Optimal. Leaf size=241 \[ \frac {1}{2} \sqrt {\frac {1}{2} \left (2-\sqrt {2}\right )} \tan ^{-1}\left (\frac {\sqrt {2-\sqrt {2}} x \sqrt [4]{x^6+1}}{\sqrt {x^6+1}-x^2}\right )-\frac {1}{2} \sqrt {\frac {1}{2} \left (2+\sqrt {2}\right )} \tan ^{-1}\left (\frac {\sqrt {2+\sqrt {2}} x \sqrt [4]{x^6+1}}{\sqrt {x^6+1}-x^2}\right )+\frac {1}{2} \sqrt {\frac {1}{2} \left (2-\sqrt {2}\right )} \tanh ^{-1}\left (\frac {\sqrt {2-\sqrt {2}} x \sqrt [4]{x^6+1}}{\sqrt {x^6+1}+x^2}\right )-\frac {1}{2} \sqrt {\frac {1}{2} \left (2+\sqrt {2}\right )} \tanh ^{-1}\left (\frac {\sqrt {2+\sqrt {2}} x \sqrt [4]{x^6+1}}{\sqrt {x^6+1}+x^2}\right ) \]
________________________________________________________________________________________
Rubi [F] time = 1.40, 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 (-2+x^6\right ) \left (1-x^4+x^6\right )}{\sqrt [4]{1+x^6} \left (1+2 x^6+x^8+x^{12}\right )} \, dx \end {gather*}
Verification is not applicable to the result.
[In]
[Out]
Rubi steps
\begin {align*} \int \frac {\left (-2+x^6\right ) \left (1-x^4+x^6\right )}{\sqrt [4]{1+x^6} \left (1+2 x^6+x^8+x^{12}\right )} \, dx &=\int \left (\frac {1}{\sqrt [4]{1+x^6}}-\frac {3-2 x^4+3 x^6+x^8+x^{10}}{\sqrt [4]{1+x^6} \left (1+2 x^6+x^8+x^{12}\right )}\right ) \, dx\\ &=\int \frac {1}{\sqrt [4]{1+x^6}} \, dx-\int \frac {3-2 x^4+3 x^6+x^8+x^{10}}{\sqrt [4]{1+x^6} \left (1+2 x^6+x^8+x^{12}\right )} \, dx\\ &=x \, _2F_1\left (\frac {1}{6},\frac {1}{4};\frac {7}{6};-x^6\right )-\int \left (\frac {3}{\sqrt [4]{1+x^6} \left (1+2 x^6+x^8+x^{12}\right )}-\frac {2 x^4}{\sqrt [4]{1+x^6} \left (1+2 x^6+x^8+x^{12}\right )}+\frac {3 x^6}{\sqrt [4]{1+x^6} \left (1+2 x^6+x^8+x^{12}\right )}+\frac {x^8}{\sqrt [4]{1+x^6} \left (1+2 x^6+x^8+x^{12}\right )}+\frac {x^{10}}{\sqrt [4]{1+x^6} \left (1+2 x^6+x^8+x^{12}\right )}\right ) \, dx\\ &=x \, _2F_1\left (\frac {1}{6},\frac {1}{4};\frac {7}{6};-x^6\right )+2 \int \frac {x^4}{\sqrt [4]{1+x^6} \left (1+2 x^6+x^8+x^{12}\right )} \, dx-3 \int \frac {1}{\sqrt [4]{1+x^6} \left (1+2 x^6+x^8+x^{12}\right )} \, dx-3 \int \frac {x^6}{\sqrt [4]{1+x^6} \left (1+2 x^6+x^8+x^{12}\right )} \, dx-\int \frac {x^8}{\sqrt [4]{1+x^6} \left (1+2 x^6+x^8+x^{12}\right )} \, dx-\int \frac {x^{10}}{\sqrt [4]{1+x^6} \left (1+2 x^6+x^8+x^{12}\right )} \, dx\\ \end {align*}
________________________________________________________________________________________
Mathematica [F] time = 0.35, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\left (-2+x^6\right ) \left (1-x^4+x^6\right )}{\sqrt [4]{1+x^6} \left (1+2 x^6+x^8+x^{12}\right )} \, dx \end {gather*}
Verification is not applicable to the result.
[In]
[Out]
________________________________________________________________________________________
IntegrateAlgebraic [A] time = 16.14, size = 241, normalized size = 1.00 \begin {gather*} \frac {1}{2} \sqrt {\frac {1}{2} \left (2-\sqrt {2}\right )} \tan ^{-1}\left (\frac {\sqrt {2-\sqrt {2}} x \sqrt [4]{1+x^6}}{-x^2+\sqrt {1+x^6}}\right )-\frac {1}{2} \sqrt {\frac {1}{2} \left (2+\sqrt {2}\right )} \tan ^{-1}\left (\frac {\sqrt {2+\sqrt {2}} x \sqrt [4]{1+x^6}}{-x^2+\sqrt {1+x^6}}\right )+\frac {1}{2} \sqrt {\frac {1}{2} \left (2-\sqrt {2}\right )} \tanh ^{-1}\left (\frac {\sqrt {2-\sqrt {2}} x \sqrt [4]{1+x^6}}{x^2+\sqrt {1+x^6}}\right )-\frac {1}{2} \sqrt {\frac {1}{2} \left (2+\sqrt {2}\right )} \tanh ^{-1}\left (\frac {\sqrt {2+\sqrt {2}} x \sqrt [4]{1+x^6}}{x^2+\sqrt {1+x^6}}\right ) \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
fricas [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]
________________________________________________________________________________________
giac [F] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {{\left (x^{6} - x^{4} + 1\right )} {\left (x^{6} - 2\right )}}{{\left (x^{12} + x^{8} + 2 \, x^{6} + 1\right )} {\left (x^{6} + 1\right )}^{\frac {1}{4}}}\,{d x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [C] time = 31.68, size = 679, normalized size = 2.82
method | result | size |
trager | \(-\frac {\RootOf \left (\textit {\_Z}^{8}+16\right )^{5} \ln \left (\frac {\RootOf \left (\textit {\_Z}^{8}+16\right )^{8} x^{4}-4 \RootOf \left (\textit {\_Z}^{8}+16\right )^{6} \sqrt {x^{6}+1}\, x^{2}+4 \RootOf \left (\textit {\_Z}^{8}+16\right )^{4} x^{6}-8 \RootOf \left (\textit {\_Z}^{8}+16\right )^{5} \left (x^{6}+1\right )^{\frac {1}{4}} x^{3}-4 \RootOf \left (\textit {\_Z}^{8}+16\right )^{4} x^{4}+16 \RootOf \left (\textit {\_Z}^{8}+16\right )^{3} \left (x^{6}+1\right )^{\frac {3}{4}} x +16 \RootOf \left (\textit {\_Z}^{8}+16\right )^{2} \sqrt {x^{6}+1}\, x^{2}-16 x^{6}+4 \RootOf \left (\textit {\_Z}^{8}+16\right )^{4}-16}{\RootOf \left (\textit {\_Z}^{8}+16\right )^{4} x^{4}-4 x^{6}-4}\right )}{16}-\frac {\RootOf \left (\textit {\_Z}^{8}+16\right ) \ln \left (\frac {\RootOf \left (\textit {\_Z}^{8}+16\right )^{8} x^{4}+4 \RootOf \left (\textit {\_Z}^{8}+16\right )^{6} \sqrt {x^{6}+1}\, x^{2}+4 \RootOf \left (\textit {\_Z}^{8}+16\right )^{4} x^{6}+8 \RootOf \left (\textit {\_Z}^{8}+16\right )^{5} \left (x^{6}+1\right )^{\frac {1}{4}} x^{3}+4 \RootOf \left (\textit {\_Z}^{8}+16\right )^{4} x^{4}+16 \RootOf \left (\textit {\_Z}^{8}+16\right )^{3} \left (x^{6}+1\right )^{\frac {3}{4}} x +16 \RootOf \left (\textit {\_Z}^{8}+16\right )^{2} \sqrt {x^{6}+1}\, x^{2}+16 x^{6}+4 \RootOf \left (\textit {\_Z}^{8}+16\right )^{4}+16}{\RootOf \left (\textit {\_Z}^{8}+16\right )^{4} x^{4}-4 x^{6}-4}\right )}{4}-\frac {\RootOf \left (\textit {\_Z}^{8}+16\right )^{3} \ln \left (\frac {\RootOf \left (\textit {\_Z}^{8}+16\right )^{10} x^{4}-4 \RootOf \left (\textit {\_Z}^{8}+16\right )^{6} x^{6}-4 x^{4} \RootOf \left (\textit {\_Z}^{8}+16\right )^{6}+16 \RootOf \left (\textit {\_Z}^{8}+16\right )^{4} \sqrt {x^{6}+1}\, x^{2}+16 \RootOf \left (\textit {\_Z}^{8}+16\right )^{2} x^{6}-32 \RootOf \left (\textit {\_Z}^{8}+16\right )^{3} \left (x^{6}+1\right )^{\frac {3}{4}} x -4 \RootOf \left (\textit {\_Z}^{8}+16\right )^{6}+64 \RootOf \left (\textit {\_Z}^{8}+16\right ) \left (x^{6}+1\right )^{\frac {1}{4}} x^{3}-64 x^{2} \sqrt {x^{6}+1}+16 \RootOf \left (\textit {\_Z}^{8}+16\right )^{2}}{\RootOf \left (\textit {\_Z}^{8}+16\right )^{4} x^{4}+4 x^{6}+4}\right )}{8}-\frac {\RootOf \left (\textit {\_Z}^{8}+16\right )^{7} \ln \left (-\frac {\RootOf \left (\textit {\_Z}^{8}+16\right )^{10} x^{4}-4 \RootOf \left (\textit {\_Z}^{8}+16\right )^{6} x^{6}+4 x^{4} \RootOf \left (\textit {\_Z}^{8}+16\right )^{6}-16 \RootOf \left (\textit {\_Z}^{8}+16\right )^{4} \sqrt {x^{6}+1}\, x^{2}-16 \RootOf \left (\textit {\_Z}^{8}+16\right )^{2} x^{6}+32 \RootOf \left (\textit {\_Z}^{8}+16\right )^{3} \left (x^{6}+1\right )^{\frac {3}{4}} x -4 \RootOf \left (\textit {\_Z}^{8}+16\right )^{6}+64 \RootOf \left (\textit {\_Z}^{8}+16\right ) \left (x^{6}+1\right )^{\frac {1}{4}} x^{3}-64 x^{2} \sqrt {x^{6}+1}-16 \RootOf \left (\textit {\_Z}^{8}+16\right )^{2}}{\RootOf \left (\textit {\_Z}^{8}+16\right )^{4} x^{4}+4 x^{6}+4}\right )}{32}\) | \(679\) |
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^{6} - x^{4} + 1\right )} {\left (x^{6} - 2\right )}}{{\left (x^{12} + x^{8} + 2 \, x^{6} + 1\right )} {\left (x^{6} + 1\right )}^{\frac {1}{4}}}\,{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 = -0.00 \begin {gather*} \int \frac {\left (x^6-2\right )\,\left (x^6-x^4+1\right )}{{\left (x^6+1\right )}^{1/4}\,\left (x^{12}+x^8+2\,x^6+1\right )} \,d x \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^{6} - 2\right ) \left (x^{6} - x^{4} + 1\right )}{\sqrt [4]{\left (x^{2} + 1\right ) \left (x^{4} - x^{2} + 1\right )} \left (x^{12} + x^{8} + 2 x^{6} + 1\right )}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________