Optimal. Leaf size=74 \[ \frac {\left (x^4+1\right )^{3/4} x}{8 \left (x^4+2\right )}+\frac {3 \tan ^{-1}\left (\frac {x}{\sqrt [4]{2} \sqrt [4]{x^4+1}}\right )}{16\ 2^{3/4}}+\frac {3 \tanh ^{-1}\left (\frac {x}{\sqrt [4]{2} \sqrt [4]{x^4+1}}\right )}{16\ 2^{3/4}} \]
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Rubi [A] time = 0.02, antiderivative size = 74, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 5, integrand size = 17, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.294, Rules used = {378, 377, 212, 206, 203} \[ \frac {\left (x^4+1\right )^{3/4} x}{8 \left (x^4+2\right )}+\frac {3 \tan ^{-1}\left (\frac {x}{\sqrt [4]{2} \sqrt [4]{x^4+1}}\right )}{16\ 2^{3/4}}+\frac {3 \tanh ^{-1}\left (\frac {x}{\sqrt [4]{2} \sqrt [4]{x^4+1}}\right )}{16\ 2^{3/4}} \]
Antiderivative was successfully verified.
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Rule 203
Rule 206
Rule 212
Rule 377
Rule 378
Rubi steps
\begin {align*} \int \frac {\left (1+x^4\right )^{3/4}}{\left (2+x^4\right )^2} \, dx &=\frac {x \left (1+x^4\right )^{3/4}}{8 \left (2+x^4\right )}+\frac {3}{8} \int \frac {1}{\sqrt [4]{1+x^4} \left (2+x^4\right )} \, dx\\ &=\frac {x \left (1+x^4\right )^{3/4}}{8 \left (2+x^4\right )}+\frac {3}{8} \operatorname {Subst}\left (\int \frac {1}{2-x^4} \, dx,x,\frac {x}{\sqrt [4]{1+x^4}}\right )\\ &=\frac {x \left (1+x^4\right )^{3/4}}{8 \left (2+x^4\right )}+\frac {3 \operatorname {Subst}\left (\int \frac {1}{\sqrt {2}-x^2} \, dx,x,\frac {x}{\sqrt [4]{1+x^4}}\right )}{16 \sqrt {2}}+\frac {3 \operatorname {Subst}\left (\int \frac {1}{\sqrt {2}+x^2} \, dx,x,\frac {x}{\sqrt [4]{1+x^4}}\right )}{16 \sqrt {2}}\\ &=\frac {x \left (1+x^4\right )^{3/4}}{8 \left (2+x^4\right )}+\frac {3 \tan ^{-1}\left (\frac {x}{\sqrt [4]{2} \sqrt [4]{1+x^4}}\right )}{16\ 2^{3/4}}+\frac {3 \tanh ^{-1}\left (\frac {x}{\sqrt [4]{2} \sqrt [4]{1+x^4}}\right )}{16\ 2^{3/4}}\\ \end {align*}
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Mathematica [C] time = 0.01, size = 41, normalized size = 0.55 \[ \frac {x \, _2F_1\left (-\frac {3}{4},\frac {1}{4};\frac {5}{4};-\frac {x^4}{x^4+2}\right )}{2\ 2^{3/4} \sqrt [4]{x^4+2}} \]
Warning: Unable to verify antiderivative.
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IntegrateAlgebraic [A] time = 0.22, size = 74, normalized size = 1.00 \[ \frac {\left (x^4+1\right )^{3/4} x}{8 \left (x^4+2\right )}+\frac {3 \tan ^{-1}\left (\frac {x}{\sqrt [4]{2} \sqrt [4]{x^4+1}}\right )}{16\ 2^{3/4}}+\frac {3 \tanh ^{-1}\left (\frac {x}{\sqrt [4]{2} \sqrt [4]{x^4+1}}\right )}{16\ 2^{3/4}} \]
Antiderivative was successfully verified.
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fricas [B] time = 11.89, size = 242, normalized size = 3.27 \[ -\frac {12 \cdot 8^{\frac {3}{4}} {\left (x^{4} + 2\right )} \arctan \left (-\frac {8^{\frac {3}{4}} {\left (x^{4} + 1\right )}^{\frac {1}{4}} x^{3} + 4 \cdot 8^{\frac {1}{4}} {\left (x^{4} + 1\right )}^{\frac {3}{4}} x - 2^{\frac {1}{4}} {\left (8^{\frac {3}{4}} \sqrt {x^{4} + 1} x^{2} + 8^{\frac {1}{4}} {\left (3 \, x^{4} + 2\right )}\right )}}{2 \, {\left (x^{4} + 2\right )}}\right ) - 3 \cdot 8^{\frac {3}{4}} {\left (x^{4} + 2\right )} \log \left (\frac {8 \, \sqrt {2} {\left (x^{4} + 1\right )}^{\frac {1}{4}} x^{3} + 8 \cdot 8^{\frac {1}{4}} \sqrt {x^{4} + 1} x^{2} + 8^{\frac {3}{4}} {\left (3 \, x^{4} + 2\right )} + 16 \, {\left (x^{4} + 1\right )}^{\frac {3}{4}} x}{x^{4} + 2}\right ) + 3 \cdot 8^{\frac {3}{4}} {\left (x^{4} + 2\right )} \log \left (\frac {8 \, \sqrt {2} {\left (x^{4} + 1\right )}^{\frac {1}{4}} x^{3} - 8 \cdot 8^{\frac {1}{4}} \sqrt {x^{4} + 1} x^{2} - 8^{\frac {3}{4}} {\left (3 \, x^{4} + 2\right )} + 16 \, {\left (x^{4} + 1\right )}^{\frac {3}{4}} x}{x^{4} + 2}\right ) - 64 \, {\left (x^{4} + 1\right )}^{\frac {3}{4}} x}{512 \, {\left (x^{4} + 2\right )}} \]
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 \[ \int \frac {{\left (x^{4} + 1\right )}^{\frac {3}{4}}}{{\left (x^{4} + 2\right )}^{2}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [C] time = 2.83, size = 228, normalized size = 3.08
method | result | size |
risch | \(\frac {\left (x^{4}+1\right )^{\frac {3}{4}} x}{8 x^{4}+16}+\frac {3 \RootOf \left (\textit {\_Z}^{2}+\RootOf \left (\textit {\_Z}^{4}-2\right )^{2}\right ) \ln \left (\frac {2 \sqrt {x^{4}+1}\, \RootOf \left (\textit {\_Z}^{4}-2\right )^{2} \RootOf \left (\textit {\_Z}^{2}+\RootOf \left (\textit {\_Z}^{4}-2\right )^{2}\right ) x^{2}+2 \left (x^{4}+1\right )^{\frac {1}{4}} \RootOf \left (\textit {\_Z}^{4}-2\right )^{2} x^{3}-3 \RootOf \left (\textit {\_Z}^{2}+\RootOf \left (\textit {\_Z}^{4}-2\right )^{2}\right ) x^{4}-4 \left (x^{4}+1\right )^{\frac {3}{4}} x -2 \RootOf \left (\textit {\_Z}^{2}+\RootOf \left (\textit {\_Z}^{4}-2\right )^{2}\right )}{x^{4}+2}\right )}{64}+\frac {3 \RootOf \left (\textit {\_Z}^{4}-2\right ) \ln \left (-\frac {2 \sqrt {x^{4}+1}\, \RootOf \left (\textit {\_Z}^{4}-2\right )^{3} x^{2}+2 \left (x^{4}+1\right )^{\frac {1}{4}} \RootOf \left (\textit {\_Z}^{4}-2\right )^{2} x^{3}+3 \RootOf \left (\textit {\_Z}^{4}-2\right ) x^{4}+4 \left (x^{4}+1\right )^{\frac {3}{4}} x +2 \RootOf \left (\textit {\_Z}^{4}-2\right )}{x^{4}+2}\right )}{64}\) | \(228\) |
trager | \(\frac {\left (x^{4}+1\right )^{\frac {3}{4}} x}{8 x^{4}+16}-\frac {3 \RootOf \left (\textit {\_Z}^{2}+\RootOf \left (\textit {\_Z}^{4}-2\right )^{2}\right ) \ln \left (-\frac {2 \sqrt {x^{4}+1}\, \RootOf \left (\textit {\_Z}^{4}-2\right )^{2} \RootOf \left (\textit {\_Z}^{2}+\RootOf \left (\textit {\_Z}^{4}-2\right )^{2}\right ) x^{2}-2 \left (x^{4}+1\right )^{\frac {1}{4}} \RootOf \left (\textit {\_Z}^{4}-2\right )^{2} x^{3}-3 \RootOf \left (\textit {\_Z}^{2}+\RootOf \left (\textit {\_Z}^{4}-2\right )^{2}\right ) x^{4}+4 \left (x^{4}+1\right )^{\frac {3}{4}} x -2 \RootOf \left (\textit {\_Z}^{2}+\RootOf \left (\textit {\_Z}^{4}-2\right )^{2}\right )}{x^{4}+2}\right )}{64}-\frac {3 \RootOf \left (\textit {\_Z}^{4}-2\right ) \ln \left (-\frac {-2 \sqrt {x^{4}+1}\, \RootOf \left (\textit {\_Z}^{4}-2\right )^{3} x^{2}+2 \left (x^{4}+1\right )^{\frac {1}{4}} \RootOf \left (\textit {\_Z}^{4}-2\right )^{2} x^{3}-3 \RootOf \left (\textit {\_Z}^{4}-2\right ) x^{4}+4 \left (x^{4}+1\right )^{\frac {3}{4}} x -2 \RootOf \left (\textit {\_Z}^{4}-2\right )}{x^{4}+2}\right )}{64}\) | \(229\) |
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 \[ \int \frac {{\left (x^{4} + 1\right )}^{\frac {3}{4}}}{{\left (x^{4} + 2\right )}^{2}}\,{d x} \]
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 \[ \int \frac {{\left (x^4+1\right )}^{3/4}}{{\left (x^4+2\right )}^2} \,d x \]
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 \[ \int \frac {\left (x^{4} + 1\right )^{\frac {3}{4}}}{\left (x^{4} + 2\right )^{2}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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