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

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

________________________________________________________________________________________

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 {gather*} -\frac {80 (-16 x (5+\log (\log (4)))+119+32 \log (\log (4)))}{41 \left (16 x^2-32 x+41\right ) \left (281+16 \log ^2(\log (4))+128 \log (\log (4))\right )}-\frac {16 (4+\log (\log (4)))^2}{5 (3+\log (\log (4))) \left (281+16 \log ^2(\log (4))+128 \log (\log (4))\right ) (x+3+\log (\log (4)))}+\frac {16}{205 x (3+\log (\log (4)))} \end {gather*}

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 {gather*} \begin {aligned} \text {integral} &=\int \left (-\frac {16}{205 x^2 (3+\log (\log (4)))}-\frac {1280 (5+\log (\log (4)))}{41 \left (41-32 x+16 x^2\right ) \left (281+128 \log (\log (4))+16 \log ^2(\log (4))\right )}+\frac {16 (4+\log (\log (4)))^2}{5 (3+\log (\log (4))) (3+x+\log (\log (4)))^2 \left (281+128 \log (\log (4))+16 \log ^2(\log (4))\right )}+\frac {2560 (86+9 \log (\log (4))+x (39+16 \log (\log (4))))}{41 \left (41-32 x+16 x^2\right )^2 \left (281+128 \log (\log (4))+16 \log ^2(\log (4))\right )}\right ) \, dx\\ &=\frac {16}{205 x (3+\log (\log (4)))}-\frac {16 (4+\log (\log (4)))^2}{5 (3+\log (\log (4))) (3+x+\log (\log (4))) \left (281+128 \log (\log (4))+16 \log ^2(\log (4))\right )}+\frac {2560 \int \frac {86+9 \log (\log (4))+x (39+16 \log (\log (4)))}{\left (41-32 x+16 x^2\right )^2} \, dx}{41 \left (281+128 \log (\log (4))+16 \log ^2(\log (4))\right )}-\frac {(1280 (5+\log (\log (4)))) \int \frac {1}{41-32 x+16 x^2} \, dx}{41 \left (281+128 \log (\log (4))+16 \log ^2(\log (4))\right )}\\ &=\frac {16}{205 x (3+\log (\log (4)))}-\frac {16 (4+\log (\log (4)))^2}{5 (3+\log (\log (4))) (3+x+\log (\log (4))) \left (281+128 \log (\log (4))+16 \log ^2(\log (4))\right )}-\frac {80 (119+32 \log (\log (4))-16 x (5+\log (\log (4))))}{41 \left (41-32 x+16 x^2\right ) \left (281+128 \log (\log (4))+16 \log ^2(\log (4))\right )}+\frac {(1280 (5+\log (\log (4)))) \int \frac {1}{41-32 x+16 x^2} \, dx}{41 \left (281+128 \log (\log (4))+16 \log ^2(\log (4))\right )}+\frac {(2560 (5+\log (\log (4)))) \operatorname {Subst}\left (\int \frac {1}{-1600-x^2} \, dx,x,-32+32 x\right )}{41 \left (281+128 \log (\log (4))+16 \log ^2(\log (4))\right )}\\ &=\frac {16}{205 x (3+\log (\log (4)))}+\frac {64 \tan ^{-1}\left (\frac {4 (1-x)}{5}\right ) (5+\log (\log (4)))}{41 \left (281+128 \log (\log (4))+16 \log ^2(\log (4))\right )}-\frac {16 (4+\log (\log (4)))^2}{5 (3+\log (\log (4))) (3+x+\log (\log (4))) \left (281+128 \log (\log (4))+16 \log ^2(\log (4))\right )}-\frac {80 (119+32 \log (\log (4))-16 x (5+\log (\log (4))))}{41 \left (41-32 x+16 x^2\right ) \left (281+128 \log (\log (4))+16 \log ^2(\log (4))\right )}-\frac {(2560 (5+\log (\log (4)))) \operatorname {Subst}\left (\int \frac {1}{-1600-x^2} \, dx,x,-32+32 x\right )}{41 \left (281+128 \log (\log (4))+16 \log ^2(\log (4))\right )}\\ &=\frac {16}{205 x (3+\log (\log (4)))}-\frac {16 (4+\log (\log (4)))^2}{5 (3+\log (\log (4))) (3+x+\log (\log (4))) \left (281+128 \log (\log (4))+16 \log ^2(\log (4))\right )}-\frac {80 (119+32 \log (\log (4))-16 x (5+\log (\log (4))))}{41 \left (41-32 x+16 x^2\right ) \left (281+128 \log (\log (4))+16 \log ^2(\log (4))\right )}\\ \end {aligned} \end {gather*}

________________________________________________________________________________________

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

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]]))

________________________________________________________________________________________

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

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)

________________________________________________________________________________________

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

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)

________________________________________________________________________________________

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

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

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)))

________________________________________________________________________________________

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

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))

________________________________________________________________________________________

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

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)

________________________________________________________________________________________

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

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))

________________________________________________________________________________________