3.69.5 \(\int (-36 x+12 e^3 x^2-e^6 x^3+(-12 x+2 e^3 x^2) \log (4)-x \log ^2(4)+(-72 x+36 e^3 x^2-4 e^6 x^3+(-24 x+6 e^3 x^2) \log (4)-2 x \log ^2(4)) \log (x)) \, dx\)

Optimal. Leaf size=21 \[ 5-x^2 \left (6-e^3 x+\log (4)\right )^2 \log (x) \]

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Rubi [B]  time = 0.09, antiderivative size = 105, normalized size of antiderivative = 5.00, number of steps used = 9, number of rules used = 3, integrand size = 87, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.034, Rules used = {6, 2356, 2304} \begin {gather*} -e^6 x^4 \log (x)+4 e^3 x^3+2 e^3 x^3 (6+\log (4)) \log (x)-\frac {2}{3} e^3 x^3 (6+\log (4))+\frac {2}{3} e^3 x^3 \log (4)-\frac {1}{2} x^2 \left (36+\log ^2(4)\right )-x^2 (6+\log (4))^2 \log (x)+\frac {1}{2} x^2 (6+\log (4))^2-6 x^2 \log (4) \end {gather*}

Antiderivative was successfully verified.

[In]

Int[-36*x + 12*E^3*x^2 - E^6*x^3 + (-12*x + 2*E^3*x^2)*Log[4] - x*Log[4]^2 + (-72*x + 36*E^3*x^2 - 4*E^6*x^3 +
(-24*x + 6*E^3*x^2)*Log[4] - 2*x*Log[4]^2)*Log[x],x]

[Out]

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

Rule 6

Int[(u_.)*((w_.) + (a_.)*(v_) + (b_.)*(v_))^(p_.), x_Symbol] :> Int[u*((a + b)*v + w)^p, x] /; FreeQ[{a, b}, x
] &&  !FreeQ[v, 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} &=\int \left (12 e^3 x^2-e^6 x^3+\left (-12 x+2 e^3 x^2\right ) \log (4)+x \left (-36-\log ^2(4)\right )+\left (-72 x+36 e^3 x^2-4 e^6 x^3+\left (-24 x+6 e^3 x^2\right ) \log (4)-2 x \log ^2(4)\right ) \log (x)\right ) \, dx\\ &=4 e^3 x^3-\frac {e^6 x^4}{4}-\frac {1}{2} x^2 \left (36+\log ^2(4)\right )+\log (4) \int \left (-12 x+2 e^3 x^2\right ) \, dx+\int \left (-72 x+36 e^3 x^2-4 e^6 x^3+\left (-24 x+6 e^3 x^2\right ) \log (4)-2 x \log ^2(4)\right ) \log (x) \, dx\\ &=4 e^3 x^3-\frac {e^6 x^4}{4}-6 x^2 \log (4)+\frac {2}{3} e^3 x^3 \log (4)-\frac {1}{2} x^2 \left (36+\log ^2(4)\right )+\int \left (36 e^3 x^2-4 e^6 x^3+\left (-24 x+6 e^3 x^2\right ) \log (4)+x \left (-72-2 \log ^2(4)\right )\right ) \log (x) \, dx\\ &=4 e^3 x^3-\frac {e^6 x^4}{4}-6 x^2 \log (4)+\frac {2}{3} e^3 x^3 \log (4)-\frac {1}{2} x^2 \left (36+\log ^2(4)\right )+\int \left (-4 e^6 x^3 \log (x)+6 e^3 x^2 (6+\log (4)) \log (x)-2 x (6+\log (4))^2 \log (x)\right ) \, dx\\ &=4 e^3 x^3-\frac {e^6 x^4}{4}-6 x^2 \log (4)+\frac {2}{3} e^3 x^3 \log (4)-\frac {1}{2} x^2 \left (36+\log ^2(4)\right )-\left (4 e^6\right ) \int x^3 \log (x) \, dx+\left (6 e^3 (6+\log (4))\right ) \int x^2 \log (x) \, dx-\left (2 (6+\log (4))^2\right ) \int x \log (x) \, dx\\ &=4 e^3 x^3-6 x^2 \log (4)+\frac {2}{3} e^3 x^3 \log (4)-\frac {2}{3} e^3 x^3 (6+\log (4))+\frac {1}{2} x^2 (6+\log (4))^2-\frac {1}{2} x^2 \left (36+\log ^2(4)\right )-e^6 x^4 \log (x)+2 e^3 x^3 (6+\log (4)) \log (x)-x^2 (6+\log (4))^2 \log (x)\\ \end {aligned} \end {gather*}

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Mathematica [B]  time = 0.09, size = 69, normalized size = 3.29 \begin {gather*} -36 x^2 \log (x)+12 e^3 x^3 \log (x)-e^6 x^4 \log (x)-12 x^2 \log (2) \log (x)-6 x^2 \log (4) \log (x)+2 e^3 x^3 \log (4) \log (x)-2 x^2 \log (2) \log (4) \log (x) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[-36*x + 12*E^3*x^2 - E^6*x^3 + (-12*x + 2*E^3*x^2)*Log[4] - x*Log[4]^2 + (-72*x + 36*E^3*x^2 - 4*E^6
*x^3 + (-24*x + 6*E^3*x^2)*Log[4] - 2*x*Log[4]^2)*Log[x],x]

[Out]

-36*x^2*Log[x] + 12*E^3*x^3*Log[x] - E^6*x^4*Log[x] - 12*x^2*Log[2]*Log[x] - 6*x^2*Log[4]*Log[x] + 2*E^3*x^3*L
og[4]*Log[x] - 2*x^2*Log[2]*Log[4]*Log[x]

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fricas [B]  time = 0.63, size = 48, normalized size = 2.29 \begin {gather*} -{\left (x^{4} e^{6} - 12 \, x^{3} e^{3} + 4 \, x^{2} \log \relax (2)^{2} + 36 \, x^{2} - 4 \, {\left (x^{3} e^{3} - 6 \, x^{2}\right )} \log \relax (2)\right )} \log \relax (x) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-8*x*log(2)^2+2*(6*x^2*exp(3)-24*x)*log(2)-4*x^3*exp(3)^2+36*x^2*exp(3)-72*x)*log(x)-4*x*log(2)^2+2
*(2*x^2*exp(3)-12*x)*log(2)-x^3*exp(3)^2+12*x^2*exp(3)-36*x,x, algorithm="fricas")

[Out]

-(x^4*e^6 - 12*x^3*e^3 + 4*x^2*log(2)^2 + 36*x^2 - 4*(x^3*e^3 - 6*x^2)*log(2))*log(x)

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giac [B]  time = 0.19, size = 89, normalized size = 4.24 \begin {gather*} -x^{4} e^{6} \log \relax (x) + 4 \, x^{3} e^{3} \log \relax (2) \log \relax (x) - \frac {4}{3} \, x^{3} e^{3} \log \relax (2) + 12 \, x^{3} e^{3} \log \relax (x) - 4 \, x^{2} \log \relax (2)^{2} \log \relax (x) - 24 \, x^{2} \log \relax (2) \log \relax (x) + 12 \, x^{2} \log \relax (2) - 36 \, x^{2} \log \relax (x) + \frac {4}{3} \, {\left (x^{3} e^{3} - 9 \, x^{2}\right )} \log \relax (2) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-8*x*log(2)^2+2*(6*x^2*exp(3)-24*x)*log(2)-4*x^3*exp(3)^2+36*x^2*exp(3)-72*x)*log(x)-4*x*log(2)^2+2
*(2*x^2*exp(3)-12*x)*log(2)-x^3*exp(3)^2+12*x^2*exp(3)-36*x,x, algorithm="giac")

[Out]

-x^4*e^6*log(x) + 4*x^3*e^3*log(2)*log(x) - 4/3*x^3*e^3*log(2) + 12*x^3*e^3*log(x) - 4*x^2*log(2)^2*log(x) - 2
4*x^2*log(2)*log(x) + 12*x^2*log(2) - 36*x^2*log(x) + 4/3*(x^3*e^3 - 9*x^2)*log(2)

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maple [A]  time = 0.04, size = 38, normalized size = 1.81

method	result	size
risch	\(-x^{2} \left (x^{2} {\mathrm e}^{6}-4 \,{\mathrm e}^{3} \ln \relax (2) x -12 x \,{\mathrm e}^{3}+4 \ln \relax (2)^{2}+24 \ln \relax (2)+36\right ) \ln \relax (x )\)	\(38\)
norman	\(\left (4 \,{\mathrm e}^{3} \ln \relax (2)+12 \,{\mathrm e}^{3}\right ) x^{3} \ln \relax (x )+\left (-4 \ln \relax (2)^{2}-24 \ln \relax (2)-36\right ) x^{2} \ln \relax (x )-{\mathrm e}^{6} x^{4} \ln \relax (x )\)	\(48\)
default	\(-{\mathrm e}^{6} x^{4} \ln \relax (x )+4 \,{\mathrm e}^{3} \ln \relax (2) x^{3} \ln \relax (x )+12 \,{\mathrm e}^{3} x^{3} \ln \relax (x )-4 \ln \relax (2)^{2} \ln \relax (x ) x^{2}-24 x^{2} \ln \relax (2) \ln \relax (x )-36 x^{2} \ln \relax (x )\)	\(60\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((-8*x*ln(2)^2+2*(6*x^2*exp(3)-24*x)*ln(2)-4*x^3*exp(3)^2+36*x^2*exp(3)-72*x)*ln(x)-4*x*ln(2)^2+2*(2*x^2*ex
p(3)-12*x)*ln(2)-x^3*exp(3)^2+12*x^2*exp(3)-36*x,x,method=_RETURNVERBOSE)

[Out]

-x^2*(x^2*exp(6)-4*exp(3)*ln(2)*x-12*x*exp(3)+4*ln(2)^2+24*ln(2)+36)*ln(x)

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maxima [B]  time = 0.37, size = 116, normalized size = 5.52 \begin {gather*} -\frac {4}{3} \, {\left (e^{3} \log \relax (2) + 3 \, e^{3}\right )} x^{3} + 4 \, x^{3} e^{3} - 2 \, x^{2} \log \relax (2)^{2} + 2 \, {\left (\log \relax (2)^{2} + 6 \, \log \relax (2) + 9\right )} x^{2} - 18 \, x^{2} + \frac {4}{3} \, {\left (x^{3} e^{3} - 9 \, x^{2}\right )} \log \relax (2) - {\left (x^{4} e^{6} - 12 \, x^{3} e^{3} + 4 \, x^{2} \log \relax (2)^{2} + 36 \, x^{2} - 4 \, {\left (x^{3} e^{3} - 6 \, x^{2}\right )} \log \relax (2)\right )} \log \relax (x) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-8*x*log(2)^2+2*(6*x^2*exp(3)-24*x)*log(2)-4*x^3*exp(3)^2+36*x^2*exp(3)-72*x)*log(x)-4*x*log(2)^2+2
*(2*x^2*exp(3)-12*x)*log(2)-x^3*exp(3)^2+12*x^2*exp(3)-36*x,x, algorithm="maxima")

[Out]

-4/3*(e^3*log(2) + 3*e^3)*x^3 + 4*x^3*e^3 - 2*x^2*log(2)^2 + 2*(log(2)^2 + 6*log(2) + 9)*x^2 - 18*x^2 + 4/3*(x
^3*e^3 - 9*x^2)*log(2) - (x^4*e^6 - 12*x^3*e^3 + 4*x^2*log(2)^2 + 36*x^2 - 4*(x^3*e^3 - 6*x^2)*log(2))*log(x)

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mupad [B]  time = 4.56, size = 18, normalized size = 0.86 \begin {gather*} -x^2\,\ln \relax (x)\,{\left (\ln \relax (4)-x\,{\mathrm {e}}^3+6\right )}^2 \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(12*x^2*exp(3) - 36*x - x^3*exp(6) - 2*log(2)*(12*x - 2*x^2*exp(3)) - 4*x*log(2)^2 - log(x)*(72*x - 36*x^2*
exp(3) + 4*x^3*exp(6) + 2*log(2)*(24*x - 6*x^2*exp(3)) + 8*x*log(2)^2),x)

[Out]

-x^2*log(x)*(log(4) - x*exp(3) + 6)^2

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sympy [B]  time = 0.17, size = 53, normalized size = 2.52 \begin {gather*} \left (- x^{4} e^{6} + 4 x^{3} e^{3} \log {\relax (2 )} + 12 x^{3} e^{3} - 36 x^{2} - 24 x^{2} \log {\relax (2 )} - 4 x^{2} \log {\relax (2 )}^{2}\right ) \log {\relax (x )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-8*x*ln(2)**2+2*(6*x**2*exp(3)-24*x)*ln(2)-4*x**3*exp(3)**2+36*x**2*exp(3)-72*x)*ln(x)-4*x*ln(2)**2
+2*(2*x**2*exp(3)-12*x)*ln(2)-x**3*exp(3)**2+12*x**2*exp(3)-36*x,x)

[Out]

(-x**4*exp(6) + 4*x**3*exp(3)*log(2) + 12*x**3*exp(3) - 36*x**2 - 24*x**2*log(2) - 4*x**2*log(2)**2)*log(x)

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