3.93 \(\int \frac{1}{(1+x) \sqrt [3]{2+x^3}} \, dx\)

Optimal. Leaf size=108 \[ \frac{3}{4} \log \left (-\sqrt [3]{x^3+2}+x+2\right )-\frac{1}{4} \log \left (\sqrt [3]{x^3+2}-x\right )+\frac{\tan ^{-1}\left (\frac{\frac{2 x}{\sqrt [3]{x^3+2}}+1}{\sqrt{3}}\right )}{2 \sqrt{3}}-\frac{1}{2} \sqrt{3} \tan ^{-1}\left (\frac{\frac{2 (x+2)}{\sqrt [3]{x^3+2}}+1}{\sqrt{3}}\right )-\frac{1}{2} \log (x+1) \]

[Out]

ArcTan[(1 + (2*x)/(2 + x^3)^(1/3))/Sqrt[3]]/(2*Sqrt[3]) - (Sqrt[3]*ArcTan[(1 + (2*(2 + x))/(2 + x^3)^(1/3))/Sq
rt[3]])/2 - Log[1 + x]/2 + (3*Log[2 + x - (2 + x^3)^(1/3)])/4 - Log[-x + (2 + x^3)^(1/3)]/4

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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.

[In]

Int[1/((1 + x)*(2 + x^3)^(1/3)),x]

[Out]

ArcTan[(1 + (2*x)/(2 + x^3)^(1/3))/Sqrt[3]]/(2*Sqrt[3]) - (Sqrt[3]*ArcTan[(1 + (2*(2 + x))/(2 + x^3)^(1/3))/Sq
rt[3]])/2 - Log[1 + x]/2 + (3*Log[2 + x - (2 + x^3)^(1/3)])/4 - Log[-x + (2 + x^3)^(1/3)]/4

Rule 2149

Int[1/(((c_) + (d_.)*(x_))*((a_) + (b_.)*(x_)^3)^(1/3)), x_Symbol] :> Dist[1/(2*c), Int[1/(a + b*x^3)^(1/3), x
], x] + Dist[1/(2*c), Int[(c - d*x)/((c + d*x)*(a + b*x^3)^(1/3)), x], x] /; FreeQ[{a, b, c, d}, x] && EqQ[2*b
*c^3 - a*d^3, 0]

Rule 239

Int[((a_) + (b_.)*(x_)^3)^(-1/3), x_Symbol] :> Simp[ArcTan[(1 + (2*Rt[b, 3]*x)/(a + b*x^3)^(1/3))/Sqrt[3]]/(Sq
rt[3]*Rt[b, 3]), x] - Simp[Log[(a + b*x^3)^(1/3) - Rt[b, 3]*x]/(2*Rt[b, 3]), x] /; FreeQ[{a, b}, x]

Rule 2151

Int[((e_) + (f_.)*(x_))/(((c_) + (d_.)*(x_))*((a_) + (b_.)*(x_)^3)^(1/3)), x_Symbol] :> Simp[(Sqrt[3]*f*ArcTan
[(1 + (2*Rt[b, 3]*(2*c + d*x))/(d*(a + b*x^3)^(1/3)))/Sqrt[3]])/(Rt[b, 3]*d), x] + (Simp[(f*Log[c + d*x])/(Rt[
b, 3]*d), x] - Simp[(3*f*Log[Rt[b, 3]*(2*c + d*x) - d*(a + b*x^3)^(1/3)])/(2*Rt[b, 3]*d), x]) /; FreeQ[{a, b,
c, d, e, f}, x] && EqQ[d*e + c*f, 0] && EqQ[2*b*c^3 - a*d^3, 0]

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.

[In]

Integrate[1/((1 + x)*(2 + x^3)^(1/3)),x]

[Out]

Integrate[1/((1 + x)*(2 + x^3)^(1/3)), x]

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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.

[In]

int(1/(1+x)/(x^3+2)^(1/3),x)

[Out]

int(1/(1+x)/(x^3+2)^(1/3),x)

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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.

[In]

integrate(1/(1+x)/(x^3+2)^(1/3),x, algorithm="maxima")

[Out]

integrate(1/((x^3 + 2)^(1/3)*(x + 1)), x)

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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.

[In]

integrate(1/(1+x)/(x^3+2)^(1/3),x, algorithm="fricas")

[Out]

1/6*sqrt(3)*arctan(1/3*(13910019318573948542*sqrt(3)*(7114781247*x^4 + 13663058416*x^3 - 46178206896*x^2 - 126
842559344*x - 77084338088)*(x^3 + 2)^(2/3) - 27820038637147897084*sqrt(3)*(1625757424*x^5 + 16302821713*x^4 +
26102613730*x^3 - 26431113242*x^2 - 80188343316*x - 42779182428)*(x^3 + 2)^(1/3) + sqrt(3)*(932925708335594356
63132301885*x^6 + 382151535711085278859235047618*x^5 + 673924074224408772959625384792*x^4 + 889426563183087468
015580290048*x^3 + 888876515195959220955879945824*x^2 + 351260598258508240019971964880*x - 4767400099559721105
7816884304))/(78905434814564721745708464883*x^6 + 337746705836458222863347934450*x^5 + 15598952776058587894336
070976*x^4 - 895430525315100108684787964824*x^3 + 361667862240477028869533375352*x^2 + 25418023010116325106459
72090336*x + 1554815286823334092314485968880)) + 1/12*log((22*x^6 + 6*x^5 - 48*x^4 + 44*x^3 + 24*x^2 + 3*(7*x^
4 - 2*x^3 - 32*x^2 - 20*x + 4)*(x^3 + 2)^(2/3) + 3*(7*x^5 - 16*x^3 + 34*x^2 + 76*x + 32)*(x^3 + 2)^(1/3) - 192
*x - 140)/(x^6 + 6*x^5 + 15*x^4 + 20*x^3 + 15*x^2 + 6*x + 1))

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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.

[In]

integrate(1/(1+x)/(x**3+2)**(1/3),x)

[Out]

Integral(1/((x + 1)*(x**3 + 2)**(1/3)), x)

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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.

[In]

integrate(1/(1+x)/(x^3+2)^(1/3),x, algorithm="giac")

[Out]

integrate(1/((x^3 + 2)^(1/3)*(x + 1)), x)