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

Optimal. Leaf size=74 \[ \log \left (\sqrt [3]{x^4+1}-x\right )-\sqrt {3} \tan ^{-1}\left (\frac {\sqrt {3} x}{2 \sqrt [3]{x^4+1}+x}\right )-\frac {1}{2} \log \left (\sqrt [3]{x^4+1} x+\left (x^4+1\right )^{2/3}+x^2\right ) \]

________________________________________________________________________________________

Rubi [F]  time = 0.59, 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 {-3+x^4}{\sqrt [3]{1+x^4} \left (1-x^3+x^4\right )} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

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

[Out]

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

Rubi steps

\begin {align*} \int \frac {-3+x^4}{\sqrt [3]{1+x^4} \left (1-x^3+x^4\right )} \, dx &=\int \left (\frac {1}{\sqrt [3]{1+x^4}}-\frac {4-x^3}{\sqrt [3]{1+x^4} \left (1-x^3+x^4\right )}\right ) \, dx\\ &=\int \frac {1}{\sqrt [3]{1+x^4}} \, dx-\int \frac {4-x^3}{\sqrt [3]{1+x^4} \left (1-x^3+x^4\right )} \, dx\\ &=x \, _2F_1\left (\frac {1}{4},\frac {1}{3};\frac {5}{4};-x^4\right )-\int \left (\frac {4}{\sqrt [3]{1+x^4} \left (1-x^3+x^4\right )}-\frac {x^3}{\sqrt [3]{1+x^4} \left (1-x^3+x^4\right )}\right ) \, dx\\ &=x \, _2F_1\left (\frac {1}{4},\frac {1}{3};\frac {5}{4};-x^4\right )-4 \int \frac {1}{\sqrt [3]{1+x^4} \left (1-x^3+x^4\right )} \, dx+\int \frac {x^3}{\sqrt [3]{1+x^4} \left (1-x^3+x^4\right )} \, dx\\ \end {align*}

________________________________________________________________________________________

Mathematica [F]  time = 0.14, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {-3+x^4}{\sqrt [3]{1+x^4} \left (1-x^3+x^4\right )} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

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

[Out]

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

________________________________________________________________________________________

IntegrateAlgebraic [A]  time = 1.25, size = 74, 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[(-3 + x^4)/((1 + x^4)^(1/3)*(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

________________________________________________________________________________________

fricas [A]  time = 1.74, size = 110, normalized size = 1.49 \begin {gather*} -\sqrt {3} \arctan \left (-\frac {2 \, \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} - x^{3} + 1\right )}}{3 \, {\left (x^{4} + x^{3} + 1\right )}}\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^4-3)/(x^4+1)^(1/3)/(x^4-x^3+1),x, algorithm="fricas")

[Out]

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

________________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

maple [C]  time = 3.77, size = 260, normalized size = 3.51

method	result	size
trager	\(\ln \left (-\frac {-\RootOf \left (\textit {\_Z}^{2}+\textit {\_Z} +1\right )^{2} x^{3}-\RootOf \left (\textit {\_Z}^{2}+\textit {\_Z} +1\right ) x^{4}+\RootOf \left (\textit {\_Z}^{2}+\textit {\_Z} +1\right ) \left (x^{4}+1\right )^{\frac {2}{3}} x +\left (x^{4}+1\right )^{\frac {1}{3}} \RootOf \left (\textit {\_Z}^{2}+\textit {\_Z} +1\right ) x^{2}-2 \RootOf \left (\textit {\_Z}^{2}+\textit {\_Z} +1\right ) x^{3}-x^{4}+2 \left (x^{4}+1\right )^{\frac {2}{3}} x -x^{2} \left (x^{4}+1\right )^{\frac {1}{3}}-x^{3}-\RootOf \left (\textit {\_Z}^{2}+\textit {\_Z} +1\right )-1}{x^{4}-x^{3}+1}\right )+\RootOf \left (\textit {\_Z}^{2}+\textit {\_Z} +1\right ) \ln \left (\frac {-\RootOf \left (\textit {\_Z}^{2}+\textit {\_Z} +1\right )^{2} x^{3}+\RootOf \left (\textit {\_Z}^{2}+\textit {\_Z} +1\right ) x^{4}+\RootOf \left (\textit {\_Z}^{2}+\textit {\_Z} +1\right ) \left (x^{4}+1\right )^{\frac {2}{3}} x -2 \left (x^{4}+1\right )^{\frac {1}{3}} \RootOf \left (\textit {\_Z}^{2}+\textit {\_Z} +1\right ) x^{2}-\RootOf \left (\textit {\_Z}^{2}+\textit {\_Z} +1\right ) x^{3}+x^{4}-\left (x^{4}+1\right )^{\frac {2}{3}} x -x^{2} \left (x^{4}+1\right )^{\frac {1}{3}}+\RootOf \left (\textit {\_Z}^{2}+\textit {\_Z} +1\right )+1}{x^{4}-x^{3}+1}\right )\)	\(260\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

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

[Out]

Timed out

________________________________________________________________________________________