3.6.61 \(\int \frac {2+x^3+x^6}{x \sqrt [4]{1+x^6} (-4+5 x^3-4 x^6+x^9)} \, dx\)

Optimal. Leaf size=43 \[ \frac {1}{3} \tan ^{-1}\left (\frac {1-x^3}{\sqrt [4]{x^6+1}}\right )-\frac {1}{3} \tanh ^{-1}\left (\frac {x^3-1}{\sqrt [4]{x^6+1}}\right ) \]

________________________________________________________________________________________

Rubi [F]  time = 1.14, 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 {2+x^3+x^6}{x \sqrt [4]{1+x^6} \left (-4+5 x^3-4 x^6+x^9\right )} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Int[(2 + x^3 + x^6)/(x*(1 + x^6)^(1/4)*(-4 + 5*x^3 - 4*x^6 + x^9)),x]

[Out]

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

Rubi steps

\begin {align*} \int \frac {2+x^3+x^6}{x \sqrt [4]{1+x^6} \left (-4+5 x^3-4 x^6+x^9\right )} \, dx &=\frac {1}{3} \operatorname {Subst}\left (\int \frac {2+x+x^2}{x \sqrt [4]{1+x^2} \left (-4+5 x-4 x^2+x^3\right )} \, dx,x,x^3\right )\\ &=\frac {1}{3} \operatorname {Subst}\left (\int \left (-\frac {1}{2 x \sqrt [4]{1+x^2}}+\frac {7-2 x+x^2}{2 \sqrt [4]{1+x^2} \left (-4+5 x-4 x^2+x^3\right )}\right ) \, dx,x,x^3\right )\\ &=-\left (\frac {1}{6} \operatorname {Subst}\left (\int \frac {1}{x \sqrt [4]{1+x^2}} \, dx,x,x^3\right )\right )+\frac {1}{6} \operatorname {Subst}\left (\int \frac {7-2 x+x^2}{\sqrt [4]{1+x^2} \left (-4+5 x-4 x^2+x^3\right )} \, dx,x,x^3\right )\\ &=-\left (\frac {1}{12} \operatorname {Subst}\left (\int \frac {1}{x \sqrt [4]{1+x}} \, dx,x,x^6\right )\right )+\frac {1}{6} \operatorname {Subst}\left (\int \left (\frac {7}{\sqrt [4]{1+x^2} \left (-4+5 x-4 x^2+x^3\right )}-\frac {2 x}{\sqrt [4]{1+x^2} \left (-4+5 x-4 x^2+x^3\right )}+\frac {x^2}{\sqrt [4]{1+x^2} \left (-4+5 x-4 x^2+x^3\right )}\right ) \, dx,x,x^3\right )\\ &=\frac {1}{6} \operatorname {Subst}\left (\int \frac {x^2}{\sqrt [4]{1+x^2} \left (-4+5 x-4 x^2+x^3\right )} \, dx,x,x^3\right )-\frac {1}{3} \operatorname {Subst}\left (\int \frac {x}{\sqrt [4]{1+x^2} \left (-4+5 x-4 x^2+x^3\right )} \, dx,x,x^3\right )-\frac {1}{3} \operatorname {Subst}\left (\int \frac {x^2}{-1+x^4} \, dx,x,\sqrt [4]{1+x^6}\right )+\frac {7}{6} \operatorname {Subst}\left (\int \frac {1}{\sqrt [4]{1+x^2} \left (-4+5 x-4 x^2+x^3\right )} \, dx,x,x^3\right )\\ &=\frac {1}{6} \operatorname {Subst}\left (\int \frac {1}{1-x^2} \, dx,x,\sqrt [4]{1+x^6}\right )-\frac {1}{6} \operatorname {Subst}\left (\int \frac {1}{1+x^2} \, dx,x,\sqrt [4]{1+x^6}\right )+\frac {1}{6} \operatorname {Subst}\left (\int \frac {x^2}{\sqrt [4]{1+x^2} \left (-4+5 x-4 x^2+x^3\right )} \, dx,x,x^3\right )-\frac {1}{3} \operatorname {Subst}\left (\int \frac {x}{\sqrt [4]{1+x^2} \left (-4+5 x-4 x^2+x^3\right )} \, dx,x,x^3\right )+\frac {7}{6} \operatorname {Subst}\left (\int \frac {1}{\sqrt [4]{1+x^2} \left (-4+5 x-4 x^2+x^3\right )} \, dx,x,x^3\right )\\ &=-\frac {1}{6} \tan ^{-1}\left (\sqrt [4]{1+x^6}\right )+\frac {1}{6} \tanh ^{-1}\left (\sqrt [4]{1+x^6}\right )+\frac {1}{6} \operatorname {Subst}\left (\int \frac {x^2}{\sqrt [4]{1+x^2} \left (-4+5 x-4 x^2+x^3\right )} \, dx,x,x^3\right )-\frac {1}{3} \operatorname {Subst}\left (\int \frac {x}{\sqrt [4]{1+x^2} \left (-4+5 x-4 x^2+x^3\right )} \, dx,x,x^3\right )+\frac {7}{6} \operatorname {Subst}\left (\int \frac {1}{\sqrt [4]{1+x^2} \left (-4+5 x-4 x^2+x^3\right )} \, dx,x,x^3\right )\\ \end {align*}

________________________________________________________________________________________

Mathematica [F]  time = 0.24, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {2+x^3+x^6}{x \sqrt [4]{1+x^6} \left (-4+5 x^3-4 x^6+x^9\right )} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Integrate[(2 + x^3 + x^6)/(x*(1 + x^6)^(1/4)*(-4 + 5*x^3 - 4*x^6 + x^9)),x]

[Out]

Integrate[(2 + x^3 + x^6)/(x*(1 + x^6)^(1/4)*(-4 + 5*x^3 - 4*x^6 + x^9)), x]

________________________________________________________________________________________

IntegrateAlgebraic [A]  time = 15.61, size = 43, normalized size = 1.00 \begin {gather*} \frac {1}{3} \tan ^{-1}\left (\frac {1-x^3}{\sqrt [4]{1+x^6}}\right )-\frac {1}{3} \tanh ^{-1}\left (\frac {-1+x^3}{\sqrt [4]{1+x^6}}\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

IntegrateAlgebraic[(2 + x^3 + x^6)/(x*(1 + x^6)^(1/4)*(-4 + 5*x^3 - 4*x^6 + x^9)),x]

[Out]

ArcTan[(1 - x^3)/(1 + x^6)^(1/4)]/3 - ArcTanh[(-1 + x^3)/(1 + x^6)^(1/4)]/3

________________________________________________________________________________________

fricas [B]  time = 12.67, size = 167, normalized size = 3.88 \begin {gather*} \frac {1}{6} \, \arctan \left (\frac {2 \, {\left ({\left (x^{6} + 1\right )}^{\frac {3}{4}} {\left (x^{3} - 1\right )} + {\left (x^{9} - 3 \, x^{6} + 3 \, x^{3} - 1\right )} {\left (x^{6} + 1\right )}^{\frac {1}{4}}\right )}}{x^{12} - 4 \, x^{9} + 5 \, x^{6} - 4 \, x^{3}}\right ) + \frac {1}{6} \, \log \left (-\frac {x^{12} - 4 \, x^{9} + 7 \, x^{6} - 4 \, x^{3} - 2 \, {\left (x^{6} + 1\right )}^{\frac {3}{4}} {\left (x^{3} - 1\right )} + 2 \, {\left (x^{6} - 2 \, x^{3} + 1\right )} \sqrt {x^{6} + 1} - 2 \, {\left (x^{9} - 3 \, x^{6} + 3 \, x^{3} - 1\right )} {\left (x^{6} + 1\right )}^{\frac {1}{4}} + 2}{x^{12} - 4 \, x^{9} + 5 \, x^{6} - 4 \, x^{3}}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((x^6+x^3+2)/x/(x^6+1)^(1/4)/(x^9-4*x^6+5*x^3-4),x, algorithm="fricas")

[Out]

1/6*arctan(2*((x^6 + 1)^(3/4)*(x^3 - 1) + (x^9 - 3*x^6 + 3*x^3 - 1)*(x^6 + 1)^(1/4))/(x^12 - 4*x^9 + 5*x^6 - 4
*x^3)) + 1/6*log(-(x^12 - 4*x^9 + 7*x^6 - 4*x^3 - 2*(x^6 + 1)^(3/4)*(x^3 - 1) + 2*(x^6 - 2*x^3 + 1)*sqrt(x^6 +
1) - 2*(x^9 - 3*x^6 + 3*x^3 - 1)*(x^6 + 1)^(1/4) + 2)/(x^12 - 4*x^9 + 5*x^6 - 4*x^3))

________________________________________________________________________________________

giac [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {x^{6} + x^{3} + 2}{{\left (x^{9} - 4 \, x^{6} + 5 \, x^{3} - 4\right )} {\left (x^{6} + 1\right )}^{\frac {1}{4}} x}\,{d x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((x^6+x^3+2)/x/(x^6+1)^(1/4)/(x^9-4*x^6+5*x^3-4),x, algorithm="giac")

[Out]

integrate((x^6 + x^3 + 2)/((x^9 - 4*x^6 + 5*x^3 - 4)*(x^6 + 1)^(1/4)*x), x)

________________________________________________________________________________________

maple [C]  time = 122.00, size = 347, normalized size = 8.07

method	result	size
trager	\(-\frac {\ln \left (-\frac {x^{12}+2 \left (x^{6}+1\right )^{\frac {1}{4}} x^{9}-4 x^{9}+2 \sqrt {x^{6}+1}\, x^{6}-6 \left (x^{6}+1\right )^{\frac {1}{4}} x^{6}+2 \left (x^{6}+1\right )^{\frac {3}{4}} x^{3}+7 x^{6}-4 x^{3} \sqrt {x^{6}+1}+6 \left (x^{6}+1\right )^{\frac {1}{4}} x^{3}-2 \left (x^{6}+1\right )^{\frac {3}{4}}-4 x^{3}+2 \sqrt {x^{6}+1}-2 \left (x^{6}+1\right )^{\frac {1}{4}}+2}{x^{3} \left (x^{9}-4 x^{6}+5 x^{3}-4\right )}\right )}{6}+\frac {\RootOf \left (\textit {\_Z}^{2}+1\right ) \ln \left (-\frac {-\RootOf \left (\textit {\_Z}^{2}+1\right ) x^{12}+4 \RootOf \left (\textit {\_Z}^{2}+1\right ) x^{9}-2 \left (x^{6}+1\right )^{\frac {1}{4}} x^{9}+2 \RootOf \left (\textit {\_Z}^{2}+1\right ) \sqrt {x^{6}+1}\, x^{6}-7 \RootOf \left (\textit {\_Z}^{2}+1\right ) x^{6}+6 \left (x^{6}+1\right )^{\frac {1}{4}} x^{6}-4 \RootOf \left (\textit {\_Z}^{2}+1\right ) \sqrt {x^{6}+1}\, x^{3}+2 \left (x^{6}+1\right )^{\frac {3}{4}} x^{3}+4 \RootOf \left (\textit {\_Z}^{2}+1\right ) x^{3}-6 \left (x^{6}+1\right )^{\frac {1}{4}} x^{3}+2 \sqrt {x^{6}+1}\, \RootOf \left (\textit {\_Z}^{2}+1\right )-2 \left (x^{6}+1\right )^{\frac {3}{4}}-2 \RootOf \left (\textit {\_Z}^{2}+1\right )+2 \left (x^{6}+1\right )^{\frac {1}{4}}}{x^{3} \left (x^{9}-4 x^{6}+5 x^{3}-4\right )}\right )}{6}\)	\(347\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((x^6+x^3+2)/x/(x^6+1)^(1/4)/(x^9-4*x^6+5*x^3-4),x,method=_RETURNVERBOSE)

[Out]

-1/6*ln(-(x^12+2*(x^6+1)^(1/4)*x^9-4*x^9+2*(x^6+1)^(1/2)*x^6-6*(x^6+1)^(1/4)*x^6+2*(x^6+1)^(3/4)*x^3+7*x^6-4*x
^3*(x^6+1)^(1/2)+6*(x^6+1)^(1/4)*x^3-2*(x^6+1)^(3/4)-4*x^3+2*(x^6+1)^(1/2)-2*(x^6+1)^(1/4)+2)/x^3/(x^9-4*x^6+5
*x^3-4))+1/6*RootOf(_Z^2+1)*ln(-(-RootOf(_Z^2+1)*x^12+4*RootOf(_Z^2+1)*x^9-2*(x^6+1)^(1/4)*x^9+2*RootOf(_Z^2+1
)*(x^6+1)^(1/2)*x^6-7*RootOf(_Z^2+1)*x^6+6*(x^6+1)^(1/4)*x^6-4*RootOf(_Z^2+1)*(x^6+1)^(1/2)*x^3+2*(x^6+1)^(3/4
)*x^3+4*RootOf(_Z^2+1)*x^3-6*(x^6+1)^(1/4)*x^3+2*(x^6+1)^(1/2)*RootOf(_Z^2+1)-2*(x^6+1)^(3/4)-2*RootOf(_Z^2+1)
+2*(x^6+1)^(1/4))/x^3/(x^9-4*x^6+5*x^3-4))

________________________________________________________________________________________

maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {x^{6} + x^{3} + 2}{{\left (x^{9} - 4 \, x^{6} + 5 \, x^{3} - 4\right )} {\left (x^{6} + 1\right )}^{\frac {1}{4}} x}\,{d x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((x^6+x^3+2)/x/(x^6+1)^(1/4)/(x^9-4*x^6+5*x^3-4),x, algorithm="maxima")

[Out]

integrate((x^6 + x^3 + 2)/((x^9 - 4*x^6 + 5*x^3 - 4)*(x^6 + 1)^(1/4)*x), x)

________________________________________________________________________________________

mupad [F]  time = 0.00, size = -1, normalized size = -0.02 \begin {gather*} \int \frac {x^6+x^3+2}{x\,{\left (x^6+1\right )}^{1/4}\,\left (x^9-4\,x^6+5\,x^3-4\right )} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((x^3 + x^6 + 2)/(x*(x^6 + 1)^(1/4)*(5*x^3 - 4*x^6 + x^9 - 4)),x)

[Out]

int((x^3 + x^6 + 2)/(x*(x^6 + 1)^(1/4)*(5*x^3 - 4*x^6 + x^9 - 4)), 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+x**3+2)/x/(x**6+1)**(1/4)/(x**9-4*x**6+5*x**3-4),x)

[Out]

Timed out

________________________________________________________________________________________