Optimal. Leaf size=53 \[ \frac {\tan ^{-1}\left (\frac {\sqrt {2} x}{\sqrt [4]{x^4-1}}\right )}{2 \sqrt {2}}+\frac {\tanh ^{-1}\left (\frac {\sqrt {2} x}{\sqrt [4]{x^4-1}}\right )}{2 \sqrt {2}} \]
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Rubi [A] time = 0.01, antiderivative size = 53, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, integrand size = 19, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.210, Rules used = {377, 212, 206, 203} \begin {gather*} \frac {\tan ^{-1}\left (\frac {\sqrt {2} x}{\sqrt [4]{x^4-1}}\right )}{2 \sqrt {2}}+\frac {\tanh ^{-1}\left (\frac {\sqrt {2} x}{\sqrt [4]{x^4-1}}\right )}{2 \sqrt {2}} \end {gather*}
Antiderivative was successfully verified.
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Rule 203
Rule 206
Rule 212
Rule 377
Rubi steps
\begin {align*} \int \frac {1}{\sqrt [4]{-1+x^4} \left (1+3 x^4\right )} \, dx &=\operatorname {Subst}\left (\int \frac {1}{1-4 x^4} \, dx,x,\frac {x}{\sqrt [4]{-1+x^4}}\right )\\ &=\frac {1}{2} \operatorname {Subst}\left (\int \frac {1}{1-2 x^2} \, dx,x,\frac {x}{\sqrt [4]{-1+x^4}}\right )+\frac {1}{2} \operatorname {Subst}\left (\int \frac {1}{1+2 x^2} \, dx,x,\frac {x}{\sqrt [4]{-1+x^4}}\right )\\ &=\frac {\tan ^{-1}\left (\frac {\sqrt {2} x}{\sqrt [4]{-1+x^4}}\right )}{2 \sqrt {2}}+\frac {\tanh ^{-1}\left (\frac {\sqrt {2} x}{\sqrt [4]{-1+x^4}}\right )}{2 \sqrt {2}}\\ \end {align*}
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Mathematica [A] time = 0.01, size = 44, normalized size = 0.83 \begin {gather*} \frac {\tan ^{-1}\left (\frac {\sqrt {2} x}{\sqrt [4]{x^4-1}}\right )+\tanh ^{-1}\left (\frac {\sqrt {2} x}{\sqrt [4]{x^4-1}}\right )}{2 \sqrt {2}} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [A] time = 0.22, size = 53, normalized size = 1.00 \begin {gather*} \frac {\tan ^{-1}\left (\frac {\sqrt {2} x}{\sqrt [4]{-1+x^4}}\right )}{2 \sqrt {2}}+\frac {\tanh ^{-1}\left (\frac {\sqrt {2} x}{\sqrt [4]{-1+x^4}}\right )}{2 \sqrt {2}} \end {gather*}
Antiderivative was successfully verified.
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fricas [B] time = 7.66, size = 144, normalized size = 2.72 \begin {gather*} -\frac {1}{8} \, \sqrt {2} \arctan \left (\frac {2 \, {\left (2 \, \sqrt {2} {\left (x^{4} - 1\right )}^{\frac {1}{4}} x^{3} + \sqrt {2} {\left (x^{4} - 1\right )}^{\frac {3}{4}} x\right )}}{3 \, x^{4} + 1}\right ) + \frac {1}{16} \, \sqrt {2} \log \left (-\frac {73 \, x^{8} - 58 \, x^{4} + 4 \, \sqrt {2} {\left (13 \, x^{5} - x\right )} {\left (x^{4} - 1\right )}^{\frac {3}{4}} + 8 \, \sqrt {2} {\left (7 \, x^{7} - 3 \, x^{3}\right )} {\left (x^{4} - 1\right )}^{\frac {1}{4}} + 16 \, {\left (5 \, x^{6} - x^{2}\right )} \sqrt {x^{4} - 1} + 1}{9 \, x^{8} + 6 \, 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 {1}{{\left (3 \, x^{4} + 1\right )} {\left (x^{4} - 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 = 1.10, size = 160, normalized size = 3.02
method | result | size |
trager | \(\frac {\RootOf \left (\textit {\_Z}^{2}-2\right ) \ln \left (\frac {4 \RootOf \left (\textit {\_Z}^{2}-2\right ) \sqrt {x^{4}-1}\, x^{2}+5 \RootOf \left (\textit {\_Z}^{2}-2\right ) x^{4}+4 \left (x^{4}-1\right )^{\frac {3}{4}} x +8 x^{3} \left (x^{4}-1\right )^{\frac {1}{4}}-\RootOf \left (\textit {\_Z}^{2}-2\right )}{3 x^{4}+1}\right )}{8}+\frac {\RootOf \left (\textit {\_Z}^{2}+2\right ) \ln \left (\frac {-4 \RootOf \left (\textit {\_Z}^{2}+2\right ) \sqrt {x^{4}-1}\, x^{2}+5 \RootOf \left (\textit {\_Z}^{2}+2\right ) x^{4}+4 \left (x^{4}-1\right )^{\frac {3}{4}} x -8 x^{3} \left (x^{4}-1\right )^{\frac {1}{4}}-\RootOf \left (\textit {\_Z}^{2}+2\right )}{3 x^{4}+1}\right )}{8}\) | \(160\) |
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 {1}{{\left (3 \, x^{4} + 1\right )} {\left (x^{4} - 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 = -0.02 \begin {gather*} \int \frac {1}{{\left (x^4-1\right )}^{1/4}\,\left (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] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {1}{\sqrt [4]{\left (x - 1\right ) \left (x + 1\right ) \left (x^{2} + 1\right )} \left (3 x^{4} + 1\right )}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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