Optimal. Leaf size=217 \[ \tan ^{-1}\left (\frac {x}{\sqrt [4]{x^4+1}}\right )-\frac {1}{2} \sqrt {\frac {1}{10} \left (\sqrt {5}-1\right )} \tan ^{-1}\left (\frac {\sqrt {\frac {\sqrt {5}}{2}-\frac {1}{2}} x}{\sqrt [4]{x^4+1}}\right )-\frac {1}{2} \sqrt {\frac {1}{10} \left (1+\sqrt {5}\right )} \tan ^{-1}\left (\frac {\sqrt {\frac {1}{2}+\frac {\sqrt {5}}{2}} x}{\sqrt [4]{x^4+1}}\right )+\tanh ^{-1}\left (\frac {x}{\sqrt [4]{x^4+1}}\right )-\frac {1}{2} \sqrt {\frac {1}{10} \left (\sqrt {5}-1\right )} \tanh ^{-1}\left (\frac {\sqrt {\frac {\sqrt {5}}{2}-\frac {1}{2}} x}{\sqrt [4]{x^4+1}}\right )-\frac {1}{2} \sqrt {\frac {1}{10} \left (1+\sqrt {5}\right )} \tanh ^{-1}\left (\frac {\sqrt {\frac {1}{2}+\frac {\sqrt {5}}{2}} x}{\sqrt [4]{x^4+1}}\right ) \]
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Rubi [A] time = 0.30, antiderivative size = 233, normalized size of antiderivative = 1.07, number of steps used = 15, number of rules used = 7, integrand size = 32, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.219, Rules used = {6728, 240, 212, 206, 203, 1428, 377} \begin {gather*} \tan ^{-1}\left (\frac {x}{\sqrt [4]{x^4+1}}\right )-\frac {\sqrt [4]{\frac {1}{2} \left (3-\sqrt {5}\right )} \tan ^{-1}\left (\frac {\sqrt [4]{\frac {2}{3+\sqrt {5}}} x}{\sqrt [4]{x^4+1}}\right )}{2 \sqrt {5}}-\frac {\sqrt [4]{\frac {1}{2} \left (3+\sqrt {5}\right )} \tan ^{-1}\left (\frac {\sqrt [4]{\frac {1}{2} \left (3+\sqrt {5}\right )} x}{\sqrt [4]{x^4+1}}\right )}{2 \sqrt {5}}+\tanh ^{-1}\left (\frac {x}{\sqrt [4]{x^4+1}}\right )-\frac {\sqrt [4]{\frac {1}{2} \left (3-\sqrt {5}\right )} \tanh ^{-1}\left (\frac {\sqrt [4]{\frac {2}{3+\sqrt {5}}} x}{\sqrt [4]{x^4+1}}\right )}{2 \sqrt {5}}-\frac {\sqrt [4]{\frac {1}{2} \left (3+\sqrt {5}\right )} \tanh ^{-1}\left (\frac {\sqrt [4]{\frac {1}{2} \left (3+\sqrt {5}\right )} x}{\sqrt [4]{x^4+1}}\right )}{2 \sqrt {5}} \end {gather*}
Warning: Unable to verify antiderivative.
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Rule 203
Rule 206
Rule 212
Rule 240
Rule 377
Rule 1428
Rule 6728
Rubi steps
\begin {align*} \int \frac {-1+2 x^4+2 x^8}{\sqrt [4]{1+x^4} \left (-1+x^4+x^8\right )} \, dx &=\int \left (\frac {2}{\sqrt [4]{1+x^4}}+\frac {1}{\sqrt [4]{1+x^4} \left (-1+x^4+x^8\right )}\right ) \, dx\\ &=2 \int \frac {1}{\sqrt [4]{1+x^4}} \, dx+\int \frac {1}{\sqrt [4]{1+x^4} \left (-1+x^4+x^8\right )} \, dx\\ &=2 \operatorname {Subst}\left (\int \frac {1}{1-x^4} \, dx,x,\frac {x}{\sqrt [4]{1+x^4}}\right )+\frac {2 \int \frac {1}{\sqrt [4]{1+x^4} \left (1-\sqrt {5}+2 x^4\right )} \, dx}{\sqrt {5}}-\frac {2 \int \frac {1}{\sqrt [4]{1+x^4} \left (1+\sqrt {5}+2 x^4\right )} \, dx}{\sqrt {5}}\\ &=\frac {2 \operatorname {Subst}\left (\int \frac {1}{1-\sqrt {5}-\left (-1-\sqrt {5}\right ) x^4} \, dx,x,\frac {x}{\sqrt [4]{1+x^4}}\right )}{\sqrt {5}}-\frac {2 \operatorname {Subst}\left (\int \frac {1}{1+\sqrt {5}-\left (-1+\sqrt {5}\right ) x^4} \, dx,x,\frac {x}{\sqrt [4]{1+x^4}}\right )}{\sqrt {5}}+\operatorname {Subst}\left (\int \frac {1}{1-x^2} \, dx,x,\frac {x}{\sqrt [4]{1+x^4}}\right )+\operatorname {Subst}\left (\int \frac {1}{1+x^2} \, dx,x,\frac {x}{\sqrt [4]{1+x^4}}\right )\\ &=\tan ^{-1}\left (\frac {x}{\sqrt [4]{1+x^4}}\right )+\tanh ^{-1}\left (\frac {x}{\sqrt [4]{1+x^4}}\right )-\frac {\operatorname {Subst}\left (\int \frac {1}{\sqrt {3-\sqrt {5}}-\sqrt {2} x^2} \, dx,x,\frac {x}{\sqrt [4]{1+x^4}}\right )}{\sqrt {10}}-\frac {\operatorname {Subst}\left (\int \frac {1}{\sqrt {3+\sqrt {5}}-\sqrt {2} x^2} \, dx,x,\frac {x}{\sqrt [4]{1+x^4}}\right )}{\sqrt {10}}-\frac {\operatorname {Subst}\left (\int \frac {1}{\sqrt {3-\sqrt {5}}+\sqrt {2} x^2} \, dx,x,\frac {x}{\sqrt [4]{1+x^4}}\right )}{\sqrt {10}}-\frac {\operatorname {Subst}\left (\int \frac {1}{\sqrt {3+\sqrt {5}}+\sqrt {2} x^2} \, dx,x,\frac {x}{\sqrt [4]{1+x^4}}\right )}{\sqrt {10}}\\ &=\tan ^{-1}\left (\frac {x}{\sqrt [4]{1+x^4}}\right )-\frac {\tan ^{-1}\left (\frac {\sqrt [4]{\frac {2}{3+\sqrt {5}}} x}{\sqrt [4]{1+x^4}}\right )}{2^{3/4} \sqrt {5} \sqrt [4]{3+\sqrt {5}}}-\frac {\sqrt [4]{\frac {1}{2} \left (3+\sqrt {5}\right )} \tan ^{-1}\left (\frac {\sqrt [4]{\frac {1}{2} \left (3+\sqrt {5}\right )} x}{\sqrt [4]{1+x^4}}\right )}{2 \sqrt {5}}+\tanh ^{-1}\left (\frac {x}{\sqrt [4]{1+x^4}}\right )-\frac {\tanh ^{-1}\left (\frac {\sqrt [4]{\frac {2}{3+\sqrt {5}}} x}{\sqrt [4]{1+x^4}}\right )}{2^{3/4} \sqrt {5} \sqrt [4]{3+\sqrt {5}}}-\frac {\sqrt [4]{\frac {1}{2} \left (3+\sqrt {5}\right )} \tanh ^{-1}\left (\frac {\sqrt [4]{\frac {1}{2} \left (3+\sqrt {5}\right )} x}{\sqrt [4]{1+x^4}}\right )}{2 \sqrt {5}}\\ \end {align*}
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Mathematica [A] time = 0.49, size = 239, normalized size = 1.10 \begin {gather*} \tan ^{-1}\left (\frac {x}{\sqrt [4]{x^4+1}}\right )-\frac {\sqrt [4]{\frac {1}{2} \left (3+\sqrt {5}\right )} \tan ^{-1}\left (\frac {\sqrt [4]{\frac {2}{3+\sqrt {5}}} x}{\sqrt [4]{x^4+1}}\right )}{5+\sqrt {5}}+\frac {\sqrt [4]{\frac {1}{2} \left (3-\sqrt {5}\right )} \tan ^{-1}\left (\frac {\sqrt [4]{\frac {1}{2} \left (3+\sqrt {5}\right )} x}{\sqrt [4]{x^4+1}}\right )}{\sqrt {5}-5}+\tanh ^{-1}\left (\frac {x}{\sqrt [4]{x^4+1}}\right )-\frac {\sqrt [4]{\frac {1}{2} \left (3+\sqrt {5}\right )} \tanh ^{-1}\left (\frac {\sqrt [4]{\frac {2}{3+\sqrt {5}}} x}{\sqrt [4]{x^4+1}}\right )}{5+\sqrt {5}}+\frac {\sqrt [4]{\frac {1}{2} \left (3-\sqrt {5}\right )} \tanh ^{-1}\left (\frac {\sqrt [4]{\frac {1}{2} \left (3+\sqrt {5}\right )} x}{\sqrt [4]{x^4+1}}\right )}{\sqrt {5}-5} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [A] time = 0.66, size = 217, normalized size = 1.00 \begin {gather*} \tan ^{-1}\left (\frac {x}{\sqrt [4]{1+x^4}}\right )-\frac {1}{2} \sqrt {\frac {1}{10} \left (-1+\sqrt {5}\right )} \tan ^{-1}\left (\frac {\sqrt {-\frac {1}{2}+\frac {\sqrt {5}}{2}} x}{\sqrt [4]{1+x^4}}\right )-\frac {1}{2} \sqrt {\frac {1}{10} \left (1+\sqrt {5}\right )} \tan ^{-1}\left (\frac {\sqrt {\frac {1}{2}+\frac {\sqrt {5}}{2}} x}{\sqrt [4]{1+x^4}}\right )+\tanh ^{-1}\left (\frac {x}{\sqrt [4]{1+x^4}}\right )-\frac {1}{2} \sqrt {\frac {1}{10} \left (-1+\sqrt {5}\right )} \tanh ^{-1}\left (\frac {\sqrt {-\frac {1}{2}+\frac {\sqrt {5}}{2}} x}{\sqrt [4]{1+x^4}}\right )-\frac {1}{2} \sqrt {\frac {1}{10} \left (1+\sqrt {5}\right )} \tanh ^{-1}\left (\frac {\sqrt {\frac {1}{2}+\frac {\sqrt {5}}{2}} x}{\sqrt [4]{1+x^4}}\right ) \end {gather*}
Antiderivative was successfully verified.
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fricas [B] time = 0.53, size = 406, normalized size = 1.87 \begin {gather*} \frac {1}{10} \, \sqrt {10} \sqrt {\sqrt {5} + 1} \arctan \left (\frac {\sqrt {10} \sqrt {2} {\left (\sqrt {5} x - 5 \, x\right )} \sqrt {\sqrt {5} + 1} \sqrt {\frac {\sqrt {5} x^{2} + x^{2} + 2 \, \sqrt {x^{4} + 1}}{x^{2}}} - 2 \, \sqrt {10} {\left (x^{4} + 1\right )}^{\frac {1}{4}} \sqrt {\sqrt {5} + 1} {\left (\sqrt {5} - 5\right )}}{40 \, x}\right ) - \frac {1}{10} \, \sqrt {10} \sqrt {\sqrt {5} - 1} \arctan \left (\frac {\sqrt {10} \sqrt {2} {\left (\sqrt {5} x + 5 \, x\right )} \sqrt {\sqrt {5} - 1} \sqrt {\frac {\sqrt {5} x^{2} - x^{2} + 2 \, \sqrt {x^{4} + 1}}{x^{2}}} - 2 \, \sqrt {10} {\left (x^{4} + 1\right )}^{\frac {1}{4}} {\left (\sqrt {5} + 5\right )} \sqrt {\sqrt {5} - 1}}{40 \, x}\right ) - \frac {1}{40} \, \sqrt {10} \sqrt {\sqrt {5} + 1} \log \left (\frac {\sqrt {10} \sqrt {5} x \sqrt {\sqrt {5} + 1} + 10 \, {\left (x^{4} + 1\right )}^{\frac {1}{4}}}{x}\right ) + \frac {1}{40} \, \sqrt {10} \sqrt {\sqrt {5} + 1} \log \left (-\frac {\sqrt {10} \sqrt {5} x \sqrt {\sqrt {5} + 1} - 10 \, {\left (x^{4} + 1\right )}^{\frac {1}{4}}}{x}\right ) - \frac {1}{40} \, \sqrt {10} \sqrt {\sqrt {5} - 1} \log \left (\frac {\sqrt {10} \sqrt {5} x \sqrt {\sqrt {5} - 1} + 10 \, {\left (x^{4} + 1\right )}^{\frac {1}{4}}}{x}\right ) + \frac {1}{40} \, \sqrt {10} \sqrt {\sqrt {5} - 1} \log \left (-\frac {\sqrt {10} \sqrt {5} x \sqrt {\sqrt {5} - 1} - 10 \, {\left (x^{4} + 1\right )}^{\frac {1}{4}}}{x}\right ) - \arctan \left (\frac {{\left (x^{4} + 1\right )}^{\frac {1}{4}}}{x}\right ) + \frac {1}{2} \, \log \left (\frac {x + {\left (x^{4} + 1\right )}^{\frac {1}{4}}}{x}\right ) - \frac {1}{2} \, \log \left (-\frac {x - {\left (x^{4} + 1\right )}^{\frac {1}{4}}}{x}\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [F(-2)] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: TypeError} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.03, size = 0, normalized size = 0.00 \[\int \frac {2 x^{8}+2 x^{4}-1}{\left (x^{4}+1\right )^{\frac {1}{4}} \left (x^{8}+x^{4}-1\right )}\, dx\]
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 {2 \, x^{8} + 2 \, x^{4} - 1}{{\left (x^{8} + 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 = -0.00 \begin {gather*} \int \frac {2\,x^8+2\,x^4-1}{{\left (x^4+1\right )}^{1/4}\,\left (x^8+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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