\(\int \log ^{1-x}(2) (-10+66 x-120 x^2+64 x^3+(-1+10 x-33 x^2+40 x^3-16 x^4) \log (\log (2))) \, dx\) [2390]

Optimal result

Integrand size = 48, antiderivative size = 24 \[ \int \log ^{1-x}(2) \left (-10+66 x-120 x^2+64 x^3+\left (-1+10 x-33 x^2+40 x^3-16 x^4\right ) \log (\log (2))\right ) \, dx=1+\left ((1-2 x)^2-x\right )^2 \log ^{1-x}(2) \]

[Out]

1+exp((-x+3)*ln(ln(2)))/ln(2)^2*((1-2*x)^2-x)^2

Rubi [B] (verified)

Leaf count is larger than twice the leaf count of optimal. \(59\) vs. \(2(24)=48\).

Time = 0.26 (sec) , antiderivative size = 59, normalized size of antiderivative = 2.46, number of steps used = 29, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.062, Rules used = {2227, 2225, 2207} \[ \int \log ^{1-x}(2) \left (-10+66 x-120 x^2+64 x^3+\left (-1+10 x-33 x^2+40 x^3-16 x^4\right ) \log (\log (2))\right ) \, dx=16 x^4 \log ^{1-x}(2)-40 x^3 \log ^{1-x}(2)+33 x^2 \log ^{1-x}(2)-10 x \log ^{1-x}(2)+\log ^{1-x}(2) \]

[In]

Int[Log[2]^(1 - x)*(-10 + 66*x - 120*x^2 + 64*x^3 + (-1 + 10*x - 33*x^2 + 40*x^3 - 16*x^4)*Log[Log[2]]),x]

[Out]

Log[2]^(1 - x) - 10*x*Log[2]^(1 - x) + 33*x^2*Log[2]^(1 - x) - 40*x^3*Log[2]^(1 - x) + 16*x^4*Log[2]^(1 - x)

Rule 2207

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] &&  !TrueQ[$UseGamma]

Rule 2225

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 2227

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] &&  !TrueQ[$UseGamma]

Rubi steps \begin{align*} \text {integral}& = \int \left (-10 \log ^{1-x}(2)+66 x \log ^{1-x}(2)-120 x^2 \log ^{1-x}(2)+64 x^3 \log ^{1-x}(2)-\left (1-5 x+4 x^2\right )^2 \log ^{1-x}(2) \log (\log (2))\right ) \, dx \\ & = -\left (10 \int \log ^{1-x}(2) \, dx\right )+64 \int x^3 \log ^{1-x}(2) \, dx+66 \int x \log ^{1-x}(2) \, dx-120 \int x^2 \log ^{1-x}(2) \, dx-\log (\log (2)) \int \left (1-5 x+4 x^2\right )^2 \log ^{1-x}(2) \, dx \\ & = \frac {10 \log ^{1-x}(2)}{\log (\log (2))}-\frac {66 x \log ^{1-x}(2)}{\log (\log (2))}+\frac {120 x^2 \log ^{1-x}(2)}{\log (\log (2))}-\frac {64 x^3 \log ^{1-x}(2)}{\log (\log (2))}+\frac {66 \int \log ^{1-x}(2) \, dx}{\log (\log (2))}+\frac {192 \int x^2 \log ^{1-x}(2) \, dx}{\log (\log (2))}-\frac {240 \int x \log ^{1-x}(2) \, dx}{\log (\log (2))}-\log (\log (2)) \int \left (\log ^{1-x}(2)-10 x \log ^{1-x}(2)+33 x^2 \log ^{1-x}(2)-40 x^3 \log ^{1-x}(2)+16 x^4 \log ^{1-x}(2)\right ) \, dx \\ & = -\frac {66 \log ^{1-x}(2)}{\log ^2(\log (2))}+\frac {240 x \log ^{1-x}(2)}{\log ^2(\log (2))}-\frac {192 x^2 \log ^{1-x}(2)}{\log ^2(\log (2))}+\frac {10 \log ^{1-x}(2)}{\log (\log (2))}-\frac {66 x \log ^{1-x}(2)}{\log (\log (2))}+\frac {120 x^2 \log ^{1-x}(2)}{\log (\log (2))}-\frac {64 x^3 \log ^{1-x}(2)}{\log (\log (2))}-\frac {240 \int \log ^{1-x}(2) \, dx}{\log ^2(\log (2))}+\frac {384 \int x \log ^{1-x}(2) \, dx}{\log ^2(\log (2))}-\log (\log (2)) \int \log ^{1-x}(2) \, dx+(10 \log (\log (2))) \int x \log ^{1-x}(2) \, dx-(16 \log (\log (2))) \int x^4 \log ^{1-x}(2) \, dx-(33 \log (\log (2))) \int x^2 \log ^{1-x}(2) \, dx+(40 \log (\log (2))) \int x^3 \log ^{1-x}(2) \, dx \\ & = \log ^{1-x}(2)-10 x \log ^{1-x}(2)+33 x^2 \log ^{1-x}(2)-40 x^3 \log ^{1-x}(2)+16 x^4 \log ^{1-x}(2)+\frac {240 \log ^{1-x}(2)}{\log ^3(\log (2))}-\frac {384 x \log ^{1-x}(2)}{\log ^3(\log (2))}-\frac {66 \log ^{1-x}(2)}{\log ^2(\log (2))}+\frac {240 x \log ^{1-x}(2)}{\log ^2(\log (2))}-\frac {192 x^2 \log ^{1-x}(2)}{\log ^2(\log (2))}+\frac {10 \log ^{1-x}(2)}{\log (\log (2))}-\frac {66 x \log ^{1-x}(2)}{\log (\log (2))}+\frac {120 x^2 \log ^{1-x}(2)}{\log (\log (2))}-\frac {64 x^3 \log ^{1-x}(2)}{\log (\log (2))}+10 \int \log ^{1-x}(2) \, dx-64 \int x^3 \log ^{1-x}(2) \, dx-66 \int x \log ^{1-x}(2) \, dx+120 \int x^2 \log ^{1-x}(2) \, dx+\frac {384 \int \log ^{1-x}(2) \, dx}{\log ^3(\log (2))} \\ & = \log ^{1-x}(2)-10 x \log ^{1-x}(2)+33 x^2 \log ^{1-x}(2)-40 x^3 \log ^{1-x}(2)+16 x^4 \log ^{1-x}(2)-\frac {384 \log ^{1-x}(2)}{\log ^4(\log (2))}+\frac {240 \log ^{1-x}(2)}{\log ^3(\log (2))}-\frac {384 x \log ^{1-x}(2)}{\log ^3(\log (2))}-\frac {66 \log ^{1-x}(2)}{\log ^2(\log (2))}+\frac {240 x \log ^{1-x}(2)}{\log ^2(\log (2))}-\frac {192 x^2 \log ^{1-x}(2)}{\log ^2(\log (2))}-\frac {66 \int \log ^{1-x}(2) \, dx}{\log (\log (2))}-\frac {192 \int x^2 \log ^{1-x}(2) \, dx}{\log (\log (2))}+\frac {240 \int x \log ^{1-x}(2) \, dx}{\log (\log (2))} \\ & = \log ^{1-x}(2)-10 x \log ^{1-x}(2)+33 x^2 \log ^{1-x}(2)-40 x^3 \log ^{1-x}(2)+16 x^4 \log ^{1-x}(2)-\frac {384 \log ^{1-x}(2)}{\log ^4(\log (2))}+\frac {240 \log ^{1-x}(2)}{\log ^3(\log (2))}-\frac {384 x \log ^{1-x}(2)}{\log ^3(\log (2))}+\frac {240 \int \log ^{1-x}(2) \, dx}{\log ^2(\log (2))}-\frac {384 \int x \log ^{1-x}(2) \, dx}{\log ^2(\log (2))} \\ & = \log ^{1-x}(2)-10 x \log ^{1-x}(2)+33 x^2 \log ^{1-x}(2)-40 x^3 \log ^{1-x}(2)+16 x^4 \log ^{1-x}(2)-\frac {384 \log ^{1-x}(2)}{\log ^4(\log (2))}-\frac {384 \int \log ^{1-x}(2) \, dx}{\log ^3(\log (2))} \\ & = \log ^{1-x}(2)-10 x \log ^{1-x}(2)+33 x^2 \log ^{1-x}(2)-40 x^3 \log ^{1-x}(2)+16 x^4 \log ^{1-x}(2) \\ \end{align*}

Mathematica [A] (verified)

Time = 0.22 (sec) , antiderivative size = 21, normalized size of antiderivative = 0.88 \[ \int \log ^{1-x}(2) \left (-10+66 x-120 x^2+64 x^3+\left (-1+10 x-33 x^2+40 x^3-16 x^4\right ) \log (\log (2))\right ) \, dx=\left (1-5 x+4 x^2\right )^2 \log ^{1-x}(2) \]

[In]

Integrate[Log[2]^(1 - x)*(-10 + 66*x - 120*x^2 + 64*x^3 + (-1 + 10*x - 33*x^2 + 40*x^3 - 16*x^4)*Log[Log[2]]),
x]

[Out]

(1 - 5*x + 4*x^2)^2*Log[2]^(1 - x)

Maple [A] (verified)

Time = 0.21 (sec) , antiderivative size = 34, normalized size of antiderivative = 1.42

method	result	size
risch	\(\frac {\left (16 x^{4}-40 x^{3}+33 x^{2}-10 x +1\right ) \ln \left (2\right )^{-x +3}}{\ln \left (2\right )^{2}}\)	\(34\)
gosper	\(\frac {{\mathrm e}^{-\left (-3+x \right ) \ln \left (\ln \left (2\right )\right )} \left (-1+4 x \right ) \left (4 x^{3}-9 x^{2}+6 x -1\right )}{\ln \left (2\right )^{2}}\)	\(35\)
parallelrisch	\(\frac {16 \,{\mathrm e}^{\left (-x +3\right ) \ln \left (\ln \left (2\right )\right )} x^{4}-40 x^{3} {\mathrm e}^{\left (-x +3\right ) \ln \left (\ln \left (2\right )\right )}+33 \,{\mathrm e}^{\left (-x +3\right ) \ln \left (\ln \left (2\right )\right )} x^{2}-10 x \,{\mathrm e}^{\left (-x +3\right ) \ln \left (\ln \left (2\right )\right )}+{\mathrm e}^{\left (-x +3\right ) \ln \left (\ln \left (2\right )\right )}}{\ln \left (2\right )^{2}}\)	\(75\)
norman	\(\frac {\frac {{\mathrm e}^{\left (-x +3\right ) \ln \left (\ln \left (2\right )\right )}}{\ln \left (2\right )}-\frac {10 x \,{\mathrm e}^{\left (-x +3\right ) \ln \left (\ln \left (2\right )\right )}}{\ln \left (2\right )}+\frac {33 x^{2} {\mathrm e}^{\left (-x +3\right ) \ln \left (\ln \left (2\right )\right )}}{\ln \left (2\right )}-\frac {40 x^{3} {\mathrm e}^{\left (-x +3\right ) \ln \left (\ln \left (2\right )\right )}}{\ln \left (2\right )}+\frac {16 x^{4} {\mathrm e}^{\left (-x +3\right ) \ln \left (\ln \left (2\right )\right )}}{\ln \left (2\right )}}{\ln \left (2\right )}\)	\(96\)
meijerg	\(-\frac {16 \ln \left (2\right ) \left (24-\frac {\left (5 x^{4} \ln \left (\ln \left (2\right )\right )^{4}+20 x^{3} \ln \left (\ln \left (2\right )\right )^{3}+60 x^{2} \ln \left (\ln \left (2\right )\right )^{2}+120 x \ln \left (\ln \left (2\right )\right )+120\right ) {\mathrm e}^{-x \ln \left (\ln \left (2\right )\right )}}{5}\right )}{\ln \left (\ln \left (2\right )\right )^{4}}+\frac {\left (40 \ln \left (\ln \left (2\right )\right )+64\right ) \ln \left (2\right ) \left (6-\frac {\left (4 x^{3} \ln \left (\ln \left (2\right )\right )^{3}+12 x^{2} \ln \left (\ln \left (2\right )\right )^{2}+24 x \ln \left (\ln \left (2\right )\right )+24\right ) {\mathrm e}^{-x \ln \left (\ln \left (2\right )\right )}}{4}\right )}{\ln \left (\ln \left (2\right )\right )^{4}}+\frac {\left (-33 \ln \left (\ln \left (2\right )\right )-120\right ) \ln \left (2\right ) \left (2-\frac {\left (3 x^{2} \ln \left (\ln \left (2\right )\right )^{2}+6 x \ln \left (\ln \left (2\right )\right )+6\right ) {\mathrm e}^{-x \ln \left (\ln \left (2\right )\right )}}{3}\right )}{\ln \left (\ln \left (2\right )\right )^{3}}+\frac {\left (10 \ln \left (\ln \left (2\right )\right )+66\right ) \ln \left (2\right ) \left (1-\frac {\left (2+2 x \ln \left (\ln \left (2\right )\right )\right ) {\mathrm e}^{-x \ln \left (\ln \left (2\right )\right )}}{2}\right )}{\ln \left (\ln \left (2\right )\right )^{2}}-\ln \left (2\right ) \left (1-{\mathrm e}^{-x \ln \left (\ln \left (2\right )\right )}\right )-\frac {10 \ln \left (2\right ) \left (1-{\mathrm e}^{-x \ln \left (\ln \left (2\right )\right )}\right )}{\ln \left (\ln \left (2\right )\right )}\)	\(227\)
parts	\(\text {Expression too large to display}\)	\(551\)
derivativedivides	\(\text {Expression too large to display}\)	\(613\)
default	\(\text {Expression too large to display}\)	\(613\)

[In]

int(((-16*x^4+40*x^3-33*x^2+10*x-1)*ln(ln(2))+64*x^3-120*x^2+66*x-10)*exp((-x+3)*ln(ln(2)))/ln(2)^2,x,method=_
RETURNVERBOSE)

[Out]

1/ln(2)^2*(16*x^4-40*x^3+33*x^2-10*x+1)*ln(2)^(-x+3)

Fricas [A] (verification not implemented)

none

Time = 0.26 (sec) , antiderivative size = 33, normalized size of antiderivative = 1.38 \[ \int \log ^{1-x}(2) \left (-10+66 x-120 x^2+64 x^3+\left (-1+10 x-33 x^2+40 x^3-16 x^4\right ) \log (\log (2))\right ) \, dx=\frac {{\left (16 \, x^{4} - 40 \, x^{3} + 33 \, x^{2} - 10 \, x + 1\right )} \log \left (2\right )^{-x + 3}}{\log \left (2\right )^{2}} \]

[In]

integrate(((-16*x^4+40*x^3-33*x^2+10*x-1)*log(log(2))+64*x^3-120*x^2+66*x-10)*exp((-x+3)*log(log(2)))/log(2)^2
,x, algorithm="fricas")

[Out]

(16*x^4 - 40*x^3 + 33*x^2 - 10*x + 1)*log(2)^(-x + 3)/log(2)^2

Sympy [A] (verification not implemented)

Time = 0.08 (sec) , antiderivative size = 34, normalized size of antiderivative = 1.42 \[ \int \log ^{1-x}(2) \left (-10+66 x-120 x^2+64 x^3+\left (-1+10 x-33 x^2+40 x^3-16 x^4\right ) \log (\log (2))\right ) \, dx=\frac {\left (16 x^{4} - 40 x^{3} + 33 x^{2} - 10 x + 1\right ) e^{\left (3 - x\right ) \log {\left (\log {\left (2 \right )} \right )}}}{\log {\left (2 \right )}^{2}} \]

[In]

integrate(((-16*x**4+40*x**3-33*x**2+10*x-1)*ln(ln(2))+64*x**3-120*x**2+66*x-10)*exp((-x+3)*ln(ln(2)))/ln(2)**
2,x)

[Out]

(16*x**4 - 40*x**3 + 33*x**2 - 10*x + 1)*exp((3 - x)*log(log(2)))/log(2)**2

Maxima [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 303 vs. \(2 (28) = 56\).

Time = 0.28 (sec) , antiderivative size = 303, normalized size of antiderivative = 12.62 \[ \int \log ^{1-x}(2) \left (-10+66 x-120 x^2+64 x^3+\left (-1+10 x-33 x^2+40 x^3-16 x^4\right ) \log (\log (2))\right ) \, dx=\log \left (2\right )^{-x + 1} - \frac {10 \, {\left (x \log \left (2\right ) \log \left (\log \left (2\right )\right ) + \log \left (2\right )\right )} \log \left (2\right )^{-x}}{\log \left (\log \left (2\right )\right )} + \frac {33 \, {\left (x^{2} \log \left (2\right ) \log \left (\log \left (2\right )\right )^{2} + 2 \, x \log \left (2\right ) \log \left (\log \left (2\right )\right ) + 2 \, \log \left (2\right )\right )} \log \left (2\right )^{-x}}{\log \left (\log \left (2\right )\right )^{2}} - \frac {66 \, {\left (x \log \left (2\right ) \log \left (\log \left (2\right )\right ) + \log \left (2\right )\right )} \log \left (2\right )^{-x}}{\log \left (\log \left (2\right )\right )^{2}} + \frac {10 \, \log \left (2\right )^{-x + 1}}{\log \left (\log \left (2\right )\right )} - \frac {40 \, {\left (x^{3} \log \left (2\right ) \log \left (\log \left (2\right )\right )^{3} + 3 \, x^{2} \log \left (2\right ) \log \left (\log \left (2\right )\right )^{2} + 6 \, x \log \left (2\right ) \log \left (\log \left (2\right )\right ) + 6 \, \log \left (2\right )\right )} \log \left (2\right )^{-x}}{\log \left (\log \left (2\right )\right )^{3}} + \frac {120 \, {\left (x^{2} \log \left (2\right ) \log \left (\log \left (2\right )\right )^{2} + 2 \, x \log \left (2\right ) \log \left (\log \left (2\right )\right ) + 2 \, \log \left (2\right )\right )} \log \left (2\right )^{-x}}{\log \left (\log \left (2\right )\right )^{3}} + \frac {16 \, {\left (x^{4} \log \left (2\right ) \log \left (\log \left (2\right )\right )^{4} + 4 \, x^{3} \log \left (2\right ) \log \left (\log \left (2\right )\right )^{3} + 12 \, x^{2} \log \left (2\right ) \log \left (\log \left (2\right )\right )^{2} + 24 \, x \log \left (2\right ) \log \left (\log \left (2\right )\right ) + 24 \, \log \left (2\right )\right )} \log \left (2\right )^{-x}}{\log \left (\log \left (2\right )\right )^{4}} - \frac {64 \, {\left (x^{3} \log \left (2\right ) \log \left (\log \left (2\right )\right )^{3} + 3 \, x^{2} \log \left (2\right ) \log \left (\log \left (2\right )\right )^{2} + 6 \, x \log \left (2\right ) \log \left (\log \left (2\right )\right ) + 6 \, \log \left (2\right )\right )} \log \left (2\right )^{-x}}{\log \left (\log \left (2\right )\right )^{4}} \]

[In]

integrate(((-16*x^4+40*x^3-33*x^2+10*x-1)*log(log(2))+64*x^3-120*x^2+66*x-10)*exp((-x+3)*log(log(2)))/log(2)^2
,x, algorithm="maxima")

[Out]

log(2)^(-x + 1) - 10*(x*log(2)*log(log(2)) + log(2))*log(2)^(-x)/log(log(2)) + 33*(x^2*log(2)*log(log(2))^2 +
2*x*log(2)*log(log(2)) + 2*log(2))*log(2)^(-x)/log(log(2))^2 - 66*(x*log(2)*log(log(2)) + log(2))*log(2)^(-x)/
log(log(2))^2 + 10*log(2)^(-x + 1)/log(log(2)) - 40*(x^3*log(2)*log(log(2))^3 + 3*x^2*log(2)*log(log(2))^2 + 6
*x*log(2)*log(log(2)) + 6*log(2))*log(2)^(-x)/log(log(2))^3 + 120*(x^2*log(2)*log(log(2))^2 + 2*x*log(2)*log(l
og(2)) + 2*log(2))*log(2)^(-x)/log(log(2))^3 + 16*(x^4*log(2)*log(log(2))^4 + 4*x^3*log(2)*log(log(2))^3 + 12*
x^2*log(2)*log(log(2))^2 + 24*x*log(2)*log(log(2)) + 24*log(2))*log(2)^(-x)/log(log(2))^4 - 64*(x^3*log(2)*log
(log(2))^3 + 3*x^2*log(2)*log(log(2))^2 + 6*x*log(2)*log(log(2)) + 6*log(2))*log(2)^(-x)/log(log(2))^4

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 67 vs. \(2 (28) = 56\).

Time = 0.26 (sec) , antiderivative size = 67, normalized size of antiderivative = 2.79 \[ \int \log ^{1-x}(2) \left (-10+66 x-120 x^2+64 x^3+\left (-1+10 x-33 x^2+40 x^3-16 x^4\right ) \log (\log (2))\right ) \, dx=\frac {{\left (16 \, x^{4} \log \left (\log \left (2\right )\right )^{5} - 40 \, x^{3} \log \left (\log \left (2\right )\right )^{5} + 33 \, x^{2} \log \left (\log \left (2\right )\right )^{5} - 10 \, x \log \left (\log \left (2\right )\right )^{5} + \log \left (\log \left (2\right )\right )^{5}\right )} e^{\left (-x \log \left (\log \left (2\right )\right ) + 3 \, \log \left (\log \left (2\right )\right )\right )}}{\log \left (2\right )^{2} \log \left (\log \left (2\right )\right )^{5}} \]

[In]

integrate(((-16*x^4+40*x^3-33*x^2+10*x-1)*log(log(2))+64*x^3-120*x^2+66*x-10)*exp((-x+3)*log(log(2)))/log(2)^2
,x, algorithm="giac")

[Out]

(16*x^4*log(log(2))^5 - 40*x^3*log(log(2))^5 + 33*x^2*log(log(2))^5 - 10*x*log(log(2))^5 + log(log(2))^5)*e^(-
x*log(log(2)) + 3*log(log(2)))/(log(2)^2*log(log(2))^5)

Mupad [B] (verification not implemented)

Time = 0.16 (sec) , antiderivative size = 21, normalized size of antiderivative = 0.88 \[ \int \log ^{1-x}(2) \left (-10+66 x-120 x^2+64 x^3+\left (-1+10 x-33 x^2+40 x^3-16 x^4\right ) \log (\log (2))\right ) \, dx={\ln \left (2\right )}^{1-x}\,{\left (4\,x^2-5\,x+1\right )}^2 \]

[In]

int(-(exp(-log(log(2))*(x - 3))*(log(log(2))*(33*x^2 - 10*x - 40*x^3 + 16*x^4 + 1) - 66*x + 120*x^2 - 64*x^3 +
10))/log(2)^2,x)

[Out]

log(2)^(1 - x)*(4*x^2 - 5*x + 1)^2