\(\int \frac {\sqrt [4]{-x^3+x^4}}{x^2 (-1+x^2)} \, dx\) [975]

Optimal result

Integrand size = 24, antiderivative size = 74 \[ \int \frac {\sqrt [4]{-x^3+x^4}}{x^2 \left (-1+x^2\right )} \, dx=\frac {4 \sqrt [4]{-x^3+x^4}}{x}+\sqrt [4]{2} \arctan \left (\frac {\sqrt [4]{2} x}{\sqrt [4]{-x^3+x^4}}\right )-\sqrt [4]{2} \text {arctanh}\left (\frac {\sqrt [4]{2} x}{\sqrt [4]{-x^3+x^4}}\right ) \] Output:

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))
 

Mathematica [A] (verified)

Time = 0.34 (sec) , antiderivative size = 91, normalized size of antiderivative = 1.23 \[ \int \frac {\sqrt [4]{-x^3+x^4}}{x^2 \left (-1+x^2\right )} \, dx=\frac {(-1+x)^{3/4} x^2 \left (4 \sqrt [4]{-1+x}+\sqrt [4]{2} \sqrt [4]{x} \arctan \left (\frac {\sqrt [4]{2}}{\sqrt [4]{\frac {-1+x}{x}}}\right )-\sqrt [4]{2} \sqrt [4]{x} \text {arctanh}\left (\frac {\sqrt [4]{2}}{\sqrt [4]{\frac {-1+x}{x}}}\right )\right )}{\left ((-1+x) x^3\right )^{3/4}} \] Input:

Integrate[(-x^3 + x^4)^(1/4)/(x^2*(-1 + x^2)),x]
 

Output:

((-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)
 

Rubi [A] (verified)

Time = 0.31 (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

\(\displaystyle -\frac {\sqrt [4]{x^4-x^3} \left (4 \left (\frac {\text {arctanh}\left (\frac {\sqrt [4]{2} \sqrt [4]{x}}{\sqrt [4]{x-1}}\right )}{2\ 2^{3/4}}-\frac {\arctan \left (\frac {\sqrt [4]{2} \sqrt [4]{x}}{\sqrt [4]{x-1}}\right )}{2\ 2^{3/4}}\right )-\frac {4 \sqrt [4]{x-1}}{\sqrt [4]{x}}\right )}{\sqrt [4]{x-1} x^{3/4}}\)

Input:

Int[(-x^3 + x^4)^(1/4)/(x^2*(-1 + x^2)),x]
 

Output:

-(((-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)))
 

Defintions of rubi rules used

rule 25

Int[-(Fx_), x_Symbol] :> Simp[Identity[-1]   Int[Fx, x], x]
 

rule 104

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]
 

rule 107

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])
 

rule 216

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])
 

rule 219

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])
 

rule 516

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]))
 

rule 827

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]
 

rule 2467

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]
 

Maple [A] (verified)

Time = 2.18 (sec) , antiderivative size = 87, normalized size of antiderivative = 1.18

method	result	size
pseudoelliptic	\(\frac {-\ln \left (\frac {-2^{\frac {1}{4}} x -\left (x^{3} \left (-1+x \right )\right )^{\frac {1}{4}}}{2^{\frac {1}{4}} x -\left (x^{3} \left (-1+x \right )\right )^{\frac {1}{4}}}\right ) 2^{\frac {1}{4}} x -2 \arctan \left (\frac {2^{\frac {3}{4}} \left (x^{3} \left (-1+x \right )\right )^{\frac {1}{4}}}{2 x}\right ) 2^{\frac {1}{4}} x +8 \left (x^{3} \left (-1+x \right )\right )^{\frac {1}{4}}}{2 x}\)	\(87\)
trager	\(\frac {4 \left (x^{4}-x^{3}\right )^{\frac {1}{4}}}{x}+\frac {\operatorname {RootOf}\left (\textit {\_Z}^{4}-2\right ) \ln \left (-\frac {3 \operatorname {RootOf}\left (\textit {\_Z}^{4}-2\right )^{3} x^{3}-\operatorname {RootOf}\left (\textit {\_Z}^{4}-2\right )^{3} x^{2}-4 \operatorname {RootOf}\left (\textit {\_Z}^{4}-2\right )^{2} \left (x^{4}-x^{3}\right )^{\frac {1}{4}} x^{2}+4 \sqrt {x^{4}-x^{3}}\, \operatorname {RootOf}\left (\textit {\_Z}^{4}-2\right ) x -4 \left (x^{4}-x^{3}\right )^{\frac {3}{4}}}{x^{2} \left (1+x \right )}\right )}{2}-\frac {\operatorname {RootOf}\left (\textit {\_Z}^{2}+\operatorname {RootOf}\left (\textit {\_Z}^{4}-2\right )^{2}\right ) \ln \left (-\frac {3 \operatorname {RootOf}\left (\textit {\_Z}^{2}+\operatorname {RootOf}\left (\textit {\_Z}^{4}-2\right )^{2}\right ) \operatorname {RootOf}\left (\textit {\_Z}^{4}-2\right )^{2} x^{3}-\operatorname {RootOf}\left (\textit {\_Z}^{2}+\operatorname {RootOf}\left (\textit {\_Z}^{4}-2\right )^{2}\right ) \operatorname {RootOf}\left (\textit {\_Z}^{4}-2\right )^{2} x^{2}+4 \operatorname {RootOf}\left (\textit {\_Z}^{4}-2\right )^{2} \left (x^{4}-x^{3}\right )^{\frac {1}{4}} x^{2}-4 \sqrt {x^{4}-x^{3}}\, \operatorname {RootOf}\left (\textit {\_Z}^{2}+\operatorname {RootOf}\left (\textit {\_Z}^{4}-2\right )^{2}\right ) x -4 \left (x^{4}-x^{3}\right )^{\frac {3}{4}}}{x^{2} \left (1+x \right )}\right )}{2}\)	\(264\)
risch	\(\frac {4 \left (x^{3} \left (-1+x \right )\right )^{\frac {1}{4}}}{x}+\frac {\left (-\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 (-1+x \right )^{2} \left (1+x \right )}\right )}{2}-\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}-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}^{2}+\operatorname {RootOf}\left (\textit {\_Z}^{4}-2\right )^{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}^{2}+\operatorname {RootOf}\left (\textit {\_Z}^{4}-2\right )^{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}^{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 (-1+x \right )^{2} \left (1+x \right )}\right )}{2}\right ) \left (x^{3} \left (-1+x \right )\right )^{\frac {1}{4}} \left (x \left (-1+x \right )^{3}\right )^{\frac {1}{4}}}{x \left (-1+x \right )}\)	\(566\)

Input:

int((x^4-x^3)^(1/4)/x^2/(x^2-1),x,method=_RETURNVERBOSE)
 

Output:

1/2*(-ln((-2^(1/4)*x-(x^3*(-1+x))^(1/4))/(2^(1/4)*x-(x^3*(-1+x))^(1/4)))*2 
^(1/4)*x-2*arctan(1/2*2^(3/4)/x*(x^3*(-1+x))^(1/4))*2^(1/4)*x+8*(x^3*(-1+x 
))^(1/4))/x
 

Fricas [A] (verification not implemented)

Time = 0.09 (sec) , antiderivative size = 113, normalized size of antiderivative = 1.53 \[ \int \frac {\sqrt [4]{-x^3+x^4}}{x^2 \left (-1+x^2\right )} \, dx=\frac {2 \cdot 8^{\frac {3}{4}} x \arctan \left (\frac {8^{\frac {3}{4}} {\left (x^{4} - x^{3}\right )}^{\frac {3}{4}}}{4 \, {\left (x^{3} - x^{2}\right )}}\right ) - 8^{\frac {3}{4}} x \log \left (\frac {8^{\frac {3}{4}} x + 4 \, {\left (x^{4} - x^{3}\right )}^{\frac {1}{4}}}{x}\right ) + 8^{\frac {3}{4}} x \log \left (-\frac {8^{\frac {3}{4}} x - 4 \, {\left (x^{4} - x^{3}\right )}^{\frac {1}{4}}}{x}\right ) + 32 \, {\left (x^{4} - x^{3}\right )}^{\frac {1}{4}}}{8 \, x} \] Input:

integrate((x^4-x^3)^(1/4)/x^2/(x^2-1),x, algorithm="fricas")
 

Output:

1/8*(2*8^(3/4)*x*arctan(1/4*8^(3/4)*(x^4 - x^3)^(3/4)/(x^3 - x^2)) - 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) + 32*(x^4 - x^3)^(1/4))/x
 

Sympy [F]

\[ \int \frac {\sqrt [4]{-x^3+x^4}}{x^2 \left (-1+x^2\right )} \, dx=\int \frac {\sqrt [4]{x^{3} \left (x - 1\right )}}{x^{2} \left (x - 1\right ) \left (x + 1\right )}\, dx \] Input:

integrate((x**4-x**3)**(1/4)/x**2/(x**2-1),x)
 

Output:

Integral((x**3*(x - 1))**(1/4)/(x**2*(x - 1)*(x + 1)), x)
 

Maxima [F]

\[ \int \frac {\sqrt [4]{-x^3+x^4}}{x^2 \left (-1+x^2\right )} \, dx=\int { \frac {{\left (x^{4} - x^{3}\right )}^{\frac {1}{4}}}{{\left (x^{2} - 1\right )} x^{2}} \,d x } \] Input:

integrate((x^4-x^3)^(1/4)/x^2/(x^2-1),x, algorithm="maxima")
 

Output:

integrate((x^4 - x^3)^(1/4)/((x^2 - 1)*x^2), x)
 

Giac [A] (verification not implemented)

Time = 0.32 (sec) , antiderivative size = 73, normalized size of antiderivative = 0.99 \[ \int \frac {\sqrt [4]{-x^3+x^4}}{x^2 \left (-1+x^2\right )} \, dx=-2^{\frac {1}{4}} \arctan \left (\frac {1}{2} \cdot 2^{\frac {3}{4}} {\left (-\frac {1}{x} + 1\right )}^{\frac {1}{4}}\right ) - \frac {1}{2} \cdot 2^{\frac {1}{4}} \log \left (2^{\frac {1}{4}} + {\left (-\frac {1}{x} + 1\right )}^{\frac {1}{4}}\right ) + \frac {1}{2} \cdot 2^{\frac {1}{4}} \log \left ({\left | -2^{\frac {1}{4}} + {\left (-\frac {1}{x} + 1\right )}^{\frac {1}{4}} \right |}\right ) + 4 \, {\left (-\frac {1}{x} + 1\right )}^{\frac {1}{4}} \] Input:

integrate((x^4-x^3)^(1/4)/x^2/(x^2-1),x, algorithm="giac")
 

Output:

-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)
 

Mupad [F(-1)]

Timed out. \[ \int \frac {\sqrt [4]{-x^3+x^4}}{x^2 \left (-1+x^2\right )} \, dx=-\int \frac {{\left (x^4-x^3\right )}^{1/4}}{x^2-x^4} \,d x \] Input:

int((x^4 - x^3)^(1/4)/(x^2*(x^2 - 1)),x)
 

Output:

-int((x^4 - x^3)^(1/4)/(x^2 - x^4), x)
 

Reduce [F]

\[ \int \frac {\sqrt [4]{-x^3+x^4}}{x^2 \left (-1+x^2\right )} \, dx=\int \frac {x^{\frac {3}{4}} \left (x -1\right )^{\frac {1}{4}}}{x^{4}-x^{2}}d x \] Input:

int((x^4-x^3)^(1/4)/x^2/(x^2-1),x)
 

Output:

int((x**(3/4)*(x - 1)**(1/4))/(x**4 - x**2),x)