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

Optimal. Leaf size=95 \[ \frac {2 \left (x^6+1\right )^{3/4}}{3 x^3}+\sqrt {2} \tan ^{-1}\left (\frac {\sqrt {2} x \sqrt [4]{x^6+1}}{\sqrt {x^6+1}-x^2}\right )+\sqrt {2} \tanh ^{-1}\left (\frac {\sqrt {2} x \sqrt [4]{x^6+1}}{\sqrt {x^6+1}+x^2}\right ) \]

________________________________________________________________________________________

Rubi [F]  time = 1.03, 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 )}{x^4 \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 - x^4 + x^6))/(x^4*(1 + x^6)^(1/4)*(1 + x^4 + x^6)),x]

[Out]

(2*(1 + x^6)^(3/4))/(3*x^3) - 2*x*Hypergeometric2F1[1/6, 1/4, 7/6, -x^6] + 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-x^4+x^6\right )}{x^4 \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^4 \sqrt [4]{1+x^6}}+\frac {x^2}{\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\\ &=-\left (2 \int \frac {1}{\sqrt [4]{1+x^6}} \, dx\right )-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 {x^2}{\sqrt [4]{1+x^6}} \, dx\\ &=-2 x \, _2F_1\left (\frac {1}{6},\frac {1}{4};\frac {7}{6};-x^6\right )+\frac {1}{3} \operatorname {Subst}\left (\int \frac {1}{\sqrt [4]{1+x^2}} \, dx,x,x^3\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 {2 x^3}{3 \sqrt [4]{1+x^6}}+\frac {2 \left (1+x^6\right )^{3/4}}{3 x^3}-2 x \, _2F_1\left (\frac {1}{6},\frac {1}{4};\frac {7}{6};-x^6\right )-\frac {1}{3} \operatorname {Subst}\left (\int \frac {1}{\left (1+x^2\right )^{5/4}} \, dx,x,x^3\right )-\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 {2 \left (1+x^6\right )^{3/4}}{3 x^3}-\frac {2}{3} E\left (\left .\frac {1}{2} \tan ^{-1}\left (x^3\right )\right |2\right )-2 x \, _2F_1\left (\frac {1}{6},\frac {1}{4};\frac {7}{6};-x^6\right )+\frac {1}{3} \operatorname {Subst}\left (\int \frac {1}{\left (1+x^2\right )^{5/4}} \, 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 {2 \left (1+x^6\right )^{3/4}}{3 x^3}-2 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+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*}

________________________________________________________________________________________

Mathematica [F]  time = 0.27, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\left (-2+x^6\right ) \left (1-x^4+x^6\right )}{x^4 \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 - x^4 + x^6))/(x^4*(1 + x^6)^(1/4)*(1 + x^4 + x^6)),x]

[Out]

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

________________________________________________________________________________________

IntegrateAlgebraic [A]  time = 7.03, size = 95, normalized size = 1.00 \begin {gather*} \frac {2 \left (1+x^6\right )^{3/4}}{3 x^3}+\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 - x^4 + x^6))/(x^4*(1 + x^6)^(1/4)*(1 + x^4 + x^6)),x]

[Out]

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

________________________________________________________________________________________

fricas [B]  time = 173.50, size = 702, normalized size = 7.39 \begin {gather*} \frac {12 \, \sqrt {2} x^{3} \arctan \left (\frac {x^{12} + 2 \, x^{10} + x^{8} + 2 \, x^{6} + 2 \, x^{4} + 2 \, \sqrt {2} {\left (x^{7} - 3 \, x^{5} + x\right )} {\left (x^{6} + 1\right )}^{\frac {3}{4}} + 2 \, \sqrt {2} {\left (3 \, x^{9} - x^{7} + 3 \, x^{3}\right )} {\left (x^{6} + 1\right )}^{\frac {1}{4}} + 4 \, {\left (x^{8} + x^{6} + x^{2}\right )} \sqrt {x^{6} + 1} + {\left (16 \, {\left (x^{6} + 1\right )}^{\frac {3}{4}} x^{5} + 2 \, \sqrt {2} {\left (x^{8} - 3 \, x^{6} + x^{2}\right )} \sqrt {x^{6} + 1} + \sqrt {2} {\left (x^{12} - 8 \, x^{10} - x^{8} + 2 \, x^{6} - 8 \, x^{4} + 1\right )} + 4 \, {\left (x^{9} + x^{7} + x^{3}\right )} {\left (x^{6} + 1\right )}^{\frac {1}{4}}\right )} \sqrt {\frac {x^{6} + x^{4} + 2 \, \sqrt {2} {\left (x^{6} + 1\right )}^{\frac {1}{4}} x^{3} + 4 \, \sqrt {x^{6} + 1} x^{2} + 2 \, \sqrt {2} {\left (x^{6} + 1\right )}^{\frac {3}{4}} x + 1}{x^{6} + x^{4} + 1}} + 1}{x^{12} - 14 \, x^{10} + x^{8} + 2 \, x^{6} - 14 \, x^{4} + 1}\right ) - 12 \, \sqrt {2} x^{3} \arctan \left (\frac {x^{12} + 2 \, x^{10} + x^{8} + 2 \, x^{6} + 2 \, x^{4} - 2 \, \sqrt {2} {\left (x^{7} - 3 \, x^{5} + x\right )} {\left (x^{6} + 1\right )}^{\frac {3}{4}} - 2 \, \sqrt {2} {\left (3 \, x^{9} - x^{7} + 3 \, x^{3}\right )} {\left (x^{6} + 1\right )}^{\frac {1}{4}} + 4 \, {\left (x^{8} + x^{6} + x^{2}\right )} \sqrt {x^{6} + 1} + {\left (16 \, {\left (x^{6} + 1\right )}^{\frac {3}{4}} x^{5} - 2 \, \sqrt {2} {\left (x^{8} - 3 \, x^{6} + x^{2}\right )} \sqrt {x^{6} + 1} - \sqrt {2} {\left (x^{12} - 8 \, x^{10} - x^{8} + 2 \, x^{6} - 8 \, x^{4} + 1\right )} + 4 \, {\left (x^{9} + x^{7} + x^{3}\right )} {\left (x^{6} + 1\right )}^{\frac {1}{4}}\right )} \sqrt {\frac {x^{6} + x^{4} - 2 \, \sqrt {2} {\left (x^{6} + 1\right )}^{\frac {1}{4}} x^{3} + 4 \, \sqrt {x^{6} + 1} x^{2} - 2 \, \sqrt {2} {\left (x^{6} + 1\right )}^{\frac {3}{4}} x + 1}{x^{6} + x^{4} + 1}} + 1}{x^{12} - 14 \, x^{10} + x^{8} + 2 \, x^{6} - 14 \, x^{4} + 1}\right ) + 3 \, \sqrt {2} x^{3} \log \left (\frac {4 \, {\left (x^{6} + x^{4} + 2 \, \sqrt {2} {\left (x^{6} + 1\right )}^{\frac {1}{4}} x^{3} + 4 \, \sqrt {x^{6} + 1} x^{2} + 2 \, \sqrt {2} {\left (x^{6} + 1\right )}^{\frac {3}{4}} x + 1\right )}}{x^{6} + x^{4} + 1}\right ) - 3 \, \sqrt {2} x^{3} \log \left (\frac {4 \, {\left (x^{6} + x^{4} - 2 \, \sqrt {2} {\left (x^{6} + 1\right )}^{\frac {1}{4}} x^{3} + 4 \, \sqrt {x^{6} + 1} x^{2} - 2 \, \sqrt {2} {\left (x^{6} + 1\right )}^{\frac {3}{4}} x + 1\right )}}{x^{6} + x^{4} + 1}\right ) + 8 \, {\left (x^{6} + 1\right )}^{\frac {3}{4}}}{12 \, x^{3}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/12*(12*sqrt(2)*x^3*arctan((x^12 + 2*x^10 + x^8 + 2*x^6 + 2*x^4 + 2*sqrt(2)*(x^7 - 3*x^5 + x)*(x^6 + 1)^(3/4)
+ 2*sqrt(2)*(3*x^9 - x^7 + 3*x^3)*(x^6 + 1)^(1/4) + 4*(x^8 + x^6 + x^2)*sqrt(x^6 + 1) + (16*(x^6 + 1)^(3/4)*x
^5 + 2*sqrt(2)*(x^8 - 3*x^6 + x^2)*sqrt(x^6 + 1) + sqrt(2)*(x^12 - 8*x^10 - x^8 + 2*x^6 - 8*x^4 + 1) + 4*(x^9
+ x^7 + x^3)*(x^6 + 1)^(1/4))*sqrt((x^6 + x^4 + 2*sqrt(2)*(x^6 + 1)^(1/4)*x^3 + 4*sqrt(x^6 + 1)*x^2 + 2*sqrt(2
)*(x^6 + 1)^(3/4)*x + 1)/(x^6 + x^4 + 1)) + 1)/(x^12 - 14*x^10 + x^8 + 2*x^6 - 14*x^4 + 1)) - 12*sqrt(2)*x^3*a
rctan((x^12 + 2*x^10 + x^8 + 2*x^6 + 2*x^4 - 2*sqrt(2)*(x^7 - 3*x^5 + x)*(x^6 + 1)^(3/4) - 2*sqrt(2)*(3*x^9 -
x^7 + 3*x^3)*(x^6 + 1)^(1/4) + 4*(x^8 + x^6 + x^2)*sqrt(x^6 + 1) + (16*(x^6 + 1)^(3/4)*x^5 - 2*sqrt(2)*(x^8 -
3*x^6 + x^2)*sqrt(x^6 + 1) - sqrt(2)*(x^12 - 8*x^10 - x^8 + 2*x^6 - 8*x^4 + 1) + 4*(x^9 + x^7 + x^3)*(x^6 + 1)
^(1/4))*sqrt((x^6 + x^4 - 2*sqrt(2)*(x^6 + 1)^(1/4)*x^3 + 4*sqrt(x^6 + 1)*x^2 - 2*sqrt(2)*(x^6 + 1)^(3/4)*x +
1)/(x^6 + x^4 + 1)) + 1)/(x^12 - 14*x^10 + x^8 + 2*x^6 - 14*x^4 + 1)) + 3*sqrt(2)*x^3*log(4*(x^6 + x^4 + 2*sqr
t(2)*(x^6 + 1)^(1/4)*x^3 + 4*sqrt(x^6 + 1)*x^2 + 2*sqrt(2)*(x^6 + 1)^(3/4)*x + 1)/(x^6 + x^4 + 1)) - 3*sqrt(2)
*x^3*log(4*(x^6 + x^4 - 2*sqrt(2)*(x^6 + 1)^(1/4)*x^3 + 4*sqrt(x^6 + 1)*x^2 - 2*sqrt(2)*(x^6 + 1)^(3/4)*x + 1)
/(x^6 + x^4 + 1)) + 8*(x^6 + 1)^(3/4))/x^3

________________________________________________________________________________________

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^{6} + x^{4} + 1\right )} {\left (x^{6} + 1\right )}^{\frac {1}{4}} x^{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^4/(x^6+1)^(1/4)/(x^6+x^4+1),x, algorithm="giac")

[Out]

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

________________________________________________________________________________________

maple [C]  time = 31.98, size = 216, normalized size = 2.27

method	result	size
trager	\(\frac {2 \left (x^{6}+1\right )^{\frac {3}{4}}}{3 x^{3}}+\RootOf \left (\textit {\_Z}^{4}+1\right ) \ln \left (-\frac {-2 \sqrt {x^{6}+1}\, \RootOf \left (\textit {\_Z}^{4}+1\right )^{3} x^{2}+\RootOf \left (\textit {\_Z}^{4}+1\right ) x^{6}-2 \left (x^{6}+1\right )^{\frac {1}{4}} \RootOf \left (\textit {\_Z}^{4}+1\right )^{2} x^{3}-\RootOf \left (\textit {\_Z}^{4}+1\right ) x^{4}+2 \left (x^{6}+1\right )^{\frac {3}{4}} x +\RootOf \left (\textit {\_Z}^{4}+1\right )}{x^{6}+x^{4}+1}\right )+\RootOf \left (\textit {\_Z}^{4}+1\right )^{3} \ln \left (-\frac {\RootOf \left (\textit {\_Z}^{4}+1\right )^{3} x^{6}-\RootOf \left (\textit {\_Z}^{4}+1\right )^{3} x^{4}+2 \left (x^{6}+1\right )^{\frac {1}{4}} \RootOf \left (\textit {\_Z}^{4}+1\right )^{2} x^{3}-2 \sqrt {x^{6}+1}\, \RootOf \left (\textit {\_Z}^{4}+1\right ) x^{2}+2 \left (x^{6}+1\right )^{\frac {3}{4}} x +\RootOf \left (\textit {\_Z}^{4}+1\right )^{3}}{x^{6}+x^{4}+1}\right )\)	\(216\)
risch	\(\frac {2 \left (x^{6}+1\right )^{\frac {3}{4}}}{3 x^{3}}+\RootOf \left (\textit {\_Z}^{4}+1\right )^{3} \ln \left (\frac {\RootOf \left (\textit {\_Z}^{4}+1\right )^{3} x^{6}-\RootOf \left (\textit {\_Z}^{4}+1\right )^{3} x^{4}+2 \left (x^{6}+1\right )^{\frac {1}{4}} \RootOf \left (\textit {\_Z}^{4}+1\right )^{2} x^{3}-2 \sqrt {x^{6}+1}\, \RootOf \left (\textit {\_Z}^{4}+1\right ) x^{2}+2 \left (x^{6}+1\right )^{\frac {3}{4}} x +\RootOf \left (\textit {\_Z}^{4}+1\right )^{3}}{x^{6}+x^{4}+1}\right )-\RootOf \left (\textit {\_Z}^{4}+1\right ) \ln \left (\frac {2 \sqrt {x^{6}+1}\, \RootOf \left (\textit {\_Z}^{4}+1\right )^{3} x^{2}-\RootOf \left (\textit {\_Z}^{4}+1\right ) x^{6}-2 \left (x^{6}+1\right )^{\frac {1}{4}} \RootOf \left (\textit {\_Z}^{4}+1\right )^{2} x^{3}+\RootOf \left (\textit {\_Z}^{4}+1\right ) x^{4}+2 \left (x^{6}+1\right )^{\frac {3}{4}} x -\RootOf \left (\textit {\_Z}^{4}+1\right )}{x^{6}+x^{4}+1}\right )\)	\(217\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2/3*(x^6+1)^(3/4)/x^3+RootOf(_Z^4+1)*ln(-(-2*(x^6+1)^(1/2)*RootOf(_Z^4+1)^3*x^2+RootOf(_Z^4+1)*x^6-2*(x^6+1)^(
1/4)*RootOf(_Z^4+1)^2*x^3-RootOf(_Z^4+1)*x^4+2*(x^6+1)^(3/4)*x+RootOf(_Z^4+1))/(x^6+x^4+1))+RootOf(_Z^4+1)^3*l
n(-(RootOf(_Z^4+1)^3*x^6-RootOf(_Z^4+1)^3*x^4+2*(x^6+1)^(1/4)*RootOf(_Z^4+1)^2*x^3-2*(x^6+1)^(1/2)*RootOf(_Z^4
+1)*x^2+2*(x^6+1)^(3/4)*x+RootOf(_Z^4+1)^3)/(x^6+x^4+1))

________________________________________________________________________________________

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^{6} + x^{4} + 1\right )} {\left (x^{6} + 1\right )}^{\frac {1}{4}} x^{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^4/(x^6+1)^(1/4)/(x^6+x^4+1),x, algorithm="maxima")

[Out]

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

________________________________________________________________________________________

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

[Out]

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

[Out]

Timed out

________________________________________________________________________________________