Optimal. Leaf size=125 \[ -\frac {\log \left (2^{2/3} \sqrt [3]{x^2+2 x-1}+2\right )}{2 \sqrt [3]{2}}+\frac {\log \left (-\sqrt [3]{2} \left (x^2+2 x-1\right )^{2/3}+2^{2/3} \sqrt [3]{x^2+2 x-1}-2\right )}{4 \sqrt [3]{2}}-\frac {\sqrt {3} \tan ^{-1}\left (\frac {1}{\sqrt {3}}-\frac {2^{2/3} \sqrt [3]{x^2+2 x-1}}{\sqrt {3}}\right )}{2 \sqrt [3]{2}} \]
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Rubi [A] time = 0.06, antiderivative size = 82, normalized size of antiderivative = 0.66, number of steps used = 6, number of rules used = 6, integrand size = 18, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.333, Rules used = {694, 266, 56, 617, 204, 31} \begin {gather*} \frac {\log (x+1)}{2 \sqrt [3]{2}}-\frac {3 \log \left (\sqrt [3]{(x+1)^2-2}+\sqrt [3]{2}\right )}{4 \sqrt [3]{2}}-\frac {\sqrt {3} \tan ^{-1}\left (\frac {1-2^{2/3} \sqrt [3]{(x+1)^2-2}}{\sqrt {3}}\right )}{2 \sqrt [3]{2}} \end {gather*}
Antiderivative was successfully verified.
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Rule 31
Rule 56
Rule 204
Rule 266
Rule 617
Rule 694
Rubi steps
\begin {align*} \int \frac {1}{(1+x) \sqrt [3]{-1+2 x+x^2}} \, dx &=\operatorname {Subst}\left (\int \frac {1}{x \sqrt [3]{-2+x^2}} \, dx,x,1+x\right )\\ &=\frac {1}{2} \operatorname {Subst}\left (\int \frac {1}{\sqrt [3]{-2+x} x} \, dx,x,(1+x)^2\right )\\ &=\frac {\log (1+x)}{2 \sqrt [3]{2}}+\frac {3}{4} \operatorname {Subst}\left (\int \frac {1}{2^{2/3}-\sqrt [3]{2} x+x^2} \, dx,x,\sqrt [3]{-2+(1+x)^2}\right )-\frac {3 \operatorname {Subst}\left (\int \frac {1}{\sqrt [3]{2}+x} \, dx,x,\sqrt [3]{-2+(1+x)^2}\right )}{4 \sqrt [3]{2}}\\ &=\frac {\log (1+x)}{2 \sqrt [3]{2}}-\frac {3 \log \left (\sqrt [3]{2}+\sqrt [3]{-2+(1+x)^2}\right )}{4 \sqrt [3]{2}}+\frac {3 \operatorname {Subst}\left (\int \frac {1}{-3-x^2} \, dx,x,1-2^{2/3} \sqrt [3]{-2+(1+x)^2}\right )}{2 \sqrt [3]{2}}\\ &=-\frac {\sqrt {3} \tan ^{-1}\left (\frac {1-2^{2/3} \sqrt [3]{-2+(1+x)^2}}{\sqrt {3}}\right )}{2 \sqrt [3]{2}}+\frac {\log (1+x)}{2 \sqrt [3]{2}}-\frac {3 \log \left (\sqrt [3]{2}+\sqrt [3]{-2+(1+x)^2}\right )}{4 \sqrt [3]{2}}\\ \end {align*}
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Mathematica [C] time = 0.01, size = 36, normalized size = 0.29 \begin {gather*} \frac {3}{8} \left ((x+1)^2-2\right )^{2/3} \, _2F_1\left (\frac {2}{3},1;\frac {5}{3};\frac {1}{2} \left (2-(x+1)^2\right )\right ) \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [A] time = 0.22, size = 125, normalized size = 1.00 \begin {gather*} -\frac {\log \left (2^{2/3} \sqrt [3]{x^2+2 x-1}+2\right )}{2 \sqrt [3]{2}}+\frac {\log \left (-\sqrt [3]{2} \left (x^2+2 x-1\right )^{2/3}+2^{2/3} \sqrt [3]{x^2+2 x-1}-2\right )}{4 \sqrt [3]{2}}-\frac {\sqrt {3} \tan ^{-1}\left (\frac {1}{\sqrt {3}}-\frac {2^{2/3} \sqrt [3]{x^2+2 x-1}}{\sqrt {3}}\right )}{2 \sqrt [3]{2}} \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 0.40, size = 116, normalized size = 0.93 \begin {gather*} \frac {1}{4} \, \sqrt {3} 2^{\frac {2}{3}} \left (-1\right )^{\frac {1}{3}} \arctan \left (\frac {1}{6} \, \sqrt {3} 2^{\frac {1}{6}} {\left (2 \, \sqrt {2} \left (-1\right )^{\frac {1}{3}} {\left (x^{2} + 2 \, x - 1\right )}^{\frac {1}{3}} + 2^{\frac {5}{6}}\right )}\right ) - \frac {1}{8} \cdot 2^{\frac {2}{3}} \left (-1\right )^{\frac {1}{3}} \log \left (-2^{\frac {1}{3}} \left (-1\right )^{\frac {2}{3}} {\left (x^{2} + 2 \, x - 1\right )}^{\frac {1}{3}} - 2^{\frac {2}{3}} \left (-1\right )^{\frac {1}{3}} + {\left (x^{2} + 2 \, x - 1\right )}^{\frac {2}{3}}\right ) + \frac {1}{4} \cdot 2^{\frac {2}{3}} \left (-1\right )^{\frac {1}{3}} \log \left (2^{\frac {1}{3}} \left (-1\right )^{\frac {2}{3}} + {\left (x^{2} + 2 \, x - 1\right )}^{\frac {1}{3}}\right ) \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 {1}{{\left (x^{2} + 2 \, x - 1\right )}^{\frac {1}{3}} {\left (x + 1\right )}}\,{d x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [C] time = 7.33, size = 802, normalized size = 6.42
result too large to display
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 {1}{{\left (x^{2} + 2 \, x - 1\right )}^{\frac {1}{3}} {\left (x + 1\right )}}\,{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 = -0.01 \begin {gather*} \int \frac {1}{\left (x+1\right )\,{\left (x^2+2\,x-1\right )}^{1/3}} \,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 {1}{\left (x + 1\right ) \sqrt [3]{x^{2} + 2 x - 1}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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