Integrand size = 24, antiderivative size = 74 \[ \int \frac {\sqrt [4]{-x^3+x^4}}{x^2 \left (-1+x^2\right )} \, dx=\frac {4 \sqrt [4]{-x^3+x^4}}{x}+\sqrt [4]{2} \arctan \left (\frac {\sqrt [4]{2} x}{\sqrt [4]{-x^3+x^4}}\right )-\sqrt [4]{2} \text {arctanh}\left (\frac {\sqrt [4]{2} x}{\sqrt [4]{-x^3+x^4}}\right ) \]
4*(x^4-x^3)^(1/4)/x+2^(1/4)*arctan(2^(1/4)*x/(x^4-x^3)^(1/4))-2^(1/4)*arct anh(2^(1/4)*x/(x^4-x^3)^(1/4))
Time = 0.28 (sec) , antiderivative size = 91, normalized size of antiderivative = 1.23 \[ \int \frac {\sqrt [4]{-x^3+x^4}}{x^2 \left (-1+x^2\right )} \, dx=\frac {(-1+x)^{3/4} x^2 \left (4 \sqrt [4]{-1+x}+\sqrt [4]{2} \sqrt [4]{x} \arctan \left (\frac {\sqrt [4]{2}}{\sqrt [4]{\frac {-1+x}{x}}}\right )-\sqrt [4]{2} \sqrt [4]{x} \text {arctanh}\left (\frac {\sqrt [4]{2}}{\sqrt [4]{\frac {-1+x}{x}}}\right )\right )}{\left ((-1+x) x^3\right )^{3/4}} \]
((-1 + x)^(3/4)*x^2*(4*(-1 + x)^(1/4) + 2^(1/4)*x^(1/4)*ArcTan[2^(1/4)/((- 1 + x)/x)^(1/4)] - 2^(1/4)*x^(1/4)*ArcTanh[2^(1/4)/((-1 + x)/x)^(1/4)]))/( (-1 + x)*x^3)^(3/4)
Time = 0.32 (sec) , antiderivative size = 101, normalized size of antiderivative = 1.36, number of steps used = 10, number of rules used = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.375, Rules used = {2467, 25, 516, 107, 25, 104, 827, 216, 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.
| \(\displaystyle \int \frac {\sqrt [4]{x^4-x^3}}{x^2 \left (x^2-1\right )} \, dx\) |
| \(\Big \downarrow \) 2467 |
| \(\displaystyle \frac {\sqrt [4]{x^4-x^3} \int -\frac {\sqrt [4]{x-1}}{x^{5/4} \left (1-x^2\right )}dx}{\sqrt [4]{x-1} x^{3/4}}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle -\frac {\sqrt [4]{x^4-x^3} \int \frac {\sqrt [4]{x-1}}{x^{5/4} \left (1-x^2\right )}dx}{\sqrt [4]{x-1} x^{3/4}}\) |
| \(\Big \downarrow \) 516 |
| \(\displaystyle -\frac {\sqrt [4]{x^4-x^3} \int \frac {1}{(-x-1) (x-1)^{3/4} x^{5/4}}dx}{\sqrt [4]{x-1} x^{3/4}}\) |
| \(\Big \downarrow \) 107 |
| \(\displaystyle -\frac {\sqrt [4]{x^4-x^3} \left (-\int -\frac {1}{(x-1)^{3/4} \sqrt [4]{x} (x+1)}dx-\frac {4 \sqrt [4]{x-1}}{\sqrt [4]{x}}\right )}{\sqrt [4]{x-1} x^{3/4}}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle -\frac {\sqrt [4]{x^4-x^3} \left (\int \frac {1}{(x-1)^{3/4} \sqrt [4]{x} (x+1)}dx-\frac {4 \sqrt [4]{x-1}}{\sqrt [4]{x}}\right )}{\sqrt [4]{x-1} x^{3/4}}\) |
| \(\Big \downarrow \) 104 |
| \(\displaystyle -\frac {\sqrt [4]{x^4-x^3} \left (4 \int \frac {\sqrt {x}}{\sqrt {x-1} \left (1-\frac {2 x}{x-1}\right )}d\frac {\sqrt [4]{x}}{\sqrt [4]{x-1}}-\frac {4 \sqrt [4]{x-1}}{\sqrt [4]{x}}\right )}{\sqrt [4]{x-1} x^{3/4}}\) |
| \(\Big \downarrow \) 827 |
| \(\displaystyle -\frac {\sqrt [4]{x^4-x^3} \left (4 \left (\frac {\int \frac {1}{1-\frac {\sqrt {2} \sqrt {x}}{\sqrt {x-1}}}d\frac {\sqrt [4]{x}}{\sqrt [4]{x-1}}}{2 \sqrt {2}}-\frac {\int \frac {1}{\frac {\sqrt {2} \sqrt {x}}{\sqrt {x-1}}+1}d\frac {\sqrt [4]{x}}{\sqrt [4]{x-1}}}{2 \sqrt {2}}\right )-\frac {4 \sqrt [4]{x-1}}{\sqrt [4]{x}}\right )}{\sqrt [4]{x-1} x^{3/4}}\) |
| \(\Big \downarrow \) 216 |
| \(\displaystyle -\frac {\sqrt [4]{x^4-x^3} \left (4 \left (\frac {\int \frac {1}{1-\frac {\sqrt {2} \sqrt {x}}{\sqrt {x-1}}}d\frac {\sqrt [4]{x}}{\sqrt [4]{x-1}}}{2 \sqrt {2}}-\frac {\arctan \left (\frac {\sqrt [4]{2} \sqrt [4]{x}}{\sqrt [4]{x-1}}\right )}{2\ 2^{3/4}}\right )-\frac {4 \sqrt [4]{x-1}}{\sqrt [4]{x}}\right )}{\sqrt [4]{x-1} x^{3/4}}\) |
| \(\Big \downarrow \) 219 |
| \(\displaystyle -\frac {\sqrt [4]{x^4-x^3} \left (4 \left (\frac {\text {arctanh}\left (\frac {\sqrt [4]{2} \sqrt [4]{x}}{\sqrt [4]{x-1}}\right )}{2\ 2^{3/4}}-\frac {\arctan \left (\frac {\sqrt [4]{2} \sqrt [4]{x}}{\sqrt [4]{x-1}}\right )}{2\ 2^{3/4}}\right )-\frac {4 \sqrt [4]{x-1}}{\sqrt [4]{x}}\right )}{\sqrt [4]{x-1} x^{3/4}}\) |
-(((-x^3 + x^4)^(1/4)*((-4*(-1 + x)^(1/4))/x^(1/4) + 4*(-1/2*ArcTan[(2^(1/ 4)*x^(1/4))/(-1 + x)^(1/4)]/2^(3/4) + ArcTanh[(2^(1/4)*x^(1/4))/(-1 + x)^( 1/4)]/(2*2^(3/4)))))/((-1 + x)^(1/4)*x^(3/4)))
3.10.75.3.1 Defintions of rubi rules used
Int[(((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_))/((e_.) + (f_.)*(x _)), x_] :> With[{q = Denominator[m]}, Simp[q Subst[Int[x^(q*(m + 1) - 1) /(b*e - a*f - (d*e - c*f)*x^q), x], x, (a + b*x)^(1/q)/(c + d*x)^(1/q)], x] ] /; FreeQ[{a, b, c, d, e, f}, x] && EqQ[m + n + 1, 0] && RationalQ[n] && L tQ[-1, m, 0] && SimplerQ[a + b*x, c + d*x]
Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_)*((e_.) + (f_.)*(x_) )^(p_), x_] :> Simp[b*(a + b*x)^(m + 1)*(c + d*x)^(n + 1)*((e + f*x)^(p + 1 )/((m + 1)*(b*c - a*d)*(b*e - a*f))), x] + Simp[(a*d*f*(m + 1) + b*c*f*(n + 1) + b*d*e*(p + 1))/((m + 1)*(b*c - a*d)*(b*e - a*f)) Int[(a + b*x)^(m + 1)*(c + d*x)^n*(e + f*x)^p, x], x] /; FreeQ[{a, b, c, d, e, f, m, n, p}, x ] && EqQ[Simplify[m + n + p + 3], 0] && (LtQ[m, -1] || SumSimplerQ[m, 1])
Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1/(Rt[a, 2]*Rt[b, 2]))*A rcTan[Rt[b, 2]*(x/Rt[a, 2])], x] /; FreeQ[{a, b}, x] && PosQ[a/b] && (GtQ[a , 0] || GtQ[b, 0])
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[((e_.)*(x_))^(m_.)*((c_) + (d_.)*(x_))^(n_.)*((a_) + (b_.)*(x_)^2)^(p_. ), x_Symbol] :> Int[(e*x)^m*(c + d*x)^(n + p)*(a/c + (b/d)*x)^p, x] /; Free Q[{a, b, c, d, e, m, n, p}, x] && EqQ[b*c^2 + a*d^2, 0] && (IntegerQ[p] || (GtQ[a, 0] && GtQ[c, 0] && !IntegerQ[n]))
Int[(x_)^2/((a_) + (b_.)*(x_)^4), x_Symbol] :> With[{r = Numerator[Rt[-a/b, 2]], s = Denominator[Rt[-a/b, 2]]}, Simp[s/(2*b) Int[1/(r + s*x^2), x], x] - Simp[s/(2*b) Int[1/(r - s*x^2), x], x]] /; FreeQ[{a, b}, x] && !GtQ [a/b, 0]
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]
Time = 6.40 (sec) , antiderivative size = 87, normalized size of antiderivative = 1.18
| method | result | size |
| pseudoelliptic | \(\frac {-2 \arctan \left (\frac {2^{\frac {3}{4}} \left (x^{3} \left (x -1\right )\right )^{\frac {1}{4}}}{2 x}\right ) 2^{\frac {1}{4}} x -\ln \left (\frac {-2^{\frac {1}{4}} x -\left (x^{3} \left (x -1\right )\right )^{\frac {1}{4}}}{2^{\frac {1}{4}} x -\left (x^{3} \left (x -1\right )\right )^{\frac {1}{4}}}\right ) 2^{\frac {1}{4}} x +8 \left (x^{3} \left (x -1\right )\right )^{\frac {1}{4}}}{2 x}\) | \(87\) |
| trager | \(\frac {4 \left (x^{4}-x^{3}\right )^{\frac {1}{4}}}{x}+\frac {\operatorname {RootOf}\left (\textit {\_Z}^{2}+\operatorname {RootOf}\left (\textit {\_Z}^{4}-2\right )^{2}\right ) \ln \left (\frac {3 \operatorname {RootOf}\left (\textit {\_Z}^{2}+\operatorname {RootOf}\left (\textit {\_Z}^{4}-2\right )^{2}\right ) \operatorname {RootOf}\left (\textit {\_Z}^{4}-2\right )^{2} x^{3}-4 \left (x^{4}-x^{3}\right )^{\frac {1}{4}} \operatorname {RootOf}\left (\textit {\_Z}^{4}-2\right )^{2} x^{2}-\operatorname {RootOf}\left (\textit {\_Z}^{2}+\operatorname {RootOf}\left (\textit {\_Z}^{4}-2\right )^{2}\right ) \operatorname {RootOf}\left (\textit {\_Z}^{4}-2\right )^{2} x^{2}-4 \sqrt {x^{4}-x^{3}}\, \operatorname {RootOf}\left (\textit {\_Z}^{2}+\operatorname {RootOf}\left (\textit {\_Z}^{4}-2\right )^{2}\right ) x +4 \left (x^{4}-x^{3}\right )^{\frac {3}{4}}}{x^{2} \left (1+x \right )}\right )}{2}-\frac {\operatorname {RootOf}\left (\textit {\_Z}^{4}-2\right ) \ln \left (\frac {3 \operatorname {RootOf}\left (\textit {\_Z}^{4}-2\right )^{3} x^{3}+4 \left (x^{4}-x^{3}\right )^{\frac {1}{4}} \operatorname {RootOf}\left (\textit {\_Z}^{4}-2\right )^{2} x^{2}-\operatorname {RootOf}\left (\textit {\_Z}^{4}-2\right )^{3} x^{2}+4 \sqrt {x^{4}-x^{3}}\, \operatorname {RootOf}\left (\textit {\_Z}^{4}-2\right ) x +4 \left (x^{4}-x^{3}\right )^{\frac {3}{4}}}{x^{2} \left (1+x \right )}\right )}{2}\) | \(262\) |
| risch | \(\frac {4 \left (x^{3} \left (x -1\right )\right )^{\frac {1}{4}}}{x}+\frac {\left (-\frac {\operatorname {RootOf}\left (\textit {\_Z}^{2}+\operatorname {RootOf}\left (\textit {\_Z}^{4}-2\right )^{2}\right ) \ln \left (-\frac {2 \sqrt {x^{4}-3 x^{3}+3 x^{2}-x}\, \operatorname {RootOf}\left (\textit {\_Z}^{4}-2\right )^{2} \operatorname {RootOf}\left (\textit {\_Z}^{2}+\operatorname {RootOf}\left (\textit {\_Z}^{4}-2\right )^{2}\right ) x -2 \sqrt {x^{4}-3 x^{3}+3 x^{2}-x}\, \operatorname {RootOf}\left (\textit {\_Z}^{4}-2\right )^{2} \operatorname {RootOf}\left (\textit {\_Z}^{2}+\operatorname {RootOf}\left (\textit {\_Z}^{4}-2\right )^{2}\right )+2 \operatorname {RootOf}\left (\textit {\_Z}^{4}-2\right )^{2} \left (x^{4}-3 x^{3}+3 x^{2}-x \right )^{\frac {1}{4}} x^{2}-3 \operatorname {RootOf}\left (\textit {\_Z}^{2}+\operatorname {RootOf}\left (\textit {\_Z}^{4}-2\right )^{2}\right ) x^{3}-4 \left (x^{4}-3 x^{3}+3 x^{2}-x \right )^{\frac {1}{4}} \operatorname {RootOf}\left (\textit {\_Z}^{4}-2\right )^{2} x +7 \operatorname {RootOf}\left (\textit {\_Z}^{2}+\operatorname {RootOf}\left (\textit {\_Z}^{4}-2\right )^{2}\right ) x^{2}+2 \left (x^{4}-3 x^{3}+3 x^{2}-x \right )^{\frac {1}{4}} \operatorname {RootOf}\left (\textit {\_Z}^{4}-2\right )^{2}-4 \left (x^{4}-3 x^{3}+3 x^{2}-x \right )^{\frac {3}{4}}-5 \operatorname {RootOf}\left (\textit {\_Z}^{2}+\operatorname {RootOf}\left (\textit {\_Z}^{4}-2\right )^{2}\right ) x +\operatorname {RootOf}\left (\textit {\_Z}^{2}+\operatorname {RootOf}\left (\textit {\_Z}^{4}-2\right )^{2}\right )}{\left (x -1\right )^{2} \left (1+x \right )}\right )}{2}+\frac {\operatorname {RootOf}\left (\textit {\_Z}^{4}-2\right ) \ln \left (-\frac {2 \sqrt {x^{4}-3 x^{3}+3 x^{2}-x}\, \operatorname {RootOf}\left (\textit {\_Z}^{4}-2\right )^{3} x -2 \sqrt {x^{4}-3 x^{3}+3 x^{2}-x}\, \operatorname {RootOf}\left (\textit {\_Z}^{4}-2\right )^{3}-2 \operatorname {RootOf}\left (\textit {\_Z}^{4}-2\right )^{2} \left (x^{4}-3 x^{3}+3 x^{2}-x \right )^{\frac {1}{4}} x^{2}+4 \left (x^{4}-3 x^{3}+3 x^{2}-x \right )^{\frac {1}{4}} \operatorname {RootOf}\left (\textit {\_Z}^{4}-2\right )^{2} x +3 \operatorname {RootOf}\left (\textit {\_Z}^{4}-2\right ) x^{3}-2 \left (x^{4}-3 x^{3}+3 x^{2}-x \right )^{\frac {1}{4}} \operatorname {RootOf}\left (\textit {\_Z}^{4}-2\right )^{2}-7 \operatorname {RootOf}\left (\textit {\_Z}^{4}-2\right ) x^{2}-4 \left (x^{4}-3 x^{3}+3 x^{2}-x \right )^{\frac {3}{4}}+5 \operatorname {RootOf}\left (\textit {\_Z}^{4}-2\right ) x -\operatorname {RootOf}\left (\textit {\_Z}^{4}-2\right )}{\left (x -1\right )^{2} \left (1+x \right )}\right )}{2}\right ) \left (x^{3} \left (x -1\right )\right )^{\frac {1}{4}} \left (x \left (x -1\right )^{3}\right )^{\frac {1}{4}}}{x \left (x -1\right )}\) | \(567\) |
1/2*(-2*arctan(1/2*2^(3/4)/x*(x^3*(x-1))^(1/4))*2^(1/4)*x-ln((-2^(1/4)*x-( x^3*(x-1))^(1/4))/(2^(1/4)*x-(x^3*(x-1))^(1/4)))*2^(1/4)*x+8*(x^3*(x-1))^( 1/4))/x
Result contains complex when optimal does not.
Time = 0.26 (sec) , antiderivative size = 141, normalized size of antiderivative = 1.91 \[ \int \frac {\sqrt [4]{-x^3+x^4}}{x^2 \left (-1+x^2\right )} \, dx=-\frac {8^{\frac {3}{4}} x \log \left (\frac {8^{\frac {3}{4}} x + 4 \, {\left (x^{4} - x^{3}\right )}^{\frac {1}{4}}}{x}\right ) - 8^{\frac {3}{4}} x \log \left (-\frac {8^{\frac {3}{4}} x - 4 \, {\left (x^{4} - x^{3}\right )}^{\frac {1}{4}}}{x}\right ) + i \cdot 8^{\frac {3}{4}} x \log \left (\frac {i \cdot 8^{\frac {3}{4}} x + 4 \, {\left (x^{4} - x^{3}\right )}^{\frac {1}{4}}}{x}\right ) - i \cdot 8^{\frac {3}{4}} x \log \left (\frac {-i \cdot 8^{\frac {3}{4}} x + 4 \, {\left (x^{4} - x^{3}\right )}^{\frac {1}{4}}}{x}\right ) - 32 \, {\left (x^{4} - x^{3}\right )}^{\frac {1}{4}}}{8 \, x} \]
-1/8*(8^(3/4)*x*log((8^(3/4)*x + 4*(x^4 - x^3)^(1/4))/x) - 8^(3/4)*x*log(- (8^(3/4)*x - 4*(x^4 - x^3)^(1/4))/x) + I*8^(3/4)*x*log((I*8^(3/4)*x + 4*(x ^4 - x^3)^(1/4))/x) - I*8^(3/4)*x*log((-I*8^(3/4)*x + 4*(x^4 - x^3)^(1/4)) /x) - 32*(x^4 - x^3)^(1/4))/x
\[ \int \frac {\sqrt [4]{-x^3+x^4}}{x^2 \left (-1+x^2\right )} \, dx=\int \frac {\sqrt [4]{x^{3} \left (x - 1\right )}}{x^{2} \left (x - 1\right ) \left (x + 1\right )}\, dx \]
\[ \int \frac {\sqrt [4]{-x^3+x^4}}{x^2 \left (-1+x^2\right )} \, dx=\int { \frac {{\left (x^{4} - x^{3}\right )}^{\frac {1}{4}}}{{\left (x^{2} - 1\right )} x^{2}} \,d x } \]
Time = 0.29 (sec) , antiderivative size = 73, normalized size of antiderivative = 0.99 \[ \int \frac {\sqrt [4]{-x^3+x^4}}{x^2 \left (-1+x^2\right )} \, dx=-2^{\frac {1}{4}} \arctan \left (\frac {1}{2} \cdot 2^{\frac {3}{4}} {\left (-\frac {1}{x} + 1\right )}^{\frac {1}{4}}\right ) - \frac {1}{2} \cdot 2^{\frac {1}{4}} \log \left (2^{\frac {1}{4}} + {\left (-\frac {1}{x} + 1\right )}^{\frac {1}{4}}\right ) + \frac {1}{2} \cdot 2^{\frac {1}{4}} \log \left ({\left | -2^{\frac {1}{4}} + {\left (-\frac {1}{x} + 1\right )}^{\frac {1}{4}} \right |}\right ) + 4 \, {\left (-\frac {1}{x} + 1\right )}^{\frac {1}{4}} \]
-2^(1/4)*arctan(1/2*2^(3/4)*(-1/x + 1)^(1/4)) - 1/2*2^(1/4)*log(2^(1/4) + (-1/x + 1)^(1/4)) + 1/2*2^(1/4)*log(abs(-2^(1/4) + (-1/x + 1)^(1/4))) + 4* (-1/x + 1)^(1/4)
Timed out. \[ \int \frac {\sqrt [4]{-x^3+x^4}}{x^2 \left (-1+x^2\right )} \, dx=-\int \frac {{\left (x^4-x^3\right )}^{1/4}}{x^2-x^4} \,d x \]