3.18.88 \(\int \frac {-160-8 x^3+(160+8 x^3) \log (\frac {1600-16000 x+40000 x^2-80 x^3+400 x^4+x^6}{400 x^2})}{-40 x+200 x^2+x^4} \, dx\)

Optimal. Leaf size=21 \[ \left (-1+\log \left (\left (10-\frac {2}{x}+\frac {x^2}{20}\right )^2\right )\right )^2 \]

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Rubi [F]  time = 13.28, 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 {-160-8 x^3+\left (160+8 x^3\right ) \log \left (\frac {1600-16000 x+40000 x^2-80 x^3+400 x^4+x^6}{400 x^2}\right )}{-40 x+200 x^2+x^4} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Int[(-160 - 8*x^3 + (160 + 8*x^3)*Log[(1600 - 16000*x + 40000*x^2 - 80*x^3 + 400*x^4 + x^6)/(400*x^2)])/(-40*x
+ 200*x^2 + x^4),x]

[Out]

(1200*15^(1/3)*(2/(9 + Sqrt[60081]))^(1/6)*(10*2^(2/3)*15^(1/3) - (9 + Sqrt[60081])^(2/3))*ArcTan[((2*(9 + Sqr
t[60081]))^(1/6)*((10*30^(2/3))/(9 + Sqrt[60081])^(1/3) - (15*(9 + Sqrt[60081]))^(1/3) - 3*2^(1/3)*x))/(3*Sqrt
[2^(1/3)*3^(1/6)*5^(2/3)*(3*Sqrt[3] + Sqrt[20027]) + 1000*2^(2/3)*(15/(9 + Sqrt[60081]))^(1/3) + 200*(9 + Sqrt
[60081])^(1/3)])])/(Sqrt[2^(1/3)*3^(1/6)*5^(2/3)*(3*Sqrt[3] + Sqrt[20027]) + 1000*2^(2/3)*(15/(9 + Sqrt[60081]
))^(1/3) + 200*(9 + Sqrt[60081])^(1/3)]*(300 - 3000*15^(1/3)*(2/(9 + Sqrt[60081]))^(2/3) - 2^(1/3)*(15*(9 + Sq
rt[60081]))^(2/3))) - (600*5^(1/3)*6^(5/6)*(20*(15/(9 + Sqrt[60081]))^(1/3) - (2*(9 + Sqrt[60081]))^(1/3))*Arc
Tan[(20*5^(2/3)*(3/(9 + Sqrt[60081]))^(1/3) - (10*(9 + Sqrt[60081]))^(1/3) - 6^(2/3)*x)/(6^(1/6)*Sqrt[600 + 30
00*15^(1/3)*(2/(9 + Sqrt[60081]))^(2/3) + 2^(1/3)*(15*(9 + Sqrt[60081]))^(2/3)])])/((300 - 3000*15^(1/3)*(2/(9
+ Sqrt[60081]))^(2/3) - 2^(1/3)*(15*(9 + Sqrt[60081]))^(2/3))*Sqrt[600 + 3000*15^(1/3)*(2/(9 + Sqrt[60081]))^
(2/3) + 2^(1/3)*(15*(9 + Sqrt[60081]))^(2/3)]) - 4*Log[x]^2 + (1200*Log[30^(1/3)*(20*(15/(9 + Sqrt[60081]))^(1
/3) - (2*(9 + Sqrt[60081]))^(1/3)) + 3*x])/(300 - 3000*15^(1/3)*(2/(9 + Sqrt[60081]))^(2/3) - 2^(1/3)*(15*(9 +
Sqrt[60081]))^(2/3)) - (4*(600 - 2^(1/3)*((3000*30^(1/3))/(9 + Sqrt[60081])^(2/3) + (15*(9 + Sqrt[60081]))^(2
/3)))*Log[30^(1/3)*(20*(15/(9 + Sqrt[60081]))^(1/3) - (2*(9 + Sqrt[60081]))^(1/3)) + 3*x])/(300 - 3000*15^(1/3
)*(2/(9 + Sqrt[60081]))^(2/3) - 2^(1/3)*(15*(9 + Sqrt[60081]))^(2/3)) - (600*Log[2*(300 + 3000*15^(1/3)*(2/(9
+ Sqrt[60081]))^(2/3) + 2^(1/3)*(15*(9 + Sqrt[60081]))^(2/3)) - 60*15^(2/3)*(2/(9 + Sqrt[60081]))^(1/3)*x + 3*
2^(2/3)*(15*(9 + Sqrt[60081]))^(1/3)*x + 9*x^2])/(300 - 3000*15^(1/3)*(2/(9 + Sqrt[60081]))^(2/3) - 2^(1/3)*(1
5*(9 + Sqrt[60081]))^(2/3)) - (4*(150 - 2^(1/3)*((3000*30^(1/3))/(9 + Sqrt[60081])^(2/3) + (15*(9 + Sqrt[60081
]))^(2/3)))*Log[2*(2^(1/3)*3^(1/6)*5^(2/3)*(3*Sqrt[3] + Sqrt[20027]) + 1000*2^(2/3)*(15/(9 + Sqrt[60081]))^(1/
3) + 100*(9 + Sqrt[60081])^(1/3)) - 20*2^(1/3)*15^(2/3)*x + 15^(1/3)*(2*(9 + Sqrt[60081]))^(2/3)*x + 3*(9 + Sq
rt[60081])^(1/3)*x^2])/(300 - 3000*15^(1/3)*(2/(9 + Sqrt[60081]))^(2/3) - 2^(1/3)*(15*(9 + Sqrt[60081]))^(2/3)
) + 4*Log[x]*(1 - Log[(40 - 200*x - x^3)^2/(400*x^2)]) + 1600*Defer[Int][Log[x]/(-40 + 200*x + x^3), x] + 24*D
efer[Int][(x^2*Log[x])/(-40 + 200*x + x^3), x] + 800*Defer[Int][Log[(-40 + 200*x + x^3)^2/(400*x^2)]/(-40 + 20
0*x + x^3), x] + 12*Defer[Int][(x^2*Log[(-40 + 200*x + x^3)^2/(400*x^2)])/(-40 + 200*x + x^3), x]

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\int \frac {-160-8 x^3+\left (160+8 x^3\right ) \log \left (\frac {1600-16000 x+40000 x^2-80 x^3+400 x^4+x^6}{400 x^2}\right )}{x \left (-40+200 x+x^3\right )} \, dx\\ &=\int \frac {8 \left (20+x^3\right ) \left (1-\log \left (\frac {\left (-40+200 x+x^3\right )^2}{400 x^2}\right )\right )}{x \left (40-200 x-x^3\right )} \, dx\\ &=8 \int \frac {\left (20+x^3\right ) \left (1-\log \left (\frac {\left (-40+200 x+x^3\right )^2}{400 x^2}\right )\right )}{x \left (40-200 x-x^3\right )} \, dx\\ &=8 \int \left (\frac {1-\log \left (\frac {\left (-40+200 x+x^3\right )^2}{400 x^2}\right )}{2 x}+\frac {\left (-200-3 x^2\right ) \left (1-\log \left (\frac {\left (-40+200 x+x^3\right )^2}{400 x^2}\right )\right )}{2 \left (-40+200 x+x^3\right )}\right ) \, dx\\ &=4 \int \frac {1-\log \left (\frac {\left (-40+200 x+x^3\right )^2}{400 x^2}\right )}{x} \, dx+4 \int \frac {\left (-200-3 x^2\right ) \left (1-\log \left (\frac {\left (-40+200 x+x^3\right )^2}{400 x^2}\right )\right )}{-40+200 x+x^3} \, dx\\ &=4 \log (x) \left (1-\log \left (\frac {\left (40-200 x-x^3\right )^2}{400 x^2}\right )\right )+4 \int \frac {400 x^2 \left (\frac {\left (200+3 x^2\right ) \left (-40+200 x+x^3\right )}{200 x^2}-\frac {\left (-40+200 x+x^3\right )^2}{200 x^3}\right ) \log (x)}{\left (-40+200 x+x^3\right )^2} \, dx+4 \int \left (-\frac {200 \left (1-\log \left (\frac {\left (-40+200 x+x^3\right )^2}{400 x^2}\right )\right )}{-40+200 x+x^3}-\frac {3 x^2 \left (1-\log \left (\frac {\left (-40+200 x+x^3\right )^2}{400 x^2}\right )\right )}{-40+200 x+x^3}\right ) \, dx\\ &=4 \log (x) \left (1-\log \left (\frac {\left (40-200 x-x^3\right )^2}{400 x^2}\right )\right )-12 \int \frac {x^2 \left (1-\log \left (\frac {\left (-40+200 x+x^3\right )^2}{400 x^2}\right )\right )}{-40+200 x+x^3} \, dx-800 \int \frac {1-\log \left (\frac {\left (-40+200 x+x^3\right )^2}{400 x^2}\right )}{-40+200 x+x^3} \, dx+1600 \int \frac {x^2 \left (\frac {\left (200+3 x^2\right ) \left (-40+200 x+x^3\right )}{200 x^2}-\frac {\left (-40+200 x+x^3\right )^2}{200 x^3}\right ) \log (x)}{\left (-40+200 x+x^3\right )^2} \, dx\\ &=4 \log (x) \left (1-\log \left (\frac {\left (40-200 x-x^3\right )^2}{400 x^2}\right )\right )-12 \int \left (\frac {x^2}{-40+200 x+x^3}-\frac {x^2 \log \left (\frac {\left (-40+200 x+x^3\right )^2}{400 x^2}\right )}{-40+200 x+x^3}\right ) \, dx-800 \int \left (\frac {1}{-40+200 x+x^3}-\frac {\log \left (\frac {\left (-40+200 x+x^3\right )^2}{400 x^2}\right )}{-40+200 x+x^3}\right ) \, dx+1600 \int \left (-\frac {\log (x)}{200 x}+\frac {\left (200+3 x^2\right ) \log (x)}{200 \left (-40+200 x+x^3\right )}\right ) \, dx\\ &=4 \log (x) \left (1-\log \left (\frac {\left (40-200 x-x^3\right )^2}{400 x^2}\right )\right )-8 \int \frac {\log (x)}{x} \, dx+8 \int \frac {\left (200+3 x^2\right ) \log (x)}{-40+200 x+x^3} \, dx-12 \int \frac {x^2}{-40+200 x+x^3} \, dx+12 \int \frac {x^2 \log \left (\frac {\left (-40+200 x+x^3\right )^2}{400 x^2}\right )}{-40+200 x+x^3} \, dx-800 \int \frac {1}{-40+200 x+x^3} \, dx+800 \int \frac {\log \left (\frac {\left (-40+200 x+x^3\right )^2}{400 x^2}\right )}{-40+200 x+x^3} \, dx\\ &=-4 \log ^2(x)+4 \log (x) \left (1-\log \left (\frac {\left (40-200 x-x^3\right )^2}{400 x^2}\right )\right )+8 \int \left (\frac {200 \log (x)}{-40+200 x+x^3}+\frac {3 x^2 \log (x)}{-40+200 x+x^3}\right ) \, dx-12 \int \frac {x^2}{\left (\frac {\sqrt [3]{10} \left (20 \sqrt [3]{\frac {15}{9+\sqrt {60081}}}-\sqrt [3]{2 \left (9+\sqrt {60081}\right )}\right )}{3^{2/3}}+x\right ) \left (\frac {2}{9} \left (300+3000 \sqrt [3]{15} \left (\frac {2}{9+\sqrt {60081}}\right )^{2/3}+\sqrt [3]{2} \left (15 \left (9+\sqrt {60081}\right )\right )^{2/3}\right )-\frac {\sqrt [3]{10} \left (20 \sqrt [3]{\frac {15}{9+\sqrt {60081}}}-\sqrt [3]{2 \left (9+\sqrt {60081}\right )}\right ) x}{3^{2/3}}+x^2\right )} \, dx+12 \int \frac {x^2 \log \left (\frac {\left (-40+200 x+x^3\right )^2}{400 x^2}\right )}{-40+200 x+x^3} \, dx-800 \int \frac {1}{\left (\frac {\sqrt [3]{10} \left (20 \sqrt [3]{\frac {15}{9+\sqrt {60081}}}-\sqrt [3]{2 \left (9+\sqrt {60081}\right )}\right )}{3^{2/3}}+x\right ) \left (\frac {2}{9} \left (300+3000 \sqrt [3]{15} \left (\frac {2}{9+\sqrt {60081}}\right )^{2/3}+\sqrt [3]{2} \left (15 \left (9+\sqrt {60081}\right )\right )^{2/3}\right )-\frac {\sqrt [3]{10} \left (20 \sqrt [3]{\frac {15}{9+\sqrt {60081}}}-\sqrt [3]{2 \left (9+\sqrt {60081}\right )}\right ) x}{3^{2/3}}+x^2\right )} \, dx+800 \int \frac {\log \left (\frac {\left (-40+200 x+x^3\right )^2}{400 x^2}\right )}{-40+200 x+x^3} \, dx\\ &=\text {Rest of rules removed due to large latex content} \end {aligned} \end {gather*}

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Mathematica [B]  time = 0.16, size = 49, normalized size = 2.33 \begin {gather*} 8 \left (-\frac {1}{4} \log \left (\frac {\left (-40+200 x+x^3\right )^2}{400 x^2}\right )+\frac {1}{8} \log ^2\left (\frac {\left (-40+200 x+x^3\right )^2}{400 x^2}\right )\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(-160 - 8*x^3 + (160 + 8*x^3)*Log[(1600 - 16000*x + 40000*x^2 - 80*x^3 + 400*x^4 + x^6)/(400*x^2)])/
(-40*x + 200*x^2 + x^4),x]

[Out]

8*(-1/4*Log[(-40 + 200*x + x^3)^2/(400*x^2)] + Log[(-40 + 200*x + x^3)^2/(400*x^2)]^2/8)

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fricas [B]  time = 0.64, size = 63, normalized size = 3.00 \begin {gather*} \log \left (\frac {x^{6} + 400 \, x^{4} - 80 \, x^{3} + 40000 \, x^{2} - 16000 \, x + 1600}{400 \, x^{2}}\right )^{2} - 2 \, \log \left (\frac {x^{6} + 400 \, x^{4} - 80 \, x^{3} + 40000 \, x^{2} - 16000 \, x + 1600}{400 \, x^{2}}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((8*x^3+160)*log(1/400*(x^6+400*x^4-80*x^3+40000*x^2-16000*x+1600)/x^2)-8*x^3-160)/(x^4+200*x^2-40*x
),x, algorithm="fricas")

[Out]

log(1/400*(x^6 + 400*x^4 - 80*x^3 + 40000*x^2 - 16000*x + 1600)/x^2)^2 - 2*log(1/400*(x^6 + 400*x^4 - 80*x^3 +
40000*x^2 - 16000*x + 1600)/x^2)

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giac [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int -\frac {8 \, {\left (x^{3} - {\left (x^{3} + 20\right )} \log \left (\frac {x^{6} + 400 \, x^{4} - 80 \, x^{3} + 40000 \, x^{2} - 16000 \, x + 1600}{400 \, x^{2}}\right ) + 20\right )}}{x^{4} + 200 \, x^{2} - 40 \, x}\,{d x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((8*x^3+160)*log(1/400*(x^6+400*x^4-80*x^3+40000*x^2-16000*x+1600)/x^2)-8*x^3-160)/(x^4+200*x^2-40*x
),x, algorithm="giac")

[Out]

integrate(-8*(x^3 - (x^3 + 20)*log(1/400*(x^6 + 400*x^4 - 80*x^3 + 40000*x^2 - 16000*x + 1600)/x^2) + 20)/(x^4
+ 200*x^2 - 40*x), x)

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maple [B]  time = 0.24, size = 48, normalized size = 2.29

method	result	size
norman	\(\ln \left (\frac {x^{6}+400 x^{4}-80 x^{3}+40000 x^{2}-16000 x +1600}{400 x^{2}}\right )^{2}-4 \ln \left (x^{3}+200 x -40\right )+4 \ln \relax (x )\)	\(48\)
default	\(-4 \ln \left (x^{3}+200 x -40\right )+4 \ln \relax (x )+8 \ln \left (20\right ) \ln \relax (x )-8 \ln \left (20\right ) \ln \left (x^{3}+200 x -40\right )+8 \left (\munderset {\underline {\hspace {1.25 ex}}\alpha =\RootOf \left (\textit {\_Z}^{3}+200 \textit {\_Z} -40\right )}{\sum }\left (\frac {\ln \left (x -\underline {\hspace {1.25 ex}}\alpha \right ) \ln \left (\frac {x^{6}+400 x^{4}-80 x^{3}+40000 x^{2}-16000 x +1600}{x^{2}}\right )}{2}-\ln \left (x -\underline {\hspace {1.25 ex}}\alpha \right ) \ln \left (\frac {i \sqrt {3 \underline {\hspace {1.25 ex}}\alpha ^{2}+800}-\underline {\hspace {1.25 ex}}\alpha -2 x}{i \sqrt {3 \underline {\hspace {1.25 ex}}\alpha ^{2}+800}-3 \underline {\hspace {1.25 ex}}\alpha }\right )-\ln \left (x -\underline {\hspace {1.25 ex}}\alpha \right ) \ln \left (\frac {i \sqrt {3 \underline {\hspace {1.25 ex}}\alpha ^{2}+800}+\underline {\hspace {1.25 ex}}\alpha +2 x}{i \sqrt {3 \underline {\hspace {1.25 ex}}\alpha ^{2}+800}+3 \underline {\hspace {1.25 ex}}\alpha }\right )-\dilog \left (\frac {i \sqrt {3 \underline {\hspace {1.25 ex}}\alpha ^{2}+800}-\underline {\hspace {1.25 ex}}\alpha -2 x}{i \sqrt {3 \underline {\hspace {1.25 ex}}\alpha ^{2}+800}-3 \underline {\hspace {1.25 ex}}\alpha }\right )-\dilog \left (\frac {i \sqrt {3 \underline {\hspace {1.25 ex}}\alpha ^{2}+800}+\underline {\hspace {1.25 ex}}\alpha +2 x}{i \sqrt {3 \underline {\hspace {1.25 ex}}\alpha ^{2}+800}+3 \underline {\hspace {1.25 ex}}\alpha }\right )+\dilog \left (\frac {x}{\underline {\hspace {1.25 ex}}\alpha }\right )+\ln \left (x -\underline {\hspace {1.25 ex}}\alpha \right ) \ln \left (\frac {x}{\underline {\hspace {1.25 ex}}\alpha }\right )-\frac {\ln \left (x -\underline {\hspace {1.25 ex}}\alpha \right )^{2}}{2}\right )\right )-4 \ln \relax (x ) \ln \left (\frac {x^{6}+400 x^{4}-80 x^{3}+40000 x^{2}-16000 x +1600}{x^{2}}\right )+8 \ln \relax (x ) \ln \left (\frac {\RootOf \left (\textit {\_Z}^{3}+200 \textit {\_Z} -40, \mathit {index} =1\right )-x}{\RootOf \left (\textit {\_Z}^{3}+200 \textit {\_Z} -40, \mathit {index} =1\right )}\right )+8 \dilog \left (\frac {\RootOf \left (\textit {\_Z}^{3}+200 \textit {\_Z} -40, \mathit {index} =1\right )-x}{\RootOf \left (\textit {\_Z}^{3}+200 \textit {\_Z} -40, \mathit {index} =1\right )}\right )+8 \ln \relax (x ) \ln \left (\frac {\RootOf \left (\textit {\_Z}^{3}+200 \textit {\_Z} -40, \mathit {index} =2\right )-x}{\RootOf \left (\textit {\_Z}^{3}+200 \textit {\_Z} -40, \mathit {index} =2\right )}\right )+8 \dilog \left (\frac {\RootOf \left (\textit {\_Z}^{3}+200 \textit {\_Z} -40, \mathit {index} =2\right )-x}{\RootOf \left (\textit {\_Z}^{3}+200 \textit {\_Z} -40, \mathit {index} =2\right )}\right )+8 \ln \relax (x ) \ln \left (\frac {\RootOf \left (\textit {\_Z}^{3}+200 \textit {\_Z} -40, \mathit {index} =3\right )-x}{\RootOf \left (\textit {\_Z}^{3}+200 \textit {\_Z} -40, \mathit {index} =3\right )}\right )+8 \dilog \left (\frac {\RootOf \left (\textit {\_Z}^{3}+200 \textit {\_Z} -40, \mathit {index} =3\right )-x}{\RootOf \left (\textit {\_Z}^{3}+200 \textit {\_Z} -40, \mathit {index} =3\right )}\right )-4 \ln \relax (x )^{2}\)	\(536\)
risch	\(\text {Expression too large to display}\)	\(4105\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((8*x^3+160)*ln(1/400*(x^6+400*x^4-80*x^3+40000*x^2-16000*x+1600)/x^2)-8*x^3-160)/(x^4+200*x^2-40*x),x,met
hod=_RETURNVERBOSE)

[Out]

ln(1/400*(x^6+400*x^4-80*x^3+40000*x^2-16000*x+1600)/x^2)^2-4*ln(x^3+200*x-40)+4*ln(x)

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maxima [B]  time = 0.48, size = 59, normalized size = 2.81 \begin {gather*} -4 \, {\left (2 \, \log \relax (5) + 4 \, \log \relax (2) + 2 \, \log \relax (x) + 1\right )} \log \left (x^{3} + 200 \, x - 40\right ) + 4 \, \log \left (x^{3} + 200 \, x - 40\right )^{2} + 4 \, {\left (2 \, \log \relax (5) + 4 \, \log \relax (2) + 1\right )} \log \relax (x) + 4 \, \log \relax (x)^{2} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((8*x^3+160)*log(1/400*(x^6+400*x^4-80*x^3+40000*x^2-16000*x+1600)/x^2)-8*x^3-160)/(x^4+200*x^2-40*x
),x, algorithm="maxima")

[Out]

-4*(2*log(5) + 4*log(2) + 2*log(x) + 1)*log(x^3 + 200*x - 40) + 4*log(x^3 + 200*x - 40)^2 + 4*(2*log(5) + 4*lo
g(2) + 1)*log(x) + 4*log(x)^2

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mupad [B]  time = 1.30, size = 46, normalized size = 2.19 \begin {gather*} {\ln \left (\frac {\frac {x^6}{400}+x^4-\frac {x^3}{5}+100\,x^2-40\,x+4}{x^2}\right )}^2-4\,\ln \left (x^3+200\,x-40\right )+4\,\ln \relax (x) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(-(8*x^3 - log((100*x^2 - 40*x - x^3/5 + x^4 + x^6/400 + 4)/x^2)*(8*x^3 + 160) + 160)/(200*x^2 - 40*x + x^4
),x)

[Out]

4*log(x) - 4*log(200*x + x^3 - 40) + log((100*x^2 - 40*x - x^3/5 + x^4 + x^6/400 + 4)/x^2)^2

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sympy [B]  time = 0.21, size = 46, normalized size = 2.19 \begin {gather*} 4 \log {\relax (x )} + \log {\left (\frac {\frac {x^{6}}{400} + x^{4} - \frac {x^{3}}{5} + 100 x^{2} - 40 x + 4}{x^{2}} \right )}^{2} - 4 \log {\left (x^{3} + 200 x - 40 \right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((8*x**3+160)*ln(1/400*(x**6+400*x**4-80*x**3+40000*x**2-16000*x+1600)/x**2)-8*x**3-160)/(x**4+200*x
**2-40*x),x)

[Out]

4*log(x) + log((x**6/400 + x**4 - x**3/5 + 100*x**2 - 40*x + 4)/x**2)**2 - 4*log(x**3 + 200*x - 40)

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