3.17.94 \(\int \frac {x^6 (4+x^3)}{(1+x^3)^{3/4} (-1-2 x^3-x^6+x^8)} \, dx\)

Optimal. Leaf size=113 \[ \tan ^{-1}\left (\frac {x}{\sqrt [4]{x^3+1}}\right )-\tanh ^{-1}\left (\frac {x}{\sqrt [4]{x^3+1}}\right )-\frac {\tan ^{-1}\left (\frac {\frac {\sqrt {x^3+1}}{\sqrt {2}}-\frac {x^2}{\sqrt {2}}}{x \sqrt [4]{x^3+1}}\right )}{\sqrt {2}}-\frac {\tanh ^{-1}\left (\frac {\sqrt {2} x \sqrt [4]{x^3+1}}{\sqrt {x^3+1}+x^2}\right )}{\sqrt {2}} \]

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Rubi [F]  time = 2.18, 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 {x^6 \left (4+x^3\right )}{\left (1+x^3\right )^{3/4} \left (-1-2 x^3-x^6+x^8\right )} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

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

[Out]

(x^2*Hypergeometric2F1[2/3, 3/4, 5/3, -x^3])/2 + Defer[Int][1/((1 + x^3)^(3/4)*(-1 - x^3 + x^4)), x]/2 + Defer
[Int][x/((1 + x^3)^(3/4)*(-1 - x^3 + x^4)), x]/2 + 2*Defer[Int][x^2/((1 + x^3)^(3/4)*(-1 - x^3 + x^4)), x] + D
efer[Int][x^3/((1 + x^3)^(3/4)*(-1 - x^3 + x^4)), x]/2 + Defer[Int][1/((1 + x^3)^(3/4)*(1 + x^3 + x^4)), x]/2
- Defer[Int][x/((1 + x^3)^(3/4)*(1 + x^3 + x^4)), x]/2 + 2*Defer[Int][x^2/((1 + x^3)^(3/4)*(1 + x^3 + x^4)), x
] + Defer[Int][x^3/((1 + x^3)^(3/4)*(1 + x^3 + x^4)), x]/2

Rubi steps

\begin {align*} \int \frac {x^6 \left (4+x^3\right )}{\left (1+x^3\right )^{3/4} \left (-1-2 x^3-x^6+x^8\right )} \, dx &=\int \left (\frac {x}{\left (1+x^3\right )^{3/4}}+\frac {x \left (1+2 x^3+4 x^5+x^6\right )}{\left (1+x^3\right )^{3/4} \left (-1-2 x^3-x^6+x^8\right )}\right ) \, dx\\ &=\int \frac {x}{\left (1+x^3\right )^{3/4}} \, dx+\int \frac {x \left (1+2 x^3+4 x^5+x^6\right )}{\left (1+x^3\right )^{3/4} \left (-1-2 x^3-x^6+x^8\right )} \, dx\\ &=\frac {1}{2} x^2 \, _2F_1\left (\frac {2}{3},\frac {3}{4};\frac {5}{3};-x^3\right )+\int \left (\frac {1+x+4 x^2+x^3}{2 \left (1+x^3\right )^{3/4} \left (-1-x^3+x^4\right )}+\frac {1-x+4 x^2+x^3}{2 \left (1+x^3\right )^{3/4} \left (1+x^3+x^4\right )}\right ) \, dx\\ &=\frac {1}{2} x^2 \, _2F_1\left (\frac {2}{3},\frac {3}{4};\frac {5}{3};-x^3\right )+\frac {1}{2} \int \frac {1+x+4 x^2+x^3}{\left (1+x^3\right )^{3/4} \left (-1-x^3+x^4\right )} \, dx+\frac {1}{2} \int \frac {1-x+4 x^2+x^3}{\left (1+x^3\right )^{3/4} \left (1+x^3+x^4\right )} \, dx\\ &=\frac {1}{2} x^2 \, _2F_1\left (\frac {2}{3},\frac {3}{4};\frac {5}{3};-x^3\right )+\frac {1}{2} \int \left (\frac {1}{\left (1+x^3\right )^{3/4} \left (-1-x^3+x^4\right )}+\frac {x}{\left (1+x^3\right )^{3/4} \left (-1-x^3+x^4\right )}+\frac {4 x^2}{\left (1+x^3\right )^{3/4} \left (-1-x^3+x^4\right )}+\frac {x^3}{\left (1+x^3\right )^{3/4} \left (-1-x^3+x^4\right )}\right ) \, dx+\frac {1}{2} \int \left (\frac {1}{\left (1+x^3\right )^{3/4} \left (1+x^3+x^4\right )}-\frac {x}{\left (1+x^3\right )^{3/4} \left (1+x^3+x^4\right )}+\frac {4 x^2}{\left (1+x^3\right )^{3/4} \left (1+x^3+x^4\right )}+\frac {x^3}{\left (1+x^3\right )^{3/4} \left (1+x^3+x^4\right )}\right ) \, dx\\ &=\frac {1}{2} x^2 \, _2F_1\left (\frac {2}{3},\frac {3}{4};\frac {5}{3};-x^3\right )+\frac {1}{2} \int \frac {1}{\left (1+x^3\right )^{3/4} \left (-1-x^3+x^4\right )} \, dx+\frac {1}{2} \int \frac {x}{\left (1+x^3\right )^{3/4} \left (-1-x^3+x^4\right )} \, dx+\frac {1}{2} \int \frac {x^3}{\left (1+x^3\right )^{3/4} \left (-1-x^3+x^4\right )} \, dx+\frac {1}{2} \int \frac {1}{\left (1+x^3\right )^{3/4} \left (1+x^3+x^4\right )} \, dx-\frac {1}{2} \int \frac {x}{\left (1+x^3\right )^{3/4} \left (1+x^3+x^4\right )} \, dx+\frac {1}{2} \int \frac {x^3}{\left (1+x^3\right )^{3/4} \left (1+x^3+x^4\right )} \, dx+2 \int \frac {x^2}{\left (1+x^3\right )^{3/4} \left (-1-x^3+x^4\right )} \, dx+2 \int \frac {x^2}{\left (1+x^3\right )^{3/4} \left (1+x^3+x^4\right )} \, dx\\ \end {align*}

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Mathematica [F]  time = 0.36, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {x^6 \left (4+x^3\right )}{\left (1+x^3\right )^{3/4} \left (-1-2 x^3-x^6+x^8\right )} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

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

[Out]

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

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IntegrateAlgebraic [A]  time = 3.72, size = 113, normalized size = 1.00 \begin {gather*} \tan ^{-1}\left (\frac {x}{\sqrt [4]{1+x^3}}\right )-\frac {\tan ^{-1}\left (\frac {-\frac {x^2}{\sqrt {2}}+\frac {\sqrt {1+x^3}}{\sqrt {2}}}{x \sqrt [4]{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^6*(4 + x^3))/((1 + x^3)^(3/4)*(-1 - 2*x^3 - x^6 + x^8)),x]

[Out]

ArcTan[x/(1 + x^3)^(1/4)] - ArcTan[(-(x^2/Sqrt[2]) + Sqrt[1 + x^3]/Sqrt[2])/(x*(1 + x^3)^(1/4))]/Sqrt[2] - Arc
Tanh[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 = 1.23, size = 236, normalized size = 2.09 \begin {gather*} \sqrt {2} \arctan \left (\frac {\sqrt {2} x \sqrt {\frac {x^{2} + \sqrt {2} {\left (x^{3} + 1\right )}^{\frac {1}{4}} x + \sqrt {x^{3} + 1}}{x^{2}}} - x - \sqrt {2} {\left (x^{3} + 1\right )}^{\frac {1}{4}}}{x}\right ) + \sqrt {2} \arctan \left (\frac {\sqrt {2} x \sqrt {\frac {x^{2} - \sqrt {2} {\left (x^{3} + 1\right )}^{\frac {1}{4}} x + \sqrt {x^{3} + 1}}{x^{2}}} + x - \sqrt {2} {\left (x^{3} + 1\right )}^{\frac {1}{4}}}{x}\right ) - \frac {1}{4} \, \sqrt {2} \log \left (\frac {4 \, {\left (x^{2} + \sqrt {2} {\left (x^{3} + 1\right )}^{\frac {1}{4}} x + \sqrt {x^{3} + 1}\right )}}{x^{2}}\right ) + \frac {1}{4} \, \sqrt {2} \log \left (\frac {4 \, {\left (x^{2} - \sqrt {2} {\left (x^{3} + 1\right )}^{\frac {1}{4}} x + \sqrt {x^{3} + 1}\right )}}{x^{2}}\right ) - \arctan \left (\frac {{\left (x^{3} + 1\right )}^{\frac {1}{4}}}{x}\right ) - \frac {1}{2} \, \log \left (\frac {x + {\left (x^{3} + 1\right )}^{\frac {1}{4}}}{x}\right ) + \frac {1}{2} \, \log \left (-\frac {x - {\left (x^{3} + 1\right )}^{\frac {1}{4}}}{x}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maple [C]  time = 3.07, size = 444, normalized size = 3.93

method	result	size
trager	\(\frac {\RootOf \left (\textit {\_Z}^{2}+1\right ) \ln \left (-\frac {-2 \RootOf \left (\textit {\_Z}^{2}+1\right ) \sqrt {x^{3}+1}\, x^{2}+\RootOf \left (\textit {\_Z}^{2}+1\right ) x^{4}+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}^{2}+1\right ) x^{3}+\RootOf \left (\textit {\_Z}^{2}+1\right )}{x^{4}-x^{3}-1}\right )}{2}+\frac {\RootOf \left (\textit {\_Z}^{2}-\RootOf \left (\textit {\_Z}^{2}+1\right )\right ) \ln \left (\frac {-\RootOf \left (\textit {\_Z}^{2}-\RootOf \left (\textit {\_Z}^{2}+1\right )\right ) \RootOf \left (\textit {\_Z}^{2}+1\right ) x^{4}-2 \RootOf \left (\textit {\_Z}^{2}-\RootOf \left (\textit {\_Z}^{2}+1\right )\right ) \sqrt {x^{3}+1}\, x^{2}+2 \RootOf \left (\textit {\_Z}^{2}+1\right ) \left (x^{3}+1\right )^{\frac {1}{4}} x^{3}+\RootOf \left (\textit {\_Z}^{2}-\RootOf \left (\textit {\_Z}^{2}+1\right )\right ) \RootOf \left (\textit {\_Z}^{2}+1\right ) x^{3}+2 \left (x^{3}+1\right )^{\frac {3}{4}} x +\RootOf \left (\textit {\_Z}^{2}+1\right ) \RootOf \left (\textit {\_Z}^{2}-\RootOf \left (\textit {\_Z}^{2}+1\right )\right )}{x^{4}+x^{3}+1}\right )}{2}+\frac {\RootOf \left (\textit {\_Z}^{2}+1\right ) \RootOf \left (\textit {\_Z}^{2}-\RootOf \left (\textit {\_Z}^{2}+1\right )\right ) \ln \left (\frac {-2 \RootOf \left (\textit {\_Z}^{2}+1\right ) \RootOf \left (\textit {\_Z}^{2}-\RootOf \left (\textit {\_Z}^{2}+1\right )\right ) \sqrt {x^{3}+1}\, x^{2}-2 \RootOf \left (\textit {\_Z}^{2}+1\right ) \left (x^{3}+1\right )^{\frac {1}{4}} x^{3}-\RootOf \left (\textit {\_Z}^{2}-\RootOf \left (\textit {\_Z}^{2}+1\right )\right ) x^{4}+2 \left (x^{3}+1\right )^{\frac {3}{4}} x +\RootOf \left (\textit {\_Z}^{2}-\RootOf \left (\textit {\_Z}^{2}+1\right )\right ) x^{3}+\RootOf \left (\textit {\_Z}^{2}-\RootOf \left (\textit {\_Z}^{2}+1\right )\right )}{x^{4}+x^{3}+1}\right )}{2}+\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}\)	\(444\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

[Out]

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

[Out]

Timed out

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