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

Optimal. Leaf size=24 \[ 1+\left ((1-2 x)^2-x\right )^2 \log ^{1-x}(2) \]

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Rubi [B]  time = 0.49, antiderivative size = 59, normalized size of antiderivative = 2.46, number of steps used = 29, number of rules used = 3, integrand size = 48, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.062, Rules used = {2196, 2194, 2176} \begin {gather*} 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) \end {gather*}

Antiderivative was successfully verified.

[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 2176

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] &&  !$UseGamma === True

Rule 2194

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 2196

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] &&  !$UseGamma === True

Rubi steps

\begin {gather*} \begin {aligned} \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 {aligned} \end {gather*}

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Mathematica [A]  time = 0.20, size = 21, normalized size = 0.88 \begin {gather*} \left (1-5 x+4 x^2\right )^2 \log ^{1-x}(2) \end {gather*}

Antiderivative was successfully verified.

[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)

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fricas [A]  time = 0.51, size = 33, normalized size = 1.38 \begin {gather*} \frac {{\left (16 \, x^{4} - 40 \, x^{3} + 33 \, x^{2} - 10 \, x + 1\right )} \log \relax (2)^{-x + 3}}{\log \relax (2)^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[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((3-x)*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

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giac [B]  time = 0.22, size = 67, normalized size = 2.79 \begin {gather*} \frac {{\left (16 \, x^{4} \log \left (\log \relax (2)\right )^{5} - 40 \, x^{3} \log \left (\log \relax (2)\right )^{5} + 33 \, x^{2} \log \left (\log \relax (2)\right )^{5} - 10 \, x \log \left (\log \relax (2)\right )^{5} + \log \left (\log \relax (2)\right )^{5}\right )} e^{\left (-x \log \left (\log \relax (2)\right ) + 3 \, \log \left (\log \relax (2)\right )\right )}}{\log \relax (2)^{2} \log \left (\log \relax (2)\right )^{5}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[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((3-x)*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)

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maple [A]  time = 0.14, size = 34, normalized size = 1.42

method	result	size
risch	\(\frac {\left (16 x^{4}-40 x^{3}+33 x^{2}-10 x +1\right ) \ln \relax (2)^{3-x}}{\ln \relax (2)^{2}}\)	\(34\)
gosper	\(\frac {{\mathrm e}^{-\left (x -3\right ) \ln \left (\ln \relax (2)\right )} \left (4 x -1\right ) \left (4 x^{3}-9 x^{2}+6 x -1\right )}{\ln \relax (2)^{2}}\)	\(35\)
norman	\(\frac {\frac {{\mathrm e}^{\left (3-x \right ) \ln \left (\ln \relax (2)\right )}}{\ln \relax (2)}-\frac {10 x \,{\mathrm e}^{\left (3-x \right ) \ln \left (\ln \relax (2)\right )}}{\ln \relax (2)}+\frac {33 x^{2} {\mathrm e}^{\left (3-x \right ) \ln \left (\ln \relax (2)\right )}}{\ln \relax (2)}-\frac {40 x^{3} {\mathrm e}^{\left (3-x \right ) \ln \left (\ln \relax (2)\right )}}{\ln \relax (2)}+\frac {16 x^{4} {\mathrm e}^{\left (3-x \right ) \ln \left (\ln \relax (2)\right )}}{\ln \relax (2)}}{\ln \relax (2)}\)	\(96\)
derivativedivides	\(-\frac {836 \,{\mathrm e}^{-x \ln \left (\ln \relax (2)\right )+3 \ln \left (\ln \relax (2)\right )} \left (-x \ln \left (\ln \relax (2)\right )+3 \ln \left (\ln \relax (2)\right )\right )-484 \,{\mathrm e}^{-x \ln \left (\ln \relax (2)\right )+3 \ln \left (\ln \relax (2)\right )} \ln \left (\ln \relax (2)\right )-\frac {16 \,{\mathrm e}^{-x \ln \left (\ln \relax (2)\right )+3 \ln \left (\ln \relax (2)\right )} \left (-x \ln \left (\ln \relax (2)\right )+3 \ln \left (\ln \relax (2)\right )\right )^{4}}{\ln \left (\ln \relax (2)\right )^{3}}+\frac {152 \,{\mathrm e}^{-x \ln \left (\ln \relax (2)\right )+3 \ln \left (\ln \relax (2)\right )} \left (-x \ln \left (\ln \relax (2)\right )+3 \ln \left (\ln \relax (2)\right )\right )^{3}}{\ln \left (\ln \relax (2)\right )^{2}}-\frac {537 \,{\mathrm e}^{-x \ln \left (\ln \relax (2)\right )+3 \ln \left (\ln \relax (2)\right )} \left (-x \ln \left (\ln \relax (2)\right )+3 \ln \left (\ln \relax (2)\right )\right )^{2}}{\ln \left (\ln \relax (2)\right )}}{\ln \relax (2)^{2} \ln \left (\ln \relax (2)\right )}\)	\(160\)
default	\(-\frac {836 \,{\mathrm e}^{-x \ln \left (\ln \relax (2)\right )+3 \ln \left (\ln \relax (2)\right )} \left (-x \ln \left (\ln \relax (2)\right )+3 \ln \left (\ln \relax (2)\right )\right )-484 \,{\mathrm e}^{-x \ln \left (\ln \relax (2)\right )+3 \ln \left (\ln \relax (2)\right )} \ln \left (\ln \relax (2)\right )-\frac {16 \,{\mathrm e}^{-x \ln \left (\ln \relax (2)\right )+3 \ln \left (\ln \relax (2)\right )} \left (-x \ln \left (\ln \relax (2)\right )+3 \ln \left (\ln \relax (2)\right )\right )^{4}}{\ln \left (\ln \relax (2)\right )^{3}}+\frac {152 \,{\mathrm e}^{-x \ln \left (\ln \relax (2)\right )+3 \ln \left (\ln \relax (2)\right )} \left (-x \ln \left (\ln \relax (2)\right )+3 \ln \left (\ln \relax (2)\right )\right )^{3}}{\ln \left (\ln \relax (2)\right )^{2}}-\frac {537 \,{\mathrm e}^{-x \ln \left (\ln \relax (2)\right )+3 \ln \left (\ln \relax (2)\right )} \left (-x \ln \left (\ln \relax (2)\right )+3 \ln \left (\ln \relax (2)\right )\right )^{2}}{\ln \left (\ln \relax (2)\right )}}{\ln \relax (2)^{2} \ln \left (\ln \relax (2)\right )}\)	\(160\)
meijerg	\(-\frac {16 \ln \relax (2) \left (24-\frac {\left (5 x^{4} \ln \left (\ln \relax (2)\right )^{4}+20 x^{3} \ln \left (\ln \relax (2)\right )^{3}+60 x^{2} \ln \left (\ln \relax (2)\right )^{2}+120 x \ln \left (\ln \relax (2)\right )+120\right ) {\mathrm e}^{-x \ln \left (\ln \relax (2)\right )}}{5}\right )}{\ln \left (\ln \relax (2)\right )^{4}}+\frac {\left (40 \ln \left (\ln \relax (2)\right )+64\right ) \ln \relax (2) \left (6-\frac {\left (4 x^{3} \ln \left (\ln \relax (2)\right )^{3}+12 x^{2} \ln \left (\ln \relax (2)\right )^{2}+24 x \ln \left (\ln \relax (2)\right )+24\right ) {\mathrm e}^{-x \ln \left (\ln \relax (2)\right )}}{4}\right )}{\ln \left (\ln \relax (2)\right )^{4}}+\frac {\left (-33 \ln \left (\ln \relax (2)\right )-120\right ) \ln \relax (2) \left (2-\frac {\left (3 x^{2} \ln \left (\ln \relax (2)\right )^{2}+6 x \ln \left (\ln \relax (2)\right )+6\right ) {\mathrm e}^{-x \ln \left (\ln \relax (2)\right )}}{3}\right )}{\ln \left (\ln \relax (2)\right )^{3}}+\frac {\left (10 \ln \left (\ln \relax (2)\right )+66\right ) \ln \relax (2) \left (1-\frac {\left (2+2 x \ln \left (\ln \relax (2)\right )\right ) {\mathrm e}^{-x \ln \left (\ln \relax (2)\right )}}{2}\right )}{\ln \left (\ln \relax (2)\right )^{2}}-\ln \relax (2) \left (1-{\mathrm e}^{-x \ln \left (\ln \relax (2)\right )}\right )-\frac {10 \ln \relax (2) \left (1-{\mathrm e}^{-x \ln \left (\ln \relax (2)\right )}\right )}{\ln \left (\ln \relax (2)\right )}\)	\(227\)

Verification of antiderivative is not currently implemented for this CAS.

[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((3-x)*ln(ln(2)))/ln(2)^2,x,method=_R
ETURNVERBOSE)

[Out]

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

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maxima [B]  time = 0.69, size = 303, normalized size = 12.62 \begin {gather*} \log \relax (2)^{-x + 1} - \frac {10 \, {\left (x \log \relax (2) \log \left (\log \relax (2)\right ) + \log \relax (2)\right )} \log \relax (2)^{-x}}{\log \left (\log \relax (2)\right )} + \frac {33 \, {\left (x^{2} \log \relax (2) \log \left (\log \relax (2)\right )^{2} + 2 \, x \log \relax (2) \log \left (\log \relax (2)\right ) + 2 \, \log \relax (2)\right )} \log \relax (2)^{-x}}{\log \left (\log \relax (2)\right )^{2}} - \frac {66 \, {\left (x \log \relax (2) \log \left (\log \relax (2)\right ) + \log \relax (2)\right )} \log \relax (2)^{-x}}{\log \left (\log \relax (2)\right )^{2}} + \frac {10 \, \log \relax (2)^{-x + 1}}{\log \left (\log \relax (2)\right )} - \frac {40 \, {\left (x^{3} \log \relax (2) \log \left (\log \relax (2)\right )^{3} + 3 \, x^{2} \log \relax (2) \log \left (\log \relax (2)\right )^{2} + 6 \, x \log \relax (2) \log \left (\log \relax (2)\right ) + 6 \, \log \relax (2)\right )} \log \relax (2)^{-x}}{\log \left (\log \relax (2)\right )^{3}} + \frac {120 \, {\left (x^{2} \log \relax (2) \log \left (\log \relax (2)\right )^{2} + 2 \, x \log \relax (2) \log \left (\log \relax (2)\right ) + 2 \, \log \relax (2)\right )} \log \relax (2)^{-x}}{\log \left (\log \relax (2)\right )^{3}} + \frac {16 \, {\left (x^{4} \log \relax (2) \log \left (\log \relax (2)\right )^{4} + 4 \, x^{3} \log \relax (2) \log \left (\log \relax (2)\right )^{3} + 12 \, x^{2} \log \relax (2) \log \left (\log \relax (2)\right )^{2} + 24 \, x \log \relax (2) \log \left (\log \relax (2)\right ) + 24 \, \log \relax (2)\right )} \log \relax (2)^{-x}}{\log \left (\log \relax (2)\right )^{4}} - \frac {64 \, {\left (x^{3} \log \relax (2) \log \left (\log \relax (2)\right )^{3} + 3 \, x^{2} \log \relax (2) \log \left (\log \relax (2)\right )^{2} + 6 \, x \log \relax (2) \log \left (\log \relax (2)\right ) + 6 \, \log \relax (2)\right )} \log \relax (2)^{-x}}{\log \left (\log \relax (2)\right )^{4}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[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((3-x)*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

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mupad [B]  time = 0.11, size = 21, normalized size = 0.88 \begin {gather*} {\ln \relax (2)}^{1-x}\,{\left (4\,x^2-5\,x+1\right )}^2 \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[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

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sympy [A]  time = 0.17, size = 34, normalized size = 1.42 \begin {gather*} \frac {\left (16 x^{4} - 40 x^{3} + 33 x^{2} - 10 x + 1\right ) e^{\left (3 - x\right ) \log {\left (\log {\relax (2 )} \right )}}}{\log {\relax (2 )}^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[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((3-x)*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

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