3.22.77 \(\int \frac {-4-2 x+2 x^2+x^4}{x (-2+x^2) \sqrt [4]{\frac {2+x^2}{-2+x^2}} (8-10 x+4 x^2+4 x^3-4 x^4+x^5)} \, dx\)

Optimal. Leaf size=161 \[ \frac {\tanh ^{-1}\left (\frac {\frac {x^2}{\sqrt {2}}+\frac {\sqrt {\frac {x^2+2}{x^2-2}}}{\sqrt {2}}-\sqrt {2} x+\frac {1}{\sqrt {2}}}{(x-1) \sqrt [4]{\frac {x^2+2}{x^2-2}}}\right )}{2 \sqrt {2}}-\frac {\tan ^{-1}\left (\frac {-\frac {x^2}{\sqrt {2}}+\frac {\sqrt {\frac {x^2+2}{x^2-2}}}{\sqrt {2}}+\sqrt {2} x-\frac {1}{\sqrt {2}}}{(x-1) \sqrt [4]{\frac {x^2+2}{x^2-2}}}\right )}{2 \sqrt {2}} \]

________________________________________________________________________________________

Rubi [F]  time = 6.13, antiderivative size = 0, normalized size of antiderivative = 0.00, number of steps used = 0, number of rules used = 0, integrand size = 0, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.000, Rules used = {} \begin {gather*} \int \frac {-4-2 x+2 x^2+x^4}{x \left (-2+x^2\right ) \sqrt [4]{\frac {2+x^2}{-2+x^2}} \left (8-10 x+4 x^2+4 x^3-4 x^4+x^5\right )} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Int[(-4 - 2*x + 2*x^2 + x^4)/(x*(-2 + x^2)*((2 + x^2)/(-2 + x^2))^(1/4)*(8 - 10*x + 4*x^2 + 4*x^3 - 4*x^4 + x^
5)),x]

[Out]

-1/4*((2 + x^2)^(1/4)*ArcTan[1 - (Sqrt[2]*(2 + x^2)^(1/4))/(-2 + x^2)^(1/4)])/(Sqrt[2]*(-2 + x^2)^(1/4)*(-((2
+ x^2)/(2 - x^2)))^(1/4)) + ((2 + x^2)^(1/4)*ArcTan[1 + (Sqrt[2]*(2 + x^2)^(1/4))/(-2 + x^2)^(1/4)])/(4*Sqrt[2
]*(-2 + x^2)^(1/4)*(-((2 + x^2)/(2 - x^2)))^(1/4)) + ((2 + x^2)^(1/4)*Log[1 - (Sqrt[2]*(2 + x^2)^(1/4))/(-2 +
x^2)^(1/4) + Sqrt[2 + x^2]/Sqrt[-2 + x^2]])/(8*Sqrt[2]*(-2 + x^2)^(1/4)*(-((2 + x^2)/(2 - x^2)))^(1/4)) - ((2
+ x^2)^(1/4)*Log[1 + (Sqrt[2]*(2 + x^2)^(1/4))/(-2 + x^2)^(1/4) + Sqrt[2 + x^2]/Sqrt[-2 + x^2]])/(8*Sqrt[2]*(-
2 + x^2)^(1/4)*(-((2 + x^2)/(2 - x^2)))^(1/4)) - (7*(2 + x^2)^(1/4)*Defer[Int][1/((-2 + x^2)^(3/4)*(2 + x^2)^(
1/4)*(8 - 10*x + 4*x^2 + 4*x^3 - 4*x^4 + x^5)), x])/((-2 + x^2)^(1/4)*(-((2 + x^2)/(2 - x^2)))^(1/4)) + (4*(2
+ x^2)^(1/4)*Defer[Int][x/((-2 + x^2)^(3/4)*(2 + x^2)^(1/4)*(8 - 10*x + 4*x^2 + 4*x^3 - 4*x^4 + x^5)), x])/((-
2 + x^2)^(1/4)*(-((2 + x^2)/(2 - x^2)))^(1/4)) + (2*(2 + x^2)^(1/4)*Defer[Int][x^2/((-2 + x^2)^(3/4)*(2 + x^2)
^(1/4)*(8 - 10*x + 4*x^2 + 4*x^3 - 4*x^4 + x^5)), x])/((-2 + x^2)^(1/4)*(-((2 + x^2)/(2 - x^2)))^(1/4)) - ((2
+ x^2)^(1/4)*Defer[Int][x^3/((-2 + x^2)^(3/4)*(2 + x^2)^(1/4)*(8 - 10*x + 4*x^2 + 4*x^3 - 4*x^4 + x^5)), x])/(
(-2 + x^2)^(1/4)*(-((2 + x^2)/(2 - x^2)))^(1/4)) + ((2 + x^2)^(1/4)*Defer[Int][x^4/((-2 + x^2)^(3/4)*(2 + x^2)
^(1/4)*(8 - 10*x + 4*x^2 + 4*x^3 - 4*x^4 + x^5)), x])/(2*(-2 + x^2)^(1/4)*(-((2 + x^2)/(2 - x^2)))^(1/4))

Rubi steps

\begin {align*} \int \frac {-4-2 x+2 x^2+x^4}{x \left (-2+x^2\right ) \sqrt [4]{\frac {2+x^2}{-2+x^2}} \left (8-10 x+4 x^2+4 x^3-4 x^4+x^5\right )} \, dx &=\frac {\sqrt [4]{2+x^2} \int \frac {-4-2 x+2 x^2+x^4}{x \left (-2+x^2\right )^{3/4} \sqrt [4]{2+x^2} \left (8-10 x+4 x^2+4 x^3-4 x^4+x^5\right )} \, dx}{\sqrt [4]{-2+x^2} \sqrt [4]{\frac {2+x^2}{-2+x^2}}}\\ &=\frac {\sqrt [4]{2+x^2} \int \left (-\frac {1}{2 x \left (-2+x^2\right )^{3/4} \sqrt [4]{2+x^2}}+\frac {-14+8 x+4 x^2-2 x^3+x^4}{2 \left (-2+x^2\right )^{3/4} \sqrt [4]{2+x^2} \left (8-10 x+4 x^2+4 x^3-4 x^4+x^5\right )}\right ) \, dx}{\sqrt [4]{-2+x^2} \sqrt [4]{\frac {2+x^2}{-2+x^2}}}\\ &=-\frac {\sqrt [4]{2+x^2} \int \frac {1}{x \left (-2+x^2\right )^{3/4} \sqrt [4]{2+x^2}} \, dx}{2 \sqrt [4]{-2+x^2} \sqrt [4]{\frac {2+x^2}{-2+x^2}}}+\frac {\sqrt [4]{2+x^2} \int \frac {-14+8 x+4 x^2-2 x^3+x^4}{\left (-2+x^2\right )^{3/4} \sqrt [4]{2+x^2} \left (8-10 x+4 x^2+4 x^3-4 x^4+x^5\right )} \, dx}{2 \sqrt [4]{-2+x^2} \sqrt [4]{\frac {2+x^2}{-2+x^2}}}\\ &=-\frac {\sqrt [4]{2+x^2} \operatorname {Subst}\left (\int \frac {1}{(-2+x)^{3/4} x \sqrt [4]{2+x}} \, dx,x,x^2\right )}{4 \sqrt [4]{-2+x^2} \sqrt [4]{\frac {2+x^2}{-2+x^2}}}+\frac {\sqrt [4]{2+x^2} \int \left (-\frac {14}{\left (-2+x^2\right )^{3/4} \sqrt [4]{2+x^2} \left (8-10 x+4 x^2+4 x^3-4 x^4+x^5\right )}+\frac {8 x}{\left (-2+x^2\right )^{3/4} \sqrt [4]{2+x^2} \left (8-10 x+4 x^2+4 x^3-4 x^4+x^5\right )}+\frac {4 x^2}{\left (-2+x^2\right )^{3/4} \sqrt [4]{2+x^2} \left (8-10 x+4 x^2+4 x^3-4 x^4+x^5\right )}-\frac {2 x^3}{\left (-2+x^2\right )^{3/4} \sqrt [4]{2+x^2} \left (8-10 x+4 x^2+4 x^3-4 x^4+x^5\right )}+\frac {x^4}{\left (-2+x^2\right )^{3/4} \sqrt [4]{2+x^2} \left (8-10 x+4 x^2+4 x^3-4 x^4+x^5\right )}\right ) \, dx}{2 \sqrt [4]{-2+x^2} \sqrt [4]{\frac {2+x^2}{-2+x^2}}}\\ &=\frac {\sqrt [4]{2+x^2} \int \frac {x^4}{\left (-2+x^2\right )^{3/4} \sqrt [4]{2+x^2} \left (8-10 x+4 x^2+4 x^3-4 x^4+x^5\right )} \, dx}{2 \sqrt [4]{-2+x^2} \sqrt [4]{\frac {2+x^2}{-2+x^2}}}-\frac {\sqrt [4]{2+x^2} \int \frac {x^3}{\left (-2+x^2\right )^{3/4} \sqrt [4]{2+x^2} \left (8-10 x+4 x^2+4 x^3-4 x^4+x^5\right )} \, dx}{\sqrt [4]{-2+x^2} \sqrt [4]{\frac {2+x^2}{-2+x^2}}}-\frac {\sqrt [4]{2+x^2} \operatorname {Subst}\left (\int \frac {x^2}{-2-2 x^4} \, dx,x,\frac {\sqrt [4]{2+x^2}}{\sqrt [4]{-2+x^2}}\right )}{\sqrt [4]{-2+x^2} \sqrt [4]{\frac {2+x^2}{-2+x^2}}}+\frac {\left (2 \sqrt [4]{2+x^2}\right ) \int \frac {x^2}{\left (-2+x^2\right )^{3/4} \sqrt [4]{2+x^2} \left (8-10 x+4 x^2+4 x^3-4 x^4+x^5\right )} \, dx}{\sqrt [4]{-2+x^2} \sqrt [4]{\frac {2+x^2}{-2+x^2}}}+\frac {\left (4 \sqrt [4]{2+x^2}\right ) \int \frac {x}{\left (-2+x^2\right )^{3/4} \sqrt [4]{2+x^2} \left (8-10 x+4 x^2+4 x^3-4 x^4+x^5\right )} \, dx}{\sqrt [4]{-2+x^2} \sqrt [4]{\frac {2+x^2}{-2+x^2}}}-\frac {\left (7 \sqrt [4]{2+x^2}\right ) \int \frac {1}{\left (-2+x^2\right )^{3/4} \sqrt [4]{2+x^2} \left (8-10 x+4 x^2+4 x^3-4 x^4+x^5\right )} \, dx}{\sqrt [4]{-2+x^2} \sqrt [4]{\frac {2+x^2}{-2+x^2}}}\\ &=\frac {\sqrt [4]{2+x^2} \int \frac {x^4}{\left (-2+x^2\right )^{3/4} \sqrt [4]{2+x^2} \left (8-10 x+4 x^2+4 x^3-4 x^4+x^5\right )} \, dx}{2 \sqrt [4]{-2+x^2} \sqrt [4]{\frac {2+x^2}{-2+x^2}}}+\frac {\sqrt [4]{2+x^2} \operatorname {Subst}\left (\int \frac {1-x^2}{-2-2 x^4} \, dx,x,\frac {\sqrt [4]{2+x^2}}{\sqrt [4]{-2+x^2}}\right )}{2 \sqrt [4]{-2+x^2} \sqrt [4]{\frac {2+x^2}{-2+x^2}}}-\frac {\sqrt [4]{2+x^2} \operatorname {Subst}\left (\int \frac {1+x^2}{-2-2 x^4} \, dx,x,\frac {\sqrt [4]{2+x^2}}{\sqrt [4]{-2+x^2}}\right )}{2 \sqrt [4]{-2+x^2} \sqrt [4]{\frac {2+x^2}{-2+x^2}}}-\frac {\sqrt [4]{2+x^2} \int \frac {x^3}{\left (-2+x^2\right )^{3/4} \sqrt [4]{2+x^2} \left (8-10 x+4 x^2+4 x^3-4 x^4+x^5\right )} \, dx}{\sqrt [4]{-2+x^2} \sqrt [4]{\frac {2+x^2}{-2+x^2}}}+\frac {\left (2 \sqrt [4]{2+x^2}\right ) \int \frac {x^2}{\left (-2+x^2\right )^{3/4} \sqrt [4]{2+x^2} \left (8-10 x+4 x^2+4 x^3-4 x^4+x^5\right )} \, dx}{\sqrt [4]{-2+x^2} \sqrt [4]{\frac {2+x^2}{-2+x^2}}}+\frac {\left (4 \sqrt [4]{2+x^2}\right ) \int \frac {x}{\left (-2+x^2\right )^{3/4} \sqrt [4]{2+x^2} \left (8-10 x+4 x^2+4 x^3-4 x^4+x^5\right )} \, dx}{\sqrt [4]{-2+x^2} \sqrt [4]{\frac {2+x^2}{-2+x^2}}}-\frac {\left (7 \sqrt [4]{2+x^2}\right ) \int \frac {1}{\left (-2+x^2\right )^{3/4} \sqrt [4]{2+x^2} \left (8-10 x+4 x^2+4 x^3-4 x^4+x^5\right )} \, dx}{\sqrt [4]{-2+x^2} \sqrt [4]{\frac {2+x^2}{-2+x^2}}}\\ &=\frac {\sqrt [4]{2+x^2} \operatorname {Subst}\left (\int \frac {1}{1-\sqrt {2} x+x^2} \, dx,x,\frac {\sqrt [4]{2+x^2}}{\sqrt [4]{-2+x^2}}\right )}{8 \sqrt [4]{-2+x^2} \sqrt [4]{\frac {2+x^2}{-2+x^2}}}+\frac {\sqrt [4]{2+x^2} \operatorname {Subst}\left (\int \frac {1}{1+\sqrt {2} x+x^2} \, dx,x,\frac {\sqrt [4]{2+x^2}}{\sqrt [4]{-2+x^2}}\right )}{8 \sqrt [4]{-2+x^2} \sqrt [4]{\frac {2+x^2}{-2+x^2}}}+\frac {\sqrt [4]{2+x^2} \int \frac {x^4}{\left (-2+x^2\right )^{3/4} \sqrt [4]{2+x^2} \left (8-10 x+4 x^2+4 x^3-4 x^4+x^5\right )} \, dx}{2 \sqrt [4]{-2+x^2} \sqrt [4]{\frac {2+x^2}{-2+x^2}}}-\frac {\sqrt [4]{2+x^2} \int \frac {x^3}{\left (-2+x^2\right )^{3/4} \sqrt [4]{2+x^2} \left (8-10 x+4 x^2+4 x^3-4 x^4+x^5\right )} \, dx}{\sqrt [4]{-2+x^2} \sqrt [4]{\frac {2+x^2}{-2+x^2}}}+\frac {\left (2 \sqrt [4]{2+x^2}\right ) \int \frac {x^2}{\left (-2+x^2\right )^{3/4} \sqrt [4]{2+x^2} \left (8-10 x+4 x^2+4 x^3-4 x^4+x^5\right )} \, dx}{\sqrt [4]{-2+x^2} \sqrt [4]{\frac {2+x^2}{-2+x^2}}}+\frac {\left (4 \sqrt [4]{2+x^2}\right ) \int \frac {x}{\left (-2+x^2\right )^{3/4} \sqrt [4]{2+x^2} \left (8-10 x+4 x^2+4 x^3-4 x^4+x^5\right )} \, dx}{\sqrt [4]{-2+x^2} \sqrt [4]{\frac {2+x^2}{-2+x^2}}}-\frac {\left (7 \sqrt [4]{2+x^2}\right ) \int \frac {1}{\left (-2+x^2\right )^{3/4} \sqrt [4]{2+x^2} \left (8-10 x+4 x^2+4 x^3-4 x^4+x^5\right )} \, dx}{\sqrt [4]{-2+x^2} \sqrt [4]{\frac {2+x^2}{-2+x^2}}}+\frac {\sqrt [4]{2+x^2} \operatorname {Subst}\left (\int \frac {\sqrt {2}+2 x}{-1-\sqrt {2} x-x^2} \, dx,x,\frac {\sqrt [4]{2+x^2}}{\sqrt [4]{-2+x^2}}\right )}{8 \sqrt {2} \sqrt [4]{-2+x^2} \sqrt [4]{\frac {2+x^2}{-2+x^2}}}+\frac {\sqrt [4]{2+x^2} \operatorname {Subst}\left (\int \frac {\sqrt {2}-2 x}{-1+\sqrt {2} x-x^2} \, dx,x,\frac {\sqrt [4]{2+x^2}}{\sqrt [4]{-2+x^2}}\right )}{8 \sqrt {2} \sqrt [4]{-2+x^2} \sqrt [4]{\frac {2+x^2}{-2+x^2}}}\\ &=\frac {\sqrt [4]{2+x^2} \log \left (1-\frac {\sqrt {2} \sqrt [4]{2+x^2}}{\sqrt [4]{-2+x^2}}+\frac {\sqrt {2+x^2}}{\sqrt {-2+x^2}}\right )}{8 \sqrt {2} \sqrt [4]{-2+x^2} \sqrt [4]{-\frac {2+x^2}{2-x^2}}}-\frac {\sqrt [4]{2+x^2} \log \left (1+\frac {\sqrt {2} \sqrt [4]{2+x^2}}{\sqrt [4]{-2+x^2}}+\frac {\sqrt {2+x^2}}{\sqrt {-2+x^2}}\right )}{8 \sqrt {2} \sqrt [4]{-2+x^2} \sqrt [4]{-\frac {2+x^2}{2-x^2}}}+\frac {\sqrt [4]{2+x^2} \int \frac {x^4}{\left (-2+x^2\right )^{3/4} \sqrt [4]{2+x^2} \left (8-10 x+4 x^2+4 x^3-4 x^4+x^5\right )} \, dx}{2 \sqrt [4]{-2+x^2} \sqrt [4]{\frac {2+x^2}{-2+x^2}}}-\frac {\sqrt [4]{2+x^2} \int \frac {x^3}{\left (-2+x^2\right )^{3/4} \sqrt [4]{2+x^2} \left (8-10 x+4 x^2+4 x^3-4 x^4+x^5\right )} \, dx}{\sqrt [4]{-2+x^2} \sqrt [4]{\frac {2+x^2}{-2+x^2}}}+\frac {\left (2 \sqrt [4]{2+x^2}\right ) \int \frac {x^2}{\left (-2+x^2\right )^{3/4} \sqrt [4]{2+x^2} \left (8-10 x+4 x^2+4 x^3-4 x^4+x^5\right )} \, dx}{\sqrt [4]{-2+x^2} \sqrt [4]{\frac {2+x^2}{-2+x^2}}}+\frac {\left (4 \sqrt [4]{2+x^2}\right ) \int \frac {x}{\left (-2+x^2\right )^{3/4} \sqrt [4]{2+x^2} \left (8-10 x+4 x^2+4 x^3-4 x^4+x^5\right )} \, dx}{\sqrt [4]{-2+x^2} \sqrt [4]{\frac {2+x^2}{-2+x^2}}}-\frac {\left (7 \sqrt [4]{2+x^2}\right ) \int \frac {1}{\left (-2+x^2\right )^{3/4} \sqrt [4]{2+x^2} \left (8-10 x+4 x^2+4 x^3-4 x^4+x^5\right )} \, dx}{\sqrt [4]{-2+x^2} \sqrt [4]{\frac {2+x^2}{-2+x^2}}}+\frac {\sqrt [4]{2+x^2} \operatorname {Subst}\left (\int \frac {1}{-1-x^2} \, dx,x,1-\frac {\sqrt {2} \sqrt [4]{2+x^2}}{\sqrt [4]{-2+x^2}}\right )}{4 \sqrt {2} \sqrt [4]{-2+x^2} \sqrt [4]{\frac {2+x^2}{-2+x^2}}}-\frac {\sqrt [4]{2+x^2} \operatorname {Subst}\left (\int \frac {1}{-1-x^2} \, dx,x,1+\frac {\sqrt {2} \sqrt [4]{2+x^2}}{\sqrt [4]{-2+x^2}}\right )}{4 \sqrt {2} \sqrt [4]{-2+x^2} \sqrt [4]{\frac {2+x^2}{-2+x^2}}}\\ &=-\frac {\sqrt [4]{2+x^2} \tan ^{-1}\left (1-\frac {\sqrt {2} \sqrt [4]{2+x^2}}{\sqrt [4]{-2+x^2}}\right )}{4 \sqrt {2} \sqrt [4]{-2+x^2} \sqrt [4]{-\frac {2+x^2}{2-x^2}}}+\frac {\sqrt [4]{2+x^2} \tan ^{-1}\left (1+\frac {\sqrt {2} \sqrt [4]{2+x^2}}{\sqrt [4]{-2+x^2}}\right )}{4 \sqrt {2} \sqrt [4]{-2+x^2} \sqrt [4]{-\frac {2+x^2}{2-x^2}}}+\frac {\sqrt [4]{2+x^2} \log \left (1-\frac {\sqrt {2} \sqrt [4]{2+x^2}}{\sqrt [4]{-2+x^2}}+\frac {\sqrt {2+x^2}}{\sqrt {-2+x^2}}\right )}{8 \sqrt {2} \sqrt [4]{-2+x^2} \sqrt [4]{-\frac {2+x^2}{2-x^2}}}-\frac {\sqrt [4]{2+x^2} \log \left (1+\frac {\sqrt {2} \sqrt [4]{2+x^2}}{\sqrt [4]{-2+x^2}}+\frac {\sqrt {2+x^2}}{\sqrt {-2+x^2}}\right )}{8 \sqrt {2} \sqrt [4]{-2+x^2} \sqrt [4]{-\frac {2+x^2}{2-x^2}}}+\frac {\sqrt [4]{2+x^2} \int \frac {x^4}{\left (-2+x^2\right )^{3/4} \sqrt [4]{2+x^2} \left (8-10 x+4 x^2+4 x^3-4 x^4+x^5\right )} \, dx}{2 \sqrt [4]{-2+x^2} \sqrt [4]{\frac {2+x^2}{-2+x^2}}}-\frac {\sqrt [4]{2+x^2} \int \frac {x^3}{\left (-2+x^2\right )^{3/4} \sqrt [4]{2+x^2} \left (8-10 x+4 x^2+4 x^3-4 x^4+x^5\right )} \, dx}{\sqrt [4]{-2+x^2} \sqrt [4]{\frac {2+x^2}{-2+x^2}}}+\frac {\left (2 \sqrt [4]{2+x^2}\right ) \int \frac {x^2}{\left (-2+x^2\right )^{3/4} \sqrt [4]{2+x^2} \left (8-10 x+4 x^2+4 x^3-4 x^4+x^5\right )} \, dx}{\sqrt [4]{-2+x^2} \sqrt [4]{\frac {2+x^2}{-2+x^2}}}+\frac {\left (4 \sqrt [4]{2+x^2}\right ) \int \frac {x}{\left (-2+x^2\right )^{3/4} \sqrt [4]{2+x^2} \left (8-10 x+4 x^2+4 x^3-4 x^4+x^5\right )} \, dx}{\sqrt [4]{-2+x^2} \sqrt [4]{\frac {2+x^2}{-2+x^2}}}-\frac {\left (7 \sqrt [4]{2+x^2}\right ) \int \frac {1}{\left (-2+x^2\right )^{3/4} \sqrt [4]{2+x^2} \left (8-10 x+4 x^2+4 x^3-4 x^4+x^5\right )} \, dx}{\sqrt [4]{-2+x^2} \sqrt [4]{\frac {2+x^2}{-2+x^2}}}\\ \end {align*}

________________________________________________________________________________________

Mathematica [F]  time = 0.56, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {-4-2 x+2 x^2+x^4}{x \left (-2+x^2\right ) \sqrt [4]{\frac {2+x^2}{-2+x^2}} \left (8-10 x+4 x^2+4 x^3-4 x^4+x^5\right )} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Integrate[(-4 - 2*x + 2*x^2 + x^4)/(x*(-2 + x^2)*((2 + x^2)/(-2 + x^2))^(1/4)*(8 - 10*x + 4*x^2 + 4*x^3 - 4*x^
4 + x^5)),x]

[Out]

Integrate[(-4 - 2*x + 2*x^2 + x^4)/(x*(-2 + x^2)*((2 + x^2)/(-2 + x^2))^(1/4)*(8 - 10*x + 4*x^2 + 4*x^3 - 4*x^
4 + x^5)), x]

________________________________________________________________________________________

IntegrateAlgebraic [A]  time = 0.52, size = 161, normalized size = 1.00 \begin {gather*} -\frac {\tan ^{-1}\left (\frac {-\frac {1}{\sqrt {2}}+\sqrt {2} x-\frac {x^2}{\sqrt {2}}+\frac {\sqrt {\frac {2+x^2}{-2+x^2}}}{\sqrt {2}}}{(-1+x) \sqrt [4]{\frac {2+x^2}{-2+x^2}}}\right )}{2 \sqrt {2}}+\frac {\tanh ^{-1}\left (\frac {\frac {1}{\sqrt {2}}-\sqrt {2} x+\frac {x^2}{\sqrt {2}}+\frac {\sqrt {\frac {2+x^2}{-2+x^2}}}{\sqrt {2}}}{(-1+x) \sqrt [4]{\frac {2+x^2}{-2+x^2}}}\right )}{2 \sqrt {2}} \end {gather*}

Antiderivative was successfully verified.

[In]

IntegrateAlgebraic[(-4 - 2*x + 2*x^2 + x^4)/(x*(-2 + x^2)*((2 + x^2)/(-2 + x^2))^(1/4)*(8 - 10*x + 4*x^2 + 4*x
^3 - 4*x^4 + x^5)),x]

[Out]

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

________________________________________________________________________________________

fricas [B]  time = 46.72, size = 1858, normalized size = 11.54

result too large to display

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((x^4+2*x^2-2*x-4)/x/(x^2-2)/((x^2+2)/(x^2-2))^(1/4)/(x^5-4*x^4+4*x^3+4*x^2-10*x+8),x, algorithm="fri
cas")

[Out]

1/4*sqrt(2)*arctan(-(x^12 - 8*x^11 + 24*x^10 - 24*x^9 - 36*x^8 + 128*x^7 - 128*x^6 - 16*x^5 + 164*x^4 - 160*x^
3 + 2*sqrt(2)*(3*x^9 - 15*x^8 + 18*x^7 + 30*x^6 - 94*x^5 + 58*x^4 + 60*x^3 - 108*x^2 + 64*x - 16)*((x^2 + 2)/(
x^2 - 2))^(3/4) + 64*x^2 + 2*sqrt(2)*(x^11 - 7*x^10 + 17*x^9 - 7*x^8 - 48*x^7 + 100*x^6 - 58*x^5 - 54*x^4 + 12
4*x^3 - 116*x^2 + 64*x - 16)*((x^2 + 2)/(x^2 - 2))^(1/4) - (2*sqrt(2)*(3*x^10 - 18*x^9 + 33*x^8 + 12*x^7 - 124
*x^6 + 152*x^5 + 2*x^4 - 168*x^3 + 172*x^2 - 80*x + 16)*sqrt((x^2 + 2)/(x^2 - 2)) + 16*(x^9 - 5*x^8 + 6*x^7 +
10*x^6 - 31*x^5 + 19*x^4 + 20*x^3 - 36*x^2 + 20*x - 4)*((x^2 + 2)/(x^2 - 2))^(3/4) + sqrt(2)*(x^12 - 8*x^11 +
24*x^10 - 24*x^9 - 30*x^8 + 104*x^7 - 92*x^6 - 40*x^5 + 144*x^4 - 64*x^3 - 88*x^2 + 96*x - 32) + 4*(x^11 - 7*x
^10 + 17*x^9 - 7*x^8 - 44*x^7 + 88*x^6 - 46*x^5 - 58*x^4 + 108*x^3 - 68*x^2 + 16*x)*((x^2 + 2)/(x^2 - 2))^(1/4
))*sqrt((x^6 - 4*x^5 + 4*x^4 + 4*x^3 - 2*sqrt(2)*(x^3 - x^2 - 2*x + 2)*((x^2 + 2)/(x^2 - 2))^(3/4) - 10*x^2 -
2*sqrt(2)*(x^5 - 3*x^4 + x^3 + 5*x^2 - 6*x + 2)*((x^2 + 2)/(x^2 - 2))^(1/4) + 4*(x^4 - 2*x^3 - x^2 + 4*x - 2)*
sqrt((x^2 + 2)/(x^2 - 2)) + 8*x)/(x^6 - 4*x^5 + 4*x^4 + 4*x^3 - 10*x^2 + 8*x)) + 4*(x^10 - 6*x^9 + 11*x^8 + 4*
x^7 - 40*x^6 + 48*x^5 + 2*x^4 - 56*x^3 + 52*x^2 - 16*x)*sqrt((x^2 + 2)/(x^2 - 2)))/(x^12 - 8*x^11 + 24*x^10 -
24*x^9 - 52*x^8 + 192*x^7 - 224*x^6 + 48*x^5 + 212*x^4 - 416*x^3 + 448*x^2 - 256*x + 64)) - 1/4*sqrt(2)*arctan
(-(x^12 - 8*x^11 + 24*x^10 - 24*x^9 - 36*x^8 + 128*x^7 - 128*x^6 - 16*x^5 + 164*x^4 - 160*x^3 - 2*sqrt(2)*(3*x
^9 - 15*x^8 + 18*x^7 + 30*x^6 - 94*x^5 + 58*x^4 + 60*x^3 - 108*x^2 + 64*x - 16)*((x^2 + 2)/(x^2 - 2))^(3/4) +
64*x^2 - 2*sqrt(2)*(x^11 - 7*x^10 + 17*x^9 - 7*x^8 - 48*x^7 + 100*x^6 - 58*x^5 - 54*x^4 + 124*x^3 - 116*x^2 +
64*x - 16)*((x^2 + 2)/(x^2 - 2))^(1/4) + (2*sqrt(2)*(3*x^10 - 18*x^9 + 33*x^8 + 12*x^7 - 124*x^6 + 152*x^5 + 2
*x^4 - 168*x^3 + 172*x^2 - 80*x + 16)*sqrt((x^2 + 2)/(x^2 - 2)) - 16*(x^9 - 5*x^8 + 6*x^7 + 10*x^6 - 31*x^5 +
19*x^4 + 20*x^3 - 36*x^2 + 20*x - 4)*((x^2 + 2)/(x^2 - 2))^(3/4) + sqrt(2)*(x^12 - 8*x^11 + 24*x^10 - 24*x^9 -
30*x^8 + 104*x^7 - 92*x^6 - 40*x^5 + 144*x^4 - 64*x^3 - 88*x^2 + 96*x - 32) - 4*(x^11 - 7*x^10 + 17*x^9 - 7*x
^8 - 44*x^7 + 88*x^6 - 46*x^5 - 58*x^4 + 108*x^3 - 68*x^2 + 16*x)*((x^2 + 2)/(x^2 - 2))^(1/4))*sqrt((x^6 - 4*x
^5 + 4*x^4 + 4*x^3 + 2*sqrt(2)*(x^3 - x^2 - 2*x + 2)*((x^2 + 2)/(x^2 - 2))^(3/4) - 10*x^2 + 2*sqrt(2)*(x^5 - 3
*x^4 + x^3 + 5*x^2 - 6*x + 2)*((x^2 + 2)/(x^2 - 2))^(1/4) + 4*(x^4 - 2*x^3 - x^2 + 4*x - 2)*sqrt((x^2 + 2)/(x^
2 - 2)) + 8*x)/(x^6 - 4*x^5 + 4*x^4 + 4*x^3 - 10*x^2 + 8*x)) + 4*(x^10 - 6*x^9 + 11*x^8 + 4*x^7 - 40*x^6 + 48*
x^5 + 2*x^4 - 56*x^3 + 52*x^2 - 16*x)*sqrt((x^2 + 2)/(x^2 - 2)))/(x^12 - 8*x^11 + 24*x^10 - 24*x^9 - 52*x^8 +
192*x^7 - 224*x^6 + 48*x^5 + 212*x^4 - 416*x^3 + 448*x^2 - 256*x + 64)) + 1/16*sqrt(2)*log(4*(x^6 - 4*x^5 + 4*
x^4 + 4*x^3 + 2*sqrt(2)*(x^3 - x^2 - 2*x + 2)*((x^2 + 2)/(x^2 - 2))^(3/4) - 10*x^2 + 2*sqrt(2)*(x^5 - 3*x^4 +
x^3 + 5*x^2 - 6*x + 2)*((x^2 + 2)/(x^2 - 2))^(1/4) + 4*(x^4 - 2*x^3 - x^2 + 4*x - 2)*sqrt((x^2 + 2)/(x^2 - 2))
+ 8*x)/(x^6 - 4*x^5 + 4*x^4 + 4*x^3 - 10*x^2 + 8*x)) - 1/16*sqrt(2)*log(4*(x^6 - 4*x^5 + 4*x^4 + 4*x^3 - 2*sq
rt(2)*(x^3 - x^2 - 2*x + 2)*((x^2 + 2)/(x^2 - 2))^(3/4) - 10*x^2 - 2*sqrt(2)*(x^5 - 3*x^4 + x^3 + 5*x^2 - 6*x
+ 2)*((x^2 + 2)/(x^2 - 2))^(1/4) + 4*(x^4 - 2*x^3 - x^2 + 4*x - 2)*sqrt((x^2 + 2)/(x^2 - 2)) + 8*x)/(x^6 - 4*x
^5 + 4*x^4 + 4*x^3 - 10*x^2 + 8*x))

________________________________________________________________________________________

giac [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {x^{4} + 2 \, x^{2} - 2 \, x - 4}{{\left (x^{5} - 4 \, x^{4} + 4 \, x^{3} + 4 \, x^{2} - 10 \, x + 8\right )} {\left (x^{2} - 2\right )} x \left (\frac {x^{2} + 2}{x^{2} - 2}\right )^{\frac {1}{4}}}\,{d x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((x^4+2*x^2-2*x-4)/x/(x^2-2)/((x^2+2)/(x^2-2))^(1/4)/(x^5-4*x^4+4*x^3+4*x^2-10*x+8),x, algorithm="gia
c")

[Out]

integrate((x^4 + 2*x^2 - 2*x - 4)/((x^5 - 4*x^4 + 4*x^3 + 4*x^2 - 10*x + 8)*(x^2 - 2)*x*((x^2 + 2)/(x^2 - 2))^
(1/4)), x)

________________________________________________________________________________________

maple [C]  time = 6.79, size = 1065, normalized size = 6.61

method	result	size
trager	\(\frac {\RootOf \left (\textit {\_Z}^{4}+1\right ) \ln \left (\frac {-2 \sqrt {-\frac {-x^{2}-2}{x^{2}-2}}\, x^{4} \RootOf \left (\textit {\_Z}^{4}+1\right )^{3}+4 \sqrt {-\frac {-x^{2}-2}{x^{2}-2}}\, x^{3} \RootOf \left (\textit {\_Z}^{4}+1\right )^{3}-2 \left (-\frac {-x^{2}-2}{x^{2}-2}\right )^{\frac {1}{4}} x^{5} \RootOf \left (\textit {\_Z}^{4}+1\right )^{2}+2 \sqrt {-\frac {-x^{2}-2}{x^{2}-2}}\, x^{2} \RootOf \left (\textit {\_Z}^{4}+1\right )^{3}+6 \left (-\frac {-x^{2}-2}{x^{2}-2}\right )^{\frac {1}{4}} x^{4} \RootOf \left (\textit {\_Z}^{4}+1\right )^{2}-\RootOf \left (\textit {\_Z}^{4}+1\right ) x^{6}+2 \left (-\frac {-x^{2}-2}{x^{2}-2}\right )^{\frac {3}{4}} x^{3}-8 \sqrt {-\frac {-x^{2}-2}{x^{2}-2}}\, \RootOf \left (\textit {\_Z}^{4}+1\right )^{3} x -2 \left (-\frac {-x^{2}-2}{x^{2}-2}\right )^{\frac {1}{4}} x^{3} \RootOf \left (\textit {\_Z}^{4}+1\right )^{2}+4 \RootOf \left (\textit {\_Z}^{4}+1\right ) x^{5}-2 \left (-\frac {-x^{2}-2}{x^{2}-2}\right )^{\frac {3}{4}} x^{2}+4 \sqrt {-\frac {-x^{2}-2}{x^{2}-2}}\, \RootOf \left (\textit {\_Z}^{4}+1\right )^{3}-10 \left (-\frac {-x^{2}-2}{x^{2}-2}\right )^{\frac {1}{4}} \RootOf \left (\textit {\_Z}^{4}+1\right )^{2} x^{2}-4 \RootOf \left (\textit {\_Z}^{4}+1\right ) x^{4}-4 \left (-\frac {-x^{2}-2}{x^{2}-2}\right )^{\frac {3}{4}} x +12 \left (-\frac {-x^{2}-2}{x^{2}-2}\right )^{\frac {1}{4}} x \RootOf \left (\textit {\_Z}^{4}+1\right )^{2}-4 \RootOf \left (\textit {\_Z}^{4}+1\right ) x^{3}+4 \left (-\frac {-x^{2}-2}{x^{2}-2}\right )^{\frac {3}{4}}-4 \left (-\frac {-x^{2}-2}{x^{2}-2}\right )^{\frac {1}{4}} \RootOf \left (\textit {\_Z}^{4}+1\right )^{2}+12 \RootOf \left (\textit {\_Z}^{4}+1\right ) x^{2}-8 \RootOf \left (\textit {\_Z}^{4}+1\right ) x +4 \RootOf \left (\textit {\_Z}^{4}+1\right )}{x \left (x^{5}-4 x^{4}+4 x^{3}+4 x^{2}-10 x +8\right )}\right )}{4}-\frac {\RootOf \left (\textit {\_Z}^{4}+1\right )^{3} \ln \left (\frac {\RootOf \left (\textit {\_Z}^{4}+1\right )^{3} x^{6}+2 \left (-\frac {-x^{2}-2}{x^{2}-2}\right )^{\frac {1}{4}} x^{5} \RootOf \left (\textit {\_Z}^{4}+1\right )^{2}-4 \RootOf \left (\textit {\_Z}^{4}+1\right )^{3} x^{5}+2 x^{4} \sqrt {-\frac {-x^{2}-2}{x^{2}-2}}\, \RootOf \left (\textit {\_Z}^{4}+1\right )-6 \left (-\frac {-x^{2}-2}{x^{2}-2}\right )^{\frac {1}{4}} x^{4} \RootOf \left (\textit {\_Z}^{4}+1\right )^{2}+4 \RootOf \left (\textit {\_Z}^{4}+1\right )^{3} x^{4}+2 \left (-\frac {-x^{2}-2}{x^{2}-2}\right )^{\frac {3}{4}} x^{3}-4 x^{3} \sqrt {-\frac {-x^{2}-2}{x^{2}-2}}\, \RootOf \left (\textit {\_Z}^{4}+1\right )+2 \left (-\frac {-x^{2}-2}{x^{2}-2}\right )^{\frac {1}{4}} x^{3} \RootOf \left (\textit {\_Z}^{4}+1\right )^{2}+4 \RootOf \left (\textit {\_Z}^{4}+1\right )^{3} x^{3}-2 \left (-\frac {-x^{2}-2}{x^{2}-2}\right )^{\frac {3}{4}} x^{2}-2 x^{2} \sqrt {-\frac {-x^{2}-2}{x^{2}-2}}\, \RootOf \left (\textit {\_Z}^{4}+1\right )+10 \left (-\frac {-x^{2}-2}{x^{2}-2}\right )^{\frac {1}{4}} \RootOf \left (\textit {\_Z}^{4}+1\right )^{2} x^{2}-12 \RootOf \left (\textit {\_Z}^{4}+1\right )^{3} x^{2}-4 \left (-\frac {-x^{2}-2}{x^{2}-2}\right )^{\frac {3}{4}} x +8 \sqrt {-\frac {-x^{2}-2}{x^{2}-2}}\, \RootOf \left (\textit {\_Z}^{4}+1\right ) x -12 \left (-\frac {-x^{2}-2}{x^{2}-2}\right )^{\frac {1}{4}} x \RootOf \left (\textit {\_Z}^{4}+1\right )^{2}+8 \RootOf \left (\textit {\_Z}^{4}+1\right )^{3} x +4 \left (-\frac {-x^{2}-2}{x^{2}-2}\right )^{\frac {3}{4}}-4 \sqrt {-\frac {-x^{2}-2}{x^{2}-2}}\, \RootOf \left (\textit {\_Z}^{4}+1\right )+4 \left (-\frac {-x^{2}-2}{x^{2}-2}\right )^{\frac {1}{4}} \RootOf \left (\textit {\_Z}^{4}+1\right )^{2}-4 \RootOf \left (\textit {\_Z}^{4}+1\right )^{3}}{x \left (x^{5}-4 x^{4}+4 x^{3}+4 x^{2}-10 x +8\right )}\right )}{4}\)	\(1065\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((x^4+2*x^2-2*x-4)/x/(x^2-2)/((x^2+2)/(x^2-2))^(1/4)/(x^5-4*x^4+4*x^3+4*x^2-10*x+8),x,method=_RETURNVERBOSE
)

[Out]

1/4*RootOf(_Z^4+1)*ln((-2*(-(-x^2-2)/(x^2-2))^(1/2)*x^4*RootOf(_Z^4+1)^3+4*(-(-x^2-2)/(x^2-2))^(1/2)*x^3*RootO
f(_Z^4+1)^3-2*(-(-x^2-2)/(x^2-2))^(1/4)*x^5*RootOf(_Z^4+1)^2+2*(-(-x^2-2)/(x^2-2))^(1/2)*x^2*RootOf(_Z^4+1)^3+
6*(-(-x^2-2)/(x^2-2))^(1/4)*x^4*RootOf(_Z^4+1)^2-RootOf(_Z^4+1)*x^6+2*(-(-x^2-2)/(x^2-2))^(3/4)*x^3-8*(-(-x^2-
2)/(x^2-2))^(1/2)*RootOf(_Z^4+1)^3*x-2*(-(-x^2-2)/(x^2-2))^(1/4)*x^3*RootOf(_Z^4+1)^2+4*RootOf(_Z^4+1)*x^5-2*(
-(-x^2-2)/(x^2-2))^(3/4)*x^2+4*(-(-x^2-2)/(x^2-2))^(1/2)*RootOf(_Z^4+1)^3-10*(-(-x^2-2)/(x^2-2))^(1/4)*RootOf(
_Z^4+1)^2*x^2-4*RootOf(_Z^4+1)*x^4-4*(-(-x^2-2)/(x^2-2))^(3/4)*x+12*(-(-x^2-2)/(x^2-2))^(1/4)*x*RootOf(_Z^4+1)
^2-4*RootOf(_Z^4+1)*x^3+4*(-(-x^2-2)/(x^2-2))^(3/4)-4*(-(-x^2-2)/(x^2-2))^(1/4)*RootOf(_Z^4+1)^2+12*RootOf(_Z^
4+1)*x^2-8*RootOf(_Z^4+1)*x+4*RootOf(_Z^4+1))/x/(x^5-4*x^4+4*x^3+4*x^2-10*x+8))-1/4*RootOf(_Z^4+1)^3*ln((RootO
f(_Z^4+1)^3*x^6+2*(-(-x^2-2)/(x^2-2))^(1/4)*x^5*RootOf(_Z^4+1)^2-4*RootOf(_Z^4+1)^3*x^5+2*x^4*(-(-x^2-2)/(x^2-
2))^(1/2)*RootOf(_Z^4+1)-6*(-(-x^2-2)/(x^2-2))^(1/4)*x^4*RootOf(_Z^4+1)^2+4*RootOf(_Z^4+1)^3*x^4+2*(-(-x^2-2)/
(x^2-2))^(3/4)*x^3-4*x^3*(-(-x^2-2)/(x^2-2))^(1/2)*RootOf(_Z^4+1)+2*(-(-x^2-2)/(x^2-2))^(1/4)*x^3*RootOf(_Z^4+
1)^2+4*RootOf(_Z^4+1)^3*x^3-2*(-(-x^2-2)/(x^2-2))^(3/4)*x^2-2*x^2*(-(-x^2-2)/(x^2-2))^(1/2)*RootOf(_Z^4+1)+10*
(-(-x^2-2)/(x^2-2))^(1/4)*RootOf(_Z^4+1)^2*x^2-12*RootOf(_Z^4+1)^3*x^2-4*(-(-x^2-2)/(x^2-2))^(3/4)*x+8*(-(-x^2
-2)/(x^2-2))^(1/2)*RootOf(_Z^4+1)*x-12*(-(-x^2-2)/(x^2-2))^(1/4)*x*RootOf(_Z^4+1)^2+8*RootOf(_Z^4+1)^3*x+4*(-(
-x^2-2)/(x^2-2))^(3/4)-4*(-(-x^2-2)/(x^2-2))^(1/2)*RootOf(_Z^4+1)+4*(-(-x^2-2)/(x^2-2))^(1/4)*RootOf(_Z^4+1)^2
-4*RootOf(_Z^4+1)^3)/x/(x^5-4*x^4+4*x^3+4*x^2-10*x+8))

________________________________________________________________________________________

maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {x^{4} + 2 \, x^{2} - 2 \, x - 4}{{\left (x^{5} - 4 \, x^{4} + 4 \, x^{3} + 4 \, x^{2} - 10 \, x + 8\right )} {\left (x^{2} - 2\right )} x \left (\frac {x^{2} + 2}{x^{2} - 2}\right )^{\frac {1}{4}}}\,{d x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((x^4+2*x^2-2*x-4)/x/(x^2-2)/((x^2+2)/(x^2-2))^(1/4)/(x^5-4*x^4+4*x^3+4*x^2-10*x+8),x, algorithm="max
ima")

[Out]

integrate((x^4 + 2*x^2 - 2*x - 4)/((x^5 - 4*x^4 + 4*x^3 + 4*x^2 - 10*x + 8)*(x^2 - 2)*x*((x^2 + 2)/(x^2 - 2))^
(1/4)), x)

________________________________________________________________________________________

mupad [F]  time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \int -\frac {-x^4-2\,x^2+2\,x+4}{x\,{\left (\frac {x^2+2}{x^2-2}\right )}^{1/4}\,\left (x^2-2\right )\,\left (x^5-4\,x^4+4\,x^3+4\,x^2-10\,x+8\right )} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(-(2*x - 2*x^2 - x^4 + 4)/(x*((x^2 + 2)/(x^2 - 2))^(1/4)*(x^2 - 2)*(4*x^2 - 10*x + 4*x^3 - 4*x^4 + x^5 + 8)
),x)

[Out]

int(-(2*x - 2*x^2 - x^4 + 4)/(x*((x^2 + 2)/(x^2 - 2))^(1/4)*(x^2 - 2)*(4*x^2 - 10*x + 4*x^3 - 4*x^4 + x^5 + 8)
), x)

________________________________________________________________________________________

sympy [F(-1)]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((x**4+2*x**2-2*x-4)/x/(x**2-2)/((x**2+2)/(x**2-2))**(1/4)/(x**5-4*x**4+4*x**3+4*x**2-10*x+8),x)

[Out]

Timed out

________________________________________________________________________________________