3.81.58 \(\int \frac {-32-80 x+2 x^2-16 x \log (x)}{(1680 x+800 x^2+95 x^3+(656 x+320 x^2+39 x^3) \log (x)+(64 x+32 x^2+4 x^3) \log ^2(x)) \log (\frac {28224+12768 x+1444 x^2+(10752+5120 x+608 x^2) \log (x)+(1024+512 x+64 x^2) \log ^2(x)}{400+200 x+25 x^2+(160+80 x+10 x^2) \log (x)+(16+8 x+x^2) \log ^2(x)})} \, dx\)

Optimal. Leaf size=24 \[ \log \left (\log \left (4 \left (4-\frac {-4+x}{(4+x) (5+\log (x))}\right )^2\right )\right ) \]

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Rubi [F]  time = 12.35, 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 {-32-80 x+2 x^2-16 x \log (x)}{\left (1680 x+800 x^2+95 x^3+\left (656 x+320 x^2+39 x^3\right ) \log (x)+\left (64 x+32 x^2+4 x^3\right ) \log ^2(x)\right ) \log \left (\frac {28224+12768 x+1444 x^2+\left (10752+5120 x+608 x^2\right ) \log (x)+\left (1024+512 x+64 x^2\right ) \log ^2(x)}{400+200 x+25 x^2+\left (160+80 x+10 x^2\right ) \log (x)+\left (16+8 x+x^2\right ) \log ^2(x)}\right )} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Int[(-32 - 80*x + 2*x^2 - 16*x*Log[x])/((1680*x + 800*x^2 + 95*x^3 + (656*x + 320*x^2 + 39*x^3)*Log[x] + (64*x
+ 32*x^2 + 4*x^3)*Log[x]^2)*Log[(28224 + 12768*x + 1444*x^2 + (10752 + 5120*x + 608*x^2)*Log[x] + (1024 + 512
*x + 64*x^2)*Log[x]^2)/(400 + 200*x + 25*x^2 + (160 + 80*x + 10*x^2)*Log[x] + (16 + 8*x + x^2)*Log[x]^2)]),x]

[Out]

2*Defer[Int][1/((5 + Log[x])*(84 + 19*x + 16*Log[x] + 4*x*Log[x])*Log[(4*(84 + 19*x + 4*(4 + x)*Log[x])^2)/((4
+ x)^2*(5 + Log[x])^2)]), x] - 8*Defer[Int][1/(x*(5 + Log[x])*(84 + 19*x + 16*Log[x] + 4*x*Log[x])*Log[(4*(84
+ 19*x + 4*(4 + x)*Log[x])^2)/((4 + x)^2*(5 + Log[x])^2)]), x] - 80*Defer[Int][1/((4 + x)*(5 + Log[x])*(84 +
19*x + 16*Log[x] + 4*x*Log[x])*Log[(4*(84 + 19*x + 4*(4 + x)*Log[x])^2)/((4 + x)^2*(5 + Log[x])^2)]), x] - 16*
Defer[Int][Log[x]/((4 + x)*(5 + Log[x])*(84 + 19*x + 16*Log[x] + 4*x*Log[x])*Log[(4*(84 + 19*x + 4*(4 + x)*Log
[x])^2)/((4 + x)^2*(5 + Log[x])^2)]), x]

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\int \frac {2 \left (-16-40 x+x^2-8 x \log (x)\right )}{x (4+x) (5+\log (x)) (84+19 x+4 (4+x) \log (x)) \log \left (\frac {4 (84+19 x+4 (4+x) \log (x))^2}{(4+x)^2 (5+\log (x))^2}\right )} \, dx\\ &=2 \int \frac {-16-40 x+x^2-8 x \log (x)}{x (4+x) (5+\log (x)) (84+19 x+4 (4+x) \log (x)) \log \left (\frac {4 (84+19 x+4 (4+x) \log (x))^2}{(4+x)^2 (5+\log (x))^2}\right )} \, dx\\ &=2 \int \left (\frac {-16-40 x+x^2-8 x \log (x)}{4 x (5+\log (x)) (84+19 x+16 \log (x)+4 x \log (x)) \log \left (\frac {4 (84+19 x+4 (4+x) \log (x))^2}{(4+x)^2 (5+\log (x))^2}\right )}+\frac {16+40 x-x^2+8 x \log (x)}{4 (4+x) (5+\log (x)) (84+19 x+16 \log (x)+4 x \log (x)) \log \left (\frac {4 (84+19 x+4 (4+x) \log (x))^2}{(4+x)^2 (5+\log (x))^2}\right )}\right ) \, dx\\ &=\frac {1}{2} \int \frac {-16-40 x+x^2-8 x \log (x)}{x (5+\log (x)) (84+19 x+16 \log (x)+4 x \log (x)) \log \left (\frac {4 (84+19 x+4 (4+x) \log (x))^2}{(4+x)^2 (5+\log (x))^2}\right )} \, dx+\frac {1}{2} \int \frac {16+40 x-x^2+8 x \log (x)}{(4+x) (5+\log (x)) (84+19 x+16 \log (x)+4 x \log (x)) \log \left (\frac {4 (84+19 x+4 (4+x) \log (x))^2}{(4+x)^2 (5+\log (x))^2}\right )} \, dx\\ &=\frac {1}{2} \int \left (-\frac {40}{(5+\log (x)) (84+19 x+16 \log (x)+4 x \log (x)) \log \left (\frac {4 (84+19 x+4 (4+x) \log (x))^2}{(4+x)^2 (5+\log (x))^2}\right )}-\frac {16}{x (5+\log (x)) (84+19 x+16 \log (x)+4 x \log (x)) \log \left (\frac {4 (84+19 x+4 (4+x) \log (x))^2}{(4+x)^2 (5+\log (x))^2}\right )}+\frac {x}{(5+\log (x)) (84+19 x+16 \log (x)+4 x \log (x)) \log \left (\frac {4 (84+19 x+4 (4+x) \log (x))^2}{(4+x)^2 (5+\log (x))^2}\right )}-\frac {8 \log (x)}{(5+\log (x)) (84+19 x+16 \log (x)+4 x \log (x)) \log \left (\frac {4 (84+19 x+4 (4+x) \log (x))^2}{(4+x)^2 (5+\log (x))^2}\right )}\right ) \, dx+\frac {1}{2} \int \left (\frac {16}{(4+x) (5+\log (x)) (84+19 x+16 \log (x)+4 x \log (x)) \log \left (\frac {4 (84+19 x+4 (4+x) \log (x))^2}{(4+x)^2 (5+\log (x))^2}\right )}+\frac {40 x}{(4+x) (5+\log (x)) (84+19 x+16 \log (x)+4 x \log (x)) \log \left (\frac {4 (84+19 x+4 (4+x) \log (x))^2}{(4+x)^2 (5+\log (x))^2}\right )}-\frac {x^2}{(4+x) (5+\log (x)) (84+19 x+16 \log (x)+4 x \log (x)) \log \left (\frac {4 (84+19 x+4 (4+x) \log (x))^2}{(4+x)^2 (5+\log (x))^2}\right )}+\frac {8 x \log (x)}{(4+x) (5+\log (x)) (84+19 x+16 \log (x)+4 x \log (x)) \log \left (\frac {4 (84+19 x+4 (4+x) \log (x))^2}{(4+x)^2 (5+\log (x))^2}\right )}\right ) \, dx\\ &=\frac {1}{2} \int \frac {x}{(5+\log (x)) (84+19 x+16 \log (x)+4 x \log (x)) \log \left (\frac {4 (84+19 x+4 (4+x) \log (x))^2}{(4+x)^2 (5+\log (x))^2}\right )} \, dx-\frac {1}{2} \int \frac {x^2}{(4+x) (5+\log (x)) (84+19 x+16 \log (x)+4 x \log (x)) \log \left (\frac {4 (84+19 x+4 (4+x) \log (x))^2}{(4+x)^2 (5+\log (x))^2}\right )} \, dx-4 \int \frac {\log (x)}{(5+\log (x)) (84+19 x+16 \log (x)+4 x \log (x)) \log \left (\frac {4 (84+19 x+4 (4+x) \log (x))^2}{(4+x)^2 (5+\log (x))^2}\right )} \, dx+4 \int \frac {x \log (x)}{(4+x) (5+\log (x)) (84+19 x+16 \log (x)+4 x \log (x)) \log \left (\frac {4 (84+19 x+4 (4+x) \log (x))^2}{(4+x)^2 (5+\log (x))^2}\right )} \, dx-8 \int \frac {1}{x (5+\log (x)) (84+19 x+16 \log (x)+4 x \log (x)) \log \left (\frac {4 (84+19 x+4 (4+x) \log (x))^2}{(4+x)^2 (5+\log (x))^2}\right )} \, dx+8 \int \frac {1}{(4+x) (5+\log (x)) (84+19 x+16 \log (x)+4 x \log (x)) \log \left (\frac {4 (84+19 x+4 (4+x) \log (x))^2}{(4+x)^2 (5+\log (x))^2}\right )} \, dx-20 \int \frac {1}{(5+\log (x)) (84+19 x+16 \log (x)+4 x \log (x)) \log \left (\frac {4 (84+19 x+4 (4+x) \log (x))^2}{(4+x)^2 (5+\log (x))^2}\right )} \, dx+20 \int \frac {x}{(4+x) (5+\log (x)) (84+19 x+16 \log (x)+4 x \log (x)) \log \left (\frac {4 (84+19 x+4 (4+x) \log (x))^2}{(4+x)^2 (5+\log (x))^2}\right )} \, dx\\ &=-\left (\frac {1}{2} \int \left (-\frac {4}{(5+\log (x)) (84+19 x+16 \log (x)+4 x \log (x)) \log \left (\frac {4 (84+19 x+4 (4+x) \log (x))^2}{(4+x)^2 (5+\log (x))^2}\right )}+\frac {x}{(5+\log (x)) (84+19 x+16 \log (x)+4 x \log (x)) \log \left (\frac {4 (84+19 x+4 (4+x) \log (x))^2}{(4+x)^2 (5+\log (x))^2}\right )}+\frac {16}{(4+x) (5+\log (x)) (84+19 x+16 \log (x)+4 x \log (x)) \log \left (\frac {4 (84+19 x+4 (4+x) \log (x))^2}{(4+x)^2 (5+\log (x))^2}\right )}\right ) \, dx\right )+\frac {1}{2} \int \frac {x}{(5+\log (x)) (84+19 x+16 \log (x)+4 x \log (x)) \log \left (\frac {4 (84+19 x+4 (4+x) \log (x))^2}{(4+x)^2 (5+\log (x))^2}\right )} \, dx+4 \int \left (\frac {\log (x)}{(5+\log (x)) (84+19 x+16 \log (x)+4 x \log (x)) \log \left (\frac {4 (84+19 x+4 (4+x) \log (x))^2}{(4+x)^2 (5+\log (x))^2}\right )}-\frac {4 \log (x)}{(4+x) (5+\log (x)) (84+19 x+16 \log (x)+4 x \log (x)) \log \left (\frac {4 (84+19 x+4 (4+x) \log (x))^2}{(4+x)^2 (5+\log (x))^2}\right )}\right ) \, dx-4 \int \frac {\log (x)}{(5+\log (x)) (84+19 x+16 \log (x)+4 x \log (x)) \log \left (\frac {4 (84+19 x+4 (4+x) \log (x))^2}{(4+x)^2 (5+\log (x))^2}\right )} \, dx-8 \int \frac {1}{x (5+\log (x)) (84+19 x+16 \log (x)+4 x \log (x)) \log \left (\frac {4 (84+19 x+4 (4+x) \log (x))^2}{(4+x)^2 (5+\log (x))^2}\right )} \, dx+8 \int \frac {1}{(4+x) (5+\log (x)) (84+19 x+16 \log (x)+4 x \log (x)) \log \left (\frac {4 (84+19 x+4 (4+x) \log (x))^2}{(4+x)^2 (5+\log (x))^2}\right )} \, dx+20 \int \left (\frac {1}{(5+\log (x)) (84+19 x+16 \log (x)+4 x \log (x)) \log \left (\frac {4 (84+19 x+4 (4+x) \log (x))^2}{(4+x)^2 (5+\log (x))^2}\right )}-\frac {4}{(4+x) (5+\log (x)) (84+19 x+16 \log (x)+4 x \log (x)) \log \left (\frac {4 (84+19 x+4 (4+x) \log (x))^2}{(4+x)^2 (5+\log (x))^2}\right )}\right ) \, dx-20 \int \frac {1}{(5+\log (x)) (84+19 x+16 \log (x)+4 x \log (x)) \log \left (\frac {4 (84+19 x+4 (4+x) \log (x))^2}{(4+x)^2 (5+\log (x))^2}\right )} \, dx\\ &=2 \int \frac {1}{(5+\log (x)) (84+19 x+16 \log (x)+4 x \log (x)) \log \left (\frac {4 (84+19 x+4 (4+x) \log (x))^2}{(4+x)^2 (5+\log (x))^2}\right )} \, dx-8 \int \frac {1}{x (5+\log (x)) (84+19 x+16 \log (x)+4 x \log (x)) \log \left (\frac {4 (84+19 x+4 (4+x) \log (x))^2}{(4+x)^2 (5+\log (x))^2}\right )} \, dx-16 \int \frac {\log (x)}{(4+x) (5+\log (x)) (84+19 x+16 \log (x)+4 x \log (x)) \log \left (\frac {4 (84+19 x+4 (4+x) \log (x))^2}{(4+x)^2 (5+\log (x))^2}\right )} \, dx-80 \int \frac {1}{(4+x) (5+\log (x)) (84+19 x+16 \log (x)+4 x \log (x)) \log \left (\frac {4 (84+19 x+4 (4+x) \log (x))^2}{(4+x)^2 (5+\log (x))^2}\right )} \, dx\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.41, size = 29, normalized size = 1.21 \begin {gather*} \log \left (\log \left (\frac {4 (84+19 x+4 (4+x) \log (x))^2}{(4+x)^2 (5+\log (x))^2}\right )\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(-32 - 80*x + 2*x^2 - 16*x*Log[x])/((1680*x + 800*x^2 + 95*x^3 + (656*x + 320*x^2 + 39*x^3)*Log[x] +
(64*x + 32*x^2 + 4*x^3)*Log[x]^2)*Log[(28224 + 12768*x + 1444*x^2 + (10752 + 5120*x + 608*x^2)*Log[x] + (1024
+ 512*x + 64*x^2)*Log[x]^2)/(400 + 200*x + 25*x^2 + (160 + 80*x + 10*x^2)*Log[x] + (16 + 8*x + x^2)*Log[x]^2)
]),x]

[Out]

Log[Log[(4*(84 + 19*x + 4*(4 + x)*Log[x])^2)/((4 + x)^2*(5 + Log[x])^2)]]

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fricas [B]  time = 0.97, size = 79, normalized size = 3.29 \begin {gather*} \log \left (\log \left (\frac {4 \, {\left (16 \, {\left (x^{2} + 8 \, x + 16\right )} \log \relax (x)^{2} + 361 \, x^{2} + 8 \, {\left (19 \, x^{2} + 160 \, x + 336\right )} \log \relax (x) + 3192 \, x + 7056\right )}}{{\left (x^{2} + 8 \, x + 16\right )} \log \relax (x)^{2} + 25 \, x^{2} + 10 \, {\left (x^{2} + 8 \, x + 16\right )} \log \relax (x) + 200 \, x + 400}\right )\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-16*x*log(x)+2*x^2-80*x-32)/((4*x^3+32*x^2+64*x)*log(x)^2+(39*x^3+320*x^2+656*x)*log(x)+95*x^3+800*
x^2+1680*x)/log(((64*x^2+512*x+1024)*log(x)^2+(608*x^2+5120*x+10752)*log(x)+1444*x^2+12768*x+28224)/((x^2+8*x+
16)*log(x)^2+(10*x^2+80*x+160)*log(x)+25*x^2+200*x+400)),x, algorithm="fricas")

[Out]

log(log(4*(16*(x^2 + 8*x + 16)*log(x)^2 + 361*x^2 + 8*(19*x^2 + 160*x + 336)*log(x) + 3192*x + 7056)/((x^2 + 8
*x + 16)*log(x)^2 + 25*x^2 + 10*(x^2 + 8*x + 16)*log(x) + 200*x + 400)))

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giac [B]  time = 1.44, size = 101, normalized size = 4.21 \begin {gather*} \log \left (-\log \left (64 \, x^{2} \log \relax (x)^{2} + 608 \, x^{2} \log \relax (x) + 512 \, x \log \relax (x)^{2} + 1444 \, x^{2} + 5120 \, x \log \relax (x) + 1024 \, \log \relax (x)^{2} + 12768 \, x + 10752 \, \log \relax (x) + 28224\right ) + \log \left (x^{2} \log \relax (x)^{2} + 10 \, x^{2} \log \relax (x) + 8 \, x \log \relax (x)^{2} + 25 \, x^{2} + 80 \, x \log \relax (x) + 16 \, \log \relax (x)^{2} + 200 \, x + 160 \, \log \relax (x) + 400\right )\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-16*x*log(x)+2*x^2-80*x-32)/((4*x^3+32*x^2+64*x)*log(x)^2+(39*x^3+320*x^2+656*x)*log(x)+95*x^3+800*
x^2+1680*x)/log(((64*x^2+512*x+1024)*log(x)^2+(608*x^2+5120*x+10752)*log(x)+1444*x^2+12768*x+28224)/((x^2+8*x+
16)*log(x)^2+(10*x^2+80*x+160)*log(x)+25*x^2+200*x+400)),x, algorithm="giac")

[Out]

log(-log(64*x^2*log(x)^2 + 608*x^2*log(x) + 512*x*log(x)^2 + 1444*x^2 + 5120*x*log(x) + 1024*log(x)^2 + 12768*
x + 10752*log(x) + 28224) + log(x^2*log(x)^2 + 10*x^2*log(x) + 8*x*log(x)^2 + 25*x^2 + 80*x*log(x) + 16*log(x)
^2 + 200*x + 160*log(x) + 400))

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maple [C]  time = 0.70, size = 622, normalized size = 25.92

method	result	size
risch	\(\ln \left (\ln \left (\left (\ln \relax (x )+\frac {19}{4}\right ) x +4 \ln \relax (x )+21\right )-\frac {i \left (\pi \,\mathrm {csgn}\left (\frac {i}{\left (4+x \right )^{2}}\right ) \mathrm {csgn}\left (\frac {i \left (\left (\ln \relax (x )+\frac {19}{4}\right ) x +4 \ln \relax (x )+21\right )^{2}}{\left (5+\ln \relax (x )\right )^{2}}\right ) \mathrm {csgn}\left (\frac {i \left (\left (\ln \relax (x )+\frac {19}{4}\right ) x +4 \ln \relax (x )+21\right )^{2}}{\left (4+x \right )^{2} \left (5+\ln \relax (x )\right )^{2}}\right )-\pi \,\mathrm {csgn}\left (\frac {i}{\left (4+x \right )^{2}}\right ) \mathrm {csgn}\left (\frac {i \left (\left (\ln \relax (x )+\frac {19}{4}\right ) x +4 \ln \relax (x )+21\right )^{2}}{\left (4+x \right )^{2} \left (5+\ln \relax (x )\right )^{2}}\right )^{2}+\pi \,\mathrm {csgn}\left (\frac {i}{\left (5+\ln \relax (x )\right )^{2}}\right ) \mathrm {csgn}\left (i \left (\left (\ln \relax (x )+\frac {19}{4}\right ) x +4 \ln \relax (x )+21\right )^{2}\right ) \mathrm {csgn}\left (\frac {i \left (\left (\ln \relax (x )+\frac {19}{4}\right ) x +4 \ln \relax (x )+21\right )^{2}}{\left (5+\ln \relax (x )\right )^{2}}\right )-\pi \,\mathrm {csgn}\left (\frac {i}{\left (5+\ln \relax (x )\right )^{2}}\right ) \mathrm {csgn}\left (\frac {i \left (\left (\ln \relax (x )+\frac {19}{4}\right ) x +4 \ln \relax (x )+21\right )^{2}}{\left (5+\ln \relax (x )\right )^{2}}\right )^{2}-\pi \mathrm {csgn}\left (i \left (5+\ln \relax (x )\right )\right )^{2} \mathrm {csgn}\left (i \left (5+\ln \relax (x )\right )^{2}\right )+2 \pi \,\mathrm {csgn}\left (i \left (5+\ln \relax (x )\right )\right ) \mathrm {csgn}\left (i \left (5+\ln \relax (x )\right )^{2}\right )^{2}-\pi \mathrm {csgn}\left (i \left (5+\ln \relax (x )\right )^{2}\right )^{3}+\pi \mathrm {csgn}\left (i \left (\left (\ln \relax (x )+\frac {19}{4}\right ) x +4 \ln \relax (x )+21\right )\right )^{2} \mathrm {csgn}\left (i \left (\left (\ln \relax (x )+\frac {19}{4}\right ) x +4 \ln \relax (x )+21\right )^{2}\right )-2 \pi \,\mathrm {csgn}\left (i \left (\left (\ln \relax (x )+\frac {19}{4}\right ) x +4 \ln \relax (x )+21\right )\right ) \mathrm {csgn}\left (i \left (\left (\ln \relax (x )+\frac {19}{4}\right ) x +4 \ln \relax (x )+21\right )^{2}\right )^{2}+\pi \mathrm {csgn}\left (i \left (\left (\ln \relax (x )+\frac {19}{4}\right ) x +4 \ln \relax (x )+21\right )^{2}\right )^{3}-\pi \,\mathrm {csgn}\left (i \left (\left (\ln \relax (x )+\frac {19}{4}\right ) x +4 \ln \relax (x )+21\right )^{2}\right ) \mathrm {csgn}\left (\frac {i \left (\left (\ln \relax (x )+\frac {19}{4}\right ) x +4 \ln \relax (x )+21\right )^{2}}{\left (5+\ln \relax (x )\right )^{2}}\right )^{2}+\pi \mathrm {csgn}\left (\frac {i \left (\left (\ln \relax (x )+\frac {19}{4}\right ) x +4 \ln \relax (x )+21\right )^{2}}{\left (5+\ln \relax (x )\right )^{2}}\right )^{3}-\pi \,\mathrm {csgn}\left (\frac {i \left (\left (\ln \relax (x )+\frac {19}{4}\right ) x +4 \ln \relax (x )+21\right )^{2}}{\left (5+\ln \relax (x )\right )^{2}}\right ) \mathrm {csgn}\left (\frac {i \left (\left (\ln \relax (x )+\frac {19}{4}\right ) x +4 \ln \relax (x )+21\right )^{2}}{\left (4+x \right )^{2} \left (5+\ln \relax (x )\right )^{2}}\right )^{2}+\pi \mathrm {csgn}\left (\frac {i \left (\left (\ln \relax (x )+\frac {19}{4}\right ) x +4 \ln \relax (x )+21\right )^{2}}{\left (4+x \right )^{2} \left (5+\ln \relax (x )\right )^{2}}\right )^{3}-\pi \mathrm {csgn}\left (i \left (4+x \right )\right )^{2} \mathrm {csgn}\left (i \left (4+x \right )^{2}\right )+2 \pi \,\mathrm {csgn}\left (i \left (4+x \right )\right ) \mathrm {csgn}\left (i \left (4+x \right )^{2}\right )^{2}-\pi \mathrm {csgn}\left (i \left (4+x \right )^{2}\right )^{3}+12 i \ln \relax (2)-4 i \ln \left (5+\ln \relax (x )\right )-4 i \ln \left (4+x \right )\right )}{4}\right )\)	\(622\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((-16*x*ln(x)+2*x^2-80*x-32)/((4*x^3+32*x^2+64*x)*ln(x)^2+(39*x^3+320*x^2+656*x)*ln(x)+95*x^3+800*x^2+1680*
x)/ln(((64*x^2+512*x+1024)*ln(x)^2+(608*x^2+5120*x+10752)*ln(x)+1444*x^2+12768*x+28224)/((x^2+8*x+16)*ln(x)^2+
(10*x^2+80*x+160)*ln(x)+25*x^2+200*x+400)),x,method=_RETURNVERBOSE)

[Out]

ln(ln((ln(x)+19/4)*x+4*ln(x)+21)-1/4*I*(Pi*csgn(I/(4+x)^2)*csgn(I/(5+ln(x))^2*((ln(x)+19/4)*x+4*ln(x)+21)^2)*c
sgn(I/(4+x)^2*((ln(x)+19/4)*x+4*ln(x)+21)^2/(5+ln(x))^2)-Pi*csgn(I/(4+x)^2)*csgn(I/(4+x)^2*((ln(x)+19/4)*x+4*l
n(x)+21)^2/(5+ln(x))^2)^2+Pi*csgn(I/(5+ln(x))^2)*csgn(I*((ln(x)+19/4)*x+4*ln(x)+21)^2)*csgn(I/(5+ln(x))^2*((ln
(x)+19/4)*x+4*ln(x)+21)^2)-Pi*csgn(I/(5+ln(x))^2)*csgn(I/(5+ln(x))^2*((ln(x)+19/4)*x+4*ln(x)+21)^2)^2-Pi*csgn(
I*(5+ln(x)))^2*csgn(I*(5+ln(x))^2)+2*Pi*csgn(I*(5+ln(x)))*csgn(I*(5+ln(x))^2)^2-Pi*csgn(I*(5+ln(x))^2)^3+Pi*cs
gn(I*((ln(x)+19/4)*x+4*ln(x)+21))^2*csgn(I*((ln(x)+19/4)*x+4*ln(x)+21)^2)-2*Pi*csgn(I*((ln(x)+19/4)*x+4*ln(x)+
21))*csgn(I*((ln(x)+19/4)*x+4*ln(x)+21)^2)^2+Pi*csgn(I*((ln(x)+19/4)*x+4*ln(x)+21)^2)^3-Pi*csgn(I*((ln(x)+19/4
)*x+4*ln(x)+21)^2)*csgn(I/(5+ln(x))^2*((ln(x)+19/4)*x+4*ln(x)+21)^2)^2+Pi*csgn(I/(5+ln(x))^2*((ln(x)+19/4)*x+4
*ln(x)+21)^2)^3-Pi*csgn(I/(5+ln(x))^2*((ln(x)+19/4)*x+4*ln(x)+21)^2)*csgn(I/(4+x)^2*((ln(x)+19/4)*x+4*ln(x)+21
)^2/(5+ln(x))^2)^2+Pi*csgn(I/(4+x)^2*((ln(x)+19/4)*x+4*ln(x)+21)^2/(5+ln(x))^2)^3-Pi*csgn(I*(4+x))^2*csgn(I*(4
+x)^2)+2*Pi*csgn(I*(4+x))*csgn(I*(4+x)^2)^2-Pi*csgn(I*(4+x)^2)^3+12*I*ln(2)-4*I*ln(5+ln(x))-4*I*ln(4+x)))

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maxima [A]  time = 1.30, size = 30, normalized size = 1.25 \begin {gather*} \log \left (\log \relax (2) + \log \left (4 \, {\left (x + 4\right )} \log \relax (x) + 19 \, x + 84\right ) - \log \left (x + 4\right ) - \log \left (\log \relax (x) + 5\right )\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-16*x*log(x)+2*x^2-80*x-32)/((4*x^3+32*x^2+64*x)*log(x)^2+(39*x^3+320*x^2+656*x)*log(x)+95*x^3+800*
x^2+1680*x)/log(((64*x^2+512*x+1024)*log(x)^2+(608*x^2+5120*x+10752)*log(x)+1444*x^2+12768*x+28224)/((x^2+8*x+
16)*log(x)^2+(10*x^2+80*x+160)*log(x)+25*x^2+200*x+400)),x, algorithm="maxima")

[Out]

log(log(2) + log(4*(x + 4)*log(x) + 19*x + 84) - log(x + 4) - log(log(x) + 5))

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mupad [B]  time = 9.17, size = 79, normalized size = 3.29 \begin {gather*} \ln \left (\ln \left (\frac {12768\,x+{\ln \relax (x)}^2\,\left (64\,x^2+512\,x+1024\right )+\ln \relax (x)\,\left (608\,x^2+5120\,x+10752\right )+1444\,x^2+28224}{200\,x+\ln \relax (x)\,\left (10\,x^2+80\,x+160\right )+{\ln \relax (x)}^2\,\left (x^2+8\,x+16\right )+25\,x^2+400}\right )\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(-(80*x + 16*x*log(x) - 2*x^2 + 32)/(log((12768*x + log(x)^2*(512*x + 64*x^2 + 1024) + log(x)*(5120*x + 608
*x^2 + 10752) + 1444*x^2 + 28224)/(200*x + log(x)*(80*x + 10*x^2 + 160) + log(x)^2*(8*x + x^2 + 16) + 25*x^2 +
400))*(1680*x + log(x)^2*(64*x + 32*x^2 + 4*x^3) + 800*x^2 + 95*x^3 + log(x)*(656*x + 320*x^2 + 39*x^3))),x)

[Out]

log(log((12768*x + log(x)^2*(512*x + 64*x^2 + 1024) + log(x)*(5120*x + 608*x^2 + 10752) + 1444*x^2 + 28224)/(2
00*x + log(x)*(80*x + 10*x^2 + 160) + log(x)^2*(8*x + x^2 + 16) + 25*x^2 + 400)))

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sympy [B]  time = 1.76, size = 78, normalized size = 3.25 \begin {gather*} \log {\left (\log {\left (\frac {1444 x^{2} + 12768 x + \left (64 x^{2} + 512 x + 1024\right ) \log {\relax (x )}^{2} + \left (608 x^{2} + 5120 x + 10752\right ) \log {\relax (x )} + 28224}{25 x^{2} + 200 x + \left (x^{2} + 8 x + 16\right ) \log {\relax (x )}^{2} + \left (10 x^{2} + 80 x + 160\right ) \log {\relax (x )} + 400} \right )} \right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-16*x*ln(x)+2*x**2-80*x-32)/((4*x**3+32*x**2+64*x)*ln(x)**2+(39*x**3+320*x**2+656*x)*ln(x)+95*x**3+
800*x**2+1680*x)/ln(((64*x**2+512*x+1024)*ln(x)**2+(608*x**2+5120*x+10752)*ln(x)+1444*x**2+12768*x+28224)/((x*
*2+8*x+16)*ln(x)**2+(10*x**2+80*x+160)*ln(x)+25*x**2+200*x+400)),x)

[Out]

log(log((1444*x**2 + 12768*x + (64*x**2 + 512*x + 1024)*log(x)**2 + (608*x**2 + 5120*x + 10752)*log(x) + 28224
)/(25*x**2 + 200*x + (x**2 + 8*x + 16)*log(x)**2 + (10*x**2 + 80*x + 160)*log(x) + 400)))

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