Integrand size = 38, antiderivative size = 162 \[ \int \frac {d+e x+f x^2+g x^3+h x^4+i x^5}{\left (4-5 x^2+x^4\right )^2} \, dx=\frac {x \left (17 d+20 f+32 h-(5 d+8 f+20 h) x^2\right )}{72 \left (4-5 x^2+x^4\right )}+\frac {5 e+8 g+20 i-(2 e+5 g+17 i) x^2}{18 \left (4-5 x^2+x^4\right )}+\frac {1}{432} (19 d+52 f+112 h) \text {arctanh}\left (\frac {x}{2}\right )-\frac {1}{54} (d+7 f+13 h) \text {arctanh}(x)+\frac {1}{54} (2 e+5 g+8 i) \log \left (1-x^2\right )-\frac {1}{54} (2 e+5 g+8 i) \log \left (4-x^2\right ) \] Output:
x*(17*d+20*f+32*h-(5*d+8*f+20*h)*x^2)/(72*x^4-360*x^2+288)+(5*e+8*g+20*i-( 2*e+5*g+17*i)*x^2)/(18*x^4-90*x^2+72)+1/432*(19*d+52*f+112*h)*arctanh(1/2* x)-1/54*(d+7*f+13*h)*arctanh(x)+1/54*(2*e+5*g+8*i)*ln(-x^2+1)-1/54*(2*e+5* g+8*i)*ln(-x^2+4)
Time = 0.09 (sec) , antiderivative size = 185, normalized size of antiderivative = 1.14 \[ \int \frac {d+e x+f x^2+g x^3+h x^4+i x^5}{\left (4-5 x^2+x^4\right )^2} \, dx=\frac {20 e+32 g+80 i+17 d x+20 f x+32 h x-8 e x^2-20 g x^2-68 i x^2-5 d x^3-8 f x^3-20 h x^3}{72 \left (4-5 x^2+x^4\right )}+\frac {1}{108} (d+4 e+7 f+10 g+13 h+16 i) \log (1-x)+\frac {1}{864} (-19 d-32 e-52 f-80 g-112 h-128 i) \log (2-x)+\frac {1}{108} (-d+4 e-7 f+10 g-13 h+16 i) \log (1+x)+\frac {1}{864} (19 d-32 e+52 f-80 g+112 h-128 i) \log (2+x) \] Input:
Integrate[(d + e*x + f*x^2 + g*x^3 + h*x^4 + i*x^5)/(4 - 5*x^2 + x^4)^2,x]
Output:
(20*e + 32*g + 80*i + 17*d*x + 20*f*x + 32*h*x - 8*e*x^2 - 20*g*x^2 - 68*i *x^2 - 5*d*x^3 - 8*f*x^3 - 20*h*x^3)/(72*(4 - 5*x^2 + x^4)) + ((d + 4*e + 7*f + 10*g + 13*h + 16*i)*Log[1 - x])/108 + ((-19*d - 32*e - 52*f - 80*g - 112*h - 128*i)*Log[2 - x])/864 + ((-d + 4*e - 7*f + 10*g - 13*h + 16*i)*L og[1 + x])/108 + ((19*d - 32*e + 52*f - 80*g + 112*h - 128*i)*Log[2 + x])/ 864
Time = 0.57 (sec) , antiderivative size = 167, normalized size of antiderivative = 1.03, number of steps used = 11, number of rules used = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.263, Rules used = {2202, 2194, 2191, 27, 1081, 2009, 2206, 25, 1480, 220}
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.
| \(\displaystyle \int \frac {d+e x+f x^2+g x^3+h x^4+i x^5}{\left (x^4-5 x^2+4\right )^2} \, dx\) |
| \(\Big \downarrow \) 2202 |
| \(\displaystyle \int \frac {h x^4+f x^2+d}{\left (x^4-5 x^2+4\right )^2}dx+\int \frac {x \left (i x^4+g x^2+e\right )}{\left (x^4-5 x^2+4\right )^2}dx\) |
| \(\Big \downarrow \) 2194 |
| \(\displaystyle \int \frac {h x^4+f x^2+d}{\left (x^4-5 x^2+4\right )^2}dx+\frac {1}{2} \int \frac {i x^4+g x^2+e}{\left (x^4-5 x^2+4\right )^2}dx^2\) |
| \(\Big \downarrow \) 2191 |
| \(\displaystyle \int \frac {h x^4+f x^2+d}{\left (x^4-5 x^2+4\right )^2}dx+\frac {1}{2} \left (\frac {-\left (x^2 (2 e+5 g+17 i)\right )+5 e+8 g+20 i}{9 \left (x^4-5 x^2+4\right )}-\frac {1}{9} \int \frac {2 e+5 g+8 i}{x^4-5 x^2+4}dx^2\right )\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \int \frac {h x^4+f x^2+d}{\left (x^4-5 x^2+4\right )^2}dx+\frac {1}{2} \left (\frac {-\left (x^2 (2 e+5 g+17 i)\right )+5 e+8 g+20 i}{9 \left (x^4-5 x^2+4\right )}-\frac {1}{9} (2 e+5 g+8 i) \int \frac {1}{x^4-5 x^2+4}dx^2\right )\) |
| \(\Big \downarrow \) 1081 |
| \(\displaystyle \int \frac {h x^4+f x^2+d}{\left (x^4-5 x^2+4\right )^2}dx+\frac {1}{2} \left (\frac {-\left (x^2 (2 e+5 g+17 i)\right )+5 e+8 g+20 i}{9 \left (x^4-5 x^2+4\right )}-\frac {1}{9} (2 e+5 g+8 i) \int \left (\frac {1}{3 \left (1-x^2\right )}-\frac {1}{3 \left (4-x^2\right )}\right )dx^2\right )\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \int \frac {h x^4+f x^2+d}{\left (x^4-5 x^2+4\right )^2}dx+\frac {1}{2} \left (\frac {-\left (x^2 (2 e+5 g+17 i)\right )+5 e+8 g+20 i}{9 \left (x^4-5 x^2+4\right )}-\frac {1}{9} \left (\frac {1}{3} \log \left (4-x^2\right )-\frac {1}{3} \log \left (1-x^2\right )\right ) (2 e+5 g+8 i)\right )\) |
| \(\Big \downarrow \) 2206 |
| \(\displaystyle -\frac {1}{72} \int -\frac {-\left ((5 d+8 f+20 h) x^2\right )+d-20 f-32 h}{x^4-5 x^2+4}dx+\frac {x \left (-\left (x^2 (5 d+8 f+20 h)\right )+17 d+20 f+32 h\right )}{72 \left (x^4-5 x^2+4\right )}+\frac {1}{2} \left (\frac {-\left (x^2 (2 e+5 g+17 i)\right )+5 e+8 g+20 i}{9 \left (x^4-5 x^2+4\right )}-\frac {1}{9} \left (\frac {1}{3} \log \left (4-x^2\right )-\frac {1}{3} \log \left (1-x^2\right )\right ) (2 e+5 g+8 i)\right )\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {1}{72} \int \frac {-\left ((5 d+8 f+20 h) x^2\right )+d-20 f-32 h}{x^4-5 x^2+4}dx+\frac {x \left (-\left (x^2 (5 d+8 f+20 h)\right )+17 d+20 f+32 h\right )}{72 \left (x^4-5 x^2+4\right )}+\frac {1}{2} \left (\frac {-\left (x^2 (2 e+5 g+17 i)\right )+5 e+8 g+20 i}{9 \left (x^4-5 x^2+4\right )}-\frac {1}{9} \left (\frac {1}{3} \log \left (4-x^2\right )-\frac {1}{3} \log \left (1-x^2\right )\right ) (2 e+5 g+8 i)\right )\) |
| \(\Big \downarrow \) 1480 |
| \(\displaystyle \frac {1}{72} \left (\frac {4}{3} (d+7 f+13 h) \int \frac {1}{x^2-1}dx-\frac {1}{3} (19 d+52 f+112 h) \int \frac {1}{x^2-4}dx\right )+\frac {x \left (-\left (x^2 (5 d+8 f+20 h)\right )+17 d+20 f+32 h\right )}{72 \left (x^4-5 x^2+4\right )}+\frac {1}{2} \left (\frac {-\left (x^2 (2 e+5 g+17 i)\right )+5 e+8 g+20 i}{9 \left (x^4-5 x^2+4\right )}-\frac {1}{9} \left (\frac {1}{3} \log \left (4-x^2\right )-\frac {1}{3} \log \left (1-x^2\right )\right ) (2 e+5 g+8 i)\right )\) |
| \(\Big \downarrow \) 220 |
| \(\displaystyle \frac {1}{72} \left (\frac {1}{6} \text {arctanh}\left (\frac {x}{2}\right ) (19 d+52 f+112 h)-\frac {4}{3} \text {arctanh}(x) (d+7 f+13 h)\right )+\frac {x \left (-\left (x^2 (5 d+8 f+20 h)\right )+17 d+20 f+32 h\right )}{72 \left (x^4-5 x^2+4\right )}+\frac {1}{2} \left (\frac {-\left (x^2 (2 e+5 g+17 i)\right )+5 e+8 g+20 i}{9 \left (x^4-5 x^2+4\right )}-\frac {1}{9} \left (\frac {1}{3} \log \left (4-x^2\right )-\frac {1}{3} \log \left (1-x^2\right )\right ) (2 e+5 g+8 i)\right )\) |
Input:
Int[(d + e*x + f*x^2 + g*x^3 + h*x^4 + i*x^5)/(4 - 5*x^2 + x^4)^2,x]
Output:
(x*(17*d + 20*f + 32*h - (5*d + 8*f + 20*h)*x^2))/(72*(4 - 5*x^2 + x^4)) + (((19*d + 52*f + 112*h)*ArcTanh[x/2])/6 - (4*(d + 7*f + 13*h)*ArcTanh[x]) /3)/72 + ((5*e + 8*g + 20*i - (2*e + 5*g + 17*i)*x^2)/(9*(4 - 5*x^2 + x^4) ) - ((2*e + 5*g + 8*i)*(-1/3*Log[1 - x^2] + Log[4 - x^2]/3))/9)/2
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(-(Rt[-a, 2]*Rt[b, 2])^(- 1))*ArcTanh[Rt[b, 2]*(x/Rt[-a, 2])], x] /; FreeQ[{a, b}, x] && NegQ[a/b] && (LtQ[a, 0] || GtQ[b, 0])
Int[((a_) + (b_.)*(x_) + (c_.)*(x_)^2)^(-1), x_Symbol] :> With[{q = Rt[b^2 - 4*a*c, 2]}, Simp[c Int[ExpandIntegrand[1/((b/2 - q/2 + c*x)*(b/2 + q/2 + c*x)), x], x], x]] /; FreeQ[{a, b, c}, x] && NiceSqrtQ[b^2 - 4*a*c]
Int[((d_) + (e_.)*(x_)^2)/((a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4), x_Symbol] : > With[{q = Rt[b^2 - 4*a*c, 2]}, Simp[(e/2 + (2*c*d - b*e)/(2*q)) Int[1/( b/2 - q/2 + c*x^2), x], x] + Simp[(e/2 - (2*c*d - b*e)/(2*q)) Int[1/(b/2 + q/2 + c*x^2), x], x]] /; FreeQ[{a, b, c, d, e}, x] && NeQ[b^2 - 4*a*c, 0] && NeQ[c*d^2 - a*e^2, 0] && PosQ[b^2 - 4*a*c]
Int[(Pq_)*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> With[{Q = PolynomialQuotient[Pq, a + b*x + c*x^2, x], f = Coeff[PolynomialRemainder[P q, a + b*x + c*x^2, x], x, 0], g = Coeff[PolynomialRemainder[Pq, a + b*x + c*x^2, x], x, 1]}, Simp[(b*f - 2*a*g + (2*c*f - b*g)*x)*((a + b*x + c*x^2)^ (p + 1)/((p + 1)*(b^2 - 4*a*c))), x] + Simp[1/((p + 1)*(b^2 - 4*a*c)) Int [(a + b*x + c*x^2)^(p + 1)*ExpandToSum[(p + 1)*(b^2 - 4*a*c)*Q - (2*p + 3)* (2*c*f - b*g), x], x], x]] /; FreeQ[{a, b, c}, x] && PolyQ[Pq, x] && NeQ[b^ 2 - 4*a*c, 0] && LtQ[p, -1]
Int[(Pq_)*(x_)^(m_.)*((a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4)^(p_), x_Symbol] : > Simp[1/2 Subst[Int[x^((m - 1)/2)*SubstFor[x^2, Pq, x]*(a + b*x + c*x^2) ^p, x], x, x^2], x] /; FreeQ[{a, b, c, p}, x] && PolyQ[Pq, x^2] && IntegerQ [(m - 1)/2]
Int[(Pn_)*((a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4)^(p_), x_Symbol] :> Module[{n = Expon[Pn, x], k}, Int[Sum[Coeff[Pn, x, 2*k]*x^(2*k), {k, 0, n/2}]*(a + b *x^2 + c*x^4)^p, x] + Int[x*Sum[Coeff[Pn, x, 2*k + 1]*x^(2*k), {k, 0, (n - 1)/2}]*(a + b*x^2 + c*x^4)^p, x]] /; FreeQ[{a, b, c, p}, x] && PolyQ[Pn, x] && !PolyQ[Pn, x^2]
Int[(Px_)*((a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4)^(p_), x_Symbol] :> With[{d = Coeff[PolynomialRemainder[Px, a + b*x^2 + c*x^4, x], x, 0], e = Coeff[Poly nomialRemainder[Px, a + b*x^2 + c*x^4, x], x, 2]}, Simp[x*(a + b*x^2 + c*x^ 4)^(p + 1)*((a*b*e - d*(b^2 - 2*a*c) - c*(b*d - 2*a*e)*x^2)/(2*a*(p + 1)*(b ^2 - 4*a*c))), x] + Simp[1/(2*a*(p + 1)*(b^2 - 4*a*c)) Int[(a + b*x^2 + c *x^4)^(p + 1)*ExpandToSum[2*a*(p + 1)*(b^2 - 4*a*c)*PolynomialQuotient[Px, a + b*x^2 + c*x^4, x] + b^2*d*(2*p + 3) - 2*a*c*d*(4*p + 5) - a*b*e + c*(4* p + 7)*(b*d - 2*a*e)*x^2, x], x], x]] /; FreeQ[{a, b, c}, x] && PolyQ[Px, x ^2] && Expon[Px, x^2] > 1 && NeQ[b^2 - 4*a*c, 0] && LtQ[p, -1]
Time = 0.10 (sec) , antiderivative size = 161, normalized size of antiderivative = 0.99
| method | result | size |
| norman | \(\frac {\left (-\frac {5 d}{72}-\frac {f}{9}-\frac {5 h}{18}\right ) x^{3}+\left (\frac {17 d}{72}+\frac {5 f}{18}+\frac {4 h}{9}\right ) x +\left (-\frac {5 g}{18}-\frac {e}{9}-\frac {17 i}{18}\right ) x^{2}+\frac {4 g}{9}+\frac {5 e}{18}+\frac {10 i}{9}}{x^{4}-5 x^{2}+4}+\left (-\frac {19 d}{864}-\frac {e}{27}-\frac {13 f}{216}-\frac {5 g}{54}-\frac {7 h}{54}-\frac {4 i}{27}\right ) \ln \left (x -2\right )+\left (-\frac {d}{108}+\frac {e}{27}-\frac {7 f}{108}+\frac {5 g}{54}-\frac {13 h}{108}+\frac {4 i}{27}\right ) \ln \left (1+x \right )+\left (\frac {d}{108}+\frac {e}{27}+\frac {7 f}{108}+\frac {5 g}{54}+\frac {13 h}{108}+\frac {4 i}{27}\right ) \ln \left (x -1\right )+\left (\frac {19 d}{864}-\frac {e}{27}+\frac {13 f}{216}-\frac {5 g}{54}+\frac {7 h}{54}-\frac {4 i}{27}\right ) \ln \left (x +2\right )\) | \(161\) |
| default | \(\left (-\frac {19 d}{864}-\frac {e}{27}-\frac {13 f}{216}-\frac {5 g}{54}-\frac {7 h}{54}-\frac {4 i}{27}\right ) \ln \left (x -2\right )-\frac {\frac {d}{144}+\frac {e}{72}+\frac {f}{36}+\frac {g}{18}+\frac {h}{9}+\frac {2 i}{9}}{x -2}+\left (-\frac {d}{108}+\frac {e}{27}-\frac {7 f}{108}+\frac {5 g}{54}-\frac {13 h}{108}+\frac {4 i}{27}\right ) \ln \left (1+x \right )-\frac {\frac {d}{36}-\frac {e}{36}+\frac {f}{36}-\frac {g}{36}+\frac {h}{36}-\frac {i}{36}}{1+x}-\frac {\frac {d}{36}+\frac {e}{36}+\frac {f}{36}+\frac {g}{36}+\frac {h}{36}+\frac {i}{36}}{x -1}+\left (\frac {d}{108}+\frac {e}{27}+\frac {7 f}{108}+\frac {5 g}{54}+\frac {13 h}{108}+\frac {4 i}{27}\right ) \ln \left (x -1\right )-\frac {\frac {d}{144}-\frac {e}{72}+\frac {f}{36}-\frac {g}{18}+\frac {h}{9}-\frac {2 i}{9}}{x +2}+\left (\frac {19 d}{864}-\frac {e}{27}+\frac {13 f}{216}-\frac {5 g}{54}+\frac {7 h}{54}-\frac {4 i}{27}\right ) \ln \left (x +2\right )\) | \(202\) |
| risch | \(-\frac {\ln \left (2-x \right ) e}{27}-\frac {\ln \left (x +2\right ) e}{27}+\frac {\ln \left (1-x \right ) d}{108}-\frac {19 \ln \left (2-x \right ) d}{864}-\frac {13 \ln \left (2-x \right ) f}{216}-\frac {\ln \left (1+x \right ) d}{108}+\frac {13 \ln \left (x +2\right ) f}{216}+\frac {19 \ln \left (x +2\right ) d}{864}+\frac {\left (-\frac {5 d}{72}-\frac {f}{9}-\frac {5 h}{18}\right ) x^{3}+\left (\frac {17 d}{72}+\frac {5 f}{18}+\frac {4 h}{9}\right ) x +\left (-\frac {5 g}{18}-\frac {e}{9}-\frac {17 i}{18}\right ) x^{2}+\frac {4 g}{9}+\frac {5 e}{18}+\frac {10 i}{9}}{x^{4}-5 x^{2}+4}+\frac {7 \ln \left (1-x \right ) f}{108}+\frac {\ln \left (1-x \right ) e}{27}-\frac {7 \ln \left (1+x \right ) f}{108}+\frac {\ln \left (1+x \right ) e}{27}-\frac {13 \ln \left (1+x \right ) h}{108}+\frac {4 \ln \left (1+x \right ) i}{27}+\frac {5 \ln \left (1+x \right ) g}{54}-\frac {5 \ln \left (x +2\right ) g}{54}+\frac {7 \ln \left (x +2\right ) h}{54}-\frac {4 \ln \left (x +2\right ) i}{27}+\frac {5 \ln \left (1-x \right ) g}{54}+\frac {13 \ln \left (1-x \right ) h}{108}+\frac {4 \ln \left (1-x \right ) i}{27}-\frac {5 \ln \left (2-x \right ) g}{54}-\frac {7 \ln \left (2-x \right ) h}{54}-\frac {4 \ln \left (2-x \right ) i}{27}\) | \(257\) |
| parallelrisch | \(-\frac {-960 i -384 g -204 d x -240 f x -240 e +128 \ln \left (x -2\right ) e +816 i \,x^{2}-128 \ln \left (x -1\right ) x^{4} i -640 \ln \left (x +2\right ) x^{2} i +128 \ln \left (x +2\right ) e +40 \ln \left (x -1\right ) x^{2} d +160 \ln \left (x -1\right ) x^{2} e +60 d \,x^{3}+400 \ln \left (1+x \right ) x^{2} g +96 e \,x^{2}+208 \ln \left (x -2\right ) f +56 \ln \left (1+x \right ) x^{4} f -19 \ln \left (x +2\right ) x^{4} d +32 \ln \left (1+x \right ) d -208 \ln \left (x +2\right ) f -32 \ln \left (x -1\right ) d +128 \ln \left (x +2\right ) x^{4} i +640 \ln \left (1+x \right ) x^{2} i -400 \ln \left (x +2\right ) x^{2} g +560 \ln \left (x +2\right ) x^{2} h -400 \ln \left (x -2\right ) x^{2} g -560 \ln \left (x -2\right ) x^{2} h -76 \ln \left (x +2\right ) d +76 \ln \left (x -2\right ) d +80 \ln \left (x -2\right ) x^{4} g +112 \ln \left (x -2\right ) x^{4} h +128 \ln \left (x -2\right ) x^{4} i -128 \ln \left (x -1\right ) e +224 \ln \left (1+x \right ) f -128 \ln \left (1+x \right ) e -224 \ln \left (x -1\right ) f +32 \ln \left (x -2\right ) x^{4} e +52 \ln \left (x -2\right ) x^{4} f -32 \ln \left (x -1\right ) x^{4} e -56 \ln \left (x -1\right ) x^{4} f +8 \ln \left (1+x \right ) x^{4} d -32 \ln \left (1+x \right ) x^{4} e +416 \ln \left (1+x \right ) h -512 \ln \left (1+x \right ) i -320 \ln \left (1+x \right ) g +240 g \,x^{2}+240 h \,x^{3}+96 f \,x^{3}+280 \ln \left (x -1\right ) x^{2} f -40 \ln \left (1+x \right ) x^{2} d +160 \ln \left (1+x \right ) x^{2} e -95 \ln \left (x -2\right ) x^{2} d -160 \ln \left (x -2\right ) x^{2} e +19 \ln \left (x -2\right ) x^{4} d -280 \ln \left (1+x \right ) x^{2} f +95 \ln \left (x +2\right ) x^{2} d -160 \ln \left (x +2\right ) x^{2} e +260 \ln \left (x +2\right ) x^{2} f -80 \ln \left (x -1\right ) x^{4} g -104 \ln \left (x -1\right ) x^{4} h +640 \ln \left (x -1\right ) x^{2} i -384 h x -640 \ln \left (x -2\right ) x^{2} i -520 \ln \left (1+x \right ) x^{2} h +320 \ln \left (x +2\right ) g -448 \ln \left (x +2\right ) h +512 \ln \left (x +2\right ) i -80 \ln \left (1+x \right ) x^{4} g +104 \ln \left (1+x \right ) x^{4} h -128 \ln \left (1+x \right ) x^{4} i +80 \ln \left (x +2\right ) x^{4} g -112 \ln \left (x +2\right ) x^{4} h +400 \ln \left (x -1\right ) x^{2} g +520 \ln \left (x -1\right ) x^{2} h -8 \ln \left (x -1\right ) x^{4} d -52 \ln \left (x +2\right ) x^{4} f -260 \ln \left (x -2\right ) x^{2} f +320 \ln \left (x -2\right ) g +448 \ln \left (x -2\right ) h +512 \ln \left (x -2\right ) i -320 \ln \left (x -1\right ) g -416 \ln \left (x -1\right ) h -512 \ln \left (x -1\right ) i +32 \ln \left (x +2\right ) x^{4} e}{864 \left (x^{4}-5 x^{2}+4\right )}\) | \(721\) |
Input:
int((i*x^5+h*x^4+g*x^3+f*x^2+e*x+d)/(x^4-5*x^2+4)^2,x,method=_RETURNVERBOS E)
Output:
((-5/72*d-1/9*f-5/18*h)*x^3+(17/72*d+5/18*f+4/9*h)*x+(-5/18*g-1/9*e-17/18* i)*x^2+4/9*g+5/18*e+10/9*i)/(x^4-5*x^2+4)+(-19/864*d-1/27*e-13/216*f-5/54* g-7/54*h-4/27*i)*ln(x-2)+(-1/108*d+1/27*e-7/108*f+5/54*g-13/108*h+4/27*i)* ln(1+x)+(1/108*d+1/27*e+7/108*f+5/54*g+13/108*h+4/27*i)*ln(x-1)+(19/864*d- 1/27*e+13/216*f-5/54*g+7/54*h-4/27*i)*ln(x+2)
Leaf count of result is larger than twice the leaf count of optimal. 346 vs. \(2 (146) = 292\).
Time = 5.90 (sec) , antiderivative size = 346, normalized size of antiderivative = 2.14 \[ \int \frac {d+e x+f x^2+g x^3+h x^4+i x^5}{\left (4-5 x^2+x^4\right )^2} \, dx=-\frac {12 \, {\left (5 \, d + 8 \, f + 20 \, h\right )} x^{3} + 48 \, {\left (2 \, e + 5 \, g + 17 \, i\right )} x^{2} - 12 \, {\left (17 \, d + 20 \, f + 32 \, h\right )} x - {\left ({\left (19 \, d - 32 \, e + 52 \, f - 80 \, g + 112 \, h - 128 \, i\right )} x^{4} - 5 \, {\left (19 \, d - 32 \, e + 52 \, f - 80 \, g + 112 \, h - 128 \, i\right )} x^{2} + 76 \, d - 128 \, e + 208 \, f - 320 \, g + 448 \, h - 512 \, i\right )} \log \left (x + 2\right ) + 8 \, {\left ({\left (d - 4 \, e + 7 \, f - 10 \, g + 13 \, h - 16 \, i\right )} x^{4} - 5 \, {\left (d - 4 \, e + 7 \, f - 10 \, g + 13 \, h - 16 \, i\right )} x^{2} + 4 \, d - 16 \, e + 28 \, f - 40 \, g + 52 \, h - 64 \, i\right )} \log \left (x + 1\right ) - 8 \, {\left ({\left (d + 4 \, e + 7 \, f + 10 \, g + 13 \, h + 16 \, i\right )} x^{4} - 5 \, {\left (d + 4 \, e + 7 \, f + 10 \, g + 13 \, h + 16 \, i\right )} x^{2} + 4 \, d + 16 \, e + 28 \, f + 40 \, g + 52 \, h + 64 \, i\right )} \log \left (x - 1\right ) + {\left ({\left (19 \, d + 32 \, e + 52 \, f + 80 \, g + 112 \, h + 128 \, i\right )} x^{4} - 5 \, {\left (19 \, d + 32 \, e + 52 \, f + 80 \, g + 112 \, h + 128 \, i\right )} x^{2} + 76 \, d + 128 \, e + 208 \, f + 320 \, g + 448 \, h + 512 \, i\right )} \log \left (x - 2\right ) - 240 \, e - 384 \, g - 960 \, i}{864 \, {\left (x^{4} - 5 \, x^{2} + 4\right )}} \] Input:
integrate((i*x^5+h*x^4+g*x^3+f*x^2+e*x+d)/(x^4-5*x^2+4)^2,x, algorithm="fr icas")
Output:
-1/864*(12*(5*d + 8*f + 20*h)*x^3 + 48*(2*e + 5*g + 17*i)*x^2 - 12*(17*d + 20*f + 32*h)*x - ((19*d - 32*e + 52*f - 80*g + 112*h - 128*i)*x^4 - 5*(19 *d - 32*e + 52*f - 80*g + 112*h - 128*i)*x^2 + 76*d - 128*e + 208*f - 320* g + 448*h - 512*i)*log(x + 2) + 8*((d - 4*e + 7*f - 10*g + 13*h - 16*i)*x^ 4 - 5*(d - 4*e + 7*f - 10*g + 13*h - 16*i)*x^2 + 4*d - 16*e + 28*f - 40*g + 52*h - 64*i)*log(x + 1) - 8*((d + 4*e + 7*f + 10*g + 13*h + 16*i)*x^4 - 5*(d + 4*e + 7*f + 10*g + 13*h + 16*i)*x^2 + 4*d + 16*e + 28*f + 40*g + 52 *h + 64*i)*log(x - 1) + ((19*d + 32*e + 52*f + 80*g + 112*h + 128*i)*x^4 - 5*(19*d + 32*e + 52*f + 80*g + 112*h + 128*i)*x^2 + 76*d + 128*e + 208*f + 320*g + 448*h + 512*i)*log(x - 2) - 240*e - 384*g - 960*i)/(x^4 - 5*x^2 + 4)
Timed out. \[ \int \frac {d+e x+f x^2+g x^3+h x^4+i x^5}{\left (4-5 x^2+x^4\right )^2} \, dx=\text {Timed out} \] Input:
integrate((i*x**5+h*x**4+g*x**3+f*x**2+e*x+d)/(x**4-5*x**2+4)**2,x)
Output:
Timed out
Time = 0.03 (sec) , antiderivative size = 163, normalized size of antiderivative = 1.01 \[ \int \frac {d+e x+f x^2+g x^3+h x^4+i x^5}{\left (4-5 x^2+x^4\right )^2} \, dx=\frac {1}{864} \, {\left (19 \, d - 32 \, e + 52 \, f - 80 \, g + 112 \, h - 128 \, i\right )} \log \left (x + 2\right ) - \frac {1}{108} \, {\left (d - 4 \, e + 7 \, f - 10 \, g + 13 \, h - 16 \, i\right )} \log \left (x + 1\right ) + \frac {1}{108} \, {\left (d + 4 \, e + 7 \, f + 10 \, g + 13 \, h + 16 \, i\right )} \log \left (x - 1\right ) - \frac {1}{864} \, {\left (19 \, d + 32 \, e + 52 \, f + 80 \, g + 112 \, h + 128 \, i\right )} \log \left (x - 2\right ) - \frac {{\left (5 \, d + 8 \, f + 20 \, h\right )} x^{3} + 4 \, {\left (2 \, e + 5 \, g + 17 \, i\right )} x^{2} - {\left (17 \, d + 20 \, f + 32 \, h\right )} x - 20 \, e - 32 \, g - 80 \, i}{72 \, {\left (x^{4} - 5 \, x^{2} + 4\right )}} \] Input:
integrate((i*x^5+h*x^4+g*x^3+f*x^2+e*x+d)/(x^4-5*x^2+4)^2,x, algorithm="ma xima")
Output:
1/864*(19*d - 32*e + 52*f - 80*g + 112*h - 128*i)*log(x + 2) - 1/108*(d - 4*e + 7*f - 10*g + 13*h - 16*i)*log(x + 1) + 1/108*(d + 4*e + 7*f + 10*g + 13*h + 16*i)*log(x - 1) - 1/864*(19*d + 32*e + 52*f + 80*g + 112*h + 128* i)*log(x - 2) - 1/72*((5*d + 8*f + 20*h)*x^3 + 4*(2*e + 5*g + 17*i)*x^2 - (17*d + 20*f + 32*h)*x - 20*e - 32*g - 80*i)/(x^4 - 5*x^2 + 4)
Time = 0.13 (sec) , antiderivative size = 173, normalized size of antiderivative = 1.07 \[ \int \frac {d+e x+f x^2+g x^3+h x^4+i x^5}{\left (4-5 x^2+x^4\right )^2} \, dx=\frac {1}{864} \, {\left (19 \, d - 32 \, e + 52 \, f - 80 \, g + 112 \, h - 128 \, i\right )} \log \left ({\left | x + 2 \right |}\right ) - \frac {1}{108} \, {\left (d - 4 \, e + 7 \, f - 10 \, g + 13 \, h - 16 \, i\right )} \log \left ({\left | x + 1 \right |}\right ) + \frac {1}{108} \, {\left (d + 4 \, e + 7 \, f + 10 \, g + 13 \, h + 16 \, i\right )} \log \left ({\left | x - 1 \right |}\right ) - \frac {1}{864} \, {\left (19 \, d + 32 \, e + 52 \, f + 80 \, g + 112 \, h + 128 \, i\right )} \log \left ({\left | x - 2 \right |}\right ) - \frac {5 \, d x^{3} + 8 \, f x^{3} + 20 \, h x^{3} + 8 \, e x^{2} + 20 \, g x^{2} + 68 \, i x^{2} - 17 \, d x - 20 \, f x - 32 \, h x - 20 \, e - 32 \, g - 80 \, i}{72 \, {\left (x^{4} - 5 \, x^{2} + 4\right )}} \] Input:
integrate((i*x^5+h*x^4+g*x^3+f*x^2+e*x+d)/(x^4-5*x^2+4)^2,x, algorithm="gi ac")
Output:
1/864*(19*d - 32*e + 52*f - 80*g + 112*h - 128*i)*log(abs(x + 2)) - 1/108* (d - 4*e + 7*f - 10*g + 13*h - 16*i)*log(abs(x + 1)) + 1/108*(d + 4*e + 7* f + 10*g + 13*h + 16*i)*log(abs(x - 1)) - 1/864*(19*d + 32*e + 52*f + 80*g + 112*h + 128*i)*log(abs(x - 2)) - 1/72*(5*d*x^3 + 8*f*x^3 + 20*h*x^3 + 8 *e*x^2 + 20*g*x^2 + 68*i*x^2 - 17*d*x - 20*f*x - 32*h*x - 20*e - 32*g - 80 *i)/(x^4 - 5*x^2 + 4)
Time = 18.41 (sec) , antiderivative size = 164, normalized size of antiderivative = 1.01 \[ \int \frac {d+e x+f x^2+g x^3+h x^4+i x^5}{\left (4-5 x^2+x^4\right )^2} \, dx=\frac {\left (-\frac {5\,d}{72}-\frac {f}{9}-\frac {5\,h}{18}\right )\,x^3+\left (-\frac {e}{9}-\frac {5\,g}{18}-\frac {17\,i}{18}\right )\,x^2+\left (\frac {17\,d}{72}+\frac {5\,f}{18}+\frac {4\,h}{9}\right )\,x+\frac {5\,e}{18}+\frac {4\,g}{9}+\frac {10\,i}{9}}{x^4-5\,x^2+4}+\ln \left (x-1\right )\,\left (\frac {d}{108}+\frac {e}{27}+\frac {7\,f}{108}+\frac {5\,g}{54}+\frac {13\,h}{108}+\frac {4\,i}{27}\right )-\ln \left (x+1\right )\,\left (\frac {d}{108}-\frac {e}{27}+\frac {7\,f}{108}-\frac {5\,g}{54}+\frac {13\,h}{108}-\frac {4\,i}{27}\right )-\ln \left (x-2\right )\,\left (\frac {19\,d}{864}+\frac {e}{27}+\frac {13\,f}{216}+\frac {5\,g}{54}+\frac {7\,h}{54}+\frac {4\,i}{27}\right )+\ln \left (x+2\right )\,\left (\frac {19\,d}{864}-\frac {e}{27}+\frac {13\,f}{216}-\frac {5\,g}{54}+\frac {7\,h}{54}-\frac {4\,i}{27}\right ) \] Input:
int((d + e*x + f*x^2 + g*x^3 + h*x^4 + i*x^5)/(x^4 - 5*x^2 + 4)^2,x)
Output:
((5*e)/18 + (4*g)/9 + (10*i)/9 + x*((17*d)/72 + (5*f)/18 + (4*h)/9) - x^3* ((5*d)/72 + f/9 + (5*h)/18) - x^2*(e/9 + (5*g)/18 + (17*i)/18))/(x^4 - 5*x ^2 + 4) + log(x - 1)*(d/108 + e/27 + (7*f)/108 + (5*g)/54 + (13*h)/108 + ( 4*i)/27) - log(x + 1)*(d/108 - e/27 + (7*f)/108 - (5*g)/54 + (13*h)/108 - (4*i)/27) - log(x - 2)*((19*d)/864 + e/27 + (13*f)/216 + (5*g)/54 + (7*h)/ 54 + (4*i)/27) + log(x + 2)*((19*d)/864 - e/27 + (13*f)/216 - (5*g)/54 + ( 7*h)/54 - (4*i)/27)
Time = 0.16 (sec) , antiderivative size = 721, normalized size of antiderivative = 4.45 \[ \int \frac {d+e x+f x^2+g x^3+h x^4+i x^5}{\left (4-5 x^2+x^4\right )^2} \, dx =\text {Too large to display} \] Input:
int((i*x^5+h*x^4+g*x^3+f*x^2+e*x+d)/(x^4-5*x^2+4)^2,x)
Output:
( - 95*log(x - 2)*d*x**4 + 475*log(x - 2)*d*x**2 - 380*log(x - 2)*d - 160* log(x - 2)*e*x**4 + 800*log(x - 2)*e*x**2 - 640*log(x - 2)*e - 260*log(x - 2)*f*x**4 + 1300*log(x - 2)*f*x**2 - 1040*log(x - 2)*f - 400*log(x - 2)*g *x**4 + 2000*log(x - 2)*g*x**2 - 1600*log(x - 2)*g - 560*log(x - 2)*h*x**4 + 2800*log(x - 2)*h*x**2 - 2240*log(x - 2)*h - 640*log(x - 2)*i*x**4 + 32 00*log(x - 2)*i*x**2 - 2560*log(x - 2)*i + 40*log(x - 1)*d*x**4 - 200*log( x - 1)*d*x**2 + 160*log(x - 1)*d + 160*log(x - 1)*e*x**4 - 800*log(x - 1)* e*x**2 + 640*log(x - 1)*e + 280*log(x - 1)*f*x**4 - 1400*log(x - 1)*f*x**2 + 1120*log(x - 1)*f + 400*log(x - 1)*g*x**4 - 2000*log(x - 1)*g*x**2 + 16 00*log(x - 1)*g + 520*log(x - 1)*h*x**4 - 2600*log(x - 1)*h*x**2 + 2080*lo g(x - 1)*h + 640*log(x - 1)*i*x**4 - 3200*log(x - 1)*i*x**2 + 2560*log(x - 1)*i + 95*log(x + 2)*d*x**4 - 475*log(x + 2)*d*x**2 + 380*log(x + 2)*d - 160*log(x + 2)*e*x**4 + 800*log(x + 2)*e*x**2 - 640*log(x + 2)*e + 260*log (x + 2)*f*x**4 - 1300*log(x + 2)*f*x**2 + 1040*log(x + 2)*f - 400*log(x + 2)*g*x**4 + 2000*log(x + 2)*g*x**2 - 1600*log(x + 2)*g + 560*log(x + 2)*h* x**4 - 2800*log(x + 2)*h*x**2 + 2240*log(x + 2)*h - 640*log(x + 2)*i*x**4 + 3200*log(x + 2)*i*x**2 - 2560*log(x + 2)*i - 40*log(x + 1)*d*x**4 + 200* log(x + 1)*d*x**2 - 160*log(x + 1)*d + 160*log(x + 1)*e*x**4 - 800*log(x + 1)*e*x**2 + 640*log(x + 1)*e - 280*log(x + 1)*f*x**4 + 1400*log(x + 1)*f* x**2 - 1120*log(x + 1)*f + 400*log(x + 1)*g*x**4 - 2000*log(x + 1)*g*x*...