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

Optimal. Leaf size=112 \[ \frac {3 \sqrt [3]{-x^4+x^3-1}}{x}+\log \left (\sqrt [3]{-x^4+x^3-1}-x\right )+\sqrt {3} \tan ^{-1}\left (\frac {\sqrt {3} x}{2 \sqrt [3]{-x^4+x^3-1}+x}\right )-\frac {1}{2} \log \left (x^2+\sqrt [3]{-x^4+x^3-1} x+\left (-x^4+x^3-1\right )^{2/3}\right ) \]

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Rubi [F]  time = 1.18, 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^3-x^4} \left (-3+x^4\right )}{x^2 \left (1+x^4\right )} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

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

[Out]

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

Rubi steps

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

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Mathematica [F]  time = 0.55, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\sqrt [3]{-1+x^3-x^4} \left (-3+x^4\right )}{x^2 \left (1+x^4\right )} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

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

[Out]

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

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IntegrateAlgebraic [A]  time = 0.63, size = 112, normalized size = 1.00 \begin {gather*} \frac {3 \sqrt [3]{-1+x^3-x^4}}{x}+\sqrt {3} \tan ^{-1}\left (\frac {\sqrt {3} x}{x+2 \sqrt [3]{-1+x^3-x^4}}\right )+\log \left (-x+\sqrt [3]{-1+x^3-x^4}\right )-\frac {1}{2} \log \left (x^2+x \sqrt [3]{-1+x^3-x^4}+\left (-1+x^3-x^4\right )^{2/3}\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

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fricas [A]  time = 2.17, size = 137, normalized size = 1.22 \begin {gather*} -\frac {2 \, \sqrt {3} x \arctan \left (\frac {\sqrt {3} x^{3} - 2 \, \sqrt {3} {\left (-x^{4} + x^{3} - 1\right )}^{\frac {1}{3}} x^{2} + 4 \, \sqrt {3} {\left (-x^{4} + x^{3} - 1\right )}^{\frac {2}{3}} x}{8 \, x^{4} - 9 \, x^{3} + 8}\right ) - x \log \left (\frac {x^{4} - 3 \, {\left (-x^{4} + x^{3} - 1\right )}^{\frac {1}{3}} x^{2} + 3 \, {\left (-x^{4} + x^{3} - 1\right )}^{\frac {2}{3}} x + 1}{x^{4} + 1}\right ) - 6 \, {\left (-x^{4} + x^{3} - 1\right )}^{\frac {1}{3}}}{2 \, x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/2*(2*sqrt(3)*x*arctan((sqrt(3)*x^3 - 2*sqrt(3)*(-x^4 + x^3 - 1)^(1/3)*x^2 + 4*sqrt(3)*(-x^4 + x^3 - 1)^(2/3
)*x)/(8*x^4 - 9*x^3 + 8)) - x*log((x^4 - 3*(-x^4 + x^3 - 1)^(1/3)*x^2 + 3*(-x^4 + x^3 - 1)^(2/3)*x + 1)/(x^4 +
1)) - 6*(-x^4 + x^3 - 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} + x^{3} - 1\right )}^{\frac {1}{3}}}{{\left (x^{4} + 1\right )} x^{2}}\,{d x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maple [C]  time = 6.81, size = 650, normalized size = 5.80

method	result	size
trager	\(\frac {3 \left (-x^{4}+x^{3}-1\right )^{\frac {1}{3}}}{x}-3 \ln \left (\frac {153 \RootOf \left (9 \textit {\_Z}^{2}+3 \textit {\_Z} +1\right )^{2} x^{4}-306 \RootOf \left (9 \textit {\_Z}^{2}+3 \textit {\_Z} +1\right )^{2} x^{3}+6 \RootOf \left (9 \textit {\_Z}^{2}+3 \textit {\_Z} +1\right ) x^{4}+141 \RootOf \left (9 \textit {\_Z}^{2}+3 \textit {\_Z} +1\right ) \left (-x^{4}+x^{3}-1\right )^{\frac {2}{3}} x +141 \RootOf \left (9 \textit {\_Z}^{2}+3 \textit {\_Z} +1\right ) \left (-x^{4}+x^{3}-1\right )^{\frac {1}{3}} x^{2}+39 \RootOf \left (9 \textit {\_Z}^{2}+3 \textit {\_Z} +1\right ) x^{3}-15 x^{4}+49 \left (-x^{4}+x^{3}-1\right )^{\frac {2}{3}} x +49 x^{2} \left (-x^{4}+x^{3}-1\right )^{\frac {1}{3}}+15 x^{3}+153 \RootOf \left (9 \textit {\_Z}^{2}+3 \textit {\_Z} +1\right )^{2}+6 \RootOf \left (9 \textit {\_Z}^{2}+3 \textit {\_Z} +1\right )-15}{x^{4}+1}\right ) \RootOf \left (9 \textit {\_Z}^{2}+3 \textit {\_Z} +1\right )+3 \RootOf \left (9 \textit {\_Z}^{2}+3 \textit {\_Z} +1\right ) \ln \left (\frac {153 \RootOf \left (9 \textit {\_Z}^{2}+3 \textit {\_Z} +1\right )^{2} x^{4}-306 \RootOf \left (9 \textit {\_Z}^{2}+3 \textit {\_Z} +1\right )^{2} x^{3}+96 \RootOf \left (9 \textit {\_Z}^{2}+3 \textit {\_Z} +1\right ) x^{4}-141 \RootOf \left (9 \textit {\_Z}^{2}+3 \textit {\_Z} +1\right ) \left (-x^{4}+x^{3}-1\right )^{\frac {2}{3}} x -141 \RootOf \left (9 \textit {\_Z}^{2}+3 \textit {\_Z} +1\right ) \left (-x^{4}+x^{3}-1\right )^{\frac {1}{3}} x^{2}-243 \RootOf \left (9 \textit {\_Z}^{2}+3 \textit {\_Z} +1\right ) x^{3}+2 \left (-x^{4}+x^{3}-1\right )^{\frac {2}{3}} x +2 x^{2} \left (-x^{4}+x^{3}-1\right )^{\frac {1}{3}}-32 x^{3}+153 \RootOf \left (9 \textit {\_Z}^{2}+3 \textit {\_Z} +1\right )^{2}+96 \RootOf \left (9 \textit {\_Z}^{2}+3 \textit {\_Z} +1\right )}{x^{4}+1}\right )-\ln \left (\frac {153 \RootOf \left (9 \textit {\_Z}^{2}+3 \textit {\_Z} +1\right )^{2} x^{4}-306 \RootOf \left (9 \textit {\_Z}^{2}+3 \textit {\_Z} +1\right )^{2} x^{3}+6 \RootOf \left (9 \textit {\_Z}^{2}+3 \textit {\_Z} +1\right ) x^{4}+141 \RootOf \left (9 \textit {\_Z}^{2}+3 \textit {\_Z} +1\right ) \left (-x^{4}+x^{3}-1\right )^{\frac {2}{3}} x +141 \RootOf \left (9 \textit {\_Z}^{2}+3 \textit {\_Z} +1\right ) \left (-x^{4}+x^{3}-1\right )^{\frac {1}{3}} x^{2}+39 \RootOf \left (9 \textit {\_Z}^{2}+3 \textit {\_Z} +1\right ) x^{3}-15 x^{4}+49 \left (-x^{4}+x^{3}-1\right )^{\frac {2}{3}} x +49 x^{2} \left (-x^{4}+x^{3}-1\right )^{\frac {1}{3}}+15 x^{3}+153 \RootOf \left (9 \textit {\_Z}^{2}+3 \textit {\_Z} +1\right )^{2}+6 \RootOf \left (9 \textit {\_Z}^{2}+3 \textit {\_Z} +1\right )-15}{x^{4}+1}\right )\)	\(650\)
risch	\(-\frac {3 \left (x^{4}-x^{3}+1\right )}{x \left (-x^{4}+x^{3}-1\right )^{\frac {2}{3}}}+\frac {\left (\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}-\RootOf \left (\textit {\_Z}^{2}+\textit {\_Z} +1\right )^{2} x^{6}-3 x^{7} \RootOf \left (\textit {\_Z}^{2}+\textit {\_Z} +1\right )-x^{8}+2 x^{6} \RootOf \left (\textit {\_Z}^{2}+\textit {\_Z} +1\right )+2 x^{7}-3 \left (x^{8}-2 x^{7}+x^{6}+2 x^{4}-2 x^{3}+1\right )^{\frac {1}{3}} x^{5}-x^{6}+\RootOf \left (\textit {\_Z}^{2}+\textit {\_Z} +1\right )^{2} x^{3}-3 \left (x^{8}-2 x^{7}+x^{6}+2 x^{4}-2 x^{3}+1\right )^{\frac {2}{3}} \RootOf \left (\textit {\_Z}^{2}+\textit {\_Z} +1\right ) x^{2}+2 \RootOf \left (\textit {\_Z}^{2}+\textit {\_Z} +1\right ) x^{4}+3 \left (x^{8}-2 x^{7}+x^{6}+2 x^{4}-2 x^{3}+1\right )^{\frac {1}{3}} x^{4}-3 \RootOf \left (\textit {\_Z}^{2}+\textit {\_Z} +1\right ) x^{3}-3 \left (x^{8}-2 x^{7}+x^{6}+2 x^{4}-2 x^{3}+1\right )^{\frac {2}{3}} x^{2}-2 x^{4}+2 x^{3}-3 \left (x^{8}-2 x^{7}+x^{6}+2 x^{4}-2 x^{3}+1\right )^{\frac {1}{3}} x +\RootOf \left (\textit {\_Z}^{2}+\textit {\_Z} +1\right )-1}{\left (x^{4}-x^{3}+1\right ) \left (x^{4}+1\right )}\right )+\RootOf \left (\textit {\_Z}^{2}+\textit {\_Z} +1\right ) \ln \left (\frac {2 \RootOf \left (\textit {\_Z}^{2}+\textit {\_Z} +1\right )^{2} x^{7}-2 \RootOf \left (\textit {\_Z}^{2}+\textit {\_Z} +1\right ) x^{8}-2 \RootOf \left (\textit {\_Z}^{2}+\textit {\_Z} +1\right )^{2} x^{6}+7 x^{7} \RootOf \left (\textit {\_Z}^{2}+\textit {\_Z} +1\right )-x^{8}-5 x^{6} \RootOf \left (\textit {\_Z}^{2}+\textit {\_Z} +1\right )+3 x^{7}+3 \left (x^{8}-2 x^{7}+x^{6}+2 x^{4}-2 x^{3}+1\right )^{\frac {1}{3}} x^{5}-2 x^{6}+2 \RootOf \left (\textit {\_Z}^{2}+\textit {\_Z} +1\right )^{2} x^{3}+3 \left (x^{8}-2 x^{7}+x^{6}+2 x^{4}-2 x^{3}+1\right )^{\frac {2}{3}} \RootOf \left (\textit {\_Z}^{2}+\textit {\_Z} +1\right ) x^{2}-4 \RootOf \left (\textit {\_Z}^{2}+\textit {\_Z} +1\right ) x^{4}-3 \left (x^{8}-2 x^{7}+x^{6}+2 x^{4}-2 x^{3}+1\right )^{\frac {1}{3}} x^{4}+7 \RootOf \left (\textit {\_Z}^{2}+\textit {\_Z} +1\right ) x^{3}+3 \left (x^{8}-2 x^{7}+x^{6}+2 x^{4}-2 x^{3}+1\right )^{\frac {2}{3}} x^{2}-2 x^{4}+3 x^{3}+3 \left (x^{8}-2 x^{7}+x^{6}+2 x^{4}-2 x^{3}+1\right )^{\frac {1}{3}} x -2 \RootOf \left (\textit {\_Z}^{2}+\textit {\_Z} +1\right )-1}{\left (x^{4}-x^{3}+1\right ) \left (x^{4}+1\right )}\right )\right ) \left (\left (x^{4}-x^{3}+1\right )^{2}\right )^{\frac {1}{3}}}{\left (-x^{4}+x^{3}-1\right )^{\frac {2}{3}}}\)	\(689\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

3*(-x^4+x^3-1)^(1/3)/x-3*ln((153*RootOf(9*_Z^2+3*_Z+1)^2*x^4-306*RootOf(9*_Z^2+3*_Z+1)^2*x^3+6*RootOf(9*_Z^2+3
*_Z+1)*x^4+141*RootOf(9*_Z^2+3*_Z+1)*(-x^4+x^3-1)^(2/3)*x+141*RootOf(9*_Z^2+3*_Z+1)*(-x^4+x^3-1)^(1/3)*x^2+39*
RootOf(9*_Z^2+3*_Z+1)*x^3-15*x^4+49*(-x^4+x^3-1)^(2/3)*x+49*x^2*(-x^4+x^3-1)^(1/3)+15*x^3+153*RootOf(9*_Z^2+3*
_Z+1)^2+6*RootOf(9*_Z^2+3*_Z+1)-15)/(x^4+1))*RootOf(9*_Z^2+3*_Z+1)+3*RootOf(9*_Z^2+3*_Z+1)*ln((153*RootOf(9*_Z
^2+3*_Z+1)^2*x^4-306*RootOf(9*_Z^2+3*_Z+1)^2*x^3+96*RootOf(9*_Z^2+3*_Z+1)*x^4-141*RootOf(9*_Z^2+3*_Z+1)*(-x^4+
x^3-1)^(2/3)*x-141*RootOf(9*_Z^2+3*_Z+1)*(-x^4+x^3-1)^(1/3)*x^2-243*RootOf(9*_Z^2+3*_Z+1)*x^3+2*(-x^4+x^3-1)^(
2/3)*x+2*x^2*(-x^4+x^3-1)^(1/3)-32*x^3+153*RootOf(9*_Z^2+3*_Z+1)^2+96*RootOf(9*_Z^2+3*_Z+1))/(x^4+1))-ln((153*
RootOf(9*_Z^2+3*_Z+1)^2*x^4-306*RootOf(9*_Z^2+3*_Z+1)^2*x^3+6*RootOf(9*_Z^2+3*_Z+1)*x^4+141*RootOf(9*_Z^2+3*_Z
+1)*(-x^4+x^3-1)^(2/3)*x+141*RootOf(9*_Z^2+3*_Z+1)*(-x^4+x^3-1)^(1/3)*x^2+39*RootOf(9*_Z^2+3*_Z+1)*x^3-15*x^4+
49*(-x^4+x^3-1)^(2/3)*x+49*x^2*(-x^4+x^3-1)^(1/3)+15*x^3+153*RootOf(9*_Z^2+3*_Z+1)^2+6*RootOf(9*_Z^2+3*_Z+1)-1
5)/(x^4+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} + x^{3} - 1\right )}^{\frac {1}{3}}}{{\left (x^{4} + 1\right )} x^{2}}\,{d x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate((x^4 - 3)*(-x^4 + x^3 - 1)^(1/3)/((x^4 + 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^4-3\right )\,{\left (-x^4+x^3-1\right )}^{1/3}}{x^2\,\left (x^4+1\right )} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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