\(\int \frac {\sqrt {-1+x^3} (2+x^3)}{x^2 (-2-4 x^2+2 x^3)} \, dx\) [480]

Optimal result

Integrand size = 32, antiderivative size = 38 \[ \int \frac {\sqrt {-1+x^3} \left (2+x^3\right )}{x^2 \left (-2-4 x^2+2 x^3\right )} \, dx=\frac {\sqrt {-1+x^3}}{x}-\sqrt {2} \text {arctanh}\left (\frac {\sqrt {2} x}{\sqrt {-1+x^3}}\right ) \] Output:

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

Mathematica [A] (verified)

Time = 0.70 (sec) , antiderivative size = 38, normalized size of antiderivative = 1.00 \[ \int \frac {\sqrt {-1+x^3} \left (2+x^3\right )}{x^2 \left (-2-4 x^2+2 x^3\right )} \, dx=\frac {\sqrt {-1+x^3}}{x}-\sqrt {2} \text {arctanh}\left (\frac {\sqrt {2} x}{\sqrt {-1+x^3}}\right ) \] Input:

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

Output:

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

Rubi [F]

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.

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

\(\Big \downarrow \) 7293

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

\(\Big \downarrow \) 2009

\(\displaystyle -2 \int \frac {\sqrt {x^3-1}}{x^3-2 x^2-1}dx+\frac {3}{2} \int \frac {x \sqrt {x^3-1}}{x^3-2 x^2-1}dx+\frac {\sqrt {2} 3^{3/4} (1-x) \sqrt {\frac {x^2+x+1}{\left (-x-\sqrt {3}+1\right )^2}} \operatorname {EllipticF}\left (\arcsin \left (\frac {-x+\sqrt {3}+1}{-x-\sqrt {3}+1}\right ),-7+4 \sqrt {3}\right )}{\sqrt {-\frac {1-x}{\left (-x-\sqrt {3}+1\right )^2}} \sqrt {x^3-1}}-\frac {3 \sqrt [4]{3} \sqrt {2+\sqrt {3}} (1-x) \sqrt {\frac {x^2+x+1}{\left (-x-\sqrt {3}+1\right )^2}} E\left (\arcsin \left (\frac {-x+\sqrt {3}+1}{-x-\sqrt {3}+1}\right )|-7+4 \sqrt {3}\right )}{2 \sqrt {-\frac {1-x}{\left (-x-\sqrt {3}+1\right )^2}} \sqrt {x^3-1}}+\frac {3 \sqrt {x^3-1}}{-x-\sqrt {3}+1}+\frac {\sqrt {x^3-1}}{x}\)

Input:

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

Output:

$Aborted
 

Maple [A] (verified)

Time = 0.58 (sec) , antiderivative size = 34, normalized size of antiderivative = 0.89

method	result	size
risch	\(\frac {\sqrt {x^{3}-1}}{x}-\sqrt {2}\, \operatorname {arctanh}\left (\frac {\sqrt {x^{3}-1}\, \sqrt {2}}{2 x}\right )\)	\(34\)
pseudoelliptic	\(\frac {-\sqrt {2}\, \operatorname {arctanh}\left (\frac {\sqrt {x^{3}-1}\, \sqrt {2}}{2 x}\right ) x +\sqrt {x^{3}-1}}{x}\)	\(35\)
default	\(\frac {-2 \sqrt {2}\, \operatorname {arctanh}\left (\frac {\sqrt {x^{3}-1}\, \sqrt {2}}{2 x}\right ) x +2 \sqrt {x^{3}-1}}{2 x}\)	\(38\)
trager	\(\frac {\sqrt {x^{3}-1}}{x}+\frac {\operatorname {RootOf}\left (\textit {\_Z}^{2}-2\right ) \ln \left (-\frac {-\operatorname {RootOf}\left (\textit {\_Z}^{2}-2\right ) x^{3}-2 \operatorname {RootOf}\left (\textit {\_Z}^{2}-2\right ) x^{2}+4 \sqrt {x^{3}-1}\, x +\operatorname {RootOf}\left (\textit {\_Z}^{2}-2\right )}{x^{3}-2 x^{2}-1}\right )}{2}\)	\(75\)
elliptic	\(\frac {\sqrt {x^{3}-1}}{x}+\frac {2 \left (-\frac {3}{2}-\frac {i \sqrt {3}}{2}\right ) \sqrt {\frac {-1+x}{-\frac {3}{2}-\frac {i \sqrt {3}}{2}}}\, \sqrt {\frac {x +\frac {1}{2}-\frac {i \sqrt {3}}{2}}{\frac {3}{2}-\frac {i \sqrt {3}}{2}}}\, \sqrt {\frac {x +\frac {1}{2}+\frac {i \sqrt {3}}{2}}{\frac {3}{2}+\frac {i \sqrt {3}}{2}}}\, \operatorname {EllipticF}\left (\sqrt {\frac {-1+x}{-\frac {3}{2}-\frac {i \sqrt {3}}{2}}}, \sqrt {\frac {\frac {3}{2}+\frac {i \sqrt {3}}{2}}{\frac {3}{2}-\frac {i \sqrt {3}}{2}}}\right )}{\sqrt {x^{3}-1}}+\frac {\sqrt {2}\, \left (\munderset {\underline {\hspace {1.25 ex}}\alpha =\operatorname {RootOf}\left (\textit {\_Z}^{3}-2 \textit {\_Z}^{2}-1\right )}{\sum }\frac {\underline {\hspace {1.25 ex}}\alpha \left (-\underline {\hspace {1.25 ex}}\alpha ^{2}+\underline {\hspace {1.25 ex}}\alpha +1\right ) \left (-3-i \sqrt {3}\right ) \sqrt {\frac {-1+x}{-3-i \sqrt {3}}}\, \sqrt {\frac {-i \sqrt {3}+2 x +1}{3-i \sqrt {3}}}\, \sqrt {\frac {i \sqrt {3}+2 x +1}{i \sqrt {3}+3}}\, \operatorname {EllipticPi}\left (\sqrt {\frac {-1+x}{-\frac {3}{2}-\frac {i \sqrt {3}}{2}}}, -\frac {3 \underline {\hspace {1.25 ex}}\alpha ^{2}}{4}+\frac {3 \underline {\hspace {1.25 ex}}\alpha }{4}+\frac {3}{4}-\frac {i \underline {\hspace {1.25 ex}}\alpha ^{2} \sqrt {3}}{4}+\frac {i \sqrt {3}\, \underline {\hspace {1.25 ex}}\alpha }{4}+\frac {i \sqrt {3}}{4}, \sqrt {\frac {\frac {3}{2}+\frac {i \sqrt {3}}{2}}{\frac {3}{2}-\frac {i \sqrt {3}}{2}}}\right )}{\sqrt {x^{3}-1}}\right )}{2}\)	\(306\)

Input:

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

Output:

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

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 99 vs. \(2 (30) = 60\).

Time = 0.14 (sec) , antiderivative size = 99, normalized size of antiderivative = 2.61 \[ \int \frac {\sqrt {-1+x^3} \left (2+x^3\right )}{x^2 \left (-2-4 x^2+2 x^3\right )} \, dx=\frac {\sqrt {2} x \log \left (-\frac {x^{6} + 12 \, x^{5} + 4 \, x^{4} - 2 \, x^{3} - 4 \, \sqrt {2} {\left (x^{4} + 2 \, x^{3} - x\right )} \sqrt {x^{3} - 1} - 12 \, x^{2} + 1}{x^{6} - 4 \, x^{5} + 4 \, x^{4} - 2 \, x^{3} + 4 \, x^{2} + 1}\right ) + 4 \, \sqrt {x^{3} - 1}}{4 \, x} \] Input:

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

Output:

1/4*(sqrt(2)*x*log(-(x^6 + 12*x^5 + 4*x^4 - 2*x^3 - 4*sqrt(2)*(x^4 + 2*x^3 
 - x)*sqrt(x^3 - 1) - 12*x^2 + 1)/(x^6 - 4*x^5 + 4*x^4 - 2*x^3 + 4*x^2 + 1 
)) + 4*sqrt(x^3 - 1))/x
 

Sympy [F]

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

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

Output:

(Integral(2*sqrt(x**3 - 1)/(x**5 - 2*x**4 - x**2), x) + Integral(x**3*sqrt 
(x**3 - 1)/(x**5 - 2*x**4 - x**2), x))/2
 

Maxima [F]

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

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

Output:

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

Giac [F]

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

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

Output:

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

Mupad [B] (verification not implemented)

Time = 7.76 (sec) , antiderivative size = 56, normalized size of antiderivative = 1.47 \[ \int \frac {\sqrt {-1+x^3} \left (2+x^3\right )}{x^2 \left (-2-4 x^2+2 x^3\right )} \, dx=\frac {\sqrt {x^3-1}}{x}+\frac {\sqrt {2}\,\ln \left (\frac {2\,x^2+x^3-2\,\sqrt {2}\,x\,\sqrt {x^3-1}-1}{-8\,x^3+16\,x^2+8}\right )}{2} \] Input:

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

Output:

(x^3 - 1)^(1/2)/x + (2^(1/2)*log((2*x^2 + x^3 - 2*2^(1/2)*x*(x^3 - 1)^(1/2 
) - 1)/(16*x^2 - 8*x^3 + 8)))/2
 

Reduce [F]

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

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

Output:

(sqrt(x**3 - 1) + 2*int(sqrt(x**3 - 1)/(x**6 - 2*x**5 - 2*x**3 + 2*x**2 + 
1),x)*x + int((sqrt(x**3 - 1)*x**3)/(x**6 - 2*x**5 - 2*x**3 + 2*x**2 + 1), 
x)*x)/x