Optimal. Leaf size=275 \[ -\sqrt {\frac {1}{2} \left (2+\sqrt {2}\right )} \tan ^{-1}\left (\frac {\frac {\sqrt {2-\sqrt {2}} x^2}{\sqrt {2}-2}-\frac {\sqrt {2-\sqrt {2}} \sqrt {x^3+1}}{\sqrt {2}-2}}{x \sqrt [4]{x^3+1}}\right )-\sqrt {\frac {1}{2} \left (2-\sqrt {2}\right )} \tan ^{-1}\left (\frac {\frac {\sqrt {x^3+1}}{\sqrt {2+\sqrt {2}}}-\frac {x^2}{\sqrt {2+\sqrt {2}}}}{x \sqrt [4]{x^3+1}}\right )+\sqrt {\frac {1}{2} \left (2+\sqrt {2}\right )} \tanh ^{-1}\left (\frac {\sqrt {2-\sqrt {2}} x \sqrt [4]{x^3+1}}{\sqrt {x^3+1}+x^2}\right )+\sqrt {\frac {1}{2} \left (2-\sqrt {2}\right )} \tanh ^{-1}\left (\frac {\sqrt {2+\sqrt {2}} x \sqrt [4]{x^3+1}}{\sqrt {x^3+1}+x^2}\right ) \]
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Rubi [F] time = 1.23, 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 (4+x^3\right ) \left (1+x^3+x^4\right )}{\sqrt [4]{1+x^3} \left (1+2 x^3+x^6+x^8\right )} \, dx \end {gather*}
Verification is not applicable to the result.
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\begin {align*} \int \frac {\left (4+x^3\right ) \left (1+x^3+x^4\right )}{\sqrt [4]{1+x^3} \left (1+2 x^3+x^6+x^8\right )} \, dx &=\int \left (\frac {4}{\sqrt [4]{1+x^3} \left (1+2 x^3+x^6+x^8\right )}+\frac {5 x^3}{\sqrt [4]{1+x^3} \left (1+2 x^3+x^6+x^8\right )}+\frac {4 x^4}{\sqrt [4]{1+x^3} \left (1+2 x^3+x^6+x^8\right )}+\frac {x^6}{\sqrt [4]{1+x^3} \left (1+2 x^3+x^6+x^8\right )}+\frac {x^7}{\sqrt [4]{1+x^3} \left (1+2 x^3+x^6+x^8\right )}\right ) \, dx\\ &=4 \int \frac {1}{\sqrt [4]{1+x^3} \left (1+2 x^3+x^6+x^8\right )} \, dx+4 \int \frac {x^4}{\sqrt [4]{1+x^3} \left (1+2 x^3+x^6+x^8\right )} \, dx+5 \int \frac {x^3}{\sqrt [4]{1+x^3} \left (1+2 x^3+x^6+x^8\right )} \, dx+\int \frac {x^6}{\sqrt [4]{1+x^3} \left (1+2 x^3+x^6+x^8\right )} \, dx+\int \frac {x^7}{\sqrt [4]{1+x^3} \left (1+2 x^3+x^6+x^8\right )} \, dx\\ \end {align*}
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Mathematica [F] time = 0.25, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\left (4+x^3\right ) \left (1+x^3+x^4\right )}{\sqrt [4]{1+x^3} \left (1+2 x^3+x^6+x^8\right )} \, dx \end {gather*}
Verification is not applicable to the result.
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IntegrateAlgebraic [A] time = 2.33, size = 257, normalized size = 0.93 \begin {gather*} -\sqrt {\frac {1}{2} \left (2-\sqrt {2}\right )} \tan ^{-1}\left (\frac {-\sqrt {1-\frac {1}{\sqrt {2}}} x^2+\sqrt {1-\frac {1}{\sqrt {2}}} \sqrt {1+x^3}}{x \sqrt [4]{1+x^3}}\right )-\sqrt {\frac {1}{2} \left (2+\sqrt {2}\right )} \tan ^{-1}\left (\frac {-\sqrt {1+\frac {1}{\sqrt {2}}} x^2+\sqrt {1+\frac {1}{\sqrt {2}}} \sqrt {1+x^3}}{x \sqrt [4]{1+x^3}}\right )+\sqrt {\frac {1}{2} \left (2+\sqrt {2}\right )} \tanh ^{-1}\left (\frac {\sqrt {2-\sqrt {2}} x \sqrt [4]{1+x^3}}{x^2+\sqrt {1+x^3}}\right )+\sqrt {\frac {1}{2} \left (2-\sqrt {2}\right )} \tanh ^{-1}\left (\frac {\sqrt {2+\sqrt {2}} x \sqrt [4]{1+x^3}}{x^2+\sqrt {1+x^3}}\right ) \end {gather*}
Antiderivative was successfully verified.
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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.
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giac [F] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {{\left (x^{4} + x^{3} + 1\right )} {\left (x^{3} + 4\right )}}{{\left (x^{8} + x^{6} + 2 \, x^{3} + 1\right )} {\left (x^{3} + 1\right )}^{\frac {1}{4}}}\,{d x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [C] time = 14.73, size = 693, normalized size = 2.52
method | result | size |
trager | \(-\frac {\RootOf \left (\textit {\_Z}^{8}+16\right )^{7} \ln \left (-\frac {\RootOf \left (\textit {\_Z}^{8}+16\right )^{11} x^{4}+4 x^{4} \RootOf \left (\textit {\_Z}^{8}+16\right )^{7}+16 \RootOf \left (\textit {\_Z}^{8}+16\right )^{6} \left (x^{3}+1\right )^{\frac {1}{4}} x^{3}-4 \RootOf \left (\textit {\_Z}^{8}+16\right )^{7} x^{3}+16 \sqrt {x^{3}+1}\, \RootOf \left (\textit {\_Z}^{8}+16\right )^{5} x^{2}-4 \RootOf \left (\textit {\_Z}^{8}+16\right )^{7}-16 \RootOf \left (\textit {\_Z}^{8}+16\right )^{3} x^{3}+64 \sqrt {x^{3}+1}\, \RootOf \left (\textit {\_Z}^{8}+16\right ) x^{2}+128 \left (x^{3}+1\right )^{\frac {3}{4}} x -16 \RootOf \left (\textit {\_Z}^{8}+16\right )^{3}}{\RootOf \left (\textit {\_Z}^{8}+16\right )^{4} x^{4}+4 x^{3}+4}\right )}{16}-\frac {\RootOf \left (\textit {\_Z}^{8}+16\right ) \ln \left (-\frac {-\RootOf \left (\textit {\_Z}^{8}+16\right )^{9} x^{4}+4 \sqrt {x^{3}+1}\, \RootOf \left (\textit {\_Z}^{8}+16\right )^{7} x^{2}-4 \RootOf \left (\textit {\_Z}^{8}+16\right )^{5} x^{4}-4 \RootOf \left (\textit {\_Z}^{8}+16\right )^{5} x^{3}+16 \sqrt {x^{3}+1}\, \RootOf \left (\textit {\_Z}^{8}+16\right )^{3} x^{2}-32 \left (x^{3}+1\right )^{\frac {1}{4}} \RootOf \left (\textit {\_Z}^{8}+16\right )^{2} x^{3}-4 \RootOf \left (\textit {\_Z}^{8}+16\right )^{5}+64 \left (x^{3}+1\right )^{\frac {3}{4}} x -16 x^{3} \RootOf \left (\textit {\_Z}^{8}+16\right )-16 \RootOf \left (\textit {\_Z}^{8}+16\right )}{\RootOf \left (\textit {\_Z}^{8}+16\right )^{4} x^{4}-4 x^{3}-4}\right )}{2}+\frac {\RootOf \left (\textit {\_Z}^{8}+16\right )^{3} \ln \left (-\frac {\RootOf \left (\textit {\_Z}^{8}+16\right )^{11} x^{4}-4 x^{4} \RootOf \left (\textit {\_Z}^{8}+16\right )^{7}-16 \RootOf \left (\textit {\_Z}^{8}+16\right )^{6} \left (x^{3}+1\right )^{\frac {1}{4}} x^{3}-4 \RootOf \left (\textit {\_Z}^{8}+16\right )^{7} x^{3}-16 \sqrt {x^{3}+1}\, \RootOf \left (\textit {\_Z}^{8}+16\right )^{5} x^{2}-4 \RootOf \left (\textit {\_Z}^{8}+16\right )^{7}+16 \RootOf \left (\textit {\_Z}^{8}+16\right )^{3} x^{3}+64 \sqrt {x^{3}+1}\, \RootOf \left (\textit {\_Z}^{8}+16\right ) x^{2}+128 \left (x^{3}+1\right )^{\frac {3}{4}} x +16 \RootOf \left (\textit {\_Z}^{8}+16\right )^{3}}{\RootOf \left (\textit {\_Z}^{8}+16\right )^{4} x^{4}+4 x^{3}+4}\right )}{4}+\frac {\RootOf \left (\textit {\_Z}^{8}+16\right )^{5} \ln \left (-\frac {\RootOf \left (\textit {\_Z}^{8}+16\right )^{9} x^{4}+4 \sqrt {x^{3}+1}\, \RootOf \left (\textit {\_Z}^{8}+16\right )^{7} x^{2}-4 \RootOf \left (\textit {\_Z}^{8}+16\right )^{5} x^{4}+4 \RootOf \left (\textit {\_Z}^{8}+16\right )^{5} x^{3}-16 \sqrt {x^{3}+1}\, \RootOf \left (\textit {\_Z}^{8}+16\right )^{3} x^{2}+32 \left (x^{3}+1\right )^{\frac {1}{4}} \RootOf \left (\textit {\_Z}^{8}+16\right )^{2} x^{3}+4 \RootOf \left (\textit {\_Z}^{8}+16\right )^{5}+64 \left (x^{3}+1\right )^{\frac {3}{4}} x -16 x^{3} \RootOf \left (\textit {\_Z}^{8}+16\right )-16 \RootOf \left (\textit {\_Z}^{8}+16\right )}{\RootOf \left (\textit {\_Z}^{8}+16\right )^{4} x^{4}-4 x^{3}-4}\right )}{8}\) | \(693\) |
Verification of antiderivative is not currently implemented for this CAS.
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maxima [F] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {{\left (x^{4} + x^{3} + 1\right )} {\left (x^{3} + 4\right )}}{{\left (x^{8} + x^{6} + 2 \, x^{3} + 1\right )} {\left (x^{3} + 1\right )}^{\frac {1}{4}}}\,{d x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.00 \begin {gather*} \int \frac {\left (x^3+4\right )\,\left (x^4+x^3+1\right )}{{\left (x^3+1\right )}^{1/4}\,\left (x^8+x^6+2\,x^3+1\right )} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\left (x^{3} + 4\right ) \left (x^{4} + x^{3} + 1\right )}{\sqrt [4]{\left (x + 1\right ) \left (x^{2} - x + 1\right )} \left (x^{8} + x^{6} + 2 x^{3} + 1\right )}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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