Integrand size = 141, antiderivative size = 23 \[ \int \frac {4+4 x+(-4-6 x) \log (x)+(-2-2 x) \log (x) \log (1+x)+(-2-2 x) \log (x) \log \left (\frac {x^2}{\log ^2(x)}\right )}{\left (x^3+x^4\right ) \log (x) \log ^3(1+x)+\left (3 x^3+3 x^4\right ) \log (x) \log ^2(1+x) \log \left (\frac {x^2}{\log ^2(x)}\right )+\left (3 x^3+3 x^4\right ) \log (x) \log (1+x) \log ^2\left (\frac {x^2}{\log ^2(x)}\right )+\left (x^3+x^4\right ) \log (x) \log ^3\left (\frac {x^2}{\log ^2(x)}\right )} \, dx=\frac {1}{\left (-x+x \left (1+\log (1+x)+\log \left (\frac {x^2}{\log ^2(x)}\right )\right )\right )^2} \] Output:
1/(x*(ln(x^2/ln(x)^2)+1+ln(1+x))-x)^2
Time = 0.25 (sec) , antiderivative size = 20, normalized size of antiderivative = 0.87 \[ \int \frac {4+4 x+(-4-6 x) \log (x)+(-2-2 x) \log (x) \log (1+x)+(-2-2 x) \log (x) \log \left (\frac {x^2}{\log ^2(x)}\right )}{\left (x^3+x^4\right ) \log (x) \log ^3(1+x)+\left (3 x^3+3 x^4\right ) \log (x) \log ^2(1+x) \log \left (\frac {x^2}{\log ^2(x)}\right )+\left (3 x^3+3 x^4\right ) \log (x) \log (1+x) \log ^2\left (\frac {x^2}{\log ^2(x)}\right )+\left (x^3+x^4\right ) \log (x) \log ^3\left (\frac {x^2}{\log ^2(x)}\right )} \, dx=\frac {1}{x^2 \left (\log (1+x)+\log \left (\frac {x^2}{\log ^2(x)}\right )\right )^2} \] Input:
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]
Output:
1/(x^2*(Log[1 + x] + Log[x^2/Log[x]^2])^2)
Time = 0.62 (sec) , antiderivative size = 20, normalized size of antiderivative = 0.87, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.014, Rules used = {7239, 7238}
Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.
| \(\displaystyle \int \frac {(-2 x-2) \log (x) \log \left (\frac {x^2}{\log ^2(x)}\right )+4 x+(-6 x-4) \log (x)+(-2 x-2) \log (x) \log (x+1)+4}{\left (x^4+x^3\right ) \log (x) \log ^3(x+1)+\left (3 x^4+3 x^3\right ) \log (x) \log \left (\frac {x^2}{\log ^2(x)}\right ) \log ^2(x+1)+\left (3 x^4+3 x^3\right ) \log (x) \log ^2\left (\frac {x^2}{\log ^2(x)}\right ) \log (x+1)+\left (x^4+x^3\right ) \log (x) \log ^3\left (\frac {x^2}{\log ^2(x)}\right )} \, dx\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle \int \frac {4 (x+1)-2 \log (x) \left ((x+1) \log \left (\frac {x^2}{\log ^2(x)}\right )+3 x+(x+1) \log (x+1)+2\right )}{x^3 (x+1) \log (x) \left (\log \left (\frac {x^2}{\log ^2(x)}\right )+\log (x+1)\right )^3}dx\) |
| \(\Big \downarrow \) 7238 |
| \(\displaystyle \frac {1}{x^2 \left (\log \left (\frac {x^2}{\log ^2(x)}\right )+\log (x+1)\right )^2}\) |
Input:
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]
Output:
1/(x^2*(Log[1 + x] + Log[x^2/Log[x]^2])^2)
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]
Int[u_, x_Symbol] :> With[{v = SimplifyIntegrand[u, x]}, Int[v, x] /; Simpl erIntegrandQ[v, u, x]]
Time = 49.72 (sec) , antiderivative size = 40, normalized size of antiderivative = 1.74
| method | result | size |
| parallelrisch | \(\frac {1}{x^{2} \left (\ln \left (1+x \right )^{2}+2 \ln \left (1+x \right ) \ln \left (\frac {x^{2}}{\ln \left (x \right )^{2}}\right )+\ln \left (\frac {x^{2}}{\ln \left (x \right )^{2}}\right )^{2}\right )}\) | \(40\) |
| risch | \(-\frac {4}{x^{2} {\left (-\pi \,\operatorname {csgn}\left (\frac {i}{\ln \left (x \right )^{2}}\right ) \operatorname {csgn}\left (i x^{2}\right ) \operatorname {csgn}\left (\frac {i x^{2}}{\ln \left (x \right )^{2}}\right )+\pi \,\operatorname {csgn}\left (\frac {i}{\ln \left (x \right )^{2}}\right ) \operatorname {csgn}\left (\frac {i x^{2}}{\ln \left (x \right )^{2}}\right )^{2}+\pi \operatorname {csgn}\left (i \ln \left (x \right )\right )^{2} \operatorname {csgn}\left (i \ln \left (x \right )^{2}\right )-2 \pi \,\operatorname {csgn}\left (i \ln \left (x \right )\right ) \operatorname {csgn}\left (i \ln \left (x \right )^{2}\right )^{2}+\pi \operatorname {csgn}\left (i \ln \left (x \right )^{2}\right )^{3}-\pi \operatorname {csgn}\left (i x \right )^{2} \operatorname {csgn}\left (i x^{2}\right )+2 \pi \,\operatorname {csgn}\left (i x \right ) \operatorname {csgn}\left (i x^{2}\right )^{2}-\pi \operatorname {csgn}\left (i x^{2}\right )^{3}+\pi \,\operatorname {csgn}\left (i x^{2}\right ) \operatorname {csgn}\left (\frac {i x^{2}}{\ln \left (x \right )^{2}}\right )^{2}-\pi \operatorname {csgn}\left (\frac {i x^{2}}{\ln \left (x \right )^{2}}\right )^{3}+4 i \ln \left (\ln \left (x \right )\right )-2 i \ln \left (1+x \right )-4 i \ln \left (x \right )\right )}^{2}}\) | \(212\) |
| default | \(\text {Expression too large to display}\) | \(4667\) |
| parts | \(\text {Expression too large to display}\) | \(4667\) |
Input:
int(((-2-2*x)*ln(x)*ln(x^2/ln(x)^2)+(-2-2*x)*ln(x)*ln(1+x)+(-4-6*x)*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(1+x)*ln(x ^2/ln(x)^2)^2+(3*x^4+3*x^3)*ln(x)*ln(1+x)^2*ln(x^2/ln(x)^2)+(x^4+x^3)*ln(x )*ln(1+x)^3),x,method=_RETURNVERBOSE)
Output:
1/x^2/(ln(1+x)^2+2*ln(1+x)*ln(x^2/ln(x)^2)+ln(x^2/ln(x)^2)^2)
Time = 0.08 (sec) , antiderivative size = 46, normalized size of antiderivative = 2.00 \[ \int \frac {4+4 x+(-4-6 x) \log (x)+(-2-2 x) \log (x) \log (1+x)+(-2-2 x) \log (x) \log \left (\frac {x^2}{\log ^2(x)}\right )}{\left (x^3+x^4\right ) \log (x) \log ^3(1+x)+\left (3 x^3+3 x^4\right ) \log (x) \log ^2(1+x) \log \left (\frac {x^2}{\log ^2(x)}\right )+\left (3 x^3+3 x^4\right ) \log (x) \log (1+x) \log ^2\left (\frac {x^2}{\log ^2(x)}\right )+\left (x^3+x^4\right ) \log (x) \log ^3\left (\frac {x^2}{\log ^2(x)}\right )} \, dx=\frac {1}{x^{2} \log \left (x + 1\right )^{2} + 2 \, x^{2} \log \left (x + 1\right ) \log \left (\frac {x^{2}}{\log \left (x\right )^{2}}\right ) + x^{2} \log \left (\frac {x^{2}}{\log \left (x\right )^{2}}\right )^{2}} \] Input:
integrate(((-2-2*x)*log(x)*log(x^2/log(x)^2)+(-2-2*x)*log(x)*log(1+x)+(-4- 6*x)*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(1+x)*log(x^2/log(x)^2)^2+(3*x^4+3*x^3)*log(x)*log(1+x)^2*log(x^2/l og(x)^2)+(x^4+x^3)*log(x)*log(1+x)^3),x, algorithm="fricas")
Output:
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)
Leaf count of result is larger than twice the leaf count of optimal. 46 vs. \(2 (22) = 44\).
Time = 0.20 (sec) , antiderivative size = 46, normalized size of antiderivative = 2.00 \[ \int \frac {4+4 x+(-4-6 x) \log (x)+(-2-2 x) \log (x) \log (1+x)+(-2-2 x) \log (x) \log \left (\frac {x^2}{\log ^2(x)}\right )}{\left (x^3+x^4\right ) \log (x) \log ^3(1+x)+\left (3 x^3+3 x^4\right ) \log (x) \log ^2(1+x) \log \left (\frac {x^2}{\log ^2(x)}\right )+\left (3 x^3+3 x^4\right ) \log (x) \log (1+x) \log ^2\left (\frac {x^2}{\log ^2(x)}\right )+\left (x^3+x^4\right ) \log (x) \log ^3\left (\frac {x^2}{\log ^2(x)}\right )} \, dx=\frac {1}{x^{2} \log {\left (\frac {x^{2}}{\log {\left (x \right )}^{2}} \right )}^{2} + 2 x^{2} \log {\left (\frac {x^{2}}{\log {\left (x \right )}^{2}} \right )} \log {\left (x + 1 \right )} + x^{2} \log {\left (x + 1 \right )}^{2}} \] Input:
integrate(((-2-2*x)*ln(x)*ln(x**2/ln(x)**2)+(-2-2*x)*ln(x)*ln(1+x)+(-4-6*x )*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(1+x)*ln(x**2/ln(x)**2)**2+(3*x**4+3*x**3)*ln(x)*ln(1+x)**2*ln(x**2/l n(x)**2)+(x**4+x**3)*ln(x)*ln(1+x)**3),x)
Output:
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)
Leaf count of result is larger than twice the leaf count of optimal. 63 vs. \(2 (23) = 46\).
Time = 0.12 (sec) , antiderivative size = 63, normalized size of antiderivative = 2.74 \[ \int \frac {4+4 x+(-4-6 x) \log (x)+(-2-2 x) \log (x) \log (1+x)+(-2-2 x) \log (x) \log \left (\frac {x^2}{\log ^2(x)}\right )}{\left (x^3+x^4\right ) \log (x) \log ^3(1+x)+\left (3 x^3+3 x^4\right ) \log (x) \log ^2(1+x) \log \left (\frac {x^2}{\log ^2(x)}\right )+\left (3 x^3+3 x^4\right ) \log (x) \log (1+x) \log ^2\left (\frac {x^2}{\log ^2(x)}\right )+\left (x^3+x^4\right ) \log (x) \log ^3\left (\frac {x^2}{\log ^2(x)}\right )} \, dx=\frac {1}{x^{2} \log \left (x + 1\right )^{2} + 4 \, x^{2} \log \left (x\right )^{2} - 8 \, x^{2} \log \left (x\right ) \log \left (\log \left (x\right )\right ) + 4 \, x^{2} \log \left (\log \left (x\right )\right )^{2} + 4 \, {\left (x^{2} \log \left (x\right ) - x^{2} \log \left (\log \left (x\right )\right )\right )} \log \left (x + 1\right )} \] Input:
integrate(((-2-2*x)*log(x)*log(x^2/log(x)^2)+(-2-2*x)*log(x)*log(1+x)+(-4- 6*x)*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(1+x)*log(x^2/log(x)^2)^2+(3*x^4+3*x^3)*log(x)*log(1+x)^2*log(x^2/l og(x)^2)+(x^4+x^3)*log(x)*log(1+x)^3),x, algorithm="maxima")
Output:
1/(x^2*log(x + 1)^2 + 4*x^2*log(x)^2 - 8*x^2*log(x)*log(log(x)) + 4*x^2*lo g(log(x))^2 + 4*(x^2*log(x) - x^2*log(log(x)))*log(x + 1))
Leaf count of result is larger than twice the leaf count of optimal. 315 vs. \(2 (23) = 46\).
Time = 1.35 (sec) , antiderivative size = 315, normalized size of antiderivative = 13.70 \[ \int \frac {4+4 x+(-4-6 x) \log (x)+(-2-2 x) \log (x) \log (1+x)+(-2-2 x) \log (x) \log \left (\frac {x^2}{\log ^2(x)}\right )}{\left (x^3+x^4\right ) \log (x) \log ^3(1+x)+\left (3 x^3+3 x^4\right ) \log (x) \log ^2(1+x) \log \left (\frac {x^2}{\log ^2(x)}\right )+\left (3 x^3+3 x^4\right ) \log (x) \log (1+x) \log ^2\left (\frac {x^2}{\log ^2(x)}\right )+\left (x^3+x^4\right ) \log (x) \log ^3\left (\frac {x^2}{\log ^2(x)}\right )} \, dx=-\frac {3 \, x \log \left (x\right ) - 2 \, x + 2 \, \log \left (x\right ) - 2}{6 \, x^{3} \log \left (x + 1\right ) \log \left (x\right ) \log \left (\log \left (x\right )^{2}\right ) + 12 \, x^{3} \log \left (x\right )^{2} \log \left (\log \left (x\right )^{2}\right ) - 3 \, x^{3} \log \left (x\right ) \log \left (\log \left (x\right )^{2}\right )^{2} - 3 \, x^{3} \log \left (x + 1\right )^{2} \log \left (x\right ) - 12 \, x^{3} \log \left (x + 1\right ) \log \left (x\right )^{2} - 12 \, x^{3} \log \left (x\right )^{3} - 8 \, x^{3} \log \left (x\right ) \log \left (\log \left (x\right )^{2}\right ) + 4 \, x^{2} \log \left (x + 1\right ) \log \left (x\right ) \log \left (\log \left (x\right )^{2}\right ) + 8 \, x^{2} \log \left (x\right )^{2} \log \left (\log \left (x\right )^{2}\right ) + 2 \, x^{3} \log \left (\log \left (x\right )^{2}\right )^{2} - 2 \, x^{2} \log \left (x\right ) \log \left (\log \left (x\right )^{2}\right )^{2} - 4 \, x^{3} \log \left (\log \left (x\right )^{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 \left (x\right ) - 2 \, x^{2} \log \left (x + 1\right )^{2} \log \left (x\right ) + 8 \, x^{3} \log \left (x\right )^{2} - 8 \, x^{2} \log \left (x + 1\right ) \log \left (x\right )^{2} - 8 \, x^{2} \log \left (x\right )^{3} - 8 \, x^{2} \log \left (x\right ) \log \left (\log \left (x\right )^{2}\right ) + 2 \, x^{2} \log \left (\log \left (x\right )^{2}\right )^{2} - 4 \, x^{2} \log \left (\log \left (x\right )^{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 \left (x\right ) + 8 \, x^{2} \log \left (x\right )^{2}} \] Input:
integrate(((-2-2*x)*log(x)*log(x^2/log(x)^2)+(-2-2*x)*log(x)*log(1+x)+(-4- 6*x)*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(1+x)*log(x^2/log(x)^2)^2+(3*x^4+3*x^3)*log(x)*log(1+x)^2*log(x^2/l og(x)^2)+(x^4+x^3)*log(x)*log(1+x)^3),x, algorithm="giac")
Output:
-(3*x*log(x) - 2*x + 2*log(x) - 2)/(6*x^3*log(x + 1)*log(x)*log(log(x)^2) + 12*x^3*log(x)^2*log(log(x)^2) - 3*x^3*log(x)*log(log(x)^2)^2 - 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*lo g(x)*log(log(x)^2) + 4*x^2*log(x + 1)*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) + 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)
Timed out. \[ \int \frac {4+4 x+(-4-6 x) \log (x)+(-2-2 x) \log (x) \log (1+x)+(-2-2 x) \log (x) \log \left (\frac {x^2}{\log ^2(x)}\right )}{\left (x^3+x^4\right ) \log (x) \log ^3(1+x)+\left (3 x^3+3 x^4\right ) \log (x) \log ^2(1+x) \log \left (\frac {x^2}{\log ^2(x)}\right )+\left (3 x^3+3 x^4\right ) \log (x) \log (1+x) \log ^2\left (\frac {x^2}{\log ^2(x)}\right )+\left (x^3+x^4\right ) \log (x) \log ^3\left (\frac {x^2}{\log ^2(x)}\right )} \, dx=\int -\frac {\ln \left (x\right )\,\left (6\,x+4\right )-4\,x+\ln \left (x\right )\,\ln \left (\frac {x^2}{{\ln \left (x\right )}^2}\right )\,\left (2\,x+2\right )+\ln \left (x+1\right )\,\ln \left (x\right )\,\left (2\,x+2\right )-4}{\ln \left (x\right )\,\left (x^4+x^3\right )\,{\ln \left (x+1\right )}^3+\ln \left (x\right )\,\left (3\,x^4+3\,x^3\right )\,{\ln \left (x+1\right )}^2\,\ln \left (\frac {x^2}{{\ln \left (x\right )}^2}\right )+\ln \left (x\right )\,\left (3\,x^4+3\,x^3\right )\,\ln \left (x+1\right )\,{\ln \left (\frac {x^2}{{\ln \left (x\right )}^2}\right )}^2+\ln \left (x\right )\,\left (x^4+x^3\right )\,{\ln \left (\frac {x^2}{{\ln \left (x\right )}^2}\right )}^3} \,d x \] Input:
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)*log(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)
Output:
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)*log(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)
Time = 0.19 (sec) , antiderivative size = 39, normalized size of antiderivative = 1.70 \[ \int \frac {4+4 x+(-4-6 x) \log (x)+(-2-2 x) \log (x) \log (1+x)+(-2-2 x) \log (x) \log \left (\frac {x^2}{\log ^2(x)}\right )}{\left (x^3+x^4\right ) \log (x) \log ^3(1+x)+\left (3 x^3+3 x^4\right ) \log (x) \log ^2(1+x) \log \left (\frac {x^2}{\log ^2(x)}\right )+\left (3 x^3+3 x^4\right ) \log (x) \log (1+x) \log ^2\left (\frac {x^2}{\log ^2(x)}\right )+\left (x^3+x^4\right ) \log (x) \log ^3\left (\frac {x^2}{\log ^2(x)}\right )} \, dx=\frac {1}{x^{2} \left (\mathrm {log}\left (x +1\right )^{2}+2 \,\mathrm {log}\left (x +1\right ) \mathrm {log}\left (\frac {x^{2}}{\mathrm {log}\left (x \right )^{2}}\right )+\mathrm {log}\left (\frac {x^{2}}{\mathrm {log}\left (x \right )^{2}}\right )^{2}\right )} \] Input:
int(((-2-2*x)*log(x)*log(x^2/log(x)^2)+(-2-2*x)*log(x)*log(1+x)+(-4-6*x)*l og(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)*lo g(1+x)*log(x^2/log(x)^2)^2+(3*x^4+3*x^3)*log(x)*log(1+x)^2*log(x^2/log(x)^ 2)+(x^4+x^3)*log(x)*log(1+x)^3),x)
Output:
1/(x**2*(log(x + 1)**2 + 2*log(x + 1)*log(x**2/log(x)**2) + log(x**2/log(x )**2)**2))