\(\int \frac {-1+x^2}{(1+x^2) \sqrt [3]{x+x^5}} \, dx\) [1792]

Optimal result

Integrand size = 22, antiderivative size = 121 \[ \int \frac {-1+x^2}{\left (1+x^2\right ) \sqrt [3]{x+x^5}} \, dx=-\frac {\sqrt {3} \arctan \left (\frac {\sqrt {3} x}{-x+2^{2/3} \sqrt [3]{x+x^5}}\right )}{2 \sqrt [3]{2}}-\frac {\log \left (2 x+2^{2/3} \sqrt [3]{x+x^5}\right )}{2 \sqrt [3]{2}}+\frac {\log \left (-2 x^2+2^{2/3} x \sqrt [3]{x+x^5}-\sqrt [3]{2} \left (x+x^5\right )^{2/3}\right )}{4 \sqrt [3]{2}} \] Output:

-1/4*3^(1/2)*arctan(3^(1/2)*x/(-x+2^(2/3)*(x^5+x)^(1/3)))*2^(2/3)-1/4*ln(2 
*x+2^(2/3)*(x^5+x)^(1/3))*2^(2/3)+1/8*ln(-2*x^2+2^(2/3)*x*(x^5+x)^(1/3)-2^ 
(1/3)*(x^5+x)^(2/3))*2^(2/3)
                                                                                    
                                                                                    
 

Mathematica [A] (verified)

Time = 15.32 (sec) , antiderivative size = 106, normalized size of antiderivative = 0.88 \[ \int \frac {-1+x^2}{\left (1+x^2\right ) \sqrt [3]{x+x^5}} \, dx=\frac {2 \sqrt {3} \arctan \left (\frac {\sqrt {3} x}{x-2^{2/3} \sqrt [3]{x+x^5}}\right )-2 \log \left (2 x+2^{2/3} \sqrt [3]{x+x^5}\right )+\log \left (-2 x^2+2^{2/3} x \sqrt [3]{x+x^5}-\sqrt [3]{2} \left (x+x^5\right )^{2/3}\right )}{4 \sqrt [3]{2}} \] Input:

Integrate[(-1 + x^2)/((1 + x^2)*(x + x^5)^(1/3)),x]
 

Output:

(2*Sqrt[3]*ArcTan[(Sqrt[3]*x)/(x - 2^(2/3)*(x + x^5)^(1/3))] - 2*Log[2*x + 
 2^(2/3)*(x + x^5)^(1/3)] + Log[-2*x^2 + 2^(2/3)*x*(x + x^5)^(1/3) - 2^(1/ 
3)*(x + x^5)^(2/3)])/(4*2^(1/3))
 

Rubi [C] (warning: unable to verify)

Result contains higher order function than in optimal. Order 6 vs. order 3 in optimal.

Time = 1.05 (sec) , antiderivative size = 320, normalized size of antiderivative = 2.64, number of steps used = 7, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.273, Rules used = {2467, 25, 2035, 7266, 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.

\(\displaystyle \int \frac {x^2-1}{\left (x^2+1\right ) \sqrt [3]{x^5+x}} \, dx\)

\(\Big \downarrow \) 2467

\(\displaystyle \frac {\sqrt [3]{x} \sqrt [3]{x^4+1} \int -\frac {1-x^2}{\sqrt [3]{x} \left (x^2+1\right ) \sqrt [3]{x^4+1}}dx}{\sqrt [3]{x^5+x}}\)

\(\Big \downarrow \) 25

\(\displaystyle -\frac {\sqrt [3]{x} \sqrt [3]{x^4+1} \int \frac {1-x^2}{\sqrt [3]{x} \left (x^2+1\right ) \sqrt [3]{x^4+1}}dx}{\sqrt [3]{x^5+x}}\)

\(\Big \downarrow \) 2035

\(\displaystyle -\frac {3 \sqrt [3]{x} \sqrt [3]{x^4+1} \int \frac {\sqrt [3]{x} \left (1-x^2\right )}{\left (x^2+1\right ) \sqrt [3]{x^4+1}}d\sqrt [3]{x}}{\sqrt [3]{x^5+x}}\)

\(\Big \downarrow \) 7266

\(\displaystyle -\frac {3 \sqrt [3]{x} \sqrt [3]{x^4+1} \int \frac {1-x}{(x+1) \sqrt [3]{x^2+1}}dx^{2/3}}{2 \sqrt [3]{x^5+x}}\)

\(\Big \downarrow \) 7276

\(\displaystyle -\frac {3 \sqrt [3]{x} \sqrt [3]{x^4+1} \int \left (\frac {2}{(x+1) \sqrt [3]{x^2+1}}-\frac {1}{\sqrt [3]{x^2+1}}\right )dx^{2/3}}{2 \sqrt [3]{x^5+x}}\)

\(\Big \downarrow \) 2009

\(\displaystyle -\frac {3 \sqrt [3]{x} \sqrt [3]{x^4+1} \left (2 x^{2/3} \operatorname {AppellF1}\left (\frac {1}{6},1,\frac {1}{3},\frac {7}{6},x^2,-x^2\right )-\frac {\arctan \left (\frac {1-\frac {2 \sqrt [3]{2} \left (x^{2/3}+1\right )}{\sqrt [3]{x^2+1}}}{\sqrt {3}}\right )}{\sqrt [3]{2} \sqrt {3}}-\frac {\arctan \left (\frac {\frac {\sqrt [3]{2} \left (x^{2/3}+1\right )}{\sqrt [3]{x^2+1}}+1}{\sqrt {3}}\right )}{2 \sqrt [3]{2} \sqrt {3}}-x^{2/3} \operatorname {Hypergeometric2F1}\left (\frac {1}{6},\frac {1}{3},\frac {7}{6},-x^2\right )-\frac {\log \left (\left (1-x^{2/3}\right )^2 \left (x^{2/3}+1\right )\right )}{12 \sqrt [3]{2}}-\frac {\log \left (\frac {2^{2/3} \left (x^{2/3}+1\right )^2}{\left (x^2+1\right )^{2/3}}-\frac {\sqrt [3]{2} \left (x^{2/3}+1\right )}{\sqrt [3]{x^2+1}}+1\right )}{6 \sqrt [3]{2}}+\frac {\log \left (\frac {\sqrt [3]{2} \left (x^{2/3}+1\right )}{\sqrt [3]{x^2+1}}+1\right )}{3 \sqrt [3]{2}}+\frac {\log \left (x^{2/3}-2^{2/3} \sqrt [3]{x^2+1}+1\right )}{4 \sqrt [3]{2}}\right )}{2 \sqrt [3]{x^5+x}}\)

Input:

Int[(-1 + x^2)/((1 + x^2)*(x + x^5)^(1/3)),x]
 

Output:

(-3*x^(1/3)*(1 + x^4)^(1/3)*(2*x^(2/3)*AppellF1[1/6, 1, 1/3, 7/6, x^2, -x^ 
2] - ArcTan[(1 - (2*2^(1/3)*(1 + x^(2/3)))/(1 + x^2)^(1/3))/Sqrt[3]]/(2^(1 
/3)*Sqrt[3]) - ArcTan[(1 + (2^(1/3)*(1 + x^(2/3)))/(1 + x^2)^(1/3))/Sqrt[3 
]]/(2*2^(1/3)*Sqrt[3]) - x^(2/3)*Hypergeometric2F1[1/6, 1/3, 7/6, -x^2] - 
Log[(1 - x^(2/3))^2*(1 + x^(2/3))]/(12*2^(1/3)) - Log[1 + (2^(2/3)*(1 + x^ 
(2/3))^2)/(1 + x^2)^(2/3) - (2^(1/3)*(1 + x^(2/3)))/(1 + x^2)^(1/3)]/(6*2^ 
(1/3)) + Log[1 + (2^(1/3)*(1 + x^(2/3)))/(1 + x^2)^(1/3)]/(3*2^(1/3)) + Lo 
g[1 + x^(2/3) - 2^(2/3)*(1 + x^2)^(1/3)]/(4*2^(1/3))))/(2*(x + x^5)^(1/3))
 

Defintions of rubi rules used

rule 25

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

rule 2009

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

rule 2035

Int[(Fx_)*(x_)^(m_), x_Symbol] :> With[{k = Denominator[m]}, Simp[k   Subst 
[Int[x^(k*(m + 1) - 1)*SubstPower[Fx, x, k], x], x, x^(1/k)], x]] /; Fracti 
onQ[m] && AlgebraicFunctionQ[Fx, x]
 

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]
 

rule 7266

Int[(u_)*(x_)^(m_.), x_Symbol] :> Simp[1/(m + 1)   Subst[Int[SubstFor[x^(m 
+ 1), u, x], x], x, x^(m + 1)], x] /; FreeQ[m, x] && NeQ[m, -1] && Function 
OfQ[x^(m + 1), u, x]
 

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 [A] (verified)

Time = 13.90 (sec) , antiderivative size = 95, normalized size of antiderivative = 0.79

method	result	size
pseudoelliptic	\(-\frac {2^{\frac {2}{3}} \left (\sqrt {3}\, \arctan \left (\frac {\sqrt {3}\, \left (-2^{\frac {2}{3}} {\left (x \left (x^{4}+1\right )\right )}^{\frac {1}{3}}+x \right )}{3 x}\right )+\ln \left (\frac {2^{\frac {1}{3}} x +{\left (x \left (x^{4}+1\right )\right )}^{\frac {1}{3}}}{x}\right )-\frac {\ln \left (\frac {2^{\frac {2}{3}} x^{2}-2^{\frac {1}{3}} {\left (x \left (x^{4}+1\right )\right )}^{\frac {1}{3}} x +{\left (x \left (x^{4}+1\right )\right )}^{\frac {2}{3}}}{x^{2}}\right )}{2}\right )}{4}\)	\(95\)
trager	\(\text {Expression too large to display}\)	\(1205\)

Input:

int((x^2-1)/(x^2+1)/(x^5+x)^(1/3),x,method=_RETURNVERBOSE)
 

Output:

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

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 289 vs. \(2 (89) = 178\).

Time = 2.24 (sec) , antiderivative size = 289, normalized size of antiderivative = 2.39 \[ \int \frac {-1+x^2}{\left (1+x^2\right ) \sqrt [3]{x+x^5}} \, dx=-\frac {1}{6} \cdot 2^{\frac {1}{6}} \sqrt {\frac {3}{2}} \arctan \left (\frac {2^{\frac {1}{6}} \sqrt {\frac {3}{2}} {\left (6 \cdot 2^{\frac {2}{3}} {\left (x^{8} - 14 \, x^{6} + 6 \, x^{4} - 14 \, x^{2} + 1\right )} {\left (x^{5} + x\right )}^{\frac {2}{3}} + 2^{\frac {1}{3}} {\left (x^{12} + 24 \, x^{10} - 57 \, x^{8} + 56 \, x^{6} - 57 \, x^{4} + 24 \, x^{2} + 1\right )} + 24 \, {\left (x^{9} + x^{7} + x^{3} + x\right )} {\left (x^{5} + x\right )}^{\frac {1}{3}}\right )}}{3 \, {\left (x^{12} - 48 \, x^{10} + 15 \, x^{8} - 88 \, x^{6} + 15 \, x^{4} - 48 \, x^{2} + 1\right )}}\right ) + \frac {1}{24} \cdot 2^{\frac {2}{3}} \log \left (\frac {2^{\frac {2}{3}} {\left (x^{8} - 14 \, x^{6} + 6 \, x^{4} - 14 \, x^{2} + 1\right )} + 12 \cdot 2^{\frac {1}{3}} {\left (x^{5} - x^{3} + x\right )} {\left (x^{5} + x\right )}^{\frac {1}{3}} - 6 \, {\left (x^{5} + x\right )}^{\frac {2}{3}} {\left (x^{4} - 4 \, x^{2} + 1\right )}}{x^{8} + 4 \, x^{6} + 6 \, x^{4} + 4 \, x^{2} + 1}\right ) - \frac {1}{12} \cdot 2^{\frac {2}{3}} \log \left (-\frac {3 \cdot 2^{\frac {2}{3}} {\left (x^{5} + x\right )}^{\frac {2}{3}} + 2^{\frac {1}{3}} {\left (x^{4} + 2 \, x^{2} + 1\right )} + 6 \, {\left (x^{5} + x\right )}^{\frac {1}{3}} x}{x^{4} + 2 \, x^{2} + 1}\right ) \] Input:

integrate((x^2-1)/(x^2+1)/(x^5+x)^(1/3),x, algorithm="fricas")
 

Output:

-1/6*2^(1/6)*sqrt(3/2)*arctan(1/3*2^(1/6)*sqrt(3/2)*(6*2^(2/3)*(x^8 - 14*x 
^6 + 6*x^4 - 14*x^2 + 1)*(x^5 + x)^(2/3) + 2^(1/3)*(x^12 + 24*x^10 - 57*x^ 
8 + 56*x^6 - 57*x^4 + 24*x^2 + 1) + 24*(x^9 + x^7 + x^3 + x)*(x^5 + x)^(1/ 
3))/(x^12 - 48*x^10 + 15*x^8 - 88*x^6 + 15*x^4 - 48*x^2 + 1)) + 1/24*2^(2/ 
3)*log((2^(2/3)*(x^8 - 14*x^6 + 6*x^4 - 14*x^2 + 1) + 12*2^(1/3)*(x^5 - x^ 
3 + x)*(x^5 + x)^(1/3) - 6*(x^5 + x)^(2/3)*(x^4 - 4*x^2 + 1))/(x^8 + 4*x^6 
 + 6*x^4 + 4*x^2 + 1)) - 1/12*2^(2/3)*log(-(3*2^(2/3)*(x^5 + x)^(2/3) + 2^ 
(1/3)*(x^4 + 2*x^2 + 1) + 6*(x^5 + x)^(1/3)*x)/(x^4 + 2*x^2 + 1))
 

Sympy [F]

\[ \int \frac {-1+x^2}{\left (1+x^2\right ) \sqrt [3]{x+x^5}} \, dx=\int \frac {\left (x - 1\right ) \left (x + 1\right )}{\sqrt [3]{x \left (x^{4} + 1\right )} \left (x^{2} + 1\right )}\, dx \] Input:

integrate((x**2-1)/(x**2+1)/(x**5+x)**(1/3),x)
 

Output:

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

Maxima [F]

\[ \int \frac {-1+x^2}{\left (1+x^2\right ) \sqrt [3]{x+x^5}} \, dx=\int { \frac {x^{2} - 1}{{\left (x^{5} + x\right )}^{\frac {1}{3}} {\left (x^{2} + 1\right )}} \,d x } \] Input:

integrate((x^2-1)/(x^2+1)/(x^5+x)^(1/3),x, algorithm="maxima")
 

Output:

integrate((x^2 - 1)/((x^5 + x)^(1/3)*(x^2 + 1)), x)
 

Giac [F]

\[ \int \frac {-1+x^2}{\left (1+x^2\right ) \sqrt [3]{x+x^5}} \, dx=\int { \frac {x^{2} - 1}{{\left (x^{5} + x\right )}^{\frac {1}{3}} {\left (x^{2} + 1\right )}} \,d x } \] Input:

integrate((x^2-1)/(x^2+1)/(x^5+x)^(1/3),x, algorithm="giac")
 

Output:

integrate((x^2 - 1)/((x^5 + x)^(1/3)*(x^2 + 1)), x)
 

Mupad [F(-1)]

Timed out. \[ \int \frac {-1+x^2}{\left (1+x^2\right ) \sqrt [3]{x+x^5}} \, dx=\int \frac {x^2-1}{\left (x^2+1\right )\,{\left (x^5+x\right )}^{1/3}} \,d x \] Input:

int((x^2 - 1)/((x^2 + 1)*(x + x^5)^(1/3)),x)
 

Output:

int((x^2 - 1)/((x^2 + 1)*(x + x^5)^(1/3)), x)
                                                                                    
                                                                                    
 

Reduce [F]

\[ \int \frac {-1+x^2}{\left (1+x^2\right ) \sqrt [3]{x+x^5}} \, dx=\int \frac {x^{2}}{x^{\frac {7}{3}} \left (x^{4}+1\right )^{\frac {1}{3}}+x^{\frac {1}{3}} \left (x^{4}+1\right )^{\frac {1}{3}}}d x -\left (\int \frac {1}{x^{\frac {7}{3}} \left (x^{4}+1\right )^{\frac {1}{3}}+x^{\frac {1}{3}} \left (x^{4}+1\right )^{\frac {1}{3}}}d x \right ) \] Input:

int((x^2-1)/(x^2+1)/(x^5+x)^(1/3),x)
 

Output:

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