\(\int \frac {\sqrt {-1+x^6} (2+x^6)}{x^3 (-1-x^4+x^6)} \, dx\) [370]

Optimal result

Integrand size = 30, antiderivative size = 30 \[ \int \frac {\sqrt {-1+x^6} \left (2+x^6\right )}{x^3 \left (-1-x^4+x^6\right )} \, dx=\frac {\sqrt {-1+x^6}}{x^2}-\text {arctanh}\left (\frac {x^2}{\sqrt {-1+x^6}}\right ) \]

[Out]

(x^6-1)^(1/2)/x^2-arctanh(x^2/(x^6-1)^(1/2))

Rubi [F]

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

[In]

Int[(Sqrt[-1 + x^6]*(2 + x^6))/(x^3*(-1 - x^4 + x^6)),x]

[Out]

Sqrt[-1 + x^6]/x^2 + (3*Sqrt[-1 + x^6])/(1 - Sqrt[3] - x^2) - (3*3^(1/4)*Sqrt[2 + Sqrt[3]]*(1 - x^2)*Sqrt[(1 +
x^2 + x^4)/(1 - Sqrt[3] - x^2)^2]*EllipticE[ArcSin[(1 + Sqrt[3] - x^2)/(1 - Sqrt[3] - x^2)], -7 + 4*Sqrt[3]])
/(2*Sqrt[-((1 - x^2)/(1 - Sqrt[3] - x^2)^2)]*Sqrt[-1 + x^6]) + (Sqrt[2]*3^(3/4)*(1 - x^2)*Sqrt[(1 + x^2 + x^4)
/(1 - Sqrt[3] - x^2)^2]*EllipticF[ArcSin[(1 + Sqrt[3] - x^2)/(1 - Sqrt[3] - x^2)], -7 + 4*Sqrt[3]])/(Sqrt[-((1
- x^2)/(1 - Sqrt[3] - x^2)^2)]*Sqrt[-1 + x^6]) - Defer[Subst][Defer[Int][Sqrt[-1 + x^3]/(-1 - x^2 + x^3), x],
x, x^2] + (3*Defer[Subst][Defer[Int][(x*Sqrt[-1 + x^3])/(-1 - x^2 + x^3), x], x, x^2])/2

Rubi steps \begin{align*} \text {integral}& = \int \left (-\frac {2 \sqrt {-1+x^6}}{x^3}+\frac {x \left (2-3 x^2\right ) \sqrt {-1+x^6}}{1+x^4-x^6}\right ) \, dx \\ & = -\left (2 \int \frac {\sqrt {-1+x^6}}{x^3} \, dx\right )+\int \frac {x \left (2-3 x^2\right ) \sqrt {-1+x^6}}{1+x^4-x^6} \, dx \\ & = \frac {1}{2} \text {Subst}\left (\int \frac {(2-3 x) \sqrt {-1+x^3}}{1+x^2-x^3} \, dx,x,x^2\right )-\text {Subst}\left (\int \frac {\sqrt {-1+x^3}}{x^2} \, dx,x,x^2\right ) \\ & = \frac {\sqrt {-1+x^6}}{x^2}+\frac {1}{2} \text {Subst}\left (\int \left (-\frac {2 \sqrt {-1+x^3}}{-1-x^2+x^3}+\frac {3 x \sqrt {-1+x^3}}{-1-x^2+x^3}\right ) \, dx,x,x^2\right )-\frac {3}{2} \text {Subst}\left (\int \frac {x}{\sqrt {-1+x^3}} \, dx,x,x^2\right ) \\ & = \frac {\sqrt {-1+x^6}}{x^2}+\frac {3}{2} \text {Subst}\left (\int \frac {1+\sqrt {3}-x}{\sqrt {-1+x^3}} \, dx,x,x^2\right )+\frac {3}{2} \text {Subst}\left (\int \frac {x \sqrt {-1+x^3}}{-1-x^2+x^3} \, dx,x,x^2\right )-\frac {1}{2} \left (3 \left (1+\sqrt {3}\right )\right ) \text {Subst}\left (\int \frac {1}{\sqrt {-1+x^3}} \, dx,x,x^2\right )-\text {Subst}\left (\int \frac {\sqrt {-1+x^3}}{-1-x^2+x^3} \, dx,x,x^2\right ) \\ & = \frac {\sqrt {-1+x^6}}{x^2}+\frac {3 \sqrt {-1+x^6}}{1-\sqrt {3}-x^2}-\frac {3 \sqrt [4]{3} \sqrt {2+\sqrt {3}} \left (1-x^2\right ) \sqrt {\frac {1+x^2+x^4}{\left (1-\sqrt {3}-x^2\right )^2}} E\left (\arcsin \left (\frac {1+\sqrt {3}-x^2}{1-\sqrt {3}-x^2}\right )|-7+4 \sqrt {3}\right )}{2 \sqrt {-\frac {1-x^2}{\left (1-\sqrt {3}-x^2\right )^2}} \sqrt {-1+x^6}}+\frac {\sqrt {2} 3^{3/4} \left (1-x^2\right ) \sqrt {\frac {1+x^2+x^4}{\left (1-\sqrt {3}-x^2\right )^2}} \operatorname {EllipticF}\left (\arcsin \left (\frac {1+\sqrt {3}-x^2}{1-\sqrt {3}-x^2}\right ),-7+4 \sqrt {3}\right )}{\sqrt {-\frac {1-x^2}{\left (1-\sqrt {3}-x^2\right )^2}} \sqrt {-1+x^6}}+\frac {3}{2} \text {Subst}\left (\int \frac {x \sqrt {-1+x^3}}{-1-x^2+x^3} \, dx,x,x^2\right )-\text {Subst}\left (\int \frac {\sqrt {-1+x^3}}{-1-x^2+x^3} \, dx,x,x^2\right ) \\ \end{align*}

Mathematica [A] (verified)

Time = 5.32 (sec) , antiderivative size = 30, normalized size of antiderivative = 1.00 \[ \int \frac {\sqrt {-1+x^6} \left (2+x^6\right )}{x^3 \left (-1-x^4+x^6\right )} \, dx=\frac {\sqrt {-1+x^6}}{x^2}-\text {arctanh}\left (\frac {x^2}{\sqrt {-1+x^6}}\right ) \]

[In]

Integrate[(Sqrt[-1 + x^6]*(2 + x^6))/(x^3*(-1 - x^4 + x^6)),x]

[Out]

Sqrt[-1 + x^6]/x^2 - ArcTanh[x^2/Sqrt[-1 + x^6]]

Maple [A] (verified)

Time = 1.97 (sec) , antiderivative size = 49, normalized size of antiderivative = 1.63

method	result	size
trager	\(\frac {\sqrt {x^{6}-1}}{x^{2}}-\frac {\ln \left (\frac {x^{6}+x^{4}+2 x^{2} \sqrt {x^{6}-1}-1}{x^{6}-x^{4}-1}\right )}{2}\)	\(49\)
risch	\(\frac {\sqrt {x^{6}-1}}{x^{2}}+\frac {\ln \left (-\frac {-x^{6}-x^{4}+2 x^{2} \sqrt {x^{6}-1}+1}{x^{6}-x^{4}-1}\right )}{2}\)	\(54\)
pseudoelliptic	\(\frac {\ln \left (\frac {-x^{2}+\sqrt {x^{6}-1}}{x^{2}}\right ) x^{2}-\ln \left (\frac {x^{2}+\sqrt {x^{6}-1}}{x^{2}}\right ) x^{2}+2 \sqrt {x^{6}-1}}{2 x^{2}}\)	\(59\)

[In]

int((x^6-1)^(1/2)*(x^6+2)/x^3/(x^6-x^4-1),x,method=_RETURNVERBOSE)

[Out]

(x^6-1)^(1/2)/x^2-1/2*ln((x^6+x^4+2*x^2*(x^6-1)^(1/2)-1)/(x^6-x^4-1))

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 53 vs. \(2 (26) = 52\).

Time = 0.27 (sec) , antiderivative size = 53, normalized size of antiderivative = 1.77 \[ \int \frac {\sqrt {-1+x^6} \left (2+x^6\right )}{x^3 \left (-1-x^4+x^6\right )} \, dx=\frac {x^{2} \log \left (\frac {x^{6} + x^{4} - 2 \, \sqrt {x^{6} - 1} x^{2} - 1}{x^{6} - x^{4} - 1}\right ) + 2 \, \sqrt {x^{6} - 1}}{2 \, x^{2}} \]

[In]

integrate((x^6-1)^(1/2)*(x^6+2)/x^3/(x^6-x^4-1),x, algorithm="fricas")

[Out]

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

Sympy [F(-1)]

Timed out. \[ \int \frac {\sqrt {-1+x^6} \left (2+x^6\right )}{x^3 \left (-1-x^4+x^6\right )} \, dx=\text {Timed out} \]

[In]

integrate((x**6-1)**(1/2)*(x**6+2)/x**3/(x**6-x**4-1),x)

[Out]

Timed out

Maxima [F]

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

[In]

integrate((x^6-1)^(1/2)*(x^6+2)/x^3/(x^6-x^4-1),x, algorithm="maxima")

[Out]

integrate((x^6 + 2)*sqrt(x^6 - 1)/((x^6 - x^4 - 1)*x^3), x)

Giac [F]

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

[In]

integrate((x^6-1)^(1/2)*(x^6+2)/x^3/(x^6-x^4-1),x, algorithm="giac")

[Out]

integrate((x^6 + 2)*sqrt(x^6 - 1)/((x^6 - x^4 - 1)*x^3), x)

Mupad [F(-1)]

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

[In]

int(-((x^6 - 1)^(1/2)*(x^6 + 2))/(x^3*(x^4 - x^6 + 1)),x)

[Out]

int(-((x^6 - 1)^(1/2)*(x^6 + 2))/(x^3*(x^4 - x^6 + 1)), x)