∫ 32768−8192x2+1536x3−128x4−128x5+16x6−2x7+e25+log2(2) (2048−1536x+384x2−64x3) 1024_________________________________ −16384x3−4096x4+3072x5+768x6−192x7−48x8+4x9+x10+e50+2 log2(2)(−64x3+48x4−12x5+x6) ____________________________________________________1048576+e25+log2(2)(−2048x3+512x4+256x5−64x6−8x7+2x8) ________________________________________________________________________

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

Optimal. Leaf size=38 \[ \frac {\left (\frac {4}{4-x}+\frac {4-x}{x}\right )^2}{e^{(-5+\log (2))^2}+(4+x)^2} \]

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Rubi [B] time = 0.86, antiderivative size = 194, normalized size of antiderivative = 5.11, number of steps used = 7, number of rules used = 4, integrand size = 177, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.023, Rules used = {2074, 638, 618, 204} \begin {gather*} \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)}+9126805504 e^{50+2 \log ^2(2)}+e^{4 \left (25+\log ^2(2)\right )}+277076930199552 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 )} \end {gather*}

Antiderivative was successfully veriﬁed.

[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 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 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 {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 )}+1073741824 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 )}+7516192768 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 )}+1073741824 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 )}+7516192768 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 )}+9126805504 e^{50+2 \log ^2(2)}+131072 e^{75+3 \log ^2(2)}+402653184 \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 ) \operatorname {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 )}+9126805504 e^{50+2 \log ^2(2)}+131072 e^{75+3 \log ^2(2)}+402653184 \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 ) \operatorname {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 )}+9126805504 e^{50+2 \log ^2(2)}+131072 e^{75+3 \log ^2(2)}+402653184 \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 {aligned} \end {gather*}

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Mathematica [A] time = 0.32, size = 38, normalized size = 1.00 \begin {gather*} \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 )} \end {gather*}

Antiderivative was successfully veriﬁed.

[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))

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fricas [A] time = 0.59, size = 61, normalized size = 1.61 \begin {gather*} \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 \relax (2)^{2} - 10 \, \log \relax (2) + 25\right )}} \end {gather*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[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))

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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(((-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]

Timed out

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maple [A] time = 1.30, size = 49, normalized size = 1.29


method	result	size

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

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[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)

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maxima [B] time = 1.01, size = 67, normalized size = 1.76 \begin {gather*} \frac {1024 \, {\left (x^{4} - 8 \, x^{3} + 48 \, x^{2} - 128 \, x + 256\right )}}{1024 \, x^{6} + x^{4} {\left (e^{\left (\log \relax (2)^{2} + 25\right )} - 32768\right )} - 8 \, x^{3} e^{\left (\log \relax (2)^{2} + 25\right )} + 16 \, x^{2} {\left (e^{\left (\log \relax (2)^{2} + 25\right )} + 16384\right )}} \end {gather*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[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))

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mupad [B] time = 2.41, size = 69, normalized size = 1.82 \begin {gather*} \frac {1024\,x^4-8192\,x^3+49152\,x^2-131072\,x+262144}{1024\,x^6+\left ({\mathrm {e}}^{{\ln \relax (2)}^2+25}-32768\right )\,x^4-8\,{\mathrm {e}}^{{\ln \relax (2)}^2+25}\,x^3+\left (16\,{\mathrm {e}}^{{\ln \relax (2)}^2+25}+262144\right )\,x^2} \end {gather*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[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)

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sympy [B] time = 11.46, size = 73, normalized size = 1.92 \begin {gather*} - \frac {- 1024 x^{4} + 8192 x^{3} - 49152 x^{2} + 131072 x - 262144}{1024 x^{6} + x^{4} \left (-32768 + e^{25} e^{\log {\relax (2 )}^{2}}\right ) - 8 x^{3} e^{25} e^{\log {\relax (2 )}^{2}} + x^{2} \left (262144 + 16 e^{25} e^{\log {\relax (2 )}^{2}}\right )} \end {gather*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[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)))

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