\(\int \frac {-1+2 x^6}{\sqrt {1+x^6} (1-x^2+x^6)} \, dx\) [63]

Optimal result

Integrand size = 29, antiderivative size = 14 \[ \int \frac {-1+2 x^6}{\sqrt {1+x^6} \left (1-x^2+x^6\right )} \, dx=-\text {arctanh}\left (\frac {x}{\sqrt {1+x^6}}\right ) \]

[Out]

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

Rubi [F]

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

[In]

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

[Out]

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

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

Mathematica [A] (verified)

Time = 3.17 (sec) , antiderivative size = 14, normalized size of antiderivative = 1.00 \[ \int \frac {-1+2 x^6}{\sqrt {1+x^6} \left (1-x^2+x^6\right )} \, dx=-\text {arctanh}\left (\frac {x}{\sqrt {1+x^6}}\right ) \]

[In]

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

[Out]

-ArcTanh[x/Sqrt[1 + x^6]]

Maple [A] (verified)

Time = 1.19 (sec) , antiderivative size = 15, normalized size of antiderivative = 1.07

method	result	size
pseudoelliptic	\(-\operatorname {arctanh}\left (\frac {\sqrt {x^{6}+1}}{x}\right )\)	\(15\)
trager	\(\frac {\ln \left (-\frac {-x^{6}+2 x \sqrt {x^{6}+1}-x^{2}-1}{x^{6}-x^{2}+1}\right )}{2}\)	\(40\)

[In]

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

[Out]

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

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 34 vs. \(2 (12) = 24\).

Time = 0.27 (sec) , antiderivative size = 34, normalized size of antiderivative = 2.43 \[ \int \frac {-1+2 x^6}{\sqrt {1+x^6} \left (1-x^2+x^6\right )} \, dx=\frac {1}{2} \, \log \left (\frac {x^{6} + x^{2} - 2 \, \sqrt {x^{6} + 1} x + 1}{x^{6} - x^{2} + 1}\right ) \]

[In]

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

[Out]

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

Sympy [F(-1)]

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

[In]

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

[Out]

Timed out

Maxima [F]

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

[In]

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

[Out]

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

Giac [F]

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

[In]

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

[Out]

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

Mupad [F(-1)]

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

[In]

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

[Out]

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