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

Optimal. Leaf size=90 \[ \frac {3 \sqrt [3]{x^4-1}}{x}-\log \left (\sqrt [3]{x^4-1}+x\right )+\sqrt {3} \tan ^{-1}\left (\frac {\sqrt {3} x}{2 \sqrt [3]{x^4-1}-x}\right )+\frac {1}{2} \log \left (-\sqrt [3]{x^4-1} x+\left (x^4-1\right )^{2/3}+x^2\right ) \]

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Rubi [F]  time = 0.78, 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 {\sqrt [3]{-1+x^4} \left (3+x^4\right )}{x^2 \left (-1+x^3+x^4\right )} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

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

[Out]

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

Rubi steps

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

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Mathematica [F]  time = 0.30, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\sqrt [3]{-1+x^4} \left (3+x^4\right )}{x^2 \left (-1+x^3+x^4\right )} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

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

[Out]

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

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IntegrateAlgebraic [A]  time = 0.85, size = 90, normalized size = 1.00 \begin {gather*} \frac {3 \sqrt [3]{-1+x^4}}{x}+\sqrt {3} \tan ^{-1}\left (\frac {\sqrt {3} x}{-x+2 \sqrt [3]{-1+x^4}}\right )-\log \left (x+\sqrt [3]{-1+x^4}\right )+\frac {1}{2} \log \left (x^2-x \sqrt [3]{-1+x^4}+\left (-1+x^4\right )^{2/3}\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

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fricas [A]  time = 4.24, size = 128, normalized size = 1.42 \begin {gather*} \frac {2 \, \sqrt {3} x \arctan \left (-\frac {33798185694614068 \, \sqrt {3} {\left (x^{4} - 1\right )}^{\frac {1}{3}} x^{2} - 35774000716806898 \, \sqrt {3} {\left (x^{4} - 1\right )}^{\frac {2}{3}} x + \sqrt {3} {\left (18215948833549379 \, x^{4} - 16570144372161104 \, x^{3} - 18215948833549379\right )}}{18912305915671589 \, x^{4} + 15948583382382344 \, x^{3} - 18912305915671589}\right ) - x \log \left (\frac {x^{4} + x^{3} + 3 \, {\left (x^{4} - 1\right )}^{\frac {1}{3}} x^{2} + 3 \, {\left (x^{4} - 1\right )}^{\frac {2}{3}} x - 1}{x^{4} + x^{3} - 1}\right ) + 6 \, {\left (x^{4} - 1\right )}^{\frac {1}{3}}}{2 \, x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/2*(2*sqrt(3)*x*arctan(-(33798185694614068*sqrt(3)*(x^4 - 1)^(1/3)*x^2 - 35774000716806898*sqrt(3)*(x^4 - 1)^
(2/3)*x + sqrt(3)*(18215948833549379*x^4 - 16570144372161104*x^3 - 18215948833549379))/(18912305915671589*x^4
+ 15948583382382344*x^3 - 18912305915671589)) - x*log((x^4 + x^3 + 3*(x^4 - 1)^(1/3)*x^2 + 3*(x^4 - 1)^(2/3)*x
- 1)/(x^4 + x^3 - 1)) + 6*(x^4 - 1)^(1/3))/x

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maple [C]  time = 9.65, size = 402, normalized size = 4.47

method	result	size
trager	\(\frac {3 \left (x^{4}-1\right )^{\frac {1}{3}}}{x}-\ln \left (-\frac {10242144 \RootOf \left (36 \textit {\_Z}^{2}-6 \textit {\_Z} +1\right )^{2} x^{4}-19204020 \RootOf \left (36 \textit {\_Z}^{2}-6 \textit {\_Z} +1\right )^{2} x^{3}-108450 \RootOf \left (36 \textit {\_Z}^{2}-6 \textit {\_Z} +1\right ) x^{4}+4707432 \RootOf \left (36 \textit {\_Z}^{2}-6 \textit {\_Z} +1\right ) \left (x^{4}-1\right )^{\frac {2}{3}} x +4619142 \left (x^{4}-1\right )^{\frac {1}{3}} \RootOf \left (36 \textit {\_Z}^{2}-6 \textit {\_Z} +1\right ) x^{2}+4710954 \RootOf \left (36 \textit {\_Z}^{2}-6 \textit {\_Z} +1\right ) x^{3}-233639 x^{4}-769857 \left (x^{4}-1\right )^{\frac {2}{3}} x +14715 \left (x^{4}-1\right )^{\frac {1}{3}} x^{2}-267016 x^{3}-10242144 \RootOf \left (36 \textit {\_Z}^{2}-6 \textit {\_Z} +1\right )^{2}+108450 \RootOf \left (36 \textit {\_Z}^{2}-6 \textit {\_Z} +1\right )+233639}{x^{4}+x^{3}-1}\right )+6 \RootOf \left (36 \textit {\_Z}^{2}-6 \textit {\_Z} +1\right ) \ln \left (\frac {629568 \RootOf \left (36 \textit {\_Z}^{2}-6 \textit {\_Z} +1\right )^{2} x^{4}-1180440 \RootOf \left (36 \textit {\_Z}^{2}-6 \textit {\_Z} +1\right )^{2} x^{3}+1405356 \RootOf \left (36 \textit {\_Z}^{2}-6 \textit {\_Z} +1\right ) x^{4}+4707432 \RootOf \left (36 \textit {\_Z}^{2}-6 \textit {\_Z} +1\right ) \left (x^{4}-1\right )^{\frac {2}{3}} x +88290 \left (x^{4}-1\right )^{\frac {1}{3}} \RootOf \left (36 \textit {\_Z}^{2}-6 \textit {\_Z} +1\right ) x^{2}-2912118 \RootOf \left (36 \textit {\_Z}^{2}-6 \textit {\_Z} +1\right ) x^{3}-500655 x^{4}-14715 \left (x^{4}-1\right )^{\frac {2}{3}} x +769857 \left (x^{4}-1\right )^{\frac {1}{3}} x^{2}+233639 x^{3}-629568 \RootOf \left (36 \textit {\_Z}^{2}-6 \textit {\_Z} +1\right )^{2}-1405356 \RootOf \left (36 \textit {\_Z}^{2}-6 \textit {\_Z} +1\right )+500655}{x^{4}+x^{3}-1}\right )\)	\(402\)
risch	\(\frac {3 \left (x^{4}-1\right )^{\frac {1}{3}}}{x}+\frac {\left (-\ln \left (-\frac {\RootOf \left (\textit {\_Z}^{2}-\textit {\_Z} +1\right ) x^{7}-x^{8}+\RootOf \left (\textit {\_Z}^{2}-\textit {\_Z} +1\right ) \left (x^{8}-2 x^{4}+1\right )^{\frac {1}{3}} x^{5}-2 \left (x^{8}-2 x^{4}+1\right )^{\frac {1}{3}} x^{5}+2 \RootOf \left (\textit {\_Z}^{2}-\textit {\_Z} +1\right ) \left (x^{8}-2 x^{4}+1\right )^{\frac {2}{3}} x^{2}-\left (x^{8}-2 x^{4}+1\right )^{\frac {2}{3}} x^{2}-\RootOf \left (\textit {\_Z}^{2}-\textit {\_Z} +1\right ) x^{3}+2 x^{4}-\RootOf \left (\textit {\_Z}^{2}-\textit {\_Z} +1\right ) \left (x^{8}-2 x^{4}+1\right )^{\frac {1}{3}} x +2 \left (x^{8}-2 x^{4}+1\right )^{\frac {1}{3}} x -1}{\left (x^{4}+x^{3}-1\right ) \left (-1+x \right ) \left (1+x \right ) \left (x^{2}+1\right )}\right )+\RootOf \left (\textit {\_Z}^{2}-\textit {\_Z} +1\right ) \ln \left (-\frac {-\RootOf \left (\textit {\_Z}^{2}-\textit {\_Z} +1\right )^{2} x^{7}-\RootOf \left (\textit {\_Z}^{2}-\textit {\_Z} +1\right ) x^{8}+2 \RootOf \left (\textit {\_Z}^{2}-\textit {\_Z} +1\right ) x^{7}+x^{8}-\RootOf \left (\textit {\_Z}^{2}-\textit {\_Z} +1\right ) \left (x^{8}-2 x^{4}+1\right )^{\frac {1}{3}} x^{5}-x^{7}-\left (x^{8}-2 x^{4}+1\right )^{\frac {1}{3}} x^{5}+\RootOf \left (\textit {\_Z}^{2}-\textit {\_Z} +1\right ) \left (x^{8}-2 x^{4}+1\right )^{\frac {2}{3}} x^{2}+\RootOf \left (\textit {\_Z}^{2}-\textit {\_Z} +1\right )^{2} x^{3}+2 \RootOf \left (\textit {\_Z}^{2}-\textit {\_Z} +1\right ) x^{4}-2 \left (x^{8}-2 x^{4}+1\right )^{\frac {2}{3}} x^{2}-2 \RootOf \left (\textit {\_Z}^{2}-\textit {\_Z} +1\right ) x^{3}-2 x^{4}+\RootOf \left (\textit {\_Z}^{2}-\textit {\_Z} +1\right ) \left (x^{8}-2 x^{4}+1\right )^{\frac {1}{3}} x +x^{3}+\left (x^{8}-2 x^{4}+1\right )^{\frac {1}{3}} x -\RootOf \left (\textit {\_Z}^{2}-\textit {\_Z} +1\right )+1}{\left (x^{4}+x^{3}-1\right ) \left (-1+x \right ) \left (1+x \right ) \left (x^{2}+1\right )}\right )\right ) \left (\left (x^{4}-1\right )^{2}\right )^{\frac {1}{3}}}{\left (x^{4}-1\right )^{\frac {2}{3}}}\)	\(505\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

3*(x^4-1)^(1/3)/x-ln(-(10242144*RootOf(36*_Z^2-6*_Z+1)^2*x^4-19204020*RootOf(36*_Z^2-6*_Z+1)^2*x^3-108450*Root
Of(36*_Z^2-6*_Z+1)*x^4+4707432*RootOf(36*_Z^2-6*_Z+1)*(x^4-1)^(2/3)*x+4619142*(x^4-1)^(1/3)*RootOf(36*_Z^2-6*_
Z+1)*x^2+4710954*RootOf(36*_Z^2-6*_Z+1)*x^3-233639*x^4-769857*(x^4-1)^(2/3)*x+14715*(x^4-1)^(1/3)*x^2-267016*x
^3-10242144*RootOf(36*_Z^2-6*_Z+1)^2+108450*RootOf(36*_Z^2-6*_Z+1)+233639)/(x^4+x^3-1))+6*RootOf(36*_Z^2-6*_Z+
1)*ln((629568*RootOf(36*_Z^2-6*_Z+1)^2*x^4-1180440*RootOf(36*_Z^2-6*_Z+1)^2*x^3+1405356*RootOf(36*_Z^2-6*_Z+1)
*x^4+4707432*RootOf(36*_Z^2-6*_Z+1)*(x^4-1)^(2/3)*x+88290*(x^4-1)^(1/3)*RootOf(36*_Z^2-6*_Z+1)*x^2-2912118*Roo
tOf(36*_Z^2-6*_Z+1)*x^3-500655*x^4-14715*(x^4-1)^(2/3)*x+769857*(x^4-1)^(1/3)*x^2+233639*x^3-629568*RootOf(36*
_Z^2-6*_Z+1)^2-1405356*RootOf(36*_Z^2-6*_Z+1)+500655)/(x^4+x^3-1))

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

[Out]

Timed out

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