3.58.43 \(\int \frac {-800 x-40 x^5+(800 x+800 x^3-280 x^5-40 x^7) \log (x)+(-100-200 x^2+75 x^4+10 x^6) \log ^3(x)+(-80 x^5 \log (x)+20 x^4 \log ^3(x)) \log (\frac {-4 x^2+x \log ^2(x)}{\log ^2(x)})}{(-1600 x^2-160 x^6-4 x^{10}) \log (x)+(400 x+40 x^5+x^9) \log ^3(x)} \, dx\) [5743]

Optimal. Leaf size=33 \[ \frac {-4-x^2-\log \left (x-\frac {4 x^2}{\log ^2(x)}\right )}{4+\frac {x^4}{5}} \]

[Out]

(-4-x^2-ln(x-4*x^2/ln(x)^2))/(4+1/5*x^4)

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Rubi [F]
time = 7.63, 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 {-800 x-40 x^5+\left (800 x+800 x^3-280 x^5-40 x^7\right ) \log (x)+\left (-100-200 x^2+75 x^4+10 x^6\right ) \log ^3(x)+\left (-80 x^5 \log (x)+20 x^4 \log ^3(x)\right ) \log \left (\frac {-4 x^2+x \log ^2(x)}{\log ^2(x)}\right )}{\left (-1600 x^2-160 x^6-4 x^{10}\right ) \log (x)+\left (400 x+40 x^5+x^9\right ) \log ^3(x)} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Int[(-800*x - 40*x^5 + (800*x + 800*x^3 - 280*x^5 - 40*x^7)*Log[x] + (-100 - 200*x^2 + 75*x^4 + 10*x^6)*Log[x]
^3 + (-80*x^5*Log[x] + 20*x^4*Log[x]^3)*Log[(-4*x^2 + x*Log[x]^2)/Log[x]^2])/((-1600*x^2 - 160*x^6 - 4*x^10)*L
og[x] + (400*x + 40*x^5 + x^9)*Log[x]^3),x]

[Out]

-((x*(5*x - x^3))/(20 + x^4)) - Log[x]/4 + Log[20 + x^4]/16 - Log[-4*x + Log[x]^2]/4 + 200*Defer[Int][1/(x*(20
+ x^4)^2*Log[x]), x] + 10*Defer[Int][x^3/((20 + x^4)^2*Log[x]), x] - (1/4 - I/4)*5^(1/4)*Defer[Int][1/(((1 -
I)*5^(1/4) - x)*(4*x - Log[x]^2)), x] - (1/4 + I/4)*5^(1/4)*Defer[Int][1/(((1 + I)*5^(1/4) - x)*(4*x - Log[x]^
2)), x] - (1/4 - I/4)*5^(1/4)*Defer[Int][1/(((1 - I)*5^(1/4) + x)*(4*x - Log[x]^2)), x] - (1/4 + I/4)*5^(1/4)*
Defer[Int][1/(((1 + I)*5^(1/4) + x)*(4*x - Log[x]^2)), x] + Defer[Int][Log[x]/(((1 - I)*5^(1/4) - x)*(4*x - Lo
g[x]^2)), x]/8 + Defer[Int][Log[x]/(((1 + I)*5^(1/4) - x)*(4*x - Log[x]^2)), x]/8 - Defer[Int][Log[x]/(((1 - I
)*5^(1/4) + x)*(4*x - Log[x]^2)), x]/8 - Defer[Int][Log[x]/(((1 + I)*5^(1/4) + x)*(4*x - Log[x]^2)), x]/8 - De
fer[Int][(-4*x + Log[x]^2)^(-1), x] + 20*Defer[Int][(x^3*Log[x - (4*x^2)/Log[x]^2])/(20 + x^4)^2, x]

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\int \frac {800 x+40 x^5-\left (800 x+800 x^3-280 x^5-40 x^7\right ) \log (x)-\left (-100-200 x^2+75 x^4+10 x^6\right ) \log ^3(x)-\left (-80 x^5 \log (x)+20 x^4 \log ^3(x)\right ) \log \left (\frac {-4 x^2+x \log ^2(x)}{\log ^2(x)}\right )}{x \left (20+x^4\right )^2 \log (x) \left (4 x-\log ^2(x)\right )} \, dx\\ &=\int \left (\frac {40 \left (-20-20 x^2+7 x^4+x^6\right )}{\left (20+x^4\right )^2 \left (4 x-\log ^2(x)\right )}+\frac {800}{\left (20+x^4\right )^2 \log (x) \left (4 x-\log ^2(x)\right )}+\frac {40 x^4}{\left (20+x^4\right )^2 \log (x) \left (4 x-\log ^2(x)\right )}-\frac {5 \left (-20-40 x^2+15 x^4+2 x^6\right ) \log ^2(x)}{x \left (20+x^4\right )^2 \left (4 x-\log ^2(x)\right )}+\frac {20 x^3 \log \left (x-\frac {4 x^2}{\log ^2(x)}\right )}{\left (20+x^4\right )^2}\right ) \, dx\\ &=-\left (5 \int \frac {\left (-20-40 x^2+15 x^4+2 x^6\right ) \log ^2(x)}{x \left (20+x^4\right )^2 \left (4 x-\log ^2(x)\right )} \, dx\right )+20 \int \frac {x^3 \log \left (x-\frac {4 x^2}{\log ^2(x)}\right )}{\left (20+x^4\right )^2} \, dx+40 \int \frac {-20-20 x^2+7 x^4+x^6}{\left (20+x^4\right )^2 \left (4 x-\log ^2(x)\right )} \, dx+40 \int \frac {x^4}{\left (20+x^4\right )^2 \log (x) \left (4 x-\log ^2(x)\right )} \, dx+800 \int \frac {1}{\left (20+x^4\right )^2 \log (x) \left (4 x-\log ^2(x)\right )} \, dx\\ &=-\left (5 \int \left (\frac {20+40 x^2-15 x^4-2 x^6}{x \left (20+x^4\right )^2}+\frac {4 \left (-20-40 x^2+15 x^4+2 x^6\right )}{\left (20+x^4\right )^2 \left (4 x-\log ^2(x)\right )}\right ) \, dx\right )+20 \int \frac {x^3 \log \left (x-\frac {4 x^2}{\log ^2(x)}\right )}{\left (20+x^4\right )^2} \, dx+40 \int \left (-\frac {40 \left (4+x^2\right )}{\left (20+x^4\right )^2 \left (4 x-\log ^2(x)\right )}+\frac {7+x^2}{\left (20+x^4\right ) \left (4 x-\log ^2(x)\right )}\right ) \, dx+40 \int \left (\frac {x^3}{4 \left (20+x^4\right )^2 \log (x)}+\frac {x^3 \log (x)}{4 \left (20+x^4\right )^2 \left (4 x-\log ^2(x)\right )}\right ) \, dx+800 \int \left (\frac {1}{4 x \left (20+x^4\right )^2 \log (x)}+\frac {\log (x)}{4 x \left (20+x^4\right )^2 \left (4 x-\log ^2(x)\right )}\right ) \, dx\\ &=-\left (5 \int \frac {20+40 x^2-15 x^4-2 x^6}{x \left (20+x^4\right )^2} \, dx\right )+10 \int \frac {x^3}{\left (20+x^4\right )^2 \log (x)} \, dx+10 \int \frac {x^3 \log (x)}{\left (20+x^4\right )^2 \left (4 x-\log ^2(x)\right )} \, dx-20 \int \frac {-20-40 x^2+15 x^4+2 x^6}{\left (20+x^4\right )^2 \left (4 x-\log ^2(x)\right )} \, dx+20 \int \frac {x^3 \log \left (x-\frac {4 x^2}{\log ^2(x)}\right )}{\left (20+x^4\right )^2} \, dx+40 \int \frac {7+x^2}{\left (20+x^4\right ) \left (4 x-\log ^2(x)\right )} \, dx+200 \int \frac {1}{x \left (20+x^4\right )^2 \log (x)} \, dx+200 \int \frac {\log (x)}{x \left (20+x^4\right )^2 \left (4 x-\log ^2(x)\right )} \, dx-1600 \int \frac {4+x^2}{\left (20+x^4\right )^2 \left (4 x-\log ^2(x)\right )} \, dx\\ &=-\frac {x \left (5 x-x^3\right )}{20+x^4}+\frac {1}{16} \int -\frac {80}{x \left (20+x^4\right )} \, dx+10 \int \frac {x^3}{\left (20+x^4\right )^2 \log (x)} \, dx+10 \int \frac {x^3 \log (x)}{\left (20+x^4\right )^2 \left (4 x-\log ^2(x)\right )} \, dx-20 \int \left (-\frac {80 \left (4+x^2\right )}{\left (20+x^4\right )^2 \left (4 x-\log ^2(x)\right )}+\frac {15+2 x^2}{\left (20+x^4\right ) \left (4 x-\log ^2(x)\right )}\right ) \, dx+20 \int \frac {x^3 \log \left (x-\frac {4 x^2}{\log ^2(x)}\right )}{\left (20+x^4\right )^2} \, dx+40 \int \left (\frac {7}{\left (20+x^4\right ) \left (4 x-\log ^2(x)\right )}+\frac {x^2}{\left (20+x^4\right ) \left (4 x-\log ^2(x)\right )}\right ) \, dx+200 \int \frac {1}{x \left (20+x^4\right )^2 \log (x)} \, dx+200 \int \left (-\frac {x^3 \log (x)}{20 \left (20+x^4\right )^2 \left (4 x-\log ^2(x)\right )}-\frac {x^3 \log (x)}{400 \left (20+x^4\right ) \left (4 x-\log ^2(x)\right )}-\frac {\log (x)}{400 x \left (-4 x+\log ^2(x)\right )}\right ) \, dx-1600 \int \left (\frac {4}{\left (20+x^4\right )^2 \left (4 x-\log ^2(x)\right )}+\frac {x^2}{\left (20+x^4\right )^2 \left (4 x-\log ^2(x)\right )}\right ) \, dx\\ &=-\frac {x \left (5 x-x^3\right )}{20+x^4}-\frac {1}{2} \int \frac {x^3 \log (x)}{\left (20+x^4\right ) \left (4 x-\log ^2(x)\right )} \, dx-\frac {1}{2} \int \frac {\log (x)}{x \left (-4 x+\log ^2(x)\right )} \, dx-5 \int \frac {1}{x \left (20+x^4\right )} \, dx+10 \int \frac {x^3}{\left (20+x^4\right )^2 \log (x)} \, dx-20 \int \frac {15+2 x^2}{\left (20+x^4\right ) \left (4 x-\log ^2(x)\right )} \, dx+20 \int \frac {x^3 \log \left (x-\frac {4 x^2}{\log ^2(x)}\right )}{\left (20+x^4\right )^2} \, dx+40 \int \frac {x^2}{\left (20+x^4\right ) \left (4 x-\log ^2(x)\right )} \, dx+200 \int \frac {1}{x \left (20+x^4\right )^2 \log (x)} \, dx+280 \int \frac {1}{\left (20+x^4\right ) \left (4 x-\log ^2(x)\right )} \, dx-1600 \int \frac {x^2}{\left (20+x^4\right )^2 \left (4 x-\log ^2(x)\right )} \, dx+1600 \int \frac {4+x^2}{\left (20+x^4\right )^2 \left (4 x-\log ^2(x)\right )} \, dx-6400 \int \frac {1}{\left (20+x^4\right )^2 \left (4 x-\log ^2(x)\right )} \, dx\\ &=-\frac {x \left (5 x-x^3\right )}{20+x^4}-\frac {1}{4} \log \left (-4 x+\log ^2(x)\right )-\frac {1}{2} \int \left (\frac {x \log (x)}{2 \left (-2 i \sqrt {5}+x^2\right ) \left (4 x-\log ^2(x)\right )}+\frac {x \log (x)}{2 \left (2 i \sqrt {5}+x^2\right ) \left (4 x-\log ^2(x)\right )}\right ) \, dx-\frac {5}{4} \text {Subst}\left (\int \frac {1}{x (20+x)} \, dx,x,x^4\right )+10 \int \frac {x^3}{\left (20+x^4\right )^2 \log (x)} \, dx-20 \int \left (\frac {15}{\left (20+x^4\right ) \left (4 x-\log ^2(x)\right )}+\frac {2 x^2}{\left (20+x^4\right ) \left (4 x-\log ^2(x)\right )}\right ) \, dx+20 \int \frac {x^3 \log \left (x-\frac {4 x^2}{\log ^2(x)}\right )}{\left (20+x^4\right )^2} \, dx+40 \int \left (-\frac {1}{2 \left (2 i \sqrt {5}-x^2\right ) \left (4 x-\log ^2(x)\right )}+\frac {1}{2 \left (2 i \sqrt {5}+x^2\right ) \left (4 x-\log ^2(x)\right )}\right ) \, dx+200 \int \frac {1}{x \left (20+x^4\right )^2 \log (x)} \, dx+280 \int \left (\frac {i}{4 \sqrt {5} \left (2 i \sqrt {5}-x^2\right ) \left (4 x-\log ^2(x)\right )}+\frac {i}{4 \sqrt {5} \left (2 i \sqrt {5}+x^2\right ) \left (4 x-\log ^2(x)\right )}\right ) \, dx-1600 \int \frac {x^2}{\left (20+x^4\right )^2 \left (4 x-\log ^2(x)\right )} \, dx+1600 \int \left (\frac {4}{\left (20+x^4\right )^2 \left (4 x-\log ^2(x)\right )}+\frac {x^2}{\left (20+x^4\right )^2 \left (4 x-\log ^2(x)\right )}\right ) \, dx-6400 \int \frac {1}{\left (20+x^4\right )^2 \left (4 x-\log ^2(x)\right )} \, dx-\int \frac {1}{-4 x+\log ^2(x)} \, dx\\ &=-\frac {x \left (5 x-x^3\right )}{20+x^4}-\frac {1}{4} \log \left (-4 x+\log ^2(x)\right )-\frac {1}{16} \text {Subst}\left (\int \frac {1}{x} \, dx,x,x^4\right )+\frac {1}{16} \text {Subst}\left (\int \frac {1}{20+x} \, dx,x,x^4\right )-\frac {1}{4} \int \frac {x \log (x)}{\left (-2 i \sqrt {5}+x^2\right ) \left (4 x-\log ^2(x)\right )} \, dx-\frac {1}{4} \int \frac {x \log (x)}{\left (2 i \sqrt {5}+x^2\right ) \left (4 x-\log ^2(x)\right )} \, dx+10 \int \frac {x^3}{\left (20+x^4\right )^2 \log (x)} \, dx-20 \int \frac {1}{\left (2 i \sqrt {5}-x^2\right ) \left (4 x-\log ^2(x)\right )} \, dx+20 \int \frac {1}{\left (2 i \sqrt {5}+x^2\right ) \left (4 x-\log ^2(x)\right )} \, dx+20 \int \frac {x^3 \log \left (x-\frac {4 x^2}{\log ^2(x)}\right )}{\left (20+x^4\right )^2} \, dx-40 \int \frac {x^2}{\left (20+x^4\right ) \left (4 x-\log ^2(x)\right )} \, dx+200 \int \frac {1}{x \left (20+x^4\right )^2 \log (x)} \, dx-300 \int \frac {1}{\left (20+x^4\right ) \left (4 x-\log ^2(x)\right )} \, dx+\left (14 i \sqrt {5}\right ) \int \frac {1}{\left (2 i \sqrt {5}-x^2\right ) \left (4 x-\log ^2(x)\right )} \, dx+\left (14 i \sqrt {5}\right ) \int \frac {1}{\left (2 i \sqrt {5}+x^2\right ) \left (4 x-\log ^2(x)\right )} \, dx-\int \frac {1}{-4 x+\log ^2(x)} \, dx\\ &=\text {Rest of rules removed due to large latex content} \end {aligned} \end {gather*}

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Mathematica [A]
time = 0.08, size = 26, normalized size = 0.79 \begin {gather*} -\frac {5 \left (4+x^2+\log \left (x-\frac {4 x^2}{\log ^2(x)}\right )\right )}{20+x^4} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(-800*x - 40*x^5 + (800*x + 800*x^3 - 280*x^5 - 40*x^7)*Log[x] + (-100 - 200*x^2 + 75*x^4 + 10*x^6)*
Log[x]^3 + (-80*x^5*Log[x] + 20*x^4*Log[x]^3)*Log[(-4*x^2 + x*Log[x]^2)/Log[x]^2])/((-1600*x^2 - 160*x^6 - 4*x
^10)*Log[x] + (400*x + 40*x^5 + x^9)*Log[x]^3),x]

[Out]

(-5*(4 + x^2 + Log[x - (4*x^2)/Log[x]^2]))/(20 + x^4)

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Maple [C] Result contains higher order function than in optimal. Order 9 vs. order 3.
time = 15.15, size = 414, normalized size = 12.55

method	result	size
risch	\(-\frac {5 \ln \left (-\frac {\ln \left (x \right )^{2}}{4}+x \right )}{x^{4}+20}-\frac {5 \left (-2 i \pi \mathrm {csgn}\left (\frac {i \left (\frac {\ln \left (x \right )^{2}}{4}-x \right ) x}{\ln \left (x \right )^{2}}\right )^{2}-2 i \pi \,\mathrm {csgn}\left (i \ln \left (x \right )\right ) \mathrm {csgn}\left (i \ln \left (x \right )^{2}\right )^{2}-i \pi \mathrm {csgn}\left (\frac {i \left (\frac {\ln \left (x \right )^{2}}{4}-x \right )}{\ln \left (x \right )^{2}}\right )^{2} \mathrm {csgn}\left (i \left (\frac {\ln \left (x \right )^{2}}{4}-x \right )\right )+i \pi \,\mathrm {csgn}\left (i x \right ) \mathrm {csgn}\left (\frac {i \left (\frac {\ln \left (x \right )^{2}}{4}-x \right ) x}{\ln \left (x \right )^{2}}\right )^{2}-i \pi \,\mathrm {csgn}\left (\frac {i}{\ln \left (x \right )^{2}}\right ) \mathrm {csgn}\left (\frac {i \left (\frac {\ln \left (x \right )^{2}}{4}-x \right )}{\ln \left (x \right )^{2}}\right ) \mathrm {csgn}\left (i \left (\frac {\ln \left (x \right )^{2}}{4}-x \right )\right )-i \pi \,\mathrm {csgn}\left (i x \right ) \mathrm {csgn}\left (\frac {i \left (\frac {\ln \left (x \right )^{2}}{4}-x \right )}{\ln \left (x \right )^{2}}\right ) \mathrm {csgn}\left (\frac {i \left (\frac {\ln \left (x \right )^{2}}{4}-x \right ) x}{\ln \left (x \right )^{2}}\right )+2 i \pi -i \pi \mathrm {csgn}\left (\frac {i \left (\frac {\ln \left (x \right )^{2}}{4}-x \right ) x}{\ln \left (x \right )^{2}}\right )^{3}+i \pi \,\mathrm {csgn}\left (\frac {i}{\ln \left (x \right )^{2}}\right ) \mathrm {csgn}\left (\frac {i \left (\frac {\ln \left (x \right )^{2}}{4}-x \right )}{\ln \left (x \right )^{2}}\right )^{2}-i \pi \,\mathrm {csgn}\left (\frac {i \left (\frac {\ln \left (x \right )^{2}}{4}-x \right )}{\ln \left (x \right )^{2}}\right ) \mathrm {csgn}\left (\frac {i \left (\frac {\ln \left (x \right )^{2}}{4}-x \right ) x}{\ln \left (x \right )^{2}}\right )^{2}+i \pi \mathrm {csgn}\left (i \ln \left (x \right )^{2}\right )^{3}+i \pi \mathrm {csgn}\left (i \ln \left (x \right )\right )^{2} \mathrm {csgn}\left (i \ln \left (x \right )^{2}\right )+i \pi \mathrm {csgn}\left (\frac {i \left (\frac {\ln \left (x \right )^{2}}{4}-x \right )}{\ln \left (x \right )^{2}}\right )^{3}+8+2 x^{2}+4 \ln \left (2\right )+2 \ln \left (x \right )-4 \ln \left (\ln \left (x \right )\right )\right )}{2 \left (x^{4}+20\right )}\)	\(414\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((20*x^4*ln(x)^3-80*x^5*ln(x))*ln((x*ln(x)^2-4*x^2)/ln(x)^2)+(10*x^6+75*x^4-200*x^2-100)*ln(x)^3+(-40*x^7-
280*x^5+800*x^3+800*x)*ln(x)-40*x^5-800*x)/((x^9+40*x^5+400*x)*ln(x)^3+(-4*x^10-160*x^6-1600*x^2)*ln(x)),x,met
hod=_RETURNVERBOSE)

[Out]

-5/(x^4+20)*ln(-1/4*ln(x)^2+x)-5/2*(-2*I*Pi*csgn(I*(1/4*ln(x)^2-x)/ln(x)^2*x)^2-2*I*Pi*csgn(I*ln(x))*csgn(I*ln
(x)^2)^2-I*Pi*csgn(I/ln(x)^2*(1/4*ln(x)^2-x))^2*csgn(I*(1/4*ln(x)^2-x))+I*Pi*csgn(I*x)*csgn(I*(1/4*ln(x)^2-x)/
ln(x)^2*x)^2-I*Pi*csgn(I/ln(x)^2)*csgn(I/ln(x)^2*(1/4*ln(x)^2-x))*csgn(I*(1/4*ln(x)^2-x))-I*Pi*csgn(I*x)*csgn(
I/ln(x)^2*(1/4*ln(x)^2-x))*csgn(I*(1/4*ln(x)^2-x)/ln(x)^2*x)+2*I*Pi-I*Pi*csgn(I*(1/4*ln(x)^2-x)/ln(x)^2*x)^3+I
*Pi*csgn(I/ln(x)^2)*csgn(I/ln(x)^2*(1/4*ln(x)^2-x))^2-I*Pi*csgn(I/ln(x)^2*(1/4*ln(x)^2-x))*csgn(I*(1/4*ln(x)^2
-x)/ln(x)^2*x)^2+I*Pi*csgn(I*ln(x)^2)^3+I*Pi*csgn(I*ln(x))^2*csgn(I*ln(x)^2)+I*Pi*csgn(I/ln(x)^2*(1/4*ln(x)^2-
x))^3+8+2*x^2+4*ln(2)+2*ln(x)-4*ln(ln(x)))/(x^4+20)

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Maxima [A]
time = 0.66, size = 30, normalized size = 0.91 \begin {gather*} -\frac {5 \, {\left (x^{2} + \log \left (\log \left (x\right )^{2} - 4 \, x\right ) + \log \left (x\right ) - 2 \, \log \left (\log \left (x\right )\right ) + 4\right )}}{x^{4} + 20} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((20*x^4*log(x)^3-80*x^5*log(x))*log((x*log(x)^2-4*x^2)/log(x)^2)+(10*x^6+75*x^4-200*x^2-100)*log(x)
^3+(-40*x^7-280*x^5+800*x^3+800*x)*log(x)-40*x^5-800*x)/((x^9+40*x^5+400*x)*log(x)^3+(-4*x^10-160*x^6-1600*x^2
)*log(x)),x, algorithm="maxima")

[Out]

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

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Fricas [A]
time = 0.36, size = 32, normalized size = 0.97 \begin {gather*} -\frac {5 \, {\left (x^{2} + \log \left (\frac {x \log \left (x\right )^{2} - 4 \, x^{2}}{\log \left (x\right )^{2}}\right ) + 4\right )}}{x^{4} + 20} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((20*x^4*log(x)^3-80*x^5*log(x))*log((x*log(x)^2-4*x^2)/log(x)^2)+(10*x^6+75*x^4-200*x^2-100)*log(x)
^3+(-40*x^7-280*x^5+800*x^3+800*x)*log(x)-40*x^5-800*x)/((x^9+40*x^5+400*x)*log(x)^3+(-4*x^10-160*x^6-1600*x^2
)*log(x)),x, algorithm="fricas")

[Out]

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

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Sympy [A]
time = 0.23, size = 37, normalized size = 1.12 \begin {gather*} \frac {- 5 x^{2} - 20}{x^{4} + 20} - \frac {5 \log {\left (\frac {- 4 x^{2} + x \log {\left (x \right )}^{2}}{\log {\left (x \right )}^{2}} \right )}}{x^{4} + 20} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((20*x**4*ln(x)**3-80*x**5*ln(x))*ln((x*ln(x)**2-4*x**2)/ln(x)**2)+(10*x**6+75*x**4-200*x**2-100)*ln
(x)**3+(-40*x**7-280*x**5+800*x**3+800*x)*ln(x)-40*x**5-800*x)/((x**9+40*x**5+400*x)*ln(x)**3+(-4*x**10-160*x*
*6-1600*x**2)*ln(x)),x)

[Out]

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

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Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 58 vs. \(2 (26) = 52\).
time = 0.44, size = 58, normalized size = 1.76 \begin {gather*} -\frac {5 \, {\left (x^{2} + 4\right )}}{x^{4} + 20} - \frac {5 \, \log \left (\log \left (x\right )^{2} - 4 \, x\right )}{x^{4} + 20} + \frac {5 \, \log \left (\log \left (x\right )^{2}\right )}{x^{4} + 20} - \frac {5 \, \log \left (x\right )}{x^{4} + 20} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((20*x^4*log(x)^3-80*x^5*log(x))*log((x*log(x)^2-4*x^2)/log(x)^2)+(10*x^6+75*x^4-200*x^2-100)*log(x)
^3+(-40*x^7-280*x^5+800*x^3+800*x)*log(x)-40*x^5-800*x)/((x^9+40*x^5+400*x)*log(x)^3+(-4*x^10-160*x^6-1600*x^2
)*log(x)),x, algorithm="giac")

[Out]

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

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Mupad [B]
time = 3.91, size = 44, normalized size = 1.33 \begin {gather*} -\frac {5\,x^2+20}{x^4+20}-\frac {5\,\ln \left (\frac {x\,{\ln \left (x\right )}^2-4\,x^2}{{\ln \left (x\right )}^2}\right )}{x^4+20} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((800*x + log((x*log(x)^2 - 4*x^2)/log(x)^2)*(80*x^5*log(x) - 20*x^4*log(x)^3) - log(x)*(800*x + 800*x^3 -
280*x^5 - 40*x^7) + log(x)^3*(200*x^2 - 75*x^4 - 10*x^6 + 100) + 40*x^5)/(log(x)*(1600*x^2 + 160*x^6 + 4*x^10)
- log(x)^3*(400*x + 40*x^5 + x^9)),x)

[Out]

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

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