\(\int \frac {(-2+x^3) \sqrt {1-x^2+x^3}}{(1+x^3)^2} \, dx\) [568]

Optimal result

Integrand size = 27, antiderivative size = 44 \[ \int \frac {\left (-2+x^3\right ) \sqrt {1-x^2+x^3}}{\left (1+x^3\right )^2} \, dx=-\frac {x \sqrt {1-x^2+x^3}}{1+x^3}-\arctan \left (\frac {x}{\sqrt {1-x^2+x^3}}\right ) \] Output:

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

Mathematica [A] (verified)

Time = 0.25 (sec) , antiderivative size = 44, normalized size of antiderivative = 1.00 \[ \int \frac {\left (-2+x^3\right ) \sqrt {1-x^2+x^3}}{\left (1+x^3\right )^2} \, dx=-\frac {x \sqrt {1-x^2+x^3}}{1+x^3}-\arctan \left (\frac {x}{\sqrt {1-x^2+x^3}}\right ) \] Input:

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

Output:

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

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

\(\Big \downarrow \) 7293

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

\(\Big \downarrow \) 7239

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

\(\Big \downarrow \) 7293

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

\(\Big \downarrow \) 7239

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

\(\Big \downarrow \) 7293

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

\(\Big \downarrow \) 7239

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

\(\Big \downarrow \) 7293

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

\(\Big \downarrow \) 7239

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

\(\Big \downarrow \) 7293

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

\(\Big \downarrow \) 7239

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

\(\Big \downarrow \) 7293

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

\(\Big \downarrow \) 7239

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

\(\Big \downarrow \) 7293

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

\(\Big \downarrow \) 7239

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

\(\Big \downarrow \) 7293

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

\(\Big \downarrow \) 7239

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

\(\Big \downarrow \) 7293

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

\(\Big \downarrow \) 7239

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

\(\Big \downarrow \) 7293

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

\(\Big \downarrow \) 7239

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

\(\Big \downarrow \) 7293

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

\(\Big \downarrow \) 7239

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

\(\Big \downarrow \) 7293

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

\(\Big \downarrow \) 7239

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

\(\Big \downarrow \) 7293

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

\(\Big \downarrow \) 7239

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

\(\Big \downarrow \) 7293

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

\(\Big \downarrow \) 7239

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

\(\Big \downarrow \) 7293

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

\(\Big \downarrow \) 7239

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

Input:

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

Output:

$Aborted
 

Maple [A] (verified)

Time = 1.92 (sec) , antiderivative size = 41, normalized size of antiderivative = 0.93

method	result	size
risch	\(-\frac {x \sqrt {x^{3}-x^{2}+1}}{x^{3}+1}+\arctan \left (\frac {\sqrt {x^{3}-x^{2}+1}}{x}\right )\)	\(41\)
default	\(\frac {\arctan \left (\frac {\sqrt {x^{3}-x^{2}+1}}{x}\right ) x^{3}-x \sqrt {x^{3}-x^{2}+1}+\arctan \left (\frac {\sqrt {x^{3}-x^{2}+1}}{x}\right )}{x^{3}+1}\)	\(63\)
pseudoelliptic	\(\frac {\arctan \left (\frac {\sqrt {x^{3}-x^{2}+1}}{x}\right ) x^{3}-x \sqrt {x^{3}-x^{2}+1}+\arctan \left (\frac {\sqrt {x^{3}-x^{2}+1}}{x}\right )}{x^{3}+1}\)	\(63\)
trager	\(-\frac {x \sqrt {x^{3}-x^{2}+1}}{x^{3}+1}+\frac {\operatorname {RootOf}\left (\textit {\_Z}^{2}+1\right ) \ln \left (-\frac {-\operatorname {RootOf}\left (\textit {\_Z}^{2}+1\right ) x^{3}+2 \operatorname {RootOf}\left (\textit {\_Z}^{2}+1\right ) x^{2}+2 x \sqrt {x^{3}-x^{2}+1}-\operatorname {RootOf}\left (\textit {\_Z}^{2}+1\right )}{\left (1+x \right ) \left (x^{2}-x +1\right )}\right )}{2}\)	\(96\)
elliptic	\(\text {Expression too large to display}\)	\(50736\)

Input:

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

Output:

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

Fricas [A] (verification not implemented)

Time = 0.11 (sec) , antiderivative size = 68, normalized size of antiderivative = 1.55 \[ \int \frac {\left (-2+x^3\right ) \sqrt {1-x^2+x^3}}{\left (1+x^3\right )^2} \, dx=\frac {{\left (x^{3} + 1\right )} \arctan \left (\frac {\sqrt {x^{3} - x^{2} + 1} {\left (x^{3} - 2 \, x^{2} + 1\right )}}{2 \, {\left (x^{4} - x^{3} + x\right )}}\right ) - 2 \, \sqrt {x^{3} - x^{2} + 1} x}{2 \, {\left (x^{3} + 1\right )}} \] Input:

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

Output:

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

Sympy [F]

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

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

Output:

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

Maxima [F]

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

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

Output:

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

Giac [F]

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

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

Output:

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

Mupad [B] (verification not implemented)

Time = 8.90 (sec) , antiderivative size = 61, normalized size of antiderivative = 1.39 \[ \int \frac {\left (-2+x^3\right ) \sqrt {1-x^2+x^3}}{\left (1+x^3\right )^2} \, dx=-\frac {x\,\sqrt {x^3-x^2+1}}{x^3+1}+\frac {\ln \left (\frac {x^3-2\,x^2+1+x\,\sqrt {x^3-x^2+1}\,2{}\mathrm {i}}{x^3+1}\right )\,1{}\mathrm {i}}{2} \] Input:

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

Output:

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

Reduce [F]

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

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

Output:

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