3.11.82 \(\int \frac {3+x^4}{\sqrt [3]{-1+x^4} (-1-8 x^3+x^4)} \, dx\)

Optimal. Leaf size=81 \[ \frac {1}{2} \log \left (\sqrt [3]{x^4-1}-2 x\right )-\frac {1}{2} \sqrt {3} \tan ^{-1}\left (\frac {\sqrt {3} x}{\sqrt [3]{x^4-1}+x}\right )-\frac {1}{4} \log \left (2 \sqrt [3]{x^4-1} x+\left (x^4-1\right )^{2/3}+4 x^2\right ) \]

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Rubi [F]  time = 0.64, 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 {3+x^4}{\sqrt [3]{-1+x^4} \left (-1-8 x^3+x^4\right )} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Int[(3 + x^4)/((-1 + x^4)^(1/3)*(-1 - 8*x^3 + x^4)),x]

[Out]

(x*(1 - x^4)^(1/3)*Hypergeometric2F1[1/4, 1/3, 5/4, x^4])/(-1 + x^4)^(1/3) + 4*Defer[Int][1/((-1 + x^4)^(1/3)*
(-1 - 8*x^3 + x^4)), x] + 8*Defer[Int][x^3/((-1 + x^4)^(1/3)*(-1 - 8*x^3 + x^4)), x]

Rubi steps

\begin {align*} \int \frac {3+x^4}{\sqrt [3]{-1+x^4} \left (-1-8 x^3+x^4\right )} \, dx &=\int \left (\frac {1}{\sqrt [3]{-1+x^4}}+\frac {4 \left (1+2 x^3\right )}{\sqrt [3]{-1+x^4} \left (-1-8 x^3+x^4\right )}\right ) \, dx\\ &=4 \int \frac {1+2 x^3}{\sqrt [3]{-1+x^4} \left (-1-8 x^3+x^4\right )} \, dx+\int \frac {1}{\sqrt [3]{-1+x^4}} \, dx\\ &=4 \int \left (\frac {1}{\sqrt [3]{-1+x^4} \left (-1-8 x^3+x^4\right )}+\frac {2 x^3}{\sqrt [3]{-1+x^4} \left (-1-8 x^3+x^4\right )}\right ) \, dx+\frac {\sqrt [3]{1-x^4} \int \frac {1}{\sqrt [3]{1-x^4}} \, dx}{\sqrt [3]{-1+x^4}}\\ &=\frac {x \sqrt [3]{1-x^4} \, _2F_1\left (\frac {1}{4},\frac {1}{3};\frac {5}{4};x^4\right )}{\sqrt [3]{-1+x^4}}+4 \int \frac {1}{\sqrt [3]{-1+x^4} \left (-1-8 x^3+x^4\right )} \, dx+8 \int \frac {x^3}{\sqrt [3]{-1+x^4} \left (-1-8 x^3+x^4\right )} \, dx\\ \end {align*}

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Mathematica [F]  time = 0.15, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {3+x^4}{\sqrt [3]{-1+x^4} \left (-1-8 x^3+x^4\right )} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Integrate[(3 + x^4)/((-1 + x^4)^(1/3)*(-1 - 8*x^3 + x^4)),x]

[Out]

Integrate[(3 + x^4)/((-1 + x^4)^(1/3)*(-1 - 8*x^3 + x^4)), x]

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IntegrateAlgebraic [A]  time = 1.13, size = 81, normalized size = 1.00 \begin {gather*} -\frac {1}{2} \sqrt {3} \tan ^{-1}\left (\frac {\sqrt {3} x}{x+\sqrt [3]{-1+x^4}}\right )+\frac {1}{2} \log \left (-2 x+\sqrt [3]{-1+x^4}\right )-\frac {1}{4} \log \left (4 x^2+2 x \sqrt [3]{-1+x^4}+\left (-1+x^4\right )^{2/3}\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

IntegrateAlgebraic[(3 + x^4)/((-1 + x^4)^(1/3)*(-1 - 8*x^3 + x^4)),x]

[Out]

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

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fricas [A]  time = 2.41, size = 112, normalized size = 1.38 \begin {gather*} -\frac {1}{2} \, \sqrt {3} \arctan \left (-\frac {8 \, \sqrt {3} {\left (x^{4} - 1\right )}^{\frac {1}{3}} x^{2} - 4 \, \sqrt {3} {\left (x^{4} - 1\right )}^{\frac {2}{3}} x + \sqrt {3} {\left (x^{4} - 8 \, x^{3} - 1\right )}}{3 \, {\left (x^{4} + 8 \, x^{3} - 1\right )}}\right ) + \frac {1}{4} \, \log \left (\frac {x^{4} - 8 \, x^{3} + 12 \, {\left (x^{4} - 1\right )}^{\frac {1}{3}} x^{2} - 6 \, {\left (x^{4} - 1\right )}^{\frac {2}{3}} x - 1}{x^{4} - 8 \, x^{3} - 1}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((x^4+3)/(x^4-1)^(1/3)/(x^4-8*x^3-1),x, algorithm="fricas")

[Out]

-1/2*sqrt(3)*arctan(-1/3*(8*sqrt(3)*(x^4 - 1)^(1/3)*x^2 - 4*sqrt(3)*(x^4 - 1)^(2/3)*x + sqrt(3)*(x^4 - 8*x^3 -
1))/(x^4 + 8*x^3 - 1)) + 1/4*log((x^4 - 8*x^3 + 12*(x^4 - 1)^(1/3)*x^2 - 6*(x^4 - 1)^(2/3)*x - 1)/(x^4 - 8*x^
3 - 1))

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giac [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {x^{4} + 3}{{\left (x^{4} - 8 \, x^{3} - 1\right )} {\left (x^{4} - 1\right )}^{\frac {1}{3}}}\,{d x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((x^4+3)/(x^4-1)^(1/3)/(x^4-8*x^3-1),x, algorithm="giac")

[Out]

integrate((x^4 + 3)/((x^4 - 8*x^3 - 1)*(x^4 - 1)^(1/3)), x)

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maple [C]  time = 2.75, size = 343, normalized size = 4.23

method	result	size
trager	\(\RootOf \left (4 \textit {\_Z}^{2}+2 \textit {\_Z} +1\right ) \ln \left (\frac {4 \left (x^{4}-1\right )^{\frac {2}{3}} \RootOf \left (4 \textit {\_Z}^{2}+2 \textit {\_Z} +1\right ) x +8 \left (x^{4}-1\right )^{\frac {1}{3}} \RootOf \left (4 \textit {\_Z}^{2}+2 \textit {\_Z} +1\right ) x^{2}+16 x^{3} \RootOf \left (4 \textit {\_Z}^{2}+2 \textit {\_Z} +1\right )+x^{4}+4 \left (x^{4}-1\right )^{\frac {2}{3}} x +8 \left (x^{4}-1\right )^{\frac {1}{3}} x^{2}+8 x^{3}-1}{x^{4}-8 x^{3}-1}\right )-\frac {\ln \left (-\frac {4 \left (x^{4}-1\right )^{\frac {2}{3}} \RootOf \left (4 \textit {\_Z}^{2}+2 \textit {\_Z} +1\right ) x +8 \left (x^{4}-1\right )^{\frac {1}{3}} \RootOf \left (4 \textit {\_Z}^{2}+2 \textit {\_Z} +1\right ) x^{2}+16 x^{3} \RootOf \left (4 \textit {\_Z}^{2}+2 \textit {\_Z} +1\right )-x^{4}-2 \left (x^{4}-1\right )^{\frac {2}{3}} x -4 \left (x^{4}-1\right )^{\frac {1}{3}} x^{2}+1}{x^{4}-8 x^{3}-1}\right )}{2}-\ln \left (-\frac {4 \left (x^{4}-1\right )^{\frac {2}{3}} \RootOf \left (4 \textit {\_Z}^{2}+2 \textit {\_Z} +1\right ) x +8 \left (x^{4}-1\right )^{\frac {1}{3}} \RootOf \left (4 \textit {\_Z}^{2}+2 \textit {\_Z} +1\right ) x^{2}+16 x^{3} \RootOf \left (4 \textit {\_Z}^{2}+2 \textit {\_Z} +1\right )-x^{4}-2 \left (x^{4}-1\right )^{\frac {2}{3}} x -4 \left (x^{4}-1\right )^{\frac {1}{3}} x^{2}+1}{x^{4}-8 x^{3}-1}\right ) \RootOf \left (4 \textit {\_Z}^{2}+2 \textit {\_Z} +1\right )\)	\(343\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((x^4+3)/(x^4-1)^(1/3)/(x^4-8*x^3-1),x,method=_RETURNVERBOSE)

[Out]

RootOf(4*_Z^2+2*_Z+1)*ln((4*(x^4-1)^(2/3)*RootOf(4*_Z^2+2*_Z+1)*x+8*(x^4-1)^(1/3)*RootOf(4*_Z^2+2*_Z+1)*x^2+16
*x^3*RootOf(4*_Z^2+2*_Z+1)+x^4+4*(x^4-1)^(2/3)*x+8*(x^4-1)^(1/3)*x^2+8*x^3-1)/(x^4-8*x^3-1))-1/2*ln(-(4*(x^4-1
)^(2/3)*RootOf(4*_Z^2+2*_Z+1)*x+8*(x^4-1)^(1/3)*RootOf(4*_Z^2+2*_Z+1)*x^2+16*x^3*RootOf(4*_Z^2+2*_Z+1)-x^4-2*(
x^4-1)^(2/3)*x-4*(x^4-1)^(1/3)*x^2+1)/(x^4-8*x^3-1))-ln(-(4*(x^4-1)^(2/3)*RootOf(4*_Z^2+2*_Z+1)*x+8*(x^4-1)^(1
/3)*RootOf(4*_Z^2+2*_Z+1)*x^2+16*x^3*RootOf(4*_Z^2+2*_Z+1)-x^4-2*(x^4-1)^(2/3)*x-4*(x^4-1)^(1/3)*x^2+1)/(x^4-8
*x^3-1))*RootOf(4*_Z^2+2*_Z+1)

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maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {x^{4} + 3}{{\left (x^{4} - 8 \, x^{3} - 1\right )} {\left (x^{4} - 1\right )}^{\frac {1}{3}}}\,{d x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((x^4+3)/(x^4-1)^(1/3)/(x^4-8*x^3-1),x, algorithm="maxima")

[Out]

integrate((x^4 + 3)/((x^4 - 8*x^3 - 1)*(x^4 - 1)^(1/3)), x)

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mupad [F]  time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \int -\frac {x^4+3}{{\left (x^4-1\right )}^{1/3}\,\left (-x^4+8\,x^3+1\right )} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(-(x^4 + 3)/((x^4 - 1)^(1/3)*(8*x^3 - x^4 + 1)),x)

[Out]

int(-(x^4 + 3)/((x^4 - 1)^(1/3)*(8*x^3 - 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((x**4+3)/(x**4-1)**(1/3)/(x**4-8*x**3-1),x)

[Out]

Timed out

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