\(\int \frac {1}{(1+x) \sqrt [4]{1+6 x^2+x^4}} \, dx\) [1080]

Optimal result

Integrand size = 20, antiderivative size = 81 \[ \int \frac {1}{(1+x) \sqrt [4]{1+6 x^2+x^4}} \, dx=\frac {\arctan \left (\frac {-\frac {1}{\sqrt [4]{2}}+\frac {x}{\sqrt [4]{2}}}{\sqrt [4]{1+6 x^2+x^4}}\right )}{2\ 2^{3/4}}+\frac {\text {arctanh}\left (\frac {-\frac {1}{\sqrt [4]{2}}+\frac {x}{\sqrt [4]{2}}}{\sqrt [4]{1+6 x^2+x^4}}\right )}{2\ 2^{3/4}} \]

[Out]

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

Rubi [F]

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

[In]

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

[Out]

Defer[Int][1/((1 + x)*(1 + 6*x^2 + x^4)^(1/4)), x]

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

Mathematica [A] (verified)

Time = 0.87 (sec) , antiderivative size = 58, normalized size of antiderivative = 0.72 \[ \int \frac {1}{(1+x) \sqrt [4]{1+6 x^2+x^4}} \, dx=\frac {\arctan \left (\frac {-1+x}{\sqrt [4]{2} \sqrt [4]{1+6 x^2+x^4}}\right )+\text {arctanh}\left (\frac {-1+x}{\sqrt [4]{2} \sqrt [4]{1+6 x^2+x^4}}\right )}{2\ 2^{3/4}} \]

[In]

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

[Out]

(ArcTan[(-1 + x)/(2^(1/4)*(1 + 6*x^2 + x^4)^(1/4))] + ArcTanh[(-1 + x)/(2^(1/4)*(1 + 6*x^2 + x^4)^(1/4))])/(2*
2^(3/4))

Maple [F]

\[\int \frac {1}{\left (1+x \right ) \left (x^{4}+6 x^{2}+1\right )^{\frac {1}{4}}}d x\]

[In]

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

[Out]

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

Fricas [C] (verification not implemented)

Result contains complex when optimal does not.

Time = 4.71 (sec) , antiderivative size = 502, normalized size of antiderivative = 6.20 \[ \int \frac {1}{(1+x) \sqrt [4]{1+6 x^2+x^4}} \, dx=\frac {1}{64} \cdot 8^{\frac {3}{4}} \log \left (\frac {8^{\frac {3}{4}} {\left (3 \, x^{4} - 4 \, x^{3} + 18 \, x^{2} - 4 \, x + 3\right )} + 8 \, \sqrt {2} {\left (x^{4} + 6 \, x^{2} + 1\right )}^{\frac {1}{4}} {\left (x^{3} - 3 \, x^{2} + 3 \, x - 1\right )} + 8 \cdot 8^{\frac {1}{4}} \sqrt {x^{4} + 6 \, x^{2} + 1} {\left (x^{2} - 2 \, x + 1\right )} + 16 \, {\left (x^{4} + 6 \, x^{2} + 1\right )}^{\frac {3}{4}} {\left (x - 1\right )}}{x^{4} + 4 \, x^{3} + 6 \, x^{2} + 4 \, x + 1}\right ) - \frac {1}{64} \cdot 8^{\frac {3}{4}} \log \left (-\frac {8^{\frac {3}{4}} {\left (3 \, x^{4} - 4 \, x^{3} + 18 \, x^{2} - 4 \, x + 3\right )} - 8 \, \sqrt {2} {\left (x^{4} + 6 \, x^{2} + 1\right )}^{\frac {1}{4}} {\left (x^{3} - 3 \, x^{2} + 3 \, x - 1\right )} + 8 \cdot 8^{\frac {1}{4}} \sqrt {x^{4} + 6 \, x^{2} + 1} {\left (x^{2} - 2 \, x + 1\right )} - 16 \, {\left (x^{4} + 6 \, x^{2} + 1\right )}^{\frac {3}{4}} {\left (x - 1\right )}}{x^{4} + 4 \, x^{3} + 6 \, x^{2} + 4 \, x + 1}\right ) + \frac {1}{64} i \cdot 8^{\frac {3}{4}} \log \left (\frac {8^{\frac {3}{4}} {\left (3 i \, x^{4} - 4 i \, x^{3} + 18 i \, x^{2} - 4 i \, x + 3 i\right )} - 8 \, \sqrt {2} {\left (x^{4} + 6 \, x^{2} + 1\right )}^{\frac {1}{4}} {\left (x^{3} - 3 \, x^{2} + 3 \, x - 1\right )} - 8 \cdot 8^{\frac {1}{4}} \sqrt {x^{4} + 6 \, x^{2} + 1} {\left (i \, x^{2} - 2 i \, x + i\right )} + 16 \, {\left (x^{4} + 6 \, x^{2} + 1\right )}^{\frac {3}{4}} {\left (x - 1\right )}}{x^{4} + 4 \, x^{3} + 6 \, x^{2} + 4 \, x + 1}\right ) - \frac {1}{64} i \cdot 8^{\frac {3}{4}} \log \left (\frac {8^{\frac {3}{4}} {\left (-3 i \, x^{4} + 4 i \, x^{3} - 18 i \, x^{2} + 4 i \, x - 3 i\right )} - 8 \, \sqrt {2} {\left (x^{4} + 6 \, x^{2} + 1\right )}^{\frac {1}{4}} {\left (x^{3} - 3 \, x^{2} + 3 \, x - 1\right )} - 8 \cdot 8^{\frac {1}{4}} \sqrt {x^{4} + 6 \, x^{2} + 1} {\left (-i \, x^{2} + 2 i \, x - i\right )} + 16 \, {\left (x^{4} + 6 \, x^{2} + 1\right )}^{\frac {3}{4}} {\left (x - 1\right )}}{x^{4} + 4 \, x^{3} + 6 \, x^{2} + 4 \, x + 1}\right ) \]

[In]

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

[Out]

1/64*8^(3/4)*log((8^(3/4)*(3*x^4 - 4*x^3 + 18*x^2 - 4*x + 3) + 8*sqrt(2)*(x^4 + 6*x^2 + 1)^(1/4)*(x^3 - 3*x^2
+ 3*x - 1) + 8*8^(1/4)*sqrt(x^4 + 6*x^2 + 1)*(x^2 - 2*x + 1) + 16*(x^4 + 6*x^2 + 1)^(3/4)*(x - 1))/(x^4 + 4*x^
3 + 6*x^2 + 4*x + 1)) - 1/64*8^(3/4)*log(-(8^(3/4)*(3*x^4 - 4*x^3 + 18*x^2 - 4*x + 3) - 8*sqrt(2)*(x^4 + 6*x^2
+ 1)^(1/4)*(x^3 - 3*x^2 + 3*x - 1) + 8*8^(1/4)*sqrt(x^4 + 6*x^2 + 1)*(x^2 - 2*x + 1) - 16*(x^4 + 6*x^2 + 1)^(
3/4)*(x - 1))/(x^4 + 4*x^3 + 6*x^2 + 4*x + 1)) + 1/64*I*8^(3/4)*log((8^(3/4)*(3*I*x^4 - 4*I*x^3 + 18*I*x^2 - 4
*I*x + 3*I) - 8*sqrt(2)*(x^4 + 6*x^2 + 1)^(1/4)*(x^3 - 3*x^2 + 3*x - 1) - 8*8^(1/4)*sqrt(x^4 + 6*x^2 + 1)*(I*x
^2 - 2*I*x + I) + 16*(x^4 + 6*x^2 + 1)^(3/4)*(x - 1))/(x^4 + 4*x^3 + 6*x^2 + 4*x + 1)) - 1/64*I*8^(3/4)*log((8
^(3/4)*(-3*I*x^4 + 4*I*x^3 - 18*I*x^2 + 4*I*x - 3*I) - 8*sqrt(2)*(x^4 + 6*x^2 + 1)^(1/4)*(x^3 - 3*x^2 + 3*x -
1) - 8*8^(1/4)*sqrt(x^4 + 6*x^2 + 1)*(-I*x^2 + 2*I*x - I) + 16*(x^4 + 6*x^2 + 1)^(3/4)*(x - 1))/(x^4 + 4*x^3 +
6*x^2 + 4*x + 1))

Sympy [F]

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

[In]

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

[Out]

Integral(1/((x + 1)*(x**4 + 6*x**2 + 1)**(1/4)), x)

Maxima [F]

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

[In]

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

[Out]

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

Giac [F]

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

[In]

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

[Out]

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

Mupad [F(-1)]

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

[In]

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

[Out]

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