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

Optimal. Leaf size=241 \[ \frac {1}{2} \sqrt {\frac {1}{2} \left (2-\sqrt {2}\right )} \tan ^{-1}\left (\frac {\sqrt {2-\sqrt {2}} x \sqrt [4]{x^6+1}}{\sqrt {x^6+1}-x^2}\right )-\frac {1}{2} \sqrt {\frac {1}{2} \left (2+\sqrt {2}\right )} \tan ^{-1}\left (\frac {\sqrt {2+\sqrt {2}} x \sqrt [4]{x^6+1}}{\sqrt {x^6+1}-x^2}\right )+\frac {1}{2} \sqrt {\frac {1}{2} \left (2-\sqrt {2}\right )} \tanh ^{-1}\left (\frac {\sqrt {2-\sqrt {2}} x \sqrt [4]{x^6+1}}{\sqrt {x^6+1}+x^2}\right )-\frac {1}{2} \sqrt {\frac {1}{2} \left (2+\sqrt {2}\right )} \tanh ^{-1}\left (\frac {\sqrt {2+\sqrt {2}} x \sqrt [4]{x^6+1}}{\sqrt {x^6+1}+x^2}\right ) \]

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Rubi [F]  time = 1.40, 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*Hypergeometric2F1[1/6, 1/4, 7/6, -x^6] - 3*Defer[Int][1/((1 + x^6)^(1/4)*(1 + 2*x^6 + x^8 + x^12)), x] + 2*D
efer[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\\ &=x \, _2F_1\left (\frac {1}{6},\frac {1}{4};\frac {7}{6};-x^6\right )-\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\\ &=x \, _2F_1\left (\frac {1}{6},\frac {1}{4};\frac {7}{6};-x^6\right )+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*}

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Mathematica [F]  time = 0.35, 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]

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IntegrateAlgebraic [A]  time = 16.14, size = 241, normalized size = 1.00 \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]

(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 + Sq
rt[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]*Arc
Tanh[(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

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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

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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)

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maple [C]  time = 31.68, size = 679, normalized size = 2.82

method	result	size
trager	\(-\frac {\RootOf \left (\textit {\_Z}^{8}+16\right )^{5} \ln \left (\frac {\RootOf \left (\textit {\_Z}^{8}+16\right )^{8} x^{4}-4 \RootOf \left (\textit {\_Z}^{8}+16\right )^{6} \sqrt {x^{6}+1}\, x^{2}+4 \RootOf \left (\textit {\_Z}^{8}+16\right )^{4} x^{6}-8 \RootOf \left (\textit {\_Z}^{8}+16\right )^{5} \left (x^{6}+1\right )^{\frac {1}{4}} 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 \RootOf \left (\textit {\_Z}^{8}+16\right )^{6} \sqrt {x^{6}+1}\, x^{2}+4 \RootOf \left (\textit {\_Z}^{8}+16\right )^{4} x^{6}+8 \RootOf \left (\textit {\_Z}^{8}+16\right )^{5} \left (x^{6}+1\right )^{\frac {1}{4}} 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 )^{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 )^{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}\)	\(679\)

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]

-1/16*RootOf(_Z^8+16)^5*ln((RootOf(_Z^8+16)^8*x^4-4*RootOf(_Z^8+16)^6*(x^6+1)^(1/2)*x^2+4*RootOf(_Z^8+16)^4*x^
6-8*RootOf(_Z^8+16)^5*(x^6+1)^(1/4)*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))-1/4*RootOf(_Z^8+
16)*ln((RootOf(_Z^8+16)^8*x^4+4*RootOf(_Z^8+16)^6*(x^6+1)^(1/2)*x^2+4*RootOf(_Z^8+16)^4*x^6+8*RootOf(_Z^8+16)^
5*(x^6+1)^(1/4)*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))-1/8*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^3-6
4*x^2*(x^6+1)^(1/2)+16*RootOf(_Z^8+16)^2)/(RootOf(_Z^8+16)^4*x^4+4*x^6+4))-1/32*RootOf(_Z^8+16)^7*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*Root
Of(_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))

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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)

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

[Out]

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

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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^{2} + 1\right ) \left (x^{4} - x^{2} + 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**2 + 1)*(x**4 - x**2 + 1))**(1/4)*(x**12 + x**8 + 2*x**6 + 1)), x)

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