Optimal. Leaf size=86 \[ \frac {1}{2} \log \left (\sqrt [3]{x^2+2 x+2}-1\right )-\frac {1}{4} \log \left (\left (x^2+2 x+2\right )^{2/3}+\sqrt [3]{x^2+2 x+2}+1\right )+\frac {1}{2} \sqrt {3} \tan ^{-1}\left (\frac {2 \sqrt [3]{x^2+2 x+2}}{\sqrt {3}}+\frac {1}{\sqrt {3}}\right ) \]
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Rubi [A] time = 0.05, antiderivative size = 60, normalized size of antiderivative = 0.70, 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, 55, 618, 204, 31} \begin {gather*} -\frac {1}{2} \log (x+1)+\frac {3}{4} \log \left (1-\sqrt [3]{(x+1)^2+1}\right )+\frac {1}{2} \sqrt {3} \tan ^{-1}\left (\frac {2 \sqrt [3]{(x+1)^2+1}+1}{\sqrt {3}}\right ) \end {gather*}
Antiderivative was successfully verified.
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Rule 31
Rule 55
Rule 204
Rule 266
Rule 618
Rule 694
Rubi steps
\begin {align*} \int \frac {1}{(1+x) \sqrt [3]{2+2 x+x^2}} \, dx &=\operatorname {Subst}\left (\int \frac {1}{x \sqrt [3]{1+x^2}} \, dx,x,1+x\right )\\ &=\frac {1}{2} \operatorname {Subst}\left (\int \frac {1}{x \sqrt [3]{1+x}} \, dx,x,(1+x)^2\right )\\ &=-\frac {1}{2} \log (1+x)-\frac {3}{4} \operatorname {Subst}\left (\int \frac {1}{1-x} \, dx,x,\sqrt [3]{1+(1+x)^2}\right )+\frac {3}{4} \operatorname {Subst}\left (\int \frac {1}{1+x+x^2} \, dx,x,\sqrt [3]{1+(1+x)^2}\right )\\ &=-\frac {1}{2} \log (1+x)+\frac {3}{4} \log \left (1-\sqrt [3]{1+(1+x)^2}\right )-\frac {3}{2} \operatorname {Subst}\left (\int \frac {1}{-3-x^2} \, dx,x,1+2 \sqrt [3]{1+(1+x)^2}\right )\\ &=\frac {1}{2} \sqrt {3} \tan ^{-1}\left (\frac {1+2 \sqrt [3]{1+(1+x)^2}}{\sqrt {3}}\right )-\frac {1}{2} \log (1+x)+\frac {3}{4} \log \left (1-\sqrt [3]{1+(1+x)^2}\right )\\ \end {align*}
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Mathematica [A] time = 0.02, size = 59, normalized size = 0.69 \begin {gather*} \frac {1}{2} \left (-\log (x+1)+\frac {3}{2} \log \left (1-\sqrt [3]{(x+1)^2+1}\right )+\sqrt {3} \tan ^{-1}\left (\frac {2 \sqrt [3]{(x+1)^2+1}+1}{\sqrt {3}}\right )\right ) \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [A] time = 0.09, size = 86, normalized size = 1.00 \begin {gather*} \frac {1}{2} \sqrt {3} \tan ^{-1}\left (\frac {1}{\sqrt {3}}+\frac {2 \sqrt [3]{2+2 x+x^2}}{\sqrt {3}}\right )+\frac {1}{2} \log \left (-1+\sqrt [3]{2+2 x+x^2}\right )-\frac {1}{4} \log \left (1+\sqrt [3]{2+2 x+x^2}+\left (2+2 x+x^2\right )^{2/3}\right ) \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 0.44, size = 68, normalized size = 0.79 \begin {gather*} \frac {1}{2} \, \sqrt {3} \arctan \left (\frac {2}{3} \, \sqrt {3} {\left (x^{2} + 2 \, x + 2\right )}^{\frac {1}{3}} + \frac {1}{3} \, \sqrt {3}\right ) - \frac {1}{4} \, \log \left ({\left (x^{2} + 2 \, x + 2\right )}^{\frac {2}{3}} + {\left (x^{2} + 2 \, x + 2\right )}^{\frac {1}{3}} + 1\right ) + \frac {1}{2} \, \log \left ({\left (x^{2} + 2 \, x + 2\right )}^{\frac {1}{3}} - 1\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 + 2\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 = 1.74, size = 433, normalized size = 5.03
method | result | size |
trager | \(\frac {\RootOf \left (\textit {\_Z}^{2}+\textit {\_Z} +1\right ) \ln \left (-\frac {4 \RootOf \left (\textit {\_Z}^{2}+\textit {\_Z} +1\right )^{2} x^{2}+8 \RootOf \left (\textit {\_Z}^{2}+\textit {\_Z} +1\right )^{2} x -15 \RootOf \left (\textit {\_Z}^{2}+\textit {\_Z} +1\right ) \left (x^{2}+2 x +2\right )^{\frac {2}{3}}-9 \RootOf \left (\textit {\_Z}^{2}+\textit {\_Z} +1\right ) x^{2}-15 \left (x^{2}+2 x +2\right )^{\frac {1}{3}} \RootOf \left (\textit {\_Z}^{2}+\textit {\_Z} +1\right )-18 \RootOf \left (\textit {\_Z}^{2}+\textit {\_Z} +1\right ) x +9 \left (x^{2}+2 x +2\right )^{\frac {2}{3}}+2 x^{2}-28 \RootOf \left (\textit {\_Z}^{2}+\textit {\_Z} +1\right )+9 \left (x^{2}+2 x +2\right )^{\frac {1}{3}}+4 x +7}{\left (1+x \right )^{2}}\right )}{2}-\frac {\ln \left (-\frac {4 \RootOf \left (\textit {\_Z}^{2}+\textit {\_Z} +1\right )^{2} x^{2}+8 \RootOf \left (\textit {\_Z}^{2}+\textit {\_Z} +1\right )^{2} x +15 \RootOf \left (\textit {\_Z}^{2}+\textit {\_Z} +1\right ) \left (x^{2}+2 x +2\right )^{\frac {2}{3}}+17 \RootOf \left (\textit {\_Z}^{2}+\textit {\_Z} +1\right ) x^{2}+15 \left (x^{2}+2 x +2\right )^{\frac {1}{3}} \RootOf \left (\textit {\_Z}^{2}+\textit {\_Z} +1\right )+34 \RootOf \left (\textit {\_Z}^{2}+\textit {\_Z} +1\right ) x +24 \left (x^{2}+2 x +2\right )^{\frac {2}{3}}+15 x^{2}+28 \RootOf \left (\textit {\_Z}^{2}+\textit {\_Z} +1\right )+24 \left (x^{2}+2 x +2\right )^{\frac {1}{3}}+30 x +35}{\left (1+x \right )^{2}}\right ) \RootOf \left (\textit {\_Z}^{2}+\textit {\_Z} +1\right )}{2}-\frac {\ln \left (-\frac {4 \RootOf \left (\textit {\_Z}^{2}+\textit {\_Z} +1\right )^{2} x^{2}+8 \RootOf \left (\textit {\_Z}^{2}+\textit {\_Z} +1\right )^{2} x +15 \RootOf \left (\textit {\_Z}^{2}+\textit {\_Z} +1\right ) \left (x^{2}+2 x +2\right )^{\frac {2}{3}}+17 \RootOf \left (\textit {\_Z}^{2}+\textit {\_Z} +1\right ) x^{2}+15 \left (x^{2}+2 x +2\right )^{\frac {1}{3}} \RootOf \left (\textit {\_Z}^{2}+\textit {\_Z} +1\right )+34 \RootOf \left (\textit {\_Z}^{2}+\textit {\_Z} +1\right ) x +24 \left (x^{2}+2 x +2\right )^{\frac {2}{3}}+15 x^{2}+28 \RootOf \left (\textit {\_Z}^{2}+\textit {\_Z} +1\right )+24 \left (x^{2}+2 x +2\right )^{\frac {1}{3}}+30 x +35}{\left (1+x \right )^{2}}\right )}{2}\) | \(433\) |
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 + 2\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+2\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 + 2}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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