3.15.71 \(\int \frac {16-76 x-52 x^2+22 x^4+8 x^5+(-40 x-26 x^2-4 x^3+10 x^4+4 x^5) \log (\frac {4+x-x^3}{x})+(-8 x-2 x^2+2 x^4+(-4 x-x^2+x^4) \log (\frac {4+x-x^3}{x})) \log (2+\log (\frac {4+x-x^3}{x}))}{-8 x-2 x^2+2 x^4+(-4 x-x^2+x^4) \log (\frac {4+x-x^3}{x})} \, dx\)

Optimal. Leaf size=25 \[ (4+x) \left (2+2 x+\log \left (2+\log \left (\frac {4+x-x^3}{x}\right )\right )\right ) \]

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Rubi [F]  time = 2.47, 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 {16-76 x-52 x^2+22 x^4+8 x^5+\left (-40 x-26 x^2-4 x^3+10 x^4+4 x^5\right ) \log \left (\frac {4+x-x^3}{x}\right )+\left (-8 x-2 x^2+2 x^4+\left (-4 x-x^2+x^4\right ) \log \left (\frac {4+x-x^3}{x}\right )\right ) \log \left (2+\log \left (\frac {4+x-x^3}{x}\right )\right )}{-8 x-2 x^2+2 x^4+\left (-4 x-x^2+x^4\right ) \log \left (\frac {4+x-x^3}{x}\right )} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Int[(16 - 76*x - 52*x^2 + 22*x^4 + 8*x^5 + (-40*x - 26*x^2 - 4*x^3 + 10*x^4 + 4*x^5)*Log[(4 + x - x^3)/x] + (-
8*x - 2*x^2 + 2*x^4 + (-4*x - x^2 + x^4)*Log[(4 + x - x^3)/x])*Log[2 + Log[(4 + x - x^3)/x]])/(-8*x - 2*x^2 +
2*x^4 + (-4*x - x^2 + x^4)*Log[(4 + x - x^3)/x]),x]

[Out]

22*Defer[Int][(2 + Log[(4 + x - x^3)/x])^(-1), x] - 4*Defer[Int][1/(x*(2 + Log[(4 + x - x^3)/x])), x] + 8*Defe
r[Int][x/(2 + Log[(4 + x - x^3)/x]), x] + 8*Defer[Int][1/((-4 - x + x^3)*(2 + Log[(4 + x - x^3)/x])), x] + 2*D
efer[Int][x/((-4 - x + x^3)*(2 + Log[(4 + x - x^3)/x])), x] + 12*Defer[Int][x^2/((-4 - x + x^3)*(2 + Log[(4 +
x - x^3)/x])), x] + 10*Defer[Int][Log[1 + 4/x - x^2]/(2 + Log[(4 + x - x^3)/x]), x] + 4*Defer[Int][(x*Log[1 +
4/x - x^2])/(2 + Log[(4 + x - x^3)/x]), x] + Defer[Int][Log[2 + Log[(4 + x - x^3)/x]], x]

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\int \frac {-16+76 x+52 x^2-22 x^4-8 x^5-\left (-40 x-26 x^2-4 x^3+10 x^4+4 x^5\right ) \log \left (\frac {4+x-x^3}{x}\right )-\left (-8 x-2 x^2+2 x^4+\left (-4 x-x^2+x^4\right ) \log \left (\frac {4+x-x^3}{x}\right )\right ) \log \left (2+\log \left (\frac {4+x-x^3}{x}\right )\right )}{x \left (4+x-x^3\right ) \left (2+\log \left (\frac {4+x-x^3}{x}\right )\right )} \, dx\\ &=\int \left (-\frac {76}{\left (-4-x+x^3\right ) \left (2+\log \left (\frac {4+x-x^3}{x}\right )\right )}+\frac {16}{x \left (-4-x+x^3\right ) \left (2+\log \left (\frac {4+x-x^3}{x}\right )\right )}-\frac {52 x}{\left (-4-x+x^3\right ) \left (2+\log \left (\frac {4+x-x^3}{x}\right )\right )}+\frac {22 x^3}{\left (-4-x+x^3\right ) \left (2+\log \left (\frac {4+x-x^3}{x}\right )\right )}+\frac {8 x^4}{\left (-4-x+x^3\right ) \left (2+\log \left (\frac {4+x-x^3}{x}\right )\right )}+\frac {2 (5+2 x) \log \left (1+\frac {4}{x}-x^2\right )}{2+\log \left (\frac {4+x-x^3}{x}\right )}+\log \left (2+\log \left (\frac {4+x-x^3}{x}\right )\right )\right ) \, dx\\ &=2 \int \frac {(5+2 x) \log \left (1+\frac {4}{x}-x^2\right )}{2+\log \left (\frac {4+x-x^3}{x}\right )} \, dx+8 \int \frac {x^4}{\left (-4-x+x^3\right ) \left (2+\log \left (\frac {4+x-x^3}{x}\right )\right )} \, dx+16 \int \frac {1}{x \left (-4-x+x^3\right ) \left (2+\log \left (\frac {4+x-x^3}{x}\right )\right )} \, dx+22 \int \frac {x^3}{\left (-4-x+x^3\right ) \left (2+\log \left (\frac {4+x-x^3}{x}\right )\right )} \, dx-52 \int \frac {x}{\left (-4-x+x^3\right ) \left (2+\log \left (\frac {4+x-x^3}{x}\right )\right )} \, dx-76 \int \frac {1}{\left (-4-x+x^3\right ) \left (2+\log \left (\frac {4+x-x^3}{x}\right )\right )} \, dx+\int \log \left (2+\log \left (\frac {4+x-x^3}{x}\right )\right ) \, dx\\ &=2 \int \left (\frac {5 \log \left (1+\frac {4}{x}-x^2\right )}{2+\log \left (\frac {4+x-x^3}{x}\right )}+\frac {2 x \log \left (1+\frac {4}{x}-x^2\right )}{2+\log \left (\frac {4+x-x^3}{x}\right )}\right ) \, dx+8 \int \left (\frac {x}{2+\log \left (\frac {4+x-x^3}{x}\right )}+\frac {x (4+x)}{\left (-4-x+x^3\right ) \left (2+\log \left (\frac {4+x-x^3}{x}\right )\right )}\right ) \, dx+16 \int \left (-\frac {1}{4 x \left (2+\log \left (\frac {4+x-x^3}{x}\right )\right )}+\frac {-1+x^2}{4 \left (-4-x+x^3\right ) \left (2+\log \left (\frac {4+x-x^3}{x}\right )\right )}\right ) \, dx+22 \int \left (\frac {1}{2+\log \left (\frac {4+x-x^3}{x}\right )}+\frac {4+x}{\left (-4-x+x^3\right ) \left (2+\log \left (\frac {4+x-x^3}{x}\right )\right )}\right ) \, dx-52 \int \frac {x}{\left (-4-x+x^3\right ) \left (2+\log \left (\frac {4+x-x^3}{x}\right )\right )} \, dx-76 \int \frac {1}{\left (-4-x+x^3\right ) \left (2+\log \left (\frac {4+x-x^3}{x}\right )\right )} \, dx+\int \log \left (2+\log \left (\frac {4+x-x^3}{x}\right )\right ) \, dx\\ &=-\left (4 \int \frac {1}{x \left (2+\log \left (\frac {4+x-x^3}{x}\right )\right )} \, dx\right )+4 \int \frac {-1+x^2}{\left (-4-x+x^3\right ) \left (2+\log \left (\frac {4+x-x^3}{x}\right )\right )} \, dx+4 \int \frac {x \log \left (1+\frac {4}{x}-x^2\right )}{2+\log \left (\frac {4+x-x^3}{x}\right )} \, dx+8 \int \frac {x}{2+\log \left (\frac {4+x-x^3}{x}\right )} \, dx+8 \int \frac {x (4+x)}{\left (-4-x+x^3\right ) \left (2+\log \left (\frac {4+x-x^3}{x}\right )\right )} \, dx+10 \int \frac {\log \left (1+\frac {4}{x}-x^2\right )}{2+\log \left (\frac {4+x-x^3}{x}\right )} \, dx+22 \int \frac {1}{2+\log \left (\frac {4+x-x^3}{x}\right )} \, dx+22 \int \frac {4+x}{\left (-4-x+x^3\right ) \left (2+\log \left (\frac {4+x-x^3}{x}\right )\right )} \, dx-52 \int \frac {x}{\left (-4-x+x^3\right ) \left (2+\log \left (\frac {4+x-x^3}{x}\right )\right )} \, dx-76 \int \frac {1}{\left (-4-x+x^3\right ) \left (2+\log \left (\frac {4+x-x^3}{x}\right )\right )} \, dx+\int \log \left (2+\log \left (\frac {4+x-x^3}{x}\right )\right ) \, dx\\ &=-\left (4 \int \frac {1}{x \left (2+\log \left (\frac {4+x-x^3}{x}\right )\right )} \, dx\right )+4 \int \frac {x \log \left (1+\frac {4}{x}-x^2\right )}{2+\log \left (\frac {4+x-x^3}{x}\right )} \, dx+4 \int \left (-\frac {1}{\left (-4-x+x^3\right ) \left (2+\log \left (\frac {4+x-x^3}{x}\right )\right )}+\frac {x^2}{\left (-4-x+x^3\right ) \left (2+\log \left (\frac {4+x-x^3}{x}\right )\right )}\right ) \, dx+8 \int \frac {x}{2+\log \left (\frac {4+x-x^3}{x}\right )} \, dx+8 \int \left (\frac {4 x}{\left (-4-x+x^3\right ) \left (2+\log \left (\frac {4+x-x^3}{x}\right )\right )}+\frac {x^2}{\left (-4-x+x^3\right ) \left (2+\log \left (\frac {4+x-x^3}{x}\right )\right )}\right ) \, dx+10 \int \frac {\log \left (1+\frac {4}{x}-x^2\right )}{2+\log \left (\frac {4+x-x^3}{x}\right )} \, dx+22 \int \frac {1}{2+\log \left (\frac {4+x-x^3}{x}\right )} \, dx+22 \int \left (\frac {4}{\left (-4-x+x^3\right ) \left (2+\log \left (\frac {4+x-x^3}{x}\right )\right )}+\frac {x}{\left (-4-x+x^3\right ) \left (2+\log \left (\frac {4+x-x^3}{x}\right )\right )}\right ) \, dx-52 \int \frac {x}{\left (-4-x+x^3\right ) \left (2+\log \left (\frac {4+x-x^3}{x}\right )\right )} \, dx-76 \int \frac {1}{\left (-4-x+x^3\right ) \left (2+\log \left (\frac {4+x-x^3}{x}\right )\right )} \, dx+\int \log \left (2+\log \left (\frac {4+x-x^3}{x}\right )\right ) \, dx\\ &=-\left (4 \int \frac {1}{x \left (2+\log \left (\frac {4+x-x^3}{x}\right )\right )} \, dx\right )-4 \int \frac {1}{\left (-4-x+x^3\right ) \left (2+\log \left (\frac {4+x-x^3}{x}\right )\right )} \, dx+4 \int \frac {x^2}{\left (-4-x+x^3\right ) \left (2+\log \left (\frac {4+x-x^3}{x}\right )\right )} \, dx+4 \int \frac {x \log \left (1+\frac {4}{x}-x^2\right )}{2+\log \left (\frac {4+x-x^3}{x}\right )} \, dx+8 \int \frac {x}{2+\log \left (\frac {4+x-x^3}{x}\right )} \, dx+8 \int \frac {x^2}{\left (-4-x+x^3\right ) \left (2+\log \left (\frac {4+x-x^3}{x}\right )\right )} \, dx+10 \int \frac {\log \left (1+\frac {4}{x}-x^2\right )}{2+\log \left (\frac {4+x-x^3}{x}\right )} \, dx+22 \int \frac {1}{2+\log \left (\frac {4+x-x^3}{x}\right )} \, dx+22 \int \frac {x}{\left (-4-x+x^3\right ) \left (2+\log \left (\frac {4+x-x^3}{x}\right )\right )} \, dx+32 \int \frac {x}{\left (-4-x+x^3\right ) \left (2+\log \left (\frac {4+x-x^3}{x}\right )\right )} \, dx-52 \int \frac {x}{\left (-4-x+x^3\right ) \left (2+\log \left (\frac {4+x-x^3}{x}\right )\right )} \, dx-76 \int \frac {1}{\left (-4-x+x^3\right ) \left (2+\log \left (\frac {4+x-x^3}{x}\right )\right )} \, dx+88 \int \frac {1}{\left (-4-x+x^3\right ) \left (2+\log \left (\frac {4+x-x^3}{x}\right )\right )} \, dx+\int \log \left (2+\log \left (\frac {4+x-x^3}{x}\right )\right ) \, dx\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.09, size = 45, normalized size = 1.80 \begin {gather*} 10 x+2 x^2+4 \log \left (2+\log \left (\frac {4+x-x^3}{x}\right )\right )+x \log \left (2+\log \left (\frac {4+x-x^3}{x}\right )\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(16 - 76*x - 52*x^2 + 22*x^4 + 8*x^5 + (-40*x - 26*x^2 - 4*x^3 + 10*x^4 + 4*x^5)*Log[(4 + x - x^3)/x
] + (-8*x - 2*x^2 + 2*x^4 + (-4*x - x^2 + x^4)*Log[(4 + x - x^3)/x])*Log[2 + Log[(4 + x - x^3)/x]])/(-8*x - 2*
x^2 + 2*x^4 + (-4*x - x^2 + x^4)*Log[(4 + x - x^3)/x]),x]

[Out]

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

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fricas [A]  time = 0.60, size = 30, normalized size = 1.20 \begin {gather*} 2 \, x^{2} + {\left (x + 4\right )} \log \left (\log \left (-\frac {x^{3} - x - 4}{x}\right ) + 2\right ) + 10 \, x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((((x^4-x^2-4*x)*log((-x^3+x+4)/x)+2*x^4-2*x^2-8*x)*log(log((-x^3+x+4)/x)+2)+(4*x^5+10*x^4-4*x^3-26*x
^2-40*x)*log((-x^3+x+4)/x)+8*x^5+22*x^4-52*x^2-76*x+16)/((x^4-x^2-4*x)*log((-x^3+x+4)/x)+2*x^4-2*x^2-8*x),x, a
lgorithm="fricas")

[Out]

2*x^2 + (x + 4)*log(log(-(x^3 - x - 4)/x) + 2) + 10*x

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giac [A]  time = 0.48, size = 46, normalized size = 1.84 \begin {gather*} 2 \, x^{2} + x \log \left (\log \left (-\frac {x^{3} - x - 4}{x}\right ) + 2\right ) + 10 \, x + 4 \, \log \left (\log \left (-x^{3} + x + 4\right ) - \log \relax (x) + 2\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((((x^4-x^2-4*x)*log((-x^3+x+4)/x)+2*x^4-2*x^2-8*x)*log(log((-x^3+x+4)/x)+2)+(4*x^5+10*x^4-4*x^3-26*x
^2-40*x)*log((-x^3+x+4)/x)+8*x^5+22*x^4-52*x^2-76*x+16)/((x^4-x^2-4*x)*log((-x^3+x+4)/x)+2*x^4-2*x^2-8*x),x, a
lgorithm="giac")

[Out]

2*x^2 + x*log(log(-(x^3 - x - 4)/x) + 2) + 10*x + 4*log(log(-x^3 + x + 4) - log(x) + 2)

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maple [F]  time = 0.03, size = 0, normalized size = 0.00 \[\int \frac {\left (\left (x^{4}-x^{2}-4 x \right ) \ln \left (\frac {-x^{3}+x +4}{x}\right )+2 x^{4}-2 x^{2}-8 x \right ) \ln \left (\ln \left (\frac {-x^{3}+x +4}{x}\right )+2\right )+\left (4 x^{5}+10 x^{4}-4 x^{3}-26 x^{2}-40 x \right ) \ln \left (\frac {-x^{3}+x +4}{x}\right )+8 x^{5}+22 x^{4}-52 x^{2}-76 x +16}{\left (x^{4}-x^{2}-4 x \right ) \ln \left (\frac {-x^{3}+x +4}{x}\right )+2 x^{4}-2 x^{2}-8 x}\, dx\]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((((x^4-x^2-4*x)*ln((-x^3+x+4)/x)+2*x^4-2*x^2-8*x)*ln(ln((-x^3+x+4)/x)+2)+(4*x^5+10*x^4-4*x^3-26*x^2-40*x)*
ln((-x^3+x+4)/x)+8*x^5+22*x^4-52*x^2-76*x+16)/((x^4-x^2-4*x)*ln((-x^3+x+4)/x)+2*x^4-2*x^2-8*x),x)

[Out]

int((((x^4-x^2-4*x)*ln((-x^3+x+4)/x)+2*x^4-2*x^2-8*x)*ln(ln((-x^3+x+4)/x)+2)+(4*x^5+10*x^4-4*x^3-26*x^2-40*x)*
ln((-x^3+x+4)/x)+8*x^5+22*x^4-52*x^2-76*x+16)/((x^4-x^2-4*x)*ln((-x^3+x+4)/x)+2*x^4-2*x^2-8*x),x)

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maxima [A]  time = 0.43, size = 29, normalized size = 1.16 \begin {gather*} 2 \, x^{2} + {\left (x + 4\right )} \log \left (\log \left (-x^{3} + x + 4\right ) - \log \relax (x) + 2\right ) + 10 \, x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((((x^4-x^2-4*x)*log((-x^3+x+4)/x)+2*x^4-2*x^2-8*x)*log(log((-x^3+x+4)/x)+2)+(4*x^5+10*x^4-4*x^3-26*x
^2-40*x)*log((-x^3+x+4)/x)+8*x^5+22*x^4-52*x^2-76*x+16)/((x^4-x^2-4*x)*log((-x^3+x+4)/x)+2*x^4-2*x^2-8*x),x, a
lgorithm="maxima")

[Out]

2*x^2 + (x + 4)*log(log(-x^3 + x + 4) - log(x) + 2) + 10*x

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mupad [B]  time = 1.29, size = 45, normalized size = 1.80 \begin {gather*} 10\,x+4\,\ln \left (\ln \left (\frac {-x^3+x+4}{x}\right )+2\right )+x\,\ln \left (\ln \left (\frac {-x^3+x+4}{x}\right )+2\right )+2\,x^2 \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((76*x + log((x - x^3 + 4)/x)*(40*x + 26*x^2 + 4*x^3 - 10*x^4 - 4*x^5) + 52*x^2 - 22*x^4 - 8*x^5 + log(log(
(x - x^3 + 4)/x) + 2)*(8*x + log((x - x^3 + 4)/x)*(4*x + x^2 - x^4) + 2*x^2 - 2*x^4) - 16)/(8*x + log((x - x^3
+ 4)/x)*(4*x + x^2 - x^4) + 2*x^2 - 2*x^4),x)

[Out]

10*x + 4*log(log((x - x^3 + 4)/x) + 2) + x*log(log((x - x^3 + 4)/x) + 2) + 2*x^2

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sympy [A]  time = 1.11, size = 37, normalized size = 1.48 \begin {gather*} 2 x^{2} + x \log {\left (\log {\left (\frac {- x^{3} + x + 4}{x} \right )} + 2 \right )} + 10 x + 4 \log {\left (\log {\left (\frac {- x^{3} + x + 4}{x} \right )} + 2 \right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((((x**4-x**2-4*x)*ln((-x**3+x+4)/x)+2*x**4-2*x**2-8*x)*ln(ln((-x**3+x+4)/x)+2)+(4*x**5+10*x**4-4*x**
3-26*x**2-40*x)*ln((-x**3+x+4)/x)+8*x**5+22*x**4-52*x**2-76*x+16)/((x**4-x**2-4*x)*ln((-x**3+x+4)/x)+2*x**4-2*
x**2-8*x),x)

[Out]

2*x**2 + x*log(log((-x**3 + x + 4)/x) + 2) + 10*x + 4*log(log((-x**3 + x + 4)/x) + 2)

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