3.33.55 $\int \frac{-1968+1760x-560x^2+1280x^4-512x^5+(-656+1024x-1136x^2+1024x^3-256x^4)\log (\log (4))}{75645x^2-67650x^3+34805x^4+10880x^5-7520x^6+2560x^7+1280x^8+(50430x^2-61910x^3+43840x^4-7360x^5-2560x^6+2560x^7)\log (\log (4))+(8405x^2-13120x^3+11680x^4-5120x^5+1280x^6){\log }^2(\log (4))}dx$

Optimal. Leaf size=35 $$\frac{4}{5x(4+\frac{25x^2}{4{(x-x^2)}^2})(3+x+\log (\log (4)))}$$

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Rubi [B]  time = 0.42, antiderivative size = 104, normalized size of antiderivative = 2.97, number of steps used = 7, number of rules used = 4, integrand size = 150, $\frac{\text{number of rules}}{\text{integrand size}}$ = 0.027, Rules used = {2074, 618, 204, 638} $$\begin{array}{c}-\frac{80(-16x(5+\log (\log (4)))+119+32\log (\log (4)))}{41(16x^2-32x+41)(281+16{\log }^2(\log (4))+128\log (\log (4)))}-\frac{16(4+\log (\log (4)){)}^2}{5(3+\log (\log (4)))(281+16{\log }^2(\log (4))+128\log (\log (4)))(x+3+\log (\log (4)))}+\frac{16}{205x(3+\log (\log (4)))}\end{array}$$

Antiderivative was successfully verified.

[In]

Int[(-1968 + 1760*x - 560*x^2 + 1280*x^4 - 512*x^5 + (-656 + 1024*x - 1136*x^2 + 1024*x^3 - 256*x^4)*Log[Log[4
]])/(75645*x^2 - 67650*x^3 + 34805*x^4 + 10880*x^5 - 7520*x^6 + 2560*x^7 + 1280*x^8 + (50430*x^2 - 61910*x^3 +
43840*x^4 - 7360*x^5 - 2560*x^6 + 2560*x^7)*Log[Log[4]] + (8405*x^2 - 13120*x^3 + 11680*x^4 - 5120*x^5 + 1280
*x^6)*Log[Log[4]]^2),x]

[Out]

16/(205*x*(3 + Log[Log[4]])) - (16*(4 + Log[Log[4]])^2)/(5*(3 + Log[Log[4]])*(3 + x + Log[Log[4]])*(281 + 128*
Log[Log[4]] + 16*Log[Log[4]]^2)) - (80*(119 + 32*Log[Log[4]] - 16*x*(5 + Log[Log[4]])))/(41*(41 - 32*x + 16*x^
2)*(281 + 128*Log[Log[4]] + 16*Log[Log[4]]^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{array}{c}\begin{array}{rl}\text{integral}& =\int (-\frac{16}{205x^2(3+\log (\log (4)))}-\frac{1280(5+\log (\log (4)))}{41(41-32x+16x^2)(281+128\log (\log (4))+16{\log }^2(\log (4)))}+\frac{16(4+\log (\log (4)){)}^2}{5(3+\log (\log (4)))(3+x+\log (\log (4)){)}^2(281+128\log (\log (4))+16{\log }^2(\log (4)))}+\frac{2560(86+9\log (\log (4))+x(39+16\log (\log (4))))}{41{(41-32x+16x^2)}^2(281+128\log (\log (4))+16{\log }^2(\log (4)))})dx\\ & =\frac{16}{205x(3+\log (\log (4)))}-\frac{16(4+\log (\log (4)){)}^2}{5(3+\log (\log (4)))(3+x+\log (\log (4)))(281+128\log (\log (4))+16{\log }^2(\log (4)))}+\frac{2560\int \frac{86+9\log (\log (4))+x(39+16\log (\log (4)))}{{(41-32x+16x^2)}^2}dx}{41(281+128\log (\log (4))+16{\log }^2(\log (4)))}-\frac{(1280(5+\log (\log (4))))\int \frac{1}{41-32x+16x^2}dx}{41(281+128\log (\log (4))+16{\log }^2(\log (4)))}\\ & =\frac{16}{205x(3+\log (\log (4)))}-\frac{16(4+\log (\log (4)){)}^2}{5(3+\log (\log (4)))(3+x+\log (\log (4)))(281+128\log (\log (4))+16{\log }^2(\log (4)))}-\frac{80(119+32\log (\log (4))-16x(5+\log (\log (4))))}{41(41-32x+16x^2)(281+128\log (\log (4))+16{\log }^2(\log (4)))}+\frac{(1280(5+\log (\log (4))))\int \frac{1}{41-32x+16x^2}dx}{41(281+128\log (\log (4))+16{\log }^2(\log (4)))}+\frac{(2560(5+\log (\log (4))))\mathrm{Subst}(\int \frac{1}{-1600-x^2}dx,x,-32+32x)}{41(281+128\log (\log (4))+16{\log }^2(\log (4)))}\\ & =\frac{16}{205x(3+\log (\log (4)))}+\frac{64{\tan }^{-1}\left(\frac{4(1-x)}{5}\right)(5+\log (\log (4)))}{41(281+128\log (\log (4))+16{\log }^2(\log (4)))}-\frac{16(4+\log (\log (4)){)}^2}{5(3+\log (\log (4)))(3+x+\log (\log (4)))(281+128\log (\log (4))+16{\log }^2(\log (4)))}-\frac{80(119+32\log (\log (4))-16x(5+\log (\log (4))))}{41(41-32x+16x^2)(281+128\log (\log (4))+16{\log }^2(\log (4)))}-\frac{(2560(5+\log (\log (4))))\mathrm{Subst}(\int \frac{1}{-1600-x^2}dx,x,-32+32x)}{41(281+128\log (\log (4))+16{\log }^2(\log (4)))}\\ & =\frac{16}{205x(3+\log (\log (4)))}-\frac{16(4+\log (\log (4)){)}^2}{5(3+\log (\log (4)))(3+x+\log (\log (4)))(281+128\log (\log (4))+16{\log }^2(\log (4)))}-\frac{80(119+32\log (\log (4))-16x(5+\log (\log (4))))}{41(41-32x+16x^2)(281+128\log (\log (4))+16{\log }^2(\log (4)))}\end{array}\end{array}$$

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Mathematica [A]  time = 0.12, size = 32, normalized size = 0.91 $$\begin{array}{c}\frac{16(-1+x{)}^2}{5x(41-32x+16x^2)(3+x+\log (\log (4)))}\end{array}$$

Antiderivative was successfully verified.

[In]

Integrate[(-1968 + 1760*x - 560*x^2 + 1280*x^4 - 512*x^5 + (-656 + 1024*x - 1136*x^2 + 1024*x^3 - 256*x^4)*Log
[Log[4]])/(75645*x^2 - 67650*x^3 + 34805*x^4 + 10880*x^5 - 7520*x^6 + 2560*x^7 + 1280*x^8 + (50430*x^2 - 61910
*x^3 + 43840*x^4 - 7360*x^5 - 2560*x^6 + 2560*x^7)*Log[Log[4]] + (8405*x^2 - 13120*x^3 + 11680*x^4 - 5120*x^5
+ 1280*x^6)*Log[Log[4]]^2),x]

[Out]

(16*(-1 + x)^2)/(5*x*(41 - 32*x + 16*x^2)*(3 + x + Log[Log[4]]))

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fricas [A]  time = 0.61, size = 51, normalized size = 1.46 $$\begin{array}{c}\frac{16(x^2-2x+1)}{5(16x^4+16x^3-55x^2+(16x^3-32x^2+41x)\log (2\log (2))+123x)}\end{array}$$

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-256*x^4+1024*x^3-1136*x^2+1024*x-656)*log(2*log(2))-512*x^5+1280*x^4-560*x^2+1760*x-1968)/((1280*
x^6-5120*x^5+11680*x^4-13120*x^3+8405*x^2)*log(2*log(2))^2+(2560*x^7-2560*x^6-7360*x^5+43840*x^4-61910*x^3+504
30*x^2)*log(2*log(2))+1280*x^8+2560*x^7-7520*x^6+10880*x^5+34805*x^4-67650*x^3+75645*x^2),x, algorithm="fricas
")

[Out]

16/5*(x^2 - 2*x + 1)/(16*x^4 + 16*x^3 - 55*x^2 + (16*x^3 - 32*x^2 + 41*x)*log(2*log(2)) + 123*x)

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giac [A]  time = 0.28, size = 59, normalized size = 1.69 $$\begin{array}{c}\frac{16(x^2-2x+1)}{5(16x^4+16x^3\log (2\log (2))+16x^3-32x^2\log (2\log (2))-55x^2+41x\log (2\log (2))+123x)}\end{array}$$

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-256*x^4+1024*x^3-1136*x^2+1024*x-656)*log(2*log(2))-512*x^5+1280*x^4-560*x^2+1760*x-1968)/((1280*
x^6-5120*x^5+11680*x^4-13120*x^3+8405*x^2)*log(2*log(2))^2+(2560*x^7-2560*x^6-7360*x^5+43840*x^4-61910*x^3+504
30*x^2)*log(2*log(2))+1280*x^8+2560*x^7-7520*x^6+10880*x^5+34805*x^4-67650*x^3+75645*x^2),x, algorithm="giac")

[Out]

16/5*(x^2 - 2*x + 1)/(16*x^4 + 16*x^3*log(2*log(2)) + 16*x^3 - 32*x^2*log(2*log(2)) - 55*x^2 + 41*x*log(2*log(
2)) + 123*x)

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maple [A]  time = 0.43, size = 37, normalized size = 1.06

method	result	size
norman	$\frac{\frac{16}{5}-\frac{32}{5}x+\frac{16}{5}{x}^2}{x(16x^2-32x+41)(3+x+\ln (2\ln (2)))}$	$37$
gosper	$\frac{16{(x-1)}^2}{5x(16x^2\ln (2\ln (2))+16x^3-32x\ln (2\ln (2))+16x^2+41\ln (2\ln (2))-55x+123)}$	$53$
risch	$\frac{\frac{16}{5}-\frac{32}{5}x+\frac{16}{5}{x}^2}{(16x^2\ln (2)+16x^2\ln (\ln (2))+16x^3-32x\ln (2)-32x\ln (\ln (2))+16x^2+41\ln (2)+41\ln (\ln (2))-55x+123)x}$	$68$
default	$-\frac{1280((-\frac{5}{16}-\frac{\ln (2\ln (2))}{16})x+\frac{119}{256}+\frac{\ln (2\ln (2))}{8})}{41(16\ln {(2\ln (2))}^2+128\ln (2\ln (2))+281)(x^2-2x+\frac{41}{16})}+\frac{16}{5(41\ln (2\ln (2))+123)x}-\frac{16(\ln {(2\ln (2))}^2+8\ln (2\ln (2))+16)}{5(16\ln {(2\ln (2))}^2+128\ln (2\ln (2))+281)(\ln (2\ln (2))+3)(3+x+\ln (2\ln (2)))}$	$127$

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((-256*x^4+1024*x^3-1136*x^2+1024*x-656)*ln(2*ln(2))-512*x^5+1280*x^4-560*x^2+1760*x-1968)/((1280*x^6-5120
*x^5+11680*x^4-13120*x^3+8405*x^2)*ln(2*ln(2))^2+(2560*x^7-2560*x^6-7360*x^5+43840*x^4-61910*x^3+50430*x^2)*ln
(2*ln(2))+1280*x^8+2560*x^7-7520*x^6+10880*x^5+34805*x^4-67650*x^3+75645*x^2),x,method=_RETURNVERBOSE)

[Out]

(16/5-32/5*x+16/5*x^2)/x/(16*x^2-32*x+41)/(3+x+ln(2*ln(2)))

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maxima [A]  time = 0.55, size = 54, normalized size = 1.54 $$\begin{array}{c}\frac{16(x^2-2x+1)}{5(16x^4+16x^3(\log (2\log (2))+1)-x^2(32\log (2\log (2))+55)+41x(\log (2\log (2))+3))}\end{array}$$

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-256*x^4+1024*x^3-1136*x^2+1024*x-656)*log(2*log(2))-512*x^5+1280*x^4-560*x^2+1760*x-1968)/((1280*
x^6-5120*x^5+11680*x^4-13120*x^3+8405*x^2)*log(2*log(2))^2+(2560*x^7-2560*x^6-7360*x^5+43840*x^4-61910*x^3+504
30*x^2)*log(2*log(2))+1280*x^8+2560*x^7-7520*x^6+10880*x^5+34805*x^4-67650*x^3+75645*x^2),x, algorithm="maxima
")

[Out]

16/5*(x^2 - 2*x + 1)/(16*x^4 + 16*x^3*(log(2*log(2)) + 1) - x^2*(32*log(2*log(2)) + 55) + 41*x*(log(2*log(2))
+ 3))

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mupad [B]  time = 6.34, size = 205, normalized size = 5.86 $$\begin{array}{c}\frac{\frac{(\ln \left({\ln (4)}^{1121190}\right)+738961{\ln (\ln (4))}^2+261056{\ln (\ln (4))}^3+52256{\ln (\ln (4))}^4+5632{\ln (\ln (4))}^5+256{\ln (\ln (4))}^6+710649)x^2}{5{(\ln (\ln (4))+3)}^2(\ln \left({\ln (4)}^{71936}\right)+25376{\ln (\ln (4))}^2+4096{\ln (\ln (4))}^3+256{\ln (\ln (4))}^4+78961)}-\frac{(\ln \left({\ln (4)}^{2242380}\right)+1477922{\ln (\ln (4))}^2+522112{\ln (\ln (4))}^3+104512{\ln (\ln (4))}^4+11264{\ln (\ln (4))}^5+512{\ln (\ln (4))}^6+1421298)x}{5{(\ln (\ln (4))+3)}^2(\ln \left({\ln (4)}^{71936}\right)+25376{\ln (\ln (4))}^2+4096{\ln (\ln (4))}^3+256{\ln (\ln (4))}^4+78961)}+\frac{1}{5}}{x^4+(\ln (\ln (4))+1)x^3+(-\ln \left({\ln (4)}^2\right)-\frac{55}{16})x^2+(\ln \left({\ln (4)}^{41/16}\right)+\frac{123}{16})x}\end{array}$$

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(-(log(2*log(2))*(1136*x^2 - 1024*x - 1024*x^3 + 256*x^4 + 656) - 1760*x + 560*x^2 - 1280*x^4 + 512*x^5 + 1
968)/(log(2*log(2))*(50430*x^2 - 61910*x^3 + 43840*x^4 - 7360*x^5 - 2560*x^6 + 2560*x^7) + log(2*log(2))^2*(84
05*x^2 - 13120*x^3 + 11680*x^4 - 5120*x^5 + 1280*x^6) + 75645*x^2 - 67650*x^3 + 34805*x^4 + 10880*x^5 - 7520*x
^6 + 2560*x^7 + 1280*x^8),x)

[Out]

((x^2*(log(log(4)^1121190) + 738961*log(log(4))^2 + 261056*log(log(4))^3 + 52256*log(log(4))^4 + 5632*log(log(
4))^5 + 256*log(log(4))^6 + 710649))/(5*(log(log(4)) + 3)^2*(log(log(4)^71936) + 25376*log(log(4))^2 + 4096*lo
g(log(4))^3 + 256*log(log(4))^4 + 78961)) - (x*(log(log(4)^2242380) + 1477922*log(log(4))^2 + 522112*log(log(4
))^3 + 104512*log(log(4))^4 + 11264*log(log(4))^5 + 512*log(log(4))^6 + 1421298))/(5*(log(log(4)) + 3)^2*(log(
log(4)^71936) + 25376*log(log(4))^2 + 4096*log(log(4))^3 + 256*log(log(4))^4 + 78961)) + 1/5)/(x^3*(log(log(4)
) + 1) + x*(log(log(4)^(41/16)) + 123/16) - x^2*(log(log(4)^2) + 55/16) + x^4)

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sympy [B]  time = 76.22, size = 66, normalized size = 1.89 $$\begin{array}{c}-\frac{-16x^2+32x-16}{80x^4+x^3(80\log (\log (2))+80\log (2)+80)+x^2(-275-160\log (2)-160\log (\log (2)))+x(205\log (\log (2))+205\log (2)+615)}\end{array}$$

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-256*x**4+1024*x**3-1136*x**2+1024*x-656)*ln(2*ln(2))-512*x**5+1280*x**4-560*x**2+1760*x-1968)/((1
280*x**6-5120*x**5+11680*x**4-13120*x**3+8405*x**2)*ln(2*ln(2))**2+(2560*x**7-2560*x**6-7360*x**5+43840*x**4-6
1910*x**3+50430*x**2)*ln(2*ln(2))+1280*x**8+2560*x**7-7520*x**6+10880*x**5+34805*x**4-67650*x**3+75645*x**2),x
)

[Out]

-(-16*x**2 + 32*x - 16)/(80*x**4 + x**3*(80*log(log(2)) + 80*log(2) + 80) + x**2*(-275 - 160*log(2) - 160*log(
log(2))) + x*(205*log(log(2)) + 205*log(2) + 615))

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