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

Optimal. Leaf size=109 \[ -\sqrt {2} \tan ^{-1}\left (\frac {\sqrt {2} x \sqrt [4]{x^6-1}}{\sqrt {x^6-1}-x^2}\right )-\sqrt {2} \tanh ^{-1}\left (\frac {\sqrt {2} x \sqrt [4]{x^6-1}}{\sqrt {x^6-1}+x^2}\right )+\frac {2 \left (x^6-1\right )^{3/4} \left (3 x^6-7 x^4-3\right )}{21 x^7} \]

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Rubi [F]  time = 1.21, 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 (1-2 x^6+x^8+x^{12}\right )}{x^8 \sqrt [4]{-1+x^6} \left (-1+x^4+x^6\right )} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

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

[Out]

(-2*(-1 + x^6)^(3/4))/(3*x^3) + (2*(1 - x^6)^(1/4)*Hypergeometric2F1[-7/6, 1/4, -1/6, x^6])/(7*x^7*(-1 + x^6)^
(1/4)) - ((1 - x^6)^(1/4)*Hypergeometric2F1[-1/6, 1/4, 5/6, x^6])/(x*(-1 + x^6)^(1/4)) + (2*x*(1 - x^6)^(1/4)*
Hypergeometric2F1[1/6, 1/4, 7/6, x^6])/(-1 + x^6)^(1/4) + (x^5*(1 - x^6)^(1/4)*Hypergeometric2F1[1/4, 5/6, 11/
6, x^6])/(5*(-1 + x^6)^(1/4)) + 6*Defer[Int][1/((-1 + x^6)^(1/4)*(-1 + x^4 + x^6)), x] - 2*Defer[Int][x^4/((-1
+ x^6)^(1/4)*(-1 + x^4 + x^6)), x]

Rubi steps

\begin {align*} \int \frac {\left (2+x^6\right ) \left (1-2 x^6+x^8+x^{12}\right )}{x^8 \sqrt [4]{-1+x^6} \left (-1+x^4+x^6\right )} \, dx &=\int \left (\frac {2}{\sqrt [4]{-1+x^6}}-\frac {2}{x^8 \sqrt [4]{-1+x^6}}-\frac {2}{x^4 \sqrt [4]{-1+x^6}}+\frac {1}{x^2 \sqrt [4]{-1+x^6}}-\frac {x^2}{\sqrt [4]{-1+x^6}}+\frac {x^4}{\sqrt [4]{-1+x^6}}-\frac {2 \left (-3+x^4\right )}{\sqrt [4]{-1+x^6} \left (-1+x^4+x^6\right )}\right ) \, dx\\ &=2 \int \frac {1}{\sqrt [4]{-1+x^6}} \, dx-2 \int \frac {1}{x^8 \sqrt [4]{-1+x^6}} \, dx-2 \int \frac {1}{x^4 \sqrt [4]{-1+x^6}} \, dx-2 \int \frac {-3+x^4}{\sqrt [4]{-1+x^6} \left (-1+x^4+x^6\right )} \, dx+\int \frac {1}{x^2 \sqrt [4]{-1+x^6}} \, dx-\int \frac {x^2}{\sqrt [4]{-1+x^6}} \, dx+\int \frac {x^4}{\sqrt [4]{-1+x^6}} \, dx\\ &=-\left (\frac {1}{3} \operatorname {Subst}\left (\int \frac {1}{\sqrt [4]{-1+x^2}} \, dx,x,x^3\right )\right )-\frac {2}{3} \operatorname {Subst}\left (\int \frac {1}{x^2 \sqrt [4]{-1+x^2}} \, dx,x,x^3\right )-2 \int \left (-\frac {3}{\sqrt [4]{-1+x^6} \left (-1+x^4+x^6\right )}+\frac {x^4}{\sqrt [4]{-1+x^6} \left (-1+x^4+x^6\right )}\right ) \, dx+\frac {\sqrt [4]{1-x^6} \int \frac {1}{x^2 \sqrt [4]{1-x^6}} \, dx}{\sqrt [4]{-1+x^6}}+\frac {\sqrt [4]{1-x^6} \int \frac {x^4}{\sqrt [4]{1-x^6}} \, dx}{\sqrt [4]{-1+x^6}}+\frac {\left (2 \sqrt [4]{1-x^6}\right ) \int \frac {1}{\sqrt [4]{1-x^6}} \, dx}{\sqrt [4]{-1+x^6}}-\frac {\left (2 \sqrt [4]{1-x^6}\right ) \int \frac {1}{x^8 \sqrt [4]{1-x^6}} \, dx}{\sqrt [4]{-1+x^6}}\\ &=-\frac {2 \left (-1+x^6\right )^{3/4}}{3 x^3}+\frac {2 \sqrt [4]{1-x^6} \, _2F_1\left (-\frac {7}{6},\frac {1}{4};-\frac {1}{6};x^6\right )}{7 x^7 \sqrt [4]{-1+x^6}}-\frac {\sqrt [4]{1-x^6} \, _2F_1\left (-\frac {1}{6},\frac {1}{4};\frac {5}{6};x^6\right )}{x \sqrt [4]{-1+x^6}}+\frac {2 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}}+\frac {x^5 \sqrt [4]{1-x^6} \, _2F_1\left (\frac {1}{4},\frac {5}{6};\frac {11}{6};x^6\right )}{5 \sqrt [4]{-1+x^6}}+\frac {1}{3} \operatorname {Subst}\left (\int \frac {1}{\sqrt [4]{-1+x^2}} \, dx,x,x^3\right )-2 \int \frac {x^4}{\sqrt [4]{-1+x^6} \left (-1+x^4+x^6\right )} \, dx+6 \int \frac {1}{\sqrt [4]{-1+x^6} \left (-1+x^4+x^6\right )} \, dx-\frac {\left (2 \sqrt {x^6}\right ) \operatorname {Subst}\left (\int \frac {x^2}{\sqrt {1+x^4}} \, dx,x,\sqrt [4]{-1+x^6}\right )}{3 x^3}\\ &=-\frac {2 \left (-1+x^6\right )^{3/4}}{3 x^3}+\frac {2 \sqrt [4]{1-x^6} \, _2F_1\left (-\frac {7}{6},\frac {1}{4};-\frac {1}{6};x^6\right )}{7 x^7 \sqrt [4]{-1+x^6}}-\frac {\sqrt [4]{1-x^6} \, _2F_1\left (-\frac {1}{6},\frac {1}{4};\frac {5}{6};x^6\right )}{x \sqrt [4]{-1+x^6}}+\frac {2 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}}+\frac {x^5 \sqrt [4]{1-x^6} \, _2F_1\left (\frac {1}{4},\frac {5}{6};\frac {11}{6};x^6\right )}{5 \sqrt [4]{-1+x^6}}-2 \int \frac {x^4}{\sqrt [4]{-1+x^6} \left (-1+x^4+x^6\right )} \, dx+6 \int \frac {1}{\sqrt [4]{-1+x^6} \left (-1+x^4+x^6\right )} \, dx-\frac {\left (2 \sqrt {x^6}\right ) \operatorname {Subst}\left (\int \frac {1}{\sqrt {1+x^4}} \, dx,x,\sqrt [4]{-1+x^6}\right )}{3 x^3}+\frac {\left (2 \sqrt {x^6}\right ) \operatorname {Subst}\left (\int \frac {x^2}{\sqrt {1+x^4}} \, dx,x,\sqrt [4]{-1+x^6}\right )}{3 x^3}+\frac {\left (2 \sqrt {x^6}\right ) \operatorname {Subst}\left (\int \frac {1-x^2}{\sqrt {1+x^4}} \, dx,x,\sqrt [4]{-1+x^6}\right )}{3 x^3}\\ &=-\frac {2 \left (-1+x^6\right )^{3/4}}{3 x^3}-\frac {2 x^3 \sqrt [4]{-1+x^6}}{3 \left (1+\sqrt {-1+x^6}\right )}+\frac {2 \sqrt {\frac {x^6}{\left (1+\sqrt {-1+x^6}\right )^2}} \left (1+\sqrt {-1+x^6}\right ) E\left (2 \tan ^{-1}\left (\sqrt [4]{-1+x^6}\right )|\frac {1}{2}\right )}{3 x^3}-\frac {\sqrt {\frac {x^6}{\left (1+\sqrt {-1+x^6}\right )^2}} \left (1+\sqrt {-1+x^6}\right ) F\left (2 \tan ^{-1}\left (\sqrt [4]{-1+x^6}\right )|\frac {1}{2}\right )}{3 x^3}+\frac {2 \sqrt [4]{1-x^6} \, _2F_1\left (-\frac {7}{6},\frac {1}{4};-\frac {1}{6};x^6\right )}{7 x^7 \sqrt [4]{-1+x^6}}-\frac {\sqrt [4]{1-x^6} \, _2F_1\left (-\frac {1}{6},\frac {1}{4};\frac {5}{6};x^6\right )}{x \sqrt [4]{-1+x^6}}+\frac {2 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}}+\frac {x^5 \sqrt [4]{1-x^6} \, _2F_1\left (\frac {1}{4},\frac {5}{6};\frac {11}{6};x^6\right )}{5 \sqrt [4]{-1+x^6}}-2 \int \frac {x^4}{\sqrt [4]{-1+x^6} \left (-1+x^4+x^6\right )} \, dx+6 \int \frac {1}{\sqrt [4]{-1+x^6} \left (-1+x^4+x^6\right )} \, dx+\frac {\left (2 \sqrt {x^6}\right ) \operatorname {Subst}\left (\int \frac {1}{\sqrt {1+x^4}} \, dx,x,\sqrt [4]{-1+x^6}\right )}{3 x^3}-\frac {\left (2 \sqrt {x^6}\right ) \operatorname {Subst}\left (\int \frac {1-x^2}{\sqrt {1+x^4}} \, dx,x,\sqrt [4]{-1+x^6}\right )}{3 x^3}\\ &=-\frac {2 \left (-1+x^6\right )^{3/4}}{3 x^3}+\frac {2 \sqrt [4]{1-x^6} \, _2F_1\left (-\frac {7}{6},\frac {1}{4};-\frac {1}{6};x^6\right )}{7 x^7 \sqrt [4]{-1+x^6}}-\frac {\sqrt [4]{1-x^6} \, _2F_1\left (-\frac {1}{6},\frac {1}{4};\frac {5}{6};x^6\right )}{x \sqrt [4]{-1+x^6}}+\frac {2 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}}+\frac {x^5 \sqrt [4]{1-x^6} \, _2F_1\left (\frac {1}{4},\frac {5}{6};\frac {11}{6};x^6\right )}{5 \sqrt [4]{-1+x^6}}-2 \int \frac {x^4}{\sqrt [4]{-1+x^6} \left (-1+x^4+x^6\right )} \, dx+6 \int \frac {1}{\sqrt [4]{-1+x^6} \left (-1+x^4+x^6\right )} \, dx\\ \end {align*}

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

Verification is not applicable to the result.

[In]

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

[Out]

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

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IntegrateAlgebraic [A]  time = 15.34, size = 109, normalized size = 1.00 \begin {gather*} \frac {2 \left (-1+x^6\right )^{3/4} \left (-3-7 x^4+3 x^6\right )}{21 x^7}-\sqrt {2} \tan ^{-1}\left (\frac {\sqrt {2} x \sqrt [4]{-1+x^6}}{-x^2+\sqrt {-1+x^6}}\right )-\sqrt {2} \tanh ^{-1}\left (\frac {\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 - 2*x^6 + x^8 + x^12))/(x^8*(-1 + x^6)^(1/4)*(-1 + x^4 + x^6)),x]

[Out]

(2*(-1 + x^6)^(3/4)*(-3 - 7*x^4 + 3*x^6))/(21*x^7) - Sqrt[2]*ArcTan[(Sqrt[2]*x*(-1 + x^6)^(1/4))/(-x^2 + Sqrt[
-1 + x^6])] - Sqrt[2]*ArcTanh[(Sqrt[2]*x*(-1 + x^6)^(1/4))/(x^2 + Sqrt[-1 + x^6])]

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maple [C]  time = 75.18, size = 231, normalized size = 2.12

method	result	size
trager	\(\frac {2 \left (x^{6}-1\right )^{\frac {3}{4}} \left (3 x^{6}-7 x^{4}-3\right )}{21 x^{7}}-\RootOf \left (\textit {\_Z}^{4}+1\right ) \ln \left (-\frac {2 \RootOf \left (\textit {\_Z}^{4}+1\right )^{3} \sqrt {x^{6}-1}\, x^{2}-\RootOf \left (\textit {\_Z}^{4}+1\right ) x^{6}+2 \RootOf \left (\textit {\_Z}^{4}+1\right )^{2} \left (x^{6}-1\right )^{\frac {1}{4}} x^{3}+\RootOf \left (\textit {\_Z}^{4}+1\right ) x^{4}-2 \left (x^{6}-1\right )^{\frac {3}{4}} x +\RootOf \left (\textit {\_Z}^{4}+1\right )}{x^{6}+x^{4}-1}\right )+\RootOf \left (\textit {\_Z}^{4}+1\right )^{3} \ln \left (-\frac {\RootOf \left (\textit {\_Z}^{4}+1\right )^{3} x^{6}-\RootOf \left (\textit {\_Z}^{4}+1\right )^{3} x^{4}-2 \RootOf \left (\textit {\_Z}^{4}+1\right )^{2} \left (x^{6}-1\right )^{\frac {1}{4}} x^{3}-2 \RootOf \left (\textit {\_Z}^{4}+1\right ) \sqrt {x^{6}-1}\, x^{2}-2 \left (x^{6}-1\right )^{\frac {3}{4}} x -\RootOf \left (\textit {\_Z}^{4}+1\right )^{3}}{x^{6}+x^{4}-1}\right )\)	\(231\)
risch	\(\frac {\frac {2}{7} x^{12}-\frac {4}{7} x^{6}+\frac {2}{7}-\frac {2}{3} x^{10}+\frac {2}{3} x^{4}}{x^{7} \left (x^{6}-1\right )^{\frac {1}{4}}}+\RootOf \left (\textit {\_Z}^{4}+1\right )^{3} \ln \left (\frac {\RootOf \left (\textit {\_Z}^{4}+1\right )^{3} x^{6}-\RootOf \left (\textit {\_Z}^{4}+1\right )^{3} x^{4}-2 \RootOf \left (\textit {\_Z}^{4}+1\right )^{2} \left (x^{6}-1\right )^{\frac {1}{4}} x^{3}-2 \RootOf \left (\textit {\_Z}^{4}+1\right ) \sqrt {x^{6}-1}\, x^{2}-2 \left (x^{6}-1\right )^{\frac {3}{4}} x -\RootOf \left (\textit {\_Z}^{4}+1\right )^{3}}{x^{6}+x^{4}-1}\right )+\RootOf \left (\textit {\_Z}^{4}+1\right ) \ln \left (-\frac {2 \RootOf \left (\textit {\_Z}^{4}+1\right )^{3} \sqrt {x^{6}-1}\, x^{2}-\RootOf \left (\textit {\_Z}^{4}+1\right ) x^{6}-2 \RootOf \left (\textit {\_Z}^{4}+1\right )^{2} \left (x^{6}-1\right )^{\frac {1}{4}} x^{3}+\RootOf \left (\textit {\_Z}^{4}+1\right ) x^{4}+2 \left (x^{6}-1\right )^{\frac {3}{4}} x +\RootOf \left (\textit {\_Z}^{4}+1\right )}{x^{6}+x^{4}-1}\right )\)	\(239\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2/21*(x^6-1)^(3/4)*(3*x^6-7*x^4-3)/x^7-RootOf(_Z^4+1)*ln(-(2*RootOf(_Z^4+1)^3*(x^6-1)^(1/2)*x^2-RootOf(_Z^4+1)
*x^6+2*RootOf(_Z^4+1)^2*(x^6-1)^(1/4)*x^3+RootOf(_Z^4+1)*x^4-2*(x^6-1)^(3/4)*x+RootOf(_Z^4+1))/(x^6+x^4-1))+Ro
otOf(_Z^4+1)^3*ln(-(RootOf(_Z^4+1)^3*x^6-RootOf(_Z^4+1)^3*x^4-2*RootOf(_Z^4+1)^2*(x^6-1)^(1/4)*x^3-2*RootOf(_Z
^4+1)*(x^6-1)^(1/2)*x^2-2*(x^6-1)^(3/4)*x-RootOf(_Z^4+1)^3)/(x^6+x^4-1))

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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mupad [F]  time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \int \frac {\left (x^6+2\right )\,\left (x^{12}+x^8-2\,x^6+1\right )}{x^8\,{\left (x^6-1\right )}^{1/4}\,\left (x^6+x^4-1\right )} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

[Out]

Timed out

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