\(\int \frac {\sqrt {-1+\frac {1}{x}} \sqrt {\frac {1}{x}} \sqrt {x}}{\sqrt {1+x}} \, dx\) [536]

Optimal result

Integrand size = 29, antiderivative size = 21 \[ \int \frac {\sqrt {-1+\frac {1}{x}} \sqrt {\frac {1}{x}} \sqrt {x}}{\sqrt {1+x}} \, dx=-2 E\left (\left .\arcsin \left (\sqrt {x}\right )\right |-1\right )+4 \operatorname {EllipticF}\left (\arcsin \left (\sqrt {x}\right ),-1\right ) \] Output:

-2*EllipticE(x^(1/2),I)+4*EllipticF(x^(1/2),I)
                                                                                    
                                                                                    
 

Mathematica [C] (verified)

Result contains higher order function than in optimal. Order 5 vs. order 4 in optimal.

Time = 10.06 (sec) , antiderivative size = 66, normalized size of antiderivative = 3.14 \[ \int \frac {\sqrt {-1+\frac {1}{x}} \sqrt {\frac {1}{x}} \sqrt {x}}{\sqrt {1+x}} \, dx=-\frac {2 \sqrt {\frac {x}{1+x}} \sqrt {1-x^2} \left (-3 \operatorname {Hypergeometric2F1}\left (\frac {1}{4},\frac {1}{2},\frac {5}{4},x^2\right )+x \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {3}{4},\frac {7}{4},x^2\right )\right )}{3 \sqrt {1-x}} \] Input:

Integrate[(Sqrt[-1 + x^(-1)]*Sqrt[x^(-1)]*Sqrt[x])/Sqrt[1 + x],x]
 

Output:

(-2*Sqrt[x/(1 + x)]*Sqrt[1 - x^2]*(-3*Hypergeometric2F1[1/4, 1/2, 5/4, x^2 
] + x*Hypergeometric2F1[1/2, 3/4, 7/4, x^2]))/(3*Sqrt[1 - x])
 

Rubi [B] (verified)

Leaf count is larger than twice the leaf count of optimal. \(49\) vs. \(2(21)=42\).

Time = 0.28 (sec) , antiderivative size = 49, normalized size of antiderivative = 2.33, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.138, Rules used = {30, 942, 121, 120}

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.

\(\displaystyle \int \frac {\sqrt {\frac {1}{x}-1} \sqrt {\frac {1}{x}} \sqrt {x}}{\sqrt {x+1}} \, dx\)

\(\Big \downarrow \) 30

\(\displaystyle \sqrt {\frac {1}{x}} \sqrt {x} \int \frac {\sqrt {\frac {1}{x}-1}}{\sqrt {x+1}}dx\)

\(\Big \downarrow \) 942

\(\displaystyle \frac {\sqrt {\frac {1}{x}-1} \int \frac {\sqrt {1-x}}{\sqrt {x} \sqrt {x+1}}dx}{\sqrt {1-x} \sqrt {\frac {1}{x}}}\)

\(\Big \downarrow \) 121

\(\displaystyle \frac {\sqrt {\frac {1}{x}-1} \sqrt {\frac {1}{x}} \sqrt {-x} \sqrt {x} \int \frac {\sqrt {1-x}}{\sqrt {-x} \sqrt {x+1}}dx}{\sqrt {1-x}}\)

\(\Big \downarrow \) 120

\(\displaystyle -\frac {2 \sqrt {\frac {1}{x}-1} \sqrt {\frac {1}{x}} \sqrt {-x} \sqrt {x} E\left (\left .\arcsin \left (\sqrt {-x}\right )\right |-1\right )}{\sqrt {1-x}}\)

Input:

Int[(Sqrt[-1 + x^(-1)]*Sqrt[x^(-1)]*Sqrt[x])/Sqrt[1 + x],x]
 

Output:

(-2*Sqrt[-1 + x^(-1)]*Sqrt[-x]*EllipticE[ArcSin[Sqrt[-x]], -1])/(Sqrt[1 - 
x]*Sqrt[x^(-1)]*Sqrt[x])
 

Defintions of rubi rules used

rule 30

Int[(u_.)*((a_.)*(x_))^(m_.)*((b_.)*(x_)^(i_.))^(p_), x_Symbol] :> Simp[b^I 
ntPart[p]*((b*x^i)^FracPart[p]/(a^(i*IntPart[p])*(a*x)^(i*FracPart[p]))) 
Int[u*(a*x)^(m + i*p), x], x] /; FreeQ[{a, b, i, m, p}, x] && IntegerQ[i] & 
&  !IntegerQ[p]
 

rule 120

Int[Sqrt[(e_) + (f_.)*(x_)]/(Sqrt[(b_.)*(x_)]*Sqrt[(c_) + (d_.)*(x_)]), x_] 
 :> Simp[2*(Sqrt[e]/b)*Rt[-b/d, 2]*EllipticE[ArcSin[Sqrt[b*x]/(Sqrt[c]*Rt[- 
b/d, 2])], c*(f/(d*e))], x] /; FreeQ[{b, c, d, e, f}, x] && GtQ[c, 0] && Gt 
Q[e, 0] &&  !LtQ[-b/d, 0]
 

rule 121

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

rule 942

Int[((c_) + (d_.)*(x_)^(mn_.))^(q_)*((a_) + (b_.)*(x_)^(n_.))^(p_.), x_Symb 
ol] :> Simp[x^(n*FracPart[q])*((c + d/x^n)^FracPart[q]/(d + c*x^n)^FracPart 
[q])   Int[(a + b*x^n)^p*((d + c*x^n)^q/x^(n*q)), x], x] /; FreeQ[{a, b, c, 
 d, n, p, q}, x] && EqQ[mn, -n] &&  !IntegerQ[q] &&  !IntegerQ[p]
 

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(48\) vs. \(2(17)=34\).

Time = 0.32 (sec) , antiderivative size = 49, normalized size of antiderivative = 2.33

method	result	size
default	\(-\frac {2 \sqrt {\frac {1}{x}}\, \sqrt {x}\, \sqrt {-\frac {x -1}{x}}\, \operatorname {EllipticE}\left (\sqrt {x +1}, \frac {\sqrt {2}}{2}\right ) \sqrt {-x}\, \sqrt {-2 x +2}}{x -1}\)	\(49\)
derivativedivides	\(\frac {2 x^{\frac {5}{2}} \left (\frac {1}{x}\right )^{\frac {5}{2}} \sqrt {\left (1+\frac {1}{x}\right ) x}\, \sqrt {-1+\frac {1}{x}}\, \left (\sqrt {1+\frac {1}{x}}\, \sqrt {2}\, \sqrt {1-\frac {1}{x}}\, \sqrt {-\frac {1}{x}}\, \operatorname {EllipticE}\left (\sqrt {1+\frac {1}{x}}, \frac {\sqrt {2}}{2}\right )-\sqrt {1+\frac {1}{x}}\, \sqrt {2}\, \sqrt {1-\frac {1}{x}}\, \sqrt {-\frac {1}{x}}\, \operatorname {EllipticF}\left (\sqrt {1+\frac {1}{x}}, \frac {\sqrt {2}}{2}\right )+\frac {1}{x^{2}}-1\right )}{\frac {1}{x^{2}}-1}\)	\(120\)

Input:

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

Output:

-2*(1/x)^(1/2)*x^(1/2)*(-(x-1)/x)^(1/2)*EllipticE((x+1)^(1/2),1/2*2^(1/2)) 
*(-x)^(1/2)*(-2*x+2)^(1/2)/(x-1)
 

Fricas [A] (verification not implemented)

Time = 0.11 (sec) , antiderivative size = 16, normalized size of antiderivative = 0.76 \[ \int \frac {\sqrt {-1+\frac {1}{x}} \sqrt {\frac {1}{x}} \sqrt {x}}{\sqrt {1+x}} \, dx=-2 i \, {\rm weierstrassPInverse}\left (4, 0, x\right ) - 2 i \, {\rm weierstrassZeta}\left (4, 0, {\rm weierstrassPInverse}\left (4, 0, x\right )\right ) \] Input:

integrate((-1+1/x)^(1/2)*(1/x)^(1/2)*x^(1/2)/(1+x)^(1/2),x, algorithm="fri 
cas")
 

Output:

-2*I*weierstrassPInverse(4, 0, x) - 2*I*weierstrassZeta(4, 0, weierstrassP 
Inverse(4, 0, x))
 

Sympy [F(-2)]

Exception generated. \[ \int \frac {\sqrt {-1+\frac {1}{x}} \sqrt {\frac {1}{x}} \sqrt {x}}{\sqrt {1+x}} \, dx=\text {Exception raised: RecursionError} \] Input:

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

Output:

Exception raised: RecursionError >> maximum recursion depth exceeded in co 
mparison
 

Maxima [F]

\[ \int \frac {\sqrt {-1+\frac {1}{x}} \sqrt {\frac {1}{x}} \sqrt {x}}{\sqrt {1+x}} \, dx=\int { \frac {\sqrt {\frac {1}{x} - 1}}{\sqrt {x + 1}} \,d x } \] Input:

integrate((-1+1/x)^(1/2)*(1/x)^(1/2)*x^(1/2)/(1+x)^(1/2),x, algorithm="max 
ima")
 

Output:

integrate(sqrt(1/x - 1)/sqrt(x + 1), x)
 

Giac [F]

\[ \int \frac {\sqrt {-1+\frac {1}{x}} \sqrt {\frac {1}{x}} \sqrt {x}}{\sqrt {1+x}} \, dx=\int { \frac {\sqrt {\frac {1}{x} - 1}}{\sqrt {x + 1}} \,d x } \] Input:

integrate((-1+1/x)^(1/2)*(1/x)^(1/2)*x^(1/2)/(1+x)^(1/2),x, algorithm="gia 
c")
 

Output:

integrate(sqrt(1/x - 1)/sqrt(x + 1), x)
 

Mupad [F(-1)]

Timed out. \[ \int \frac {\sqrt {-1+\frac {1}{x}} \sqrt {\frac {1}{x}} \sqrt {x}}{\sqrt {1+x}} \, dx=\int \frac {\sqrt {x}\,\sqrt {\frac {1}{x}-1}\,\sqrt {\frac {1}{x}}}{\sqrt {x+1}} \,d x \] Input:

int((x^(1/2)*(1/x - 1)^(1/2)*(1/x)^(1/2))/(x + 1)^(1/2),x)
 

Output:

int((x^(1/2)*(1/x - 1)^(1/2)*(1/x)^(1/2))/(x + 1)^(1/2), x)
 

Reduce [F]

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

int((-1+1/x)^(1/2)*(1/x)^(1/2)*x^(1/2)/(1+x)^(1/2),x)
                                                                                    
                                                                                    
 

Output:

int((sqrt(x)*sqrt(x + 1)*sqrt( - x + 1))/(x**2 + x),x)