Optimal. Leaf size=90 \[ \log \left (\sqrt [3]{x^4+1}-x\right )-\sqrt {3} \tan ^{-1}\left (\frac {\sqrt {3} x}{2 \sqrt [3]{x^4+1}+x}\right )+\frac {3 \left (x^4+1\right )^{2/3}}{2 x^2}-\frac {1}{2} \log \left (\sqrt [3]{x^4+1} x+\left (x^4+1\right )^{2/3}+x^2\right ) \]
________________________________________________________________________________________
Rubi [F] time = 1.10, 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 {\left (-3+x^4\right ) \left (1+x^4\right )^{2/3}}{x^3 \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 {\left (-3+x^4\right ) \left (1+x^4\right )^{2/3}}{x^3 \left (1-x^3+x^4\right )} \, dx &=\int \left (-\frac {3 \left (1+x^4\right )^{2/3}}{x^3}+\frac {(-3+4 x) \left (1+x^4\right )^{2/3}}{1-x^3+x^4}\right ) \, dx\\ &=-\left (3 \int \frac {\left (1+x^4\right )^{2/3}}{x^3} \, dx\right )+\int \frac {(-3+4 x) \left (1+x^4\right )^{2/3}}{1-x^3+x^4} \, dx\\ &=-\left (\frac {3}{2} \operatorname {Subst}\left (\int \frac {\left (1+x^2\right )^{2/3}}{x^2} \, dx,x,x^2\right )\right )+\int \left (-\frac {3 \left (1+x^4\right )^{2/3}}{1-x^3+x^4}+\frac {4 x \left (1+x^4\right )^{2/3}}{1-x^3+x^4}\right ) \, dx\\ &=\frac {3 \left (1+x^4\right )^{2/3}}{2 x^2}-2 \operatorname {Subst}\left (\int \frac {1}{\sqrt [3]{1+x^2}} \, dx,x,x^2\right )-3 \int \frac {\left (1+x^4\right )^{2/3}}{1-x^3+x^4} \, dx+4 \int \frac {x \left (1+x^4\right )^{2/3}}{1-x^3+x^4} \, dx\\ &=\frac {3 \left (1+x^4\right )^{2/3}}{2 x^2}-3 \int \frac {\left (1+x^4\right )^{2/3}}{1-x^3+x^4} \, dx+4 \int \frac {x \left (1+x^4\right )^{2/3}}{1-x^3+x^4} \, dx-\frac {\left (3 \sqrt {x^4}\right ) \operatorname {Subst}\left (\int \frac {x}{\sqrt {-1+x^3}} \, dx,x,\sqrt [3]{1+x^4}\right )}{x^2}\\ &=\frac {3 \left (1+x^4\right )^{2/3}}{2 x^2}-3 \int \frac {\left (1+x^4\right )^{2/3}}{1-x^3+x^4} \, dx+4 \int \frac {x \left (1+x^4\right )^{2/3}}{1-x^3+x^4} \, dx+\frac {\left (3 \sqrt {x^4}\right ) \operatorname {Subst}\left (\int \frac {1+\sqrt {3}-x}{\sqrt {-1+x^3}} \, dx,x,\sqrt [3]{1+x^4}\right )}{x^2}-\frac {\left (3 \sqrt {2 \left (2+\sqrt {3}\right )} \sqrt {x^4}\right ) \operatorname {Subst}\left (\int \frac {1}{\sqrt {-1+x^3}} \, dx,x,\sqrt [3]{1+x^4}\right )}{x^2}\\ &=\frac {3 \left (1+x^4\right )^{2/3}}{2 x^2}+\frac {6 x^2}{1-\sqrt {3}-\sqrt [3]{1+x^4}}-\frac {3 \sqrt [4]{3} \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}} E\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 )}{x^2 \sqrt {-\frac {1-\sqrt [3]{1+x^4}}{\left (1-\sqrt {3}-\sqrt [3]{1+x^4}\right )^2}}}+\frac {2 \sqrt {2} 3^{3/4} \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 )}{x^2 \sqrt {-\frac {1-\sqrt [3]{1+x^4}}{\left (1-\sqrt {3}-\sqrt [3]{1+x^4}\right )^2}}}-3 \int \frac {\left (1+x^4\right )^{2/3}}{1-x^3+x^4} \, dx+4 \int \frac {x \left (1+x^4\right )^{2/3}}{1-x^3+x^4} \, dx\\ \end {align*}
________________________________________________________________________________________
Mathematica [F] time = 0.22, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\left (-3+x^4\right ) \left (1+x^4\right )^{2/3}}{x^3 \left (1-x^3+x^4\right )} \, dx \end {gather*}
Verification is not applicable to the result.
[In]
[Out]
________________________________________________________________________________________
IntegrateAlgebraic [A] time = 1.03, size = 90, normalized size = 1.00 \begin {gather*} \frac {3 \left (1+x^4\right )^{2/3}}{2 x^2}-\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 = 3.35, size = 134, normalized size = 1.49 \begin {gather*} -\frac {2 \, \sqrt {3} x^{2} \arctan \left (-\frac {13034 \, \sqrt {3} {\left (x^{4} + 1\right )}^{\frac {1}{3}} x^{2} - 686 \, \sqrt {3} {\left (x^{4} + 1\right )}^{\frac {2}{3}} x + \sqrt {3} {\left (37 \, x^{4} + 6137 \, x^{3} + 37\right )}}{3 \, {\left (x^{4} + 6859 \, x^{3} + 1\right )}}\right ) - x^{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 ) - 3 \, {\left (x^{4} + 1\right )}^{\frac {2}{3}}}{2 \, x^{2}} \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 {{\left (x^{4} + 1\right )}^{\frac {2}{3}} {\left (x^{4} - 3\right )}}{{\left (x^{4} - x^{3} + 1\right )} x^{3}}\,{d x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [C] time = 7.23, size = 267, normalized size = 2.97
method | result | size |
risch | \(\frac {3 \left (x^{4}+1\right )^{\frac {2}{3}}}{2 x^{2}}+\ln \left (\frac {\RootOf \left (\textit {\_Z}^{2}+\textit {\_Z} +1\right )^{2} x^{3}+\RootOf \left (\textit {\_Z}^{2}+\textit {\_Z} +1\right ) x^{4}-\RootOf \left (\textit {\_Z}^{2}+\textit {\_Z} +1\right ) \left (x^{4}+1\right )^{\frac {2}{3}} x -\left (x^{4}+1\right )^{\frac {1}{3}} \RootOf \left (\textit {\_Z}^{2}+\textit {\_Z} +1\right ) x^{2}+2 \RootOf \left (\textit {\_Z}^{2}+\textit {\_Z} +1\right ) x^{3}+x^{4}-2 \left (x^{4}+1\right )^{\frac {2}{3}} x +x^{2} \left (x^{4}+1\right )^{\frac {1}{3}}+x^{3}+\RootOf \left (\textit {\_Z}^{2}+\textit {\_Z} +1\right )+1}{x^{4}-x^{3}+1}\right )+\RootOf \left (\textit {\_Z}^{2}+\textit {\_Z} +1\right ) \ln \left (-\frac {\RootOf \left (\textit {\_Z}^{2}+\textit {\_Z} +1\right )^{2} x^{3}-\RootOf \left (\textit {\_Z}^{2}+\textit {\_Z} +1\right ) x^{4}-\RootOf \left (\textit {\_Z}^{2}+\textit {\_Z} +1\right ) \left (x^{4}+1\right )^{\frac {2}{3}} x +2 \left (x^{4}+1\right )^{\frac {1}{3}} \RootOf \left (\textit {\_Z}^{2}+\textit {\_Z} +1\right ) x^{2}+\RootOf \left (\textit {\_Z}^{2}+\textit {\_Z} +1\right ) x^{3}-x^{4}+\left (x^{4}+1\right )^{\frac {2}{3}} x +x^{2} \left (x^{4}+1\right )^{\frac {1}{3}}-\RootOf \left (\textit {\_Z}^{2}+\textit {\_Z} +1\right )-1}{x^{4}-x^{3}+1}\right )\) | \(267\) |
trager | \(\frac {3 \left (x^{4}+1\right )^{\frac {2}{3}}}{2 x^{2}}-3 \ln \left (\frac {10305 \RootOf \left (9 \textit {\_Z}^{2}+3 \textit {\_Z} +1\right )^{2} x^{4}-20610 \RootOf \left (9 \textit {\_Z}^{2}+3 \textit {\_Z} +1\right )^{2} x^{3}+17859 \RootOf \left (9 \textit {\_Z}^{2}+3 \textit {\_Z} +1\right ) x^{4}+9621 \RootOf \left (9 \textit {\_Z}^{2}+3 \textit {\_Z} +1\right ) \left (x^{4}+1\right )^{\frac {2}{3}} x +9621 \left (x^{4}+1\right )^{\frac {1}{3}} \RootOf \left (9 \textit {\_Z}^{2}+3 \textit {\_Z} +1\right ) x^{2}-11673 \RootOf \left (9 \textit {\_Z}^{2}+3 \textit {\_Z} +1\right ) x^{3}+7554 x^{4}+11217 \left (x^{4}+1\right )^{\frac {2}{3}} x +11217 x^{2} \left (x^{4}+1\right )^{\frac {1}{3}}+2518 x^{3}+10305 \RootOf \left (9 \textit {\_Z}^{2}+3 \textit {\_Z} +1\right )^{2}+17859 \RootOf \left (9 \textit {\_Z}^{2}+3 \textit {\_Z} +1\right )+7554}{x^{4}-x^{3}+1}\right ) \RootOf \left (9 \textit {\_Z}^{2}+3 \textit {\_Z} +1\right )+3 \RootOf \left (9 \textit {\_Z}^{2}+3 \textit {\_Z} +1\right ) \ln \left (-\frac {-10305 \RootOf \left (9 \textit {\_Z}^{2}+3 \textit {\_Z} +1\right )^{2} x^{4}+20610 \RootOf \left (9 \textit {\_Z}^{2}+3 \textit {\_Z} +1\right )^{2} x^{3}+10989 \RootOf \left (9 \textit {\_Z}^{2}+3 \textit {\_Z} +1\right ) x^{4}+9621 \RootOf \left (9 \textit {\_Z}^{2}+3 \textit {\_Z} +1\right ) \left (x^{4}+1\right )^{\frac {2}{3}} x +9621 \left (x^{4}+1\right )^{\frac {1}{3}} \RootOf \left (9 \textit {\_Z}^{2}+3 \textit {\_Z} +1\right ) x^{2}+2067 \RootOf \left (9 \textit {\_Z}^{2}+3 \textit {\_Z} +1\right ) x^{3}-2746 x^{4}-8010 \left (x^{4}+1\right )^{\frac {2}{3}} x -8010 x^{2} \left (x^{4}+1\right )^{\frac {1}{3}}-4119 x^{3}-10305 \RootOf \left (9 \textit {\_Z}^{2}+3 \textit {\_Z} +1\right )^{2}+10989 \RootOf \left (9 \textit {\_Z}^{2}+3 \textit {\_Z} +1\right )-2746}{x^{4}-x^{3}+1}\right )-\ln \left (\frac {10305 \RootOf \left (9 \textit {\_Z}^{2}+3 \textit {\_Z} +1\right )^{2} x^{4}-20610 \RootOf \left (9 \textit {\_Z}^{2}+3 \textit {\_Z} +1\right )^{2} x^{3}+17859 \RootOf \left (9 \textit {\_Z}^{2}+3 \textit {\_Z} +1\right ) x^{4}+9621 \RootOf \left (9 \textit {\_Z}^{2}+3 \textit {\_Z} +1\right ) \left (x^{4}+1\right )^{\frac {2}{3}} x +9621 \left (x^{4}+1\right )^{\frac {1}{3}} \RootOf \left (9 \textit {\_Z}^{2}+3 \textit {\_Z} +1\right ) x^{2}-11673 \RootOf \left (9 \textit {\_Z}^{2}+3 \textit {\_Z} +1\right ) x^{3}+7554 x^{4}+11217 \left (x^{4}+1\right )^{\frac {2}{3}} x +11217 x^{2} \left (x^{4}+1\right )^{\frac {1}{3}}+2518 x^{3}+10305 \RootOf \left (9 \textit {\_Z}^{2}+3 \textit {\_Z} +1\right )^{2}+17859 \RootOf \left (9 \textit {\_Z}^{2}+3 \textit {\_Z} +1\right )+7554}{x^{4}-x^{3}+1}\right )\) | \(607\) |
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 {{\left (x^{4} + 1\right )}^{\frac {2}{3}} {\left (x^{4} - 3\right )}}{{\left (x^{4} - x^{3} + 1\right )} x^{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 {{\left (x^4+1\right )}^{2/3}\,\left (x^4-3\right )}{x^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]
________________________________________________________________________________________