3.16.68 \(\int \frac {x^4 (-4+x^3)}{\sqrt [4]{-1+x^3} (-1+2 x^3-x^6+x^8)} \, dx\)

Optimal. Leaf size=107 \[ \tan ^{-1}\left (\frac {\sqrt [4]{x^3-1}}{x}\right )-\tanh ^{-1}\left (\frac {x}{\sqrt [4]{x^3-1}}\right )+\frac {\tan ^{-1}\left (\frac {\sqrt {2} x \sqrt [4]{x^3-1}}{\sqrt {x^3-1}-x^2}\right )}{\sqrt {2}}+\frac {\tanh ^{-1}\left (\frac {\sqrt {2} x \sqrt [4]{x^3-1}}{\sqrt {x^3-1}+x^2}\right )}{\sqrt {2}} \]

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Rubi [F]  time = 1.17, 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 {x^4 \left (-4+x^3\right )}{\sqrt [4]{-1+x^3} \left (-1+2 x^3-x^6+x^8\right )} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Int[(x^4*(-4 + x^3))/((-1 + x^3)^(1/4)*(-1 + 2*x^3 - x^6 + x^8)),x]

[Out]

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

Rubi steps

\begin {align*} \int \frac {x^4 \left (-4+x^3\right )}{\sqrt [4]{-1+x^3} \left (-1+2 x^3-x^6+x^8\right )} \, dx &=\int \left (\frac {-4+x^3}{2 \sqrt [4]{-1+x^3} \left (1-x^3+x^4\right )}+\frac {-4+x^3}{2 \sqrt [4]{-1+x^3} \left (-1+x^3+x^4\right )}\right ) \, dx\\ &=\frac {1}{2} \int \frac {-4+x^3}{\sqrt [4]{-1+x^3} \left (1-x^3+x^4\right )} \, dx+\frac {1}{2} \int \frac {-4+x^3}{\sqrt [4]{-1+x^3} \left (-1+x^3+x^4\right )} \, dx\\ &=\frac {1}{2} \int \left (-\frac {4}{\sqrt [4]{-1+x^3} \left (1-x^3+x^4\right )}+\frac {x^3}{\sqrt [4]{-1+x^3} \left (1-x^3+x^4\right )}\right ) \, dx+\frac {1}{2} \int \left (-\frac {4}{\sqrt [4]{-1+x^3} \left (-1+x^3+x^4\right )}+\frac {x^3}{\sqrt [4]{-1+x^3} \left (-1+x^3+x^4\right )}\right ) \, dx\\ &=\frac {1}{2} \int \frac {x^3}{\sqrt [4]{-1+x^3} \left (1-x^3+x^4\right )} \, dx+\frac {1}{2} \int \frac {x^3}{\sqrt [4]{-1+x^3} \left (-1+x^3+x^4\right )} \, dx-2 \int \frac {1}{\sqrt [4]{-1+x^3} \left (1-x^3+x^4\right )} \, dx-2 \int \frac {1}{\sqrt [4]{-1+x^3} \left (-1+x^3+x^4\right )} \, dx\\ \end {align*}

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Mathematica [F]  time = 0.24, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {x^4 \left (-4+x^3\right )}{\sqrt [4]{-1+x^3} \left (-1+2 x^3-x^6+x^8\right )} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Integrate[(x^4*(-4 + x^3))/((-1 + x^3)^(1/4)*(-1 + 2*x^3 - x^6 + x^8)),x]

[Out]

Integrate[(x^4*(-4 + x^3))/((-1 + x^3)^(1/4)*(-1 + 2*x^3 - x^6 + x^8)), x]

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IntegrateAlgebraic [A]  time = 1.14, size = 107, normalized size = 1.00 \begin {gather*} \tan ^{-1}\left (\frac {\sqrt [4]{-1+x^3}}{x}\right )+\frac {\tan ^{-1}\left (\frac {\sqrt {2} x \sqrt [4]{-1+x^3}}{-x^2+\sqrt {-1+x^3}}\right )}{\sqrt {2}}-\tanh ^{-1}\left (\frac {x}{\sqrt [4]{-1+x^3}}\right )+\frac {\tanh ^{-1}\left (\frac {\sqrt {2} x \sqrt [4]{-1+x^3}}{x^2+\sqrt {-1+x^3}}\right )}{\sqrt {2}} \end {gather*}

Antiderivative was successfully verified.

[In]

IntegrateAlgebraic[(x^4*(-4 + x^3))/((-1 + x^3)^(1/4)*(-1 + 2*x^3 - x^6 + x^8)),x]

[Out]

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

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fricas [B]  time = 62.93, size = 784, normalized size = 7.33 \begin {gather*} -\frac {1}{2} \, \sqrt {2} \arctan \left (\frac {x^{8} + 2 \, x^{7} + x^{6} - 2 \, x^{4} - 2 \, x^{3} + 2 \, \sqrt {2} {\left (3 \, x^{5} - x^{4} + x\right )} {\left (x^{3} - 1\right )}^{\frac {3}{4}} + 2 \, \sqrt {2} {\left (x^{7} - 3 \, x^{6} + 3 \, x^{3}\right )} {\left (x^{3} - 1\right )}^{\frac {1}{4}} + 4 \, {\left (x^{6} + x^{5} - x^{2}\right )} \sqrt {x^{3} - 1} + {\left (16 \, {\left (x^{3} - 1\right )}^{\frac {3}{4}} x^{5} + 2 \, \sqrt {2} {\left (3 \, x^{6} - x^{5} + x^{2}\right )} \sqrt {x^{3} - 1} + \sqrt {2} {\left (x^{8} + 8 \, x^{7} - x^{6} - 8 \, x^{4} + 2 \, x^{3} - 1\right )} + 4 \, {\left (x^{7} + x^{6} - x^{3}\right )} {\left (x^{3} - 1\right )}^{\frac {1}{4}}\right )} \sqrt {\frac {x^{4} - 2 \, \sqrt {2} {\left (x^{3} - 1\right )}^{\frac {1}{4}} x^{3} + x^{3} + 4 \, \sqrt {x^{3} - 1} x^{2} - 2 \, \sqrt {2} {\left (x^{3} - 1\right )}^{\frac {3}{4}} x - 1}{x^{4} + x^{3} - 1}} + 1}{x^{8} - 14 \, x^{7} + x^{6} + 14 \, x^{4} - 2 \, x^{3} + 1}\right ) + \frac {1}{2} \, \sqrt {2} \arctan \left (\frac {x^{8} + 2 \, x^{7} + x^{6} - 2 \, x^{4} - 2 \, x^{3} - 2 \, \sqrt {2} {\left (3 \, x^{5} - x^{4} + x\right )} {\left (x^{3} - 1\right )}^{\frac {3}{4}} - 2 \, \sqrt {2} {\left (x^{7} - 3 \, x^{6} + 3 \, x^{3}\right )} {\left (x^{3} - 1\right )}^{\frac {1}{4}} + 4 \, {\left (x^{6} + x^{5} - x^{2}\right )} \sqrt {x^{3} - 1} + {\left (16 \, {\left (x^{3} - 1\right )}^{\frac {3}{4}} x^{5} - 2 \, \sqrt {2} {\left (3 \, x^{6} - x^{5} + x^{2}\right )} \sqrt {x^{3} - 1} - \sqrt {2} {\left (x^{8} + 8 \, x^{7} - x^{6} - 8 \, x^{4} + 2 \, x^{3} - 1\right )} + 4 \, {\left (x^{7} + x^{6} - x^{3}\right )} {\left (x^{3} - 1\right )}^{\frac {1}{4}}\right )} \sqrt {\frac {x^{4} + 2 \, \sqrt {2} {\left (x^{3} - 1\right )}^{\frac {1}{4}} x^{3} + x^{3} + 4 \, \sqrt {x^{3} - 1} x^{2} + 2 \, \sqrt {2} {\left (x^{3} - 1\right )}^{\frac {3}{4}} x - 1}{x^{4} + x^{3} - 1}} + 1}{x^{8} - 14 \, x^{7} + x^{6} + 14 \, x^{4} - 2 \, x^{3} + 1}\right ) + \frac {1}{8} \, \sqrt {2} \log \left (\frac {4 \, {\left (x^{4} + 2 \, \sqrt {2} {\left (x^{3} - 1\right )}^{\frac {1}{4}} x^{3} + x^{3} + 4 \, \sqrt {x^{3} - 1} x^{2} + 2 \, \sqrt {2} {\left (x^{3} - 1\right )}^{\frac {3}{4}} x - 1\right )}}{x^{4} + x^{3} - 1}\right ) - \frac {1}{8} \, \sqrt {2} \log \left (\frac {4 \, {\left (x^{4} - 2 \, \sqrt {2} {\left (x^{3} - 1\right )}^{\frac {1}{4}} x^{3} + x^{3} + 4 \, \sqrt {x^{3} - 1} x^{2} - 2 \, \sqrt {2} {\left (x^{3} - 1\right )}^{\frac {3}{4}} x - 1\right )}}{x^{4} + x^{3} - 1}\right ) + \frac {1}{2} \, \arctan \left (\frac {2 \, {\left ({\left (x^{3} - 1\right )}^{\frac {1}{4}} x^{3} + {\left (x^{3} - 1\right )}^{\frac {3}{4}} x\right )}}{x^{4} - x^{3} + 1}\right ) + \frac {1}{2} \, \log \left (\frac {x^{4} - 2 \, {\left (x^{3} - 1\right )}^{\frac {1}{4}} x^{3} + x^{3} + 2 \, \sqrt {x^{3} - 1} x^{2} - 2 \, {\left (x^{3} - 1\right )}^{\frac {3}{4}} x - 1}{x^{4} - x^{3} + 1}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^4*(x^3-4)/(x^3-1)^(1/4)/(x^8-x^6+2*x^3-1),x, algorithm="fricas")

[Out]

-1/2*sqrt(2)*arctan((x^8 + 2*x^7 + x^6 - 2*x^4 - 2*x^3 + 2*sqrt(2)*(3*x^5 - x^4 + x)*(x^3 - 1)^(3/4) + 2*sqrt(
2)*(x^7 - 3*x^6 + 3*x^3)*(x^3 - 1)^(1/4) + 4*(x^6 + x^5 - x^2)*sqrt(x^3 - 1) + (16*(x^3 - 1)^(3/4)*x^5 + 2*sqr
t(2)*(3*x^6 - x^5 + x^2)*sqrt(x^3 - 1) + sqrt(2)*(x^8 + 8*x^7 - x^6 - 8*x^4 + 2*x^3 - 1) + 4*(x^7 + x^6 - x^3)
*(x^3 - 1)^(1/4))*sqrt((x^4 - 2*sqrt(2)*(x^3 - 1)^(1/4)*x^3 + x^3 + 4*sqrt(x^3 - 1)*x^2 - 2*sqrt(2)*(x^3 - 1)^
(3/4)*x - 1)/(x^4 + x^3 - 1)) + 1)/(x^8 - 14*x^7 + x^6 + 14*x^4 - 2*x^3 + 1)) + 1/2*sqrt(2)*arctan((x^8 + 2*x^
7 + x^6 - 2*x^4 - 2*x^3 - 2*sqrt(2)*(3*x^5 - x^4 + x)*(x^3 - 1)^(3/4) - 2*sqrt(2)*(x^7 - 3*x^6 + 3*x^3)*(x^3 -
1)^(1/4) + 4*(x^6 + x^5 - x^2)*sqrt(x^3 - 1) + (16*(x^3 - 1)^(3/4)*x^5 - 2*sqrt(2)*(3*x^6 - x^5 + x^2)*sqrt(x
^3 - 1) - sqrt(2)*(x^8 + 8*x^7 - x^6 - 8*x^4 + 2*x^3 - 1) + 4*(x^7 + x^6 - x^3)*(x^3 - 1)^(1/4))*sqrt((x^4 + 2
*sqrt(2)*(x^3 - 1)^(1/4)*x^3 + x^3 + 4*sqrt(x^3 - 1)*x^2 + 2*sqrt(2)*(x^3 - 1)^(3/4)*x - 1)/(x^4 + x^3 - 1)) +
1)/(x^8 - 14*x^7 + x^6 + 14*x^4 - 2*x^3 + 1)) + 1/8*sqrt(2)*log(4*(x^4 + 2*sqrt(2)*(x^3 - 1)^(1/4)*x^3 + x^3
+ 4*sqrt(x^3 - 1)*x^2 + 2*sqrt(2)*(x^3 - 1)^(3/4)*x - 1)/(x^4 + x^3 - 1)) - 1/8*sqrt(2)*log(4*(x^4 - 2*sqrt(2)
*(x^3 - 1)^(1/4)*x^3 + x^3 + 4*sqrt(x^3 - 1)*x^2 - 2*sqrt(2)*(x^3 - 1)^(3/4)*x - 1)/(x^4 + x^3 - 1)) + 1/2*arc
tan(2*((x^3 - 1)^(1/4)*x^3 + (x^3 - 1)^(3/4)*x)/(x^4 - x^3 + 1)) + 1/2*log((x^4 - 2*(x^3 - 1)^(1/4)*x^3 + x^3
+ 2*sqrt(x^3 - 1)*x^2 - 2*(x^3 - 1)^(3/4)*x - 1)/(x^4 - x^3 + 1))

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giac [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {{\left (x^{3} - 4\right )} x^{4}}{{\left (x^{8} - x^{6} + 2 \, x^{3} - 1\right )} {\left (x^{3} - 1\right )}^{\frac {1}{4}}}\,{d x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^4*(x^3-4)/(x^3-1)^(1/4)/(x^8-x^6+2*x^3-1),x, algorithm="giac")

[Out]

integrate((x^3 - 4)*x^4/((x^8 - x^6 + 2*x^3 - 1)*(x^3 - 1)^(1/4)), x)

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maple [C]  time = 3.25, size = 368, normalized size = 3.44

method	result	size
trager	\(\frac {\ln \left (\frac {2 \left (x^{3}-1\right )^{\frac {3}{4}} x -2 x^{2} \sqrt {x^{3}-1}+2 \left (x^{3}-1\right )^{\frac {1}{4}} x^{3}-x^{4}-x^{3}+1}{x^{4}-x^{3}+1}\right )}{2}-\frac {\RootOf \left (\textit {\_Z}^{4}+1\right ) \ln \left (-\frac {2 \sqrt {x^{3}-1}\, \RootOf \left (\textit {\_Z}^{4}+1\right )^{3} x^{2}-2 \left (x^{3}-1\right )^{\frac {1}{4}} \RootOf \left (\textit {\_Z}^{4}+1\right )^{2} x^{3}+\RootOf \left (\textit {\_Z}^{4}+1\right ) x^{4}-\RootOf \left (\textit {\_Z}^{4}+1\right ) x^{3}+2 \left (x^{3}-1\right )^{\frac {3}{4}} x +\RootOf \left (\textit {\_Z}^{4}+1\right )}{x^{4}+x^{3}-1}\right )}{2}+\frac {\RootOf \left (\textit {\_Z}^{4}+1\right )^{3} \ln \left (\frac {\RootOf \left (\textit {\_Z}^{4}+1\right )^{3} x^{4}-\RootOf \left (\textit {\_Z}^{4}+1\right )^{3} x^{3}-2 \left (x^{3}-1\right )^{\frac {1}{4}} \RootOf \left (\textit {\_Z}^{4}+1\right )^{2} x^{3}+2 \sqrt {x^{3}-1}\, \RootOf \left (\textit {\_Z}^{4}+1\right ) x^{2}-2 \left (x^{3}-1\right )^{\frac {3}{4}} x +\RootOf \left (\textit {\_Z}^{4}+1\right )^{3}}{x^{4}+x^{3}-1}\right )}{2}+\frac {\RootOf \left (\textit {\_Z}^{4}+1\right )^{2} \ln \left (\frac {2 \RootOf \left (\textit {\_Z}^{4}+1\right )^{2} \sqrt {x^{3}-1}\, x^{2}-\RootOf \left (\textit {\_Z}^{4}+1\right )^{2} x^{4}-x^{3} \RootOf \left (\textit {\_Z}^{4}+1\right )^{2}+2 \left (x^{3}-1\right )^{\frac {3}{4}} x -2 \left (x^{3}-1\right )^{\frac {1}{4}} x^{3}+\RootOf \left (\textit {\_Z}^{4}+1\right )^{2}}{x^{4}-x^{3}+1}\right )}{2}\)	\(368\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^4*(x^3-4)/(x^3-1)^(1/4)/(x^8-x^6+2*x^3-1),x,method=_RETURNVERBOSE)

[Out]

1/2*ln((2*(x^3-1)^(3/4)*x-2*x^2*(x^3-1)^(1/2)+2*(x^3-1)^(1/4)*x^3-x^4-x^3+1)/(x^4-x^3+1))-1/2*RootOf(_Z^4+1)*l
n(-(2*(x^3-1)^(1/2)*RootOf(_Z^4+1)^3*x^2-2*(x^3-1)^(1/4)*RootOf(_Z^4+1)^2*x^3+RootOf(_Z^4+1)*x^4-RootOf(_Z^4+1
)*x^3+2*(x^3-1)^(3/4)*x+RootOf(_Z^4+1))/(x^4+x^3-1))+1/2*RootOf(_Z^4+1)^3*ln((RootOf(_Z^4+1)^3*x^4-RootOf(_Z^4
+1)^3*x^3-2*(x^3-1)^(1/4)*RootOf(_Z^4+1)^2*x^3+2*(x^3-1)^(1/2)*RootOf(_Z^4+1)*x^2-2*(x^3-1)^(3/4)*x+RootOf(_Z^
4+1)^3)/(x^4+x^3-1))+1/2*RootOf(_Z^4+1)^2*ln((2*RootOf(_Z^4+1)^2*(x^3-1)^(1/2)*x^2-RootOf(_Z^4+1)^2*x^4-x^3*Ro
otOf(_Z^4+1)^2+2*(x^3-1)^(3/4)*x-2*(x^3-1)^(1/4)*x^3+RootOf(_Z^4+1)^2)/(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^{3} - 4\right )} x^{4}}{{\left (x^{8} - x^{6} + 2 \, x^{3} - 1\right )} {\left (x^{3} - 1\right )}^{\frac {1}{4}}}\,{d x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^4*(x^3-4)/(x^3-1)^(1/4)/(x^8-x^6+2*x^3-1),x, algorithm="maxima")

[Out]

integrate((x^3 - 4)*x^4/((x^8 - x^6 + 2*x^3 - 1)*(x^3 - 1)^(1/4)), x)

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mupad [F]  time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \int \frac {x^4\,\left (x^3-4\right )}{{\left (x^3-1\right )}^{1/4}\,\left (x^8-x^6+2\,x^3-1\right )} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((x^4*(x^3 - 4))/((x^3 - 1)^(1/4)*(2*x^3 - x^6 + x^8 - 1)),x)

[Out]

int((x^4*(x^3 - 4))/((x^3 - 1)^(1/4)*(2*x^3 - x^6 + x^8 - 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*(x**3-4)/(x**3-1)**(1/4)/(x**8-x**6+2*x**3-1),x)

[Out]

Timed out

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