Optimal. Leaf size=61 \[ \frac {\sqrt [3]{(-1+x)^2} \text {RootSum}\left [3+2 \text {$\#$1}^3+\text {$\#$1}^6\& ,\frac {\log \left (\sqrt [3]{-1+x}-\text {$\#$1}\right ) \text {$\#$1}^2}{1+\text {$\#$1}^3}\& \right ]}{2 (-1+x)^{2/3}} \]
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Rubi [C] Result contains complex when optimal does not.
time = 0.30, antiderivative size = 421, normalized size of antiderivative = 6.90, number of steps
used = 13, number of rules used = 7, integrand size = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.350, Rules used = {984, 726, 52,
58, 631, 210, 31} \begin {gather*} -\frac {i \sqrt {\frac {3}{2}} \left (1-i \sqrt {2}\right )^{2/3} \sqrt [3]{x^2-2 x+1} \text {ArcTan}\left (\frac {1-\frac {2 \sqrt [3]{x-1}}{\sqrt [3]{1-i \sqrt {2}}}}{\sqrt {3}}\right )}{2 (x-1)^{2/3}}+\frac {i \sqrt {\frac {3}{2}} \left (1+i \sqrt {2}\right )^{2/3} \sqrt [3]{x^2-2 x+1} \text {ArcTan}\left (\frac {1-\frac {2 \sqrt [3]{x-1}}{\sqrt [3]{1+i \sqrt {2}}}}{\sqrt {3}}\right )}{2 (x-1)^{2/3}}+\frac {i \left (1-i \sqrt {2}\right )^{2/3} \sqrt [3]{x^2-2 x+1} \log \left (\sqrt {2}+i x\right )}{4 \sqrt {2} (x-1)^{2/3}}-\frac {i \left (1+i \sqrt {2}\right )^{2/3} \sqrt [3]{x^2-2 x+1} \log \left (x+i \sqrt {2}\right )}{4 \sqrt {2} (x-1)^{2/3}}-\frac {3 i \left (1-i \sqrt {2}\right )^{2/3} \sqrt [3]{x^2-2 x+1} \log \left (\sqrt [3]{2 x-2}+\sqrt [3]{2 \left (1-i \sqrt {2}\right )}\right )}{4 \sqrt {2} (x-1)^{2/3}}+\frac {3 i \left (1+i \sqrt {2}\right )^{2/3} \sqrt [3]{x^2-2 x+1} \log \left (\sqrt [3]{2 x-2}+\sqrt [3]{2 \left (1+i \sqrt {2}\right )}\right )}{4 \sqrt {2} (x-1)^{2/3}} \end {gather*}
Antiderivative was successfully verified.
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Rule 31
Rule 52
Rule 58
Rule 210
Rule 631
Rule 726
Rule 984
Rubi steps
\begin {align*} \int \frac {\sqrt [3]{1-2 x+x^2}}{2+x^2} \, dx &=\frac {\sqrt [3]{1-2 x+x^2} \int \frac {(-2+2 x)^{2/3}}{2+x^2} \, dx}{(-2+2 x)^{2/3}}\\ &=\frac {\sqrt [3]{1-2 x+x^2} \int \left (\frac {i (-2+2 x)^{2/3}}{2 \sqrt {2} \left (i \sqrt {2}-x\right )}+\frac {i (-2+2 x)^{2/3}}{2 \sqrt {2} \left (i \sqrt {2}+x\right )}\right ) \, dx}{(-2+2 x)^{2/3}}\\ &=\frac {\left (i \sqrt [3]{1-2 x+x^2}\right ) \int \frac {(-2+2 x)^{2/3}}{i \sqrt {2}-x} \, dx}{2 \sqrt {2} (-2+2 x)^{2/3}}+\frac {\left (i \sqrt [3]{1-2 x+x^2}\right ) \int \frac {(-2+2 x)^{2/3}}{i \sqrt {2}+x} \, dx}{2 \sqrt {2} (-2+2 x)^{2/3}}\\ &=-\frac {\left (i \left (1-i \sqrt {2}\right ) \sqrt [3]{1-2 x+x^2}\right ) \int \frac {1}{\left (i \sqrt {2}-x\right ) \sqrt [3]{-2+2 x}} \, dx}{\sqrt {2} (-2+2 x)^{2/3}}-\frac {\left (i \left (1+i \sqrt {2}\right ) \sqrt [3]{1-2 x+x^2}\right ) \int \frac {1}{\left (i \sqrt {2}+x\right ) \sqrt [3]{-2+2 x}} \, dx}{\sqrt {2} (-2+2 x)^{2/3}}\\ &=\frac {i \left (1-i \sqrt {2}\right )^{2/3} \sqrt [3]{1-2 x+x^2} \log \left (\sqrt {2}+i x\right )}{4 \sqrt {2} (-1+x)^{2/3}}-\frac {i \left (1+i \sqrt {2}\right )^{2/3} \sqrt [3]{1-2 x+x^2} \log \left (i \sqrt {2}+x\right )}{4 \sqrt {2} (-1+x)^{2/3}}-\frac {\left (3 i \left (1-i \sqrt {2}\right )^{2/3} \sqrt [3]{1-2 x+x^2}\right ) \text {Subst}\left (\int \frac {1}{\sqrt [3]{2 \left (1-i \sqrt {2}\right )}+x} \, dx,x,\sqrt [3]{-2+2 x}\right )}{2\ 2^{5/6} (-2+2 x)^{2/3}}+\frac {\left (3 i \left (1-i \sqrt {2}\right ) \sqrt [3]{1-2 x+x^2}\right ) \text {Subst}\left (\int \frac {1}{\left (2 \left (1-i \sqrt {2}\right )\right )^{2/3}-\sqrt [3]{2 \left (1-i \sqrt {2}\right )} x+x^2} \, dx,x,\sqrt [3]{-2+2 x}\right )}{2 \sqrt {2} (-2+2 x)^{2/3}}+\frac {\left (3 i \left (1+i \sqrt {2}\right )^{2/3} \sqrt [3]{1-2 x+x^2}\right ) \text {Subst}\left (\int \frac {1}{\sqrt [3]{2 \left (1+i \sqrt {2}\right )}+x} \, dx,x,\sqrt [3]{-2+2 x}\right )}{2\ 2^{5/6} (-2+2 x)^{2/3}}-\frac {\left (3 i \left (1+i \sqrt {2}\right ) \sqrt [3]{1-2 x+x^2}\right ) \text {Subst}\left (\int \frac {1}{\left (2 \left (1+i \sqrt {2}\right )\right )^{2/3}-\sqrt [3]{2 \left (1+i \sqrt {2}\right )} x+x^2} \, dx,x,\sqrt [3]{-2+2 x}\right )}{2 \sqrt {2} (-2+2 x)^{2/3}}\\ &=\frac {i \left (1-i \sqrt {2}\right )^{2/3} \sqrt [3]{1-2 x+x^2} \log \left (\sqrt {2}+i x\right )}{4 \sqrt {2} (-1+x)^{2/3}}-\frac {i \left (1+i \sqrt {2}\right )^{2/3} \sqrt [3]{1-2 x+x^2} \log \left (i \sqrt {2}+x\right )}{4 \sqrt {2} (-1+x)^{2/3}}-\frac {3 i \left (1-i \sqrt {2}\right )^{2/3} \sqrt [3]{1-2 x+x^2} \log \left (\sqrt [3]{2 \left (1-i \sqrt {2}\right )}+\sqrt [3]{-2+2 x}\right )}{4 \sqrt {2} (-1+x)^{2/3}}+\frac {3 i \left (1+i \sqrt {2}\right )^{2/3} \sqrt [3]{1-2 x+x^2} \log \left (\sqrt [3]{2 \left (1+i \sqrt {2}\right )}+\sqrt [3]{-2+2 x}\right )}{4 \sqrt {2} (-1+x)^{2/3}}+\frac {\left (3 i \left (1-i \sqrt {2}\right )^{2/3} \sqrt [3]{1-2 x+x^2}\right ) \text {Subst}\left (\int \frac {1}{-3-x^2} \, dx,x,1-\frac {2 \sqrt [3]{-1+x}}{\sqrt [3]{1-i \sqrt {2}}}\right )}{2^{5/6} (-2+2 x)^{2/3}}-\frac {\left (3 i \left (1+i \sqrt {2}\right )^{2/3} \sqrt [3]{1-2 x+x^2}\right ) \text {Subst}\left (\int \frac {1}{-3-x^2} \, dx,x,1-\frac {2 \sqrt [3]{-1+x}}{\sqrt [3]{1+i \sqrt {2}}}\right )}{2^{5/6} (-2+2 x)^{2/3}}\\ &=-\frac {i \sqrt {\frac {3}{2}} \left (1-i \sqrt {2}\right )^{2/3} \sqrt [3]{1-2 x+x^2} \tan ^{-1}\left (\frac {1-\frac {2 \sqrt [3]{-1+x}}{\sqrt [3]{1-i \sqrt {2}}}}{\sqrt {3}}\right )}{2 (-1+x)^{2/3}}+\frac {i \sqrt {\frac {3}{2}} \left (1+i \sqrt {2}\right )^{2/3} \sqrt [3]{1-2 x+x^2} \tan ^{-1}\left (\frac {1-\frac {2 \sqrt [3]{-1+x}}{\sqrt [3]{1+i \sqrt {2}}}}{\sqrt {3}}\right )}{2 (-1+x)^{2/3}}+\frac {i \left (1-i \sqrt {2}\right )^{2/3} \sqrt [3]{1-2 x+x^2} \log \left (\sqrt {2}+i x\right )}{4 \sqrt {2} (-1+x)^{2/3}}-\frac {i \left (1+i \sqrt {2}\right )^{2/3} \sqrt [3]{1-2 x+x^2} \log \left (i \sqrt {2}+x\right )}{4 \sqrt {2} (-1+x)^{2/3}}-\frac {3 i \left (1-i \sqrt {2}\right )^{2/3} \sqrt [3]{1-2 x+x^2} \log \left (\sqrt [3]{2 \left (1-i \sqrt {2}\right )}+\sqrt [3]{-2+2 x}\right )}{4 \sqrt {2} (-1+x)^{2/3}}+\frac {3 i \left (1+i \sqrt {2}\right )^{2/3} \sqrt [3]{1-2 x+x^2} \log \left (\sqrt [3]{2 \left (1+i \sqrt {2}\right )}+\sqrt [3]{-2+2 x}\right )}{4 \sqrt {2} (-1+x)^{2/3}}\\ \end {align*}
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Mathematica [A]
time = 0.04, size = 61, normalized size = 1.00 \begin {gather*} \frac {\sqrt [3]{(-1+x)^2} \text {RootSum}\left [3+2 \text {$\#$1}^3+\text {$\#$1}^6\&,\frac {\log \left (\sqrt [3]{-1+x}-\text {$\#$1}\right ) \text {$\#$1}^2}{1+\text {$\#$1}^3}\&\right ]}{2 (-1+x)^{2/3}} \end {gather*}
Antiderivative was successfully verified.
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Maple [C] Result contains higher order function than in optimal. Order 9 vs. order
1.
time = 15.76, size = 15658, normalized size = 256.69
method | result | size |
trager | \(\text {Expression too large to display}\) | \(15658\) |
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 = 0.97, size = 2025, normalized size = 33.20 \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 {\sqrt [3]{\left (x - 1\right )^{2}}}{x^{2} + 2}\, 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 = -0.02 \begin {gather*} \int \frac {{\left (x^2-2\,x+1\right )}^{1/3}}{x^2+2} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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