Integrand size = 67, antiderivative size = 83 \[ \int \frac {\sqrt {1-4 x+x^2}+\left (1-4 x+x^2\right )^{3/2}}{\sqrt {1-4 x+x^2}+\left (1-4 x+x^2\right )^{3/2}-\left (1-4 x+x^2\right )^{5/2}} \, dx=-\frac {1}{2} \text {RootSum}\left [-1-4 \text {$\#$1}+17 \text {$\#$1}^2-8 \text {$\#$1}^3+\text {$\#$1}^4\&,\frac {2 \log (x-\text {$\#$1})-4 \log (x-\text {$\#$1}) \text {$\#$1}+\log (x-\text {$\#$1}) \text {$\#$1}^2}{-2+17 \text {$\#$1}-12 \text {$\#$1}^2+2 \text {$\#$1}^3}\&\right ] \]
Time = 0.01 (sec) , antiderivative size = 83, normalized size of antiderivative = 1.00 \[ \int \frac {\sqrt {1-4 x+x^2}+\left (1-4 x+x^2\right )^{3/2}}{\sqrt {1-4 x+x^2}+\left (1-4 x+x^2\right )^{3/2}-\left (1-4 x+x^2\right )^{5/2}} \, dx=-\frac {1}{2} \text {RootSum}\left [-1-4 \text {$\#$1}+17 \text {$\#$1}^2-8 \text {$\#$1}^3+\text {$\#$1}^4\&,\frac {2 \log (x-\text {$\#$1})-4 \log (x-\text {$\#$1}) \text {$\#$1}+\log (x-\text {$\#$1}) \text {$\#$1}^2}{-2+17 \text {$\#$1}-12 \text {$\#$1}^2+2 \text {$\#$1}^3}\&\right ] \]
Integrate[(Sqrt[1 - 4*x + x^2] + (1 - 4*x + x^2)^(3/2))/(Sqrt[1 - 4*x + x^ 2] + (1 - 4*x + x^2)^(3/2) - (1 - 4*x + x^2)^(5/2)),x]
-1/2*RootSum[-1 - 4*#1 + 17*#1^2 - 8*#1^3 + #1^4 & , (2*Log[x - #1] - 4*Lo g[x - #1]*#1 + Log[x - #1]*#1^2)/(-2 + 17*#1 - 12*#1^2 + 2*#1^3) & ]
Time = 0.44 (sec) , antiderivative size = 97, normalized size of antiderivative = 1.17, number of steps used = 5, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.060, Rules used = {7239, 2459, 1480, 219}
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.
| \(\displaystyle \int \frac {\left (x^2-4 x+1\right )^{3/2}+\sqrt {x^2-4 x+1}}{-\left (x^2-4 x+1\right )^{5/2}+\left (x^2-4 x+1\right )^{3/2}+\sqrt {x^2-4 x+1}} \, dx\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle \int \frac {x^2-4 x+2}{-x^4+8 x^3-17 x^2+4 x+1}dx\) |
| \(\Big \downarrow \) 2459 |
| \(\displaystyle \int \frac {(x-2)^2-2}{-(x-2)^4+7 (x-2)^2-11}d(x-2)\) |
| \(\Big \downarrow \) 1480 |
| \(\displaystyle \frac {1}{10} \left (5-3 \sqrt {5}\right ) \int \frac {1}{\frac {1}{2} \left (7-\sqrt {5}\right )-(x-2)^2}d(x-2)+\frac {1}{10} \left (5+3 \sqrt {5}\right ) \int \frac {1}{\frac {1}{2} \left (7+\sqrt {5}\right )-(x-2)^2}d(x-2)\) |
| \(\Big \downarrow \) 219 |
| \(\displaystyle \frac {\left (5-3 \sqrt {5}\right ) \text {arctanh}\left (\sqrt {\frac {2}{7-\sqrt {5}}} (x-2)\right )}{5 \sqrt {2 \left (7-\sqrt {5}\right )}}+\frac {\left (5+3 \sqrt {5}\right ) \text {arctanh}\left (\sqrt {\frac {2}{7+\sqrt {5}}} (x-2)\right )}{5 \sqrt {2 \left (7+\sqrt {5}\right )}}\) |
Int[(Sqrt[1 - 4*x + x^2] + (1 - 4*x + x^2)^(3/2))/(Sqrt[1 - 4*x + x^2] + ( 1 - 4*x + x^2)^(3/2) - (1 - 4*x + x^2)^(5/2)),x]
((5 - 3*Sqrt[5])*ArcTanh[Sqrt[2/(7 - Sqrt[5])]*(-2 + x)])/(5*Sqrt[2*(7 - S qrt[5])]) + ((5 + 3*Sqrt[5])*ArcTanh[Sqrt[2/(7 + Sqrt[5])]*(-2 + x)])/(5*S qrt[2*(7 + Sqrt[5])])
3.12.23.3.1 Defintions of rubi rules used
Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1/(Rt[a, 2]*Rt[-b, 2]))* ArcTanh[Rt[-b, 2]*(x/Rt[a, 2])], x] /; FreeQ[{a, b}, x] && NegQ[a/b] && (Gt Q[a, 0] || LtQ[b, 0])
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[(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])
Int[u_, x_Symbol] :> With[{v = SimplifyIntegrand[u, x]}, Int[v, x] /; Simpl erIntegrandQ[v, u, x]]
Time = 4.41 (sec) , antiderivative size = 57, normalized size of antiderivative = 0.69
| method | result | size |
| risch | \(-\frac {\left (\munderset {\textit {\_R} =\operatorname {RootOf}\left (\textit {\_Z}^{4}-8 \textit {\_Z}^{3}+17 \textit {\_Z}^{2}-4 \textit {\_Z} -1\right )}{\sum }\frac {\left (\textit {\_R}^{2}-4 \textit {\_R} +2\right ) \ln \left (x -\textit {\_R} \right )}{2 \textit {\_R}^{3}-12 \textit {\_R}^{2}+17 \textit {\_R} -2}\right )}{2}\) | \(57\) |
| default | \(\frac {\left (\munderset {\textit {\_R} =\operatorname {RootOf}\left (\textit {\_Z}^{4}-8 \textit {\_Z}^{3}+17 \textit {\_Z}^{2}-4 \textit {\_Z} -1\right )}{\sum }\frac {\left (-\textit {\_R}^{2}+4 \textit {\_R} -2\right ) \ln \left (x -\textit {\_R} \right )}{2 \textit {\_R}^{3}-12 \textit {\_R}^{2}+17 \textit {\_R} -2}\right )}{2}\) | \(59\) |
| trager | \(-\frac {\operatorname {RootOf}\left (\textit {\_Z}^{2}+12100 \operatorname {RootOf}\left (4400 \textit {\_Z}^{4}-340 \textit {\_Z}^{2}+1\right )^{2}-935\right ) \ln \left (-\frac {20 \operatorname {RootOf}\left (4400 \textit {\_Z}^{4}-340 \textit {\_Z}^{2}+1\right )^{2} \operatorname {RootOf}\left (\textit {\_Z}^{2}+12100 \operatorname {RootOf}\left (4400 \textit {\_Z}^{4}-340 \textit {\_Z}^{2}+1\right )^{2}-935\right ) x +2200 \operatorname {RootOf}\left (4400 \textit {\_Z}^{4}-340 \textit {\_Z}^{2}+1\right )^{2} x +4 \operatorname {RootOf}\left (\textit {\_Z}^{2}+12100 \operatorname {RootOf}\left (4400 \textit {\_Z}^{4}-340 \textit {\_Z}^{2}+1\right )^{2}-935\right ) x -120 x -35}{20 \operatorname {RootOf}\left (4400 \textit {\_Z}^{4}-340 \textit {\_Z}^{2}+1\right )^{2} \operatorname {RootOf}\left (\textit {\_Z}^{2}+12100 \operatorname {RootOf}\left (4400 \textit {\_Z}^{4}-340 \textit {\_Z}^{2}+1\right )^{2}-935\right ) x -2200 \operatorname {RootOf}\left (4400 \textit {\_Z}^{4}-340 \textit {\_Z}^{2}+1\right )^{2} x +4 \operatorname {RootOf}\left (\textit {\_Z}^{2}+12100 \operatorname {RootOf}\left (4400 \textit {\_Z}^{4}-340 \textit {\_Z}^{2}+1\right )^{2}-935\right ) x +120 x +35}\right )}{110}+\operatorname {RootOf}\left (4400 \textit {\_Z}^{4}-340 \textit {\_Z}^{2}+1\right ) \ln \left (-\frac {440 \operatorname {RootOf}\left (4400 \textit {\_Z}^{4}-340 \textit {\_Z}^{2}+1\right )^{3} x -440 \operatorname {RootOf}\left (4400 \textit {\_Z}^{4}-340 \textit {\_Z}^{2}+1\right )^{2} x -122 \operatorname {RootOf}\left (4400 \textit {\_Z}^{4}-340 \textit {\_Z}^{2}+1\right ) x +10 x -7}{440 \operatorname {RootOf}\left (4400 \textit {\_Z}^{4}-340 \textit {\_Z}^{2}+1\right )^{3} x +440 \operatorname {RootOf}\left (4400 \textit {\_Z}^{4}-340 \textit {\_Z}^{2}+1\right )^{2} x -122 \operatorname {RootOf}\left (4400 \textit {\_Z}^{4}-340 \textit {\_Z}^{2}+1\right ) x -10 x +7}\right )\) | \(345\) |
int(((x^2-4*x+1)^(1/2)+(x^2-4*x+1)^(3/2))/((x^2-4*x+1)^(1/2)+(x^2-4*x+1)^( 3/2)-(x^2-4*x+1)^(5/2)),x,method=_RETURNVERBOSE)
-1/2*sum((_R^2-4*_R+2)/(2*_R^3-12*_R^2+17*_R-2)*ln(x-_R),_R=RootOf(_Z^4-8* _Z^3+17*_Z^2-4*_Z-1))
Result contains higher order function than in optimal. Order 3 vs. order 1.
Time = 0.27 (sec) , antiderivative size = 163, normalized size of antiderivative = 1.96 \[ \int \frac {\sqrt {1-4 x+x^2}+\left (1-4 x+x^2\right )^{3/2}}{\sqrt {1-4 x+x^2}+\left (1-4 x+x^2\right )^{3/2}-\left (1-4 x+x^2\right )^{5/2}} \, dx=\frac {1}{220} \, \sqrt {110} \sqrt {7 \, \sqrt {5} + 17} \log \left (\sqrt {110} \sqrt {7 \, \sqrt {5} + 17} {\left (4 \, \sqrt {5} - 5\right )} + 110 \, x - 220\right ) - \frac {1}{220} \, \sqrt {110} \sqrt {7 \, \sqrt {5} + 17} \log \left (-\sqrt {110} \sqrt {7 \, \sqrt {5} + 17} {\left (4 \, \sqrt {5} - 5\right )} + 110 \, x - 220\right ) - \frac {1}{220} \, \sqrt {110} \sqrt {-7 \, \sqrt {5} + 17} \log \left (\sqrt {110} {\left (4 \, \sqrt {5} + 5\right )} \sqrt {-7 \, \sqrt {5} + 17} + 110 \, x - 220\right ) + \frac {1}{220} \, \sqrt {110} \sqrt {-7 \, \sqrt {5} + 17} \log \left (-\sqrt {110} {\left (4 \, \sqrt {5} + 5\right )} \sqrt {-7 \, \sqrt {5} + 17} + 110 \, x - 220\right ) \]
integrate(((x^2-4*x+1)^(1/2)+(x^2-4*x+1)^(3/2))/((x^2-4*x+1)^(1/2)+(x^2-4* x+1)^(3/2)-(x^2-4*x+1)^(5/2)),x, algorithm="fricas")
1/220*sqrt(110)*sqrt(7*sqrt(5) + 17)*log(sqrt(110)*sqrt(7*sqrt(5) + 17)*(4 *sqrt(5) - 5) + 110*x - 220) - 1/220*sqrt(110)*sqrt(7*sqrt(5) + 17)*log(-s qrt(110)*sqrt(7*sqrt(5) + 17)*(4*sqrt(5) - 5) + 110*x - 220) - 1/220*sqrt( 110)*sqrt(-7*sqrt(5) + 17)*log(sqrt(110)*(4*sqrt(5) + 5)*sqrt(-7*sqrt(5) + 17) + 110*x - 220) + 1/220*sqrt(110)*sqrt(-7*sqrt(5) + 17)*log(-sqrt(110) *(4*sqrt(5) + 5)*sqrt(-7*sqrt(5) + 17) + 110*x - 220)
Leaf count of result is larger than twice the leaf count of optimal. 114 vs. \(2 (0) = 0\).
Time = 15.84 (sec) , antiderivative size = 114, normalized size of antiderivative = 1.37 \[ \int \frac {\sqrt {1-4 x+x^2}+\left (1-4 x+x^2\right )^{3/2}}{\sqrt {1-4 x+x^2}+\left (1-4 x+x^2\right )^{3/2}-\left (1-4 x+x^2\right )^{5/2}} \, dx=- 2 \operatorname {RootSum} {\left (4400 t^{4} - 140 t^{2} + 1, \left ( t \mapsto t \log {\left (3080 t^{3} - 54 t + x - 2 \right )} \right )\right )} - \operatorname {RootSum} {\left (4400 t^{4} - 14340 t^{2} + 400 t + 1, \left ( t \mapsto t \log {\left (\frac {102640 t^{3}}{18179} - \frac {19200 t^{2}}{18179} - \frac {75898 t}{4081} + x + \frac {21202}{199969} \right )} \right )\right )} + 4 \operatorname {RootSum} {\left (4400 t^{4} - 1000 t^{2} + 80 t - 1, \left ( t \mapsto t \log {\left (\frac {275 t^{3}}{7} - \frac {110 t^{2}}{7} - \frac {305 t}{28} + x + \frac {9}{28} \right )} \right )\right )} \]
integrate(((x**2-4*x+1)**(1/2)+(x**2-4*x+1)**(3/2))/((x**2-4*x+1)**(1/2)+( x**2-4*x+1)**(3/2)-(x**2-4*x+1)**(5/2)),x)
-2*RootSum(4400*_t**4 - 140*_t**2 + 1, Lambda(_t, _t*log(3080*_t**3 - 54*_ t + x - 2))) - RootSum(4400*_t**4 - 14340*_t**2 + 400*_t + 1, Lambda(_t, _ t*log(102640*_t**3/18179 - 19200*_t**2/18179 - 75898*_t/4081 + x + 21202/1 99969))) + 4*RootSum(4400*_t**4 - 1000*_t**2 + 80*_t - 1, Lambda(_t, _t*lo g(275*_t**3/7 - 110*_t**2/7 - 305*_t/28 + x + 9/28)))
Not integrable
Time = 0.21 (sec) , antiderivative size = 63, normalized size of antiderivative = 0.76 \[ \int \frac {\sqrt {1-4 x+x^2}+\left (1-4 x+x^2\right )^{3/2}}{\sqrt {1-4 x+x^2}+\left (1-4 x+x^2\right )^{3/2}-\left (1-4 x+x^2\right )^{5/2}} \, dx=\int { -\frac {{\left (x^{2} - 4 \, x + 1\right )}^{\frac {3}{2}} + \sqrt {x^{2} - 4 \, x + 1}}{{\left (x^{2} - 4 \, x + 1\right )}^{\frac {5}{2}} - {\left (x^{2} - 4 \, x + 1\right )}^{\frac {3}{2}} - \sqrt {x^{2} - 4 \, x + 1}} \,d x } \]
integrate(((x^2-4*x+1)^(1/2)+(x^2-4*x+1)^(3/2))/((x^2-4*x+1)^(1/2)+(x^2-4* x+1)^(3/2)-(x^2-4*x+1)^(5/2)),x, algorithm="maxima")
-integrate(((x^2 - 4*x + 1)^(3/2) + sqrt(x^2 - 4*x + 1))/((x^2 - 4*x + 1)^ (5/2) - (x^2 - 4*x + 1)^(3/2) - sqrt(x^2 - 4*x + 1)), x)
Not integrable
Time = 0.63 (sec) , antiderivative size = 62, normalized size of antiderivative = 0.75 \[ \int \frac {\sqrt {1-4 x+x^2}+\left (1-4 x+x^2\right )^{3/2}}{\sqrt {1-4 x+x^2}+\left (1-4 x+x^2\right )^{3/2}-\left (1-4 x+x^2\right )^{5/2}} \, dx=\int { -\frac {{\left (x^{2} - 4 \, x + 1\right )}^{\frac {3}{2}} + \sqrt {x^{2} - 4 \, x + 1}}{{\left (x^{2} - 4 \, x + 1\right )}^{\frac {5}{2}} - {\left (x^{2} - 4 \, x + 1\right )}^{\frac {3}{2}} - \sqrt {x^{2} - 4 \, x + 1}} \,d x } \]
integrate(((x^2-4*x+1)^(1/2)+(x^2-4*x+1)^(3/2))/((x^2-4*x+1)^(1/2)+(x^2-4* x+1)^(3/2)-(x^2-4*x+1)^(5/2)),x, algorithm="giac")
integrate(-((x^2 - 4*x + 1)^(3/2) + sqrt(x^2 - 4*x + 1))/((x^2 - 4*x + 1)^ (5/2) - (x^2 - 4*x + 1)^(3/2) - sqrt(x^2 - 4*x + 1)), x)
Time = 5.66 (sec) , antiderivative size = 425, normalized size of antiderivative = 5.12 \[ \int \frac {\sqrt {1-4 x+x^2}+\left (1-4 x+x^2\right )^{3/2}}{\sqrt {1-4 x+x^2}+\left (1-4 x+x^2\right )^{3/2}-\left (1-4 x+x^2\right )^{5/2}} \, dx=\frac {\ln \left (x+\frac {\sqrt {2}\,\sqrt {7-\sqrt {5}}}{2}-2\right )\,\left (2\,\sqrt {2}\,\sqrt {7-\sqrt {5}}+{\left (\frac {\sqrt {2}\,\sqrt {7-\sqrt {5}}}{2}-2\right )}^2-6\right )}{17\,\sqrt {2}\,\sqrt {7-\sqrt {5}}+24\,{\left (\frac {\sqrt {2}\,\sqrt {7-\sqrt {5}}}{2}-2\right )}^2+4\,{\left (\frac {\sqrt {2}\,\sqrt {7-\sqrt {5}}}{2}-2\right )}^3-64}+\frac {\ln \left (x-\frac {\sqrt {2}\,\sqrt {\sqrt {5}+7}}{2}-2\right )\,\left (2\,\sqrt {2}\,\sqrt {\sqrt {5}+7}-{\left (\frac {\sqrt {2}\,\sqrt {\sqrt {5}+7}}{2}+2\right )}^2+6\right )}{4\,{\left (\frac {\sqrt {2}\,\sqrt {\sqrt {5}+7}}{2}+2\right )}^3-24\,{\left (\frac {\sqrt {2}\,\sqrt {\sqrt {5}+7}}{2}+2\right )}^2+17\,\sqrt {2}\,\sqrt {\sqrt {5}+7}+64}+\frac {\ln \left (x-\frac {\sqrt {2}\,\sqrt {7-\sqrt {5}}}{2}-2\right )\,\left (2\,\sqrt {2}\,\sqrt {7-\sqrt {5}}-{\left (\frac {\sqrt {2}\,\sqrt {7-\sqrt {5}}}{2}+2\right )}^2+6\right )}{17\,\sqrt {2}\,\sqrt {7-\sqrt {5}}-24\,{\left (\frac {\sqrt {2}\,\sqrt {7-\sqrt {5}}}{2}+2\right )}^2+4\,{\left (\frac {\sqrt {2}\,\sqrt {7-\sqrt {5}}}{2}+2\right )}^3+64}+\frac {\ln \left (x+\frac {\sqrt {2}\,\sqrt {\sqrt {5}+7}}{2}-2\right )\,\left ({\left (\frac {\sqrt {2}\,\sqrt {\sqrt {5}+7}}{2}-2\right )}^2+2\,\sqrt {2}\,\sqrt {\sqrt {5}+7}-6\right )}{24\,{\left (\frac {\sqrt {2}\,\sqrt {\sqrt {5}+7}}{2}-2\right )}^2+4\,{\left (\frac {\sqrt {2}\,\sqrt {\sqrt {5}+7}}{2}-2\right )}^3+17\,\sqrt {2}\,\sqrt {\sqrt {5}+7}-64} \]
int(((x^2 - 4*x + 1)^(1/2) + (x^2 - 4*x + 1)^(3/2))/((x^2 - 4*x + 1)^(1/2) + (x^2 - 4*x + 1)^(3/2) - (x^2 - 4*x + 1)^(5/2)),x)
(log(x + (2^(1/2)*(7 - 5^(1/2))^(1/2))/2 - 2)*(2*2^(1/2)*(7 - 5^(1/2))^(1/ 2) + ((2^(1/2)*(7 - 5^(1/2))^(1/2))/2 - 2)^2 - 6))/(17*2^(1/2)*(7 - 5^(1/2 ))^(1/2) + 24*((2^(1/2)*(7 - 5^(1/2))^(1/2))/2 - 2)^2 + 4*((2^(1/2)*(7 - 5 ^(1/2))^(1/2))/2 - 2)^3 - 64) + (log(x - (2^(1/2)*(5^(1/2) + 7)^(1/2))/2 - 2)*(2*2^(1/2)*(5^(1/2) + 7)^(1/2) - ((2^(1/2)*(5^(1/2) + 7)^(1/2))/2 + 2) ^2 + 6))/(4*((2^(1/2)*(5^(1/2) + 7)^(1/2))/2 + 2)^3 - 24*((2^(1/2)*(5^(1/2 ) + 7)^(1/2))/2 + 2)^2 + 17*2^(1/2)*(5^(1/2) + 7)^(1/2) + 64) + (log(x - ( 2^(1/2)*(7 - 5^(1/2))^(1/2))/2 - 2)*(2*2^(1/2)*(7 - 5^(1/2))^(1/2) - ((2^( 1/2)*(7 - 5^(1/2))^(1/2))/2 + 2)^2 + 6))/(17*2^(1/2)*(7 - 5^(1/2))^(1/2) - 24*((2^(1/2)*(7 - 5^(1/2))^(1/2))/2 + 2)^2 + 4*((2^(1/2)*(7 - 5^(1/2))^(1 /2))/2 + 2)^3 + 64) + (log(x + (2^(1/2)*(5^(1/2) + 7)^(1/2))/2 - 2)*(((2^( 1/2)*(5^(1/2) + 7)^(1/2))/2 - 2)^2 + 2*2^(1/2)*(5^(1/2) + 7)^(1/2) - 6))/( 24*((2^(1/2)*(5^(1/2) + 7)^(1/2))/2 - 2)^2 + 4*((2^(1/2)*(5^(1/2) + 7)^(1/ 2))/2 - 2)^3 + 17*2^(1/2)*(5^(1/2) + 7)^(1/2) - 64)