3.82.30 \(\int \frac {2 \log (x)+(3+x-x \log (x)) \log (x^2)+(-14 x-4 x^2) \log ^2(x^2)-3 x \log ^3(x^2)+(-\log (x^2)+4 x \log ^2(x^2)+x \log ^3(x^2)) \log (\log (x^2))}{((3 x+x^2) \log (x)+5 x \log ^2(x)) \log (x^2)+(-3 x^2-x^3-10 x^2 \log (x)) \log ^3(x^2)+5 x^3 \log ^5(x^2)+(-x \log (x) \log (x^2)+x^2 \log ^3(x^2)) \log (\log (x^2))} \, dx\)

Optimal. Leaf size=32 \[ \log \left (\frac {1}{5-\frac {3+x-\log \left (\log \left (x^2\right )\right )}{-\log (x)+x \log ^2\left (x^2\right )}}\right ) \]

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Rubi [A]  time = 1.97, antiderivative size = 40, normalized size of antiderivative = 1.25, number of steps used = 5, number of rules used = 3, integrand size = 162, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.019, Rules used = {6688, 6742, 6684} \begin {gather*} \log \left (\log (x)-x \log ^2\left (x^2\right )\right )-\log \left (-5 x \log ^2\left (x^2\right )-\log \left (\log \left (x^2\right )\right )+x+5 \log (x)+3\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(2*Log[x] + (3 + x - x*Log[x])*Log[x^2] + (-14*x - 4*x^2)*Log[x^2]^2 - 3*x*Log[x^2]^3 + (-Log[x^2] + 4*x*L
og[x^2]^2 + x*Log[x^2]^3)*Log[Log[x^2]])/(((3*x + x^2)*Log[x] + 5*x*Log[x]^2)*Log[x^2] + (-3*x^2 - x^3 - 10*x^
2*Log[x])*Log[x^2]^3 + 5*x^3*Log[x^2]^5 + (-(x*Log[x]*Log[x^2]) + x^2*Log[x^2]^3)*Log[Log[x^2]]),x]

[Out]

Log[Log[x] - x*Log[x^2]^2] - Log[3 + x + 5*Log[x] - 5*x*Log[x^2]^2 - 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 6688

Int[u_, x_Symbol] :> With[{v = SimplifyIntegrand[u, x]}, Int[v, x] /; SimplerIntegrandQ[v, u, 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 {\log (x) \left (2-x \log \left (x^2\right )\right )+\log \left (x^2\right ) \left (3+x-2 x \log \left (x^2\right ) \left (7+2 x-2 \log \left (\log \left (x^2\right )\right )\right )+x \log ^2\left (x^2\right ) \left (-3+\log \left (\log \left (x^2\right )\right )\right )-\log \left (\log \left (x^2\right )\right )\right )}{x \log \left (x^2\right ) \left (\log (x)-x \log ^2\left (x^2\right )\right ) \left (3+x+5 \log (x)-5 x \log ^2\left (x^2\right )-\log \left (\log \left (x^2\right )\right )\right )} \, dx\\ &=\int \left (\frac {-1+4 x \log \left (x^2\right )+x \log ^2\left (x^2\right )}{x \left (-\log (x)+x \log ^2\left (x^2\right )\right )}+\frac {-2+5 \log \left (x^2\right )+x \log \left (x^2\right )-20 x \log ^2\left (x^2\right )-5 x \log ^3\left (x^2\right )}{x \log \left (x^2\right ) \left (-3-x-5 \log (x)+5 x \log ^2\left (x^2\right )+\log \left (\log \left (x^2\right )\right )\right )}\right ) \, dx\\ &=\int \frac {-1+4 x \log \left (x^2\right )+x \log ^2\left (x^2\right )}{x \left (-\log (x)+x \log ^2\left (x^2\right )\right )} \, dx+\int \frac {-2+5 \log \left (x^2\right )+x \log \left (x^2\right )-20 x \log ^2\left (x^2\right )-5 x \log ^3\left (x^2\right )}{x \log \left (x^2\right ) \left (-3-x-5 \log (x)+5 x \log ^2\left (x^2\right )+\log \left (\log \left (x^2\right )\right )\right )} \, dx\\ &=\log \left (\log (x)-x \log ^2\left (x^2\right )\right )-\log \left (3+x+5 \log (x)-5 x \log ^2\left (x^2\right )-\log \left (\log \left (x^2\right )\right )\right )\\ \end {aligned} \end {gather*}

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Mathematica [F]  time = 0.62, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {2 \log (x)+(3+x-x \log (x)) \log \left (x^2\right )+\left (-14 x-4 x^2\right ) \log ^2\left (x^2\right )-3 x \log ^3\left (x^2\right )+\left (-\log \left (x^2\right )+4 x \log ^2\left (x^2\right )+x \log ^3\left (x^2\right )\right ) \log \left (\log \left (x^2\right )\right )}{\left (\left (3 x+x^2\right ) \log (x)+5 x \log ^2(x)\right ) \log \left (x^2\right )+\left (-3 x^2-x^3-10 x^2 \log (x)\right ) \log ^3\left (x^2\right )+5 x^3 \log ^5\left (x^2\right )+\left (-x \log (x) \log \left (x^2\right )+x^2 \log ^3\left (x^2\right )\right ) \log \left (\log \left (x^2\right )\right )} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Integrate[(2*Log[x] + (3 + x - x*Log[x])*Log[x^2] + (-14*x - 4*x^2)*Log[x^2]^2 - 3*x*Log[x^2]^3 + (-Log[x^2] +
4*x*Log[x^2]^2 + x*Log[x^2]^3)*Log[Log[x^2]])/(((3*x + x^2)*Log[x] + 5*x*Log[x]^2)*Log[x^2] + (-3*x^2 - x^3 -
10*x^2*Log[x])*Log[x^2]^3 + 5*x^3*Log[x^2]^5 + (-(x*Log[x]*Log[x^2]) + x^2*Log[x^2]^3)*Log[Log[x^2]]),x]

[Out]

Integrate[(2*Log[x] + (3 + x - x*Log[x])*Log[x^2] + (-14*x - 4*x^2)*Log[x^2]^2 - 3*x*Log[x^2]^3 + (-Log[x^2] +
4*x*Log[x^2]^2 + x*Log[x^2]^3)*Log[Log[x^2]])/(((3*x + x^2)*Log[x] + 5*x*Log[x]^2)*Log[x^2] + (-3*x^2 - x^3 -
10*x^2*Log[x])*Log[x^2]^3 + 5*x^3*Log[x^2]^5 + (-(x*Log[x]*Log[x^2]) + x^2*Log[x^2]^3)*Log[Log[x^2]]), x]

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fricas [A]  time = 1.29, size = 42, normalized size = 1.31 \begin {gather*} -\log \left (20 \, x \log \relax (x)^{2} - x - 5 \, \log \relax (x) + \log \left (2 \, \log \relax (x)\right ) - 3\right ) + \log \relax (x) + \log \left (\frac {4 \, x \log \relax (x) - 1}{x}\right ) + \log \left (\log \relax (x)\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((x*log(x^2)^3+4*x*log(x^2)^2-log(x^2))*log(log(x^2))-3*x*log(x^2)^3+(-4*x^2-14*x)*log(x^2)^2+(-x*lo
g(x)+3+x)*log(x^2)+2*log(x))/((x^2*log(x^2)^3-x*log(x)*log(x^2))*log(log(x^2))+5*x^3*log(x^2)^5+(-10*x^2*log(x
)-x^3-3*x^2)*log(x^2)^3+(5*x*log(x)^2+(x^2+3*x)*log(x))*log(x^2)),x, algorithm="fricas")

[Out]

-log(20*x*log(x)^2 - x - 5*log(x) + log(2*log(x)) - 3) + log(x) + log((4*x*log(x) - 1)/x) + log(log(x))

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giac [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int -\frac {3 \, x \log \left (x^{2}\right )^{3} + 2 \, {\left (2 \, x^{2} + 7 \, x\right )} \log \left (x^{2}\right )^{2} + {\left (x \log \relax (x) - x - 3\right )} \log \left (x^{2}\right ) - {\left (x \log \left (x^{2}\right )^{3} + 4 \, x \log \left (x^{2}\right )^{2} - \log \left (x^{2}\right )\right )} \log \left (\log \left (x^{2}\right )\right ) - 2 \, \log \relax (x)}{5 \, x^{3} \log \left (x^{2}\right )^{5} - {\left (x^{3} + 10 \, x^{2} \log \relax (x) + 3 \, x^{2}\right )} \log \left (x^{2}\right )^{3} + {\left (5 \, x \log \relax (x)^{2} + {\left (x^{2} + 3 \, x\right )} \log \relax (x)\right )} \log \left (x^{2}\right ) + {\left (x^{2} \log \left (x^{2}\right )^{3} - x \log \left (x^{2}\right ) \log \relax (x)\right )} \log \left (\log \left (x^{2}\right )\right )}\,{d x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((x*log(x^2)^3+4*x*log(x^2)^2-log(x^2))*log(log(x^2))-3*x*log(x^2)^3+(-4*x^2-14*x)*log(x^2)^2+(-x*lo
g(x)+3+x)*log(x^2)+2*log(x))/((x^2*log(x^2)^3-x*log(x)*log(x^2))*log(log(x^2))+5*x^3*log(x^2)^5+(-10*x^2*log(x
)-x^3-3*x^2)*log(x^2)^3+(5*x*log(x)^2+(x^2+3*x)*log(x))*log(x^2)),x, algorithm="giac")

[Out]

integrate(-(3*x*log(x^2)^3 + 2*(2*x^2 + 7*x)*log(x^2)^2 + (x*log(x) - x - 3)*log(x^2) - (x*log(x^2)^3 + 4*x*lo
g(x^2)^2 - log(x^2))*log(log(x^2)) - 2*log(x))/(5*x^3*log(x^2)^5 - (x^3 + 10*x^2*log(x) + 3*x^2)*log(x^2)^3 +
(5*x*log(x)^2 + (x^2 + 3*x)*log(x))*log(x^2) + (x^2*log(x^2)^3 - x*log(x^2)*log(x))*log(log(x^2))), x)

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maple [C]  time = 0.54, size = 740, normalized size = 23.12

method	result	size
risch	\(\ln \relax (x )+\ln \left (\ln \relax (x )^{2}-\frac {\left (2 i \pi x \mathrm {csgn}\left (i x \right )^{2} \mathrm {csgn}\left (i x^{2}\right )-4 i \pi x \,\mathrm {csgn}\left (i x \right ) \mathrm {csgn}\left (i x^{2}\right )^{2}+2 i \pi x \mathrm {csgn}\left (i x^{2}\right )^{3}+1\right ) \ln \relax (x )}{4 x}-\frac {\pi ^{2} \mathrm {csgn}\left (i x^{2}\right )^{2} \left (\mathrm {csgn}\left (i x \right )^{4}-4 \,\mathrm {csgn}\left (i x^{2}\right ) \mathrm {csgn}\left (i x \right )^{3}+6 \mathrm {csgn}\left (i x^{2}\right )^{2} \mathrm {csgn}\left (i x \right )^{2}-4 \mathrm {csgn}\left (i x^{2}\right )^{3} \mathrm {csgn}\left (i x \right )+\mathrm {csgn}\left (i x^{2}\right )^{4}\right )}{16}\right )-\ln \left (\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 )-\frac {75 x \,\pi ^{3} \mathrm {csgn}\left (i x \right )^{4} \mathrm {csgn}\left (i x^{2}\right )^{5}+4 \pi x \mathrm {csgn}\left (i x \right )^{2} \mathrm {csgn}\left (i x^{2}\right )-8 \pi x \,\mathrm {csgn}\left (i x \right ) \mathrm {csgn}\left (i x^{2}\right )^{2}+20 \ln \relax (x ) \pi \mathrm {csgn}\left (i x \right )^{2} \mathrm {csgn}\left (i x^{2}\right )-30 x \,\pi ^{3} \mathrm {csgn}\left (i x \right )^{5} \mathrm {csgn}\left (i x^{2}\right )^{4}+5 x \,\pi ^{3} \mathrm {csgn}\left (i x \right )^{6} \mathrm {csgn}\left (i x^{2}\right )^{3}-100 x \,\pi ^{3} \mathrm {csgn}\left (i x \right )^{3} \mathrm {csgn}\left (i x^{2}\right )^{6}+75 x \,\pi ^{3} \mathrm {csgn}\left (i x \right )^{2} \mathrm {csgn}\left (i x^{2}\right )^{7}-30 x \,\pi ^{3} \mathrm {csgn}\left (i x \right ) \mathrm {csgn}\left (i x^{2}\right )^{8}-240 x \ln \relax (x )^{2} \pi \mathrm {csgn}\left (i x^{2}\right )^{3}-40 \ln \relax (x ) \pi \,\mathrm {csgn}\left (i x \right ) \mathrm {csgn}\left (i x^{2}\right )^{2}+48 i \ln \relax (x )+12 \pi \mathrm {csgn}\left (i x^{2}\right )^{3}+80 i \ln \relax (x )^{2}-320 i x \ln \relax (x )^{3}+16 i x \ln \relax (x )+5 x \,\pi ^{3} \mathrm {csgn}\left (i x^{2}\right )^{9}+20 \ln \relax (x ) \pi \mathrm {csgn}\left (i x^{2}\right )^{3}+4 \pi x \mathrm {csgn}\left (i x^{2}\right )^{3}+12 \pi \mathrm {csgn}\left (i x \right )^{2} \mathrm {csgn}\left (i x^{2}\right )-24 \pi \,\mathrm {csgn}\left (i x \right ) \mathrm {csgn}\left (i x^{2}\right )^{2}+60 i \pi ^{2} x \mathrm {csgn}\left (i x^{2}\right )^{6} \ln \relax (x )-240 x \ln \relax (x )^{2} \pi \mathrm {csgn}\left (i x \right )^{2} \mathrm {csgn}\left (i x^{2}\right )+480 x \ln \relax (x )^{2} \pi \,\mathrm {csgn}\left (i x \right ) \mathrm {csgn}\left (i x^{2}\right )^{2}-240 i \pi ^{2} x \mathrm {csgn}\left (i x \right )^{3} \mathrm {csgn}\left (i x^{2}\right )^{3} \ln \relax (x )+60 i \pi ^{2} x \mathrm {csgn}\left (i x \right )^{4} \mathrm {csgn}\left (i x^{2}\right )^{2} \ln \relax (x )-240 i \pi ^{2} x \,\mathrm {csgn}\left (i x \right ) \mathrm {csgn}\left (i x^{2}\right )^{5} \ln \relax (x )+360 i \pi ^{2} x \mathrm {csgn}\left (i x \right )^{2} \mathrm {csgn}\left (i x^{2}\right )^{4} \ln \relax (x )}{4 \left (4 i \ln \relax (x )+\pi \mathrm {csgn}\left (i x^{2}\right )^{3}+\pi \mathrm {csgn}\left (i x \right )^{2} \mathrm {csgn}\left (i x^{2}\right )-2 \pi \,\mathrm {csgn}\left (i x \right ) \mathrm {csgn}\left (i x^{2}\right )^{2}\right )}\right )\)	\(740\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((x*ln(x^2)^3+4*x*ln(x^2)^2-ln(x^2))*ln(ln(x^2))-3*x*ln(x^2)^3+(-4*x^2-14*x)*ln(x^2)^2+(-x*ln(x)+3+x)*ln(x
^2)+2*ln(x))/((x^2*ln(x^2)^3-x*ln(x)*ln(x^2))*ln(ln(x^2))+5*x^3*ln(x^2)^5+(-10*x^2*ln(x)-x^3-3*x^2)*ln(x^2)^3+
(5*x*ln(x)^2+(x^2+3*x)*ln(x))*ln(x^2)),x,method=_RETURNVERBOSE)

[Out]

ln(x)+ln(ln(x)^2-1/4*(2*I*Pi*x*csgn(I*x)^2*csgn(I*x^2)-4*I*Pi*x*csgn(I*x)*csgn(I*x^2)^2+2*I*Pi*x*csgn(I*x^2)^3
+1)/x*ln(x)-1/16*Pi^2*csgn(I*x^2)^2*(csgn(I*x)^4-4*csgn(I*x^2)*csgn(I*x)^3+6*csgn(I*x^2)^2*csgn(I*x)^2-4*csgn(
I*x^2)^3*csgn(I*x)+csgn(I*x^2)^4))-ln(ln(2*ln(x)-1/2*I*Pi*csgn(I*x^2)*(-csgn(I*x^2)+csgn(I*x))^2)-1/4*(-40*ln(
x)*Pi*csgn(I*x)*csgn(I*x^2)^2-240*x*ln(x)^2*Pi*csgn(I*x^2)^3+20*ln(x)*Pi*csgn(I*x)^2*csgn(I*x^2)-30*x*Pi^3*csg
n(I*x)^5*csgn(I*x^2)^4+5*x*Pi^3*csgn(I*x)^6*csgn(I*x^2)^3-100*x*Pi^3*csgn(I*x)^3*csgn(I*x^2)^6+75*x*Pi^3*csgn(
I*x)^2*csgn(I*x^2)^7-30*x*Pi^3*csgn(I*x)*csgn(I*x^2)^8+75*x*Pi^3*csgn(I*x)^4*csgn(I*x^2)^5-320*I*x*ln(x)^3+16*
I*x*ln(x)+12*Pi*csgn(I*x^2)^3+4*Pi*x*csgn(I*x)^2*csgn(I*x^2)-8*Pi*x*csgn(I*x)*csgn(I*x^2)^2+5*x*Pi^3*csgn(I*x^
2)^9+20*ln(x)*Pi*csgn(I*x^2)^3-240*I*Pi^2*x*csgn(I*x)^3*csgn(I*x^2)^3*ln(x)+360*I*Pi^2*x*csgn(I*x)^2*csgn(I*x^
2)^4*ln(x)+60*I*Pi^2*x*csgn(I*x)^4*csgn(I*x^2)^2*ln(x)+12*Pi*csgn(I*x)^2*csgn(I*x^2)-24*Pi*csgn(I*x)*csgn(I*x^
2)^2+4*Pi*x*csgn(I*x^2)^3+60*I*Pi^2*x*csgn(I*x^2)^6*ln(x)-240*x*ln(x)^2*Pi*csgn(I*x)^2*csgn(I*x^2)+480*x*ln(x)
^2*Pi*csgn(I*x)*csgn(I*x^2)^2+80*I*ln(x)^2+48*I*ln(x)-240*I*Pi^2*x*csgn(I*x)*csgn(I*x^2)^5*ln(x))/(4*I*ln(x)+P
i*csgn(I*x^2)^3+Pi*csgn(I*x)^2*csgn(I*x^2)-2*Pi*csgn(I*x)*csgn(I*x^2)^2))

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maxima [A]  time = 0.53, size = 43, normalized size = 1.34 \begin {gather*} -\log \left (20 \, x \log \relax (x)^{2} - x + \log \relax (2) - 5 \, \log \relax (x) + \log \left (\log \relax (x)\right ) - 3\right ) + \log \relax (x) + \log \left (\frac {4 \, x \log \relax (x) - 1}{4 \, x}\right ) + \log \left (\log \relax (x)\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((x*log(x^2)^3+4*x*log(x^2)^2-log(x^2))*log(log(x^2))-3*x*log(x^2)^3+(-4*x^2-14*x)*log(x^2)^2+(-x*lo
g(x)+3+x)*log(x^2)+2*log(x))/((x^2*log(x^2)^3-x*log(x)*log(x^2))*log(log(x^2))+5*x^3*log(x^2)^5+(-10*x^2*log(x
)-x^3-3*x^2)*log(x^2)^3+(5*x*log(x)^2+(x^2+3*x)*log(x))*log(x^2)),x, algorithm="maxima")

[Out]

-log(20*x*log(x)^2 - x + log(2) - 5*log(x) + log(log(x)) - 3) + log(x) + log(1/4*(4*x*log(x) - 1)/x) + log(log
(x))

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mupad [B]  time = 5.83, size = 46, normalized size = 1.44 \begin {gather*} \ln \left (\frac {\ln \relax (x)-x\,{\ln \left (x^2\right )}^2}{x}\right )-\ln \left (\ln \left (\ln \left (x^2\right )\right )-x-5\,\ln \relax (x)+5\,x\,{\ln \left (x^2\right )}^2-3\right )+\ln \relax (x) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((2*log(x) + log(log(x^2))*(4*x*log(x^2)^2 - log(x^2) + x*log(x^2)^3) + log(x^2)*(x - x*log(x) + 3) - log(x
^2)^2*(14*x + 4*x^2) - 3*x*log(x^2)^3)/(log(log(x^2))*(x^2*log(x^2)^3 - x*log(x^2)*log(x)) + log(x^2)*(5*x*log
(x)^2 + log(x)*(3*x + x^2)) - log(x^2)^3*(10*x^2*log(x) + 3*x^2 + x^3) + 5*x^3*log(x^2)^5),x)

[Out]

log((log(x) - x*log(x^2)^2)/x) - log(log(log(x^2)) - x - 5*log(x) + 5*x*log(x^2)^2 - 3) + log(x)

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sympy [A]  time = 0.56, size = 41, normalized size = 1.28 \begin {gather*} \log {\relax (x )} + \log {\left (\log {\relax (x )}^{2} - \frac {\log {\relax (x )}}{4 x} \right )} - \log {\left (20 x \log {\relax (x )}^{2} - x - 5 \log {\relax (x )} + \log {\left (2 \log {\relax (x )} \right )} - 3 \right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((x*ln(x**2)**3+4*x*ln(x**2)**2-ln(x**2))*ln(ln(x**2))-3*x*ln(x**2)**3+(-4*x**2-14*x)*ln(x**2)**2+(-
x*ln(x)+3+x)*ln(x**2)+2*ln(x))/((x**2*ln(x**2)**3-x*ln(x)*ln(x**2))*ln(ln(x**2))+5*x**3*ln(x**2)**5+(-10*x**2*
ln(x)-x**3-3*x**2)*ln(x**2)**3+(5*x*ln(x)**2+(x**2+3*x)*ln(x))*ln(x**2)),x)

[Out]

log(x) + log(log(x)**2 - log(x)/(4*x)) - log(20*x*log(x)**2 - x - 5*log(x) + log(2*log(x)) - 3)

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