3.102.90 \(\int \frac {-40 x-22 x^2+2 x^3+(8 x+6 x^2) \log (x^3)+(200+134 x-4 x^2+(-80-52 x+2 x^2) \log (x^3)+(8+6 x) \log ^2(x^3)) \log (32 x+16 x^2+2 x^3)+(-120-30 x+(24+6 x) \log (x^3)) \log ^2(32 x+16 x^2+2 x^3)}{4 x+x^2} \, dx\)

Optimal. Leaf size=20 \[ \left (x+\left (-5+\log \left (x^3\right )\right ) \log \left (2 x (4+x)^2\right )\right )^2 \]

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Rubi [F]  time = 7.22, 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 {-40 x-22 x^2+2 x^3+\left (8 x+6 x^2\right ) \log \left (x^3\right )+\left (200+134 x-4 x^2+\left (-80-52 x+2 x^2\right ) \log \left (x^3\right )+(8+6 x) \log ^2\left (x^3\right )\right ) \log \left (32 x+16 x^2+2 x^3\right )+\left (-120-30 x+(24+6 x) \log \left (x^3\right )\right ) \log ^2\left (32 x+16 x^2+2 x^3\right )}{4 x+x^2} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Int[(-40*x - 22*x^2 + 2*x^3 + (8*x + 6*x^2)*Log[x^3] + (200 + 134*x - 4*x^2 + (-80 - 52*x + 2*x^2)*Log[x^3] +
(8 + 6*x)*Log[x^3]^2)*Log[32*x + 16*x^2 + 2*x^3] + (-120 - 30*x + (24 + 6*x)*Log[x^3])*Log[32*x + 16*x^2 + 2*x
^3]^2)/(4*x + x^2),x]

[Out]

x^2 - 100*Log[4]*Log[x] - 100*Log[1 + x/4]*Log[x] - 25*Log[x]^2 + (20*Log[1 + x/4]*Log[x^3]^2)/3 + (10*Log[x^3
]^3)/27 - (4*Log[1 + x/4]*Log[x^3]^3)/9 - Log[x^3]^4/54 - 100*Log[4 + x]^2 + 30*Log[-1/4*x]*Log[(4 + x)^2]^2 -
10*Log[x^3]*Log[(4 + x)^2]^2 + 40*Log[1 + x/4]*Log[x^3]*(Log[x] + Log[(4 + x)^2] - Log[2*x*(4 + x)^2]) - 10*x
*Log[2*x*(4 + x)^2] + 50*Log[x]*Log[2*x*(4 + x)^2] + 2*x*Log[x^3]*Log[2*x*(4 + x)^2] - (10*Log[x^3]^2*Log[2*x*
(4 + x)^2])/3 + (2*Log[x^3]^3*Log[2*x*(4 + x)^2])/9 + 100*Log[4 + x]*Log[2*x*(4 + x)^2] + 40*Log[x^3]*PolyLog[
2, -1/4*x] - 4*Log[x^3]^2*PolyLog[2, -1/4*x] + 120*(Log[x] + Log[(4 + x)^2] - Log[2*x*(4 + x)^2])*PolyLog[2, -
1/4*x] + 120*Log[(4 + x)^2]*PolyLog[2, (4 + x)/4] - 120*PolyLog[3, -1/4*x] + 24*Log[x^3]*PolyLog[3, -1/4*x] -
240*PolyLog[3, (4 + x)/4] - 72*PolyLog[4, -1/4*x] - 40*Defer[Int][(Log[x]*Log[x^3])/(4 + x), x] + 4*Defer[Int]
[(Log[x^3]^2*Log[2*x*(4 + x)^2])/(4 + x), x] + 6*Defer[Int][((-5 + Log[x^3])*Log[2*x*(4 + x)^2]^2)/x, x]

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\int \frac {-40 x-22 x^2+2 x^3+\left (8 x+6 x^2\right ) \log \left (x^3\right )+\left (200+134 x-4 x^2+\left (-80-52 x+2 x^2\right ) \log \left (x^3\right )+(8+6 x) \log ^2\left (x^3\right )\right ) \log \left (32 x+16 x^2+2 x^3\right )+\left (-120-30 x+(24+6 x) \log \left (x^3\right )\right ) \log ^2\left (32 x+16 x^2+2 x^3\right )}{x (4+x)} \, dx\\ &=\int \left (\frac {2 \left (-20-11 x+x^2+4 \log \left (x^3\right )+3 x \log \left (x^3\right )\right )}{4+x}+\frac {2 \left (100+67 x-2 x^2-40 \log \left (x^3\right )-26 x \log \left (x^3\right )+x^2 \log \left (x^3\right )+4 \log ^2\left (x^3\right )+3 x \log ^2\left (x^3\right )\right ) \log \left (2 x (4+x)^2\right )}{x (4+x)}+\frac {6 \left (-5+\log \left (x^3\right )\right ) \log ^2\left (2 x (4+x)^2\right )}{x}\right ) \, dx\\ &=2 \int \frac {-20-11 x+x^2+4 \log \left (x^3\right )+3 x \log \left (x^3\right )}{4+x} \, dx+2 \int \frac {\left (100+67 x-2 x^2-40 \log \left (x^3\right )-26 x \log \left (x^3\right )+x^2 \log \left (x^3\right )+4 \log ^2\left (x^3\right )+3 x \log ^2\left (x^3\right )\right ) \log \left (2 x (4+x)^2\right )}{x (4+x)} \, dx+6 \int \frac {\left (-5+\log \left (x^3\right )\right ) \log ^2\left (2 x (4+x)^2\right )}{x} \, dx\\ &=2 \int \left (\frac {-20-11 x+x^2}{4+x}+\frac {(4+3 x) \log \left (x^3\right )}{4+x}\right ) \, dx+2 \int \frac {\left (100+67 x-2 x^2+\left (-40-26 x+x^2\right ) \log \left (x^3\right )+(4+3 x) \log ^2\left (x^3\right )\right ) \log \left (2 x (4+x)^2\right )}{x (4+x)} \, dx+6 \int \frac {\left (-5+\log \left (x^3\right )\right ) \log ^2\left (2 x (4+x)^2\right )}{x} \, dx\\ &=2 \int \frac {-20-11 x+x^2}{4+x} \, dx+2 \int \frac {(4+3 x) \log \left (x^3\right )}{4+x} \, dx+2 \int \left (\frac {\left (100+67 x-2 x^2-40 \log \left (x^3\right )-26 x \log \left (x^3\right )+x^2 \log \left (x^3\right )+4 \log ^2\left (x^3\right )+3 x \log ^2\left (x^3\right )\right ) \log \left (2 x (4+x)^2\right )}{4 x}-\frac {\left (100+67 x-2 x^2-40 \log \left (x^3\right )-26 x \log \left (x^3\right )+x^2 \log \left (x^3\right )+4 \log ^2\left (x^3\right )+3 x \log ^2\left (x^3\right )\right ) \log \left (2 x (4+x)^2\right )}{4 (4+x)}\right ) \, dx+6 \int \frac {\left (-5+\log \left (x^3\right )\right ) \log ^2\left (2 x (4+x)^2\right )}{x} \, dx\\ &=\frac {1}{2} \int \frac {\left (100+67 x-2 x^2-40 \log \left (x^3\right )-26 x \log \left (x^3\right )+x^2 \log \left (x^3\right )+4 \log ^2\left (x^3\right )+3 x \log ^2\left (x^3\right )\right ) \log \left (2 x (4+x)^2\right )}{x} \, dx-\frac {1}{2} \int \frac {\left (100+67 x-2 x^2-40 \log \left (x^3\right )-26 x \log \left (x^3\right )+x^2 \log \left (x^3\right )+4 \log ^2\left (x^3\right )+3 x \log ^2\left (x^3\right )\right ) \log \left (2 x (4+x)^2\right )}{4+x} \, dx+2 \int \left (-15+x+\frac {40}{4+x}\right ) \, dx+2 \int \left (3 \log \left (x^3\right )-\frac {8 \log \left (x^3\right )}{4+x}\right ) \, dx+6 \int \frac {\left (-5+\log \left (x^3\right )\right ) \log ^2\left (2 x (4+x)^2\right )}{x} \, dx\\ &=-30 x+x^2+80 \log (4+x)+\frac {1}{2} \int \frac {\left (100+67 x-2 x^2+\left (-40-26 x+x^2\right ) \log \left (x^3\right )+(4+3 x) \log ^2\left (x^3\right )\right ) \log \left (2 x (4+x)^2\right )}{x} \, dx-\frac {1}{2} \int \frac {\left (100+67 x-2 x^2+\left (-40-26 x+x^2\right ) \log \left (x^3\right )+(4+3 x) \log ^2\left (x^3\right )\right ) \log \left (2 x (4+x)^2\right )}{4+x} \, dx+6 \int \log \left (x^3\right ) \, dx+6 \int \frac {\left (-5+\log \left (x^3\right )\right ) \log ^2\left (2 x (4+x)^2\right )}{x} \, dx-16 \int \frac {\log \left (x^3\right )}{4+x} \, dx\\ &=-48 x+x^2+6 x \log \left (x^3\right )-16 \log \left (1+\frac {x}{4}\right ) \log \left (x^3\right )+80 \log (4+x)+\frac {1}{2} \int \left (67 \log \left (2 x (4+x)^2\right )+\frac {100 \log \left (2 x (4+x)^2\right )}{x}-2 x \log \left (2 x (4+x)^2\right )-26 \log \left (x^3\right ) \log \left (2 x (4+x)^2\right )-\frac {40 \log \left (x^3\right ) \log \left (2 x (4+x)^2\right )}{x}+x \log \left (x^3\right ) \log \left (2 x (4+x)^2\right )+3 \log ^2\left (x^3\right ) \log \left (2 x (4+x)^2\right )+\frac {4 \log ^2\left (x^3\right ) \log \left (2 x (4+x)^2\right )}{x}\right ) \, dx-\frac {1}{2} \int \left (\frac {100 \log \left (2 x (4+x)^2\right )}{4+x}+\frac {67 x \log \left (2 x (4+x)^2\right )}{4+x}-\frac {2 x^2 \log \left (2 x (4+x)^2\right )}{4+x}-\frac {40 \log \left (x^3\right ) \log \left (2 x (4+x)^2\right )}{4+x}-\frac {26 x \log \left (x^3\right ) \log \left (2 x (4+x)^2\right )}{4+x}+\frac {x^2 \log \left (x^3\right ) \log \left (2 x (4+x)^2\right )}{4+x}+\frac {4 \log ^2\left (x^3\right ) \log \left (2 x (4+x)^2\right )}{4+x}+\frac {3 x \log ^2\left (x^3\right ) \log \left (2 x (4+x)^2\right )}{4+x}\right ) \, dx+6 \int \frac {\left (-5+\log \left (x^3\right )\right ) \log ^2\left (2 x (4+x)^2\right )}{x} \, dx+48 \int \frac {\log \left (1+\frac {x}{4}\right )}{x} \, dx\\ &=-48 x+x^2+6 x \log \left (x^3\right )-16 \log \left (1+\frac {x}{4}\right ) \log \left (x^3\right )+80 \log (4+x)-48 \text {Li}_2\left (-\frac {x}{4}\right )+\frac {1}{2} \int x \log \left (x^3\right ) \log \left (2 x (4+x)^2\right ) \, dx-\frac {1}{2} \int \frac {x^2 \log \left (x^3\right ) \log \left (2 x (4+x)^2\right )}{4+x} \, dx+\frac {3}{2} \int \log ^2\left (x^3\right ) \log \left (2 x (4+x)^2\right ) \, dx-\frac {3}{2} \int \frac {x \log ^2\left (x^3\right ) \log \left (2 x (4+x)^2\right )}{4+x} \, dx+2 \int \frac {\log ^2\left (x^3\right ) \log \left (2 x (4+x)^2\right )}{x} \, dx-2 \int \frac {\log ^2\left (x^3\right ) \log \left (2 x (4+x)^2\right )}{4+x} \, dx+6 \int \frac {\left (-5+\log \left (x^3\right )\right ) \log ^2\left (2 x (4+x)^2\right )}{x} \, dx-13 \int \log \left (x^3\right ) \log \left (2 x (4+x)^2\right ) \, dx+13 \int \frac {x \log \left (x^3\right ) \log \left (2 x (4+x)^2\right )}{4+x} \, dx-20 \int \frac {\log \left (x^3\right ) \log \left (2 x (4+x)^2\right )}{x} \, dx+20 \int \frac {\log \left (x^3\right ) \log \left (2 x (4+x)^2\right )}{4+x} \, dx+\frac {67}{2} \int \log \left (2 x (4+x)^2\right ) \, dx-\frac {67}{2} \int \frac {x \log \left (2 x (4+x)^2\right )}{4+x} \, dx+50 \int \frac {\log \left (2 x (4+x)^2\right )}{x} \, dx-50 \int \frac {\log \left (2 x (4+x)^2\right )}{4+x} \, dx-\int x \log \left (2 x (4+x)^2\right ) \, dx+\int \frac {x^2 \log \left (2 x (4+x)^2\right )}{4+x} \, dx\\ &=\text {Rest of rules removed due to large latex content} \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.09, size = 20, normalized size = 1.00 \begin {gather*} \left (x+\left (-5+\log \left (x^3\right )\right ) \log \left (2 x (4+x)^2\right )\right )^2 \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(-40*x - 22*x^2 + 2*x^3 + (8*x + 6*x^2)*Log[x^3] + (200 + 134*x - 4*x^2 + (-80 - 52*x + 2*x^2)*Log[x
^3] + (8 + 6*x)*Log[x^3]^2)*Log[32*x + 16*x^2 + 2*x^3] + (-120 - 30*x + (24 + 6*x)*Log[x^3])*Log[32*x + 16*x^2
+ 2*x^3]^2)/(4*x + x^2),x]

[Out]

(x + (-5 + Log[x^3])*Log[2*x*(4 + x)^2])^2

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fricas [B]  time = 1.10, size = 63, normalized size = 3.15 \begin {gather*} {\left (\log \left (x^{3}\right )^{2} - 10 \, \log \left (x^{3}\right ) + 25\right )} \log \left (2 \, x^{3} + 16 \, x^{2} + 32 \, x\right )^{2} + x^{2} + 2 \, {\left (x \log \left (x^{3}\right ) - 5 \, x\right )} \log \left (2 \, x^{3} + 16 \, x^{2} + 32 \, x\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((((24+6*x)*log(x^3)-30*x-120)*log(2*x^3+16*x^2+32*x)^2+((6*x+8)*log(x^3)^2+(2*x^2-52*x-80)*log(x^3)-
4*x^2+134*x+200)*log(2*x^3+16*x^2+32*x)+(6*x^2+8*x)*log(x^3)+2*x^3-22*x^2-40*x)/(x^2+4*x),x, algorithm="fricas
")

[Out]

(log(x^3)^2 - 10*log(x^3) + 25)*log(2*x^3 + 16*x^2 + 32*x)^2 + x^2 + 2*(x*log(x^3) - 5*x)*log(2*x^3 + 16*x^2 +
32*x)

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giac [B]  time = 0.37, size = 109, normalized size = 5.45 \begin {gather*} 9 \, \log \relax (x)^{4} + 3 \, {\left (3 \, \log \relax (x)^{2} - 10 \, \log \relax (x)\right )} \log \left (2 \, x^{2} + 16 \, x + 32\right )^{2} + {\left (6 \, x + 25\right )} \log \relax (x)^{2} - 30 \, \log \relax (x)^{3} + x^{2} + 2 \, {\left (9 \, \log \relax (x)^{3} + 3 \, x \log \relax (x) - 30 \, \log \relax (x)^{2} - 5 \, x + 50 \, \log \left (x + 4\right ) + 25 \, \log \relax (x)\right )} \log \left (2 \, x^{2} + 16 \, x + 32\right ) - 100 \, \log \left (x + 4\right )^{2} - 10 \, x \log \relax (x) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((((24+6*x)*log(x^3)-30*x-120)*log(2*x^3+16*x^2+32*x)^2+((6*x+8)*log(x^3)^2+(2*x^2-52*x-80)*log(x^3)-
4*x^2+134*x+200)*log(2*x^3+16*x^2+32*x)+(6*x^2+8*x)*log(x^3)+2*x^3-22*x^2-40*x)/(x^2+4*x),x, algorithm="giac")

[Out]

9*log(x)^4 + 3*(3*log(x)^2 - 10*log(x))*log(2*x^2 + 16*x + 32)^2 + (6*x + 25)*log(x)^2 - 30*log(x)^3 + x^2 + 2
*(9*log(x)^3 + 3*x*log(x) - 30*log(x)^2 - 5*x + 50*log(x + 4) + 25*log(x))*log(2*x^2 + 16*x + 32) - 100*log(x
+ 4)^2 - 10*x*log(x)

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maple [F(-1)]  time = 180.00, size = 0, normalized size = 0.00 hanged

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((((24+6*x)*ln(x^3)-30*x-120)*ln(2*x^3+16*x^2+32*x)^2+((6*x+8)*ln(x^3)^2+(2*x^2-52*x-80)*ln(x^3)-4*x^2+134*
x+200)*ln(2*x^3+16*x^2+32*x)+(6*x^2+8*x)*ln(x^3)+2*x^3-22*x^2-40*x)/(x^2+4*x),x,method=_RETURNVERBOSE)

[Out]

int((((24+6*x)*ln(x^3)-30*x-120)*ln(2*x^3+16*x^2+32*x)^2+((6*x+8)*ln(x^3)^2+(2*x^2-52*x-80)*ln(x^3)-4*x^2+134*
x+200)*ln(2*x^3+16*x^2+32*x)+(6*x^2+8*x)*ln(x^3)+2*x^3-22*x^2-40*x)/(x^2+4*x),x,method=_RETURNVERBOSE)

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maxima [B]  time = 0.50, size = 146, normalized size = 7.30 \begin {gather*} 6 \, {\left (3 \, \log \relax (2) - 5\right )} \log \relax (x)^{3} + 9 \, \log \relax (x)^{4} + 4 \, {\left (9 \, \log \relax (x)^{2} - 30 \, \log \relax (x) + 25\right )} \log \left (x + 4\right )^{2} + {\left (9 \, \log \relax (2)^{2} + 6 \, x - 60 \, \log \relax (2) + 25\right )} \log \relax (x)^{2} + x^{2} - 10 \, x {\left (\log \relax (2) - 3\right )} + 4 \, {\left (3 \, {\left (3 \, \log \relax (2) - 10\right )} \log \relax (x)^{2} + 9 \, \log \relax (x)^{3} + {\left (3 \, x - 30 \, \log \relax (2) + 25\right )} \log \relax (x) - 5 \, x + 25 \, \log \relax (2) - 20\right )} \log \left (x + 4\right ) + 2 \, {\left (x {\left (3 \, \log \relax (2) - 5\right )} - 15 \, \log \relax (2)^{2} + 25 \, \log \relax (2)\right )} \log \relax (x) - 30 \, x + 80 \, \log \left (x + 4\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((((24+6*x)*log(x^3)-30*x-120)*log(2*x^3+16*x^2+32*x)^2+((6*x+8)*log(x^3)^2+(2*x^2-52*x-80)*log(x^3)-
4*x^2+134*x+200)*log(2*x^3+16*x^2+32*x)+(6*x^2+8*x)*log(x^3)+2*x^3-22*x^2-40*x)/(x^2+4*x),x, algorithm="maxima
")

[Out]

6*(3*log(2) - 5)*log(x)^3 + 9*log(x)^4 + 4*(9*log(x)^2 - 30*log(x) + 25)*log(x + 4)^2 + (9*log(2)^2 + 6*x - 60
*log(2) + 25)*log(x)^2 + x^2 - 10*x*(log(2) - 3) + 4*(3*(3*log(2) - 10)*log(x)^2 + 9*log(x)^3 + (3*x - 30*log(
2) + 25)*log(x) - 5*x + 25*log(2) - 20)*log(x + 4) + 2*(x*(3*log(2) - 5) - 15*log(2)^2 + 25*log(2))*log(x) - 3
0*x + 80*log(x + 4)

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mupad [B]  time = 6.74, size = 41, normalized size = 2.05 \begin {gather*} {\left (x-5\,\ln \left (2\,x^3+16\,x^2+32\,x\right )+\ln \left (2\,x^3+16\,x^2+32\,x\right )\,\ln \left (x^3\right )\right )}^2 \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(-(40*x - log(x^3)*(8*x + 6*x^2) - log(32*x + 16*x^2 + 2*x^3)*(134*x - log(x^3)*(52*x - 2*x^2 + 80) + log(x
^3)^2*(6*x + 8) - 4*x^2 + 200) + 22*x^2 - 2*x^3 + log(32*x + 16*x^2 + 2*x^3)^2*(30*x - log(x^3)*(6*x + 24) + 1
20))/(4*x + x^2),x)

[Out]

(x - 5*log(32*x + 16*x^2 + 2*x^3) + log(32*x + 16*x^2 + 2*x^3)*log(x^3))^2

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sympy [B]  time = 0.62, size = 61, normalized size = 3.05 \begin {gather*} x^{2} + \left (2 x \log {\left (x^{3} \right )} - 10 x\right ) \log {\left (2 x^{3} + 16 x^{2} + 32 x \right )} + \left (\log {\left (x^{3} \right )}^{2} - 10 \log {\left (x^{3} \right )} + 25\right ) \log {\left (2 x^{3} + 16 x^{2} + 32 x \right )}^{2} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((((24+6*x)*ln(x**3)-30*x-120)*ln(2*x**3+16*x**2+32*x)**2+((6*x+8)*ln(x**3)**2+(2*x**2-52*x-80)*ln(x*
*3)-4*x**2+134*x+200)*ln(2*x**3+16*x**2+32*x)+(6*x**2+8*x)*ln(x**3)+2*x**3-22*x**2-40*x)/(x**2+4*x),x)

[Out]

x**2 + (2*x*log(x**3) - 10*x)*log(2*x**3 + 16*x**2 + 32*x) + (log(x**3)**2 - 10*log(x**3) + 25)*log(2*x**3 + 1
6*x**2 + 32*x)**2

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