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

Optimal. Leaf size=93 \[ -\log \left (\sqrt [3]{x^6-x^2}+x\right )+\frac {1}{2} \log \left (x^2-\sqrt [3]{x^6-x^2} x+\left (x^6-x^2\right )^{2/3}\right )-\sqrt {3} \tan ^{-1}\left (\frac {\sqrt {3} x}{2 \sqrt [3]{x^6-x^2}-x}\right ) \]

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Rubi [F]  time = 0.99, 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 {1+3 x^4}{\left (-1+x+x^4\right ) \sqrt [3]{-x^2+x^6}} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

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

[Out]

(9*x*(1 - x^4)^(1/3)*Hypergeometric2F1[1/12, 1/3, 13/12, x^4])/(-x^2 + x^6)^(1/3) + (12*x^(2/3)*(-1 + x^4)^(1/
3)*Defer[Subst][Defer[Int][1/((-1 + x^12)^(1/3)*(-1 + x^3 + x^12)), x], x, x^(1/3)])/(-x^2 + x^6)^(1/3) - (9*x
^(2/3)*(-1 + x^4)^(1/3)*Defer[Subst][Defer[Int][x^3/((-1 + x^12)^(1/3)*(-1 + x^3 + x^12)), x], x, x^(1/3)])/(-
x^2 + x^6)^(1/3)

Rubi steps

\begin {align*} \int \frac {1+3 x^4}{\left (-1+x+x^4\right ) \sqrt [3]{-x^2+x^6}} \, dx &=\frac {\left (x^{2/3} \sqrt [3]{-1+x^4}\right ) \int \frac {1+3 x^4}{x^{2/3} \sqrt [3]{-1+x^4} \left (-1+x+x^4\right )} \, dx}{\sqrt [3]{-x^2+x^6}}\\ &=\frac {\left (3 x^{2/3} \sqrt [3]{-1+x^4}\right ) \operatorname {Subst}\left (\int \frac {1+3 x^{12}}{\sqrt [3]{-1+x^{12}} \left (-1+x^3+x^{12}\right )} \, dx,x,\sqrt [3]{x}\right )}{\sqrt [3]{-x^2+x^6}}\\ &=\frac {\left (3 x^{2/3} \sqrt [3]{-1+x^4}\right ) \operatorname {Subst}\left (\int \left (\frac {3}{\sqrt [3]{-1+x^{12}}}+\frac {4-3 x^3}{\sqrt [3]{-1+x^{12}} \left (-1+x^3+x^{12}\right )}\right ) \, dx,x,\sqrt [3]{x}\right )}{\sqrt [3]{-x^2+x^6}}\\ &=\frac {\left (3 x^{2/3} \sqrt [3]{-1+x^4}\right ) \operatorname {Subst}\left (\int \frac {4-3 x^3}{\sqrt [3]{-1+x^{12}} \left (-1+x^3+x^{12}\right )} \, dx,x,\sqrt [3]{x}\right )}{\sqrt [3]{-x^2+x^6}}+\frac {\left (9 x^{2/3} \sqrt [3]{-1+x^4}\right ) \operatorname {Subst}\left (\int \frac {1}{\sqrt [3]{-1+x^{12}}} \, dx,x,\sqrt [3]{x}\right )}{\sqrt [3]{-x^2+x^6}}\\ &=\frac {\left (9 x^{2/3} \sqrt [3]{1-x^4}\right ) \operatorname {Subst}\left (\int \frac {1}{\sqrt [3]{1-x^{12}}} \, dx,x,\sqrt [3]{x}\right )}{\sqrt [3]{-x^2+x^6}}+\frac {\left (3 x^{2/3} \sqrt [3]{-1+x^4}\right ) \operatorname {Subst}\left (\int \left (\frac {4}{\sqrt [3]{-1+x^{12}} \left (-1+x^3+x^{12}\right )}-\frac {3 x^3}{\sqrt [3]{-1+x^{12}} \left (-1+x^3+x^{12}\right )}\right ) \, dx,x,\sqrt [3]{x}\right )}{\sqrt [3]{-x^2+x^6}}\\ &=\frac {9 x \sqrt [3]{1-x^4} \, _2F_1\left (\frac {1}{12},\frac {1}{3};\frac {13}{12};x^4\right )}{\sqrt [3]{-x^2+x^6}}-\frac {\left (9 x^{2/3} \sqrt [3]{-1+x^4}\right ) \operatorname {Subst}\left (\int \frac {x^3}{\sqrt [3]{-1+x^{12}} \left (-1+x^3+x^{12}\right )} \, dx,x,\sqrt [3]{x}\right )}{\sqrt [3]{-x^2+x^6}}+\frac {\left (12 x^{2/3} \sqrt [3]{-1+x^4}\right ) \operatorname {Subst}\left (\int \frac {1}{\sqrt [3]{-1+x^{12}} \left (-1+x^3+x^{12}\right )} \, dx,x,\sqrt [3]{x}\right )}{\sqrt [3]{-x^2+x^6}}\\ \end {align*}

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Mathematica [F]  time = 0.20, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {1+3 x^4}{\left (-1+x+x^4\right ) \sqrt [3]{-x^2+x^6}} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

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

[Out]

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

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IntegrateAlgebraic [A]  time = 2.70, size = 93, normalized size = 1.00 \begin {gather*} -\log \left (\sqrt [3]{x^6-x^2}+x\right )+\frac {1}{2} \log \left (x^2-\sqrt [3]{x^6-x^2} x+\left (x^6-x^2\right )^{2/3}\right )-\sqrt {3} \tan ^{-1}\left (\frac {\sqrt {3} x}{2 \sqrt [3]{x^6-x^2}-x}\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

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fricas [A]  time = 1.70, size = 124, normalized size = 1.33 \begin {gather*} -\sqrt {3} \arctan \left (\frac {2 \, \sqrt {3} {\left (x^{6} - x^{2}\right )}^{\frac {1}{3}} x + \sqrt {3} {\left (x^{5} + x^{2} - x\right )} + 2 \, \sqrt {3} {\left (x^{6} - x^{2}\right )}^{\frac {2}{3}}}{3 \, {\left (x^{5} - x^{2} - x\right )}}\right ) - \frac {1}{2} \, \log \left (\frac {x^{5} + x^{2} + 3 \, {\left (x^{6} - x^{2}\right )}^{\frac {1}{3}} x - x + 3 \, {\left (x^{6} - x^{2}\right )}^{\frac {2}{3}}}{x^{5} + x^{2} - x}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maple [C]  time = 5.48, size = 580, normalized size = 6.24 \begin {gather*} \frac {\RootOf \left (\textit {\_Z}^{2}-2 \textit {\_Z} +4\right ) \ln \left (-\frac {-40235029434530 \RootOf \left (\textit {\_Z}^{2}-2 \textit {\_Z} +4\right )^{2} x^{5}-213844410247318 \RootOf \left (\textit {\_Z}^{2}-2 \textit {\_Z} +4\right ) x^{5}+12260547776479020 x^{5}[phone] \RootOf \left (\textit {\_Z}^{2}-2 \textit {\_Z} +4\right )^{2} x^{2}+6505058416224948 \RootOf \left (\textit {\_Z}^{2}-2 \textit {\_Z} +4\right ) \left (x^{6}-x^{2}\right )^{\frac {2}{3}}-6505058416224948 \RootOf \left (\textit {\_Z}^{2}-2 \textit {\_Z} +4\right ) \left (x^{6}-x^{2}\right )^{\frac {1}{3}} x [phone] \RootOf \left (\textit {\_Z}^{2}-2 \textit {\_Z} +4\right )^{2} x +5607218505590620 \RootOf \left (\textit {\_Z}^{2}-2 \textit {\_Z} +4\right ) x^{2}-24254346850201524 \left (x^{6}-x^{2}\right )^{\frac {2}{3}}+24254346850201524 x \left (x^{6}-x^{2}\right )^{\frac {1}{3}}[phone] \RootOf \left (\textit {\_Z}^{2}-2 \textit {\_Z} +4\right ) x -10625808072948484 x^{2}-12260547776479020 x}{\left (x^{4}+x -1\right ) x}\right )}{2}-\frac {\ln \left (\frac {103130 \RootOf \left (\textit {\_Z}^{2}-2 \textit {\_Z} +4\right )^{2} x^{5}-226806 \RootOf \left (\textit {\_Z}^{2}-2 \textit {\_Z} +4\right ) x^{5}-20377448 x^{5}-773475 \RootOf \left (\textit {\_Z}^{2}-2 \textit {\_Z} +4\right )^{2} x^{2}[phone] \RootOf \left (\textit {\_Z}^{2}-2 \textit {\_Z} +4\right ) \left (x^{6}-x^{2}\right )^{\frac {2}{3}}-10436076 \RootOf \left (\textit {\_Z}^{2}-2 \textit {\_Z} +4\right ) \left (x^{6}-x^{2}\right )^{\frac {1}{3}} x -103130 \RootOf \left (\textit {\_Z}^{2}-2 \textit {\_Z} +4\right )^{2} x [phone] \RootOf \left (\textit {\_Z}^{2}-2 \textit {\_Z} +4\right ) x^{2}[phone] \left (x^{6}-x^{2}\right )^{\frac {2}{3}}-20748876 x \left (x^{6}-x^{2}\right )^{\frac {1}{3}}+226806 \RootOf \left (\textit {\_Z}^{2}-2 \textit {\_Z} +4\right ) x -3134992 x^{2}[phone] x}{\left (x^{4}+x -1\right ) x}\right ) \RootOf \left (\textit {\_Z}^{2}-2 \textit {\_Z} +4\right )}{2}+\ln \left (\frac {103130 \RootOf \left (\textit {\_Z}^{2}-2 \textit {\_Z} +4\right )^{2} x^{5}-226806 \RootOf \left (\textit {\_Z}^{2}-2 \textit {\_Z} +4\right ) x^{5}-20377448 x^{5}-773475 \RootOf \left (\textit {\_Z}^{2}-2 \textit {\_Z} +4\right )^{2} x^{2}[phone] \RootOf \left (\textit {\_Z}^{2}-2 \textit {\_Z} +4\right ) \left (x^{6}-x^{2}\right )^{\frac {2}{3}}-10436076 \RootOf \left (\textit {\_Z}^{2}-2 \textit {\_Z} +4\right ) \left (x^{6}-x^{2}\right )^{\frac {1}{3}} x -103130 \RootOf \left (\textit {\_Z}^{2}-2 \textit {\_Z} +4\right )^{2} x [phone] \RootOf \left (\textit {\_Z}^{2}-2 \textit {\_Z} +4\right ) x^{2}[phone] \left (x^{6}-x^{2}\right )^{\frac {2}{3}}-20748876 x \left (x^{6}-x^{2}\right )^{\frac {1}{3}}+226806 \RootOf \left (\textit {\_Z}^{2}-2 \textit {\_Z} +4\right ) x -3134992 x^{2}[phone] x}{\left (x^{4}+x -1\right ) x}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/2*RootOf(_Z^2-2*_Z+4)*ln(-(-40235029434530*RootOf(_Z^2-2*_Z+4)^2*x^5-213844410247318*RootOf(_Z^2-2*_Z+4)*x^5
+12260547776479020*x^5+301762720758975*RootOf(_Z^2-2*_Z+4)^2*x^2+6505058416224948*RootOf(_Z^2-2*_Z+4)*(x^6-x^2
)^(2/3)-6505058416224948*RootOf(_Z^2-2*_Z+4)*(x^6-x^2)^(1/3)*x+40235029434530*RootOf(_Z^2-2*_Z+4)^2*x+56072185
05590620*RootOf(_Z^2-2*_Z+4)*x^2-24254346850201524*(x^6-x^2)^(2/3)+24254346850201524*x*(x^6-x^2)^(1/3)+2138444
10247318*RootOf(_Z^2-2*_Z+4)*x-10625808072948484*x^2-12260547776479020*x)/(x^4+x-1)/x)-1/2*ln((103130*RootOf(_
Z^2-2*_Z+4)^2*x^5-226806*RootOf(_Z^2-2*_Z+4)*x^5-20377448*x^5-773475*RootOf(_Z^2-2*_Z+4)^2*x^2+10436076*RootOf
(_Z^2-2*_Z+4)*(x^6-x^2)^(2/3)-10436076*RootOf(_Z^2-2*_Z+4)*(x^6-x^2)^(1/3)*x-103130*RootOf(_Z^2-2*_Z+4)^2*x+11
962480*RootOf(_Z^2-2*_Z+4)*x^2+20748876*(x^6-x^2)^(2/3)-20748876*x*(x^6-x^2)^(1/3)+226806*RootOf(_Z^2-2*_Z+4)*
x-3134992*x^2+20377448*x)/(x^4+x-1)/x)*RootOf(_Z^2-2*_Z+4)+ln((103130*RootOf(_Z^2-2*_Z+4)^2*x^5-226806*RootOf(
_Z^2-2*_Z+4)*x^5-20377448*x^5-773475*RootOf(_Z^2-2*_Z+4)^2*x^2+10436076*RootOf(_Z^2-2*_Z+4)*(x^6-x^2)^(2/3)-10
436076*RootOf(_Z^2-2*_Z+4)*(x^6-x^2)^(1/3)*x-103130*RootOf(_Z^2-2*_Z+4)^2*x+11962480*RootOf(_Z^2-2*_Z+4)*x^2+2
0748876*(x^6-x^2)^(2/3)-20748876*x*(x^6-x^2)^(1/3)+226806*RootOf(_Z^2-2*_Z+4)*x-3134992*x^2+20377448*x)/(x^4+x
-1)/x)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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mupad [F]  time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \int \frac {3\,x^4+1}{{\left (x^6-x^2\right )}^{1/3}\,\left (x^4+x-1\right )} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

[Out]

Timed out

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