\(\int \frac {\sqrt [4]{-x+x^4}}{x^2} \, dx\) [705]

Optimal result

Integrand size = 15, antiderivative size = 55 \[ \int \frac {\sqrt [4]{-x+x^4}}{x^2} \, dx=-\frac {4 \sqrt [4]{-x+x^4}}{3 x}-\frac {2}{3} \arctan \left (\frac {x}{\sqrt [4]{-x+x^4}}\right )+\frac {2}{3} \text {arctanh}\left (\frac {x}{\sqrt [4]{-x+x^4}}\right ) \] Output:

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

Mathematica [A] (verified)

Time = 3.97 (sec) , antiderivative size = 81, normalized size of antiderivative = 1.47 \[ \int \frac {\sqrt [4]{-x+x^4}}{x^2} \, dx=-\frac {2 \left (-1+x^3\right )^{3/4} \left (2 \sqrt [4]{-1+x^3}+x^{3/4} \arctan \left (\frac {x^{3/4}}{\sqrt [4]{-1+x^3}}\right )-x^{3/4} \text {arctanh}\left (\frac {x^{3/4}}{\sqrt [4]{-1+x^3}}\right )\right )}{3 \left (x \left (-1+x^3\right )\right )^{3/4}} \] Input:

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

Output:

(-2*(-1 + x^3)^(3/4)*(2*(-1 + x^3)^(1/4) + x^(3/4)*ArcTan[x^(3/4)/(-1 + x^ 
3)^(1/4)] - x^(3/4)*ArcTanh[x^(3/4)/(-1 + x^3)^(1/4)]))/(3*(x*(-1 + x^3))^ 
(3/4))
 

Rubi [A] (warning: unable to verify)

Time = 0.24 (sec) , antiderivative size = 85, normalized size of antiderivative = 1.55, number of steps used = 9, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.533, Rules used = {1926, 1938, 851, 807, 854, 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.

\(\displaystyle \int \frac {\sqrt [4]{x^4-x}}{x^2} \, dx\)

\(\Big \downarrow \) 1926

\(\displaystyle \int \frac {x^2}{\left (x^4-x\right )^{3/4}}dx-\frac {4 \sqrt [4]{x^4-x}}{3 x}\)

\(\Big \downarrow \) 1938

\(\displaystyle \frac {x^{3/4} \left (x^3-1\right )^{3/4} \int \frac {x^{5/4}}{\left (x^3-1\right )^{3/4}}dx}{\left (x^4-x\right )^{3/4}}-\frac {4 \sqrt [4]{x^4-x}}{3 x}\)

\(\Big \downarrow \) 851

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

\(\Big \downarrow \) 807

\(\displaystyle \frac {4 x^{3/4} \left (x^3-1\right )^{3/4} \int \frac {\sqrt {x}}{(x-1)^{3/4}}dx^{3/4}}{3 \left (x^4-x\right )^{3/4}}-\frac {4 \sqrt [4]{x^4-x}}{3 x}\)

\(\Big \downarrow \) 854

\(\displaystyle \frac {4 x^{3/4} \left (x^3-1\right )^{3/4} \int \frac {\sqrt {x}}{1-x}d\frac {x^{3/4}}{\sqrt [4]{x-1}}}{3 \left (x^4-x\right )^{3/4}}-\frac {4 \sqrt [4]{x^4-x}}{3 x}\)

\(\Big \downarrow \) 827

\(\displaystyle \frac {4 x^{3/4} \left (x^3-1\right )^{3/4} \left (\frac {1}{2} \int \frac {1}{1-\sqrt {x}}d\frac {x^{3/4}}{\sqrt [4]{x-1}}-\frac {1}{2} \int \frac {1}{\sqrt {x}+1}d\frac {x^{3/4}}{\sqrt [4]{x-1}}\right )}{3 \left (x^4-x\right )^{3/4}}-\frac {4 \sqrt [4]{x^4-x}}{3 x}\)

\(\Big \downarrow \) 216

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

\(\Big \downarrow \) 219

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

Input:

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

Output:

(-4*(-x + x^4)^(1/4))/(3*x) + (4*x^(3/4)*(-1 + x^3)^(3/4)*(-1/2*ArcTan[x^( 
3/4)/(-1 + x)^(1/4)] + ArcTanh[x^(3/4)/(-1 + x)^(1/4)]/2))/(3*(-x + x^4)^( 
3/4))
 

Defintions of rubi rules used

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 807

Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> With[{k = GCD[m 
+ 1, n]}, Simp[1/k   Subst[Int[x^((m + 1)/k - 1)*(a + b*x^(n/k))^p, x], x, 
x^k], x] /; k != 1] /; FreeQ[{a, b, p}, x] && IGtQ[n, 0] && IntegerQ[m]
 

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 851

Int[((c_.)*(x_))^(m_)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> With[{k = 
 Denominator[m]}, Simp[k/c   Subst[Int[x^(k*(m + 1) - 1)*(a + b*(x^(k*n)/c^ 
n))^p, x], x, (c*x)^(1/k)], x]] /; FreeQ[{a, b, c, p}, x] && IGtQ[n, 0] && 
FractionQ[m] && IntBinomialQ[a, b, c, n, m, p, x]
 

rule 854

Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[a^(p + (m + 
 1)/n)   Subst[Int[x^m/(1 - b*x^n)^(p + (m + 1)/n + 1), x], x, x/(a + b*x^n 
)^(1/n)], x] /; FreeQ[{a, b}, x] && IGtQ[n, 0] && LtQ[-1, p, 0] && NeQ[p, - 
2^(-1)] && IntegersQ[m, p + (m + 1)/n]
 

rule 1926

Int[((c_.)*(x_))^(m_)*((a_.)*(x_)^(j_.) + (b_.)*(x_)^(n_.))^(p_), x_Symbol] 
 :> Simp[(c*x)^(m + 1)*((a*x^j + b*x^n)^p/(c*(m + j*p + 1))), x] - Simp[b*p 
*((n - j)/(c^n*(m + j*p + 1)))   Int[(c*x)^(m + n)*(a*x^j + b*x^n)^(p - 1), 
 x], x] /; FreeQ[{a, b, c}, x] &&  !IntegerQ[p] && LtQ[0, j, n] && (Integer 
sQ[j, n] || GtQ[c, 0]) && GtQ[p, 0] && LtQ[m + j*p + 1, 0]
 

rule 1938

Int[((c_.)*(x_))^(m_.)*((a_.)*(x_)^(j_.) + (b_.)*(x_)^(n_.))^(p_), x_Symbol 
] :> Simp[c^IntPart[m]*(c*x)^FracPart[m]*((a*x^j + b*x^n)^FracPart[p]/(x^(F 
racPart[m] + j*FracPart[p])*(a + b*x^(n - j))^FracPart[p]))   Int[x^(m + j* 
p)*(a + b*x^(n - j))^p, x], x] /; FreeQ[{a, b, c, j, m, n, p}, x] &&  !Inte 
gerQ[p] && NeQ[n, j] && PosQ[n - j]
 

Maple [C] (warning: unable to verify)

Result contains higher order function than in optimal. Order 9 vs. order 3.

Time = 4.29 (sec) , antiderivative size = 33, normalized size of antiderivative = 0.60

method	result	size
meijerg	\(-\frac {4 \operatorname {signum}\left (x^{3}-1\right )^{\frac {1}{4}} \operatorname {hypergeom}\left (\left [-\frac {1}{4}, -\frac {1}{4}\right ], \left [\frac {3}{4}\right ], x^{3}\right )}{3 {\left (-\operatorname {signum}\left (x^{3}-1\right )\right )}^{\frac {1}{4}} x^{\frac {3}{4}}}\)	\(33\)
pseudoelliptic	\(\frac {-4 \left (x^{4}-x \right )^{\frac {1}{4}}+x \left (2 \arctan \left (\frac {\left (x^{4}-x \right )^{\frac {1}{4}}}{x}\right )-\ln \left (\frac {\left (x^{4}-x \right )^{\frac {1}{4}}-x}{x}\right )+\ln \left (\frac {x +\left (x^{4}-x \right )^{\frac {1}{4}}}{x}\right )\right )}{3 x}\)	\(73\)
trager	\(-\frac {4 \left (x^{4}-x \right )^{\frac {1}{4}}}{3 x}-\frac {\operatorname {RootOf}\left (\textit {\_Z}^{2}+1\right ) \ln \left (2 \sqrt {x^{4}-x}\, \operatorname {RootOf}\left (\textit {\_Z}^{2}+1\right ) x -2 \operatorname {RootOf}\left (\textit {\_Z}^{2}+1\right ) x^{3}-2 \left (x^{4}-x \right )^{\frac {3}{4}}+2 x^{2} \left (x^{4}-x \right )^{\frac {1}{4}}+\operatorname {RootOf}\left (\textit {\_Z}^{2}+1\right )\right )}{3}-\frac {\ln \left (-2 \left (x^{4}-x \right )^{\frac {3}{4}}+2 x \sqrt {x^{4}-x}-2 x^{2} \left (x^{4}-x \right )^{\frac {1}{4}}+2 x^{3}-1\right )}{3}\)	\(133\)
risch	\(-\frac {4 {\left (x \left (x^{3}-1\right )\right )}^{\frac {1}{4}}}{3 x}+\frac {\left (\frac {\ln \left (\frac {2 x^{9}+2 \left (x^{12}-3 x^{9}+3 x^{6}-x^{3}\right )^{\frac {1}{4}} x^{6}-5 x^{6}+2 \sqrt {x^{12}-3 x^{9}+3 x^{6}-x^{3}}\, x^{3}-4 \left (x^{12}-3 x^{9}+3 x^{6}-x^{3}\right )^{\frac {1}{4}} x^{3}+2 \left (x^{12}-3 x^{9}+3 x^{6}-x^{3}\right )^{\frac {3}{4}}+4 x^{3}-2 \sqrt {x^{12}-3 x^{9}+3 x^{6}-x^{3}}+2 \left (x^{12}-3 x^{9}+3 x^{6}-x^{3}\right )^{\frac {1}{4}}-1}{\left (-1+x \right )^{2} \left (x^{2}+x +1\right )^{2}}\right )}{3}-\frac {\operatorname {RootOf}\left (\textit {\_Z}^{2}+1\right ) \ln \left (-\frac {2 x^{9}-2 \left (x^{12}-3 x^{9}+3 x^{6}-x^{3}\right )^{\frac {1}{4}} \operatorname {RootOf}\left (\textit {\_Z}^{2}+1\right ) x^{6}-5 x^{6}-2 \sqrt {x^{12}-3 x^{9}+3 x^{6}-x^{3}}\, x^{3}+4 \left (x^{12}-3 x^{9}+3 x^{6}-x^{3}\right )^{\frac {1}{4}} \operatorname {RootOf}\left (\textit {\_Z}^{2}+1\right ) x^{3}+2 \operatorname {RootOf}\left (\textit {\_Z}^{2}+1\right ) \left (x^{12}-3 x^{9}+3 x^{6}-x^{3}\right )^{\frac {3}{4}}+4 x^{3}+2 \sqrt {x^{12}-3 x^{9}+3 x^{6}-x^{3}}-2 \operatorname {RootOf}\left (\textit {\_Z}^{2}+1\right ) \left (x^{12}-3 x^{9}+3 x^{6}-x^{3}\right )^{\frac {1}{4}}-1}{\left (-1+x \right )^{2} \left (x^{2}+x +1\right )^{2}}\right )}{3}\right ) {\left (x \left (x^{3}-1\right )\right )}^{\frac {1}{4}} \left (x^{3} \left (x^{3}-1\right )^{3}\right )^{\frac {1}{4}}}{x \left (x^{3}-1\right )}\)	\(443\)

Input:

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

Output:

-4/3*signum(x^3-1)^(1/4)/(-signum(x^3-1))^(1/4)/x^(3/4)*hypergeom([-1/4,-1 
/4],[3/4],x^3)
 

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 93 vs. \(2 (43) = 86\).

Time = 0.60 (sec) , antiderivative size = 93, normalized size of antiderivative = 1.69 \[ \int \frac {\sqrt [4]{-x+x^4}}{x^2} \, dx=\frac {x \arctan \left (2 \, {\left (x^{4} - x\right )}^{\frac {1}{4}} x^{2} + 2 \, {\left (x^{4} - x\right )}^{\frac {3}{4}}\right ) + x \log \left (-2 \, x^{3} - 2 \, {\left (x^{4} - x\right )}^{\frac {1}{4}} x^{2} - 2 \, \sqrt {x^{4} - x} x - 2 \, {\left (x^{4} - x\right )}^{\frac {3}{4}} + 1\right ) - 4 \, {\left (x^{4} - x\right )}^{\frac {1}{4}}}{3 \, x} \] Input:

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

Output:

1/3*(x*arctan(2*(x^4 - x)^(1/4)*x^2 + 2*(x^4 - x)^(3/4)) + x*log(-2*x^3 - 
2*(x^4 - x)^(1/4)*x^2 - 2*sqrt(x^4 - x)*x - 2*(x^4 - x)^(3/4) + 1) - 4*(x^ 
4 - x)^(1/4))/x
 

Sympy [F]

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

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

Output:

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

Maxima [F]

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

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

Output:

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

Giac [A] (verification not implemented)

Time = 0.20 (sec) , antiderivative size = 53, normalized size of antiderivative = 0.96 \[ \int \frac {\sqrt [4]{-x+x^4}}{x^2} \, dx=-\frac {4}{3} \, {\left (-\frac {1}{x^{3}} + 1\right )}^{\frac {1}{4}} + \frac {2}{3} \, \arctan \left ({\left (-\frac {1}{x^{3}} + 1\right )}^{\frac {1}{4}}\right ) + \frac {1}{3} \, \log \left ({\left (-\frac {1}{x^{3}} + 1\right )}^{\frac {1}{4}} + 1\right ) - \frac {1}{3} \, \log \left ({\left | {\left (-\frac {1}{x^{3}} + 1\right )}^{\frac {1}{4}} - 1 \right |}\right ) \] Input:

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

Output:

-4/3*(-1/x^3 + 1)^(1/4) + 2/3*arctan((-1/x^3 + 1)^(1/4)) + 1/3*log((-1/x^3 
 + 1)^(1/4) + 1) - 1/3*log(abs((-1/x^3 + 1)^(1/4) - 1))
 

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

\[ \int \frac {\sqrt [4]{-x+x^4}}{x^2} \, dx=\int \frac {\left (x^{3}-1\right )^{\frac {1}{4}}}{x^{\frac {7}{4}}}d x \] Input:

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

Output:

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