 \, x^{15} e^{\left (6 \, x + 6\right )} \log \relax (x)^{3} + 73400320 \, x^{12} e^{\left (8 \, x + 8\right )} \log \relax (x)^{4} - 1879048192 \, x^{9} e^{\left (10 \, x + 10\right )} \log \relax (x)^{5} + 30064771072 \, x^{6} e^{\left (12 \, x + 12\right )} \log \relax (x)^{6} - 274877906944 \, x^{3} e^{\left (14 \, x + 14\right )} \log \relax (x)^{7} + 1099511627776 \, e^{\left (16 \, x + 16\right )} \log \relax (x)^{8}\right )} e^{\left (-16 \, x - 16\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/2097152*((1099511627776*exp(x+1)^16+(68719476736*x^4-103079215104*x^3)*exp(x+1)^14)*log(x)^7+(-240
518168576*x^3*exp(x+1)^14+(-15032385536*x^7+22548578304*x^6)*exp(x+1)^12)*log(x)^6+(22548578304*x^6*exp(x+1)^1
2+(1409286144*x^10-2113929216*x^9)*exp(x+1)^10)*log(x)^5+(-1174405120*x^9*exp(x+1)^10+(-73400320*x^13+11010048
0*x^12)*exp(x+1)^8)*log(x)^4+(36700160*x^12*exp(x+1)^8+(2293760*x^16-3440640*x^15)*exp(x+1)^6)*log(x)^3+(-6881
28*x^15*exp(x+1)^6+(-43008*x^19+64512*x^18)*exp(x+1)^4)*log(x)^2+(7168*x^18*exp(x+1)^4+(448*x^22-672*x^21)*exp
(x+1)^2)*log(x)-32*x^21*exp(x+1)^2-2*x^25+3*x^24)/x/exp(x+1)^16,x, algorithm="fricas")

[Out]

1/16777216*(x^24 - 256*x^21*e^(2*x + 2)*log(x) + 28672*x^18*e^(4*x + 4)*log(x)^2 - 1835008*x^15*e^(6*x + 6)*lo
g(x)^3 + 73400320*x^12*e^(8*x + 8)*log(x)^4 - 1879048192*x^9*e^(10*x + 10)*log(x)^5 + 30064771072*x^6*e^(12*x
+ 12)*log(x)^6 - 274877906944*x^3*e^(14*x + 14)*log(x)^7 + 1099511627776*e^(16*x + 16)*log(x)^8)*e^(-16*x - 16
)

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giac [B]  time = 0.28, size = 126, normalized size = 5.48 \begin {gather*} \frac {1}{16777216} \, {\left (x^{24} e^{\left (-16 \, x + 56\right )} - 256 \, x^{21} e^{\left (-14 \, x + 58\right )} \log \relax (x) + 28672 \, x^{18} e^{\left (-12 \, x + 60\right )} \log \relax (x)^{2} - 1835008 \, x^{15} e^{\left (-10 \, x + 62\right )} \log \relax (x)^{3} + 73400320 \, x^{12} e^{\left (-8 \, x + 64\right )} \log \relax (x)^{4} - 1879048192 \, x^{9} e^{\left (-6 \, x + 66\right )} \log \relax (x)^{5} + 30064771072 \, x^{6} e^{\left (-4 \, x + 68\right )} \log \relax (x)^{6} - 274877906944 \, x^{3} e^{\left (-2 \, x + 70\right )} \log \relax (x)^{7} + 1099511627776 \, e^{72} \log \relax (x)^{8}\right )} e^{\left (-72\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/2097152*((1099511627776*exp(x+1)^16+(68719476736*x^4-103079215104*x^3)*exp(x+1)^14)*log(x)^7+(-240
518168576*x^3*exp(x+1)^14+(-15032385536*x^7+22548578304*x^6)*exp(x+1)^12)*log(x)^6+(22548578304*x^6*exp(x+1)^1
2+(1409286144*x^10-2113929216*x^9)*exp(x+1)^10)*log(x)^5+(-1174405120*x^9*exp(x+1)^10+(-73400320*x^13+11010048
0*x^12)*exp(x+1)^8)*log(x)^4+(36700160*x^12*exp(x+1)^8+(2293760*x^16-3440640*x^15)*exp(x+1)^6)*log(x)^3+(-6881
28*x^15*exp(x+1)^6+(-43008*x^19+64512*x^18)*exp(x+1)^4)*log(x)^2+(7168*x^18*exp(x+1)^4+(448*x^22-672*x^21)*exp
(x+1)^2)*log(x)-32*x^21*exp(x+1)^2-2*x^25+3*x^24)/x/exp(x+1)^16,x, algorithm="giac")

[Out]

1/16777216*(x^24*e^(-16*x + 56) - 256*x^21*e^(-14*x + 58)*log(x) + 28672*x^18*e^(-12*x + 60)*log(x)^2 - 183500
8*x^15*e^(-10*x + 62)*log(x)^3 + 73400320*x^12*e^(-8*x + 64)*log(x)^4 - 1879048192*x^9*e^(-6*x + 66)*log(x)^5
+ 30064771072*x^6*e^(-4*x + 68)*log(x)^6 - 274877906944*x^3*e^(-2*x + 70)*log(x)^7 + 1099511627776*e^72*log(x)
^8)*e^(-72)

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maple [B]  time = 0.18, size = 122, normalized size = 5.30

method	result	size
risch	\(65536 \ln \relax (x )^{8}-16384 x^{3} {\mathrm e}^{-2 x -2} \ln \relax (x )^{7}+1792 x^{6} {\mathrm e}^{-4 x -4} \ln \relax (x )^{6}-112 x^{9} {\mathrm e}^{-6 x -6} \ln \relax (x )^{5}+\frac {35 x^{12} {\mathrm e}^{-8 x -8} \ln \relax (x )^{4}}{8}-\frac {7 x^{15} {\mathrm e}^{-10 x -10} \ln \relax (x )^{3}}{64}+\frac {7 x^{18} {\mathrm e}^{-12 x -12} \ln \relax (x )^{2}}{4096}-\frac {x^{21} {\mathrm e}^{-14 x -14} \ln \relax (x )}{65536}+\frac {x^{24} {\mathrm e}^{-16 x -16}}{16777216}\)	\(122\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/2097152*((1099511627776*exp(x+1)^16+(68719476736*x^4-103079215104*x^3)*exp(x+1)^14)*ln(x)^7+(-2405181685
76*x^3*exp(x+1)^14+(-15032385536*x^7+22548578304*x^6)*exp(x+1)^12)*ln(x)^6+(22548578304*x^6*exp(x+1)^12+(14092
86144*x^10-2113929216*x^9)*exp(x+1)^10)*ln(x)^5+(-1174405120*x^9*exp(x+1)^10+(-73400320*x^13+110100480*x^12)*e
xp(x+1)^8)*ln(x)^4+(36700160*x^12*exp(x+1)^8+(2293760*x^16-3440640*x^15)*exp(x+1)^6)*ln(x)^3+(-688128*x^15*exp
(x+1)^6+(-43008*x^19+64512*x^18)*exp(x+1)^4)*ln(x)^2+(7168*x^18*exp(x+1)^4+(448*x^22-672*x^21)*exp(x+1)^2)*ln(
x)-32*x^21*exp(x+1)^2-2*x^25+3*x^24)/x/exp(x+1)^16,x,method=_RETURNVERBOSE)

[Out]

65536*ln(x)^8-16384*x^3*exp(-2*x-2)*ln(x)^7+1792*x^6*exp(-4*x-4)*ln(x)^6-112*x^9*exp(-6*x-6)*ln(x)^5+35/8*x^12
*exp(-8*x-8)*ln(x)^4-7/64*x^15*exp(-10*x-10)*ln(x)^3+7/4096*x^18*exp(-12*x-12)*ln(x)^2-1/65536*x^21*exp(-14*x-
14)*ln(x)+1/16777216*x^24*exp(-16*x-16)

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maxima [B]  time = 0.53, size = 576, normalized size = 25.04 result too large to display

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/2097152*((1099511627776*exp(x+1)^16+(68719476736*x^4-103079215104*x^3)*exp(x+1)^14)*log(x)^7+(-240
518168576*x^3*exp(x+1)^14+(-15032385536*x^7+22548578304*x^6)*exp(x+1)^12)*log(x)^6+(22548578304*x^6*exp(x+1)^1
2+(1409286144*x^10-2113929216*x^9)*exp(x+1)^10)*log(x)^5+(-1174405120*x^9*exp(x+1)^10+(-73400320*x^13+11010048
0*x^12)*exp(x+1)^8)*log(x)^4+(36700160*x^12*exp(x+1)^8+(2293760*x^16-3440640*x^15)*exp(x+1)^6)*log(x)^3+(-6881
28*x^15*exp(x+1)^6+(-43008*x^19+64512*x^18)*exp(x+1)^4)*log(x)^2+(7168*x^18*exp(x+1)^4+(448*x^22-672*x^21)*exp
(x+1)^2)*log(x)-32*x^21*exp(x+1)^2-2*x^25+3*x^24)/x/exp(x+1)^16,x, algorithm="maxima")

[Out]

1/5976303958948914397184*(10213410086094336128*x^18*e^(-12*x + 2)*log(x)^2 - 653658245510037512192*x^15*e^(-10
*x + 4)*log(x)^3 + 26146329820401500487680*x^12*e^(-8*x + 6)*log(x)^4 - 669346043402278412484608*x^9*e^(-6*x +
8)*log(x)^5 + 10709536694436454599753728*x^6*e^(-4*x + 10)*log(x)^6 - 97915764063419013483462656*x^3*e^(-2*x
+ 12)*log(x)^7 + 391663056253676053933850624*e^14*log(x)^8 - (91191161482985144*x^21*log(x) + 6513654391641796
*x^20 + 9305220559488280*x^19 + 12628513616448380*x^18 + 16236660364005060*x^17 + 19715944727720430*x^16 + 225
32508260251920*x^15 + 24141973135984200*x^14 + 24141973135984200*x^13 + 22417546483413900*x^12 + 1921503984292
6200*x^11 + 15097531305156300*x^10 + 10783950932254500*x^9 + 6932539885020750*x^8 + 3961451362869000*x^7 + 198
0725681434500*x^6 + 848882434900500*x^5 + 303172298178750*x^4 + 86620656622500*x^3 + 18561569276250*x^2 + 2651
652753750*x + 189403768125)*e^(-14*x))*e^(-14) + 1/5976303958948914397184*(6513654391641796*x^20 + 93052205594
88280*x^19 + 12628513616448380*x^18 + 16236660364005060*x^17 + 19715944727720430*x^16 + 22532508260251920*x^15
+ 24141973135984200*x^14 + 24141973135984200*x^13 + 22417546483413900*x^12 + 19215039842926200*x^11 + 1509753
1305156300*x^10 + 10783950932254500*x^9 + 6932539885020750*x^8 + 3961451362869000*x^7 + 1980725681434500*x^6 +
848882434900500*x^5 + 303172298178750*x^4 + 86620656622500*x^3 + 18561569276250*x^2 + 2651652753750*x + 18940
3768125)*e^(-14*x - 14) + 1/316912650057057350374175801344*(18889465931478580854784*x^24 + 2833419889721787128
2176*x^23 + 40730410914750689968128*x^22 + 56004315007782198706176*x^21 + 73505663447714135801856*x^20 + 91882
079309642669752320*x^19 + 109109969180200670330880*x^18 + 122748715327725754122240*x^17 + 13042051003570861375
4880*x^16 + 130420510035708613754880*x^15 + 122269228158476825395200*x^14 + 106985574638667222220800*x^13 + 86
925779393917118054400*x^12 + 65194334545437838540800*x^11 + 44821104999988513996800*x^10 + 2801319062499282124
8000*x^9 + 15757419726558461952000*x^8 + 7878709863279230976000*x^7 + 3446935565184663552000*x^6 + 12926008369
44248832000*x^5 + 403937761545077760000*x^4 + 100984440386269440000*x^3 + 18934582572425520000*x^2 + 236682282
1553190000*x + 147926426347074375)*e^(-16*x - 16) - 3/316912650057057350374175801344*(9444732965739290427392*x
^23 + 13576803638250229989376*x^22 + 18668105002594066235392*x^21 + 24501887815904711933952*x^20 + 30627359769
880889917440*x^19 + 36369989726733556776960*x^18 + 40916238442575251374080*x^17 + 43473503345236204584960*x^16
+ 43473503345236204584960*x^15 + 40756409386158941798400*x^14 + 35661858212889074073600*x^13 + 28975259797972
372684800*x^12 + 21731444848479279513600*x^11 + 14940368333329504665600*x^10 + 9337730208330940416000*x^9 + 52
52473242186153984000*x^8 + 2626236621093076992000*x^7 + 1148978521728221184000*x^6 + 430866945648082944000*x^5
+ 134645920515025920000*x^4 + 33661480128756480000*x^3 + 6311527524141840000*x^2 + 788940940517730000*x + 493
08808782358125)*e^(-16*x - 16)

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mupad [F]  time = 0.00, size = -1, normalized size = -0.04 \begin {gather*} \int -\frac {{\mathrm {e}}^{-16\,x-16}\,\left (\frac {\ln \relax (x)\,\left ({\mathrm {e}}^{2\,x+2}\,\left (672\,x^{21}-448\,x^{22}\right )-7168\,x^{18}\,{\mathrm {e}}^{4\,x+4}\right )}{2097152}-\frac {{\ln \relax (x)}^6\,\left ({\mathrm {e}}^{12\,x+12}\,\left (22548578304\,x^6-15032385536\,x^7\right )-240518168576\,x^3\,{\mathrm {e}}^{14\,x+14}\right )}{2097152}-\frac {{\ln \relax (x)}^2\,\left ({\mathrm {e}}^{4\,x+4}\,\left (64512\,x^{18}-43008\,x^{19}\right )-688128\,x^{15}\,{\mathrm {e}}^{6\,x+6}\right )}{2097152}+\frac {{\ln \relax (x)}^3\,\left ({\mathrm {e}}^{6\,x+6}\,\left (3440640\,x^{15}-2293760\,x^{16}\right )-36700160\,x^{12}\,{\mathrm {e}}^{8\,x+8}\right )}{2097152}+\frac {{\ln \relax (x)}^5\,\left ({\mathrm {e}}^{10\,x+10}\,\left (2113929216\,x^9-1409286144\,x^{10}\right )-22548578304\,x^6\,{\mathrm {e}}^{12\,x+12}\right )}{2097152}-\frac {{\ln \relax (x)}^7\,\left (1099511627776\,{\mathrm {e}}^{16\,x+16}-{\mathrm {e}}^{14\,x+14}\,\left (103079215104\,x^3-68719476736\,x^4\right )\right )}{2097152}+\frac {x^{21}\,{\mathrm {e}}^{2\,x+2}}{65536}-\frac {3\,x^{24}}{2097152}+\frac {x^{25}}{1048576}+\frac {{\ln \relax (x)}^4\,\left (1174405120\,x^9\,{\mathrm {e}}^{10\,x+10}-{\mathrm {e}}^{8\,x+8}\,\left (110100480\,x^{12}-73400320\,x^{13}\right )\right )}{2097152}\right )}{x} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(-(exp(- 16*x - 16)*((log(x)*(exp(2*x + 2)*(672*x^21 - 448*x^22) - 7168*x^18*exp(4*x + 4)))/2097152 - (log(
x)^6*(exp(12*x + 12)*(22548578304*x^6 - 15032385536*x^7) - 240518168576*x^3*exp(14*x + 14)))/2097152 - (log(x)
^2*(exp(4*x + 4)*(64512*x^18 - 43008*x^19) - 688128*x^15*exp(6*x + 6)))/2097152 + (log(x)^3*(exp(6*x + 6)*(344
0640*x^15 - 2293760*x^16) - 36700160*x^12*exp(8*x + 8)))/2097152 + (log(x)^5*(exp(10*x + 10)*(2113929216*x^9 -
1409286144*x^10) - 22548578304*x^6*exp(12*x + 12)))/2097152 - (log(x)^7*(1099511627776*exp(16*x + 16) - exp(1
4*x + 14)*(103079215104*x^3 - 68719476736*x^4)))/2097152 + (x^21*exp(2*x + 2))/65536 - (3*x^24)/2097152 + x^25
/1048576 + (log(x)^4*(1174405120*x^9*exp(10*x + 10) - exp(8*x + 8)*(110100480*x^12 - 73400320*x^13)))/2097152)
)/x,x)

[Out]

int(-(exp(- 16*x - 16)*((log(x)*(exp(2*x + 2)*(672*x^21 - 448*x^22) - 7168*x^18*exp(4*x + 4)))/2097152 - (log(
x)^6*(exp(12*x + 12)*(22548578304*x^6 - 15032385536*x^7) - 240518168576*x^3*exp(14*x + 14)))/2097152 - (log(x)
^2*(exp(4*x + 4)*(64512*x^18 - 43008*x^19) - 688128*x^15*exp(6*x + 6)))/2097152 + (log(x)^3*(exp(6*x + 6)*(344
0640*x^15 - 2293760*x^16) - 36700160*x^12*exp(8*x + 8)))/2097152 + (log(x)^5*(exp(10*x + 10)*(2113929216*x^9 -
1409286144*x^10) - 22548578304*x^6*exp(12*x + 12)))/2097152 - (log(x)^7*(1099511627776*exp(16*x + 16) - exp(1
4*x + 14)*(103079215104*x^3 - 68719476736*x^4)))/2097152 + (x^21*exp(2*x + 2))/65536 - (3*x^24)/2097152 + x^25
/1048576 + (log(x)^4*(1174405120*x^9*exp(10*x + 10) - exp(8*x + 8)*(110100480*x^12 - 73400320*x^13)))/2097152)
)/x, x)

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sympy [B]  time = 1.34, size = 153, normalized size = 6.65 \begin {gather*} \frac {x^{24} e^{- 16 x - 16}}{16777216} - \frac {x^{21} e^{- 14 x - 14} \log {\relax (x )}}{65536} + \frac {7 x^{18} e^{- 12 x - 12} \log {\relax (x )}^{2}}{4096} - \frac {7 x^{15} e^{- 10 x - 10} \log {\relax (x )}^{3}}{64} + \frac {35 x^{12} e^{- 8 x - 8} \log {\relax (x )}^{4}}{8} - 112 x^{9} e^{- 6 x - 6} \log {\relax (x )}^{5} + 1792 x^{6} e^{- 4 x - 4} \log {\relax (x )}^{6} - 16384 x^{3} e^{- 2 x - 2} \log {\relax (x )}^{7} + 65536 \log {\relax (x )}^{8} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/2097152*((1099511627776*exp(x+1)**16+(68719476736*x**4-103079215104*x**3)*exp(x+1)**14)*ln(x)**7+(
-240518168576*x**3*exp(x+1)**14+(-15032385536*x**7+22548578304*x**6)*exp(x+1)**12)*ln(x)**6+(22548578304*x**6*
exp(x+1)**12+(1409286144*x**10-2113929216*x**9)*exp(x+1)**10)*ln(x)**5+(-1174405120*x**9*exp(x+1)**10+(-734003
20*x**13+110100480*x**12)*exp(x+1)**8)*ln(x)**4+(36700160*x**12*exp(x+1)**8+(2293760*x**16-3440640*x**15)*exp(
x+1)**6)*ln(x)**3+(-688128*x**15*exp(x+1)**6+(-43008*x**19+64512*x**18)*exp(x+1)**4)*ln(x)**2+(7168*x**18*exp(
x+1)**4+(448*x**22-672*x**21)*exp(x+1)**2)*ln(x)-32*x**21*exp(x+1)**2-2*x**25+3*x**24)/x/exp(x+1)**16,x)

[Out]

x**24*exp(-16*x - 16)/16777216 - x**21*exp(-14*x - 14)*log(x)/65536 + 7*x**18*exp(-12*x - 12)*log(x)**2/4096 -
7*x**15*exp(-10*x - 10)*log(x)**3/64 + 35*x**12*exp(-8*x - 8)*log(x)**4/8 - 112*x**9*exp(-6*x - 6)*log(x)**5
+ 1792*x**6*exp(-4*x - 4)*log(x)**6 - 16384*x**3*exp(-2*x - 2)*log(x)**7 + 65536*log(x)**8

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