3.8.48 \(\int \frac {1}{(-2+x) \sqrt [4]{-x^2+x^3}} \, dx\) [748]

3.8.48.1 Optimal result

Integrand size = 19, antiderivative size = 58 \[ \int \frac {1}{(-2+x) \sqrt [4]{-x^2+x^3}} \, dx=\frac {\arctan \left (\frac {\sqrt {2} \sqrt [4]{-x^2+x^3}}{x}\right )}{\sqrt {2}}-\frac {\text {arctanh}\left (\frac {x}{\sqrt {2} \sqrt [4]{-x^2+x^3}}\right )}{\sqrt {2}} \]

output

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

3.8.48.2 Mathematica [A] (verified)

Time = 0.10 (sec) , antiderivative size = 70, normalized size of antiderivative = 1.21 \[ \int \frac {1}{(-2+x) \sqrt [4]{-x^2+x^3}} \, dx=\frac {\sqrt [4]{-1+x} \sqrt {x} \left (\arctan \left (\frac {\sqrt {2} \sqrt [4]{-1+x}}{\sqrt {x}}\right )-\text {arctanh}\left (\frac {\sqrt {x}}{\sqrt {2} \sqrt [4]{-1+x}}\right )\right )}{\sqrt {2} \sqrt [4]{(-1+x) x^2}} \]

input

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

output

((-1 + x)^(1/4)*Sqrt[x]*(ArcTan[(Sqrt[2]*(-1 + x)^(1/4))/Sqrt[x]] - ArcTan 
h[Sqrt[x]/(Sqrt[2]*(-1 + x)^(1/4))]))/(Sqrt[2]*((-1 + x)*x^2)^(1/4))
 

3.8.48.3 Rubi [C] (verified)

Result contains higher order function than in optimal. Order 4 vs. order 3 in optimal.

Time = 0.46 (sec) , antiderivative size = 192, normalized size of antiderivative = 3.31, number of steps used = 11, number of rules used = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.526, Rules used = {2467, 25, 116, 25, 993, 1535, 761, 2213, 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.

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

\(\Big \downarrow \) 2467

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

\(\Big \downarrow \) 25

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

\(\Big \downarrow \) 116

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

\(\Big \downarrow \) 25

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

\(\Big \downarrow \) 993

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

\(\Big \downarrow \) 1535

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

\(\Big \downarrow \) 761

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

\(\Big \downarrow \) 2213

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

\(\Big \downarrow \) 216

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

\(\Big \downarrow \) 219

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

input

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

output

(4*(-1 + x)^(1/4)*Sqrt[x]*((-1/2*ArcTanh[(Sqrt[2]*(-1 + x)^(1/4))/Sqrt[x]] 
/Sqrt[2] - ((1 + Sqrt[-1 + x])*Sqrt[x/(1 + Sqrt[-1 + x])^2]*EllipticF[2*Ar 
cTan[(-1 + x)^(1/4)], 1/2])/(4*Sqrt[x]))/2 + (ArcTan[(Sqrt[2]*(-1 + x)^(1/ 
4))/Sqrt[x]]/(2*Sqrt[2]) + ((1 + Sqrt[-1 + x])*Sqrt[x/(1 + Sqrt[-1 + x])^2 
]*EllipticF[2*ArcTan[(-1 + x)^(1/4)], 1/2])/(4*Sqrt[x]))/2))/(-x^2 + x^3)^ 
(1/4)
 

3.8.48.3.1 Defintions of rubi rules used

rule 25

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

rule 116

Int[1/(((a_.) + (b_.)*(x_))*Sqrt[(c_.) + (d_.)*(x_)]*((e_.) + (f_.)*(x_))^( 
1/4)), x_] :> Simp[-4   Subst[Int[x^2/((b*e - a*f - b*x^4)*Sqrt[c - d*(e/f) 
 + d*(x^4/f)]), x], x, (e + f*x)^(1/4)], x] /; FreeQ[{a, b, c, d, e, f}, x] 
 && GtQ[-f/(d*e - c*f), 0]
 

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 761

Int[1/Sqrt[(a_) + (b_.)*(x_)^4], x_Symbol] :> With[{q = Rt[b/a, 4]}, Simp[( 
1 + q^2*x^2)*(Sqrt[(a + b*x^4)/(a*(1 + q^2*x^2)^2)]/(2*q*Sqrt[a + b*x^4]))* 
EllipticF[2*ArcTan[q*x], 1/2], x]] /; FreeQ[{a, b}, x] && PosQ[b/a]
 

rule 993

Int[(x_)^2/(((a_) + (b_.)*(x_)^4)*Sqrt[(c_) + (d_.)*(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)*Sqrt[c + d*x^4]), x], x] - Simp[s/(2*b)   Int[1/((r 
 - s*x^2)*Sqrt[c + d*x^4]), x], x]] /; FreeQ[{a, b, c, d}, x] && NeQ[b*c - 
a*d, 0]
 

rule 1535

Int[1/(((d_) + (e_.)*(x_)^2)*Sqrt[(a_) + (c_.)*(x_)^4]), x_Symbol] :> Simp[ 
1/(2*d)   Int[1/Sqrt[a + c*x^4], x], x] + Simp[1/(2*d)   Int[(d - e*x^2)/(( 
d + e*x^2)*Sqrt[a + c*x^4]), x], x] /; FreeQ[{a, c, d, e}, x] && NeQ[c*d^2 
+ a*e^2, 0] && EqQ[c*d^2 - a*e^2, 0]
 

rule 2213

Int[((A_) + (B_.)*(x_)^2)/(((d_) + (e_.)*(x_)^2)*Sqrt[(a_) + (c_.)*(x_)^4]) 
, x_Symbol] :> Simp[A   Subst[Int[1/(d + 2*a*e*x^2), x], x, x/Sqrt[a + c*x^ 
4]], x] /; FreeQ[{a, c, d, e, A, B}, x] && EqQ[c*d^2 - a*e^2, 0] && EqQ[B*d 
 + A*e, 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]
 

3.8.48.4 Maple [C] (verified)

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

Time = 2.19 (sec) , antiderivative size = 200, normalized size of antiderivative = 3.45

method	result	size
trager	\(-\frac {\operatorname {RootOf}\left (\textit {\_Z}^{2}+2\right ) \ln \left (-\frac {4 \operatorname {RootOf}\left (\textit {\_Z}^{2}+2\right ) \sqrt {x^{3}-x^{2}}\, x -\operatorname {RootOf}\left (\textit {\_Z}^{2}+2\right ) x^{3}-4 \operatorname {RootOf}\left (\textit {\_Z}^{2}+2\right ) x^{2}-8 \left (x^{3}-x^{2}\right )^{\frac {3}{4}}+4 \left (x^{3}-x^{2}\right )^{\frac {1}{4}} x^{2}+4 \operatorname {RootOf}\left (\textit {\_Z}^{2}+2\right ) x}{\left (x -2\right )^{2} x}\right )}{4}-\frac {\operatorname {RootOf}\left (\textit {\_Z}^{2}-2\right ) \ln \left (\frac {4 \operatorname {RootOf}\left (\textit {\_Z}^{2}-2\right ) \sqrt {x^{3}-x^{2}}\, x +\operatorname {RootOf}\left (\textit {\_Z}^{2}-2\right ) x^{3}+4 \operatorname {RootOf}\left (\textit {\_Z}^{2}-2\right ) x^{2}+8 \left (x^{3}-x^{2}\right )^{\frac {3}{4}}+4 \left (x^{3}-x^{2}\right )^{\frac {1}{4}} x^{2}-4 \operatorname {RootOf}\left (\textit {\_Z}^{2}-2\right ) x}{\left (x -2\right )^{2} x}\right )}{4}\)	\(200\)

input

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

output

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

3.8.48.5 Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 193 vs. \(2 (48) = 96\).

Time = 1.72 (sec) , antiderivative size = 193, normalized size of antiderivative = 3.33 \[ \int \frac {1}{(-2+x) \sqrt [4]{-x^2+x^3}} \, dx=\frac {1}{4} \, \sqrt {2} \arctan \left (\frac {2 \, {\left (\sqrt {2} {\left (x^{3} - x^{2}\right )}^{\frac {1}{4}} x^{2} + 2 \, \sqrt {2} {\left (x^{3} - x^{2}\right )}^{\frac {3}{4}}\right )}}{x^{3} - 4 \, x^{2} + 4 \, x}\right ) + \frac {1}{8} \, \sqrt {2} \log \left (-\frac {x^{5} + 56 \, x^{4} - 40 \, x^{3} - 8 \, \sqrt {2} {\left (x^{3} - x^{2}\right )}^{\frac {3}{4}} {\left (3 \, x^{2} + 4 \, x - 4\right )} - 32 \, x^{2} - 4 \, \sqrt {2} {\left (x^{4} + 12 \, x^{3} - 12 \, x^{2}\right )} {\left (x^{3} - x^{2}\right )}^{\frac {1}{4}} + 16 \, {\left (x^{3} + 4 \, x^{2} - 4 \, x\right )} \sqrt {x^{3} - x^{2}} + 16 \, x}{x^{5} - 8 \, x^{4} + 24 \, x^{3} - 32 \, x^{2} + 16 \, x}\right ) \]

input

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

output

1/4*sqrt(2)*arctan(2*(sqrt(2)*(x^3 - x^2)^(1/4)*x^2 + 2*sqrt(2)*(x^3 - x^2 
)^(3/4))/(x^3 - 4*x^2 + 4*x)) + 1/8*sqrt(2)*log(-(x^5 + 56*x^4 - 40*x^3 - 
8*sqrt(2)*(x^3 - x^2)^(3/4)*(3*x^2 + 4*x - 4) - 32*x^2 - 4*sqrt(2)*(x^4 + 
12*x^3 - 12*x^2)*(x^3 - x^2)^(1/4) + 16*(x^3 + 4*x^2 - 4*x)*sqrt(x^3 - x^2 
) + 16*x)/(x^5 - 8*x^4 + 24*x^3 - 32*x^2 + 16*x))
 

3.8.48.6 Sympy [F]

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

input

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

output

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

3.8.48.7 Maxima [F]

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

input

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

output

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

3.8.48.8 Giac [F]

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

input

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

output

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

3.8.48.9 Mupad [F(-1)]

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

input

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

output

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