Optimal. Leaf size=177 \[ \frac {\left (\left (-4+x^2\right )^4\right )^{7/8} \left (\frac {4 \text {ArcTan}\left (\frac {1}{\sqrt {3}}-\frac {x}{\sqrt {3}}+\frac {\sqrt {-4+x^2}}{\sqrt {3}}\right )}{3 \sqrt {3}}-\frac {2}{3} \text {RootSum}\left [16-8 \text {$\#$1}+12 \text {$\#$1}^2-2 \text {$\#$1}^3+\text {$\#$1}^4\& ,\frac {4 \log \left (-x+\sqrt {-4+x^2}-\text {$\#$1}\right )-\log \left (-x+\sqrt {-4+x^2}-\text {$\#$1}\right ) \text {$\#$1}+\log \left (-x+\sqrt {-4+x^2}-\text {$\#$1}\right ) \text {$\#$1}^2}{-4+12 \text {$\#$1}-3 \text {$\#$1}^2+2 \text {$\#$1}^3}\& \right ]\right )}{\left (-4+x^2\right )^{7/2}} \]
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Rubi [A]
time = 0.64, antiderivative size = 224, normalized size of antiderivative = 1.27, number of steps
used = 10, number of rules used = 6, integrand size = 35, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.171, Rules used = {6820, 1973,
6857, 739, 212, 210} \begin {gather*} -\frac {\left (1+(-1)^{2/3}\right ) \sqrt {4-x^2} \text {ArcTan}\left (\frac {x+4 \sqrt [3]{-1}}{\sqrt {1-4 (-1)^{2/3}} \sqrt {4-x^2}}\right )}{3 \sqrt {1-4 (-1)^{2/3}} \sqrt [8]{\left (x^2-4\right )^4}}+\frac {\left (1-\sqrt [3]{-1}\right ) \sqrt {4-x^2} \text {ArcTan}\left (\frac {(-1)^{2/3} \left (\sqrt [3]{-1} x+4\right )}{\sqrt {1+4 \sqrt [3]{-1}} \sqrt {4-x^2}}\right )}{3 \sqrt {1+4 \sqrt [3]{-1}} \sqrt [8]{\left (x^2-4\right )^4}}-\frac {2 \sqrt {4-x^2} \tanh ^{-1}\left (\frac {4-x}{\sqrt {3} \sqrt {4-x^2}}\right )}{3 \sqrt {3} \sqrt [8]{\left (x^2-4\right )^4}} \end {gather*}
Antiderivative was successfully verified.
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Rule 210
Rule 212
Rule 739
Rule 1973
Rule 6820
Rule 6857
Rubi steps
\begin {align*} \int \frac {1+x}{\left (-1+x^3\right ) \sqrt [8]{256-256 x^2+96 x^4-16 x^6+x^8}} \, dx &=\int \frac {-1-x}{\sqrt [8]{\left (-4+x^2\right )^4} \left (1-x^3\right )} \, dx\\ &=\frac {\sqrt {-4+x^2} \int \frac {-1-x}{\sqrt {-4+x^2} \left (1-x^3\right )} \, dx}{\sqrt [8]{\left (-4+x^2\right )^4}}\\ &=\frac {\sqrt {-4+x^2} \int \left (-\frac {2}{3 (1-x) \sqrt {-4+x^2}}+\frac {-1-(-1)^{2/3}}{3 \left (1+\sqrt [3]{-1} x\right ) \sqrt {-4+x^2}}+\frac {-1+\sqrt [3]{-1}}{3 \left (1-(-1)^{2/3} x\right ) \sqrt {-4+x^2}}\right ) \, dx}{\sqrt [8]{\left (-4+x^2\right )^4}}\\ &=-\frac {\left (2 \sqrt {-4+x^2}\right ) \int \frac {1}{(1-x) \sqrt {-4+x^2}} \, dx}{3 \sqrt [8]{\left (-4+x^2\right )^4}}+\frac {\left (\left (-1+\sqrt [3]{-1}\right ) \sqrt {-4+x^2}\right ) \int \frac {1}{\left (1-(-1)^{2/3} x\right ) \sqrt {-4+x^2}} \, dx}{3 \sqrt [8]{\left (-4+x^2\right )^4}}+\frac {\left (\left (-1-(-1)^{2/3}\right ) \sqrt {-4+x^2}\right ) \int \frac {1}{\left (1+\sqrt [3]{-1} x\right ) \sqrt {-4+x^2}} \, dx}{3 \sqrt [8]{\left (-4+x^2\right )^4}}\\ &=\frac {\left (2 \sqrt {-4+x^2}\right ) \text {Subst}\left (\int \frac {1}{-3-x^2} \, dx,x,\frac {4-x}{\sqrt {-4+x^2}}\right )}{3 \sqrt [8]{\left (-4+x^2\right )^4}}-\frac {\left (\left (-1+\sqrt [3]{-1}\right ) \sqrt {-4+x^2}\right ) \text {Subst}\left (\int \frac {1}{1+4 \sqrt [3]{-1}-x^2} \, dx,x,\frac {4 (-1)^{2/3}-x}{\sqrt {-4+x^2}}\right )}{3 \sqrt [8]{\left (-4+x^2\right )^4}}-\frac {\left (\left (-1-(-1)^{2/3}\right ) \sqrt {-4+x^2}\right ) \text {Subst}\left (\int \frac {1}{1-4 (-1)^{2/3}-x^2} \, dx,x,\frac {-4 \sqrt [3]{-1}-x}{\sqrt {-4+x^2}}\right )}{3 \sqrt [8]{\left (-4+x^2\right )^4}}\\ &=-\frac {2 \sqrt {-4+x^2} \tan ^{-1}\left (\frac {4-x}{\sqrt {3} \sqrt {-4+x^2}}\right )}{3 \sqrt {3} \sqrt [8]{\left (-4+x^2\right )^4}}+\frac {\left (1-\sqrt [3]{-1}\right ) \sqrt {-4+x^2} \tanh ^{-1}\left (\frac {(-1)^{2/3} \left (4+\sqrt [3]{-1} x\right )}{\sqrt {1+4 \sqrt [3]{-1}} \sqrt {-4+x^2}}\right )}{3 \sqrt {1+4 \sqrt [3]{-1}} \sqrt [8]{\left (-4+x^2\right )^4}}-\frac {\left (1+(-1)^{2/3}\right ) \sqrt {-4+x^2} \tanh ^{-1}\left (\frac {\sqrt [3]{-1} \left (4-(-1)^{2/3} x\right )}{\sqrt {1-4 (-1)^{2/3}} \sqrt {-4+x^2}}\right )}{3 \sqrt {1-4 (-1)^{2/3}} \sqrt [8]{\left (-4+x^2\right )^4}}\\ \end {align*}
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Mathematica [A]
time = 0.40, size = 168, normalized size = 0.95 \begin {gather*} -\frac {\sqrt {-4+x^2} \left (2 \sqrt {7} \text {ArcTan}\left (\frac {4-x}{\sqrt {3} \sqrt {-4+x^2}}\right )+\left (-1+\sqrt [3]{-1}\right ) \sqrt {1-4 (-1)^{2/3}} \tanh ^{-1}\left (\frac {4 (-1)^{2/3}-x}{\sqrt {1+4 \sqrt [3]{-1}} \sqrt {-4+x^2}}\right )+\sqrt {1+4 \sqrt [3]{-1}} \left (1+(-1)^{2/3}\right ) \tanh ^{-1}\left (\frac {4 \sqrt [3]{-1}+x}{\sqrt {1-4 (-1)^{2/3}} \sqrt {-4+x^2}}\right )\right )}{3 \sqrt {21} \sqrt [8]{\left (-4+x^2\right )^4}} \end {gather*}
Antiderivative was successfully verified.
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Maple [F]
time = 0.02, size = 0, normalized size = 0.00 \[\int \frac {1+x}{\left (x^{3}-1\right ) \left (x^{8}-16 x^{6}+96 x^{4}-256 x^{2}+256\right )^{\frac {1}{8}}}\, dx\]
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*} \text {Failed to integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [C] Result contains higher order function than in optimal. Order 3 vs. order
1.
time = 0.45, size = 1289, normalized size = 7.28 \begin {gather*} \text {Too large to display} \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 + 1}{\sqrt [8]{\left (x - 2\right )^{4} \left (x + 2\right )^{4}} \left (x - 1\right ) \left (x^{2} + x + 1\right )}\, dx \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*} \text {could not integrate} \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 {x+1}{\left (x^3-1\right )\,{\left (x^8-16\,x^6+96\,x^4-256\,x^2+256\right )}^{1/8}} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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