3.63.53 \(\int \frac {1}{3} (-12 x+6 x^2+(-4 x+2 x^2) (i \pi +\log (\log (3)))+\log (x^2) (-12 x+9 x^2+(-4 x+3 x^2) (i \pi +\log (\log (3))))) \, dx\)

Optimal. Leaf size=24 \[ \frac {1}{3} (-2+x) x^2 \log \left (x^2\right ) (3+i \pi +\log (\log (3))) \]

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Rubi [B]  time = 0.09, antiderivative size = 121, normalized size of antiderivative = 5.04, number of steps used = 7, number of rules used = 3, integrand size = 65, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.046, Rules used = {12, 2356, 2304} \begin {gather*} \frac {2 x^3}{3}-\frac {2}{9} x^3 (3+i \pi +\log (\log (3)))+\frac {2}{9} x^3 (\log (\log (3))+i \pi )-2 x^2-\frac {2}{3} x^2 (3+i \pi +\log (\log (3))) \log \left (x^2\right )+\frac {2}{3} x^2 (3+i \pi +\log (\log (3)))-\frac {2}{3} x^2 (\log (\log (3))+i \pi )+\frac {1}{3} x^3 (3+i \pi +\log (\log (3))) \log \left (x^2\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(-12*x + 6*x^2 + (-4*x + 2*x^2)*(I*Pi + Log[Log[3]]) + Log[x^2]*(-12*x + 9*x^2 + (-4*x + 3*x^2)*(I*Pi + Lo
g[Log[3]])))/3,x]

[Out]

-2*x^2 + (2*x^3)/3 - (2*x^2*(I*Pi + Log[Log[3]]))/3 + (2*x^3*(I*Pi + Log[Log[3]]))/9 + (2*x^2*(3 + I*Pi + Log[
Log[3]]))/3 - (2*x^3*(3 + I*Pi + Log[Log[3]]))/9 - (2*x^2*Log[x^2]*(3 + I*Pi + Log[Log[3]]))/3 + (x^3*Log[x^2]
*(3 + I*Pi + Log[Log[3]]))/3

Rule 12

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

Rule 2304

Int[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))*((d_.)*(x_))^(m_.), x_Symbol] :> Simp[((d*x)^(m + 1)*(a + b*Log[c*x^
n]))/(d*(m + 1)), x] - Simp[(b*n*(d*x)^(m + 1))/(d*(m + 1)^2), x] /; FreeQ[{a, b, c, d, m, n}, x] && NeQ[m, -1
]

Rule 2356

Int[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))^(p_.)*(Polyx_), x_Symbol] :> Int[ExpandIntegrand[Polyx*(a + b*Log[c*
x^n])^p, x], x] /; FreeQ[{a, b, c, n, p}, x] && PolynomialQ[Polyx, x]

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\frac {1}{3} \int \left (-12 x+6 x^2+\left (-4 x+2 x^2\right ) (i \pi +\log (\log (3)))+\log \left (x^2\right ) \left (-12 x+9 x^2+\left (-4 x+3 x^2\right ) (i \pi +\log (\log (3)))\right )\right ) \, dx\\ &=-2 x^2+\frac {2 x^3}{3}+\frac {1}{3} \int \log \left (x^2\right ) \left (-12 x+9 x^2+\left (-4 x+3 x^2\right ) (i \pi +\log (\log (3)))\right ) \, dx+\frac {1}{3} (i \pi +\log (\log (3))) \int \left (-4 x+2 x^2\right ) \, dx\\ &=-2 x^2+\frac {2 x^3}{3}-\frac {2}{3} x^2 (i \pi +\log (\log (3)))+\frac {2}{9} x^3 (i \pi +\log (\log (3)))+\frac {1}{3} \int \left (3 x^2 \log \left (x^2\right ) (3+i \pi +\log (\log (3)))-4 i x \log \left (x^2\right ) (\pi -i (3+\log (\log (3))))\right ) \, dx\\ &=-2 x^2+\frac {2 x^3}{3}-\frac {2}{3} x^2 (i \pi +\log (\log (3)))+\frac {2}{9} x^3 (i \pi +\log (\log (3)))+(3+i \pi +\log (\log (3))) \int x^2 \log \left (x^2\right ) \, dx-\frac {1}{3} (4 (3+i \pi +\log (\log (3)))) \int x \log \left (x^2\right ) \, dx\\ &=-2 x^2+\frac {2 x^3}{3}-\frac {2}{3} x^2 (i \pi +\log (\log (3)))+\frac {2}{9} x^3 (i \pi +\log (\log (3)))+\frac {2}{3} x^2 (3+i \pi +\log (\log (3)))-\frac {2}{9} x^3 (3+i \pi +\log (\log (3)))-\frac {2}{3} x^2 \log \left (x^2\right ) (3+i \pi +\log (\log (3)))+\frac {1}{3} x^3 \log \left (x^2\right ) (3+i \pi +\log (\log (3)))\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.01, size = 32, normalized size = 1.33 \begin {gather*} \frac {1}{3} \left (-2 x^2 \log \left (x^2\right )+x^3 \log \left (x^2\right )\right ) (3+i \pi +\log (\log (3))) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(-12*x + 6*x^2 + (-4*x + 2*x^2)*(I*Pi + Log[Log[3]]) + Log[x^2]*(-12*x + 9*x^2 + (-4*x + 3*x^2)*(I*P
i + Log[Log[3]])))/3,x]

[Out]

((-2*x^2*Log[x^2] + x^3*Log[x^2])*(3 + I*Pi + Log[Log[3]]))/3

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fricas [A]  time = 0.71, size = 32, normalized size = 1.33 \begin {gather*} \frac {1}{3} \, {\left (3 \, x^{3} - 6 \, x^{2} + {\left (x^{3} - 2 \, x^{2}\right )} \log \left (-\log \relax (3)\right )\right )} \log \left (x^{2}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/3*((3*x^2-4*x)*log(-log(3))+9*x^2-12*x)*log(x^2)+1/3*(2*x^2-4*x)*log(-log(3))+2*x^2-4*x,x, algorit
hm="fricas")

[Out]

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

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giac [B]  time = 0.36, size = 69, normalized size = 2.88 \begin {gather*} -\frac {2}{9} \, x^{3} \log \left (-\log \relax (3)\right ) + \frac {2}{3} \, x^{2} \log \left (-\log \relax (3)\right ) + \frac {1}{3} \, {\left (3 \, x^{3} - 6 \, x^{2} + {\left (x^{3} - 2 \, x^{2}\right )} \log \left (-\log \relax (3)\right )\right )} \log \left (x^{2}\right ) + \frac {2}{9} \, {\left (x^{3} - 3 \, x^{2}\right )} \log \left (-\log \relax (3)\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/3*((3*x^2-4*x)*log(-log(3))+9*x^2-12*x)*log(x^2)+1/3*(2*x^2-4*x)*log(-log(3))+2*x^2-4*x,x, algorit
hm="giac")

[Out]

-2/9*x^3*log(-log(3)) + 2/3*x^2*log(-log(3)) + 1/3*(3*x^3 - 6*x^2 + (x^3 - 2*x^2)*log(-log(3)))*log(x^2) + 2/9
*(x^3 - 3*x^2)*log(-log(3))

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maple [C]  time = 0.08, size = 40, normalized size = 1.67

method	result	size
norman	\(\left (-\frac {2 i \pi }{3}-\frac {2 \ln \left (\ln \relax (3)\right )}{3}-2\right ) x^{2} \ln \left (x^{2}\right )+\left (\frac {i \pi }{3}+\frac {\ln \left (\ln \relax (3)\right )}{3}+1\right ) x^{3} \ln \left (x^{2}\right )\)	\(40\)
risch	\(\frac {i \pi \,x^{3} \ln \left (x^{2}\right )}{3}-\frac {2 i x^{2} \ln \left (x^{2}\right ) \pi }{3}+\frac {\ln \left (\ln \relax (3)\right ) x^{3} \ln \left (x^{2}\right )}{3}-\frac {2 x^{2} \ln \left (x^{2}\right ) \ln \left (\ln \relax (3)\right )}{3}+x^{3} \ln \left (x^{2}\right )-2 x^{2} \ln \left (x^{2}\right )\)	\(65\)
default	\(-\frac {2 i x^{2} \ln \left (x^{2}\right ) \pi }{3}+\frac {2 i \pi \,x^{2}}{3}+\frac {i \pi \,x^{3} \ln \left (x^{2}\right )}{3}-\frac {2 i \pi \,x^{3}}{9}-2 x^{2} \ln \left (x^{2}\right )+x^{3} \ln \left (x^{2}\right )-\frac {2 x^{2} \ln \left (x^{2}\right ) \ln \left (\ln \relax (3)\right )}{3}+\frac {2 x^{2} \ln \left (\ln \relax (3)\right )}{3}+\frac {\ln \left (\ln \relax (3)\right ) x^{3} \ln \left (x^{2}\right )}{3}-\frac {2 x^{3} \ln \left (\ln \relax (3)\right )}{9}+\frac {2 \ln \left (-\ln \relax (3)\right ) x^{3}}{9}-\frac {2 \ln \left (-\ln \relax (3)\right ) x^{2}}{3}\)	\(115\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/3*((3*x^2-4*x)*ln(-ln(3))+9*x^2-12*x)*ln(x^2)+1/3*(2*x^2-4*x)*ln(-ln(3))+2*x^2-4*x,x,method=_RETURNVERBO
SE)

[Out]

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

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maxima [B]  time = 0.50, size = 83, normalized size = 3.46 \begin {gather*} -\frac {2}{9} \, x^{3} {\left (\log \left (-\log \relax (3)\right ) + 3\right )} + \frac {2}{3} \, x^{3} + \frac {2}{3} \, x^{2} {\left (\log \left (-\log \relax (3)\right ) + 3\right )} - 2 \, x^{2} + \frac {1}{3} \, {\left (3 \, x^{3} - 6 \, x^{2} + {\left (x^{3} - 2 \, x^{2}\right )} \log \left (-\log \relax (3)\right )\right )} \log \left (x^{2}\right ) + \frac {2}{9} \, {\left (x^{3} - 3 \, x^{2}\right )} \log \left (-\log \relax (3)\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/3*((3*x^2-4*x)*log(-log(3))+9*x^2-12*x)*log(x^2)+1/3*(2*x^2-4*x)*log(-log(3))+2*x^2-4*x,x, algorit
hm="maxima")

[Out]

-2/9*x^3*(log(-log(3)) + 3) + 2/3*x^3 + 2/3*x^2*(log(-log(3)) + 3) - 2*x^2 + 1/3*(3*x^3 - 6*x^2 + (x^3 - 2*x^2
)*log(-log(3)))*log(x^2) + 2/9*(x^3 - 3*x^2)*log(-log(3))

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mupad [B]  time = 4.13, size = 21, normalized size = 0.88 \begin {gather*} \frac {x^2\,\ln \left (x^2\right )\,\left (x-2\right )\,\left (\ln \left (\ln \relax (3)\right )+3+\pi \,1{}\mathrm {i}\right )}{3} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(2*x^2 - (log(-log(3))*(4*x - 2*x^2))/3 - (log(x^2)*(12*x + log(-log(3))*(4*x - 3*x^2) - 9*x^2))/3 - 4*x,x)

[Out]

(x^2*log(x^2)*(x - 2)*(pi*1i + log(log(3)) + 3))/3

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sympy [B]  time = 0.24, size = 53, normalized size = 2.21 \begin {gather*} \left (\frac {x^{3} \log {\left (\log {\relax (3 )} \right )}}{3} + x^{3} + \frac {i \pi x^{3}}{3} - 2 x^{2} - \frac {2 x^{2} \log {\left (\log {\relax (3 )} \right )}}{3} - \frac {2 i \pi x^{2}}{3}\right ) \log {\left (x^{2} \right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/3*((3*x**2-4*x)*ln(-ln(3))+9*x**2-12*x)*ln(x**2)+1/3*(2*x**2-4*x)*ln(-ln(3))+2*x**2-4*x,x)

[Out]

(x**3*log(log(3))/3 + x**3 + I*pi*x**3/3 - 2*x**2 - 2*x**2*log(log(3))/3 - 2*I*pi*x**2/3)*log(x**2)

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