3.13.20 \(\int \frac {x+\log (4)+2 x \log (x)+(3 x \log (4)+3 x^2 \log (x)) \log ^2(x \log (4)+x^2 \log (x))}{(x \log (4)+x^2 \log (x)) \log ^2(x \log (4)+x^2 \log (x))} \, dx\)

Optimal. Leaf size=19 \[ 2+3 x-\frac {1}{\log (x (\log (4)+x \log (x)))} \]

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Rubi [A]  time = 1.14, antiderivative size = 20, normalized size of antiderivative = 1.05, number of steps used = 5, number of rules used = 4, integrand size = 65, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.062, Rules used = {2561, 6691, 6742, 6686} \begin {gather*} 3 x-\frac {1}{\log \left (x^2 \log (x)+x \log (4)\right )} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

3*x - Log[x*Log[4] + x^2*Log[x]]^(-1)

Rule 2561

Int[(u_.)*((a_.)*(x_)^(m_.) + Log[(c_.)*(x_)^(n_.)]^(q_.)*(b_.)*(x_)^(r_.))^(p_.), x_Symbol] :> Int[u*x^(p*r)*
(a*x^(m - r) + b*Log[c*x^n]^q)^p, x] /; FreeQ[{a, b, c, m, n, p, q, r}, x] && IntegerQ[p]

Rule 6686

Int[(u_)*(y_)^(m_.), x_Symbol] :> With[{q = DerivativeDivides[y, u, x]}, Simp[(q*y^(m + 1))/(m + 1), x] /;  !F
alseQ[q]] /; FreeQ[m, x] && NeQ[m, -1]

Rule 6691

Int[(u_)^(m_.)*((a_.)*(u_)^(n_) + (v_))^(p_.)*(w_), x_Symbol] :> Int[u^(m + n*p)*(a + v/u^n)^p*w, x] /; FreeQ[
{a, m, n}, x] && IntegerQ[p] &&  !GtQ[n, 0] &&  !FreeQ[v, x]

Rule 6742

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

Rubi steps

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

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Mathematica [A]  time = 0.16, size = 18, normalized size = 0.95 \begin {gather*} 3 x-\frac {1}{\log (x (\log (4)+x \log (x)))} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

3*x - Log[x*(Log[4] + x*Log[x])]^(-1)

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fricas [A]  time = 0.82, size = 34, normalized size = 1.79 \begin {gather*} \frac {3 \, x \log \left (x^{2} \log \relax (x) + 2 \, x \log \relax (2)\right ) - 1}{\log \left (x^{2} \log \relax (x) + 2 \, x \log \relax (2)\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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giac [A]  time = 0.97, size = 21, normalized size = 1.11 \begin {gather*} 3 \, x - \frac {1}{\log \left (x \log \relax (x) + 2 \, \log \relax (2)\right ) + \log \relax (x)} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maple [C]  time = 0.08, size = 130, normalized size = 6.84

method	result	size
risch	\(3 x -\frac {2 i}{\pi \,\mathrm {csgn}\left (i x \right ) \mathrm {csgn}\left (i \left (\frac {x \ln \relax (x )}{2}+\ln \relax (2)\right )\right ) \mathrm {csgn}\left (i x \left (\frac {x \ln \relax (x )}{2}+\ln \relax (2)\right )\right )-\pi \,\mathrm {csgn}\left (i x \right ) \mathrm {csgn}\left (i x \left (\frac {x \ln \relax (x )}{2}+\ln \relax (2)\right )\right )^{2}-\pi \,\mathrm {csgn}\left (i \left (\frac {x \ln \relax (x )}{2}+\ln \relax (2)\right )\right ) \mathrm {csgn}\left (i x \left (\frac {x \ln \relax (x )}{2}+\ln \relax (2)\right )\right )^{2}+\pi \mathrm {csgn}\left (i x \left (\frac {x \ln \relax (x )}{2}+\ln \relax (2)\right )\right )^{3}+2 i \ln \relax (x )+2 i \ln \left (\frac {x \ln \relax (x )}{2}+\ln \relax (2)\right )}\)	\(130\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maxima [A]  time = 0.61, size = 36, normalized size = 1.89 \begin {gather*} \frac {3 \, x \log \left (x \log \relax (x) + 2 \, \log \relax (2)\right ) + 3 \, x \log \relax (x) - 1}{\log \left (x \log \relax (x) + 2 \, \log \relax (2)\right ) + \log \relax (x)} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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mupad [B]  time = 1.09, size = 21, normalized size = 1.11 \begin {gather*} 3\,x-\frac {1}{\ln \left (x^2\,\ln \relax (x)+2\,x\,\ln \relax (2)\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((x + 2*log(2) + log(x^2*log(x) + 2*x*log(2))^2*(3*x^2*log(x) + 6*x*log(2)) + 2*x*log(x))/(log(x^2*log(x) +
2*x*log(2))^2*(x^2*log(x) + 2*x*log(2))),x)

[Out]

3*x - 1/log(x^2*log(x) + 2*x*log(2))

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sympy [A]  time = 0.29, size = 19, normalized size = 1.00 \begin {gather*} 3 x - \frac {1}{\log {\left (x^{2} \log {\relax (x )} + 2 x \log {\relax (2 )} \right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((3*x**2*ln(x)+6*x*ln(2))*ln(x**2*ln(x)+2*x*ln(2))**2+2*x*ln(x)+x+2*ln(2))/(x**2*ln(x)+2*x*ln(2))/ln
(x**2*ln(x)+2*x*ln(2))**2,x)

[Out]

3*x - 1/log(x**2*log(x) + 2*x*log(2))

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