3.42.36 \(\int (30 x \log (2)+16 x \log (2) \log (4)+2 x \log (2) \log ^2(4)) \, dx\)

Optimal. Leaf size=28 \[ -3+\log (2) \left (e^4+x \left (\frac {4}{x}-x+x (4+\log (4))^2\right )\right ) \]

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Rubi [A]  time = 0.01, antiderivative size = 14, normalized size of antiderivative = 0.50, number of steps used = 4, number of rules used = 3, integrand size = 22, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.136, Rules used = {6, 12, 30} \begin {gather*} x^2 \log (2) (3+\log (4)) (5+\log (4)) \end {gather*}

Antiderivative was successfully verified.

[In]

Int[30*x*Log[2] + 16*x*Log[2]*Log[4] + 2*x*Log[2]*Log[4]^2,x]

[Out]

x^2*Log[2]*(3 + Log[4])*(5 + Log[4])

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 12

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

Rule 30

Int[(x_)^(m_.), x_Symbol] :> Simp[x^(m + 1)/(m + 1), x] /; FreeQ[m, x] && NeQ[m, -1]

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\int \left (2 x \log (2) \log ^2(4)+x \log (2) (30+16 \log (4))\right ) \, dx\\ &=\int x \log (2) \left (30+16 \log (4)+2 \log ^2(4)\right ) \, dx\\ &=(2 \log (2) (3+\log (4)) (5+\log (4))) \int x \, dx\\ &=x^2 \log (2) (3+\log (4)) (5+\log (4))\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.00, size = 27, normalized size = 0.96 \begin {gather*} 15 x^2 \log (2)+8 x^2 \log (2) \log (4)+x^2 \log (2) \log ^2(4) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[30*x*Log[2] + 16*x*Log[2]*Log[4] + 2*x*Log[2]*Log[4]^2,x]

[Out]

15*x^2*Log[2] + 8*x^2*Log[2]*Log[4] + x^2*Log[2]*Log[4]^2

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fricas [A]  time = 0.59, size = 26, normalized size = 0.93 \begin {gather*} 4 \, x^{2} \log \relax (2)^{3} + 16 \, x^{2} \log \relax (2)^{2} + 15 \, x^{2} \log \relax (2) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(8*x*log(2)^3+32*x*log(2)^2+30*x*log(2),x, algorithm="fricas")

[Out]

4*x^2*log(2)^3 + 16*x^2*log(2)^2 + 15*x^2*log(2)

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giac [A]  time = 0.12, size = 26, normalized size = 0.93 \begin {gather*} 4 \, x^{2} \log \relax (2)^{3} + 16 \, x^{2} \log \relax (2)^{2} + 15 \, x^{2} \log \relax (2) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(8*x*log(2)^3+32*x*log(2)^2+30*x*log(2),x, algorithm="giac")

[Out]

4*x^2*log(2)^3 + 16*x^2*log(2)^2 + 15*x^2*log(2)

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maple [A]  time = 0.02, size = 19, normalized size = 0.68

method	result	size
gosper	\(x^{2} \ln \relax (2) \left (4 \ln \relax (2)^{2}+16 \ln \relax (2)+15\right )\)	\(19\)
norman	\(\left (4 \ln \relax (2)^{3}+16 \ln \relax (2)^{2}+15 \ln \relax (2)\right ) x^{2}\)	\(22\)
default	\(4 x^{2} \ln \relax (2)^{3}+16 x^{2} \ln \relax (2)^{2}+15 x^{2} \ln \relax (2)\)	\(27\)
risch	\(4 x^{2} \ln \relax (2)^{3}+16 x^{2} \ln \relax (2)^{2}+15 x^{2} \ln \relax (2)\)	\(27\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(8*x*ln(2)^3+32*x*ln(2)^2+30*x*ln(2),x,method=_RETURNVERBOSE)

[Out]

x^2*ln(2)*(4*ln(2)^2+16*ln(2)+15)

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maxima [A]  time = 0.36, size = 26, normalized size = 0.93 \begin {gather*} 4 \, x^{2} \log \relax (2)^{3} + 16 \, x^{2} \log \relax (2)^{2} + 15 \, x^{2} \log \relax (2) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(8*x*log(2)^3+32*x*log(2)^2+30*x*log(2),x, algorithm="maxima")

[Out]

4*x^2*log(2)^3 + 16*x^2*log(2)^2 + 15*x^2*log(2)

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mupad [B]  time = 0.03, size = 21, normalized size = 0.75 \begin {gather*} x^2\,\left (15\,\ln \relax (2)+16\,{\ln \relax (2)}^2+4\,{\ln \relax (2)}^3\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(30*x*log(2) + 32*x*log(2)^2 + 8*x*log(2)^3,x)

[Out]

x^2*(15*log(2) + 16*log(2)^2 + 4*log(2)^3)

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sympy [A]  time = 0.06, size = 20, normalized size = 0.71 \begin {gather*} x^{2} \left (4 \log {\relax (2 )}^{3} + 16 \log {\relax (2 )}^{2} + 15 \log {\relax (2 )}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(8*x*ln(2)**3+32*x*ln(2)**2+30*x*ln(2),x)

[Out]

x**2*(4*log(2)**3 + 16*log(2)**2 + 15*log(2))

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