\(\int \frac {-1+x^3}{(1+x^3) \sqrt {1+x^4}} \, dx\) [831]

Optimal result

Integrand size = 22, antiderivative size = 63 \[ \int \frac {-1+x^3}{\left (1+x^3\right ) \sqrt {1+x^4}} \, dx=-\frac {4}{3} \arctan \left (\frac {x}{1-x+x^2+\sqrt {1+x^4}}\right )-\frac {1}{3} \sqrt {2} \text {arctanh}\left (\frac {\sqrt {2} x}{1+2 x+x^2+\sqrt {1+x^4}}\right ) \] Output:

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

Mathematica [A] (verified)

Time = 1.01 (sec) , antiderivative size = 63, normalized size of antiderivative = 1.00 \[ \int \frac {-1+x^3}{\left (1+x^3\right ) \sqrt {1+x^4}} \, dx=-\frac {4}{3} \arctan \left (\frac {x}{1-x+x^2+\sqrt {1+x^4}}\right )-\frac {1}{3} \sqrt {2} \text {arctanh}\left (\frac {\sqrt {2} x}{1+2 x+x^2+\sqrt {1+x^4}}\right ) \] Input:

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

Output:

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

Rubi [C] (warning: unable to verify)

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

Time = 0.94 (sec) , antiderivative size = 333, normalized size of antiderivative = 5.29, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.091, 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.

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

\(\Big \downarrow \) 7276

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

\(\Big \downarrow \) 2009

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

Input:

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

Output:

(-2*ArcTan[x/Sqrt[1 + x^4]])/3 - ArcTanh[(Sqrt[2]*x)/Sqrt[1 + x^4]]/(3*Sqr 
t[2]) + ArcTanh[(1 + x^2)/(Sqrt[2]*Sqrt[1 + x^4])]/(3*Sqrt[2]) - ((-1)^(1/ 
3)*ArcTanh[((-1)^(2/3) + x^2)/(Sqrt[1 - (-1)^(1/3)]*Sqrt[1 + x^4])])/(3*Sq 
rt[1 - (-1)^(1/3)]) + ((-1)^(2/3)*ArcTanh[(2*(1 + (-1)^(2/3)*x^2))/((I - S 
qrt[3])*Sqrt[1 + x^4])])/(3*Sqrt[1 + (-1)^(2/3)]) + ((1 + x^2)*Sqrt[(1 + x 
^4)/(1 + x^2)^2]*EllipticF[2*ArcTan[x], 1/2])/(3*Sqrt[1 + x^4]) - ((1 + x^ 
2)*Sqrt[(1 + x^4)/(1 + x^2)^2]*EllipticF[2*ArcTan[x], 1/2])/(3*(1 - (-1)^( 
1/3))*Sqrt[1 + x^4]) - ((1 - (-1)^(1/3))*(1 + x^2)*Sqrt[(1 + x^4)/(1 + x^2 
)^2]*EllipticF[2*ArcTan[x], 1/2])/(3*Sqrt[1 + x^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 [A] (verified)

Time = 0.99 (sec) , antiderivative size = 41, normalized size of antiderivative = 0.65

method	result	size
default	\(\frac {2 \arctan \left (\frac {\left (-1+x \right )^{2}}{\sqrt {x^{4}+1}}\right )}{3}+\frac {\sqrt {2}\, \operatorname {arctanh}\left (\frac {\left (x^{2}+x +1\right ) \sqrt {2}}{\sqrt {x^{4}+1}}\right )}{6}\)	\(41\)
pseudoelliptic	\(\frac {2 \arctan \left (\frac {\left (-1+x \right )^{2}}{\sqrt {x^{4}+1}}\right )}{3}+\frac {\sqrt {2}\, \operatorname {arctanh}\left (\frac {\left (x^{2}+x +1\right ) \sqrt {2}}{\sqrt {x^{4}+1}}\right )}{6}\)	\(41\)
elliptic	\(\frac {\sqrt {2}\, \operatorname {arctanh}\left (\frac {\left (2 x^{2}+2\right ) \sqrt {2}}{4 \sqrt {\left (x^{2}-1\right )^{2}+2 x^{2}}}\right )}{6}-\frac {\sqrt {2}\, \sqrt {\frac {2 \left (x^{2}+1\right )^{2}}{\left (-x^{2}+1\right )^{2}}+2}\, \arctan \left (\frac {\sqrt {\frac {2 \left (x^{2}+1\right )^{2}}{\left (-x^{2}+1\right )^{2}}+2}\, \left (x^{2}+1\right )}{\left (\frac {\left (x^{2}+1\right )^{2}}{\left (-x^{2}+1\right )^{2}}+1\right ) \left (-x^{2}+1\right )}\right )}{3 \sqrt {\frac {\frac {\left (x^{2}+1\right )^{2}}{\left (-x^{2}+1\right )^{2}}+1}{\left (1+\frac {x^{2}+1}{-x^{2}+1}\right )^{2}}}\, \left (1+\frac {x^{2}+1}{-x^{2}+1}\right )}+\frac {\left (-\frac {\ln \left (1+\frac {\sqrt {2}\, \sqrt {x^{4}+1}}{2 x}\right )}{6}+\frac {2 \sqrt {2}\, \arctan \left (\frac {\sqrt {x^{4}+1}}{x}\right )}{3}+\frac {\ln \left (-1+\frac {\sqrt {2}\, \sqrt {x^{4}+1}}{2 x}\right )}{6}\right ) \sqrt {2}}{2}\)	\(244\)

Input:

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

Output:

2/3*arctan((-1+x)^2/(x^4+1)^(1/2))+1/6*2^(1/2)*arctanh((x^2+x+1)*2^(1/2)/( 
x^4+1)^(1/2))
 

Fricas [A] (verification not implemented)

Time = 0.13 (sec) , antiderivative size = 88, normalized size of antiderivative = 1.40 \[ \int \frac {-1+x^3}{\left (1+x^3\right ) \sqrt {1+x^4}} \, dx=\frac {1}{12} \, \sqrt {2} \log \left (-\frac {3 \, x^{4} + 4 \, x^{3} + 2 \, \sqrt {2} \sqrt {x^{4} + 1} {\left (x^{2} + x + 1\right )} + 6 \, x^{2} + 4 \, x + 3}{x^{4} + 4 \, x^{3} + 6 \, x^{2} + 4 \, x + 1}\right ) - \frac {2}{3} \, \arctan \left (\frac {\sqrt {x^{4} + 1}}{x^{2} - 2 \, x + 1}\right ) \] Input:

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

Output:

1/12*sqrt(2)*log(-(3*x^4 + 4*x^3 + 2*sqrt(2)*sqrt(x^4 + 1)*(x^2 + x + 1) + 
 6*x^2 + 4*x + 3)/(x^4 + 4*x^3 + 6*x^2 + 4*x + 1)) - 2/3*arctan(sqrt(x^4 + 
 1)/(x^2 - 2*x + 1))
 

Sympy [F]

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

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

Output:

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

Maxima [F]

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

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

Output:

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

Giac [F]

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

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

Output:

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

\[ \int \frac {-1+x^3}{\left (1+x^3\right ) \sqrt {1+x^4}} \, dx=-\frac {\sqrt {2}\, \mathrm {log}\left (x^{2}+2 x +1\right )}{6}+\frac {\sqrt {2}\, \mathrm {log}\left (\sqrt {x^{4}+1}\, \sqrt {2}+2 x^{2}+2 x +2\right )}{6}-\frac {2 \left (\int \frac {\sqrt {x^{4}+1}}{x^{6}-x^{5}+x^{4}+x^{2}-x +1}d x \right )}{3}+\frac {2 \left (\int \frac {\sqrt {x^{4}+1}\, x^{2}}{x^{6}-x^{5}+x^{4}+x^{2}-x +1}d x \right )}{3} \] Input:

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

Output:

( - sqrt(2)*log(x**2 + 2*x + 1) + sqrt(2)*log(sqrt(x**4 + 1)*sqrt(2) + 2*x 
**2 + 2*x + 2) - 4*int(sqrt(x**4 + 1)/(x**6 - x**5 + x**4 + x**2 - x + 1), 
x) + 4*int((sqrt(x**4 + 1)*x**2)/(x**6 - x**5 + x**4 + x**2 - x + 1),x))/6