Optimal. Leaf size=90 \[ -\frac {\tanh ^{-1}\left (\frac {\sqrt {2} \sqrt [4]{3} x}{\sqrt {3 x^4+1}}\right )}{2 \sqrt {2} \sqrt [4]{3}}-\frac {\tan ^{-1}\left (\frac {\sqrt {3 x^4+1}}{\sqrt {3} x^2-\sqrt {2} \sqrt [4]{3} x+1}\right )}{\sqrt {2} \sqrt [4]{3}} \]
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Rubi [A] time = 0.03, antiderivative size = 77, normalized size of antiderivative = 0.86, number of steps used = 4, number of rules used = 4, integrand size = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.190, Rules used = {404, 212, 206, 203} \begin {gather*} -\frac {\tan ^{-1}\left (\frac {\sqrt {2} \sqrt [4]{3} x}{\sqrt {3 x^4+1}}\right )}{2 \sqrt {2} \sqrt [4]{3}}-\frac {\tanh ^{-1}\left (\frac {\sqrt {2} \sqrt [4]{3} x}{\sqrt {3 x^4+1}}\right )}{2 \sqrt {2} \sqrt [4]{3}} \end {gather*}
Antiderivative was successfully verified.
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Rule 203
Rule 206
Rule 212
Rule 404
Rubi steps
\begin {align*} \int \frac {\sqrt {1+3 x^4}}{-1+3 x^4} \, dx &=-\operatorname {Subst}\left (\int \frac {1}{1-12 x^4} \, dx,x,\frac {x}{\sqrt {1+3 x^4}}\right )\\ &=-\left (\frac {1}{2} \operatorname {Subst}\left (\int \frac {1}{1-2 \sqrt {3} x^2} \, dx,x,\frac {x}{\sqrt {1+3 x^4}}\right )\right )-\frac {1}{2} \operatorname {Subst}\left (\int \frac {1}{1+2 \sqrt {3} x^2} \, dx,x,\frac {x}{\sqrt {1+3 x^4}}\right )\\ &=-\frac {\tan ^{-1}\left (\frac {\sqrt {2} \sqrt [4]{3} x}{\sqrt {1+3 x^4}}\right )}{2 \sqrt {2} \sqrt [4]{3}}-\frac {\tanh ^{-1}\left (\frac {\sqrt {2} \sqrt [4]{3} x}{\sqrt {1+3 x^4}}\right )}{2 \sqrt {2} \sqrt [4]{3}}\\ \end {align*}
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Mathematica [C] time = 0.10, size = 120, normalized size = 1.33 \begin {gather*} \frac {5 x \sqrt {3 x^4+1} F_1\left (\frac {1}{4};-\frac {1}{2},1;\frac {5}{4};-3 x^4,3 x^4\right )}{\left (3 x^4-1\right ) \left (6 x^4 \left (2 F_1\left (\frac {5}{4};-\frac {1}{2},2;\frac {9}{4};-3 x^4,3 x^4\right )+F_1\left (\frac {5}{4};\frac {1}{2},1;\frac {9}{4};-3 x^4,3 x^4\right )\right )+5 F_1\left (\frac {1}{4};-\frac {1}{2},1;\frac {5}{4};-3 x^4,3 x^4\right )\right )} \end {gather*}
Warning: Unable to verify antiderivative.
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IntegrateAlgebraic [A] time = 0.43, size = 77, normalized size = 0.86 \begin {gather*} -\frac {\tan ^{-1}\left (\frac {\sqrt {2} \sqrt [4]{3} x}{\sqrt {1+3 x^4}}\right )}{2 \sqrt {2} \sqrt [4]{3}}-\frac {\tanh ^{-1}\left (\frac {\sqrt {2} \sqrt [4]{3} x}{\sqrt {1+3 x^4}}\right )}{2 \sqrt {2} \sqrt [4]{3}} \end {gather*}
Antiderivative was successfully verified.
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fricas [B] time = 0.52, size = 146, normalized size = 1.62 \begin {gather*} \frac {1}{12} \cdot 12^{\frac {3}{4}} \arctan \left (-\frac {12^{\frac {3}{4}} \sqrt {3} x^{2} - 12^{\frac {3}{4}} \sqrt {3 \, x^{4} + 1} + 2 \cdot 12^{\frac {1}{4}} \sqrt {3}}{12 \, x}\right ) - \frac {1}{48} \cdot 12^{\frac {3}{4}} \log \left (\frac {6 \cdot 12^{\frac {1}{4}} x^{3} + 12^{\frac {3}{4}} x + 2 \, \sqrt {3 \, x^{4} + 1} {\left (3 \, x^{2} + \sqrt {3}\right )}}{3 \, x^{4} - 1}\right ) + \frac {1}{48} \cdot 12^{\frac {3}{4}} \log \left (-\frac {6 \cdot 12^{\frac {1}{4}} x^{3} + 12^{\frac {3}{4}} x - 2 \, \sqrt {3 \, x^{4} + 1} {\left (3 \, x^{2} + \sqrt {3}\right )}}{3 \, x^{4} - 1}\right ) \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 {\sqrt {3 \, x^{4} + 1}}{3 \, x^{4} - 1}\,{d x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 1.30, size = 86, normalized size = 0.96
method | result | size |
default | \(\frac {\left (-\frac {3^{\frac {3}{4}} \ln \left (\frac {\frac {\sqrt {3 x^{4}+1}\, \sqrt {2}}{2 x}+3^{\frac {1}{4}}}{\frac {\sqrt {3 x^{4}+1}\, \sqrt {2}}{2 x}-3^{\frac {1}{4}}}\right )}{12}+\frac {3^{\frac {3}{4}} \arctan \left (\frac {\sqrt {3 x^{4}+1}\, \sqrt {2}\, 3^{\frac {3}{4}}}{6 x}\right )}{6}\right ) \sqrt {2}}{2}\) | \(86\) |
elliptic | \(\frac {\left (-\frac {3^{\frac {3}{4}} \ln \left (\frac {\frac {\sqrt {3 x^{4}+1}\, \sqrt {2}}{2 x}+3^{\frac {1}{4}}}{\frac {\sqrt {3 x^{4}+1}\, \sqrt {2}}{2 x}-3^{\frac {1}{4}}}\right )}{12}+\frac {3^{\frac {3}{4}} \arctan \left (\frac {\sqrt {3 x^{4}+1}\, \sqrt {2}\, 3^{\frac {3}{4}}}{6 x}\right )}{6}\right ) \sqrt {2}}{2}\) | \(86\) |
trager | \(\frac {\RootOf \left (\textit {\_Z}^{2}+\RootOf \left (\textit {\_Z}^{4}-108\right )^{2}\right ) \ln \left (\frac {x \RootOf \left (\textit {\_Z}^{4}-108\right )^{2} \RootOf \left (\textit {\_Z}^{2}+\RootOf \left (\textit {\_Z}^{4}-108\right )^{2}\right )+18 \sqrt {3 x^{4}+1}}{\RootOf \left (\textit {\_Z}^{4}-108\right )^{2} x^{2}+6}\right )}{12}-\frac {\RootOf \left (\textit {\_Z}^{4}-108\right ) \ln \left (\frac {x \RootOf \left (\textit {\_Z}^{4}-108\right )^{3}+18 \sqrt {3 x^{4}+1}}{\RootOf \left (\textit {\_Z}^{4}-108\right )^{2} x^{2}-6}\right )}{12}\) | \(118\) |
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 {\sqrt {3 \, x^{4} + 1}}{3 \, x^{4} - 1}\,{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 {\sqrt {3\,x^4+1}}{3\,x^4-1} \,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 {\sqrt {3 x^{4} + 1}}{3 x^{4} - 1}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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