3.16.67 \(\int \frac {780-24 x-36 x^2+(442-24 x-18 x^2) \log (x)+(900+360 x+36 x^2+(120+360 x) \log (x)) \log (x \log (x))+(-450+18 x^2) \log (x) \log ^2(x \log (x))}{9 x^2 \log (x)} \, dx\)

Optimal. Leaf size=23 \[ \frac {2 \left (\frac {13}{3}-x+(5+x) \log (x \log (x))\right )^2}{x} \]

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Rubi [F]  time = 1.51, antiderivative size = 0, normalized size of antiderivative = 0.00, number of steps used = 0, number of rules used = 0, integrand size = 0, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.000, Rules used = {} \begin {gather*} \int \frac {780-24 x-36 x^2+\left (442-24 x-18 x^2\right ) \log (x)+\left (900+360 x+36 x^2+(120+360 x) \log (x)\right ) \log (x \log (x))+\left (-450+18 x^2\right ) \log (x) \log ^2(x \log (x))}{9 x^2 \log (x)} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Int[(780 - 24*x - 36*x^2 + (442 - 24*x - 18*x^2)*Log[x] + (900 + 360*x + 36*x^2 + (120 + 360*x)*Log[x])*Log[x*
Log[x]] + (-450 + 18*x^2)*Log[x]*Log[x*Log[x]]^2)/(9*x^2*Log[x]),x]

[Out]

-562/(9*x) + 2*x - (8*Log[x])/3 - (40*ExpIntegralEi[-Log[x]]*Log[x])/3 + (40*ExpIntegralEi[-Log[x]]*(1 + Log[x
]))/3 - (40*Log[x*Log[x]])/(3*x) - 4*Log[x]*LogIntegral[x] + 4*Log[x*Log[x]]*LogIntegral[x] - (4*Defer[Int][((
5 + x)*(-13 + 3*x))/(x^2*Log[x]), x])/3 + 40*Defer[Int][Log[x*Log[x]]/x, x] + 100*Defer[Int][Log[x*Log[x]]/(x^
2*Log[x]), x] + 40*Defer[Int][Log[x*Log[x]]/(x*Log[x]), x] + 2*Defer[Int][Log[x*Log[x]]^2, x] - 50*Defer[Int][
Log[x*Log[x]]^2/x^2, x] - 4*Defer[Int][LogIntegral[x]/(x*Log[x]), x]

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\frac {1}{9} \int \frac {780-24 x-36 x^2+\left (442-24 x-18 x^2\right ) \log (x)+\left (900+360 x+36 x^2+(120+360 x) \log (x)\right ) \log (x \log (x))+\left (-450+18 x^2\right ) \log (x) \log ^2(x \log (x))}{x^2 \log (x)} \, dx\\ &=\frac {1}{9} \int \frac {2 (13-3 x+3 (5+x) \log (x \log (x))) (6 (5+x)+\log (x) (17+3 x+3 (-5+x) \log (x \log (x))))}{x^2 \log (x)} \, dx\\ &=\frac {2}{9} \int \frac {(13-3 x+3 (5+x) \log (x \log (x))) (6 (5+x)+\log (x) (17+3 x+3 (-5+x) \log (x \log (x))))}{x^2 \log (x)} \, dx\\ &=\frac {2}{9} \int \left (-\frac {(-13+3 x) (30+6 x+17 \log (x)+3 x \log (x))}{x^2 \log (x)}+\frac {6 \left (75+30 x+3 x^2+10 \log (x)+30 x \log (x)\right ) \log (x \log (x))}{x^2 \log (x)}+\frac {9 (-5+x) (5+x) \log ^2(x \log (x))}{x^2}\right ) \, dx\\ &=-\left (\frac {2}{9} \int \frac {(-13+3 x) (30+6 x+17 \log (x)+3 x \log (x))}{x^2 \log (x)} \, dx\right )+\frac {4}{3} \int \frac {\left (75+30 x+3 x^2+10 \log (x)+30 x \log (x)\right ) \log (x \log (x))}{x^2 \log (x)} \, dx+2 \int \frac {(-5+x) (5+x) \log ^2(x \log (x))}{x^2} \, dx\\ &=-\left (\frac {2}{9} \int \left (\frac {(-13+3 x) (17+3 x)}{x^2}+\frac {6 (5+x) (-13+3 x)}{x^2 \log (x)}\right ) \, dx\right )+\frac {4}{3} \int \left (\frac {10 \log (x \log (x))}{x^2}+\frac {30 \log (x \log (x))}{x}+\frac {3 \log (x \log (x))}{\log (x)}+\frac {75 \log (x \log (x))}{x^2 \log (x)}+\frac {30 \log (x \log (x))}{x \log (x)}\right ) \, dx+2 \int \left (\log ^2(x \log (x))-\frac {25 \log ^2(x \log (x))}{x^2}\right ) \, dx\\ &=-\left (\frac {2}{9} \int \frac {(-13+3 x) (17+3 x)}{x^2} \, dx\right )-\frac {4}{3} \int \frac {(5+x) (-13+3 x)}{x^2 \log (x)} \, dx+2 \int \log ^2(x \log (x)) \, dx+4 \int \frac {\log (x \log (x))}{\log (x)} \, dx+\frac {40}{3} \int \frac {\log (x \log (x))}{x^2} \, dx+40 \int \frac {\log (x \log (x))}{x} \, dx+40 \int \frac {\log (x \log (x))}{x \log (x)} \, dx-50 \int \frac {\log ^2(x \log (x))}{x^2} \, dx+100 \int \frac {\log (x \log (x))}{x^2 \log (x)} \, dx\\ &=-\frac {40 \log (x \log (x))}{3 x}+4 \log (x \log (x)) \text {li}(x)-\frac {2}{9} \int \left (9-\frac {221}{x^2}+\frac {12}{x}\right ) \, dx-\frac {4}{3} \int \frac {(5+x) (-13+3 x)}{x^2 \log (x)} \, dx+2 \int \log ^2(x \log (x)) \, dx-4 \int \frac {(1+\log (x)) \text {li}(x)}{x \log (x)} \, dx+\frac {40}{3} \int \frac {1+\log (x)}{x^2 \log (x)} \, dx+40 \int \frac {\log (x \log (x))}{x} \, dx+40 \int \frac {\log (x \log (x))}{x \log (x)} \, dx-50 \int \frac {\log ^2(x \log (x))}{x^2} \, dx+100 \int \frac {\log (x \log (x))}{x^2 \log (x)} \, dx\\ &=-\frac {442}{9 x}-2 x-\frac {8 \log (x)}{3}+\frac {40}{3} \text {Ei}(-\log (x)) (1+\log (x))-\frac {40 \log (x \log (x))}{3 x}+4 \log (x \log (x)) \text {li}(x)-\frac {4}{3} \int \frac {(5+x) (-13+3 x)}{x^2 \log (x)} \, dx+2 \int \log ^2(x \log (x)) \, dx-4 \int \left (\frac {\text {li}(x)}{x}+\frac {\text {li}(x)}{x \log (x)}\right ) \, dx-\frac {40}{3} \int \frac {\text {Ei}(-\log (x))}{x} \, dx+40 \int \frac {\log (x \log (x))}{x} \, dx+40 \int \frac {\log (x \log (x))}{x \log (x)} \, dx-50 \int \frac {\log ^2(x \log (x))}{x^2} \, dx+100 \int \frac {\log (x \log (x))}{x^2 \log (x)} \, dx\\ &=-\frac {442}{9 x}-2 x-\frac {8 \log (x)}{3}+\frac {40}{3} \text {Ei}(-\log (x)) (1+\log (x))-\frac {40 \log (x \log (x))}{3 x}+4 \log (x \log (x)) \text {li}(x)-\frac {4}{3} \int \frac {(5+x) (-13+3 x)}{x^2 \log (x)} \, dx+2 \int \log ^2(x \log (x)) \, dx-4 \int \frac {\text {li}(x)}{x} \, dx-4 \int \frac {\text {li}(x)}{x \log (x)} \, dx-\frac {40}{3} \operatorname {Subst}(\int \text {Ei}(-x) \, dx,x,\log (x))+40 \int \frac {\log (x \log (x))}{x} \, dx+40 \int \frac {\log (x \log (x))}{x \log (x)} \, dx-50 \int \frac {\log ^2(x \log (x))}{x^2} \, dx+100 \int \frac {\log (x \log (x))}{x^2 \log (x)} \, dx\\ &=-\frac {562}{9 x}+2 x-\frac {8 \log (x)}{3}-\frac {40}{3} \text {Ei}(-\log (x)) \log (x)+\frac {40}{3} \text {Ei}(-\log (x)) (1+\log (x))-\frac {40 \log (x \log (x))}{3 x}-4 \log (x) \text {li}(x)+4 \log (x \log (x)) \text {li}(x)-\frac {4}{3} \int \frac {(5+x) (-13+3 x)}{x^2 \log (x)} \, dx+2 \int \log ^2(x \log (x)) \, dx-4 \int \frac {\text {li}(x)}{x \log (x)} \, dx+40 \int \frac {\log (x \log (x))}{x} \, dx+40 \int \frac {\log (x \log (x))}{x \log (x)} \, dx-50 \int \frac {\log ^2(x \log (x))}{x^2} \, dx+100 \int \frac {\log (x \log (x))}{x^2 \log (x)} \, dx\\ \end {aligned} \end {gather*}

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Mathematica [B]  time = 0.18, size = 53, normalized size = 2.30 \begin {gather*} \frac {2 \left (169+9 x^2-12 x \log (x)-12 x \log (\log (x))+6 \left (65-3 x^2\right ) \log (x \log (x))+9 (5+x)^2 \log ^2(x \log (x))\right )}{9 x} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(780 - 24*x - 36*x^2 + (442 - 24*x - 18*x^2)*Log[x] + (900 + 360*x + 36*x^2 + (120 + 360*x)*Log[x])*
Log[x*Log[x]] + (-450 + 18*x^2)*Log[x]*Log[x*Log[x]]^2)/(9*x^2*Log[x]),x]

[Out]

(2*(169 + 9*x^2 - 12*x*Log[x] - 12*x*Log[Log[x]] + 6*(65 - 3*x^2)*Log[x*Log[x]] + 9*(5 + x)^2*Log[x*Log[x]]^2)
)/(9*x)

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fricas [B]  time = 0.69, size = 46, normalized size = 2.00 \begin {gather*} \frac {2 \, {\left (9 \, {\left (x^{2} + 10 \, x + 25\right )} \log \left (x \log \relax (x)\right )^{2} + 9 \, x^{2} - 6 \, {\left (3 \, x^{2} + 2 \, x - 65\right )} \log \left (x \log \relax (x)\right ) + 169\right )}}{9 \, x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/9*((18*x^2-450)*log(x)*log(x*log(x))^2+((360*x+120)*log(x)+36*x^2+360*x+900)*log(x*log(x))+(-18*x^
2-24*x+442)*log(x)-36*x^2-24*x+780)/x^2/log(x),x, algorithm="fricas")

[Out]

2/9*(9*(x^2 + 10*x + 25)*log(x*log(x))^2 + 9*x^2 - 6*(3*x^2 + 2*x - 65)*log(x*log(x)) + 169)/x

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giac [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {2 \, {\left (9 \, {\left (x^{2} - 25\right )} \log \left (x \log \relax (x)\right )^{2} \log \relax (x) - 18 \, x^{2} + 6 \, {\left (3 \, x^{2} + 10 \, {\left (3 \, x + 1\right )} \log \relax (x) + 30 \, x + 75\right )} \log \left (x \log \relax (x)\right ) - {\left (9 \, x^{2} + 12 \, x - 221\right )} \log \relax (x) - 12 \, x + 390\right )}}{9 \, x^{2} \log \relax (x)}\,{d x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/9*((18*x^2-450)*log(x)*log(x*log(x))^2+((360*x+120)*log(x)+36*x^2+360*x+900)*log(x*log(x))+(-18*x^
2-24*x+442)*log(x)-36*x^2-24*x+780)/x^2/log(x),x, algorithm="giac")

[Out]

integrate(2/9*(9*(x^2 - 25)*log(x*log(x))^2*log(x) - 18*x^2 + 6*(3*x^2 + 10*(3*x + 1)*log(x) + 30*x + 75)*log(
x*log(x)) - (9*x^2 + 12*x - 221)*log(x) - 12*x + 390)/(x^2*log(x)), x)

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maple [C]  time = 0.24, size = 1198, normalized size = 52.09

method	result	size
risch	\(\frac {2 \left (x^{2}+10 x +25\right ) \ln \left (\ln \relax (x )\right )^{2}}{x}+\frac {2 \left (-3 i \pi \,x^{2} \mathrm {csgn}\left (i x \right ) \mathrm {csgn}\left (i \ln \relax (x )\right ) \mathrm {csgn}\left (i x \ln \relax (x )\right )+3 i \pi \,x^{2} \mathrm {csgn}\left (i x \right ) \mathrm {csgn}\left (i x \ln \relax (x )\right )^{2}+3 i \pi \,x^{2} \mathrm {csgn}\left (i \ln \relax (x )\right ) \mathrm {csgn}\left (i x \ln \relax (x )\right )^{2}-3 i \pi \,x^{2} \mathrm {csgn}\left (i x \ln \relax (x )\right )^{3}-75 i \pi \,\mathrm {csgn}\left (i x \right ) \mathrm {csgn}\left (i \ln \relax (x )\right ) \mathrm {csgn}\left (i x \ln \relax (x )\right )+75 i \pi \,\mathrm {csgn}\left (i x \right ) \mathrm {csgn}\left (i x \ln \relax (x )\right )^{2}+75 i \pi \,\mathrm {csgn}\left (i \ln \relax (x )\right ) \mathrm {csgn}\left (i x \ln \relax (x )\right )^{2}-75 i \pi \mathrm {csgn}\left (i x \ln \relax (x )\right )^{3}+6 x^{2} \ln \relax (x )-6 x^{2}+60 x \ln \relax (x )+150 \ln \relax (x )+130\right ) \ln \left (\ln \relax (x )\right )}{3 x}+\frac {676-72 x^{2} \ln \relax (x )-48 x \ln \left (\ln \relax (x )\right )+36 x^{2} \ln \relax (x )^{2}+1560 \ln \relax (x )+900 \ln \relax (x )^{2}+36 x^{2}-36 i \pi \,x^{2} \mathrm {csgn}\left (i x \right ) \mathrm {csgn}\left (i x \ln \relax (x )\right )^{2}-36 i \pi \,x^{2} \mathrm {csgn}\left (i \ln \relax (x )\right ) \mathrm {csgn}\left (i x \ln \relax (x )\right )^{2}-780 i \pi \,\mathrm {csgn}\left (i x \right ) \mathrm {csgn}\left (i \ln \relax (x )\right ) \mathrm {csgn}\left (i x \ln \relax (x )\right )-9 \pi ^{2} x^{2} \mathrm {csgn}\left (i x \right )^{2} \mathrm {csgn}\left (i \ln \relax (x )\right )^{2} \mathrm {csgn}\left (i x \ln \relax (x )\right )^{2}+18 \pi ^{2} x^{2} \mathrm {csgn}\left (i x \right )^{2} \mathrm {csgn}\left (i \ln \relax (x )\right ) \mathrm {csgn}\left (i x \ln \relax (x )\right )^{3}+18 \pi ^{2} x^{2} \mathrm {csgn}\left (i x \right ) \mathrm {csgn}\left (i \ln \relax (x )\right )^{2} \mathrm {csgn}\left (i x \ln \relax (x )\right )^{3}-360 i \pi \ln \relax (x ) \mathrm {csgn}\left (i x \ln \relax (x )\right )^{3} x +360 x \ln \relax (x )^{2}-48 x \ln \relax (x )-9 \pi ^{2} x^{2} \mathrm {csgn}\left (i x \right )^{2} \mathrm {csgn}\left (i x \ln \relax (x )\right )^{4}+18 \pi ^{2} x^{2} \mathrm {csgn}\left (i x \right ) \mathrm {csgn}\left (i x \ln \relax (x )\right )^{5}-9 \pi ^{2} x^{2} \mathrm {csgn}\left (i \ln \relax (x )\right )^{2} \mathrm {csgn}\left (i x \ln \relax (x )\right )^{4}+18 \pi ^{2} x^{2} \mathrm {csgn}\left (i \ln \relax (x )\right ) \mathrm {csgn}\left (i x \ln \relax (x )\right )^{5}-225 \pi ^{2} \mathrm {csgn}\left (i x \right )^{2} \mathrm {csgn}\left (i \ln \relax (x )\right )^{2} \mathrm {csgn}\left (i x \ln \relax (x )\right )^{2}+450 \pi ^{2} \mathrm {csgn}\left (i x \right )^{2} \mathrm {csgn}\left (i \ln \relax (x )\right ) \mathrm {csgn}\left (i x \ln \relax (x )\right )^{3}+450 \pi ^{2} \mathrm {csgn}\left (i x \right ) \mathrm {csgn}\left (i \ln \relax (x )\right )^{2} \mathrm {csgn}\left (i x \ln \relax (x )\right )^{3}-900 \pi ^{2} \mathrm {csgn}\left (i x \right ) \mathrm {csgn}\left (i \ln \relax (x )\right ) \mathrm {csgn}\left (i x \ln \relax (x )\right )^{4}-900 i \ln \relax (x ) \pi \mathrm {csgn}\left (i x \ln \relax (x )\right )^{3}+36 i \pi \,x^{2} \mathrm {csgn}\left (i x \ln \relax (x )\right )^{3}+780 i \pi \,\mathrm {csgn}\left (i x \right ) \mathrm {csgn}\left (i x \ln \relax (x )\right )^{2}+780 i \pi \,\mathrm {csgn}\left (i \ln \relax (x )\right ) \mathrm {csgn}\left (i x \ln \relax (x )\right )^{2}-225 \pi ^{2} \mathrm {csgn}\left (i x \ln \relax (x )\right )^{6}-360 i \pi \ln \left (\ln \relax (x )\right ) \mathrm {csgn}\left (i x \ln \relax (x )\right )^{3} x +900 i \ln \relax (x ) \pi \,\mathrm {csgn}\left (i x \right ) \mathrm {csgn}\left (i x \ln \relax (x )\right )^{2}-36 i \pi \,x^{2} \mathrm {csgn}\left (i x \ln \relax (x )\right )^{3} \ln \relax (x )+900 i \ln \relax (x ) \pi \,\mathrm {csgn}\left (i \ln \relax (x )\right ) \mathrm {csgn}\left (i x \ln \relax (x )\right )^{2}-36 \pi ^{2} x^{2} \mathrm {csgn}\left (i x \right ) \mathrm {csgn}\left (i \ln \relax (x )\right ) \mathrm {csgn}\left (i x \ln \relax (x )\right )^{4}-360 i \pi \ln \relax (x ) \mathrm {csgn}\left (i x \right ) \mathrm {csgn}\left (i \ln \relax (x )\right ) \mathrm {csgn}\left (i x \ln \relax (x )\right ) x -36 i \pi \,x^{2} \mathrm {csgn}\left (i x \right ) \mathrm {csgn}\left (i \ln \relax (x )\right ) \mathrm {csgn}\left (i x \ln \relax (x )\right ) \ln \relax (x )-360 i \pi \ln \left (\ln \relax (x )\right ) \mathrm {csgn}\left (i x \right ) \mathrm {csgn}\left (i \ln \relax (x )\right ) \mathrm {csgn}\left (i x \ln \relax (x )\right ) x -780 i \pi \mathrm {csgn}\left (i x \ln \relax (x )\right )^{3}-225 \pi ^{2} \mathrm {csgn}\left (i \ln \relax (x )\right )^{2} \mathrm {csgn}\left (i x \ln \relax (x )\right )^{4}+450 \pi ^{2} \mathrm {csgn}\left (i \ln \relax (x )\right ) \mathrm {csgn}\left (i x \ln \relax (x )\right )^{5}-9 \pi ^{2} x^{2} \mathrm {csgn}\left (i x \ln \relax (x )\right )^{6}-225 \pi ^{2} \mathrm {csgn}\left (i x \right )^{2} \mathrm {csgn}\left (i x \ln \relax (x )\right )^{4}+450 \pi ^{2} \mathrm {csgn}\left (i x \right ) \mathrm {csgn}\left (i x \ln \relax (x )\right )^{5}+360 i \pi \ln \left (\ln \relax (x )\right ) \mathrm {csgn}\left (i x \right ) \mathrm {csgn}\left (i x \ln \relax (x )\right )^{2} x +360 i \pi \ln \left (\ln \relax (x )\right ) \mathrm {csgn}\left (i \ln \relax (x )\right ) \mathrm {csgn}\left (i x \ln \relax (x )\right )^{2} x -900 i \ln \relax (x ) \pi \,\mathrm {csgn}\left (i x \right ) \mathrm {csgn}\left (i \ln \relax (x )\right ) \mathrm {csgn}\left (i x \ln \relax (x )\right )+36 i \pi \,x^{2} \mathrm {csgn}\left (i x \right ) \mathrm {csgn}\left (i x \ln \relax (x )\right )^{2} \ln \relax (x )+36 i \pi \,x^{2} \mathrm {csgn}\left (i \ln \relax (x )\right ) \mathrm {csgn}\left (i x \ln \relax (x )\right )^{2} \ln \relax (x )+36 i \pi \,x^{2} \mathrm {csgn}\left (i x \right ) \mathrm {csgn}\left (i \ln \relax (x )\right ) \mathrm {csgn}\left (i x \ln \relax (x )\right )+360 i \pi \ln \relax (x ) \mathrm {csgn}\left (i x \right ) \mathrm {csgn}\left (i x \ln \relax (x )\right )^{2} x +360 i \pi \ln \relax (x ) \mathrm {csgn}\left (i \ln \relax (x )\right ) \mathrm {csgn}\left (i x \ln \relax (x )\right )^{2} x}{18 x}\)	\(1198\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/9*((18*x^2-450)*ln(x)*ln(x*ln(x))^2+((360*x+120)*ln(x)+36*x^2+360*x+900)*ln(x*ln(x))+(-18*x^2-24*x+442)*
ln(x)-36*x^2-24*x+780)/x^2/ln(x),x,method=_RETURNVERBOSE)

[Out]

2*(x^2+10*x+25)/x*ln(ln(x))^2+2/3*(-3*I*Pi*x^2*csgn(I*x)*csgn(I*ln(x))*csgn(I*x*ln(x))+3*I*Pi*x^2*csgn(I*x)*cs
gn(I*x*ln(x))^2+3*I*Pi*x^2*csgn(I*ln(x))*csgn(I*x*ln(x))^2-3*I*Pi*x^2*csgn(I*x*ln(x))^3-75*I*Pi*csgn(I*x)*csgn
(I*ln(x))*csgn(I*x*ln(x))+75*I*Pi*csgn(I*x)*csgn(I*x*ln(x))^2+75*I*Pi*csgn(I*ln(x))*csgn(I*x*ln(x))^2-75*I*Pi*
csgn(I*x*ln(x))^3+6*x^2*ln(x)-6*x^2+60*x*ln(x)+150*ln(x)+130)/x*ln(ln(x))+1/18*(676-72*x^2*ln(x)-48*x*ln(ln(x)
)+36*x^2*ln(x)^2+1560*ln(x)+900*ln(x)^2+36*x^2+360*x*ln(x)^2-48*x*ln(x)-9*Pi^2*x^2*csgn(I*x)^2*csgn(I*x*ln(x))
^4+18*Pi^2*x^2*csgn(I*x)*csgn(I*x*ln(x))^5-9*Pi^2*x^2*csgn(I*ln(x))^2*csgn(I*x*ln(x))^4+18*Pi^2*x^2*csgn(I*ln(
x))*csgn(I*x*ln(x))^5-225*Pi^2*csgn(I*x)^2*csgn(I*ln(x))^2*csgn(I*x*ln(x))^2+450*Pi^2*csgn(I*x)^2*csgn(I*ln(x)
)*csgn(I*x*ln(x))^3+450*Pi^2*csgn(I*x)*csgn(I*ln(x))^2*csgn(I*x*ln(x))^3-900*Pi^2*csgn(I*x)*csgn(I*ln(x))*csgn
(I*x*ln(x))^4+780*I*Pi*csgn(I*x)*csgn(I*x*ln(x))^2+780*I*Pi*csgn(I*ln(x))*csgn(I*x*ln(x))^2-900*I*ln(x)*Pi*csg
n(I*x*ln(x))^3+36*I*Pi*x^2*csgn(I*x*ln(x))^3-225*Pi^2*csgn(I*x*ln(x))^6+360*I*Pi*ln(x)*csgn(I*x)*csgn(I*x*ln(x
))^2*x+360*I*Pi*ln(x)*csgn(I*ln(x))*csgn(I*x*ln(x))^2*x+360*I*Pi*ln(ln(x))*csgn(I*x)*csgn(I*x*ln(x))^2*x+360*I
*Pi*ln(ln(x))*csgn(I*ln(x))*csgn(I*x*ln(x))^2*x-900*I*ln(x)*Pi*csgn(I*x)*csgn(I*ln(x))*csgn(I*x*ln(x))+36*I*Pi
*x^2*csgn(I*x)*csgn(I*x*ln(x))^2*ln(x)+36*I*Pi*x^2*csgn(I*ln(x))*csgn(I*x*ln(x))^2*ln(x)+36*I*Pi*x^2*csgn(I*x)
*csgn(I*ln(x))*csgn(I*x*ln(x))-9*Pi^2*x^2*csgn(I*x)^2*csgn(I*ln(x))^2*csgn(I*x*ln(x))^2+18*Pi^2*x^2*csgn(I*x)^
2*csgn(I*ln(x))*csgn(I*x*ln(x))^3+18*Pi^2*x^2*csgn(I*x)*csgn(I*ln(x))^2*csgn(I*x*ln(x))^3-36*I*Pi*x^2*csgn(I*x
)*csgn(I*x*ln(x))^2-36*I*Pi*x^2*csgn(I*ln(x))*csgn(I*x*ln(x))^2-780*I*Pi*csgn(I*x)*csgn(I*ln(x))*csgn(I*x*ln(x
))-360*I*Pi*ln(x)*csgn(I*x*ln(x))^3*x-360*I*Pi*ln(ln(x))*csgn(I*x*ln(x))^3*x+900*I*ln(x)*Pi*csgn(I*x)*csgn(I*x
*ln(x))^2-36*I*Pi*x^2*csgn(I*x*ln(x))^3*ln(x)+900*I*ln(x)*Pi*csgn(I*ln(x))*csgn(I*x*ln(x))^2-36*Pi^2*x^2*csgn(
I*x)*csgn(I*ln(x))*csgn(I*x*ln(x))^4-360*I*Pi*ln(x)*csgn(I*x)*csgn(I*ln(x))*csgn(I*x*ln(x))*x-36*I*Pi*x^2*csgn
(I*x)*csgn(I*ln(x))*csgn(I*x*ln(x))*ln(x)-360*I*Pi*ln(ln(x))*csgn(I*x)*csgn(I*ln(x))*csgn(I*x*ln(x))*x-780*I*P
i*csgn(I*x*ln(x))^3-225*Pi^2*csgn(I*ln(x))^2*csgn(I*x*ln(x))^4+450*Pi^2*csgn(I*ln(x))*csgn(I*x*ln(x))^5-9*Pi^2
*x^2*csgn(I*x*ln(x))^6-225*Pi^2*csgn(I*x)^2*csgn(I*x*ln(x))^4+450*Pi^2*csgn(I*x)*csgn(I*x*ln(x))^5)/x

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maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} -2 \, x + \frac {2 \, {\left (3 \, {\left (x^{2} + 10 \, x + 25\right )} \log \relax (x)^{2} + 3 \, {\left (x^{2} + 10 \, x + 25\right )} \log \left (\log \relax (x)\right )^{2} + 6 \, x^{2} - 2 \, {\left (3 \, x^{2} - 65\right )} \log \relax (x) - 2 \, {\left (3 \, x^{2} - 3 \, {\left (x^{2} + 10 \, x + 25\right )} \log \relax (x) - 65\right )} \log \left (\log \relax (x)\right ) + 130\right )}}{3 \, x} - \frac {442}{9 \, x} + \frac {260}{3} \, {\rm Ei}\left (-\log \relax (x)\right ) - 4 \, {\rm Ei}\left (\log \relax (x)\right ) + \frac {2}{9} \, \int \frac {6 \, {\left (3 \, x^{2} - 65\right )}}{x^{2} \log \relax (x)}\,{d x} - \frac {8}{3} \, \log \relax (x) - \frac {8}{3} \, \log \left (\log \relax (x)\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/9*((18*x^2-450)*log(x)*log(x*log(x))^2+((360*x+120)*log(x)+36*x^2+360*x+900)*log(x*log(x))+(-18*x^
2-24*x+442)*log(x)-36*x^2-24*x+780)/x^2/log(x),x, algorithm="maxima")

[Out]

-2*x + 2/3*(3*(x^2 + 10*x + 25)*log(x)^2 + 3*(x^2 + 10*x + 25)*log(log(x))^2 + 6*x^2 - 2*(3*x^2 - 65)*log(x) -
2*(3*x^2 - 3*(x^2 + 10*x + 25)*log(x) - 65)*log(log(x)) + 130)/x - 442/9/x + 260/3*Ei(-log(x)) - 4*Ei(log(x))
+ 2/9*integrate(6*(3*x^2 - 65)/(x^2*log(x)), x) - 8/3*log(x) - 8/3*log(log(x))

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mupad [B]  time = 1.21, size = 69, normalized size = 3.00 \begin {gather*} 2\,x+\frac {112\,\ln \left (\ln \relax (x)\right )}{3}+\frac {112\,\ln \relax (x)}{3}-\ln \left (x\,\ln \relax (x)\right )\,\left (8\,x-\frac {4\,x^2-40\,x+\frac {260}{3}}{x}\right )+\frac {338}{9\,x}+{\ln \left (x\,\ln \relax (x)\right )}^2\,\left (4\,x-\frac {2\,x^2-50}{x}+20\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(-((8*x)/3 - (log(x*log(x))*(360*x + log(x)*(360*x + 120) + 36*x^2 + 900))/9 + (log(x)*(24*x + 18*x^2 - 442
))/9 + 4*x^2 - (log(x*log(x))^2*log(x)*(18*x^2 - 450))/9 - 260/3)/(x^2*log(x)),x)

[Out]

2*x + (112*log(log(x)))/3 + (112*log(x))/3 - log(x*log(x))*(8*x - (4*x^2 - 40*x + 260/3)/x) + 338/(9*x) + log(
x*log(x))^2*(4*x - (2*x^2 - 50)/x + 20)

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sympy [B]  time = 0.52, size = 60, normalized size = 2.61 \begin {gather*} 2 x - \frac {8 \log {\relax (x )}}{3} - \frac {8 \log {\left (\log {\relax (x )} \right )}}{3} + \frac {\left (260 - 12 x^{2}\right ) \log {\left (x \log {\relax (x )} \right )}}{3 x} + \frac {\left (2 x^{2} + 20 x + 50\right ) \log {\left (x \log {\relax (x )} \right )}^{2}}{x} + \frac {338}{9 x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/9*((18*x**2-450)*ln(x)*ln(x*ln(x))**2+((360*x+120)*ln(x)+36*x**2+360*x+900)*ln(x*ln(x))+(-18*x**2-
24*x+442)*ln(x)-36*x**2-24*x+780)/x**2/ln(x),x)

[Out]

2*x - 8*log(x)/3 - 8*log(log(x))/3 + (260 - 12*x**2)*log(x*log(x))/(3*x) + (2*x**2 + 20*x + 50)*log(x*log(x))*
*2/x + 338/(9*x)

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