3.28.63 \(\int \frac {(2+x^6) (-1+x^4+x^6)}{\sqrt [4]{-1+x^6} (1-2 x^6+x^8+x^{12})} \, dx\)

Optimal. Leaf size=261 \[ -\frac {1}{2} \sqrt {\frac {1}{2} \left (2-\sqrt {2}\right )} \tan ^{-1}\left (\frac {\sqrt {2+\sqrt {2}} x \sqrt [4]{x^6-1}}{\sqrt {x^6-1}-x^2}\right )+\frac {1}{2} \sqrt {\frac {1}{2} \left (2+\sqrt {2}\right )} \tan ^{-1}\left (\frac {\left (\sqrt {\frac {2}{2-\sqrt {2}}}-\frac {2}{\sqrt {2-\sqrt {2}}}\right ) x \sqrt [4]{x^6-1}}{\sqrt {x^6-1}-x^2}\right )-\frac {1}{2} \sqrt {\frac {1}{2} \left (2+\sqrt {2}\right )} \tanh ^{-1}\left (\frac {\sqrt {2-\sqrt {2}} x \sqrt [4]{x^6-1}}{\sqrt {x^6-1}+x^2}\right )-\frac {1}{2} \sqrt {\frac {1}{2} \left (2-\sqrt {2}\right )} \tanh ^{-1}\left (\frac {\sqrt {2+\sqrt {2}} x \sqrt [4]{x^6-1}}{\sqrt {x^6-1}+x^2}\right ) \]

________________________________________________________________________________________

Rubi [F]  time = 1.38, 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 {\left (2+x^6\right ) \left (-1+x^4+x^6\right )}{\sqrt [4]{-1+x^6} \left (1-2 x^6+x^8+x^{12}\right )} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Int[((2 + x^6)*(-1 + x^4 + x^6))/((-1 + x^6)^(1/4)*(1 - 2*x^6 + x^8 + x^12)),x]

[Out]

(x*(1 - x^6)^(1/4)*Hypergeometric2F1[1/6, 1/4, 7/6, x^6])/(-1 + x^6)^(1/4) - 3*Defer[Int][1/((-1 + x^6)^(1/4)*
(1 - 2*x^6 + x^8 + x^12)), x] + 2*Defer[Int][x^4/((-1 + x^6)^(1/4)*(1 - 2*x^6 + x^8 + x^12)), x] + 3*Defer[Int
][x^6/((-1 + x^6)^(1/4)*(1 - 2*x^6 + x^8 + x^12)), x] - Defer[Int][x^8/((-1 + x^6)^(1/4)*(1 - 2*x^6 + x^8 + x^
12)), x] + Defer[Int][x^10/((-1 + x^6)^(1/4)*(1 - 2*x^6 + x^8 + x^12)), x]

Rubi steps

\begin {align*} \int \frac {\left (2+x^6\right ) \left (-1+x^4+x^6\right )}{\sqrt [4]{-1+x^6} \left (1-2 x^6+x^8+x^{12}\right )} \, dx &=\int \left (\frac {1}{\sqrt [4]{-1+x^6}}-\frac {3-2 x^4-3 x^6+x^8-x^{10}}{\sqrt [4]{-1+x^6} \left (1-2 x^6+x^8+x^{12}\right )}\right ) \, dx\\ &=\int \frac {1}{\sqrt [4]{-1+x^6}} \, dx-\int \frac {3-2 x^4-3 x^6+x^8-x^{10}}{\sqrt [4]{-1+x^6} \left (1-2 x^6+x^8+x^{12}\right )} \, dx\\ &=\frac {\sqrt [4]{1-x^6} \int \frac {1}{\sqrt [4]{1-x^6}} \, dx}{\sqrt [4]{-1+x^6}}-\int \left (\frac {3}{\sqrt [4]{-1+x^6} \left (1-2 x^6+x^8+x^{12}\right )}-\frac {2 x^4}{\sqrt [4]{-1+x^6} \left (1-2 x^6+x^8+x^{12}\right )}-\frac {3 x^6}{\sqrt [4]{-1+x^6} \left (1-2 x^6+x^8+x^{12}\right )}+\frac {x^8}{\sqrt [4]{-1+x^6} \left (1-2 x^6+x^8+x^{12}\right )}-\frac {x^{10}}{\sqrt [4]{-1+x^6} \left (1-2 x^6+x^8+x^{12}\right )}\right ) \, dx\\ &=\frac {x \sqrt [4]{1-x^6} \, _2F_1\left (\frac {1}{6},\frac {1}{4};\frac {7}{6};x^6\right )}{\sqrt [4]{-1+x^6}}+2 \int \frac {x^4}{\sqrt [4]{-1+x^6} \left (1-2 x^6+x^8+x^{12}\right )} \, dx-3 \int \frac {1}{\sqrt [4]{-1+x^6} \left (1-2 x^6+x^8+x^{12}\right )} \, dx+3 \int \frac {x^6}{\sqrt [4]{-1+x^6} \left (1-2 x^6+x^8+x^{12}\right )} \, dx-\int \frac {x^8}{\sqrt [4]{-1+x^6} \left (1-2 x^6+x^8+x^{12}\right )} \, dx+\int \frac {x^{10}}{\sqrt [4]{-1+x^6} \left (1-2 x^6+x^8+x^{12}\right )} \, dx\\ \end {align*}

________________________________________________________________________________________

Mathematica [F]  time = 0.31, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\left (2+x^6\right ) \left (-1+x^4+x^6\right )}{\sqrt [4]{-1+x^6} \left (1-2 x^6+x^8+x^{12}\right )} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Integrate[((2 + x^6)*(-1 + x^4 + x^6))/((-1 + x^6)^(1/4)*(1 - 2*x^6 + x^8 + x^12)),x]

[Out]

Integrate[((2 + x^6)*(-1 + x^4 + x^6))/((-1 + x^6)^(1/4)*(1 - 2*x^6 + x^8 + x^12)), x]

________________________________________________________________________________________

IntegrateAlgebraic [A]  time = 16.40, size = 241, normalized size = 0.92 \begin {gather*} -\frac {1}{2} \sqrt {\frac {1}{2} \left (2+\sqrt {2}\right )} \tan ^{-1}\left (\frac {\sqrt {2-\sqrt {2}} x \sqrt [4]{-1+x^6}}{-x^2+\sqrt {-1+x^6}}\right )-\frac {1}{2} \sqrt {\frac {1}{2} \left (2-\sqrt {2}\right )} \tan ^{-1}\left (\frac {\sqrt {2+\sqrt {2}} x \sqrt [4]{-1+x^6}}{-x^2+\sqrt {-1+x^6}}\right )-\frac {1}{2} \sqrt {\frac {1}{2} \left (2+\sqrt {2}\right )} \tanh ^{-1}\left (\frac {\sqrt {2-\sqrt {2}} x \sqrt [4]{-1+x^6}}{x^2+\sqrt {-1+x^6}}\right )-\frac {1}{2} \sqrt {\frac {1}{2} \left (2-\sqrt {2}\right )} \tanh ^{-1}\left (\frac {\sqrt {2+\sqrt {2}} x \sqrt [4]{-1+x^6}}{x^2+\sqrt {-1+x^6}}\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

IntegrateAlgebraic[((2 + x^6)*(-1 + x^4 + x^6))/((-1 + x^6)^(1/4)*(1 - 2*x^6 + x^8 + x^12)),x]

[Out]

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

________________________________________________________________________________________

fricas [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^6+2)*(x^6+x^4-1)/(x^6-1)^(1/4)/(x^12+x^8-2*x^6+1),x, algorithm="fricas")

[Out]

Timed out

________________________________________________________________________________________

giac [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {{\left (x^{6} + x^{4} - 1\right )} {\left (x^{6} + 2\right )}}{{\left (x^{12} + x^{8} - 2 \, x^{6} + 1\right )} {\left (x^{6} - 1\right )}^{\frac {1}{4}}}\,{d x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((x^6+2)*(x^6+x^4-1)/(x^6-1)^(1/4)/(x^12+x^8-2*x^6+1),x, algorithm="giac")

[Out]

integrate((x^6 + x^4 - 1)*(x^6 + 2)/((x^12 + x^8 - 2*x^6 + 1)*(x^6 - 1)^(1/4)), x)

________________________________________________________________________________________

maple [C]  time = 31.65, size = 684, normalized size = 2.62

method	result	size
trager	\(-\frac {\RootOf \left (\textit {\_Z}^{8}+16\right )^{3} \ln \left (-\frac {-\RootOf \left (\textit {\_Z}^{8}+16\right )^{10} x^{4}+4 \RootOf \left (\textit {\_Z}^{8}+16\right )^{6} x^{6}-4 x^{4} \RootOf \left (\textit {\_Z}^{8}+16\right )^{6}+16 \RootOf \left (\textit {\_Z}^{8}+16\right )^{4} \sqrt {x^{6}-1}\, x^{2}+16 \RootOf \left (\textit {\_Z}^{8}+16\right )^{2} x^{6}+32 \RootOf \left (\textit {\_Z}^{8}+16\right )^{3} \left (x^{6}-1\right )^{\frac {3}{4}} x -4 \RootOf \left (\textit {\_Z}^{8}+16\right )^{6}+64 \RootOf \left (\textit {\_Z}^{8}+16\right ) \left (x^{6}-1\right )^{\frac {1}{4}} x^{3}+64 x^{2} \sqrt {x^{6}-1}-16 \RootOf \left (\textit {\_Z}^{8}+16\right )^{2}}{\RootOf \left (\textit {\_Z}^{8}+16\right )^{4} x^{4}+4 x^{6}-4}\right )}{8}-\frac {\RootOf \left (\textit {\_Z}^{8}+16\right )^{5} \ln \left (\frac {-\RootOf \left (\textit {\_Z}^{8}+16\right )^{8} x^{4}-4 \sqrt {x^{6}-1}\, \RootOf \left (\textit {\_Z}^{8}+16\right )^{6} x^{2}-4 \RootOf \left (\textit {\_Z}^{8}+16\right )^{4} x^{6}+8 \left (x^{6}-1\right )^{\frac {1}{4}} \RootOf \left (\textit {\_Z}^{8}+16\right )^{5} x^{3}-4 \RootOf \left (\textit {\_Z}^{8}+16\right )^{4} x^{4}+16 \RootOf \left (\textit {\_Z}^{8}+16\right )^{3} \left (x^{6}-1\right )^{\frac {3}{4}} x -16 \RootOf \left (\textit {\_Z}^{8}+16\right )^{2} \sqrt {x^{6}-1}\, x^{2}-16 x^{6}+4 \RootOf \left (\textit {\_Z}^{8}+16\right )^{4}+16}{\RootOf \left (\textit {\_Z}^{8}+16\right )^{4} x^{4}-4 x^{6}+4}\right )}{16}+\frac {\RootOf \left (\textit {\_Z}^{8}+16\right ) \ln \left (\frac {-\RootOf \left (\textit {\_Z}^{8}+16\right )^{8} x^{4}+4 \sqrt {x^{6}-1}\, \RootOf \left (\textit {\_Z}^{8}+16\right )^{6} x^{2}-4 \RootOf \left (\textit {\_Z}^{8}+16\right )^{4} x^{6}-8 \left (x^{6}-1\right )^{\frac {1}{4}} \RootOf \left (\textit {\_Z}^{8}+16\right )^{5} x^{3}+4 \RootOf \left (\textit {\_Z}^{8}+16\right )^{4} x^{4}+16 \RootOf \left (\textit {\_Z}^{8}+16\right )^{3} \left (x^{6}-1\right )^{\frac {3}{4}} x -16 \RootOf \left (\textit {\_Z}^{8}+16\right )^{2} \sqrt {x^{6}-1}\, x^{2}+16 x^{6}+4 \RootOf \left (\textit {\_Z}^{8}+16\right )^{4}-16}{\RootOf \left (\textit {\_Z}^{8}+16\right )^{4} x^{4}-4 x^{6}+4}\right )}{4}-\frac {\RootOf \left (\textit {\_Z}^{8}+16\right )^{7} \ln \left (-\frac {-\RootOf \left (\textit {\_Z}^{8}+16\right )^{10} x^{4}+4 \RootOf \left (\textit {\_Z}^{8}+16\right )^{6} x^{6}+4 x^{4} \RootOf \left (\textit {\_Z}^{8}+16\right )^{6}-16 \RootOf \left (\textit {\_Z}^{8}+16\right )^{4} \sqrt {x^{6}-1}\, x^{2}-16 \RootOf \left (\textit {\_Z}^{8}+16\right )^{2} x^{6}+32 \RootOf \left (\textit {\_Z}^{8}+16\right )^{3} \left (x^{6}-1\right )^{\frac {3}{4}} x -4 \RootOf \left (\textit {\_Z}^{8}+16\right )^{6}-64 \RootOf \left (\textit {\_Z}^{8}+16\right ) \left (x^{6}-1\right )^{\frac {1}{4}} x^{3}+64 x^{2} \sqrt {x^{6}-1}+16 \RootOf \left (\textit {\_Z}^{8}+16\right )^{2}}{\RootOf \left (\textit {\_Z}^{8}+16\right )^{4} x^{4}+4 x^{6}-4}\right )}{32}\)	\(684\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((x^6+2)*(x^6+x^4-1)/(x^6-1)^(1/4)/(x^12+x^8-2*x^6+1),x,method=_RETURNVERBOSE)

[Out]

-1/8*RootOf(_Z^8+16)^3*ln(-(-RootOf(_Z^8+16)^10*x^4+4*RootOf(_Z^8+16)^6*x^6-4*x^4*RootOf(_Z^8+16)^6+16*RootOf(
_Z^8+16)^4*(x^6-1)^(1/2)*x^2+16*RootOf(_Z^8+16)^2*x^6+32*RootOf(_Z^8+16)^3*(x^6-1)^(3/4)*x-4*RootOf(_Z^8+16)^6
+64*RootOf(_Z^8+16)*(x^6-1)^(1/4)*x^3+64*x^2*(x^6-1)^(1/2)-16*RootOf(_Z^8+16)^2)/(RootOf(_Z^8+16)^4*x^4+4*x^6-
4))-1/16*RootOf(_Z^8+16)^5*ln((-RootOf(_Z^8+16)^8*x^4-4*(x^6-1)^(1/2)*RootOf(_Z^8+16)^6*x^2-4*RootOf(_Z^8+16)^
4*x^6+8*(x^6-1)^(1/4)*RootOf(_Z^8+16)^5*x^3-4*RootOf(_Z^8+16)^4*x^4+16*RootOf(_Z^8+16)^3*(x^6-1)^(3/4)*x-16*Ro
otOf(_Z^8+16)^2*(x^6-1)^(1/2)*x^2-16*x^6+4*RootOf(_Z^8+16)^4+16)/(RootOf(_Z^8+16)^4*x^4-4*x^6+4))+1/4*RootOf(_
Z^8+16)*ln((-RootOf(_Z^8+16)^8*x^4+4*(x^6-1)^(1/2)*RootOf(_Z^8+16)^6*x^2-4*RootOf(_Z^8+16)^4*x^6-8*(x^6-1)^(1/
4)*RootOf(_Z^8+16)^5*x^3+4*RootOf(_Z^8+16)^4*x^4+16*RootOf(_Z^8+16)^3*(x^6-1)^(3/4)*x-16*RootOf(_Z^8+16)^2*(x^
6-1)^(1/2)*x^2+16*x^6+4*RootOf(_Z^8+16)^4-16)/(RootOf(_Z^8+16)^4*x^4-4*x^6+4))-1/32*RootOf(_Z^8+16)^7*ln(-(-Ro
otOf(_Z^8+16)^10*x^4+4*RootOf(_Z^8+16)^6*x^6+4*x^4*RootOf(_Z^8+16)^6-16*RootOf(_Z^8+16)^4*(x^6-1)^(1/2)*x^2-16
*RootOf(_Z^8+16)^2*x^6+32*RootOf(_Z^8+16)^3*(x^6-1)^(3/4)*x-4*RootOf(_Z^8+16)^6-64*RootOf(_Z^8+16)*(x^6-1)^(1/
4)*x^3+64*x^2*(x^6-1)^(1/2)+16*RootOf(_Z^8+16)^2)/(RootOf(_Z^8+16)^4*x^4+4*x^6-4))

________________________________________________________________________________________

maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {{\left (x^{6} + x^{4} - 1\right )} {\left (x^{6} + 2\right )}}{{\left (x^{12} + x^{8} - 2 \, x^{6} + 1\right )} {\left (x^{6} - 1\right )}^{\frac {1}{4}}}\,{d x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((x^6+2)*(x^6+x^4-1)/(x^6-1)^(1/4)/(x^12+x^8-2*x^6+1),x, algorithm="maxima")

[Out]

integrate((x^6 + x^4 - 1)*(x^6 + 2)/((x^12 + x^8 - 2*x^6 + 1)*(x^6 - 1)^(1/4)), x)

________________________________________________________________________________________

mupad [F]  time = 0.00, size = -1, normalized size = -0.00 \begin {gather*} \int \frac {\left (x^6+2\right )\,\left (x^6+x^4-1\right )}{{\left (x^6-1\right )}^{1/4}\,\left (x^{12}+x^8-2\,x^6+1\right )} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((x^6 + 2)*(x^4 + x^6 - 1))/((x^6 - 1)^(1/4)*(x^8 - 2*x^6 + x^12 + 1)),x)

[Out]

int(((x^6 + 2)*(x^4 + x^6 - 1))/((x^6 - 1)^(1/4)*(x^8 - 2*x^6 + x^12 + 1)), x)

________________________________________________________________________________________

sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\left (x^{6} + 2\right ) \left (x^{6} + x^{4} - 1\right )}{\sqrt [4]{\left (x - 1\right ) \left (x + 1\right ) \left (x^{2} - x + 1\right ) \left (x^{2} + x + 1\right )} \left (x^{12} + x^{8} - 2 x^{6} + 1\right )}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((x**6+2)*(x**6+x**4-1)/(x**6-1)**(1/4)/(x**12+x**8-2*x**6+1),x)

[Out]

Integral((x**6 + 2)*(x**6 + x**4 - 1)/(((x - 1)*(x + 1)*(x**2 - x + 1)*(x**2 + x + 1))**(1/4)*(x**12 + x**8 -
2*x**6 + 1)), x)

________________________________________________________________________________________