Integrand size = 64, antiderivative size = 25 \[ \int \frac {436+784 x-13 x^2+4 x^3+2 x^4+\left (-16+8 x+7 x^2-2 x^3-x^4\right ) \log (3)}{16-8 x-7 x^2+2 x^3+x^4} \, dx=2 x+\frac {400+x}{4-x-x^2}-x \log (3) \] Output:
2*x+(400+x)/(-x^2-x+4)-x*ln(3)
Time = 0.02 (sec) , antiderivative size = 23, normalized size of antiderivative = 0.92 \[ \int \frac {436+784 x-13 x^2+4 x^3+2 x^4+\left (-16+8 x+7 x^2-2 x^3-x^4\right ) \log (3)}{16-8 x-7 x^2+2 x^3+x^4} \, dx=\frac {-400-x}{-4+x+x^2}+x (2-\log (3)) \] Input:
Integrate[(436 + 784*x - 13*x^2 + 4*x^3 + 2*x^4 + (-16 + 8*x + 7*x^2 - 2*x ^3 - x^4)*Log[3])/(16 - 8*x - 7*x^2 + 2*x^3 + x^4),x]
Output:
(-400 - x)/(-4 + x + x^2) + x*(2 - Log[3])
Time = 0.32 (sec) , antiderivative size = 37, normalized size of antiderivative = 1.48, number of steps used = 7, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.094, Rules used = {2459, 1380, 2345, 27, 281, 24}
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.
\(\displaystyle \int \frac {2 x^4+4 x^3-13 x^2+\left (-x^4-2 x^3+7 x^2+8 x-16\right ) \log (3)+784 x+436}{x^4+2 x^3-7 x^2-8 x+16} \, dx\) |
\(\Big \downarrow \) 2459 |
\(\displaystyle \int \frac {799 \left (x+\frac {1}{2}\right )+\left (x+\frac {1}{2}\right )^4 (2-\log (3))-\frac {1}{2} \left (x+\frac {1}{2}\right )^2 (32-17 \log (3))+\frac {17}{16} (38-17 \log (3))}{\left (x+\frac {1}{2}\right )^4-\frac {17}{2} \left (x+\frac {1}{2}\right )^2+\frac {289}{16}}d\left (x+\frac {1}{2}\right )\) |
\(\Big \downarrow \) 1380 |
\(\displaystyle \int \frac {12784 \left (x+\frac {1}{2}\right )+16 \left (x+\frac {1}{2}\right )^4 (2-\log (3))-8 \left (x+\frac {1}{2}\right )^2 (32-17 \log (3))+17 (38-17 \log (3))}{\left (17-4 \left (x+\frac {1}{2}\right )^2\right )^2}d\left (x+\frac {1}{2}\right )\) |
\(\Big \downarrow \) 2345 |
\(\displaystyle \frac {2 \left (2 \left (x+\frac {1}{2}\right )+799\right )}{17-4 \left (x+\frac {1}{2}\right )^2}-\frac {1}{34} \int -\frac {34 \left (17 (2-\log (3))-4 \left (x+\frac {1}{2}\right )^2 (2-\log (3))\right )}{17-4 \left (x+\frac {1}{2}\right )^2}d\left (x+\frac {1}{2}\right )\) |
\(\Big \downarrow \) 27 |
\(\displaystyle \int \frac {17 (2-\log (3))-4 \left (x+\frac {1}{2}\right )^2 (2-\log (3))}{17-4 \left (x+\frac {1}{2}\right )^2}d\left (x+\frac {1}{2}\right )+\frac {2 \left (2 \left (x+\frac {1}{2}\right )+799\right )}{17-4 \left (x+\frac {1}{2}\right )^2}\) |
\(\Big \downarrow \) 281 |
\(\displaystyle (2-\log (3)) \int 1d\left (x+\frac {1}{2}\right )+\frac {2 \left (2 \left (x+\frac {1}{2}\right )+799\right )}{17-4 \left (x+\frac {1}{2}\right )^2}\) |
\(\Big \downarrow \) 24 |
\(\displaystyle \frac {2 \left (2 \left (x+\frac {1}{2}\right )+799\right )}{17-4 \left (x+\frac {1}{2}\right )^2}+\left (x+\frac {1}{2}\right ) (2-\log (3))\) |
Input:
Int[(436 + 784*x - 13*x^2 + 4*x^3 + 2*x^4 + (-16 + 8*x + 7*x^2 - 2*x^3 - x ^4)*Log[3])/(16 - 8*x - 7*x^2 + 2*x^3 + x^4),x]
Output:
(2*(799 + 2*(1/2 + x)))/(17 - 4*(1/2 + x)^2) + (1/2 + x)*(2 - Log[3])
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_) + (b_.)*(x_)^(n_))^(p_.)*((c_) + (d_.)*(x_)^(n_))^(q_.), x_ Symbol] :> Simp[(b/d)^p Int[u*(c + d*x^n)^(p + q), x], x] /; FreeQ[{a, b, c, d, n, p, q}, x] && EqQ[b*c - a*d, 0] && IntegerQ[p] && !(IntegerQ[q] & & SimplerQ[a + b*x^n, c + d*x^n])
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[(Pq_)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> With[{Q = PolynomialQuot ient[Pq, a + b*x^2, x], f = Coeff[PolynomialRemainder[Pq, a + b*x^2, x], x, 0], g = Coeff[PolynomialRemainder[Pq, a + b*x^2, x], x, 1]}, Simp[(a*g - b *f*x)*((a + b*x^2)^(p + 1)/(2*a*b*(p + 1))), x] + Simp[1/(2*a*(p + 1)) In t[(a + b*x^2)^(p + 1)*ExpandToSum[2*a*(p + 1)*Q + f*(2*p + 3), x], x], x]] /; FreeQ[{a, b}, x] && PolyQ[Pq, x] && LtQ[p, -1]
Int[(Pn_)^(p_.)*(Qx_), x_Symbol] :> With[{S = Coeff[Pn, x, Expon[Pn, x] - 1 ]/(Expon[Pn, x]*Coeff[Pn, x, Expon[Pn, x]])}, Subst[Int[ExpandToSum[Pn /. x -> x - S, x]^p*ExpandToSum[Qx /. x -> x - S, x], x], x, x + S] /; Binomial Q[Pn /. x -> x - S, x] || (IntegerQ[Expon[Pn, x]/2] && TrinomialQ[Pn /. x - > x - S, x])] /; FreeQ[p, x] && PolyQ[Pn, x] && GtQ[Expon[Pn, x], 2] && NeQ [Coeff[Pn, x, Expon[Pn, x] - 1], 0] && PolyQ[Qx, x] && !(MonomialQ[Qx, x] && IGtQ[p, 0])
Time = 0.13 (sec) , antiderivative size = 24, normalized size of antiderivative = 0.96
method | result | size |
default | \(-x \ln \left (3\right )+2 x +\frac {-x -400}{x^{2}+x -4}\) | \(24\) |
risch | \(-x \ln \left (3\right )+2 x +\frac {-x -400}{x^{2}+x -4}\) | \(24\) |
norman | \(\frac {\left (-\ln \left (3\right )+2\right ) x^{3}+\left (5 \ln \left (3\right )-11\right ) x -392-4 \ln \left (3\right )}{x^{2}+x -4}\) | \(34\) |
gosper | \(-\frac {x^{3} \ln \left (3\right )-2 x^{3}-5 x \ln \left (3\right )+4 \ln \left (3\right )+11 x +392}{x^{2}+x -4}\) | \(36\) |
parallelrisch | \(-\frac {x^{3} \ln \left (3\right )-2 x^{3}-5 x \ln \left (3\right )+4 \ln \left (3\right )+11 x +392}{x^{2}+x -4}\) | \(36\) |
orering | \(\frac {\left (x^{3} \ln \left (3\right )-2 x^{3}-5 x \ln \left (3\right )+4 \ln \left (3\right )+11 x +392\right ) \left (x^{2}+x -4\right ) \left (\left (-x^{4}-2 x^{3}+7 x^{2}+8 x -16\right ) \ln \left (3\right )+2 x^{4}+4 x^{3}-13 x^{2}+784 x +436\right )}{\left (x^{4} \ln \left (3\right )+2 x^{3} \ln \left (3\right )-2 x^{4}-7 x^{2} \ln \left (3\right )-4 x^{3}-8 x \ln \left (3\right )+13 x^{2}+16 \ln \left (3\right )-784 x -436\right ) \left (x^{4}+2 x^{3}-7 x^{2}-8 x +16\right )}\) | \(147\) |
Input:
int(((-x^4-2*x^3+7*x^2+8*x-16)*ln(3)+2*x^4+4*x^3-13*x^2+784*x+436)/(x^4+2* x^3-7*x^2-8*x+16),x,method=_RETURNVERBOSE)
Output:
-x*ln(3)+2*x+(-x-400)/(x^2+x-4)
Time = 0.08 (sec) , antiderivative size = 38, normalized size of antiderivative = 1.52 \[ \int \frac {436+784 x-13 x^2+4 x^3+2 x^4+\left (-16+8 x+7 x^2-2 x^3-x^4\right ) \log (3)}{16-8 x-7 x^2+2 x^3+x^4} \, dx=\frac {2 \, x^{3} + 2 \, x^{2} - {\left (x^{3} + x^{2} - 4 \, x\right )} \log \left (3\right ) - 9 \, x - 400}{x^{2} + x - 4} \] Input:
integrate(((-x^4-2*x^3+7*x^2+8*x-16)*log(3)+2*x^4+4*x^3-13*x^2+784*x+436)/ (x^4+2*x^3-7*x^2-8*x+16),x, algorithm="fricas")
Output:
(2*x^3 + 2*x^2 - (x^3 + x^2 - 4*x)*log(3) - 9*x - 400)/(x^2 + x - 4)
Time = 0.08 (sec) , antiderivative size = 17, normalized size of antiderivative = 0.68 \[ \int \frac {436+784 x-13 x^2+4 x^3+2 x^4+\left (-16+8 x+7 x^2-2 x^3-x^4\right ) \log (3)}{16-8 x-7 x^2+2 x^3+x^4} \, dx=x \left (2 - \log {\left (3 \right )}\right ) + \frac {- x - 400}{x^{2} + x - 4} \] Input:
integrate(((-x**4-2*x**3+7*x**2+8*x-16)*ln(3)+2*x**4+4*x**3-13*x**2+784*x+ 436)/(x**4+2*x**3-7*x**2-8*x+16),x)
Output:
x*(2 - log(3)) + (-x - 400)/(x**2 + x - 4)
Time = 0.03 (sec) , antiderivative size = 21, normalized size of antiderivative = 0.84 \[ \int \frac {436+784 x-13 x^2+4 x^3+2 x^4+\left (-16+8 x+7 x^2-2 x^3-x^4\right ) \log (3)}{16-8 x-7 x^2+2 x^3+x^4} \, dx=-x {\left (\log \left (3\right ) - 2\right )} - \frac {x + 400}{x^{2} + x - 4} \] Input:
integrate(((-x^4-2*x^3+7*x^2+8*x-16)*log(3)+2*x^4+4*x^3-13*x^2+784*x+436)/ (x^4+2*x^3-7*x^2-8*x+16),x, algorithm="maxima")
Output:
-x*(log(3) - 2) - (x + 400)/(x^2 + x - 4)
Time = 0.12 (sec) , antiderivative size = 22, normalized size of antiderivative = 0.88 \[ \int \frac {436+784 x-13 x^2+4 x^3+2 x^4+\left (-16+8 x+7 x^2-2 x^3-x^4\right ) \log (3)}{16-8 x-7 x^2+2 x^3+x^4} \, dx=-x \log \left (3\right ) + 2 \, x - \frac {x + 400}{x^{2} + x - 4} \] Input:
integrate(((-x^4-2*x^3+7*x^2+8*x-16)*log(3)+2*x^4+4*x^3-13*x^2+784*x+436)/ (x^4+2*x^3-7*x^2-8*x+16),x, algorithm="giac")
Output:
-x*log(3) + 2*x - (x + 400)/(x^2 + x - 4)
Time = 2.94 (sec) , antiderivative size = 21, normalized size of antiderivative = 0.84 \[ \int \frac {436+784 x-13 x^2+4 x^3+2 x^4+\left (-16+8 x+7 x^2-2 x^3-x^4\right ) \log (3)}{16-8 x-7 x^2+2 x^3+x^4} \, dx=-\frac {x+400}{x^2+x-4}-x\,\left (\ln \left (3\right )-2\right ) \] Input:
int((784*x - log(3)*(2*x^3 - 7*x^2 - 8*x + x^4 + 16) - 13*x^2 + 4*x^3 + 2* x^4 + 436)/(2*x^3 - 7*x^2 - 8*x + x^4 + 16),x)
Output:
- (x + 400)/(x + x^2 - 4) - x*(log(3) - 2)
Time = 0.16 (sec) , antiderivative size = 39, normalized size of antiderivative = 1.56 \[ \int \frac {436+784 x-13 x^2+4 x^3+2 x^4+\left (-16+8 x+7 x^2-2 x^3-x^4\right ) \log (3)}{16-8 x-7 x^2+2 x^3+x^4} \, dx=\frac {-\mathrm {log}\left (3\right ) x^{3}-5 \,\mathrm {log}\left (3\right ) x^{2}+16 \,\mathrm {log}\left (3\right )+2 x^{3}+11 x^{2}-436}{x^{2}+x -4} \] Input:
int(((-x^4-2*x^3+7*x^2+8*x-16)*log(3)+2*x^4+4*x^3-13*x^2+784*x+436)/(x^4+2 *x^3-7*x^2-8*x+16),x)
Output:
( - log(3)*x**3 - 5*log(3)*x**2 + 16*log(3) + 2*x**3 + 11*x**2 - 436)/(x** 2 + x - 4)