Optimal. Leaf size=118 \[ \frac {3 \left (1+x^3+x^4\right )^{2/3} \left (2-3 x^3+2 x^4\right )}{10 x^5}+\sqrt {3} \text {ArcTan}\left (\frac {\sqrt {3} x}{x+2 \sqrt [3]{1+x^3+x^4}}\right )-\log \left (-x+\sqrt [3]{1+x^3+x^4}\right )+\frac {1}{2} \log \left (x^2+x \sqrt [3]{1+x^3+x^4}+\left (1+x^3+x^4\right )^{2/3}\right ) \]
[Out]
________________________________________________________________________________________
Rubi [F]
time = 1.08, 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^3+x^4\right ) \left (1+x^3+x^4\right )^{2/3}}{x^6 \left (1+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^3+x^4\right ) \left (1+x^3+x^4\right )^{2/3}}{x^6 \left (1+x^4\right )} \, dx &=\int \left (-\frac {3 \left (1+x^3+x^4\right )^{2/3}}{x^6}+\frac {3 \left (1+x^3+x^4\right )^{2/3}}{x^3}+\frac {\left (1+x^3+x^4\right )^{2/3}}{x^2}-\frac {4 x \left (1+x^3+x^4\right )^{2/3}}{1+x^4}\right ) \, dx\\ &=-\left (3 \int \frac {\left (1+x^3+x^4\right )^{2/3}}{x^6} \, dx\right )+3 \int \frac {\left (1+x^3+x^4\right )^{2/3}}{x^3} \, dx-4 \int \frac {x \left (1+x^3+x^4\right )^{2/3}}{1+x^4} \, dx+\int \frac {\left (1+x^3+x^4\right )^{2/3}}{x^2} \, dx\\ &=-\left (3 \int \frac {\left (1+x^3+x^4\right )^{2/3}}{x^6} \, dx\right )+3 \int \frac {\left (1+x^3+x^4\right )^{2/3}}{x^3} \, dx-4 \int \left (-\frac {i x \left (1+x^3+x^4\right )^{2/3}}{2 \left (-i+x^2\right )}+\frac {i x \left (1+x^3+x^4\right )^{2/3}}{2 \left (i+x^2\right )}\right ) \, dx+\int \frac {\left (1+x^3+x^4\right )^{2/3}}{x^2} \, dx\\ &=2 i \int \frac {x \left (1+x^3+x^4\right )^{2/3}}{-i+x^2} \, dx-2 i \int \frac {x \left (1+x^3+x^4\right )^{2/3}}{i+x^2} \, dx-3 \int \frac {\left (1+x^3+x^4\right )^{2/3}}{x^6} \, dx+3 \int \frac {\left (1+x^3+x^4\right )^{2/3}}{x^3} \, dx+\int \frac {\left (1+x^3+x^4\right )^{2/3}}{x^2} \, dx\\ &=2 i \int \left (-\frac {\left (1+x^3+x^4\right )^{2/3}}{2 \left (\sqrt [4]{-1}-x\right )}+\frac {\left (1+x^3+x^4\right )^{2/3}}{2 \left (\sqrt [4]{-1}+x\right )}\right ) \, dx-2 i \int \left (-\frac {\left (1+x^3+x^4\right )^{2/3}}{2 \left (-(-1)^{3/4}-x\right )}+\frac {\left (1+x^3+x^4\right )^{2/3}}{2 \left (-(-1)^{3/4}+x\right )}\right ) \, dx-3 \int \frac {\left (1+x^3+x^4\right )^{2/3}}{x^6} \, dx+3 \int \frac {\left (1+x^3+x^4\right )^{2/3}}{x^3} \, dx+\int \frac {\left (1+x^3+x^4\right )^{2/3}}{x^2} \, dx\\ &=-\left (i \int \frac {\left (1+x^3+x^4\right )^{2/3}}{\sqrt [4]{-1}-x} \, dx\right )+i \int \frac {\left (1+x^3+x^4\right )^{2/3}}{-(-1)^{3/4}-x} \, dx+i \int \frac {\left (1+x^3+x^4\right )^{2/3}}{\sqrt [4]{-1}+x} \, dx-i \int \frac {\left (1+x^3+x^4\right )^{2/3}}{-(-1)^{3/4}+x} \, dx-3 \int \frac {\left (1+x^3+x^4\right )^{2/3}}{x^6} \, dx+3 \int \frac {\left (1+x^3+x^4\right )^{2/3}}{x^3} \, dx+\int \frac {\left (1+x^3+x^4\right )^{2/3}}{x^2} \, dx\\ \end {align*}
________________________________________________________________________________________
Mathematica [A]
time = 1.24, size = 118, normalized size = 1.00 \begin {gather*} \frac {3 \left (1+x^3+x^4\right )^{2/3} \left (2-3 x^3+2 x^4\right )}{10 x^5}+\sqrt {3} \text {ArcTan}\left (\frac {\sqrt {3} x}{x+2 \sqrt [3]{1+x^3+x^4}}\right )-\log \left (-x+\sqrt [3]{1+x^3+x^4}\right )+\frac {1}{2} \log \left (x^2+x \sqrt [3]{1+x^3+x^4}+\left (1+x^3+x^4\right )^{2/3}\right ) \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [C] Result contains higher order function than in optimal. Order 9 vs. order
3.
time = 6.00, size = 442, normalized size = 3.75
method | result | size |
risch | \(\frac {\frac {3}{5} x^{8}-\frac {3}{10} x^{7}-\frac {9}{10} x^{6}+\frac {6}{5} x^{4}-\frac {3}{10} x^{3}+\frac {3}{5}}{x^{5} \left (x^{4}+x^{3}+1\right )^{\frac {1}{3}}}+\RootOf \left (\textit {\_Z}^{2}-\textit {\_Z} +1\right ) \ln \left (\frac {-\RootOf \left (\textit {\_Z}^{2}-\textit {\_Z} +1\right )^{2} x^{3}+\RootOf \left (\textit {\_Z}^{2}-\textit {\_Z} +1\right ) x^{4}+3 \left (x^{4}+x^{3}+1\right )^{\frac {2}{3}} \RootOf \left (\textit {\_Z}^{2}-\textit {\_Z} +1\right ) x +3 \left (x^{4}+x^{3}+1\right )^{\frac {1}{3}} \RootOf \left (\textit {\_Z}^{2}-\textit {\_Z} +1\right ) x^{2}+4 \RootOf \left (\textit {\_Z}^{2}-\textit {\_Z} +1\right ) x^{3}-2 x^{4}-3 x \left (x^{4}+x^{3}+1\right )^{\frac {2}{3}}-3 \left (x^{4}+x^{3}+1\right )^{\frac {1}{3}} x^{2}-4 x^{3}+\RootOf \left (\textit {\_Z}^{2}-\textit {\_Z} +1\right )-2}{x^{4}+1}\right )-\ln \left (-\frac {\RootOf \left (\textit {\_Z}^{2}-\textit {\_Z} +1\right )^{2} x^{3}+\RootOf \left (\textit {\_Z}^{2}-\textit {\_Z} +1\right ) x^{4}+3 \left (x^{4}+x^{3}+1\right )^{\frac {2}{3}} \RootOf \left (\textit {\_Z}^{2}-\textit {\_Z} +1\right ) x +3 \left (x^{4}+x^{3}+1\right )^{\frac {1}{3}} \RootOf \left (\textit {\_Z}^{2}-\textit {\_Z} +1\right ) x^{2}+2 \RootOf \left (\textit {\_Z}^{2}-\textit {\_Z} +1\right ) x^{3}+x^{4}+x^{3}+\RootOf \left (\textit {\_Z}^{2}-\textit {\_Z} +1\right )+1}{x^{4}+1}\right ) \RootOf \left (\textit {\_Z}^{2}-\textit {\_Z} +1\right )+\ln \left (-\frac {\RootOf \left (\textit {\_Z}^{2}-\textit {\_Z} +1\right )^{2} x^{3}+\RootOf \left (\textit {\_Z}^{2}-\textit {\_Z} +1\right ) x^{4}+3 \left (x^{4}+x^{3}+1\right )^{\frac {2}{3}} \RootOf \left (\textit {\_Z}^{2}-\textit {\_Z} +1\right ) x +3 \left (x^{4}+x^{3}+1\right )^{\frac {1}{3}} \RootOf \left (\textit {\_Z}^{2}-\textit {\_Z} +1\right ) x^{2}+2 \RootOf \left (\textit {\_Z}^{2}-\textit {\_Z} +1\right ) x^{3}+x^{4}+x^{3}+\RootOf \left (\textit {\_Z}^{2}-\textit {\_Z} +1\right )+1}{x^{4}+1}\right )\) | \(442\) |
trager | \(\frac {3 \left (x^{4}+x^{3}+1\right )^{\frac {2}{3}} \left (2 x^{4}-3 x^{3}+2\right )}{10 x^{5}}-3 \ln \left (-\frac {66672 \RootOf \left (9 \textit {\_Z}^{2}-3 \textit {\_Z} +1\right )^{2} x^{4}-133344 \RootOf \left (9 \textit {\_Z}^{2}-3 \textit {\_Z} +1\right )^{2} x^{3}+54915 \RootOf \left (9 \textit {\_Z}^{2}-3 \textit {\_Z} +1\right ) x^{4}-37851 \RootOf \left (9 \textit {\_Z}^{2}-3 \textit {\_Z} +1\right ) \left (x^{4}+x^{3}+1\right )^{\frac {2}{3}} x -96783 \RootOf \left (9 \textit {\_Z}^{2}-3 \textit {\_Z} +1\right ) \left (x^{4}+x^{3}+1\right )^{\frac {1}{3}} x^{2}+179082 \RootOf \left (9 \textit {\_Z}^{2}-3 \textit {\_Z} +1\right ) x^{3}-11757 x^{4}-32261 x \left (x^{4}+x^{3}+1\right )^{\frac {2}{3}}+44878 \left (x^{4}+x^{3}+1\right )^{\frac {1}{3}} x^{2}-27433 x^{3}+66672 \RootOf \left (9 \textit {\_Z}^{2}-3 \textit {\_Z} +1\right )^{2}+54915 \RootOf \left (9 \textit {\_Z}^{2}-3 \textit {\_Z} +1\right )-11757}{x^{4}+1}\right ) \RootOf \left (9 \textit {\_Z}^{2}-3 \textit {\_Z} +1\right )-\ln \left (\frac {31401 \RootOf \left (9 \textit {\_Z}^{2}-3 \textit {\_Z} +1\right )^{2} x^{4}-62802 \RootOf \left (9 \textit {\_Z}^{2}-3 \textit {\_Z} +1\right )^{2} x^{3}-30111 \RootOf \left (9 \textit {\_Z}^{2}-3 \textit {\_Z} +1\right ) x^{4}-37851 \RootOf \left (9 \textit {\_Z}^{2}-3 \textit {\_Z} +1\right ) \left (x^{4}+x^{3}+1\right )^{\frac {2}{3}} x +134634 \RootOf \left (9 \textit {\_Z}^{2}-3 \textit {\_Z} +1\right ) \left (x^{4}+x^{3}+1\right )^{\frac {1}{3}} x^{2}-75849 \RootOf \left (9 \textit {\_Z}^{2}-3 \textit {\_Z} +1\right ) x^{3}-15676 x^{4}+44878 x \left (x^{4}+x^{3}+1\right )^{\frac {2}{3}}-32261 \left (x^{4}+x^{3}+1\right )^{\frac {1}{3}} x^{2}-19595 x^{3}+31401 \RootOf \left (9 \textit {\_Z}^{2}-3 \textit {\_Z} +1\right )^{2}-30111 \RootOf \left (9 \textit {\_Z}^{2}-3 \textit {\_Z} +1\right )-15676}{x^{4}+1}\right )+\ln \left (-\frac {66672 \RootOf \left (9 \textit {\_Z}^{2}-3 \textit {\_Z} +1\right )^{2} x^{4}-133344 \RootOf \left (9 \textit {\_Z}^{2}-3 \textit {\_Z} +1\right )^{2} x^{3}+54915 \RootOf \left (9 \textit {\_Z}^{2}-3 \textit {\_Z} +1\right ) x^{4}-37851 \RootOf \left (9 \textit {\_Z}^{2}-3 \textit {\_Z} +1\right ) \left (x^{4}+x^{3}+1\right )^{\frac {2}{3}} x -96783 \RootOf \left (9 \textit {\_Z}^{2}-3 \textit {\_Z} +1\right ) \left (x^{4}+x^{3}+1\right )^{\frac {1}{3}} x^{2}+179082 \RootOf \left (9 \textit {\_Z}^{2}-3 \textit {\_Z} +1\right ) x^{3}-11757 x^{4}-32261 x \left (x^{4}+x^{3}+1\right )^{\frac {2}{3}}+44878 \left (x^{4}+x^{3}+1\right )^{\frac {1}{3}} x^{2}-27433 x^{3}+66672 \RootOf \left (9 \textit {\_Z}^{2}-3 \textit {\_Z} +1\right )^{2}+54915 \RootOf \left (9 \textit {\_Z}^{2}-3 \textit {\_Z} +1\right )-11757}{x^{4}+1}\right )\) | \(631\) |
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*} \text {Failed to integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [A]
time = 1.91, size = 153, normalized size = 1.30 \begin {gather*} \frac {10 \, \sqrt {3} x^{5} \arctan \left (-\frac {7043582 \, \sqrt {3} {\left (x^{4} + x^{3} + 1\right )}^{\frac {1}{3}} x^{2} - 984256 \, \sqrt {3} {\left (x^{4} + x^{3} + 1\right )}^{\frac {2}{3}} x + \sqrt {3} {\left (145408 \, x^{4} + 3029663 \, x^{3} + 145408\right )}}{32768 \, x^{4} + 12041757 \, x^{3} + 32768}\right ) - 5 \, x^{5} \log \left (\frac {x^{4} + 3 \, {\left (x^{4} + x^{3} + 1\right )}^{\frac {1}{3}} x^{2} - 3 \, {\left (x^{4} + x^{3} + 1\right )}^{\frac {2}{3}} x + 1}{x^{4} + 1}\right ) + 3 \, {\left (2 \, x^{4} - 3 \, x^{3} + 2\right )} {\left (x^{4} + x^{3} + 1\right )}^{\frac {2}{3}}}{10 \, x^{5}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\left (x^{4} - 3\right ) \left (x^{4} - x^{3} + 1\right ) \left (x^{4} + x^{3} + 1\right )^{\frac {2}{3}}}{x^{6} \left (x^{4} + 1\right )}\, dx \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*} \text {could not integrate} \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^4-3\right )\,{\left (x^4+x^3+1\right )}^{2/3}\,\left (x^4-x^3+1\right )}{x^6\,\left (x^4+1\right )} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________