3.21.49 \(\int \frac {1}{8} (1728 x^5-96 x^3 \log (\log (9))+x \log ^2(\log (9))) \, dx\)

Optimal. Leaf size=24 \[ -e^4+\frac {1}{16} x^2 \left (-24 x^2+\log (\log (9))\right )^2 \]

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Rubi [A]  time = 0.01, antiderivative size = 26, normalized size of antiderivative = 1.08, number of steps used = 2, number of rules used = 1, integrand size = 25, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.040, Rules used = {12} \begin {gather*} 36 x^6-3 x^4 \log (\log (9))+\frac {1}{16} x^2 \log ^2(\log (9)) \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(1728*x^5 - 96*x^3*Log[Log[9]] + x*Log[Log[9]]^2)/8,x]

[Out]

36*x^6 - 3*x^4*Log[Log[9]] + (x^2*Log[Log[9]]^2)/16

Rule 12

Int[(a_)*(u_), x_Symbol] :> Dist[a, Int[u, x], x] /; FreeQ[a, x] &&  !MatchQ[u, (b_)*(v_) /; FreeQ[b, x]]

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\frac {1}{8} \int \left (1728 x^5-96 x^3 \log (\log (9))+x \log ^2(\log (9))\right ) \, dx\\ &=36 x^6-3 x^4 \log (\log (9))+\frac {1}{16} x^2 \log ^2(\log (9))\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.00, size = 26, normalized size = 1.08 \begin {gather*} 36 x^6-3 x^4 \log (\log (9))+\frac {1}{16} x^2 \log ^2(\log (9)) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(1728*x^5 - 96*x^3*Log[Log[9]] + x*Log[Log[9]]^2)/8,x]

[Out]

36*x^6 - 3*x^4*Log[Log[9]] + (x^2*Log[Log[9]]^2)/16

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fricas [A]  time = 1.13, size = 28, normalized size = 1.17 \begin {gather*} 36 \, x^{6} - 3 \, x^{4} \log \left (2 \, \log \relax (3)\right ) + \frac {1}{16} \, x^{2} \log \left (2 \, \log \relax (3)\right )^{2} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/8*x*log(2*log(3))^2-12*x^3*log(2*log(3))+216*x^5,x, algorithm="fricas")

[Out]

36*x^6 - 3*x^4*log(2*log(3)) + 1/16*x^2*log(2*log(3))^2

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giac [A]  time = 0.25, size = 28, normalized size = 1.17 \begin {gather*} 36 \, x^{6} - 3 \, x^{4} \log \left (2 \, \log \relax (3)\right ) + \frac {1}{16} \, x^{2} \log \left (2 \, \log \relax (3)\right )^{2} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/8*x*log(2*log(3))^2-12*x^3*log(2*log(3))+216*x^5,x, algorithm="giac")

[Out]

36*x^6 - 3*x^4*log(2*log(3)) + 1/16*x^2*log(2*log(3))^2

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maple [A]  time = 0.03, size = 19, normalized size = 0.79

method	result	size
gosper	\(\frac {\left (\ln \left (2 \ln \relax (3)\right )-24 x^{2}\right )^{2} x^{2}}{16}\)	\(19\)
default	\(\frac {\ln \left (2 \ln \relax (3)\right )^{2} x^{2}}{16}-3 \ln \left (2 \ln \relax (3)\right ) x^{4}+36 x^{6}\)	\(29\)
norman	\(\left (-3 \ln \relax (2)-3 \ln \left (\ln \relax (3)\right )\right ) x^{4}+\left (\frac {\ln \relax (2)^{2}}{16}+\frac {\ln \relax (2) \ln \left (\ln \relax (3)\right )}{8}+\frac {\ln \left (\ln \relax (3)\right )^{2}}{16}\right ) x^{2}+36 x^{6}\)	\(46\)
risch	\(\frac {x^{2} \ln \relax (2)^{2}}{16}+\frac {\ln \relax (2) \ln \left (\ln \relax (3)\right ) x^{2}}{8}+\frac {\ln \left (\ln \relax (3)\right )^{2} x^{2}}{16}-3 x^{4} \ln \relax (2)-3 \ln \left (\ln \relax (3)\right ) x^{4}+36 x^{6}\)	\(51\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/8*x*ln(2*ln(3))^2-12*x^3*ln(2*ln(3))+216*x^5,x,method=_RETURNVERBOSE)

[Out]

1/16*(ln(2*ln(3))-24*x^2)^2*x^2

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maxima [A]  time = 0.39, size = 28, normalized size = 1.17 \begin {gather*} 36 \, x^{6} - 3 \, x^{4} \log \left (2 \, \log \relax (3)\right ) + \frac {1}{16} \, x^{2} \log \left (2 \, \log \relax (3)\right )^{2} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/8*x*log(2*log(3))^2-12*x^3*log(2*log(3))+216*x^5,x, algorithm="maxima")

[Out]

36*x^6 - 3*x^4*log(2*log(3)) + 1/16*x^2*log(2*log(3))^2

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mupad [B]  time = 0.04, size = 18, normalized size = 0.75 \begin {gather*} \frac {x^2\,{\left (\ln \left (2\,\ln \relax (3)\right )-24\,x^2\right )}^2}{16} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((x*log(2*log(3))^2)/8 - 12*x^3*log(2*log(3)) + 216*x^5,x)

[Out]

(x^2*(log(2*log(3)) - 24*x^2)^2)/16

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sympy [B]  time = 0.07, size = 49, normalized size = 2.04 \begin {gather*} 36 x^{6} + x^{4} \left (- 3 \log {\relax (2 )} - 3 \log {\left (\log {\relax (3 )} \right )}\right ) + x^{2} \left (\frac {\log {\left (\log {\relax (3 )} \right )}^{2}}{16} + \frac {\log {\relax (2 )} \log {\left (\log {\relax (3 )} \right )}}{8} + \frac {\log {\relax (2 )}^{2}}{16}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/8*x*ln(2*ln(3))**2-12*x**3*ln(2*ln(3))+216*x**5,x)

[Out]

36*x**6 + x**4*(-3*log(2) - 3*log(log(3))) + x**2*(log(log(3))**2/16 + log(2)*log(log(3))/8 + log(2)**2/16)

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