Optimal. Leaf size=108 \[ \frac{3}{4} \log \left (-\sqrt [3]{x^3+2}+x+2\right )-\frac{1}{4} \log \left (\sqrt [3]{x^3+2}-x\right )+\frac{\tan ^{-1}\left (\frac{\frac{2 x}{\sqrt [3]{x^3+2}}+1}{\sqrt{3}}\right )}{2 \sqrt{3}}-\frac{1}{2} \sqrt{3} \tan ^{-1}\left (\frac{\frac{2 (x+2)}{\sqrt [3]{x^3+2}}+1}{\sqrt{3}}\right )-\frac{1}{2} \log (x+1) \]
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Rubi [A] time = 0.0925279, antiderivative size = 108, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.2, Rules used = {2149, 239, 2151} \[ \frac{3}{4} \log \left (-\sqrt [3]{x^3+2}+x+2\right )-\frac{1}{4} \log \left (\sqrt [3]{x^3+2}-x\right )+\frac{\tan ^{-1}\left (\frac{\frac{2 x}{\sqrt [3]{x^3+2}}+1}{\sqrt{3}}\right )}{2 \sqrt{3}}-\frac{1}{2} \sqrt{3} \tan ^{-1}\left (\frac{\frac{2 (x+2)}{\sqrt [3]{x^3+2}}+1}{\sqrt{3}}\right )-\frac{1}{2} \log (x+1) \]
Antiderivative was successfully verified.
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Rule 2149
Rule 239
Rule 2151
Rubi steps
\begin{align*} \int \frac{1}{(1+x) \sqrt [3]{2+x^3}} \, dx &=\frac{1}{2} \int \frac{1}{\sqrt [3]{2+x^3}} \, dx+\frac{1}{2} \int \frac{1-x}{(1+x) \sqrt [3]{2+x^3}} \, dx\\ &=\frac{\tan ^{-1}\left (\frac{1+\frac{2 x}{\sqrt [3]{2+x^3}}}{\sqrt{3}}\right )}{2 \sqrt{3}}-\frac{1}{2} \sqrt{3} \tan ^{-1}\left (\frac{1+\frac{2 (2+x)}{\sqrt [3]{2+x^3}}}{\sqrt{3}}\right )-\frac{1}{2} \log (1+x)+\frac{3}{4} \log \left (2+x-\sqrt [3]{2+x^3}\right )-\frac{1}{4} \log \left (-x+\sqrt [3]{2+x^3}\right )\\ \end{align*}
Mathematica [F] time = 0.0572087, size = 0, normalized size = 0. \[ \int \frac{1}{(1+x) \sqrt [3]{2+x^3}} \, dx \]
Verification is Not applicable to the result.
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Maple [F] time = 0.023, size = 0, normalized size = 0. \begin{align*} \int{\frac{1}{1+x}{\frac{1}{\sqrt [3]{{x}^{3}+2}}}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{{\left (x^{3} + 2\right )}^{\frac{1}{3}}{\left (x + 1\right )}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] time = 9.28147, size = 1422, normalized size = 13.17 \begin{align*} \frac{1}{6} \, \sqrt{3} \arctan \left (\frac{13910019318573948542 \, \sqrt{3}{\left (7114781247 \, x^{4} + 13663058416 \, x^{3} - 46178206896 \, x^{2} - 126842559344 \, x - 77084338088\right )}{\left (x^{3} + 2\right )}^{\frac{2}{3}} - 27820038637147897084 \, \sqrt{3}{\left (1625757424 \, x^{5} + 16302821713 \, x^{4} + 26102613730 \, x^{3} - 26431113242 \, x^{2} - 80188343316 \, x - 42779182428\right )}{\left (x^{3} + 2\right )}^{\frac{1}{3}} + \sqrt{3}{\left (93292570833559435663132301885 \, x^{6} + 382151535711085278859235047618 \, x^{5} + 673924074224408772959625384792 \, x^{4} + 889426563183087468015580290048 \, x^{3} + 888876515195959220955879945824 \, x^{2} + 351260598258508240019971964880 \, x - 47674000995597211057816884304\right )}}{3 \,{\left (78905434814564721745708464883 \, x^{6} + 337746705836458222863347934450 \, x^{5} + 15598952776058587894336070976 \, x^{4} - 895430525315100108684787964824 \, x^{3} + 361667862240477028869533375352 \, x^{2} + 2541802301011632510645972090336 \, x + 1554815286823334092314485968880\right )}}\right ) + \frac{1}{12} \, \log \left (\frac{22 \, x^{6} + 6 \, x^{5} - 48 \, x^{4} + 44 \, x^{3} + 24 \, x^{2} + 3 \,{\left (7 \, x^{4} - 2 \, x^{3} - 32 \, x^{2} - 20 \, x + 4\right )}{\left (x^{3} + 2\right )}^{\frac{2}{3}} + 3 \,{\left (7 \, x^{5} - 16 \, x^{3} + 34 \, x^{2} + 76 \, x + 32\right )}{\left (x^{3} + 2\right )}^{\frac{1}{3}} - 192 \, x - 140}{x^{6} + 6 \, x^{5} + 15 \, x^{4} + 20 \, x^{3} + 15 \, x^{2} + 6 \, x + 1}\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{\left (x + 1\right ) \sqrt [3]{x^{3} + 2}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{{\left (x^{3} + 2\right )}^{\frac{1}{3}}{\left (x + 1\right )}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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