3.78.69 \(\int \frac {-2 x^3+(4 x^3-8 x \log (3)) \log (6 x)+16 x \log (3) \log ^2(6 x)}{\log ^2(3) \log ^3(6 x)} \, dx\) [7769]

Optimal. Leaf size=18 \[ \left (4+\frac {x^2}{\log (3) \log (6 x)}\right )^2 \]

[Out]

(4+x^2/ln(6*x)/ln(3))^2

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Rubi [A]
time = 0.23, antiderivative size = 30, normalized size of antiderivative = 1.67, number of steps used = 5, number of rules used = 3, integrand size = 44, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.068, Rules used = {12, 6873, 6844} \begin {gather*} \frac {x^4}{\log ^2(3) \log ^2(6 x)}+\frac {8 x^2}{\log (3) \log (6 x)} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(-2*x^3 + (4*x^3 - 8*x*Log[3])*Log[6*x] + 16*x*Log[3]*Log[6*x]^2)/(Log[3]^2*Log[6*x]^3),x]

[Out]

x^4/(Log[3]^2*Log[6*x]^2) + (8*x^2)/(Log[3]*Log[6*x])

Rule 12

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

Rule 6844

Int[(u_)*(v_)^(r_.)*((a_.)*(v_)^(p_.) + (b_.)*(w_)^(q_.))^(m_.), x_Symbol] :> With[{c = Simplify[u/(p*w*D[v, x
] - q*v*D[w, x])]}, Dist[(-c)*q, Subst[Int[(a + b*x^q)^m, x], x, v^(m*p + r + 1)*w], x] /; FreeQ[c, x]] /; Fre
eQ[{a, b, m, p, q, r}, x] && EqQ[p + q*(m*p + r + 1), 0] && IntegerQ[q] && IntegerQ[m]

Rule 6873

Int[u_, x_Symbol] :> With[{v = NormalizeIntegrand[u, x]}, Int[v, x] /; v =!= u]

Rubi steps

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

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Mathematica [A]
time = 0.05, size = 33, normalized size = 1.83 \begin {gather*} \frac {2 \left (\frac {x^4}{2 \log ^2(6 x)}+\frac {4 x^2 \log (3)}{\log (6 x)}\right )}{\log ^2(3)} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(-2*x^3 + (4*x^3 - 8*x*Log[3])*Log[6*x] + 16*x*Log[3]*Log[6*x]^2)/(Log[3]^2*Log[6*x]^3),x]

[Out]

(2*(x^4/(2*Log[6*x]^2) + (4*x^2*Log[3])/Log[6*x]))/Log[3]^2

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Maple [C] Result contains higher order function than in optimal. Order 4 vs. order 3.
time = 1.36, size = 55, normalized size = 3.06

method	result	size
risch	\(\frac {x^{2} \left (8 \ln \left (3\right ) \ln \left (6 x \right )+x^{2}\right )}{\ln \left (3\right )^{2} \ln \left (6 x \right )^{2}}\)	\(27\)
norman	\(\frac {\frac {x^{4}}{\ln \left (3\right )}+8 \ln \left (6 x \right ) x^{2}}{\ln \left (3\right ) \ln \left (6 x \right )^{2}}\)	\(30\)
default	\(\frac {-\frac {4 \ln \left (3\right ) \expIntegral \left (1, -2 \ln \left (6 x \right )\right )}{9}-\frac {2 \ln \left (3\right ) \left (-\frac {36 x^{2}}{\ln \left (6 x \right )}-2 \expIntegral \left (1, -2 \ln \left (6 x \right )\right )\right )}{9}+\frac {x^{4}}{\ln \left (6 x \right )^{2}}}{\ln \left (3\right )^{2}}\)	\(55\)
derivativedivides	\(\frac {-288 \ln \left (3\right ) \expIntegral \left (1, -2 \ln \left (6 x \right )\right )-144 \ln \left (3\right ) \left (-\frac {36 x^{2}}{\ln \left (6 x \right )}-2 \expIntegral \left (1, -2 \ln \left (6 x \right )\right )\right )+\frac {648 x^{4}}{\ln \left (6 x \right )^{2}}}{648 \ln \left (3\right )^{2}}\)	\(57\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((16*x*ln(3)*ln(6*x)^2+(-8*x*ln(3)+4*x^3)*ln(6*x)-2*x^3)/ln(3)^2/ln(6*x)^3,x,method=_RETURNVERBOSE)

[Out]

1/ln(3)^2*(-4/9*ln(3)*Ei(1,-2*ln(6*x))-2/9*ln(3)*(-36*x^2/ln(6*x)-2*Ei(1,-2*ln(6*x)))+x^4/ln(6*x)^2)

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Maxima [C] Result contains higher order function than in optimal. Order 4 vs. order 3.
time = 0.31, size = 48, normalized size = 2.67 \begin {gather*} \frac {36 \, {\rm Ei}\left (2 \, \log \left (6 \, x\right )\right ) \log \left (3\right ) - 36 \, \Gamma \left (-1, -2 \, \log \left (6 \, x\right )\right ) \log \left (3\right ) + \Gamma \left (-1, -4 \, \log \left (6 \, x\right )\right ) + 2 \, \Gamma \left (-2, -4 \, \log \left (6 \, x\right )\right )}{81 \, \log \left (3\right )^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((16*x*log(3)*log(6*x)^2+(-8*x*log(3)+4*x^3)*log(6*x)-2*x^3)/log(3)^2/log(6*x)^3,x, algorithm="maxima
")

[Out]

1/81*(36*Ei(2*log(6*x))*log(3) - 36*gamma(-1, -2*log(6*x))*log(3) + gamma(-1, -4*log(6*x)) + 2*gamma(-2, -4*lo
g(6*x)))/log(3)^2

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Fricas [A]
time = 0.51, size = 26, normalized size = 1.44 \begin {gather*} \frac {x^{4} + 8 \, x^{2} \log \left (3\right ) \log \left (6 \, x\right )}{\log \left (3\right )^{2} \log \left (6 \, x\right )^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((16*x*log(3)*log(6*x)^2+(-8*x*log(3)+4*x^3)*log(6*x)-2*x^3)/log(3)^2/log(6*x)^3,x, algorithm="fricas
")

[Out]

(x^4 + 8*x^2*log(3)*log(6*x))/(log(3)^2*log(6*x)^2)

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Sympy [A]
time = 0.04, size = 27, normalized size = 1.50 \begin {gather*} \frac {x^{4} + 8 x^{2} \log {\left (3 \right )} \log {\left (6 x \right )}}{\log {\left (3 \right )}^{2} \log {\left (6 x \right )}^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((16*x*ln(3)*ln(6*x)**2+(-8*x*ln(3)+4*x**3)*ln(6*x)-2*x**3)/ln(3)**2/ln(6*x)**3,x)

[Out]

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

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Giac [A]
time = 0.42, size = 29, normalized size = 1.61 \begin {gather*} \frac {\frac {x^{4}}{\log \left (6 \, x\right )^{2}} + \frac {8 \, x^{2} \log \left (3\right )}{\log \left (6 \, x\right )}}{\log \left (3\right )^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((16*x*log(3)*log(6*x)^2+(-8*x*log(3)+4*x^3)*log(6*x)-2*x^3)/log(3)^2/log(6*x)^3,x, algorithm="giac")

[Out]

(x^4/log(6*x)^2 + 8*x^2*log(3)/log(6*x))/log(3)^2

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Mupad [B]
time = 5.94, size = 26, normalized size = 1.44 \begin {gather*} \frac {x^2\,\left (8\,\ln \left (6\,x\right )\,\ln \left (3\right )+x^2\right )}{{\ln \left (6\,x\right )}^2\,{\ln \left (3\right )}^2} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(-(log(6*x)*(8*x*log(3) - 4*x^3) + 2*x^3 - 16*x*log(6*x)^2*log(3))/(log(6*x)^3*log(3)^2),x)

[Out]

(x^2*(8*log(6*x)*log(3) + x^2))/(log(6*x)^2*log(3)^2)

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