\(\int \frac {32768-8192 x^2+1536 x^3-128 x^4-128 x^5+16 x^6-2 x^7+\frac {e^{25+\log ^2(2)} (2048-1536 x+384 x^2-64 x^3)}{1024}}{-16384 x^3-4096 x^4+3072 x^5+768 x^6-192 x^7-48 x^8+4 x^9+x^{10}+\frac {e^{50+2 \log ^2(2)} (-64 x^3+48 x^4-12 x^5+x^6)}{1048576}+\frac {e^{25+\log ^2(2)} (-2048 x^3+512 x^4+256 x^5-64 x^6-8 x^7+2 x^8)}{1024}} \, dx\) [3227]

Optimal result

Integrand size = 177, antiderivative size = 38 \[ \int \frac {32768-8192 x^2+1536 x^3-128 x^4-128 x^5+16 x^6-2 x^7+\frac {e^{25+\log ^2(2)} \left (2048-1536 x+384 x^2-64 x^3\right )}{1024}}{-16384 x^3-4096 x^4+3072 x^5+768 x^6-192 x^7-48 x^8+4 x^9+x^{10}+\frac {e^{50+2 \log ^2(2)} \left (-64 x^3+48 x^4-12 x^5+x^6\right )}{1048576}+\frac {e^{25+\log ^2(2)} \left (-2048 x^3+512 x^4+256 x^5-64 x^6-8 x^7+2 x^8\right )}{1024}} \, dx=\frac {\left (\frac {4}{4-x}+\frac {4-x}{x}\right )^2}{e^{(-5+\log (2))^2}+(4+x)^2} \]

[Out]

((-x+4)/x+4/(-x+4))^2/(exp((ln(2)-5)^2)+(4+x)^2)

Rubi [B] (verified)

Leaf count is larger than twice the leaf count of optimal. \(194\) vs. \(2(38)=76\).

Time = 0.51 (sec) , antiderivative size = 194, normalized size of antiderivative = 5.11, number of steps used = 7, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.023, Rules used = {2099, 652, 632, 210} \[ \int \frac {32768-8192 x^2+1536 x^3-128 x^4-128 x^5+16 x^6-2 x^7+\frac {e^{25+\log ^2(2)} \left (2048-1536 x+384 x^2-64 x^3\right )}{1024}}{-16384 x^3-4096 x^4+3072 x^5+768 x^6-192 x^7-48 x^8+4 x^9+x^{10}+\frac {e^{50+2 \log ^2(2)} \left (-64 x^3+48 x^4-12 x^5+x^6\right )}{1048576}+\frac {e^{25+\log ^2(2)} \left (-2048 x^3+512 x^4+256 x^5-64 x^6-8 x^7+2 x^8\right )}{1024}} \, dx=\frac {1024 \left (402653184 x \left (1610612736+65536 e^{25+\log ^2(2)}+e^{50+2 \log ^2(2)}\right )+5188146770730811392+131072 e^{75+3 \log ^2(2)}[phone] e^{50+2 \log ^2(2)}+e^{4 \left (25+\log ^2(2)\right )}[phone] e^{25+\log ^2(2)}\right )}{\left (16384+e^{25+\log ^2(2)}\right )^2 \left (65536+e^{25+\log ^2(2)}\right )^2 \left (1024 x^2+8192 x+16384+e^{25+\log ^2(2)}\right )}+\frac {16384}{x^2 \left (16384+e^{25+\log ^2(2)}\right )}+\frac {268435456}{(4-x) \left (65536+e^{25+\log ^2(2)}\right )^2}-\frac {134217728}{x \left (16384+e^{25+\log ^2(2)}\right )^2}+\frac {16384}{(4-x)^2 \left (65536+e^{25+\log ^2(2)}\right )} \]

[In]

Int[(32768 - 8192*x^2 + 1536*x^3 - 128*x^4 - 128*x^5 + 16*x^6 - 2*x^7 + (E^(25 + Log[2]^2)*(2048 - 1536*x + 38
4*x^2 - 64*x^3))/1024)/(-16384*x^3 - 4096*x^4 + 3072*x^5 + 768*x^6 - 192*x^7 - 48*x^8 + 4*x^9 + x^10 + (E^(50
+ 2*Log[2]^2)*(-64*x^3 + 48*x^4 - 12*x^5 + x^6))/1048576 + (E^(25 + Log[2]^2)*(-2048*x^3 + 512*x^4 + 256*x^5 -
64*x^6 - 8*x^7 + 2*x^8))/1024),x]

[Out]

16384/((65536 + E^(25 + Log[2]^2))*(4 - x)^2) + 268435456/((65536 + E^(25 + Log[2]^2))^2*(4 - x)) + 16384/((16
384 + E^(25 + Log[2]^2))*x^2) - 134217728/((16384 + E^(25 + Log[2]^2))^2*x) + (1024*(5188146770730811392 + 277
076930199552*E^(25 + Log[2]^2) + E^(4*(25 + Log[2]^2)) + 9126805504*E^(50 + 2*Log[2]^2) + 131072*E^(75 + 3*Log
[2]^2) + 402653184*(1610612736 + 65536*E^(25 + Log[2]^2) + E^(50 + 2*Log[2]^2))*x))/((16384 + E^(25 + Log[2]^2
))^2*(65536 + E^(25 + Log[2]^2))^2*(16384 + E^(25 + Log[2]^2) + 8192*x + 1024*x^2))

Rule 210

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

Rule 632

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 652

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)/((p + 1)*(b^2 - 4*a*c)))*(a + b*x + c*x^2)^(p + 1), 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 2099

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{align*} \text {integral}& = \int \left (-\frac {32768}{\left (65536+e^{25+\log ^2(2)}\right ) (-4+x)^3}+\frac {268435456}{\left (65536+e^{25+\log ^2(2)}\right )^2 (-4+x)^2}-\frac {32768}{\left (16384+e^{25+\log ^2(2)}\right ) x^3}+\frac {134217728}{\left (16384+e^{25+\log ^2(2)}\right )^2 x^2}+\frac {2097152 \left (-4 \left (2594073385365405696+13194139533312 e^{25+\log ^2(2)}+e^{4 \left (25+\log ^2(2)\right )}[phone] e^{50+2 \log ^2(2)}+32768 e^{75+3 \log ^2(2)}\right )-\left (2594073385365405696+171523813933056 e^{25+\log ^2(2)}+e^{4 \left (25+\log ^2(2)\right )}[phone] e^{50+2 \log ^2(2)}+131072 e^{75+3 \log ^2(2)}\right ) x\right )}{\left (16384+e^{25+\log ^2(2)}\right )^2 \left (65536+e^{25+\log ^2(2)}\right )^2 \left (16384+e^{25+\log ^2(2)}+8192 x+1024 x^2\right )^2}-\frac {412316860416 \left (1610612736+65536 e^{25+\log ^2(2)}+e^{50+2 \log ^2(2)}\right )}{\left (16384+e^{25+\log ^2(2)}\right )^2 \left (65536+e^{25+\log ^2(2)}\right )^2 \left (16384+e^{25+\log ^2(2)}+8192 x+1024 x^2\right )}\right ) \, dx \\ & = \frac {16384}{\left (65536+e^{25+\log ^2(2)}\right ) (4-x)^2}+\frac {268435456}{\left (65536+e^{25+\log ^2(2)}\right )^2 (4-x)}+\frac {16384}{\left (16384+e^{25+\log ^2(2)}\right ) x^2}-\frac {134217728}{\left (16384+e^{25+\log ^2(2)}\right )^2 x}+\frac {2097152 \int \frac {-4 \left (2594073385365405696+13194139533312 e^{25+\log ^2(2)}+e^{4 \left (25+\log ^2(2)\right )}[phone] e^{50+2 \log ^2(2)}+32768 e^{75+3 \log ^2(2)}\right )-\left (2594073385365405696+171523813933056 e^{25+\log ^2(2)}+e^{4 \left (25+\log ^2(2)\right )}[phone] e^{50+2 \log ^2(2)}+131072 e^{75+3 \log ^2(2)}\right ) x}{\left (16384+e^{25+\log ^2(2)}+8192 x+1024 x^2\right )^2} \, dx}{\left (16384+e^{25+\log ^2(2)}\right )^2 \left (65536+e^{25+\log ^2(2)}\right )^2}-\frac {\left (412316860416 \left (1610612736+65536 e^{25+\log ^2(2)}+e^{50+2 \log ^2(2)}\right )\right ) \int \frac {1}{16384+e^{25+\log ^2(2)}+8192 x+1024 x^2} \, dx}{\left (16384+e^{25+\log ^2(2)}\right )^2 \left (65536+e^{25+\log ^2(2)}\right )^2} \\ & = \frac {16384}{\left (65536+e^{25+\log ^2(2)}\right ) (4-x)^2}+\frac {268435456}{\left (65536+e^{25+\log ^2(2)}\right )^2 (4-x)}+\frac {16384}{\left (16384+e^{25+\log ^2(2)}\right ) x^2}-\frac {134217728}{\left (16384+e^{25+\log ^2(2)}\right )^2 x}+\frac {1024 \left (5188146770730811392+277076930199552 e^{25+\log ^2(2)}+e^{4 \left (25+\log ^2(2)\right )}[phone] e^{50+2 \log ^2(2)}+131072 e^{75+3 \log ^2(2)}[phone] \left (1610612736+65536 e^{25+\log ^2(2)}+e^{50+2 \log ^2(2)}\right ) x\right )}{\left (16384+e^{25+\log ^2(2)}\right )^2 \left (65536+e^{25+\log ^2(2)}\right )^2 \left (16384+e^{25+\log ^2(2)}+8192 x+1024 x^2\right )}+\frac {\left (412316860416 \left (1610612736+65536 e^{25+\log ^2(2)}+e^{50+2 \log ^2(2)}\right )\right ) \int \frac {1}{16384+e^{25+\log ^2(2)}+8192 x+1024 x^2} \, dx}{\left (16384+e^{25+\log ^2(2)}\right )^2 \left (65536+e^{25+\log ^2(2)}\right )^2}+\frac {\left (824633720832 \left (1610612736+65536 e^{25+\log ^2(2)}+e^{50+2 \log ^2(2)}\right )\right ) \text {Subst}\left (\int \frac {1}{-4096 e^{25+\log ^2(2)}-x^2} \, dx,x,8192+2048 x\right )}{\left (16384+e^{25+\log ^2(2)}\right )^2 \left (65536+e^{25+\log ^2(2)}\right )^2} \\ & = \frac {16384}{\left (65536+e^{25+\log ^2(2)}\right ) (4-x)^2}+\frac {268435456}{\left (65536+e^{25+\log ^2(2)}\right )^2 (4-x)}+\frac {16384}{\left (16384+e^{25+\log ^2(2)}\right ) x^2}-\frac {134217728}{\left (16384+e^{25+\log ^2(2)}\right )^2 x}+\frac {1024 \left (5188146770730811392+277076930199552 e^{25+\log ^2(2)}+e^{4 \left (25+\log ^2(2)\right )}[phone] e^{50+2 \log ^2(2)}+131072 e^{75+3 \log ^2(2)}[phone] \left (1610612736+65536 e^{25+\log ^2(2)}+e^{50+2 \log ^2(2)}\right ) x\right )}{\left (16384+e^{25+\log ^2(2)}\right )^2 \left (65536+e^{25+\log ^2(2)}\right )^2 \left (16384+e^{25+\log ^2(2)}+8192 x+1024 x^2\right )}-\frac {12884901888 e^{-\frac {25}{2}-\frac {\log ^2(2)}{2}} \left (1610612736+65536 e^{25+\log ^2(2)}+e^{50+2 \log ^2(2)}\right ) \tan ^{-1}\left (32 e^{-\frac {25}{2}-\frac {\log ^2(2)}{2}} (4+x)\right )}{\left (16384+e^{25+\log ^2(2)}\right )^2 \left (65536+e^{25+\log ^2(2)}\right )^2}-\frac {\left (824633720832 \left (1610612736+65536 e^{25+\log ^2(2)}+e^{50+2 \log ^2(2)}\right )\right ) \text {Subst}\left (\int \frac {1}{-4096 e^{25+\log ^2(2)}-x^2} \, dx,x,8192+2048 x\right )}{\left (16384+e^{25+\log ^2(2)}\right )^2 \left (65536+e^{25+\log ^2(2)}\right )^2} \\ & = \frac {16384}{\left (65536+e^{25+\log ^2(2)}\right ) (4-x)^2}+\frac {268435456}{\left (65536+e^{25+\log ^2(2)}\right )^2 (4-x)}+\frac {16384}{\left (16384+e^{25+\log ^2(2)}\right ) x^2}-\frac {134217728}{\left (16384+e^{25+\log ^2(2)}\right )^2 x}+\frac {1024 \left (5188146770730811392+277076930199552 e^{25+\log ^2(2)}+e^{4 \left (25+\log ^2(2)\right )}[phone] e^{50+2 \log ^2(2)}+131072 e^{75+3 \log ^2(2)}[phone] \left (1610612736+65536 e^{25+\log ^2(2)}+e^{50+2 \log ^2(2)}\right ) x\right )}{\left (16384+e^{25+\log ^2(2)}\right )^2 \left (65536+e^{25+\log ^2(2)}\right )^2 \left (16384+e^{25+\log ^2(2)}+8192 x+1024 x^2\right )} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.34 (sec) , antiderivative size = 38, normalized size of antiderivative = 1.00 \[ \int \frac {32768-8192 x^2+1536 x^3-128 x^4-128 x^5+16 x^6-2 x^7+\frac {e^{25+\log ^2(2)} \left (2048-1536 x+384 x^2-64 x^3\right )}{1024}}{-16384 x^3-4096 x^4+3072 x^5+768 x^6-192 x^7-48 x^8+4 x^9+x^{10}+\frac {e^{50+2 \log ^2(2)} \left (-64 x^3+48 x^4-12 x^5+x^6\right )}{1048576}+\frac {e^{25+\log ^2(2)} \left (-2048 x^3+512 x^4+256 x^5-64 x^6-8 x^7+2 x^8\right )}{1024}} \, dx=\frac {1024 \left (16-4 x+x^2\right )^2}{(-4+x)^2 x^2 \left (e^{25+\log ^2(2)}+1024 (4+x)^2\right )} \]

[In]

Integrate[(32768 - 8192*x^2 + 1536*x^3 - 128*x^4 - 128*x^5 + 16*x^6 - 2*x^7 + (E^(25 + Log[2]^2)*(2048 - 1536*
x + 384*x^2 - 64*x^3))/1024)/(-16384*x^3 - 4096*x^4 + 3072*x^5 + 768*x^6 - 192*x^7 - 48*x^8 + 4*x^9 + x^10 + (
E^(50 + 2*Log[2]^2)*(-64*x^3 + 48*x^4 - 12*x^5 + x^6))/1048576 + (E^(25 + Log[2]^2)*(-2048*x^3 + 512*x^4 + 256
*x^5 - 64*x^6 - 8*x^7 + 2*x^8))/1024),x]

[Out]

(1024*(16 - 4*x + x^2)^2)/((-4 + x)^2*x^2*(E^(25 + Log[2]^2) + 1024*(4 + x)^2))

Maple [A] (verified)

Time = 8.93 (sec) , antiderivative size = 49, normalized size of antiderivative = 1.29

method	result	size
norman	\(\frac {1024 x^{4}-8192 x^{3}+49152 x^{2}-131072 x +262144}{x^{2} \left (x -4\right )^{2} \left (1024 x^{2}+{\mathrm e}^{25+\ln \left (2\right )^{2}}+8192 x +16384\right )}\)	\(49\)
gosper	\(\frac {\left (x^{2}-4 x +16\right )^{2}}{x^{2} \left (x^{4}+{\mathrm e}^{\ln \left (2\right )^{2}-10 \ln \left (2\right )+25} x^{2}-8 \,{\mathrm e}^{\ln \left (2\right )^{2}-10 \ln \left (2\right )+25} x -32 x^{2}+16 \,{\mathrm e}^{\ln \left (2\right )^{2}-10 \ln \left (2\right )+25}+256\right )}\)	\(69\)
risch	\(\frac {1024 x^{4}-8192 x^{3}+49152 x^{2}-131072 x +262144}{x^{2} \left ({\mathrm e}^{25+\ln \left (2\right )^{2}} x^{2}+1024 x^{4}-8 \,{\mathrm e}^{25+\ln \left (2\right )^{2}} x +16 \,{\mathrm e}^{25+\ln \left (2\right )^{2}}-32768 x^{2}+262144\right )}\)	\(69\)
parallelrisch	\(\frac {x^{4}-8 x^{3}+48 x^{2}-128 x +256}{x^{2} \left (x^{4}+{\mathrm e}^{\ln \left (2\right )^{2}-10 \ln \left (2\right )+25} x^{2}-8 \,{\mathrm e}^{\ln \left (2\right )^{2}-10 \ln \left (2\right )+25} x -32 x^{2}+16 \,{\mathrm e}^{\ln \left (2\right )^{2}-10 \ln \left (2\right )+25}+256\right )}\)	\(77\)

[In]

int(((-64*x^3+384*x^2-1536*x+2048)*exp(ln(2)^2-10*ln(2)+25)-2*x^7+16*x^6-128*x^5-128*x^4+1536*x^3-8192*x^2+327
68)/((x^6-12*x^5+48*x^4-64*x^3)*exp(ln(2)^2-10*ln(2)+25)^2+(2*x^8-8*x^7-64*x^6+256*x^5+512*x^4-2048*x^3)*exp(l
n(2)^2-10*ln(2)+25)+x^10+4*x^9-48*x^8-192*x^7+768*x^6+3072*x^5-4096*x^4-16384*x^3),x,method=_RETURNVERBOSE)

[Out]

(1024*x^4-8192*x^3+49152*x^2-131072*x+262144)/x^2/(x-4)^2/(1024*x^2+exp(25+ln(2)^2)+8192*x+16384)

Fricas [A] (verification not implemented)

none

Time = 0.26 (sec) , antiderivative size = 61, normalized size of antiderivative = 1.61 \[ \int \frac {32768-8192 x^2+1536 x^3-128 x^4-128 x^5+16 x^6-2 x^7+\frac {e^{25+\log ^2(2)} \left (2048-1536 x+384 x^2-64 x^3\right )}{1024}}{-16384 x^3-4096 x^4+3072 x^5+768 x^6-192 x^7-48 x^8+4 x^9+x^{10}+\frac {e^{50+2 \log ^2(2)} \left (-64 x^3+48 x^4-12 x^5+x^6\right )}{1048576}+\frac {e^{25+\log ^2(2)} \left (-2048 x^3+512 x^4+256 x^5-64 x^6-8 x^7+2 x^8\right )}{1024}} \, dx=\frac {x^{4} - 8 \, x^{3} + 48 \, x^{2} - 128 \, x + 256}{x^{6} - 32 \, x^{4} + 256 \, x^{2} + {\left (x^{4} - 8 \, x^{3} + 16 \, x^{2}\right )} e^{\left (\log \left (2\right )^{2} - 10 \, \log \left (2\right ) + 25\right )}} \]

[In]

integrate(((-64*x^3+384*x^2-1536*x+2048)*exp(log(2)^2-10*log(2)+25)-2*x^7+16*x^6-128*x^5-128*x^4+1536*x^3-8192
*x^2+32768)/((x^6-12*x^5+48*x^4-64*x^3)*exp(log(2)^2-10*log(2)+25)^2+(2*x^8-8*x^7-64*x^6+256*x^5+512*x^4-2048*
x^3)*exp(log(2)^2-10*log(2)+25)+x^10+4*x^9-48*x^8-192*x^7+768*x^6+3072*x^5-4096*x^4-16384*x^3),x, algorithm="f
ricas")

[Out]

(x^4 - 8*x^3 + 48*x^2 - 128*x + 256)/(x^6 - 32*x^4 + 256*x^2 + (x^4 - 8*x^3 + 16*x^2)*e^(log(2)^2 - 10*log(2)
+ 25))

Sympy [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 73 vs. \(2 (24) = 48\).

Time = 7.05 (sec) , antiderivative size = 73, normalized size of antiderivative = 1.92 \[ \int \frac {32768-8192 x^2+1536 x^3-128 x^4-128 x^5+16 x^6-2 x^7+\frac {e^{25+\log ^2(2)} \left (2048-1536 x+384 x^2-64 x^3\right )}{1024}}{-16384 x^3-4096 x^4+3072 x^5+768 x^6-192 x^7-48 x^8+4 x^9+x^{10}+\frac {e^{50+2 \log ^2(2)} \left (-64 x^3+48 x^4-12 x^5+x^6\right )}{1048576}+\frac {e^{25+\log ^2(2)} \left (-2048 x^3+512 x^4+256 x^5-64 x^6-8 x^7+2 x^8\right )}{1024}} \, dx=- \frac {- 1024 x^{4} + 8192 x^{3} - 49152 x^{2} + 131072 x - 262144}{1024 x^{6} + x^{4} \left (-32768 + e^{25} e^{\log {\left (2 \right )}^{2}}\right ) - 8 x^{3} e^{25} e^{\log {\left (2 \right )}^{2}} + x^{2} \cdot \left (262144 + 16 e^{25} e^{\log {\left (2 \right )}^{2}}\right )} \]

[In]

integrate(((-64*x**3+384*x**2-1536*x+2048)*exp(ln(2)**2-10*ln(2)+25)-2*x**7+16*x**6-128*x**5-128*x**4+1536*x**
3-8192*x**2+32768)/((x**6-12*x**5+48*x**4-64*x**3)*exp(ln(2)**2-10*ln(2)+25)**2+(2*x**8-8*x**7-64*x**6+256*x**
5+512*x**4-2048*x**3)*exp(ln(2)**2-10*ln(2)+25)+x**10+4*x**9-48*x**8-192*x**7+768*x**6+3072*x**5-4096*x**4-163
84*x**3),x)

[Out]

-(-1024*x**4 + 8192*x**3 - 49152*x**2 + 131072*x - 262144)/(1024*x**6 + x**4*(-32768 + exp(25)*exp(log(2)**2))
- 8*x**3*exp(25)*exp(log(2)**2) + x**2*(262144 + 16*exp(25)*exp(log(2)**2)))

Maxima [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 67 vs. \(2 (33) = 66\).

Time = 0.76 (sec) , antiderivative size = 67, normalized size of antiderivative = 1.76 \[ \int \frac {32768-8192 x^2+1536 x^3-128 x^4-128 x^5+16 x^6-2 x^7+\frac {e^{25+\log ^2(2)} \left (2048-1536 x+384 x^2-64 x^3\right )}{1024}}{-16384 x^3-4096 x^4+3072 x^5+768 x^6-192 x^7-48 x^8+4 x^9+x^{10}+\frac {e^{50+2 \log ^2(2)} \left (-64 x^3+48 x^4-12 x^5+x^6\right )}{1048576}+\frac {e^{25+\log ^2(2)} \left (-2048 x^3+512 x^4+256 x^5-64 x^6-8 x^7+2 x^8\right )}{1024}} \, dx=\frac {1024 \, {\left (x^{4} - 8 \, x^{3} + 48 \, x^{2} - 128 \, x + 256\right )}}{1024 \, x^{6} + x^{4} {\left (e^{\left (\log \left (2\right )^{2} + 25\right )} - 32768\right )} - 8 \, x^{3} e^{\left (\log \left (2\right )^{2} + 25\right )} + 16 \, x^{2} {\left (e^{\left (\log \left (2\right )^{2} + 25\right )} + 16384\right )}} \]

[In]

integrate(((-64*x^3+384*x^2-1536*x+2048)*exp(log(2)^2-10*log(2)+25)-2*x^7+16*x^6-128*x^5-128*x^4+1536*x^3-8192
*x^2+32768)/((x^6-12*x^5+48*x^4-64*x^3)*exp(log(2)^2-10*log(2)+25)^2+(2*x^8-8*x^7-64*x^6+256*x^5+512*x^4-2048*
x^3)*exp(log(2)^2-10*log(2)+25)+x^10+4*x^9-48*x^8-192*x^7+768*x^6+3072*x^5-4096*x^4-16384*x^3),x, algorithm="m
axima")

[Out]

1024*(x^4 - 8*x^3 + 48*x^2 - 128*x + 256)/(1024*x^6 + x^4*(e^(log(2)^2 + 25) - 32768) - 8*x^3*e^(log(2)^2 + 25
) + 16*x^2*(e^(log(2)^2 + 25) + 16384))

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 435 vs. \(2 (33) = 66\).

Time = 0.28 (sec) , antiderivative size = 435, normalized size of antiderivative = 11.45 \[ \int \frac {32768-8192 x^2+1536 x^3-128 x^4-128 x^5+16 x^6-2 x^7+\frac {e^{25+\log ^2(2)} \left (2048-1536 x+384 x^2-64 x^3\right )}{1024}}{-16384 x^3-4096 x^4+3072 x^5+768 x^6-192 x^7-48 x^8+4 x^9+x^{10}+\frac {e^{50+2 \log ^2(2)} \left (-64 x^3+48 x^4-12 x^5+x^6\right )}{1048576}+\frac {e^{25+\log ^2(2)} \left (-2048 x^3+512 x^4+256 x^5-64 x^6-8 x^7+2 x^8\right )}{1024}} \, dx=\frac {384 \, x e^{\left (2 \, \log \left (2\right )^{2} - 20 \, \log \left (2\right ) + 50\right )} + 24576 \, x e^{\left (\log \left (2\right )^{2} - 10 \, \log \left (2\right ) + 25\right )} + 589824 \, x + e^{\left (4 \, \log \left (2\right )^{2} - 40 \, \log \left (2\right ) + 100\right )} + 128 \, e^{\left (3 \, \log \left (2\right )^{2} - 30 \, \log \left (2\right ) + 75\right )} + 8704 \, e^{\left (2 \, \log \left (2\right )^{2} - 20 \, \log \left (2\right ) + 50\right )} + 258048 \, e^{\left (\log \left (2\right )^{2} - 10 \, \log \left (2\right ) + 25\right )} + 4718592}{{\left (x^{2} + 8 \, x + e^{\left (\log \left (2\right )^{2} - 10 \, \log \left (2\right ) + 25\right )} + 16\right )} {\left (e^{\left (4 \, \log \left (2\right )^{2} - 40 \, \log \left (2\right ) + 100\right )} + 160 \, e^{\left (3 \, \log \left (2\right )^{2} - 30 \, \log \left (2\right ) + 75\right )} + 8448 \, e^{\left (2 \, \log \left (2\right )^{2} - 20 \, \log \left (2\right ) + 50\right )} + 163840 \, e^{\left (\log \left (2\right )^{2} - 10 \, \log \left (2\right ) + 25\right )} + 1048576\right )}} - \frac {32 \, {\left (12 \, x^{3} e^{\left (2 \, \log \left (2\right )^{2} - 20 \, \log \left (2\right ) + 50\right )} + 768 \, x^{3} e^{\left (\log \left (2\right )^{2} - 10 \, \log \left (2\right ) + 25\right )} + 18432 \, x^{3} - x^{2} e^{\left (3 \, \log \left (2\right )^{2} - 30 \, \log \left (2\right ) + 75\right )} - 184 \, x^{2} e^{\left (2 \, \log \left (2\right )^{2} - 20 \, \log \left (2\right ) + 50\right )} - 9344 \, x^{2} e^{\left (\log \left (2\right )^{2} - 10 \, \log \left (2\right ) + 25\right )} - 180224 \, x^{2} + 4 \, x e^{\left (3 \, \log \left (2\right )^{2} - 30 \, \log \left (2\right ) + 75\right )} + 640 \, x e^{\left (2 \, \log \left (2\right )^{2} - 20 \, \log \left (2\right ) + 50\right )} + 32768 \, x e^{\left (\log \left (2\right )^{2} - 10 \, \log \left (2\right ) + 25\right )} + 524288 \, x - 8 \, e^{\left (3 \, \log \left (2\right )^{2} - 30 \, \log \left (2\right ) + 75\right )} - 1152 \, e^{\left (2 \, \log \left (2\right )^{2} - 20 \, \log \left (2\right ) + 50\right )} - 49152 \, e^{\left (\log \left (2\right )^{2} - 10 \, \log \left (2\right ) + 25\right )} - 524288\right )}}{{\left (x^{2} - 4 \, x\right )}^{2} {\left (e^{\left (4 \, \log \left (2\right )^{2} - 40 \, \log \left (2\right ) + 100\right )} + 160 \, e^{\left (3 \, \log \left (2\right )^{2} - 30 \, \log \left (2\right ) + 75\right )} + 8448 \, e^{\left (2 \, \log \left (2\right )^{2} - 20 \, \log \left (2\right ) + 50\right )} + 163840 \, e^{\left (\log \left (2\right )^{2} - 10 \, \log \left (2\right ) + 25\right )} + 1048576\right )}} \]

[In]

integrate(((-64*x^3+384*x^2-1536*x+2048)*exp(log(2)^2-10*log(2)+25)-2*x^7+16*x^6-128*x^5-128*x^4+1536*x^3-8192
*x^2+32768)/((x^6-12*x^5+48*x^4-64*x^3)*exp(log(2)^2-10*log(2)+25)^2+(2*x^8-8*x^7-64*x^6+256*x^5+512*x^4-2048*
x^3)*exp(log(2)^2-10*log(2)+25)+x^10+4*x^9-48*x^8-192*x^7+768*x^6+3072*x^5-4096*x^4-16384*x^3),x, algorithm="g
iac")

[Out]

(384*x*e^(2*log(2)^2 - 20*log(2) + 50) + 24576*x*e^(log(2)^2 - 10*log(2) + 25) + 589824*x + e^(4*log(2)^2 - 40
*log(2) + 100) + 128*e^(3*log(2)^2 - 30*log(2) + 75) + 8704*e^(2*log(2)^2 - 20*log(2) + 50) + 258048*e^(log(2)
^2 - 10*log(2) + 25) + 4718592)/((x^2 + 8*x + e^(log(2)^2 - 10*log(2) + 25) + 16)*(e^(4*log(2)^2 - 40*log(2) +
100) + 160*e^(3*log(2)^2 - 30*log(2) + 75) + 8448*e^(2*log(2)^2 - 20*log(2) + 50) + 163840*e^(log(2)^2 - 10*l
og(2) + 25) + 1048576)) - 32*(12*x^3*e^(2*log(2)^2 - 20*log(2) + 50) + 768*x^3*e^(log(2)^2 - 10*log(2) + 25) +
18432*x^3 - x^2*e^(3*log(2)^2 - 30*log(2) + 75) - 184*x^2*e^(2*log(2)^2 - 20*log(2) + 50) - 9344*x^2*e^(log(2
)^2 - 10*log(2) + 25) - 180224*x^2 + 4*x*e^(3*log(2)^2 - 30*log(2) + 75) + 640*x*e^(2*log(2)^2 - 20*log(2) + 5
0) + 32768*x*e^(log(2)^2 - 10*log(2) + 25) + 524288*x - 8*e^(3*log(2)^2 - 30*log(2) + 75) - 1152*e^(2*log(2)^2
- 20*log(2) + 50) - 49152*e^(log(2)^2 - 10*log(2) + 25) - 524288)/((x^2 - 4*x)^2*(e^(4*log(2)^2 - 40*log(2) +
100) + 160*e^(3*log(2)^2 - 30*log(2) + 75) + 8448*e^(2*log(2)^2 - 20*log(2) + 50) + 163840*e^(log(2)^2 - 10*l
og(2) + 25) + 1048576))

Mupad [B] (verification not implemented)

Time = 9.88 (sec) , antiderivative size = 69, normalized size of antiderivative = 1.82 \[ \int \frac {32768-8192 x^2+1536 x^3-128 x^4-128 x^5+16 x^6-2 x^7+\frac {e^{25+\log ^2(2)} \left (2048-1536 x+384 x^2-64 x^3\right )}{1024}}{-16384 x^3-4096 x^4+3072 x^5+768 x^6-192 x^7-48 x^8+4 x^9+x^{10}+\frac {e^{50+2 \log ^2(2)} \left (-64 x^3+48 x^4-12 x^5+x^6\right )}{1048576}+\frac {e^{25+\log ^2(2)} \left (-2048 x^3+512 x^4+256 x^5-64 x^6-8 x^7+2 x^8\right )}{1024}} \, dx=\frac {1024\,x^4-8192\,x^3+49152\,x^2-131072\,x+262144}{1024\,x^6+\left ({\mathrm {e}}^{{\ln \left (2\right )}^2+25}-32768\right )\,x^4-8\,{\mathrm {e}}^{{\ln \left (2\right )}^2+25}\,x^3+\left (16\,{\mathrm {e}}^{{\ln \left (2\right )}^2+25}+262144\right )\,x^2} \]

[In]

int((exp(log(2)^2 - 10*log(2) + 25)*(1536*x - 384*x^2 + 64*x^3 - 2048) + 8192*x^2 - 1536*x^3 + 128*x^4 + 128*x
^5 - 16*x^6 + 2*x^7 - 32768)/(exp(2*log(2)^2 - 20*log(2) + 50)*(64*x^3 - 48*x^4 + 12*x^5 - x^6) + exp(log(2)^2
- 10*log(2) + 25)*(2048*x^3 - 512*x^4 - 256*x^5 + 64*x^6 + 8*x^7 - 2*x^8) + 16384*x^3 + 4096*x^4 - 3072*x^5 -
768*x^6 + 192*x^7 + 48*x^8 - 4*x^9 - x^10),x)

[Out]

(49152*x^2 - 131072*x - 8192*x^3 + 1024*x^4 + 262144)/(x^4*(exp(log(2)^2 + 25) - 32768) + x^2*(16*exp(log(2)^2
+ 25) + 262144) - 8*x^3*exp(log(2)^2 + 25) + 1024*x^6)