Integrand size = 52, antiderivative size = 23 \[ \int \frac {78125+78125 x^3+e^3 \left (-15625-15625 x^3\right )+\left (-234375 x^3+46875 e^3 x^3\right ) \log \left (\frac {x}{2}\right )}{x+2 x^4+x^7} \, dx=\frac {15625 \left (5-e^3\right ) x \log \left (\frac {x}{2}\right )}{x+x^4} \] Output:
15625*ln(1/2*x)/(x^4+x)*x*(-exp(3)+5)
Time = 0.14 (sec) , antiderivative size = 20, normalized size of antiderivative = 0.87 \[ \int \frac {78125+78125 x^3+e^3 \left (-15625-15625 x^3\right )+\left (-234375 x^3+46875 e^3 x^3\right ) \log \left (\frac {x}{2}\right )}{x+2 x^4+x^7} \, dx=-\frac {15625 \left (-5+e^3\right ) \log \left (\frac {x}{2}\right )}{1+x^3} \] Input:
Integrate[(78125 + 78125*x^3 + E^3*(-15625 - 15625*x^3) + (-234375*x^3 + 4 6875*E^3*x^3)*Log[x/2])/(x + 2*x^4 + x^7),x]
Output:
(-15625*(-5 + E^3)*Log[x/2])/(1 + x^3)
Time = 0.61 (sec) , antiderivative size = 30, normalized size of antiderivative = 1.30, number of steps used = 7, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.135, Rules used = {2026, 1380, 27, 7292, 27, 7293, 2009}
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.
\(\displaystyle \int \frac {78125 x^3+e^3 \left (-15625 x^3-15625\right )+\left (46875 e^3 x^3-234375 x^3\right ) \log \left (\frac {x}{2}\right )+78125}{x^7+2 x^4+x} \, dx\) |
\(\Big \downarrow \) 2026 |
\(\displaystyle \int \frac {78125 x^3+e^3 \left (-15625 x^3-15625\right )+\left (46875 e^3 x^3-234375 x^3\right ) \log \left (\frac {x}{2}\right )+78125}{x \left (x^6+2 x^3+1\right )}dx\) |
\(\Big \downarrow \) 1380 |
\(\displaystyle \int \frac {15625 \left (5 x^3-e^3 \left (x^3+1\right )-3 \left (5 x^3-e^3 x^3\right ) \log \left (\frac {x}{2}\right )+5\right )}{x \left (x^3+1\right )^2}dx\) |
\(\Big \downarrow \) 27 |
\(\displaystyle 15625 \int \frac {5 x^3-e^3 \left (x^3+1\right )-3 \left (5 x^3-e^3 x^3\right ) \log \left (\frac {x}{2}\right )+5}{x \left (x^3+1\right )^2}dx\) |
\(\Big \downarrow \) 7292 |
\(\displaystyle 15625 \int \frac {\left (5-e^3\right ) \left (-3 \log \left (\frac {x}{2}\right ) x^3+x^3+1\right )}{x \left (x^3+1\right )^2}dx\) |
\(\Big \downarrow \) 27 |
\(\displaystyle 15625 \left (5-e^3\right ) \int \frac {-3 \log \left (\frac {x}{2}\right ) x^3+x^3+1}{x \left (x^3+1\right )^2}dx\) |
\(\Big \downarrow \) 7293 |
\(\displaystyle 15625 \left (5-e^3\right ) \int \left (-\frac {3 \log \left (\frac {x}{2}\right ) x^2}{\left (x^3+1\right )^2}+\frac {x^2}{\left (x^3+1\right )^2}+\frac {1}{\left (x^3+1\right )^2 x}\right )dx\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle 15625 \left (5-e^3\right ) \left (\log (x)-\frac {x^3 \log \left (\frac {x}{2}\right )}{x^3+1}\right )\) |
Input:
Int[(78125 + 78125*x^3 + E^3*(-15625 - 15625*x^3) + (-234375*x^3 + 46875*E ^3*x^3)*Log[x/2])/(x + 2*x^4 + x^7),x]
Output:
15625*(5 - E^3)*(-((x^3*Log[x/2])/(1 + x^3)) + Log[x])
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
Int[(u_)*((a_) + (c_.)*(x_)^(n2_.) + (b_.)*(x_)^(n_))^(p_.), x_Symbol] :> S imp[1/c^p Int[u*(b/2 + c*x^n)^(2*p), x], x] /; FreeQ[{a, b, c, n, p}, x] && EqQ[n2, 2*n] && EqQ[b^2 - 4*a*c, 0] && IntegerQ[p]
Int[(Fx_.)*(Px_)^(p_.), x_Symbol] :> With[{r = Expon[Px, x, Min]}, Int[x^(p *r)*ExpandToSum[Px/x^r, x]^p*Fx, x] /; IGtQ[r, 0]] /; PolyQ[Px, x] && Integ erQ[p] && !MonomialQ[Px, x] && (ILtQ[p, 0] || !PolyQ[u, x])
Time = 0.48 (sec) , antiderivative size = 18, normalized size of antiderivative = 0.78
method | result | size |
risch | \(-\frac {15625 \left ({\mathrm e}^{3}-5\right ) \ln \left (\frac {x}{2}\right )}{x^{3}+1}\) | \(18\) |
norman | \(\frac {\left (-15625 \,{\mathrm e}^{3}+78125\right ) \ln \left (\frac {x}{2}\right )}{x^{3}+1}\) | \(19\) |
parallelrisch | \(-\frac {15625 \ln \left (\frac {x}{2}\right ) {\mathrm e}^{3}-78125 \ln \left (\frac {x}{2}\right )}{x^{3}+1}\) | \(25\) |
orering | \(-\frac {\left (7 x^{3}+1\right ) \left (\left (46875 x^{3} {\mathrm e}^{3}-234375 x^{3}\right ) \ln \left (\frac {x}{2}\right )+\left (-15625 x^{3}-15625\right ) {\mathrm e}^{3}+78125 x^{3}+78125\right )}{9 x^{2} \left (x^{7}+2 x^{4}+x \right )}-\frac {\left (1+x \right ) \left (x^{2}-x +1\right ) \left (\frac {\left (140625 x^{2} {\mathrm e}^{3}-703125 x^{2}\right ) \ln \left (\frac {x}{2}\right )+\frac {46875 x^{3} {\mathrm e}^{3}-234375 x^{3}}{x}-46875 x^{2} {\mathrm e}^{3}+234375 x^{2}}{x^{7}+2 x^{4}+x}-\frac {\left (\left (46875 x^{3} {\mathrm e}^{3}-234375 x^{3}\right ) \ln \left (\frac {x}{2}\right )+\left (-15625 x^{3}-15625\right ) {\mathrm e}^{3}+78125 x^{3}+78125\right ) \left (7 x^{6}+8 x^{3}+1\right )}{\left (x^{7}+2 x^{4}+x \right )^{2}}\right )}{9 x}\) | \(200\) |
parts | \(\left (-15625 \,{\mathrm e}^{3}+78125\right ) \left (-\frac {\ln \left (1+x \right )}{3}+\ln \left (x \right )-\frac {\ln \left (x^{2}-x +1\right )}{3}\right )+\left (46875 \,{\mathrm e}^{3}-234375\right ) \left (-\frac {\ln \left (x^{2}-x +1\right )}{9}-\frac {\ln \left (\frac {x}{2}\right ) \left (i \ln \left (\frac {i \sqrt {3}-2 x +1}{1+i \sqrt {3}}\right ) \sqrt {3}\, x^{2}-i \ln \left (\frac {i \sqrt {3}+2 x -1}{i \sqrt {3}-1}\right ) \sqrt {3}\, x^{2}-i \ln \left (\frac {i \sqrt {3}-2 x +1}{1+i \sqrt {3}}\right ) \sqrt {3}\, x +i \ln \left (\frac {i \sqrt {3}+2 x -1}{i \sqrt {3}-1}\right ) \sqrt {3}\, x +i \sqrt {3}\, \ln \left (\frac {i \sqrt {3}-2 x +1}{1+i \sqrt {3}}\right )-i \sqrt {3}\, \ln \left (\frac {i \sqrt {3}+2 x -1}{i \sqrt {3}-1}\right )-6 x^{2}+3 x \right )}{27 \left (x^{2}-x +1\right )}-\frac {\ln \left (1+x \right )}{9}+\frac {x \ln \left (\frac {x}{2}\right )}{9+9 x}+\frac {i \sqrt {3}\, \ln \left (\frac {x}{2}\right ) \ln \left (\frac {i \sqrt {3}-2 x +1}{1+i \sqrt {3}}\right )}{27}-\frac {i \sqrt {3}\, \ln \left (\frac {x}{2}\right ) \ln \left (\frac {i \sqrt {3}+2 x -1}{i \sqrt {3}-1}\right )}{27}\right )\) | \(339\) |
derivativedivides | \(\left (-15625 \,{\mathrm e}^{3}+78125\right ) \left (\ln \left (\frac {x}{2}\right )-\frac {\ln \left (1+x \right )}{3}-\frac {\ln \left (x^{2}-x +1\right )}{3}\right )+\left (375000 \,{\mathrm e}^{3}-1875000\right ) \left (-\frac {\ln \left (1+x \right )}{72}+\frac {x \ln \left (\frac {x}{2}\right )}{72+72 x}-\frac {\ln \left (x^{2}-x +1\right )}{72}-\frac {\ln \left (\frac {x}{2}\right ) \left (i \ln \left (\frac {i \sqrt {3}-2 x +1}{1+i \sqrt {3}}\right ) \sqrt {3}\, x^{2}-i \ln \left (\frac {i \sqrt {3}+2 x -1}{i \sqrt {3}-1}\right ) \sqrt {3}\, x^{2}-i \ln \left (\frac {i \sqrt {3}-2 x +1}{1+i \sqrt {3}}\right ) \sqrt {3}\, x +i \ln \left (\frac {i \sqrt {3}+2 x -1}{i \sqrt {3}-1}\right ) \sqrt {3}\, x +i \sqrt {3}\, \ln \left (\frac {i \sqrt {3}-2 x +1}{1+i \sqrt {3}}\right )-i \sqrt {3}\, \ln \left (\frac {i \sqrt {3}+2 x -1}{i \sqrt {3}-1}\right )-6 x^{2}+3 x \right )}{216 \left (x^{2}-x +1\right )}+\frac {i \sqrt {3}\, \ln \left (\frac {x}{2}\right ) \ln \left (\frac {i \sqrt {3}-2 x +1}{1+i \sqrt {3}}\right )}{216}-\frac {i \sqrt {3}\, \ln \left (\frac {x}{2}\right ) \ln \left (\frac {i \sqrt {3}+2 x -1}{i \sqrt {3}-1}\right )}{216}\right )\) | \(341\) |
default | \(\left (-15625 \,{\mathrm e}^{3}+78125\right ) \left (\ln \left (\frac {x}{2}\right )-\frac {\ln \left (1+x \right )}{3}-\frac {\ln \left (x^{2}-x +1\right )}{3}\right )+\left (375000 \,{\mathrm e}^{3}-1875000\right ) \left (-\frac {\ln \left (1+x \right )}{72}+\frac {x \ln \left (\frac {x}{2}\right )}{72+72 x}-\frac {\ln \left (x^{2}-x +1\right )}{72}-\frac {\ln \left (\frac {x}{2}\right ) \left (i \ln \left (\frac {i \sqrt {3}-2 x +1}{1+i \sqrt {3}}\right ) \sqrt {3}\, x^{2}-i \ln \left (\frac {i \sqrt {3}+2 x -1}{i \sqrt {3}-1}\right ) \sqrt {3}\, x^{2}-i \ln \left (\frac {i \sqrt {3}-2 x +1}{1+i \sqrt {3}}\right ) \sqrt {3}\, x +i \ln \left (\frac {i \sqrt {3}+2 x -1}{i \sqrt {3}-1}\right ) \sqrt {3}\, x +i \sqrt {3}\, \ln \left (\frac {i \sqrt {3}-2 x +1}{1+i \sqrt {3}}\right )-i \sqrt {3}\, \ln \left (\frac {i \sqrt {3}+2 x -1}{i \sqrt {3}-1}\right )-6 x^{2}+3 x \right )}{216 \left (x^{2}-x +1\right )}+\frac {i \sqrt {3}\, \ln \left (\frac {x}{2}\right ) \ln \left (\frac {i \sqrt {3}-2 x +1}{1+i \sqrt {3}}\right )}{216}-\frac {i \sqrt {3}\, \ln \left (\frac {x}{2}\right ) \ln \left (\frac {i \sqrt {3}+2 x -1}{i \sqrt {3}-1}\right )}{216}\right )\) | \(341\) |
Input:
int(((46875*x^3*exp(3)-234375*x^3)*ln(1/2*x)+(-15625*x^3-15625)*exp(3)+781 25*x^3+78125)/(x^7+2*x^4+x),x,method=_RETURNVERBOSE)
Output:
-15625*(exp(3)-5)/(x^3+1)*ln(1/2*x)
Time = 0.09 (sec) , antiderivative size = 17, normalized size of antiderivative = 0.74 \[ \int \frac {78125+78125 x^3+e^3 \left (-15625-15625 x^3\right )+\left (-234375 x^3+46875 e^3 x^3\right ) \log \left (\frac {x}{2}\right )}{x+2 x^4+x^7} \, dx=-\frac {15625 \, {\left (e^{3} - 5\right )} \log \left (\frac {1}{2} \, x\right )}{x^{3} + 1} \] Input:
integrate(((46875*x^3*exp(3)-234375*x^3)*log(1/2*x)+(-15625*x^3-15625)*exp (3)+78125*x^3+78125)/(x^7+2*x^4+x),x, algorithm="fricas")
Output:
-15625*(e^3 - 5)*log(1/2*x)/(x^3 + 1)
Time = 0.10 (sec) , antiderivative size = 15, normalized size of antiderivative = 0.65 \[ \int \frac {78125+78125 x^3+e^3 \left (-15625-15625 x^3\right )+\left (-234375 x^3+46875 e^3 x^3\right ) \log \left (\frac {x}{2}\right )}{x+2 x^4+x^7} \, dx=\frac {\left (78125 - 15625 e^{3}\right ) \log {\left (\frac {x}{2} \right )}}{x^{3} + 1} \] Input:
integrate(((46875*x**3*exp(3)-234375*x**3)*ln(1/2*x)+(-15625*x**3-15625)*e xp(3)+78125*x**3+78125)/(x**7+2*x**4+x),x)
Output:
(78125 - 15625*exp(3))*log(x/2)/(x**3 + 1)
Leaf count of result is larger than twice the leaf count of optimal. 123 vs. \(2 (18) = 36\).
Time = 0.12 (sec) , antiderivative size = 123, normalized size of antiderivative = 5.35 \[ \int \frac {78125+78125 x^3+e^3 \left (-15625-15625 x^3\right )+\left (-234375 x^3+46875 e^3 x^3\right ) \log \left (\frac {x}{2}\right )}{x+2 x^4+x^7} \, dx=-\frac {15625}{3} \, {\left (\frac {3 \, \log \left (\frac {1}{2} \, x\right )}{x^{3} + 1} + \log \left (x^{3} + 1\right ) - \log \left (x^{3}\right )\right )} e^{3} - \frac {15625}{3} \, {\left (\frac {1}{x^{3} + 1} - \log \left (x^{2} - x + 1\right ) - \log \left (x + 1\right ) + 3 \, \log \left (x\right )\right )} e^{3} + \frac {15625 \, e^{3}}{3 \, {\left (x^{3} + 1\right )}} + \frac {78125 \, \log \left (\frac {1}{2} \, x\right )}{x^{3} + 1} + \frac {78125}{3} \, \log \left (x^{3} + 1\right ) - \frac {78125}{3} \, \log \left (x^{3}\right ) - \frac {78125}{3} \, \log \left (x^{2} - x + 1\right ) - \frac {78125}{3} \, \log \left (x + 1\right ) + 78125 \, \log \left (x\right ) \] Input:
integrate(((46875*x^3*exp(3)-234375*x^3)*log(1/2*x)+(-15625*x^3-15625)*exp (3)+78125*x^3+78125)/(x^7+2*x^4+x),x, algorithm="maxima")
Output:
-15625/3*(3*log(1/2*x)/(x^3 + 1) + log(x^3 + 1) - log(x^3))*e^3 - 15625/3* (1/(x^3 + 1) - log(x^2 - x + 1) - log(x + 1) + 3*log(x))*e^3 + 15625/3*e^3 /(x^3 + 1) + 78125*log(1/2*x)/(x^3 + 1) + 78125/3*log(x^3 + 1) - 78125/3*l og(x^3) - 78125/3*log(x^2 - x + 1) - 78125/3*log(x + 1) + 78125*log(x)
Time = 0.12 (sec) , antiderivative size = 23, normalized size of antiderivative = 1.00 \[ \int \frac {78125+78125 x^3+e^3 \left (-15625-15625 x^3\right )+\left (-234375 x^3+46875 e^3 x^3\right ) \log \left (\frac {x}{2}\right )}{x+2 x^4+x^7} \, dx=-\frac {15625 \, {\left (e^{3} \log \left (\frac {1}{2} \, x\right ) - 5 \, \log \left (\frac {1}{2} \, x\right )\right )}}{x^{3} + 1} \] Input:
integrate(((46875*x^3*exp(3)-234375*x^3)*log(1/2*x)+(-15625*x^3-15625)*exp (3)+78125*x^3+78125)/(x^7+2*x^4+x),x, algorithm="giac")
Output:
-15625*(e^3*log(1/2*x) - 5*log(1/2*x))/(x^3 + 1)
Time = 2.74 (sec) , antiderivative size = 24, normalized size of antiderivative = 1.04 \[ \int \frac {78125+78125 x^3+e^3 \left (-15625-15625 x^3\right )+\left (-234375 x^3+46875 e^3 x^3\right ) \log \left (\frac {x}{2}\right )}{x+2 x^4+x^7} \, dx=-\frac {x^2\,\ln \left (\frac {x}{2}\right )\,\left (15625\,{\mathrm {e}}^3-78125\right )}{x^5+x^2} \] Input:
int((log(x/2)*(46875*x^3*exp(3) - 234375*x^3) - exp(3)*(15625*x^3 + 15625) + 78125*x^3 + 78125)/(x + 2*x^4 + x^7),x)
Output:
-(x^2*log(x/2)*(15625*exp(3) - 78125))/(x^2 + x^5)
Time = 0.22 (sec) , antiderivative size = 58, normalized size of antiderivative = 2.52 \[ \int \frac {78125+78125 x^3+e^3 \left (-15625-15625 x^3\right )+\left (-234375 x^3+46875 e^3 x^3\right ) \log \left (\frac {x}{2}\right )}{x+2 x^4+x^7} \, dx=\frac {15625 \,\mathrm {log}\left (\frac {x}{2}\right ) e^{3} x^{3}-78125 \,\mathrm {log}\left (\frac {x}{2}\right ) x^{3}-15625 \,\mathrm {log}\left (x \right ) e^{3} x^{3}-15625 \,\mathrm {log}\left (x \right ) e^{3}+78125 \,\mathrm {log}\left (x \right ) x^{3}+78125 \,\mathrm {log}\left (x \right )}{x^{3}+1} \] Input:
int(((46875*x^3*exp(3)-234375*x^3)*log(1/2*x)+(-15625*x^3-15625)*exp(3)+78 125*x^3+78125)/(x^7+2*x^4+x),x)
Output:
(15625*(log(x/2)*e**3*x**3 - 5*log(x/2)*x**3 - log(x)*e**3*x**3 - log(x)*e **3 + 5*log(x)*x**3 + 5*log(x)))/(x**3 + 1)