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

Optimal. Leaf size=75 \[ \log \left (\sqrt [3]{x^4+1}+x\right )-\sqrt {3} \tan ^{-1}\left (\frac {\sqrt {3} x}{2 \sqrt [3]{x^4+1}-x}\right )-\frac {1}{2} \log \left (-\sqrt [3]{x^4+1} x+\left (x^4+1\right )^{2/3}+x^2\right ) \]

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Rubi [F]  time = 0.98, 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 \left (-3+x^4\right )}{\left (1+x^4\right )^{2/3} \left (1+x^3+x^4\right )} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

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

[Out]

-1/2*(3^(3/4)*Sqrt[2 - Sqrt[3]]*(1 - (1 + x^4)^(1/3))*Sqrt[(1 + (1 + x^4)^(1/3) + (1 + x^4)^(2/3))/(1 - Sqrt[3
] - (1 + x^4)^(1/3))^2]*EllipticF[ArcSin[(1 + Sqrt[3] - (1 + x^4)^(1/3))/(1 - Sqrt[3] - (1 + x^4)^(1/3))], -7
+ 4*Sqrt[3]])/(x^2*Sqrt[-((1 - (1 + x^4)^(1/3))/(1 - Sqrt[3] - (1 + x^4)^(1/3))^2)]) - x*Hypergeometric2F1[1/4
, 2/3, 5/4, -x^4] + Defer[Int][1/((1 + x^4)^(2/3)*(1 + x^3 + x^4)), x] - 4*Defer[Int][x/((1 + x^4)^(2/3)*(1 +
x^3 + x^4)), x] + Defer[Int][x^3/((1 + x^4)^(2/3)*(1 + x^3 + x^4)), x]

Rubi steps

\begin {align*} \int \frac {x \left (-3+x^4\right )}{\left (1+x^4\right )^{2/3} \left (1+x^3+x^4\right )} \, dx &=\int \left (-\frac {1}{\left (1+x^4\right )^{2/3}}+\frac {x}{\left (1+x^4\right )^{2/3}}+\frac {1-4 x+x^3}{\left (1+x^4\right )^{2/3} \left (1+x^3+x^4\right )}\right ) \, dx\\ &=-\int \frac {1}{\left (1+x^4\right )^{2/3}} \, dx+\int \frac {x}{\left (1+x^4\right )^{2/3}} \, dx+\int \frac {1-4 x+x^3}{\left (1+x^4\right )^{2/3} \left (1+x^3+x^4\right )} \, dx\\ &=-x \, _2F_1\left (\frac {1}{4},\frac {2}{3};\frac {5}{4};-x^4\right )+\frac {1}{2} \operatorname {Subst}\left (\int \frac {1}{\left (1+x^2\right )^{2/3}} \, dx,x,x^2\right )+\int \left (\frac {1}{\left (1+x^4\right )^{2/3} \left (1+x^3+x^4\right )}-\frac {4 x}{\left (1+x^4\right )^{2/3} \left (1+x^3+x^4\right )}+\frac {x^3}{\left (1+x^4\right )^{2/3} \left (1+x^3+x^4\right )}\right ) \, dx\\ &=-x \, _2F_1\left (\frac {1}{4},\frac {2}{3};\frac {5}{4};-x^4\right )-4 \int \frac {x}{\left (1+x^4\right )^{2/3} \left (1+x^3+x^4\right )} \, dx+\frac {\left (3 \sqrt {x^4}\right ) \operatorname {Subst}\left (\int \frac {1}{\sqrt {-1+x^3}} \, dx,x,\sqrt [3]{1+x^4}\right )}{4 x^2}+\int \frac {1}{\left (1+x^4\right )^{2/3} \left (1+x^3+x^4\right )} \, dx+\int \frac {x^3}{\left (1+x^4\right )^{2/3} \left (1+x^3+x^4\right )} \, dx\\ &=-\frac {3^{3/4} \sqrt {2-\sqrt {3}} \left (1-\sqrt [3]{1+x^4}\right ) \sqrt {\frac {1+\sqrt [3]{1+x^4}+\left (1+x^4\right )^{2/3}}{\left (1-\sqrt {3}-\sqrt [3]{1+x^4}\right )^2}} F\left (\sin ^{-1}\left (\frac {1+\sqrt {3}-\sqrt [3]{1+x^4}}{1-\sqrt {3}-\sqrt [3]{1+x^4}}\right )|-7+4 \sqrt {3}\right )}{2 x^2 \sqrt {-\frac {1-\sqrt [3]{1+x^4}}{\left (1-\sqrt {3}-\sqrt [3]{1+x^4}\right )^2}}}-x \, _2F_1\left (\frac {1}{4},\frac {2}{3};\frac {5}{4};-x^4\right )-4 \int \frac {x}{\left (1+x^4\right )^{2/3} \left (1+x^3+x^4\right )} \, dx+\int \frac {1}{\left (1+x^4\right )^{2/3} \left (1+x^3+x^4\right )} \, dx+\int \frac {x^3}{\left (1+x^4\right )^{2/3} \left (1+x^3+x^4\right )} \, dx\\ \end {align*}

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Mathematica [F]  time = 0.17, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {x \left (-3+x^4\right )}{\left (1+x^4\right )^{2/3} \left (1+x^3+x^4\right )} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

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

[Out]

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

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IntegrateAlgebraic [A]  time = 1.70, size = 75, normalized size = 1.00 \begin {gather*} -\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[(x*(-3 + x^4))/((1 + x^4)^(2/3)*(1 + x^3 + x^4)),x]

[Out]

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

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fricas [A]  time = 1.51, size = 102, normalized size = 1.36 \begin {gather*} -\sqrt {3} \arctan \left (\frac {4 \, \sqrt {3} {\left (x^{4} + 1\right )}^{\frac {1}{3}} x^{2} + 2 \, \sqrt {3} {\left (x^{4} + 1\right )}^{\frac {2}{3}} x + \sqrt {3} {\left (x^{4} + 1\right )}}{x^{4} - 8 \, x^{3} + 1}\right ) + \frac {1}{2} \, \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 ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maple [C]  time = 1.96, size = 287, normalized size = 3.83

method	result	size
trager	\(\RootOf \left (\textit {\_Z}^{2}+\textit {\_Z} +1\right ) \ln \left (\frac {\RootOf \left (\textit {\_Z}^{2}+\textit {\_Z} +1\right ) \left (x^{4}+1\right )^{\frac {2}{3}} x -\left (x^{4}+1\right )^{\frac {1}{3}} \RootOf \left (\textit {\_Z}^{2}+\textit {\_Z} +1\right ) x^{2}+\RootOf \left (\textit {\_Z}^{2}+\textit {\_Z} +1\right ) x^{3}+x^{4}-\left (x^{4}+1\right )^{\frac {2}{3}} x +x^{2} \left (x^{4}+1\right )^{\frac {1}{3}}+1}{x^{4}+x^{3}+1}\right )-\ln \left (-\frac {\RootOf \left (\textit {\_Z}^{2}+\textit {\_Z} +1\right ) \left (x^{4}+1\right )^{\frac {2}{3}} x -\left (x^{4}+1\right )^{\frac {1}{3}} \RootOf \left (\textit {\_Z}^{2}+\textit {\_Z} +1\right ) x^{2}+\RootOf \left (\textit {\_Z}^{2}+\textit {\_Z} +1\right ) x^{3}-x^{4}+2 \left (x^{4}+1\right )^{\frac {2}{3}} x -2 x^{2} \left (x^{4}+1\right )^{\frac {1}{3}}+x^{3}-1}{x^{4}+x^{3}+1}\right ) \RootOf \left (\textit {\_Z}^{2}+\textit {\_Z} +1\right )-\ln \left (-\frac {\RootOf \left (\textit {\_Z}^{2}+\textit {\_Z} +1\right ) \left (x^{4}+1\right )^{\frac {2}{3}} x -\left (x^{4}+1\right )^{\frac {1}{3}} \RootOf \left (\textit {\_Z}^{2}+\textit {\_Z} +1\right ) x^{2}+\RootOf \left (\textit {\_Z}^{2}+\textit {\_Z} +1\right ) x^{3}-x^{4}+2 \left (x^{4}+1\right )^{\frac {2}{3}} x -2 x^{2} \left (x^{4}+1\right )^{\frac {1}{3}}+x^{3}-1}{x^{4}+x^{3}+1}\right )\)	\(287\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

RootOf(_Z^2+_Z+1)*ln((RootOf(_Z^2+_Z+1)*(x^4+1)^(2/3)*x-(x^4+1)^(1/3)*RootOf(_Z^2+_Z+1)*x^2+RootOf(_Z^2+_Z+1)*
x^3+x^4-(x^4+1)^(2/3)*x+x^2*(x^4+1)^(1/3)+1)/(x^4+x^3+1))-ln(-(RootOf(_Z^2+_Z+1)*(x^4+1)^(2/3)*x-(x^4+1)^(1/3)
*RootOf(_Z^2+_Z+1)*x^2+RootOf(_Z^2+_Z+1)*x^3-x^4+2*(x^4+1)^(2/3)*x-2*x^2*(x^4+1)^(1/3)+x^3-1)/(x^4+x^3+1))*Roo
tOf(_Z^2+_Z+1)-ln(-(RootOf(_Z^2+_Z+1)*(x^4+1)^(2/3)*x-(x^4+1)^(1/3)*RootOf(_Z^2+_Z+1)*x^2+RootOf(_Z^2+_Z+1)*x^
3-x^4+2*(x^4+1)^(2/3)*x-2*x^2*(x^4+1)^(1/3)+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^{4} - 3\right )} x}{{\left (x^{4} + x^{3} + 1\right )} {\left (x^{4} + 1\right )}^{\frac {2}{3}}}\,{d x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

[Out]

Timed out

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