Optimal. Leaf size=61 \[ \frac {\sqrt [3]{(x-1)^2} \text {RootSum}\left [\text {$\#$1}^6+2 \text {$\#$1}^3+3\& ,\frac {\text {$\#$1}^2 \log \left (\sqrt [3]{x-1}-\text {$\#$1}\right )}{\text {$\#$1}^3+1}\& \right ]}{2 (x-1)^{2/3}} \]
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Rubi [C] time = 0.45, 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 = {970, 712, 50, 56, 617, 204, 31} \begin {gather*} \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}}-\frac {i \sqrt {\frac {3}{2}} \left (1-i \sqrt {2}\right )^{2/3} \sqrt [3]{x^2-2 x+1} \tan ^{-1}\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} \tan ^{-1}\left (\frac {1-\frac {2 \sqrt [3]{x-1}}{\sqrt [3]{1+i \sqrt {2}}}}{\sqrt {3}}\right )}{2 (x-1)^{2/3}} \end {gather*}
Antiderivative was successfully verified.
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Rule 31
Rule 50
Rule 56
Rule 204
Rule 617
Rule 712
Rule 970
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 ) \operatorname {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 ) \operatorname {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 ) \operatorname {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 ) \operatorname {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 ) \operatorname {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 ) \operatorname {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 [C] time = 0.08, size = 75, normalized size = 1.23 \begin {gather*} \frac {3 i \sqrt [3]{(x-1)^2} \left (\, _2F_1\left (\frac {2}{3},1;\frac {5}{3};-\frac {i (x-1)}{i+\sqrt {2}}\right )-\, _2F_1\left (\frac {2}{3},1;\frac {5}{3};\frac {i (x-1)}{-i+\sqrt {2}}\right )\right )}{4 \sqrt {2}} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [A] time = 4.19, 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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fricas [B] time = 1.53, size = 2025, normalized size = 33.20
result too large to display
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 {{\left (x^{2} - 2 \, x + 1\right )}^{\frac {1}{3}}}{x^{2} + 2}\,{d x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 17.82, size = 15639, normalized size = 256.38
method | result | size |
trager | \(\text {Expression too large to display}\) | \(15639\) |
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 {{\left (x^{2} - 2 \, x + 1\right )}^{\frac {1}{3}}}{x^{2} + 2}\,{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 {{\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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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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