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]
[Out]
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]
[Out]
________________________________________________________________________________________
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]
[Out]
________________________________________________________________________________________
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]
[Out]
________________________________________________________________________________________
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]
[Out]
________________________________________________________________________________________
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]
[Out]
________________________________________________________________________________________
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]
[Out]
________________________________________________________________________________________
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]
[Out]
________________________________________________________________________________________
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]
[Out]
________________________________________________________________________________________