Optimal. Leaf size=62 \[ -\frac {1}{2} \text {RootSum}\left [\text {$\#$1}^8-3 \text {$\#$1}^4+3\& ,\frac {\text {$\#$1}^3 \log \left (\sqrt [4]{x^4+x^2}-\text {$\#$1} x\right )-\text {$\#$1}^3 \log (x)}{2 \text {$\#$1}^4-3}\& \right ] \]
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Rubi [C] time = 0.51, antiderivative size = 477, normalized size of antiderivative = 7.69, number of steps used = 21, number of rules used = 11, integrand size = 29, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.379, Rules used = {2056, 1269, 1428, 408, 240, 212, 206, 203, 377, 208, 205} \begin {gather*} -\frac {\sqrt [4]{\frac {-\sqrt {3}+i}{-\sqrt {3}+3 i}} \left (\sqrt {3}+i\right ) \sqrt {x} \sqrt [4]{x^2+1} \tan ^{-1}\left (\frac {\sqrt {x}}{\sqrt [4]{\frac {-\sqrt {3}+i}{-\sqrt {3}+3 i}} \sqrt [4]{x^2+1}}\right )}{\left (-\sqrt {3}+i\right ) \sqrt [4]{x^4+x^2}}-\frac {\left (-\sqrt {3}+i\right ) \sqrt [4]{\frac {\sqrt {3}+i}{\sqrt {3}+3 i}} \sqrt {x} \sqrt [4]{x^2+1} \tan ^{-1}\left (\frac {\sqrt {x}}{\sqrt [4]{\frac {\sqrt {3}+i}{\sqrt {3}+3 i}} \sqrt [4]{x^2+1}}\right )}{\left (\sqrt {3}+i\right ) \sqrt [4]{x^4+x^2}}-\frac {\sqrt [4]{\frac {-\sqrt {3}+i}{-\sqrt {3}+3 i}} \left (\sqrt {3}+i\right ) \sqrt {x} \sqrt [4]{x^2+1} \tanh ^{-1}\left (\frac {\sqrt {x}}{\sqrt [4]{\frac {-\sqrt {3}+i}{-\sqrt {3}+3 i}} \sqrt [4]{x^2+1}}\right )}{\left (-\sqrt {3}+i\right ) \sqrt [4]{x^4+x^2}}-\frac {\left (-\sqrt {3}+i\right ) \sqrt [4]{\frac {\sqrt {3}+i}{\sqrt {3}+3 i}} \sqrt {x} \sqrt [4]{x^2+1} \tanh ^{-1}\left (\frac {\sqrt {x}}{\sqrt [4]{\frac {\sqrt {3}+i}{\sqrt {3}+3 i}} \sqrt [4]{x^2+1}}\right )}{\left (\sqrt {3}+i\right ) \sqrt [4]{x^4+x^2}} \end {gather*}
Antiderivative was successfully verified.
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Rule 203
Rule 205
Rule 206
Rule 208
Rule 212
Rule 240
Rule 377
Rule 408
Rule 1269
Rule 1428
Rule 2056
Rubi steps
\begin {align*} \int \frac {1+x^2}{\left (1-x^2+x^4\right ) \sqrt [4]{x^2+x^4}} \, dx &=\frac {\left (\sqrt {x} \sqrt [4]{1+x^2}\right ) \int \frac {\left (1+x^2\right )^{3/4}}{\sqrt {x} \left (1-x^2+x^4\right )} \, dx}{\sqrt [4]{x^2+x^4}}\\ &=\frac {\left (2 \sqrt {x} \sqrt [4]{1+x^2}\right ) \operatorname {Subst}\left (\int \frac {\left (1+x^4\right )^{3/4}}{1-x^4+x^8} \, dx,x,\sqrt {x}\right )}{\sqrt [4]{x^2+x^4}}\\ &=-\frac {\left (4 i \sqrt {x} \sqrt [4]{1+x^2}\right ) \operatorname {Subst}\left (\int \frac {\left (1+x^4\right )^{3/4}}{-1-i \sqrt {3}+2 x^4} \, dx,x,\sqrt {x}\right )}{\sqrt {3} \sqrt [4]{x^2+x^4}}+\frac {\left (4 i \sqrt {x} \sqrt [4]{1+x^2}\right ) \operatorname {Subst}\left (\int \frac {\left (1+x^4\right )^{3/4}}{-1+i \sqrt {3}+2 x^4} \, dx,x,\sqrt {x}\right )}{\sqrt {3} \sqrt [4]{x^2+x^4}}\\ &=-\frac {\left (2 i \left (-3+i \sqrt {3}\right ) \sqrt {x} \sqrt [4]{1+x^2}\right ) \operatorname {Subst}\left (\int \frac {1}{\sqrt [4]{1+x^4} \left (-1+i \sqrt {3}+2 x^4\right )} \, dx,x,\sqrt {x}\right )}{\sqrt {3} \sqrt [4]{x^2+x^4}}-\frac {\left (2 i \left (3+i \sqrt {3}\right ) \sqrt {x} \sqrt [4]{1+x^2}\right ) \operatorname {Subst}\left (\int \frac {1}{\sqrt [4]{1+x^4} \left (-1-i \sqrt {3}+2 x^4\right )} \, dx,x,\sqrt {x}\right )}{\sqrt {3} \sqrt [4]{x^2+x^4}}\\ &=-\frac {\left (2 i \left (-3+i \sqrt {3}\right ) \sqrt {x} \sqrt [4]{1+x^2}\right ) \operatorname {Subst}\left (\int \frac {1}{-1+i \sqrt {3}-\left (-3+i \sqrt {3}\right ) x^4} \, dx,x,\frac {\sqrt {x}}{\sqrt [4]{1+x^2}}\right )}{\sqrt {3} \sqrt [4]{x^2+x^4}}-\frac {\left (2 i \left (3+i \sqrt {3}\right ) \sqrt {x} \sqrt [4]{1+x^2}\right ) \operatorname {Subst}\left (\int \frac {1}{-1-i \sqrt {3}-\left (-3-i \sqrt {3}\right ) x^4} \, dx,x,\frac {\sqrt {x}}{\sqrt [4]{1+x^2}}\right )}{\sqrt {3} \sqrt [4]{x^2+x^4}}\\ &=-\frac {\left (\left (3+i \sqrt {3}\right ) \sqrt {x} \sqrt [4]{1+x^2}\right ) \operatorname {Subst}\left (\int \frac {1}{\sqrt {i-\sqrt {3}}-\sqrt {3 i-\sqrt {3}} x^2} \, dx,x,\frac {\sqrt {x}}{\sqrt [4]{1+x^2}}\right )}{\sqrt {3 \left (i-\sqrt {3}\right )} \sqrt [4]{x^2+x^4}}-\frac {\left (\left (3+i \sqrt {3}\right ) \sqrt {x} \sqrt [4]{1+x^2}\right ) \operatorname {Subst}\left (\int \frac {1}{\sqrt {i-\sqrt {3}}+\sqrt {3 i-\sqrt {3}} x^2} \, dx,x,\frac {\sqrt {x}}{\sqrt [4]{1+x^2}}\right )}{\sqrt {3 \left (i-\sqrt {3}\right )} \sqrt [4]{x^2+x^4}}-\frac {\left (\left (-3+i \sqrt {3}\right ) \sqrt {x} \sqrt [4]{1+x^2}\right ) \operatorname {Subst}\left (\int \frac {1}{\sqrt {i+\sqrt {3}}-\sqrt {3 i+\sqrt {3}} x^2} \, dx,x,\frac {\sqrt {x}}{\sqrt [4]{1+x^2}}\right )}{\sqrt {3 \left (i+\sqrt {3}\right )} \sqrt [4]{x^2+x^4}}-\frac {\left (\left (-3+i \sqrt {3}\right ) \sqrt {x} \sqrt [4]{1+x^2}\right ) \operatorname {Subst}\left (\int \frac {1}{\sqrt {i+\sqrt {3}}+\sqrt {3 i+\sqrt {3}} x^2} \, dx,x,\frac {\sqrt {x}}{\sqrt [4]{1+x^2}}\right )}{\sqrt {3 \left (i+\sqrt {3}\right )} \sqrt [4]{x^2+x^4}}\\ &=-\frac {\sqrt [4]{\frac {i-\sqrt {3}}{3 i-\sqrt {3}}} \left (i+\sqrt {3}\right ) \sqrt {x} \sqrt [4]{1+x^2} \tan ^{-1}\left (\frac {\sqrt {x}}{\sqrt [4]{\frac {i-\sqrt {3}}{3 i-\sqrt {3}}} \sqrt [4]{1+x^2}}\right )}{\left (i-\sqrt {3}\right ) \sqrt [4]{x^2+x^4}}-\frac {\left (i-\sqrt {3}\right ) \sqrt [4]{\frac {i+\sqrt {3}}{3 i+\sqrt {3}}} \sqrt {x} \sqrt [4]{1+x^2} \tan ^{-1}\left (\frac {\sqrt {x}}{\sqrt [4]{\frac {i+\sqrt {3}}{3 i+\sqrt {3}}} \sqrt [4]{1+x^2}}\right )}{\left (i+\sqrt {3}\right ) \sqrt [4]{x^2+x^4}}-\frac {\sqrt [4]{\frac {i-\sqrt {3}}{3 i-\sqrt {3}}} \left (i+\sqrt {3}\right ) \sqrt {x} \sqrt [4]{1+x^2} \tanh ^{-1}\left (\frac {\sqrt {x}}{\sqrt [4]{\frac {i-\sqrt {3}}{3 i-\sqrt {3}}} \sqrt [4]{1+x^2}}\right )}{\left (i-\sqrt {3}\right ) \sqrt [4]{x^2+x^4}}-\frac {\left (i-\sqrt {3}\right ) \sqrt [4]{\frac {i+\sqrt {3}}{3 i+\sqrt {3}}} \sqrt {x} \sqrt [4]{1+x^2} \tanh ^{-1}\left (\frac {\sqrt {x}}{\sqrt [4]{\frac {i+\sqrt {3}}{3 i+\sqrt {3}}} \sqrt [4]{1+x^2}}\right )}{\left (i+\sqrt {3}\right ) \sqrt [4]{x^2+x^4}}\\ \end {align*}
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Mathematica [A] time = 0.16, size = 72, normalized size = 1.16 \begin {gather*} -\frac {\left (x^4+x^2\right )^{3/4} \text {RootSum}\left [\text {$\#$1}^8-3 \text {$\#$1}^4+3\&,\frac {\text {$\#$1}^3 \log \left (\sqrt [4]{\frac {1}{x^2}+1}-\text {$\#$1}\right )}{2 \text {$\#$1}^4-3}\&\right ]}{2 \left (\frac {1}{x^2}+1\right )^{3/4} x^3} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [A] time = 0.20, size = 62, normalized size = 1.00 \begin {gather*} -\frac {1}{2} \text {RootSum}\left [3-3 \text {$\#$1}^4+\text {$\#$1}^8\&,\frac {-\log (x) \text {$\#$1}^3+\log \left (\sqrt [4]{x^2+x^4}-x \text {$\#$1}\right ) \text {$\#$1}^3}{-3+2 \text {$\#$1}^4}\&\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 {x^{2} + 1}{{\left (x^{4} + x^{2}\right )}^{\frac {1}{4}} {\left (x^{4} - x^{2} + 1\right )}}\,{d x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 43.74, size = 1629, normalized size = 26.27
method | result | size |
trager | \(\text {Expression too large to display}\) | \(1629\) |
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*} \frac {2 \, {\left (8 \, x^{5} - 7 \, {\left (x^{3} + x\right )} x^{2} + 9 \, x^{3} + x\right )}}{21 \, {\left (x^{\frac {9}{2}} - x^{\frac {5}{2}} + \sqrt {x}\right )} {\left (x^{2} + 1\right )}^{\frac {1}{4}}} + \int -\frac {4 \, {\left (16 \, x^{4} - 8 \, {\left (x^{4} + 2 \, x^{2} + 1\right )} x^{2} + 11 \, x^{2} - 5\right )}}{21 \, {\left (x^{\frac {17}{2}} - 2 \, x^{\frac {13}{2}} + 3 \, x^{\frac {9}{2}} - 2 \, x^{\frac {5}{2}} + \sqrt {x}\right )} {\left (x^{2} + 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 {x^2+1}{{\left (x^4+x^2\right )}^{1/4}\,\left (x^4-x^2+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 {x^{2} + 1}{\sqrt [4]{x^{2} \left (x^{2} + 1\right )} \left (x^{4} - x^{2} + 1\right )}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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