3.45.52 \(\int \frac {-2+(-1+3 \log (x)) \log (x^2)+\log (x^2) \log (\log (x^2))}{(2 x+3 x \log (x)) \log (x^2)+x \log (x^2) \log (\log (x^2))} \, dx\)

Optimal. Leaf size=31 \[ 3+\log \left (\frac {x}{5+e^{e^5}}\right )-\log \left (\log (x)+\frac {1}{3} \left (2+\log \left (\log \left (x^2\right )\right )\right )\right ) \]

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Rubi [A]  time = 0.35, antiderivative size = 17, normalized size of antiderivative = 0.55, number of steps used = 4, number of rules used = 3, integrand size = 52, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.058, Rules used = {6741, 6742, 6684} \begin {gather*} \log (x)-\log \left (\log \left (\log \left (x^2\right )\right )+3 \log (x)+2\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(-2 + (-1 + 3*Log[x])*Log[x^2] + Log[x^2]*Log[Log[x^2]])/((2*x + 3*x*Log[x])*Log[x^2] + x*Log[x^2]*Log[Log
[x^2]]),x]

[Out]

Log[x] - Log[2 + 3*Log[x] + Log[Log[x^2]]]

Rule 6684

Int[(u_)/(y_), x_Symbol] :> With[{q = DerivativeDivides[y, u, x]}, Simp[q*Log[RemoveContent[y, x]], x] /;  !Fa
lseQ[q]]

Rule 6741

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

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

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Mathematica [A]  time = 0.10, size = 34, normalized size = 1.10 \begin {gather*} \log (x)-\log \left (4+6 \left (\log (x)-\frac {\log \left (x^2\right )}{2}\right )+3 \log \left (x^2\right )+2 \log \left (\log \left (x^2\right )\right )\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(-2 + (-1 + 3*Log[x])*Log[x^2] + Log[x^2]*Log[Log[x^2]])/((2*x + 3*x*Log[x])*Log[x^2] + x*Log[x^2]*L
og[Log[x^2]]),x]

[Out]

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

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fricas [A]  time = 0.51, size = 17, normalized size = 0.55 \begin {gather*} \log \relax (x) - \log \left (3 \, \log \relax (x) + \log \left (2 \, \log \relax (x)\right ) + 2\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((log(x^2)*log(log(x^2))+(3*log(x)-1)*log(x^2)-2)/(x*log(x^2)*log(log(x^2))+(3*x*log(x)+2*x)*log(x^2)
),x, algorithm="fricas")

[Out]

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

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giac [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {{\left (3 \, \log \relax (x) - 1\right )} \log \left (x^{2}\right ) + \log \left (x^{2}\right ) \log \left (\log \left (x^{2}\right )\right ) - 2}{x \log \left (x^{2}\right ) \log \left (\log \left (x^{2}\right )\right ) + {\left (3 \, x \log \relax (x) + 2 \, x\right )} \log \left (x^{2}\right )}\,{d x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((log(x^2)*log(log(x^2))+(3*log(x)-1)*log(x^2)-2)/(x*log(x^2)*log(log(x^2))+(3*x*log(x)+2*x)*log(x^2)
),x, algorithm="giac")

[Out]

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

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maple [C]  time = 0.07, size = 47, normalized size = 1.52

method	result	size
risch	\(\ln \relax (x )-\ln \left (3 \ln \relax (x )+\ln \left (2 \ln \relax (x )-\frac {i \pi \,\mathrm {csgn}\left (i x^{2}\right ) \left (-\mathrm {csgn}\left (i x^{2}\right )+\mathrm {csgn}\left (i x \right )\right )^{2}}{2}\right )+2\right )\)	\(47\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((ln(x^2)*ln(ln(x^2))+(3*ln(x)-1)*ln(x^2)-2)/(x*ln(x^2)*ln(ln(x^2))+(3*x*ln(x)+2*x)*ln(x^2)),x,method=_RETU
RNVERBOSE)

[Out]

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

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maxima [A]  time = 0.48, size = 17, normalized size = 0.55 \begin {gather*} \log \relax (x) - \log \left (\log \relax (2) + 3 \, \log \relax (x) + \log \left (\log \relax (x)\right ) + 2\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((log(x^2)*log(log(x^2))+(3*log(x)-1)*log(x^2)-2)/(x*log(x^2)*log(log(x^2))+(3*x*log(x)+2*x)*log(x^2)
),x, algorithm="maxima")

[Out]

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

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mupad [B]  time = 3.57, size = 17, normalized size = 0.55 \begin {gather*} \ln \relax (x)-\ln \left (\ln \left (\ln \left (x^2\right )\right )+3\,\ln \relax (x)+2\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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sympy [A]  time = 0.31, size = 17, normalized size = 0.55 \begin {gather*} \log {\relax (x )} - \log {\left (3 \log {\relax (x )} + \log {\left (2 \log {\relax (x )} \right )} + 2 \right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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