Optimal. Leaf size=94 \[ -\frac {\sqrt [4]{x^4+1}}{x}-\frac {\tan ^{-1}\left (\frac {\sqrt {3} x \sqrt [4]{x^4+1}}{\sqrt {x^4+1}-x^2}\right )}{\sqrt {3}}+\frac {\tanh ^{-1}\left (\frac {\sqrt {3} x \sqrt [4]{x^4+1}}{\sqrt {x^4+1}+x^2}\right )}{\sqrt {3}} \]
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Rubi [C] time = 0.79, antiderivative size = 347, normalized size of antiderivative = 3.69, number of steps used = 13, number of rules used = 7, integrand size = 37, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.189, Rules used = {6728, 264, 1528, 494, 298, 205, 208} \begin {gather*} -\frac {\sqrt [4]{x^4+1}}{x}-\frac {1}{6} \left (3+i \sqrt {3}\right ) \sqrt [4]{-\frac {-\sqrt {3}+3 i}{\sqrt {3}+3 i}} \tan ^{-1}\left (\frac {x}{\sqrt [4]{-\frac {-\sqrt {3}+3 i}{\sqrt {3}+3 i}} \sqrt [4]{x^4+1}}\right )-\frac {\left (3-i \sqrt {3}\right ) \tan ^{-1}\left (\frac {\sqrt [4]{-\frac {-\sqrt {3}+3 i}{\sqrt {3}+3 i}} x}{\sqrt [4]{x^4+1}}\right )}{6 \sqrt [4]{-\frac {-\sqrt {3}+3 i}{\sqrt {3}+3 i}}}+\frac {1}{6} \left (3+i \sqrt {3}\right ) \sqrt [4]{-\frac {-\sqrt {3}+3 i}{\sqrt {3}+3 i}} \tanh ^{-1}\left (\frac {x}{\sqrt [4]{-\frac {-\sqrt {3}+3 i}{\sqrt {3}+3 i}} \sqrt [4]{x^4+1}}\right )+\frac {\left (3-i \sqrt {3}\right ) \tanh ^{-1}\left (\frac {\sqrt [4]{-\frac {-\sqrt {3}+3 i}{\sqrt {3}+3 i}} x}{\sqrt [4]{x^4+1}}\right )}{6 \sqrt [4]{-\frac {-\sqrt {3}+3 i}{\sqrt {3}+3 i}}} \end {gather*}
Warning: Unable to verify antiderivative.
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Rule 205
Rule 208
Rule 264
Rule 298
Rule 494
Rule 1528
Rule 6728
Rubi steps
\begin {align*} \int \frac {1+3 x^4+x^8}{x^2 \left (1+x^4\right )^{3/4} \left (1+3 x^4+3 x^8\right )} \, dx &=\int \left (\frac {1}{x^2 \left (1+x^4\right )^{3/4}}-\frac {2 x^6}{\left (1+x^4\right )^{3/4} \left (1+3 x^4+3 x^8\right )}\right ) \, dx\\ &=-\left (2 \int \frac {x^6}{\left (1+x^4\right )^{3/4} \left (1+3 x^4+3 x^8\right )} \, dx\right )+\int \frac {1}{x^2 \left (1+x^4\right )^{3/4}} \, dx\\ &=-\frac {\sqrt [4]{1+x^4}}{x}-2 \int \left (\frac {i \left (-3+i \sqrt {3}\right ) x^2}{\sqrt {3} \left (-3+i \sqrt {3}-6 x^4\right ) \left (1+x^4\right )^{3/4}}-\frac {i \left (3+i \sqrt {3}\right ) x^2}{\sqrt {3} \left (1+x^4\right )^{3/4} \left (3+i \sqrt {3}+6 x^4\right )}\right ) \, dx\\ &=-\frac {\sqrt [4]{1+x^4}}{x}-\left (2 \left (1-i \sqrt {3}\right )\right ) \int \frac {x^2}{\left (1+x^4\right )^{3/4} \left (3+i \sqrt {3}+6 x^4\right )} \, dx+\left (2 \left (1+i \sqrt {3}\right )\right ) \int \frac {x^2}{\left (-3+i \sqrt {3}-6 x^4\right ) \left (1+x^4\right )^{3/4}} \, dx\\ &=-\frac {\sqrt [4]{1+x^4}}{x}-\left (2 \left (1-i \sqrt {3}\right )\right ) \operatorname {Subst}\left (\int \frac {x^2}{3+i \sqrt {3}-\left (-3+i \sqrt {3}\right ) x^4} \, dx,x,\frac {x}{\sqrt [4]{1+x^4}}\right )+\left (2 \left (1+i \sqrt {3}\right )\right ) \operatorname {Subst}\left (\int \frac {x^2}{-3+i \sqrt {3}-\left (3+i \sqrt {3}\right ) x^4} \, dx,x,\frac {x}{\sqrt [4]{1+x^4}}\right )\\ &=-\frac {\sqrt [4]{1+x^4}}{x}-\frac {\left (i-\sqrt {3}\right ) \operatorname {Subst}\left (\int \frac {1}{\sqrt {3 i+\sqrt {3}}-\sqrt {-3 i+\sqrt {3}} x^2} \, dx,x,\frac {x}{\sqrt [4]{1+x^4}}\right )}{\sqrt {-3 i+\sqrt {3}}}+\frac {\left (i-\sqrt {3}\right ) \operatorname {Subst}\left (\int \frac {1}{\sqrt {3 i+\sqrt {3}}+\sqrt {-3 i+\sqrt {3}} x^2} \, dx,x,\frac {x}{\sqrt [4]{1+x^4}}\right )}{\sqrt {-3 i+\sqrt {3}}}+\frac {\left (i+\sqrt {3}\right ) \operatorname {Subst}\left (\int \frac {1}{\sqrt {-3 i+\sqrt {3}}-\sqrt {3 i+\sqrt {3}} x^2} \, dx,x,\frac {x}{\sqrt [4]{1+x^4}}\right )}{\sqrt {3 i+\sqrt {3}}}-\frac {\left (i+\sqrt {3}\right ) \operatorname {Subst}\left (\int \frac {1}{\sqrt {-3 i+\sqrt {3}}+\sqrt {3 i+\sqrt {3}} x^2} \, dx,x,\frac {x}{\sqrt [4]{1+x^4}}\right )}{\sqrt {3 i+\sqrt {3}}}\\ &=-\frac {\sqrt [4]{1+x^4}}{x}-\frac {1}{2} \left (1+\frac {i}{\sqrt {3}}\right ) \sqrt [4]{-\frac {3 i-\sqrt {3}}{3 i+\sqrt {3}}} \tan ^{-1}\left (\frac {x}{\sqrt [4]{-\frac {3 i-\sqrt {3}}{3 i+\sqrt {3}}} \sqrt [4]{1+x^4}}\right )-\frac {\left (1-\frac {i}{\sqrt {3}}\right ) \tan ^{-1}\left (\frac {\sqrt [4]{-\frac {3 i-\sqrt {3}}{3 i+\sqrt {3}}} x}{\sqrt [4]{1+x^4}}\right )}{2 \sqrt [4]{-\frac {3 i-\sqrt {3}}{3 i+\sqrt {3}}}}+\frac {1}{2} \left (1+\frac {i}{\sqrt {3}}\right ) \sqrt [4]{-\frac {3 i-\sqrt {3}}{3 i+\sqrt {3}}} \tanh ^{-1}\left (\frac {x}{\sqrt [4]{-\frac {3 i-\sqrt {3}}{3 i+\sqrt {3}}} \sqrt [4]{1+x^4}}\right )+\frac {\left (1-\frac {i}{\sqrt {3}}\right ) \tanh ^{-1}\left (\frac {\sqrt [4]{-\frac {3 i-\sqrt {3}}{3 i+\sqrt {3}}} x}{\sqrt [4]{1+x^4}}\right )}{2 \sqrt [4]{-\frac {3 i-\sqrt {3}}{3 i+\sqrt {3}}}}\\ \end {align*}
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Mathematica [F] time = 7.11, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {1+3 x^4+x^8}{x^2 \left (1+x^4\right )^{3/4} \left (1+3 x^4+3 x^8\right )} \, dx \end {gather*}
Verification is not applicable to the result.
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IntegrateAlgebraic [A] time = 0.51, size = 115, normalized size = 1.22 \begin {gather*} -\frac {\sqrt [4]{1+x^4}}{x}-\frac {\tan ^{-1}\left (\frac {\sqrt {3} x}{-x+2 \sqrt [4]{1+x^4}}\right )}{\sqrt {3}}-\frac {\tan ^{-1}\left (\frac {\sqrt {3} x}{x+2 \sqrt [4]{1+x^4}}\right )}{\sqrt {3}}+\frac {\tanh ^{-1}\left (\frac {\sqrt {3} x \sqrt [4]{1+x^4}}{x^2+\sqrt {1+x^4}}\right )}{\sqrt {3}} \end {gather*}
Antiderivative was successfully verified.
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fricas [B] time = 19.50, size = 227, normalized size = 2.41 \begin {gather*} -\frac {2 \, \sqrt {3} x \arctan \left (\frac {2 \, {\left (\sqrt {3} {\left (3 \, x^{5} - x\right )} {\left (x^{4} + 1\right )}^{\frac {3}{4}} - \sqrt {3} {\left (3 \, x^{7} + 4 \, x^{3}\right )} {\left (x^{4} + 1\right )}^{\frac {1}{4}}\right )}}{21 \, x^{8} + 21 \, x^{4} - 1}\right ) - \sqrt {3} x \log \left (-\frac {441 \, x^{16} + 882 \, x^{12} + 543 \, x^{8} + 102 \, x^{4} + 4 \, \sqrt {3} {\left (63 \, x^{13} + 78 \, x^{9} + 24 \, x^{5} + x\right )} {\left (x^{4} + 1\right )}^{\frac {3}{4}} + 4 \, \sqrt {3} {\left (63 \, x^{15} + 111 \, x^{11} + 57 \, x^{7} + 8 \, x^{3}\right )} {\left (x^{4} + 1\right )}^{\frac {1}{4}} + 24 \, {\left (18 \, x^{14} + 27 \, x^{10} + 11 \, x^{6} + x^{2}\right )} \sqrt {x^{4} + 1} + 1}{9 \, x^{16} + 18 \, x^{12} + 15 \, x^{8} + 6 \, x^{4} + 1}\right ) + 12 \, {\left (x^{4} + 1\right )}^{\frac {1}{4}}}{12 \, x} \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 {x^{8} + 3 \, x^{4} + 1}{{\left (3 \, x^{8} + 3 \, x^{4} + 1\right )} {\left (x^{4} + 1\right )}^{\frac {3}{4}} x^{2}}\,{d x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [C] time = 30.33, size = 287, normalized size = 3.05
method | result | size |
trager | \(-\frac {\left (x^{4}+1\right )^{\frac {1}{4}}}{x}+\frac {\RootOf \left (\textit {\_Z}^{2}-3\right ) \ln \left (\frac {12 \RootOf \left (\textit {\_Z}^{2}-3\right ) \sqrt {x^{4}+1}\, x^{6}+9 \RootOf \left (\textit {\_Z}^{2}-3\right ) x^{8}+18 \left (x^{4}+1\right )^{\frac {3}{4}} x^{5}+18 \left (x^{4}+1\right )^{\frac {1}{4}} x^{7}+6 \RootOf \left (\textit {\_Z}^{2}-3\right ) \sqrt {x^{4}+1}\, x^{2}+9 \RootOf \left (\textit {\_Z}^{2}-3\right ) x^{4}+6 \left (x^{4}+1\right )^{\frac {3}{4}} x +12 x^{3} \left (x^{4}+1\right )^{\frac {1}{4}}+\RootOf \left (\textit {\_Z}^{2}-3\right )}{3 x^{8}+3 x^{4}+1}\right )}{6}-\frac {\RootOf \left (\textit {\_Z}^{2}+3\right ) \ln \left (-\frac {12 \RootOf \left (\textit {\_Z}^{2}+3\right ) \sqrt {x^{4}+1}\, x^{6}-9 \RootOf \left (\textit {\_Z}^{2}+3\right ) x^{8}-18 \left (x^{4}+1\right )^{\frac {3}{4}} x^{5}+18 \left (x^{4}+1\right )^{\frac {1}{4}} x^{7}+6 \RootOf \left (\textit {\_Z}^{2}+3\right ) \sqrt {x^{4}+1}\, x^{2}-9 \RootOf \left (\textit {\_Z}^{2}+3\right ) x^{4}-6 \left (x^{4}+1\right )^{\frac {3}{4}} x +12 x^{3} \left (x^{4}+1\right )^{\frac {1}{4}}-\RootOf \left (\textit {\_Z}^{2}+3\right )}{3 x^{8}+3 x^{4}+1}\right )}{6}\) | \(287\) |
risch | \(-\frac {\left (x^{4}+1\right )^{\frac {1}{4}}}{x}+\frac {\left (-\frac {\RootOf \left (\textit {\_Z}^{2}+3\right ) \ln \left (\frac {-9 \RootOf \left (\textit {\_Z}^{2}+3\right ) x^{16}-18 \left (x^{12}+3 x^{8}+3 x^{4}+1\right )^{\frac {1}{4}} x^{13}+12 \sqrt {x^{12}+3 x^{8}+3 x^{4}+1}\, \RootOf \left (\textit {\_Z}^{2}+3\right ) x^{10}-27 \RootOf \left (\textit {\_Z}^{2}+3\right ) x^{12}+18 \left (x^{12}+3 x^{8}+3 x^{4}+1\right )^{\frac {3}{4}} x^{7}-42 \left (x^{12}+3 x^{8}+3 x^{4}+1\right )^{\frac {1}{4}} x^{9}+18 \sqrt {x^{12}+3 x^{8}+3 x^{4}+1}\, \RootOf \left (\textit {\_Z}^{2}+3\right ) x^{6}-28 \RootOf \left (\textit {\_Z}^{2}+3\right ) x^{8}+12 \left (x^{12}+3 x^{8}+3 x^{4}+1\right )^{\frac {3}{4}} x^{3}-30 \left (x^{12}+3 x^{8}+3 x^{4}+1\right )^{\frac {1}{4}} x^{5}+6 \sqrt {x^{12}+3 x^{8}+3 x^{4}+1}\, \RootOf \left (\textit {\_Z}^{2}+3\right ) x^{2}-11 \RootOf \left (\textit {\_Z}^{2}+3\right ) x^{4}-6 \left (x^{12}+3 x^{8}+3 x^{4}+1\right )^{\frac {1}{4}} x -\RootOf \left (\textit {\_Z}^{2}+3\right )}{\left (x^{4}+1\right )^{2} \left (3 x^{8}+3 x^{4}+1\right )}\right )}{6}+\frac {\RootOf \left (\textit {\_Z}^{2}-3\right ) \ln \left (\frac {9 \RootOf \left (\textit {\_Z}^{2}-3\right ) x^{16}+18 \left (x^{12}+3 x^{8}+3 x^{4}+1\right )^{\frac {1}{4}} x^{13}+12 \sqrt {x^{12}+3 x^{8}+3 x^{4}+1}\, \RootOf \left (\textit {\_Z}^{2}-3\right ) x^{10}+27 \RootOf \left (\textit {\_Z}^{2}-3\right ) x^{12}+18 \left (x^{12}+3 x^{8}+3 x^{4}+1\right )^{\frac {3}{4}} x^{7}+42 \left (x^{12}+3 x^{8}+3 x^{4}+1\right )^{\frac {1}{4}} x^{9}+18 \sqrt {x^{12}+3 x^{8}+3 x^{4}+1}\, \RootOf \left (\textit {\_Z}^{2}-3\right ) x^{6}+28 \RootOf \left (\textit {\_Z}^{2}-3\right ) x^{8}+12 \left (x^{12}+3 x^{8}+3 x^{4}+1\right )^{\frac {3}{4}} x^{3}+30 \left (x^{12}+3 x^{8}+3 x^{4}+1\right )^{\frac {1}{4}} x^{5}+6 \sqrt {x^{12}+3 x^{8}+3 x^{4}+1}\, \RootOf \left (\textit {\_Z}^{2}-3\right ) x^{2}+11 \RootOf \left (\textit {\_Z}^{2}-3\right ) x^{4}+6 \left (x^{12}+3 x^{8}+3 x^{4}+1\right )^{\frac {1}{4}} x +\RootOf \left (\textit {\_Z}^{2}-3\right )}{\left (x^{4}+1\right )^{2} \left (3 x^{8}+3 x^{4}+1\right )}\right )}{6}\right ) \left (\left (x^{4}+1\right )^{3}\right )^{\frac {1}{4}}}{\left (x^{4}+1\right )^{\frac {3}{4}}}\) | \(626\) |
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 {x^{8} + 3 \, x^{4} + 1}{{\left (3 \, x^{8} + 3 \, x^{4} + 1\right )} {\left (x^{4} + 1\right )}^{\frac {3}{4}} x^{2}}\,{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 = -0.01 \begin {gather*} \int \frac {x^8+3\,x^4+1}{x^2\,{\left (x^4+1\right )}^{3/4}\,\left (3\,x^8+3\,x^4+1\right )} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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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.
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