3.1.8 \(\int \frac {(-8-3 x) \log ^2(x)+(-4-x) \log ^2(x) \log (4 x^2+x^3)+(-8-2 x+(-8-2 x) \log (x)+(20 x^2+5 x^3) \log ^2(x)) \log ^2(4 x^2+x^3)}{(4 x+x^2) \log ^2(x) \log (4 x^2+x^3)+((8 x+2 x^2) \log (x)+(8 x^2+22 x^3+5 x^4) \log ^2(x)) \log ^2(4 x^2+x^3)} \, dx\)

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

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Rubi [F]  time = 41.13, 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 {(-8-3 x) \log ^2(x)+(-4-x) \log ^2(x) \log \left (4 x^2+x^3\right )+\left (-8-2 x+(-8-2 x) \log (x)+\left (20 x^2+5 x^3\right ) \log ^2(x)\right ) \log ^2\left (4 x^2+x^3\right )}{\left (4 x+x^2\right ) \log ^2(x) \log \left (4 x^2+x^3\right )+\left (\left (8 x+2 x^2\right ) \log (x)+\left (8 x^2+22 x^3+5 x^4\right ) \log ^2(x)\right ) \log ^2\left (4 x^2+x^3\right )} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Int[((-8 - 3*x)*Log[x]^2 + (-4 - x)*Log[x]^2*Log[4*x^2 + x^3] + (-8 - 2*x + (-8 - 2*x)*Log[x] + (20*x^2 + 5*x^
3)*Log[x]^2)*Log[4*x^2 + x^3]^2)/((4*x + x^2)*Log[x]^2*Log[4*x^2 + x^3] + ((8*x + 2*x^2)*Log[x] + (8*x^2 + 22*
x^3 + 5*x^4)*Log[x]^2)*Log[4*x^2 + x^3]^2),x]

[Out]

Log[2 + 5*x] - Log[Log[x]] + 2*Defer[Int][(2 + 2*x*Log[x] + 5*x^2*Log[x])^(-1), x] - 2*Defer[Int][1/(x*(2 + 2*
x*Log[x] + 5*x^2*Log[x])), x] + 5*Defer[Int][x/(2 + 2*x*Log[x] + 5*x^2*Log[x]), x] - 10*Defer[Int][1/((2 + 5*x
)*(2 + 2*x*Log[x] + 5*x^2*Log[x])), x] - Defer[Int][(8 + 3*x)/(x*(4 + x)*Log[x^2*(4 + x)]), x] + 2*Defer[Int][
1/(x*(2 + 2*x*Log[x] + 5*x^2*Log[x])*(Log[x] + 2*Log[x^2*(4 + x)] + 2*x*Log[x]*Log[x^2*(4 + x)] + 5*x^2*Log[x]
*Log[x^2*(4 + x)])), x] + 2*Defer[Int][Log[x]/(x*(2 + 2*x*Log[x] + 5*x^2*Log[x])*(Log[x] + 2*Log[x^2*(4 + x)]
+ 2*x*Log[x]*Log[x^2*(4 + x)] + 5*x^2*Log[x]*Log[x^2*(4 + x)])), x] - 5*Defer[Int][(x*Log[x]^2)/((2 + 2*x*Log[
x] + 5*x^2*Log[x])*(Log[x] + 2*Log[x^2*(4 + x)] + 2*x*Log[x]*Log[x^2*(4 + x)] + 5*x^2*Log[x]*Log[x^2*(4 + x)])
), x] + 4*Defer[Int][1/(x*(Log[x] + 2*Log[x^2*(4 + x)] + x*(2 + 5*x)*Log[x]*Log[x^2*(4 + x)])), x] + 2*Defer[I
nt][1/((4 + x)*(Log[x] + 2*Log[x^2*(4 + x)] + x*(2 + 5*x)*Log[x]*Log[x^2*(4 + x)])), x] - 14*Defer[Int][Log[x]
/(Log[x] + 2*Log[x^2*(4 + x)] + x*(2 + 5*x)*Log[x]*Log[x^2*(4 + x)]), x] - Defer[Int][Log[x]/(x*(Log[x] + 2*Lo
g[x^2*(4 + x)] + x*(2 + 5*x)*Log[x]*Log[x^2*(4 + x)])), x] + 15*Defer[Int][(x*Log[x])/(Log[x] + 2*Log[x^2*(4 +
x)] + x*(2 + 5*x)*Log[x]*Log[x^2*(4 + x)]), x] + 72*Defer[Int][Log[x]/((4 + x)*(Log[x] + 2*Log[x^2*(4 + x)] +
x*(2 + 5*x)*Log[x]*Log[x^2*(4 + x)])), x]

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\int \frac {(-8-3 x) \log ^2(x)+(-4-x) \log ^2(x) \log \left (4 x^2+x^3\right )+\left (-8-2 x+(-8-2 x) \log (x)+\left (20 x^2+5 x^3\right ) \log ^2(x)\right ) \log ^2\left (4 x^2+x^3\right )}{x (4+x) \log (x) \log \left (x^2 (4+x)\right ) \left (\log (x)+2 \log \left (x^2 (4+x)\right )+2 x \log (x) \log \left (x^2 (4+x)\right )+5 x^2 \log (x) \log \left (x^2 (4+x)\right )\right )} \, dx\\ &=\int \left (-\frac {\log (x)}{x \left (\log (x)+2 \log \left (x^2 (4+x)\right )+2 x \log (x) \log \left (x^2 (4+x)\right )+5 x^2 \log (x) \log \left (x^2 (4+x)\right )\right )}-\frac {(8+3 x) \log (x)}{x (4+x) \log \left (x^2 (4+x)\right ) \left (\log (x)+2 \log \left (x^2 (4+x)\right )+2 x \log (x) \log \left (x^2 (4+x)\right )+5 x^2 \log (x) \log \left (x^2 (4+x)\right )\right )}+\frac {\left (-2-2 \log (x)+5 x^2 \log ^2(x)\right ) \log \left (x^2 (4+x)\right )}{x \log (x) \left (\log (x)+2 \log \left (x^2 (4+x)\right )+2 x \log (x) \log \left (x^2 (4+x)\right )+5 x^2 \log (x) \log \left (x^2 (4+x)\right )\right )}\right ) \, dx\\ &=-\int \frac {\log (x)}{x \left (\log (x)+2 \log \left (x^2 (4+x)\right )+2 x \log (x) \log \left (x^2 (4+x)\right )+5 x^2 \log (x) \log \left (x^2 (4+x)\right )\right )} \, dx-\int \frac {(8+3 x) \log (x)}{x (4+x) \log \left (x^2 (4+x)\right ) \left (\log (x)+2 \log \left (x^2 (4+x)\right )+2 x \log (x) \log \left (x^2 (4+x)\right )+5 x^2 \log (x) \log \left (x^2 (4+x)\right )\right )} \, dx+\int \frac {\left (-2-2 \log (x)+5 x^2 \log ^2(x)\right ) \log \left (x^2 (4+x)\right )}{x \log (x) \left (\log (x)+2 \log \left (x^2 (4+x)\right )+2 x \log (x) \log \left (x^2 (4+x)\right )+5 x^2 \log (x) \log \left (x^2 (4+x)\right )\right )} \, dx\\ &=-\int \frac {\log (x)}{x \left (\log (x)+2 \log \left (x^2 (4+x)\right )+x (2+5 x) \log (x) \log \left (x^2 (4+x)\right )\right )} \, dx-\int \frac {(8+3 x) \log (x)}{x (4+x) \log \left (x^2 (4+x)\right ) \left (\log (x)+2 \log \left (x^2 (4+x)\right )+x (2+5 x) \log (x) \log \left (x^2 (4+x)\right )\right )} \, dx+\int \frac {\left (-2-2 \log (x)+5 x^2 \log ^2(x)\right ) \log \left (x^2 (4+x)\right )}{x \log (x) \left (\log (x)+2 \log \left (x^2 (4+x)\right )+x (2+5 x) \log (x) \log \left (x^2 (4+x)\right )\right )} \, dx\\ &=-\int \frac {\log (x)}{x \left (\log (x)+2 \log \left (x^2 (4+x)\right )+x (2+5 x) \log (x) \log \left (x^2 (4+x)\right )\right )} \, dx-\int \left (\frac {8+3 x}{x (4+x) \log \left (x^2 (4+x)\right )}-\frac {(8+3 x) \left (2+2 x \log (x)+5 x^2 \log (x)\right )}{x (4+x) \left (\log (x)+2 \log \left (x^2 (4+x)\right )+2 x \log (x) \log \left (x^2 (4+x)\right )+5 x^2 \log (x) \log \left (x^2 (4+x)\right )\right )}\right ) \, dx+\int \left (\frac {-2-2 \log (x)+5 x^2 \log ^2(x)}{x \log (x) \left (2+2 x \log (x)+5 x^2 \log (x)\right )}+\frac {2+2 \log (x)-5 x^2 \log ^2(x)}{x \left (2+2 x \log (x)+5 x^2 \log (x)\right ) \left (\log (x)+2 \log \left (x^2 (4+x)\right )+2 x \log (x) \log \left (x^2 (4+x)\right )+5 x^2 \log (x) \log \left (x^2 (4+x)\right )\right )}\right ) \, dx\\ &=\int \frac {-2-2 \log (x)+5 x^2 \log ^2(x)}{x \log (x) \left (2+2 x \log (x)+5 x^2 \log (x)\right )} \, dx-\int \frac {8+3 x}{x (4+x) \log \left (x^2 (4+x)\right )} \, dx+\int \frac {(8+3 x) \left (2+2 x \log (x)+5 x^2 \log (x)\right )}{x (4+x) \left (\log (x)+2 \log \left (x^2 (4+x)\right )+2 x \log (x) \log \left (x^2 (4+x)\right )+5 x^2 \log (x) \log \left (x^2 (4+x)\right )\right )} \, dx+\int \frac {2+2 \log (x)-5 x^2 \log ^2(x)}{x \left (2+2 x \log (x)+5 x^2 \log (x)\right ) \left (\log (x)+2 \log \left (x^2 (4+x)\right )+2 x \log (x) \log \left (x^2 (4+x)\right )+5 x^2 \log (x) \log \left (x^2 (4+x)\right )\right )} \, dx-\int \frac {\log (x)}{x \left (\log (x)+2 \log \left (x^2 (4+x)\right )+x (2+5 x) \log (x) \log \left (x^2 (4+x)\right )\right )} \, dx\\ &=\int \left (\frac {5}{2+5 x}-\frac {1}{x \log (x)}-\frac {4 (1+5 x)}{x (2+5 x) \left (2+2 x \log (x)+5 x^2 \log (x)\right )}+\frac {2+5 x}{2+2 x \log (x)+5 x^2 \log (x)}\right ) \, dx-\int \frac {8+3 x}{x (4+x) \log \left (x^2 (4+x)\right )} \, dx-\int \frac {\log (x)}{x \left (\log (x)+2 \log \left (x^2 (4+x)\right )+x (2+5 x) \log (x) \log \left (x^2 (4+x)\right )\right )} \, dx+\int \left (\frac {2}{x \left (2+2 x \log (x)+5 x^2 \log (x)\right ) \left (\log (x)+2 \log \left (x^2 (4+x)\right )+2 x \log (x) \log \left (x^2 (4+x)\right )+5 x^2 \log (x) \log \left (x^2 (4+x)\right )\right )}+\frac {2 \log (x)}{x \left (2+2 x \log (x)+5 x^2 \log (x)\right ) \left (\log (x)+2 \log \left (x^2 (4+x)\right )+2 x \log (x) \log \left (x^2 (4+x)\right )+5 x^2 \log (x) \log \left (x^2 (4+x)\right )\right )}-\frac {5 x \log ^2(x)}{\left (2+2 x \log (x)+5 x^2 \log (x)\right ) \left (\log (x)+2 \log \left (x^2 (4+x)\right )+2 x \log (x) \log \left (x^2 (4+x)\right )+5 x^2 \log (x) \log \left (x^2 (4+x)\right )\right )}\right ) \, dx+\int \left (\frac {2 \left (2+2 x \log (x)+5 x^2 \log (x)\right )}{x \left (\log (x)+2 \log \left (x^2 (4+x)\right )+2 x \log (x) \log \left (x^2 (4+x)\right )+5 x^2 \log (x) \log \left (x^2 (4+x)\right )\right )}+\frac {2+2 x \log (x)+5 x^2 \log (x)}{(4+x) \left (\log (x)+2 \log \left (x^2 (4+x)\right )+2 x \log (x) \log \left (x^2 (4+x)\right )+5 x^2 \log (x) \log \left (x^2 (4+x)\right )\right )}\right ) \, dx\\ &=\log (2+5 x)+2 \int \frac {1}{x \left (2+2 x \log (x)+5 x^2 \log (x)\right ) \left (\log (x)+2 \log \left (x^2 (4+x)\right )+2 x \log (x) \log \left (x^2 (4+x)\right )+5 x^2 \log (x) \log \left (x^2 (4+x)\right )\right )} \, dx+2 \int \frac {\log (x)}{x \left (2+2 x \log (x)+5 x^2 \log (x)\right ) \left (\log (x)+2 \log \left (x^2 (4+x)\right )+2 x \log (x) \log \left (x^2 (4+x)\right )+5 x^2 \log (x) \log \left (x^2 (4+x)\right )\right )} \, dx+2 \int \frac {2+2 x \log (x)+5 x^2 \log (x)}{x \left (\log (x)+2 \log \left (x^2 (4+x)\right )+2 x \log (x) \log \left (x^2 (4+x)\right )+5 x^2 \log (x) \log \left (x^2 (4+x)\right )\right )} \, dx-4 \int \frac {1+5 x}{x (2+5 x) \left (2+2 x \log (x)+5 x^2 \log (x)\right )} \, dx-5 \int \frac {x \log ^2(x)}{\left (2+2 x \log (x)+5 x^2 \log (x)\right ) \left (\log (x)+2 \log \left (x^2 (4+x)\right )+2 x \log (x) \log \left (x^2 (4+x)\right )+5 x^2 \log (x) \log \left (x^2 (4+x)\right )\right )} \, dx-\int \frac {1}{x \log (x)} \, dx+\int \frac {2+5 x}{2+2 x \log (x)+5 x^2 \log (x)} \, dx-\int \frac {8+3 x}{x (4+x) \log \left (x^2 (4+x)\right )} \, dx+\int \frac {2+2 x \log (x)+5 x^2 \log (x)}{(4+x) \left (\log (x)+2 \log \left (x^2 (4+x)\right )+2 x \log (x) \log \left (x^2 (4+x)\right )+5 x^2 \log (x) \log \left (x^2 (4+x)\right )\right )} \, dx-\int \frac {\log (x)}{x \left (\log (x)+2 \log \left (x^2 (4+x)\right )+x (2+5 x) \log (x) \log \left (x^2 (4+x)\right )\right )} \, dx\\ &=\text {Rest of rules removed due to large latex content} \end {aligned} \end {gather*}

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Mathematica [B]  time = 102.93, size = 75, normalized size = 2.27 \begin {gather*} \log (2+5 x)-\log (x (2+5 x))-\log (\log (x))-\log \left (\log \left (x^2 (4+x)\right )\right )+\log \left (\log (x)+2 \log \left (x^2 (4+x)\right )+2 x \log (x) \log \left (x^2 (4+x)\right )+5 x^2 \log (x) \log \left (x^2 (4+x)\right )\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[((-8 - 3*x)*Log[x]^2 + (-4 - x)*Log[x]^2*Log[4*x^2 + x^3] + (-8 - 2*x + (-8 - 2*x)*Log[x] + (20*x^2
+ 5*x^3)*Log[x]^2)*Log[4*x^2 + x^3]^2)/((4*x + x^2)*Log[x]^2*Log[4*x^2 + x^3] + ((8*x + 2*x^2)*Log[x] + (8*x^2
+ 22*x^3 + 5*x^4)*Log[x]^2)*Log[4*x^2 + x^3]^2),x]

[Out]

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

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fricas [B]  time = 0.71, size = 98, normalized size = 2.97 \begin {gather*} \log \left (5 \, x + 2\right ) + \log \left (\frac {{\left ({\left (5 \, x^{2} + 2 \, x\right )} \log \relax (x) + 2\right )} \log \left (x^{3} + 4 \, x^{2}\right ) + \log \relax (x)}{{\left (5 \, x^{2} + 2 \, x\right )} \log \relax (x) + 2}\right ) + \log \left (\frac {{\left (5 \, x^{2} + 2 \, x\right )} \log \relax (x) + 2}{5 \, x^{2} + 2 \, x}\right ) - \log \left (\log \left (x^{3} + 4 \, x^{2}\right )\right ) - \log \left (\log \relax (x)\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((((5*x^3+20*x^2)*log(x)^2+(-2*x-8)*log(x)-2*x-8)*log(x^3+4*x^2)^2+(-x-4)*log(x)^2*log(x^3+4*x^2)+(-3
*x-8)*log(x)^2)/(((5*x^4+22*x^3+8*x^2)*log(x)^2+(2*x^2+8*x)*log(x))*log(x^3+4*x^2)^2+(x^2+4*x)*log(x)^2*log(x^
3+4*x^2)),x, algorithm="fricas")

[Out]

log(5*x + 2) + log((((5*x^2 + 2*x)*log(x) + 2)*log(x^3 + 4*x^2) + log(x))/((5*x^2 + 2*x)*log(x) + 2)) + log(((
5*x^2 + 2*x)*log(x) + 2)/(5*x^2 + 2*x)) - log(log(x^3 + 4*x^2)) - log(log(x))

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giac [B]  time = 0.61, size = 70, normalized size = 2.12 \begin {gather*} \log \left (5 \, x^{2} \log \left (x + 4\right ) \log \relax (x) + 10 \, x^{2} \log \relax (x)^{2} + 2 \, x \log \left (x + 4\right ) \log \relax (x) + 4 \, x \log \relax (x)^{2} + 2 \, \log \left (x + 4\right ) + 5 \, \log \relax (x)\right ) - \log \relax (x) - \log \left (\log \left (x + 4\right ) + 2 \, \log \relax (x)\right ) - \log \left (\log \relax (x)\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((((5*x^3+20*x^2)*log(x)^2+(-2*x-8)*log(x)-2*x-8)*log(x^3+4*x^2)^2+(-x-4)*log(x)^2*log(x^3+4*x^2)+(-3
*x-8)*log(x)^2)/(((5*x^4+22*x^3+8*x^2)*log(x)^2+(2*x^2+8*x)*log(x))*log(x^3+4*x^2)^2+(x^2+4*x)*log(x)^2*log(x^
3+4*x^2)),x, algorithm="giac")

[Out]

log(5*x^2*log(x + 4)*log(x) + 10*x^2*log(x)^2 + 2*x*log(x + 4)*log(x) + 4*x*log(x)^2 + 2*log(x + 4) + 5*log(x)
) - log(x) - log(log(x + 4) + 2*log(x)) - log(log(x))

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maple [C]  time = 1.14, size = 675, normalized size = 20.45

method	result	size
risch	\(\ln \left (5 x +2\right )-\ln \left (\ln \relax (x )\right )+\ln \left (\ln \relax (x )+\frac {2}{\left (5 x +2\right ) x}\right )+\ln \left (\ln \left (4+x \right )-\frac {i \left (5 \pi \,x^{2} \mathrm {csgn}\left (i x \right )^{2} \mathrm {csgn}\left (i x^{2}\right ) \ln \relax (x )-10 \pi \,x^{2} \mathrm {csgn}\left (i x \right ) \mathrm {csgn}\left (i x^{2}\right )^{2} \ln \relax (x )+5 \pi \,x^{2} \mathrm {csgn}\left (i x^{2}\right )^{3} \ln \relax (x )+5 \pi \,x^{2} \mathrm {csgn}\left (i x^{2}\right ) \mathrm {csgn}\left (i \left (4+x \right )\right ) \mathrm {csgn}\left (i x^{2} \left (4+x \right )\right ) \ln \relax (x )-5 \pi \,x^{2} \mathrm {csgn}\left (i x^{2}\right ) \mathrm {csgn}\left (i x^{2} \left (4+x \right )\right )^{2} \ln \relax (x )-5 \pi \,x^{2} \mathrm {csgn}\left (i \left (4+x \right )\right ) \mathrm {csgn}\left (i x^{2} \left (4+x \right )\right )^{2} \ln \relax (x )+5 \pi \,x^{2} \mathrm {csgn}\left (i x^{2} \left (4+x \right )\right )^{3} \ln \relax (x )+2 \pi x \mathrm {csgn}\left (i x \right )^{2} \mathrm {csgn}\left (i x^{2}\right ) \ln \relax (x )-4 \pi x \,\mathrm {csgn}\left (i x \right ) \mathrm {csgn}\left (i x^{2}\right )^{2} \ln \relax (x )+2 \pi x \mathrm {csgn}\left (i x^{2}\right )^{3} \ln \relax (x )+2 \pi x \,\mathrm {csgn}\left (i x^{2}\right ) \mathrm {csgn}\left (i \left (4+x \right )\right ) \mathrm {csgn}\left (i x^{2} \left (4+x \right )\right ) \ln \relax (x )-2 \pi x \,\mathrm {csgn}\left (i x^{2}\right ) \mathrm {csgn}\left (i x^{2} \left (4+x \right )\right )^{2} \ln \relax (x )-2 \pi x \,\mathrm {csgn}\left (i \left (4+x \right )\right ) \mathrm {csgn}\left (i x^{2} \left (4+x \right )\right )^{2} \ln \relax (x )+2 \pi x \mathrm {csgn}\left (i x^{2} \left (4+x \right )\right )^{3} \ln \relax (x )+10 i \ln \relax (x )+2 \pi \mathrm {csgn}\left (i x \right )^{2} \mathrm {csgn}\left (i x^{2}\right )-4 \pi \,\mathrm {csgn}\left (i x \right ) \mathrm {csgn}\left (i x^{2}\right )^{2}+2 \pi \mathrm {csgn}\left (i x^{2}\right )^{3}+2 \pi \,\mathrm {csgn}\left (i x^{2}\right ) \mathrm {csgn}\left (i \left (4+x \right )\right ) \mathrm {csgn}\left (i x^{2} \left (4+x \right )\right )-2 \pi \,\mathrm {csgn}\left (i x^{2}\right ) \mathrm {csgn}\left (i x^{2} \left (4+x \right )\right )^{2}-2 \pi \,\mathrm {csgn}\left (i \left (4+x \right )\right ) \mathrm {csgn}\left (i x^{2} \left (4+x \right )\right )^{2}+2 \pi \mathrm {csgn}\left (i x^{2} \left (4+x \right )\right )^{3}+8 i x \ln \relax (x )^{2}+20 i x^{2} \ln \relax (x )^{2}\right )}{2 \left (5 x^{2} \ln \relax (x )+2 x \ln \relax (x )+2\right )}\right )-\ln \left (\ln \left (4+x \right )-\frac {i \left (\pi \mathrm {csgn}\left (i x \right )^{2} \mathrm {csgn}\left (i x^{2}\right )-2 \pi \,\mathrm {csgn}\left (i x \right ) \mathrm {csgn}\left (i x^{2}\right )^{2}+\pi \mathrm {csgn}\left (i x^{2}\right )^{3}+\pi \,\mathrm {csgn}\left (i x^{2}\right ) \mathrm {csgn}\left (i \left (4+x \right )\right ) \mathrm {csgn}\left (i x^{2} \left (4+x \right )\right )-\pi \,\mathrm {csgn}\left (i x^{2}\right ) \mathrm {csgn}\left (i x^{2} \left (4+x \right )\right )^{2}-\pi \,\mathrm {csgn}\left (i \left (4+x \right )\right ) \mathrm {csgn}\left (i x^{2} \left (4+x \right )\right )^{2}+\pi \mathrm {csgn}\left (i x^{2} \left (4+x \right )\right )^{3}+4 i \ln \relax (x )\right )}{2}\right )\)	\(675\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((((5*x^3+20*x^2)*ln(x)^2+(-2*x-8)*ln(x)-2*x-8)*ln(x^3+4*x^2)^2+(-x-4)*ln(x)^2*ln(x^3+4*x^2)+(-3*x-8)*ln(x)
^2)/(((5*x^4+22*x^3+8*x^2)*ln(x)^2+(2*x^2+8*x)*ln(x))*ln(x^3+4*x^2)^2+(x^2+4*x)*ln(x)^2*ln(x^3+4*x^2)),x,metho
d=_RETURNVERBOSE)

[Out]

ln(5*x+2)-ln(ln(x))+ln(ln(x)+2/(5*x+2)/x)+ln(ln(4+x)-1/2*I*(5*Pi*x^2*csgn(I*x)^2*csgn(I*x^2)*ln(x)-10*Pi*x^2*c
sgn(I*x)*csgn(I*x^2)^2*ln(x)+5*Pi*x^2*csgn(I*x^2)^3*ln(x)+5*Pi*x^2*csgn(I*x^2)*csgn(I*(4+x))*csgn(I*x^2*(4+x))
*ln(x)-5*Pi*x^2*csgn(I*x^2)*csgn(I*x^2*(4+x))^2*ln(x)-5*Pi*x^2*csgn(I*(4+x))*csgn(I*x^2*(4+x))^2*ln(x)+5*Pi*x^
2*csgn(I*x^2*(4+x))^3*ln(x)+2*Pi*x*csgn(I*x)^2*csgn(I*x^2)*ln(x)-4*Pi*x*csgn(I*x)*csgn(I*x^2)^2*ln(x)+2*Pi*x*c
sgn(I*x^2)^3*ln(x)+2*Pi*x*csgn(I*x^2)*csgn(I*(4+x))*csgn(I*x^2*(4+x))*ln(x)-2*Pi*x*csgn(I*x^2)*csgn(I*x^2*(4+x
))^2*ln(x)-2*Pi*x*csgn(I*(4+x))*csgn(I*x^2*(4+x))^2*ln(x)+2*Pi*x*csgn(I*x^2*(4+x))^3*ln(x)+10*I*ln(x)+2*Pi*csg
n(I*x)^2*csgn(I*x^2)-4*Pi*csgn(I*x)*csgn(I*x^2)^2+2*Pi*csgn(I*x^2)^3+2*Pi*csgn(I*x^2)*csgn(I*(4+x))*csgn(I*x^2
*(4+x))-2*Pi*csgn(I*x^2)*csgn(I*x^2*(4+x))^2-2*Pi*csgn(I*(4+x))*csgn(I*x^2*(4+x))^2+2*Pi*csgn(I*x^2*(4+x))^3+8
*I*x*ln(x)^2+20*I*x^2*ln(x)^2)/(5*x^2*ln(x)+2*x*ln(x)+2))-ln(ln(4+x)-1/2*I*(Pi*csgn(I*x)^2*csgn(I*x^2)-2*Pi*cs
gn(I*x)*csgn(I*x^2)^2+Pi*csgn(I*x^2)^3+Pi*csgn(I*x^2)*csgn(I*(4+x))*csgn(I*x^2*(4+x))-Pi*csgn(I*x^2)*csgn(I*x^
2*(4+x))^2-Pi*csgn(I*(4+x))*csgn(I*x^2*(4+x))^2+Pi*csgn(I*x^2*(4+x))^3+4*I*ln(x)))

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maxima [B]  time = 0.57, size = 108, normalized size = 3.27 \begin {gather*} \log \left (5 \, x + 2\right ) + \log \left (\frac {2 \, {\left (5 \, x^{2} + 2 \, x\right )} \log \relax (x)^{2} + {\left ({\left (5 \, x^{2} + 2 \, x\right )} \log \relax (x) + 2\right )} \log \left (x + 4\right ) + 5 \, \log \relax (x)}{{\left (5 \, x^{2} + 2 \, x\right )} \log \relax (x) + 2}\right ) + \log \left (\frac {{\left (5 \, x^{2} + 2 \, x\right )} \log \relax (x) + 2}{5 \, x^{2} + 2 \, x}\right ) - \log \left (\log \left (x + 4\right ) + 2 \, \log \relax (x)\right ) - \log \left (\log \relax (x)\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((((5*x^3+20*x^2)*log(x)^2+(-2*x-8)*log(x)-2*x-8)*log(x^3+4*x^2)^2+(-x-4)*log(x)^2*log(x^3+4*x^2)+(-3
*x-8)*log(x)^2)/(((5*x^4+22*x^3+8*x^2)*log(x)^2+(2*x^2+8*x)*log(x))*log(x^3+4*x^2)^2+(x^2+4*x)*log(x)^2*log(x^
3+4*x^2)),x, algorithm="maxima")

[Out]

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

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mupad [F]  time = 0.00, size = -1, normalized size = -0.03 \begin {gather*} \int -\frac {{\ln \left (x^3+4\,x^2\right )}^2\,\left (\left (-5\,x^3-20\,x^2\right )\,{\ln \relax (x)}^2+\left (2\,x+8\right )\,\ln \relax (x)+2\,x+8\right )+{\ln \relax (x)}^2\,\left (3\,x+8\right )+\ln \left (x^3+4\,x^2\right )\,{\ln \relax (x)}^2\,\left (x+4\right )}{{\ln \left (x^3+4\,x^2\right )}^2\,\left (\left (5\,x^4+22\,x^3+8\,x^2\right )\,{\ln \relax (x)}^2+\left (2\,x^2+8\,x\right )\,\ln \relax (x)\right )+\ln \left (x^3+4\,x^2\right )\,{\ln \relax (x)}^2\,\left (x^2+4\,x\right )} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(-(log(4*x^2 + x^3)^2*(2*x - log(x)^2*(20*x^2 + 5*x^3) + log(x)*(2*x + 8) + 8) + log(x)^2*(3*x + 8) + log(4
*x^2 + x^3)*log(x)^2*(x + 4))/(log(4*x^2 + x^3)^2*(log(x)^2*(8*x^2 + 22*x^3 + 5*x^4) + log(x)*(8*x + 2*x^2)) +
log(4*x^2 + x^3)*log(x)^2*(4*x + x^2)),x)

[Out]

int(-(log(4*x^2 + x^3)^2*(2*x - log(x)^2*(20*x^2 + 5*x^3) + log(x)*(2*x + 8) + 8) + log(x)^2*(3*x + 8) + log(4
*x^2 + x^3)*log(x)^2*(x + 4))/(log(4*x^2 + x^3)^2*(log(x)^2*(8*x^2 + 22*x^3 + 5*x^4) + log(x)*(8*x + 2*x^2)) +
log(4*x^2 + x^3)*log(x)^2*(4*x + x^2)), x)

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sympy [F(-2)]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: PolynomialError} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((((5*x**3+20*x**2)*ln(x)**2+(-2*x-8)*ln(x)-2*x-8)*ln(x**3+4*x**2)**2+(-x-4)*ln(x)**2*ln(x**3+4*x**2)
+(-3*x-8)*ln(x)**2)/(((5*x**4+22*x**3+8*x**2)*ln(x)**2+(2*x**2+8*x)*ln(x))*ln(x**3+4*x**2)**2+(x**2+4*x)*ln(x)
**2*ln(x**3+4*x**2)),x)

[Out]

Exception raised: PolynomialError

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