3.45.13 \(\int \frac {4+4 x+(-4-6 x) \log (x)+(-2-2 x) \log (x) \log (1+x)+(-2-2 x) \log (x) \log (\frac {x^2}{\log ^2(x)})}{(x^3+x^4) \log (x) \log ^3(1+x)+(3 x^3+3 x^4) \log (x) \log ^2(1+x) \log (\frac {x^2}{\log ^2(x)})+(3 x^3+3 x^4) \log (x) \log (1+x) \log ^2(\frac {x^2}{\log ^2(x)})+(x^3+x^4) \log (x) \log ^3(\frac {x^2}{\log ^2(x)})} \, dx\)

Optimal. Leaf size=23 \[ \frac {1}{\left (-x+x \left (1+\log (1+x)+\log \left (\frac {x^2}{\log ^2(x)}\right )\right )\right )^2} \]

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Rubi [A]  time = 0.37, antiderivative size = 20, normalized size of antiderivative = 0.87, number of steps used = 2, number of rules used = 2, integrand size = 141, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.014, Rules used = {6688, 6687} \begin {gather*} \frac {1}{x^2 \left (\log \left (\frac {x^2}{\log ^2(x)}\right )+\log (x+1)\right )^2} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(4 + 4*x + (-4 - 6*x)*Log[x] + (-2 - 2*x)*Log[x]*Log[1 + x] + (-2 - 2*x)*Log[x]*Log[x^2/Log[x]^2])/((x^3 +
x^4)*Log[x]*Log[1 + x]^3 + (3*x^3 + 3*x^4)*Log[x]*Log[1 + x]^2*Log[x^2/Log[x]^2] + (3*x^3 + 3*x^4)*Log[x]*Log
[1 + x]*Log[x^2/Log[x]^2]^2 + (x^3 + x^4)*Log[x]*Log[x^2/Log[x]^2]^3),x]

[Out]

1/(x^2*(Log[1 + x] + Log[x^2/Log[x]^2])^2)

Rule 6687

Int[(u_)*(y_)^(m_.)*(z_)^(n_.), x_Symbol] :> With[{q = DerivativeDivides[y*z, u*z^(n - m), x]}, Simp[(q*y^(m +
1)*z^(m + 1))/(m + 1), x] /;  !FalseQ[q]] /; FreeQ[{m, n}, x] && NeQ[m, -1]

Rule 6688

Int[u_, x_Symbol] :> With[{v = SimplifyIntegrand[u, x]}, Int[v, x] /; SimplerIntegrandQ[v, u, x]]

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\int \frac {4 (1+x)-2 \log (x) \left (2+3 x+(1+x) \log (1+x)+(1+x) \log \left (\frac {x^2}{\log ^2(x)}\right )\right )}{x^3 (1+x) \log (x) \left (\log (1+x)+\log \left (\frac {x^2}{\log ^2(x)}\right )\right )^3} \, dx\\ &=\frac {1}{x^2 \left (\log (1+x)+\log \left (\frac {x^2}{\log ^2(x)}\right )\right )^2}\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.30, size = 20, normalized size = 0.87 \begin {gather*} \frac {1}{x^2 \left (\log (1+x)+\log \left (\frac {x^2}{\log ^2(x)}\right )\right )^2} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(4 + 4*x + (-4 - 6*x)*Log[x] + (-2 - 2*x)*Log[x]*Log[1 + x] + (-2 - 2*x)*Log[x]*Log[x^2/Log[x]^2])/(
(x^3 + x^4)*Log[x]*Log[1 + x]^3 + (3*x^3 + 3*x^4)*Log[x]*Log[1 + x]^2*Log[x^2/Log[x]^2] + (3*x^3 + 3*x^4)*Log[
x]*Log[1 + x]*Log[x^2/Log[x]^2]^2 + (x^3 + x^4)*Log[x]*Log[x^2/Log[x]^2]^3),x]

[Out]

1/(x^2*(Log[1 + x] + Log[x^2/Log[x]^2])^2)

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fricas [A]  time = 0.49, size = 46, normalized size = 2.00 \begin {gather*} \frac {1}{x^{2} \log \left (x + 1\right )^{2} + 2 \, x^{2} \log \left (x + 1\right ) \log \left (\frac {x^{2}}{\log \relax (x)^{2}}\right ) + x^{2} \log \left (\frac {x^{2}}{\log \relax (x)^{2}}\right )^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-2*x-2)*log(x)*log(x^2/log(x)^2)+(-2*x-2)*log(x)*log(x+1)+(-6*x-4)*log(x)+4*x+4)/((x^4+x^3)*log(x)
*log(x^2/log(x)^2)^3+(3*x^4+3*x^3)*log(x)*log(x+1)*log(x^2/log(x)^2)^2+(3*x^4+3*x^3)*log(x)*log(x+1)^2*log(x^2
/log(x)^2)+(x^4+x^3)*log(x)*log(x+1)^3),x, algorithm="fricas")

[Out]

1/(x^2*log(x + 1)^2 + 2*x^2*log(x + 1)*log(x^2/log(x)^2) + x^2*log(x^2/log(x)^2)^2)

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giac [B]  time = 2.49, size = 315, normalized size = 13.70 \begin {gather*} -\frac {3 \, x \log \relax (x) - 2 \, x + 2 \, \log \relax (x) - 2}{12 \, x^{3} \log \relax (x)^{2} \log \left (\log \relax (x)^{2}\right ) - 3 \, x^{3} \log \relax (x) \log \left (\log \relax (x)^{2}\right )^{2} + 6 \, x^{3} \log \relax (x) \log \left (\log \relax (x)^{2}\right ) \log \left (x + 1\right ) - 3 \, x^{3} \log \left (x + 1\right )^{2} \log \relax (x) - 12 \, x^{3} \log \left (x + 1\right ) \log \relax (x)^{2} - 12 \, x^{3} \log \relax (x)^{3} - 8 \, x^{3} \log \relax (x) \log \left (\log \relax (x)^{2}\right ) + 8 \, x^{2} \log \relax (x)^{2} \log \left (\log \relax (x)^{2}\right ) + 2 \, x^{3} \log \left (\log \relax (x)^{2}\right )^{2} - 2 \, x^{2} \log \relax (x) \log \left (\log \relax (x)^{2}\right )^{2} - 4 \, x^{3} \log \left (\log \relax (x)^{2}\right ) \log \left (x + 1\right ) + 4 \, x^{2} \log \relax (x) \log \left (\log \relax (x)^{2}\right ) \log \left (x + 1\right ) + 2 \, x^{3} \log \left (x + 1\right )^{2} + 8 \, x^{3} \log \left (x + 1\right ) \log \relax (x) - 2 \, x^{2} \log \left (x + 1\right )^{2} \log \relax (x) + 8 \, x^{3} \log \relax (x)^{2} - 8 \, x^{2} \log \left (x + 1\right ) \log \relax (x)^{2} - 8 \, x^{2} \log \relax (x)^{3} - 8 \, x^{2} \log \relax (x) \log \left (\log \relax (x)^{2}\right ) + 2 \, x^{2} \log \left (\log \relax (x)^{2}\right )^{2} - 4 \, x^{2} \log \left (\log \relax (x)^{2}\right ) \log \left (x + 1\right ) + 2 \, x^{2} \log \left (x + 1\right )^{2} + 8 \, x^{2} \log \left (x + 1\right ) \log \relax (x) + 8 \, x^{2} \log \relax (x)^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-2*x-2)*log(x)*log(x^2/log(x)^2)+(-2*x-2)*log(x)*log(x+1)+(-6*x-4)*log(x)+4*x+4)/((x^4+x^3)*log(x)
*log(x^2/log(x)^2)^3+(3*x^4+3*x^3)*log(x)*log(x+1)*log(x^2/log(x)^2)^2+(3*x^4+3*x^3)*log(x)*log(x+1)^2*log(x^2
/log(x)^2)+(x^4+x^3)*log(x)*log(x+1)^3),x, algorithm="giac")

[Out]

-(3*x*log(x) - 2*x + 2*log(x) - 2)/(12*x^3*log(x)^2*log(log(x)^2) - 3*x^3*log(x)*log(log(x)^2)^2 + 6*x^3*log(x
)*log(log(x)^2)*log(x + 1) - 3*x^3*log(x + 1)^2*log(x) - 12*x^3*log(x + 1)*log(x)^2 - 12*x^3*log(x)^3 - 8*x^3*
log(x)*log(log(x)^2) + 8*x^2*log(x)^2*log(log(x)^2) + 2*x^3*log(log(x)^2)^2 - 2*x^2*log(x)*log(log(x)^2)^2 - 4
*x^3*log(log(x)^2)*log(x + 1) + 4*x^2*log(x)*log(log(x)^2)*log(x + 1) + 2*x^3*log(x + 1)^2 + 8*x^3*log(x + 1)*
log(x) - 2*x^2*log(x + 1)^2*log(x) + 8*x^3*log(x)^2 - 8*x^2*log(x + 1)*log(x)^2 - 8*x^2*log(x)^3 - 8*x^2*log(x
)*log(log(x)^2) + 2*x^2*log(log(x)^2)^2 - 4*x^2*log(log(x)^2)*log(x + 1) + 2*x^2*log(x + 1)^2 + 8*x^2*log(x +
1)*log(x) + 8*x^2*log(x)^2)

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maple [C]  time = 2.04, size = 212, normalized size = 9.22

method	result	size
risch	\(-\frac {4}{x^{2} \left (\pi \mathrm {csgn}\left (i x \right )^{2} \mathrm {csgn}\left (i x^{2}\right )-2 \pi \,\mathrm {csgn}\left (i x \right ) \mathrm {csgn}\left (i x^{2}\right )^{2}+\pi \mathrm {csgn}\left (i x^{2}\right )^{3}+\pi \,\mathrm {csgn}\left (i x^{2}\right ) \mathrm {csgn}\left (\frac {i}{\ln \relax (x )^{2}}\right ) \mathrm {csgn}\left (\frac {i x^{2}}{\ln \relax (x )^{2}}\right )-\pi \,\mathrm {csgn}\left (i x^{2}\right ) \mathrm {csgn}\left (\frac {i x^{2}}{\ln \relax (x )^{2}}\right )^{2}-\pi \,\mathrm {csgn}\left (\frac {i}{\ln \relax (x )^{2}}\right ) \mathrm {csgn}\left (\frac {i x^{2}}{\ln \relax (x )^{2}}\right )^{2}-\pi \mathrm {csgn}\left (i \ln \relax (x )\right )^{2} \mathrm {csgn}\left (i \ln \relax (x )^{2}\right )+2 \pi \,\mathrm {csgn}\left (i \ln \relax (x )\right ) \mathrm {csgn}\left (i \ln \relax (x )^{2}\right )^{2}-\pi \mathrm {csgn}\left (i \ln \relax (x )^{2}\right )^{3}+\pi \mathrm {csgn}\left (\frac {i x^{2}}{\ln \relax (x )^{2}}\right )^{3}+4 i \ln \relax (x )+2 i \ln \left (x +1\right )-4 i \ln \left (\ln \relax (x )\right )\right )^{2}}\)	\(212\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((-2*x-2)*ln(x)*ln(x^2/ln(x)^2)+(-2*x-2)*ln(x)*ln(x+1)+(-6*x-4)*ln(x)+4*x+4)/((x^4+x^3)*ln(x)*ln(x^2/ln(x)
^2)^3+(3*x^4+3*x^3)*ln(x)*ln(x+1)*ln(x^2/ln(x)^2)^2+(3*x^4+3*x^3)*ln(x)*ln(x+1)^2*ln(x^2/ln(x)^2)+(x^4+x^3)*ln
(x)*ln(x+1)^3),x,method=_RETURNVERBOSE)

[Out]

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

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maxima [B]  time = 0.44, size = 63, normalized size = 2.74 \begin {gather*} \frac {1}{x^{2} \log \left (x + 1\right )^{2} + 4 \, x^{2} \log \relax (x)^{2} - 8 \, x^{2} \log \relax (x) \log \left (\log \relax (x)\right ) + 4 \, x^{2} \log \left (\log \relax (x)\right )^{2} + 4 \, {\left (x^{2} \log \relax (x) - x^{2} \log \left (\log \relax (x)\right )\right )} \log \left (x + 1\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-2*x-2)*log(x)*log(x^2/log(x)^2)+(-2*x-2)*log(x)*log(x+1)+(-6*x-4)*log(x)+4*x+4)/((x^4+x^3)*log(x)
*log(x^2/log(x)^2)^3+(3*x^4+3*x^3)*log(x)*log(x+1)*log(x^2/log(x)^2)^2+(3*x^4+3*x^3)*log(x)*log(x+1)^2*log(x^2
/log(x)^2)+(x^4+x^3)*log(x)*log(x+1)^3),x, algorithm="maxima")

[Out]

1/(x^2*log(x + 1)^2 + 4*x^2*log(x)^2 - 8*x^2*log(x)*log(log(x)) + 4*x^2*log(log(x))^2 + 4*(x^2*log(x) - x^2*lo
g(log(x)))*log(x + 1))

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mupad [F]  time = 0.00, size = -1, normalized size = -0.04 \begin {gather*} \int -\frac {\ln \relax (x)\,\left (6\,x+4\right )-4\,x+\ln \relax (x)\,\ln \left (\frac {x^2}{{\ln \relax (x)}^2}\right )\,\left (2\,x+2\right )+\ln \left (x+1\right )\,\ln \relax (x)\,\left (2\,x+2\right )-4}{\ln \relax (x)\,\left (x^4+x^3\right )\,{\ln \left (x+1\right )}^3+\ln \relax (x)\,\left (3\,x^4+3\,x^3\right )\,{\ln \left (x+1\right )}^2\,\ln \left (\frac {x^2}{{\ln \relax (x)}^2}\right )+\ln \relax (x)\,\left (3\,x^4+3\,x^3\right )\,\ln \left (x+1\right )\,{\ln \left (\frac {x^2}{{\ln \relax (x)}^2}\right )}^2+\ln \relax (x)\,\left (x^4+x^3\right )\,{\ln \left (\frac {x^2}{{\ln \relax (x)}^2}\right )}^3} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(-(log(x)*(6*x + 4) - 4*x + log(x)*log(x^2/log(x)^2)*(2*x + 2) + log(x + 1)*log(x)*(2*x + 2) - 4)/(log(x)*l
og(x^2/log(x)^2)^3*(x^3 + x^4) + log(x + 1)^3*log(x)*(x^3 + x^4) + log(x + 1)*log(x)*log(x^2/log(x)^2)^2*(3*x^
3 + 3*x^4) + log(x + 1)^2*log(x)*log(x^2/log(x)^2)*(3*x^3 + 3*x^4)),x)

[Out]

int(-(log(x)*(6*x + 4) - 4*x + log(x)*log(x^2/log(x)^2)*(2*x + 2) + log(x + 1)*log(x)*(2*x + 2) - 4)/(log(x)*l
og(x^2/log(x)^2)^3*(x^3 + x^4) + log(x + 1)^3*log(x)*(x^3 + x^4) + log(x + 1)*log(x)*log(x^2/log(x)^2)^2*(3*x^
3 + 3*x^4) + log(x + 1)^2*log(x)*log(x^2/log(x)^2)*(3*x^3 + 3*x^4)), x)

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sympy [B]  time = 0.42, size = 46, normalized size = 2.00 \begin {gather*} \frac {1}{x^{2} \log {\left (\frac {x^{2}}{\log {\relax (x )}^{2}} \right )}^{2} + 2 x^{2} \log {\left (\frac {x^{2}}{\log {\relax (x )}^{2}} \right )} \log {\left (x + 1 \right )} + x^{2} \log {\left (x + 1 \right )}^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-2*x-2)*ln(x)*ln(x**2/ln(x)**2)+(-2*x-2)*ln(x)*ln(x+1)+(-6*x-4)*ln(x)+4*x+4)/((x**4+x**3)*ln(x)*ln
(x**2/ln(x)**2)**3+(3*x**4+3*x**3)*ln(x)*ln(x+1)*ln(x**2/ln(x)**2)**2+(3*x**4+3*x**3)*ln(x)*ln(x+1)**2*ln(x**2
/ln(x)**2)+(x**4+x**3)*ln(x)*ln(x+1)**3),x)

[Out]

1/(x**2*log(x**2/log(x)**2)**2 + 2*x**2*log(x**2/log(x)**2)*log(x + 1) + x**2*log(x + 1)**2)

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