Integrand size = 15, antiderivative size = 61 \[ \int \frac {1}{(-1+x) \sqrt [4]{x+x^3}} \, dx=\frac {\arctan \left (\frac {2^{3/4} \sqrt [4]{x+x^3}}{1+x}\right )}{2 \sqrt [4]{2}}-\frac {\text {arctanh}\left (\frac {2^{3/4} \sqrt [4]{x+x^3}}{1+x}\right )}{2 \sqrt [4]{2}} \]
1/4*arctan(2^(3/4)*(x^3+x)^(1/4)/(1+x))*2^(3/4)-1/4*arctanh(2^(3/4)*(x^3+x )^(1/4)/(1+x))*2^(3/4)
Time = 10.36 (sec) , antiderivative size = 54, normalized size of antiderivative = 0.89 \[ \int \frac {1}{(-1+x) \sqrt [4]{x+x^3}} \, dx=\frac {\arctan \left (\frac {2^{3/4} \sqrt [4]{x+x^3}}{1+x}\right )-\text {arctanh}\left (\frac {2^{3/4} \sqrt [4]{x+x^3}}{1+x}\right )}{2 \sqrt [4]{2}} \]
(ArcTan[(2^(3/4)*(x + x^3)^(1/4))/(1 + x)] - ArcTanh[(2^(3/4)*(x + x^3)^(1 /4))/(1 + x)])/(2*2^(1/4))
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.
| \(\displaystyle \int \frac {1}{(x-1) \sqrt [4]{x^3+x}} \, dx\) |
| \(\Big \downarrow \) 2467 |
| \(\displaystyle \frac {\sqrt [4]{x} \sqrt [4]{x^2+1} \int -\frac {1}{(1-x) \sqrt [4]{x} \sqrt [4]{x^2+1}}dx}{\sqrt [4]{x^3+x}}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle -\frac {\sqrt [4]{x} \sqrt [4]{x^2+1} \int \frac {1}{(1-x) \sqrt [4]{x} \sqrt [4]{x^2+1}}dx}{\sqrt [4]{x^3+x}}\) |
| \(\Big \downarrow \) 616 |
| \(\displaystyle -\frac {4 \sqrt [4]{x} \sqrt [4]{x^2+1} \int \frac {\sqrt {x}}{(1-x) \sqrt [4]{x^2+1}}d\sqrt [4]{x}}{\sqrt [4]{x^3+x}}\) |
| \(\Big \downarrow \) 1888 |
| \(\displaystyle -\frac {4 \sqrt [4]{x} \sqrt [4]{x^2+1} \int \frac {\sqrt {x}}{(1-x) \sqrt [4]{x^2+1}}d\sqrt [4]{x}}{\sqrt [4]{x^3+x}}\) |
3.8.95.3.1 Defintions of rubi rules used
Int[((e_.)*(x_))^(m_)*((c_) + (d_.)*(x_))^(n_)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> With[{k = Denominator[m]}, Simp[k/e Subst[Int[x^(k*(m + 1) - 1)*(c + d*(x^k/e))^n*(a + b*(x^(2*k)/e^2))^p, x], x, (e*x)^(1/k)], x]] /; FreeQ[{a, b, c, d, e, p}, x] && ILtQ[n, 0] && FractionQ[m]
Int[((f_.)*(x_))^(m_.)*((a_) + (c_.)*(x_)^(n2_.))^(p_.)*((d_) + (e_.)*(x_)^ (n_))^(q_.), x_Symbol] :> Unintegrable[(f*x)^m*(d + e*x^n)^q*(a + c*x^(2*n) )^p, x] /; FreeQ[{a, c, d, e, f, m, n, p, q}, x] && EqQ[n2, 2*n]
Int[(Fx_.)*(Px_)^(p_), x_Symbol] :> With[{r = Expon[Px, x, Min]}, Simp[Px^F racPart[p]/(x^(r*FracPart[p])*ExpandToSum[Px/x^r, x]^FracPart[p]) Int[x^( p*r)*ExpandToSum[Px/x^r, x]^p*Fx, x], x] /; IGtQ[r, 0]] /; FreeQ[p, x] && P olyQ[Px, x] && !IntegerQ[p] && !MonomialQ[Px, x] && !PolyQ[Fx, x]
Result contains higher order function than in optimal. Order 9 vs. order 3.
Time = 7.62 (sec) , antiderivative size = 507, normalized size of antiderivative = 8.31
| method | result | size |
| trager | \(-\frac {\operatorname {RootOf}\left (\textit {\_Z}^{4}-8\right ) \ln \left (\frac {2 \sqrt {x^{3}+x}\, \operatorname {RootOf}\left (\textit {\_Z}^{4}-8\right )^{3} x^{2}+4 \sqrt {x^{3}+x}\, \operatorname {RootOf}\left (\textit {\_Z}^{4}-8\right )^{3} x +2 \left (x^{3}+x \right )^{\frac {1}{4}} \operatorname {RootOf}\left (\textit {\_Z}^{4}-8\right )^{2} x^{3}+2 \sqrt {x^{3}+x}\, \operatorname {RootOf}\left (\textit {\_Z}^{4}-8\right )^{3}+6 \left (x^{3}+x \right )^{\frac {1}{4}} \operatorname {RootOf}\left (\textit {\_Z}^{4}-8\right )^{2} x^{2}+x^{4} \operatorname {RootOf}\left (\textit {\_Z}^{4}-8\right )+16 \left (x^{3}+x \right )^{\frac {3}{4}} x +6 \left (x^{3}+x \right )^{\frac {1}{4}} \operatorname {RootOf}\left (\textit {\_Z}^{4}-8\right )^{2} x +12 \operatorname {RootOf}\left (\textit {\_Z}^{4}-8\right ) x^{3}+16 \left (x^{3}+x \right )^{\frac {3}{4}}+2 \left (x^{3}+x \right )^{\frac {1}{4}} \operatorname {RootOf}\left (\textit {\_Z}^{4}-8\right )^{2}+6 \operatorname {RootOf}\left (\textit {\_Z}^{4}-8\right ) x^{2}+12 \operatorname {RootOf}\left (\textit {\_Z}^{4}-8\right ) x +\operatorname {RootOf}\left (\textit {\_Z}^{4}-8\right )}{\left (x -1\right )^{4}}\right )}{8}-\frac {\operatorname {RootOf}\left (\textit {\_Z}^{2}+\operatorname {RootOf}\left (\textit {\_Z}^{4}-8\right )^{2}\right ) \ln \left (\frac {-2 \sqrt {x^{3}+x}\, \operatorname {RootOf}\left (\textit {\_Z}^{4}-8\right )^{2} \operatorname {RootOf}\left (\textit {\_Z}^{2}+\operatorname {RootOf}\left (\textit {\_Z}^{4}-8\right )^{2}\right ) x^{2}-4 \sqrt {x^{3}+x}\, \operatorname {RootOf}\left (\textit {\_Z}^{2}+\operatorname {RootOf}\left (\textit {\_Z}^{4}-8\right )^{2}\right ) \operatorname {RootOf}\left (\textit {\_Z}^{4}-8\right )^{2} x -2 \left (x^{3}+x \right )^{\frac {1}{4}} \operatorname {RootOf}\left (\textit {\_Z}^{4}-8\right )^{2} x^{3}-2 \sqrt {x^{3}+x}\, \operatorname {RootOf}\left (\textit {\_Z}^{2}+\operatorname {RootOf}\left (\textit {\_Z}^{4}-8\right )^{2}\right ) \operatorname {RootOf}\left (\textit {\_Z}^{4}-8\right )^{2}-6 \left (x^{3}+x \right )^{\frac {1}{4}} \operatorname {RootOf}\left (\textit {\_Z}^{4}-8\right )^{2} x^{2}+\operatorname {RootOf}\left (\textit {\_Z}^{2}+\operatorname {RootOf}\left (\textit {\_Z}^{4}-8\right )^{2}\right ) x^{4}+16 \left (x^{3}+x \right )^{\frac {3}{4}} x -6 \left (x^{3}+x \right )^{\frac {1}{4}} \operatorname {RootOf}\left (\textit {\_Z}^{4}-8\right )^{2} x +12 \operatorname {RootOf}\left (\textit {\_Z}^{2}+\operatorname {RootOf}\left (\textit {\_Z}^{4}-8\right )^{2}\right ) x^{3}+16 \left (x^{3}+x \right )^{\frac {3}{4}}-2 \left (x^{3}+x \right )^{\frac {1}{4}} \operatorname {RootOf}\left (\textit {\_Z}^{4}-8\right )^{2}+6 \operatorname {RootOf}\left (\textit {\_Z}^{2}+\operatorname {RootOf}\left (\textit {\_Z}^{4}-8\right )^{2}\right ) x^{2}+12 \operatorname {RootOf}\left (\textit {\_Z}^{2}+\operatorname {RootOf}\left (\textit {\_Z}^{4}-8\right )^{2}\right ) x +\operatorname {RootOf}\left (\textit {\_Z}^{2}+\operatorname {RootOf}\left (\textit {\_Z}^{4}-8\right )^{2}\right )}{\left (x -1\right )^{4}}\right )}{8}\) | \(507\) |
-1/8*RootOf(_Z^4-8)*ln((2*(x^3+x)^(1/2)*RootOf(_Z^4-8)^3*x^2+4*(x^3+x)^(1/ 2)*RootOf(_Z^4-8)^3*x+2*(x^3+x)^(1/4)*RootOf(_Z^4-8)^2*x^3+2*(x^3+x)^(1/2) *RootOf(_Z^4-8)^3+6*(x^3+x)^(1/4)*RootOf(_Z^4-8)^2*x^2+x^4*RootOf(_Z^4-8)+ 16*(x^3+x)^(3/4)*x+6*(x^3+x)^(1/4)*RootOf(_Z^4-8)^2*x+12*RootOf(_Z^4-8)*x^ 3+16*(x^3+x)^(3/4)+2*(x^3+x)^(1/4)*RootOf(_Z^4-8)^2+6*RootOf(_Z^4-8)*x^2+1 2*RootOf(_Z^4-8)*x+RootOf(_Z^4-8))/(x-1)^4)-1/8*RootOf(_Z^2+RootOf(_Z^4-8) ^2)*ln((-2*(x^3+x)^(1/2)*RootOf(_Z^4-8)^2*RootOf(_Z^2+RootOf(_Z^4-8)^2)*x^ 2-4*(x^3+x)^(1/2)*RootOf(_Z^2+RootOf(_Z^4-8)^2)*RootOf(_Z^4-8)^2*x-2*(x^3+ x)^(1/4)*RootOf(_Z^4-8)^2*x^3-2*(x^3+x)^(1/2)*RootOf(_Z^2+RootOf(_Z^4-8)^2 )*RootOf(_Z^4-8)^2-6*(x^3+x)^(1/4)*RootOf(_Z^4-8)^2*x^2+RootOf(_Z^2+RootOf (_Z^4-8)^2)*x^4+16*(x^3+x)^(3/4)*x-6*(x^3+x)^(1/4)*RootOf(_Z^4-8)^2*x+12*R ootOf(_Z^2+RootOf(_Z^4-8)^2)*x^3+16*(x^3+x)^(3/4)-2*(x^3+x)^(1/4)*RootOf(_ Z^4-8)^2+6*RootOf(_Z^2+RootOf(_Z^4-8)^2)*x^2+12*RootOf(_Z^2+RootOf(_Z^4-8) ^2)*x+RootOf(_Z^2+RootOf(_Z^4-8)^2))/(x-1)^4)
Result contains complex when optimal does not.
Time = 3.47 (sec) , antiderivative size = 438, normalized size of antiderivative = 7.18 \[ \int \frac {1}{(-1+x) \sqrt [4]{x+x^3}} \, dx=-\frac {1}{16} \cdot 2^{\frac {3}{4}} \log \left (\frac {2^{\frac {3}{4}} {\left (x^{4} + 12 \, x^{3} + 6 \, x^{2} + 12 \, x + 1\right )} + 4 \, \sqrt {2} {\left (x^{3} + 3 \, x^{2} + 3 \, x + 1\right )} {\left (x^{3} + x\right )}^{\frac {1}{4}} + 8 \cdot 2^{\frac {1}{4}} \sqrt {x^{3} + x} {\left (x^{2} + 2 \, x + 1\right )} + 16 \, {\left (x^{3} + x\right )}^{\frac {3}{4}} {\left (x + 1\right )}}{x^{4} - 4 \, x^{3} + 6 \, x^{2} - 4 \, x + 1}\right ) + \frac {1}{16} \cdot 2^{\frac {3}{4}} \log \left (-\frac {2^{\frac {3}{4}} {\left (x^{4} + 12 \, x^{3} + 6 \, x^{2} + 12 \, x + 1\right )} - 4 \, \sqrt {2} {\left (x^{3} + 3 \, x^{2} + 3 \, x + 1\right )} {\left (x^{3} + x\right )}^{\frac {1}{4}} + 8 \cdot 2^{\frac {1}{4}} \sqrt {x^{3} + x} {\left (x^{2} + 2 \, x + 1\right )} - 16 \, {\left (x^{3} + x\right )}^{\frac {3}{4}} {\left (x + 1\right )}}{x^{4} - 4 \, x^{3} + 6 \, x^{2} - 4 \, x + 1}\right ) - \frac {1}{16} i \cdot 2^{\frac {3}{4}} \log \left (\frac {2^{\frac {3}{4}} {\left (i \, x^{4} + 12 i \, x^{3} + 6 i \, x^{2} + 12 i \, x + i\right )} - 4 \, \sqrt {2} {\left (x^{3} + 3 \, x^{2} + 3 \, x + 1\right )} {\left (x^{3} + x\right )}^{\frac {1}{4}} - 8 \cdot 2^{\frac {1}{4}} \sqrt {x^{3} + x} {\left (i \, x^{2} + 2 i \, x + i\right )} + 16 \, {\left (x^{3} + x\right )}^{\frac {3}{4}} {\left (x + 1\right )}}{x^{4} - 4 \, x^{3} + 6 \, x^{2} - 4 \, x + 1}\right ) + \frac {1}{16} i \cdot 2^{\frac {3}{4}} \log \left (\frac {2^{\frac {3}{4}} {\left (-i \, x^{4} - 12 i \, x^{3} - 6 i \, x^{2} - 12 i \, x - i\right )} - 4 \, \sqrt {2} {\left (x^{3} + 3 \, x^{2} + 3 \, x + 1\right )} {\left (x^{3} + x\right )}^{\frac {1}{4}} - 8 \cdot 2^{\frac {1}{4}} \sqrt {x^{3} + x} {\left (-i \, x^{2} - 2 i \, x - i\right )} + 16 \, {\left (x^{3} + x\right )}^{\frac {3}{4}} {\left (x + 1\right )}}{x^{4} - 4 \, x^{3} + 6 \, x^{2} - 4 \, x + 1}\right ) \]
-1/16*2^(3/4)*log((2^(3/4)*(x^4 + 12*x^3 + 6*x^2 + 12*x + 1) + 4*sqrt(2)*( x^3 + 3*x^2 + 3*x + 1)*(x^3 + x)^(1/4) + 8*2^(1/4)*sqrt(x^3 + x)*(x^2 + 2* x + 1) + 16*(x^3 + x)^(3/4)*(x + 1))/(x^4 - 4*x^3 + 6*x^2 - 4*x + 1)) + 1/ 16*2^(3/4)*log(-(2^(3/4)*(x^4 + 12*x^3 + 6*x^2 + 12*x + 1) - 4*sqrt(2)*(x^ 3 + 3*x^2 + 3*x + 1)*(x^3 + x)^(1/4) + 8*2^(1/4)*sqrt(x^3 + x)*(x^2 + 2*x + 1) - 16*(x^3 + x)^(3/4)*(x + 1))/(x^4 - 4*x^3 + 6*x^2 - 4*x + 1)) - 1/16 *I*2^(3/4)*log((2^(3/4)*(I*x^4 + 12*I*x^3 + 6*I*x^2 + 12*I*x + I) - 4*sqrt (2)*(x^3 + 3*x^2 + 3*x + 1)*(x^3 + x)^(1/4) - 8*2^(1/4)*sqrt(x^3 + x)*(I*x ^2 + 2*I*x + I) + 16*(x^3 + x)^(3/4)*(x + 1))/(x^4 - 4*x^3 + 6*x^2 - 4*x + 1)) + 1/16*I*2^(3/4)*log((2^(3/4)*(-I*x^4 - 12*I*x^3 - 6*I*x^2 - 12*I*x - I) - 4*sqrt(2)*(x^3 + 3*x^2 + 3*x + 1)*(x^3 + x)^(1/4) - 8*2^(1/4)*sqrt(x ^3 + x)*(-I*x^2 - 2*I*x - I) + 16*(x^3 + x)^(3/4)*(x + 1))/(x^4 - 4*x^3 + 6*x^2 - 4*x + 1))
\[ \int \frac {1}{(-1+x) \sqrt [4]{x+x^3}} \, dx=\int \frac {1}{\sqrt [4]{x \left (x^{2} + 1\right )} \left (x - 1\right )}\, dx \]
\[ \int \frac {1}{(-1+x) \sqrt [4]{x+x^3}} \, dx=\int { \frac {1}{{\left (x^{3} + x\right )}^{\frac {1}{4}} {\left (x - 1\right )}} \,d x } \]
\[ \int \frac {1}{(-1+x) \sqrt [4]{x+x^3}} \, dx=\int { \frac {1}{{\left (x^{3} + x\right )}^{\frac {1}{4}} {\left (x - 1\right )}} \,d x } \]
Timed out. \[ \int \frac {1}{(-1+x) \sqrt [4]{x+x^3}} \, dx=\int \frac {1}{{\left (x^3+x\right )}^{1/4}\,\left (x-1\right )} \,d x \]