\(\int \frac {2+x}{(-3+x) \sqrt [4]{1-x^2} (1+x^2)} \, dx\) [2299]

Optimal result

Integrand size = 27, antiderivative size = 176 \[ \int \frac {2+x}{(-3+x) \sqrt [4]{1-x^2} \left (1+x^2\right )} \, dx=-\frac {\arctan \left (\frac {\sqrt [4]{1-x^2}}{\sqrt [4]{2}-\sqrt [4]{2} x+\sqrt [4]{1-x^2}}\right )}{2 \sqrt [4]{2}}+\frac {\arctan \left (\frac {\sqrt [4]{1-x^2}}{-\sqrt [4]{2}+\sqrt [4]{2} x+\sqrt [4]{1-x^2}}\right )}{2 \sqrt [4]{2}}-\frac {\text {arctanh}\left (\frac {\left (-2 \sqrt [4]{2}+2 \sqrt [4]{2} x\right ) \sqrt [4]{1-x^2}}{\sqrt {2}-2 \sqrt {2} x+\sqrt {2} x^2+2 \sqrt {1-x^2}}\right )}{2 \sqrt [4]{2}} \] Output:

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

Mathematica [A] (verified)

Time = 0.26 (sec) , antiderivative size = 147, normalized size of antiderivative = 0.84 \[ \int \frac {2+x}{(-3+x) \sqrt [4]{1-x^2} \left (1+x^2\right )} \, dx=-\frac {\arctan \left (\frac {\sqrt [4]{1-x^2}}{\sqrt [4]{2}-\sqrt [4]{2} x-\sqrt [4]{1-x^2}}\right )+\arctan \left (\frac {\sqrt [4]{1-x^2}}{\sqrt [4]{2}-\sqrt [4]{2} x+\sqrt [4]{1-x^2}}\right )+\text {arctanh}\left (\frac {2 (-1+x) \sqrt [4]{2-2 x^2}}{\sqrt {2}-2 \sqrt {2} x+\sqrt {2} x^2+2 \sqrt {1-x^2}}\right )}{2 \sqrt [4]{2}} \] Input:

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

Output:

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

Rubi [C] (verified)

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

Time = 0.93 (sec) , antiderivative size = 378, normalized size of antiderivative = 2.15, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.074, Rules used = {7276, 2009}

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.

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

\(\Big \downarrow \) 7276

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

\(\Big \downarrow \) 2009

\(\displaystyle -\frac {\sqrt {x^2} \operatorname {EllipticPi}\left (-\frac {1}{\sqrt {2}},\arcsin \left (\sqrt [4]{1-x^2}\right ),-1\right )}{2 \sqrt {2} x}+\frac {3 i \sqrt {x^2} \operatorname {EllipticPi}\left (-\frac {i}{2 \sqrt {2}},\arcsin \left (\sqrt [4]{1-x^2}\right ),-1\right )}{4 \sqrt {2} x}-\frac {3 i \sqrt {x^2} \operatorname {EllipticPi}\left (\frac {i}{2 \sqrt {2}},\arcsin \left (\sqrt [4]{1-x^2}\right ),-1\right )}{4 \sqrt {2} x}+\frac {\sqrt {x^2} \operatorname {EllipticPi}\left (\frac {1}{\sqrt {2}},\arcsin \left (\sqrt [4]{1-x^2}\right ),-1\right )}{2 \sqrt {2} x}-\frac {\arctan \left (\frac {\sqrt [4]{1-x^2}}{\sqrt [4]{2}}\right )}{2 \sqrt [4]{2}}-\frac {\arctan \left (1-\frac {\sqrt [4]{1-x^2}}{\sqrt [4]{2}}\right )}{4 \sqrt [4]{2}}+\frac {\arctan \left (\frac {\sqrt [4]{1-x^2}}{\sqrt [4]{2}}+1\right )}{4 \sqrt [4]{2}}+\frac {\text {arctanh}\left (\frac {\sqrt [4]{1-x^2}}{\sqrt [4]{2}}\right )}{2 \sqrt [4]{2}}+\frac {\log \left (\sqrt {1-x^2}-2 \sqrt [4]{2} \sqrt [4]{1-x^2}+2 \sqrt {2}\right )}{8 \sqrt [4]{2}}-\frac {\log \left (\sqrt {1-x^2}+2 \sqrt [4]{2} \sqrt [4]{1-x^2}+2 \sqrt {2}\right )}{8 \sqrt [4]{2}}\)

Input:

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

Output:

-1/2*ArcTan[(1 - x^2)^(1/4)/2^(1/4)]/2^(1/4) - ArcTan[1 - (1 - x^2)^(1/4)/ 
2^(1/4)]/(4*2^(1/4)) + ArcTan[1 + (1 - x^2)^(1/4)/2^(1/4)]/(4*2^(1/4)) + A 
rcTanh[(1 - x^2)^(1/4)/2^(1/4)]/(2*2^(1/4)) - (Sqrt[x^2]*EllipticPi[-(1/Sq 
rt[2]), ArcSin[(1 - x^2)^(1/4)], -1])/(2*Sqrt[2]*x) + (((3*I)/4)*Sqrt[x^2] 
*EllipticPi[(-1/2*I)/Sqrt[2], ArcSin[(1 - x^2)^(1/4)], -1])/(Sqrt[2]*x) - 
(((3*I)/4)*Sqrt[x^2]*EllipticPi[(I/2)/Sqrt[2], ArcSin[(1 - x^2)^(1/4)], -1 
])/(Sqrt[2]*x) + (Sqrt[x^2]*EllipticPi[1/Sqrt[2], ArcSin[(1 - x^2)^(1/4)], 
 -1])/(2*Sqrt[2]*x) + Log[2*Sqrt[2] - 2*2^(1/4)*(1 - x^2)^(1/4) + Sqrt[1 - 
 x^2]]/(8*2^(1/4)) - Log[2*Sqrt[2] + 2*2^(1/4)*(1 - x^2)^(1/4) + Sqrt[1 - 
x^2]]/(8*2^(1/4))
 

Defintions of rubi rules used

rule 2009

Int[u_, x_Symbol] :> Simp[IntSum[u, x], x] /; SumQ[u]
 

rule 7276

Int[(u_)/((a_) + (b_.)*(x_)^(n_)), x_Symbol] :> With[{v = RationalFunctionE 
xpand[u/(a + b*x^n), x]}, Int[v, x] /; SumQ[v]] /; FreeQ[{a, b}, x] && IGtQ 
[n, 0]
 

Maple [C] (warning: unable to verify)

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

Time = 3.62 (sec) , antiderivative size = 406, normalized size of antiderivative = 2.31

method	result	size
trager	\(\frac {\operatorname {RootOf}\left (\textit {\_Z}^{2}+\operatorname {RootOf}\left (\textit {\_Z}^{4}+2\right )^{2}\right ) \ln \left (\frac {-2 \sqrt {-x^{2}+1}\, \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 \sqrt {-x^{2}+1}\, \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}+2 \left (-x^{2}+1\right )^{\frac {1}{4}} \operatorname {RootOf}\left (\textit {\_Z}^{4}+2\right )^{2} x^{2}-4 \left (-x^{2}+1\right )^{\frac {1}{4}} \operatorname {RootOf}\left (\textit {\_Z}^{4}+2\right )^{2} x +\operatorname {RootOf}\left (\textit {\_Z}^{2}+\operatorname {RootOf}\left (\textit {\_Z}^{4}+2\right )^{2}\right ) x^{3}+4 \left (-x^{2}+1\right )^{\frac {3}{4}}+2 \left (-x^{2}+1\right )^{\frac {1}{4}} \operatorname {RootOf}\left (\textit {\_Z}^{4}+2\right )^{2}-3 \operatorname {RootOf}\left (\textit {\_Z}^{2}+\operatorname {RootOf}\left (\textit {\_Z}^{4}+2\right )^{2}\right ) x^{2}+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 (-3+x \right ) \left (x^{2}+1\right )}\right )}{4}-\frac {\operatorname {RootOf}\left (\textit {\_Z}^{4}+2\right ) \ln \left (\frac {-2 \sqrt {-x^{2}+1}\, \operatorname {RootOf}\left (\textit {\_Z}^{4}+2\right )^{3} x +2 \sqrt {-x^{2}+1}\, \operatorname {RootOf}\left (\textit {\_Z}^{4}+2\right )^{3}-2 \left (-x^{2}+1\right )^{\frac {1}{4}} \operatorname {RootOf}\left (\textit {\_Z}^{4}+2\right )^{2} x^{2}+4 \left (-x^{2}+1\right )^{\frac {1}{4}} \operatorname {RootOf}\left (\textit {\_Z}^{4}+2\right )^{2} x -\operatorname {RootOf}\left (\textit {\_Z}^{4}+2\right ) x^{3}+4 \left (-x^{2}+1\right )^{\frac {3}{4}}-2 \left (-x^{2}+1\right )^{\frac {1}{4}} \operatorname {RootOf}\left (\textit {\_Z}^{4}+2\right )^{2}+3 \operatorname {RootOf}\left (\textit {\_Z}^{4}+2\right ) x^{2}-5 \operatorname {RootOf}\left (\textit {\_Z}^{4}+2\right ) x -\operatorname {RootOf}\left (\textit {\_Z}^{4}+2\right )}{\left (-3+x \right ) \left (x^{2}+1\right )}\right )}{4}\)	\(406\)

Input:

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

Output:

1/4*RootOf(_Z^2+RootOf(_Z^4+2)^2)*ln((-2*(-x^2+1)^(1/2)*RootOf(_Z^2+RootOf 
(_Z^4+2)^2)*RootOf(_Z^4+2)^2*x+2*(-x^2+1)^(1/2)*RootOf(_Z^2+RootOf(_Z^4+2) 
^2)*RootOf(_Z^4+2)^2+2*(-x^2+1)^(1/4)*RootOf(_Z^4+2)^2*x^2-4*(-x^2+1)^(1/4 
)*RootOf(_Z^4+2)^2*x+RootOf(_Z^2+RootOf(_Z^4+2)^2)*x^3+4*(-x^2+1)^(3/4)+2* 
(-x^2+1)^(1/4)*RootOf(_Z^4+2)^2-3*RootOf(_Z^2+RootOf(_Z^4+2)^2)*x^2+5*Root 
Of(_Z^2+RootOf(_Z^4+2)^2)*x+RootOf(_Z^2+RootOf(_Z^4+2)^2))/(-3+x)/(x^2+1)) 
-1/4*RootOf(_Z^4+2)*ln((-2*(-x^2+1)^(1/2)*RootOf(_Z^4+2)^3*x+2*(-x^2+1)^(1 
/2)*RootOf(_Z^4+2)^3-2*(-x^2+1)^(1/4)*RootOf(_Z^4+2)^2*x^2+4*(-x^2+1)^(1/4 
)*RootOf(_Z^4+2)^2*x-RootOf(_Z^4+2)*x^3+4*(-x^2+1)^(3/4)-2*(-x^2+1)^(1/4)* 
RootOf(_Z^4+2)^2+3*RootOf(_Z^4+2)*x^2-5*RootOf(_Z^4+2)*x-RootOf(_Z^4+2))/( 
-3+x)/(x^2+1))
 

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 500 vs. \(2 (138) = 276\).

Time = 4.65 (sec) , antiderivative size = 500, normalized size of antiderivative = 2.84 \[ \int \frac {2+x}{(-3+x) \sqrt [4]{1-x^2} \left (1+x^2\right )} \, dx=\frac {1}{8} \cdot 2^{\frac {3}{4}} \arctan \left (\frac {x^{6} - 6 \, x^{5} + 11 \, x^{4} - 12 \, x^{3} + 2 \cdot 2^{\frac {3}{4}} {\left (x^{5} - 5 \, x^{4} + 16 \, x^{3} - 16 \, x^{2} - x + 5\right )} {\left (-x^{2} + 1\right )}^{\frac {1}{4}} + 4 \, \sqrt {2} {\left (x^{4} - 4 \, x^{3} + 4 \, x^{2} - 4 \, x + 3\right )} \sqrt {-x^{2} + 1} + 4 \cdot 2^{\frac {1}{4}} {\left (3 \, x^{3} - 9 \, x^{2} + 11 \, x - 1\right )} {\left (-x^{2} + 1\right )}^{\frac {3}{4}} + 19 \, x^{2} - 6 \, x + 9}{x^{6} - 6 \, x^{5} + 43 \, x^{4} - 76 \, x^{3} + 19 \, x^{2} + 58 \, x - 23}\right ) + \frac {1}{8} \cdot 2^{\frac {3}{4}} \arctan \left (-\frac {x^{6} - 6 \, x^{5} + 11 \, x^{4} - 12 \, x^{3} - 2 \cdot 2^{\frac {3}{4}} {\left (x^{5} - 5 \, x^{4} + 16 \, x^{3} - 16 \, x^{2} - x + 5\right )} {\left (-x^{2} + 1\right )}^{\frac {1}{4}} + 4 \, \sqrt {2} {\left (x^{4} - 4 \, x^{3} + 4 \, x^{2} - 4 \, x + 3\right )} \sqrt {-x^{2} + 1} - 4 \cdot 2^{\frac {1}{4}} {\left (3 \, x^{3} - 9 \, x^{2} + 11 \, x - 1\right )} {\left (-x^{2} + 1\right )}^{\frac {3}{4}} + 19 \, x^{2} - 6 \, x + 9}{x^{6} - 6 \, x^{5} + 43 \, x^{4} - 76 \, x^{3} + 19 \, x^{2} + 58 \, x - 23}\right ) - \frac {1}{16} \cdot 2^{\frac {3}{4}} \log \left (\frac {\sqrt {2} {\left (x^{3} - 3 \, x^{2} + x - 3\right )} + 4 \cdot 2^{\frac {1}{4}} {\left (x^{2} - 2 \, x + 1\right )} {\left (-x^{2} + 1\right )}^{\frac {1}{4}} + 4 \cdot 2^{\frac {3}{4}} {\left (-x^{2} + 1\right )}^{\frac {3}{4}} + 8 \, \sqrt {-x^{2} + 1} {\left (x - 1\right )}}{x^{3} - 3 \, x^{2} + x - 3}\right ) + \frac {1}{16} \cdot 2^{\frac {3}{4}} \log \left (\frac {\sqrt {2} {\left (x^{3} - 3 \, x^{2} + x - 3\right )} - 4 \cdot 2^{\frac {1}{4}} {\left (x^{2} - 2 \, x + 1\right )} {\left (-x^{2} + 1\right )}^{\frac {1}{4}} - 4 \cdot 2^{\frac {3}{4}} {\left (-x^{2} + 1\right )}^{\frac {3}{4}} + 8 \, \sqrt {-x^{2} + 1} {\left (x - 1\right )}}{x^{3} - 3 \, x^{2} + x - 3}\right ) \] Input:

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

Output:

1/8*2^(3/4)*arctan((x^6 - 6*x^5 + 11*x^4 - 12*x^3 + 2*2^(3/4)*(x^5 - 5*x^4 
 + 16*x^3 - 16*x^2 - x + 5)*(-x^2 + 1)^(1/4) + 4*sqrt(2)*(x^4 - 4*x^3 + 4* 
x^2 - 4*x + 3)*sqrt(-x^2 + 1) + 4*2^(1/4)*(3*x^3 - 9*x^2 + 11*x - 1)*(-x^2 
 + 1)^(3/4) + 19*x^2 - 6*x + 9)/(x^6 - 6*x^5 + 43*x^4 - 76*x^3 + 19*x^2 + 
58*x - 23)) + 1/8*2^(3/4)*arctan(-(x^6 - 6*x^5 + 11*x^4 - 12*x^3 - 2*2^(3/ 
4)*(x^5 - 5*x^4 + 16*x^3 - 16*x^2 - x + 5)*(-x^2 + 1)^(1/4) + 4*sqrt(2)*(x 
^4 - 4*x^3 + 4*x^2 - 4*x + 3)*sqrt(-x^2 + 1) - 4*2^(1/4)*(3*x^3 - 9*x^2 + 
11*x - 1)*(-x^2 + 1)^(3/4) + 19*x^2 - 6*x + 9)/(x^6 - 6*x^5 + 43*x^4 - 76* 
x^3 + 19*x^2 + 58*x - 23)) - 1/16*2^(3/4)*log((sqrt(2)*(x^3 - 3*x^2 + x - 
3) + 4*2^(1/4)*(x^2 - 2*x + 1)*(-x^2 + 1)^(1/4) + 4*2^(3/4)*(-x^2 + 1)^(3/ 
4) + 8*sqrt(-x^2 + 1)*(x - 1))/(x^3 - 3*x^2 + x - 3)) + 1/16*2^(3/4)*log(( 
sqrt(2)*(x^3 - 3*x^2 + x - 3) - 4*2^(1/4)*(x^2 - 2*x + 1)*(-x^2 + 1)^(1/4) 
 - 4*2^(3/4)*(-x^2 + 1)^(3/4) + 8*sqrt(-x^2 + 1)*(x - 1))/(x^3 - 3*x^2 + x 
 - 3))
 

Sympy [F]

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

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

Output:

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

Maxima [F]

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

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

Output:

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

Giac [F]

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

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

Output:

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

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

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

Output:

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