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

Optimal. Leaf size=81 \[ \frac{\tan ^{-1}\left (\frac{\left (1-\sqrt [3]{3 x^2+1}\right )^2}{3 \sqrt{3} x}\right )}{4 \sqrt{3}}-\frac{1}{4} \tanh ^{-1}\left (\frac{1-\sqrt [3]{3 x^2+1}}{x}\right )+\frac{\tan ^{-1}\left (\frac{x}{\sqrt{3}}\right )}{4 \sqrt{3}} \]

[Out]

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

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Rubi [A]  time = 0.0107141, antiderivative size = 81, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.053, Rules used = {394} \[ \frac{\tan ^{-1}\left (\frac{\left (1-\sqrt [3]{3 x^2+1}\right )^2}{3 \sqrt{3} x}\right )}{4 \sqrt{3}}-\frac{1}{4} \tanh ^{-1}\left (\frac{1-\sqrt [3]{3 x^2+1}}{x}\right )+\frac{\tan ^{-1}\left (\frac{x}{\sqrt{3}}\right )}{4 \sqrt{3}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

Rule 394

Int[1/(((a_) + (b_.)*(x_)^2)^(1/3)*((c_) + (d_.)*(x_)^2)), x_Symbol] :> With[{q = Rt[b/a, 2]}, Simp[(q*ArcTan[
(q*x)/3])/(12*Rt[a, 3]*d), x] + (Simp[(q*ArcTan[(Rt[a, 3] - (a + b*x^2)^(1/3))^2/(3*Rt[a, 3]^2*q*x)])/(12*Rt[a
, 3]*d), x] - Simp[(q*ArcTanh[(Sqrt[3]*(Rt[a, 3] - (a + b*x^2)^(1/3)))/(Rt[a, 3]*q*x)])/(4*Sqrt[3]*Rt[a, 3]*d)
, x])] /; FreeQ[{a, b, c, d}, x] && NeQ[b*c - a*d, 0] && EqQ[b*c - 9*a*d, 0] && PosQ[b/a]

Rubi steps

\begin{align*} \int \frac{1}{\left (3+x^2\right ) \sqrt [3]{1+3 x^2}} \, dx &=\frac{\tan ^{-1}\left (\frac{x}{\sqrt{3}}\right )}{4 \sqrt{3}}+\frac{\tan ^{-1}\left (\frac{\left (1-\sqrt [3]{1+3 x^2}\right )^2}{3 \sqrt{3} x}\right )}{4 \sqrt{3}}-\frac{1}{4} \tanh ^{-1}\left (\frac{1-\sqrt [3]{1+3 x^2}}{x}\right )\\ \end{align*}

Mathematica [C]  time = 0.0950302, size = 126, normalized size = 1.56 \[ -\frac{9 x F_1\left (\frac{1}{2};\frac{1}{3},1;\frac{3}{2};-3 x^2,-\frac{x^2}{3}\right )}{\left (x^2+3\right ) \sqrt [3]{3 x^2+1} \left (2 x^2 \left (F_1\left (\frac{3}{2};\frac{1}{3},2;\frac{5}{2};-3 x^2,-\frac{x^2}{3}\right )+3 F_1\left (\frac{3}{2};\frac{4}{3},1;\frac{5}{2};-3 x^2,-\frac{x^2}{3}\right )\right )-9 F_1\left (\frac{1}{2};\frac{1}{3},1;\frac{3}{2};-3 x^2,-\frac{x^2}{3}\right )\right )} \]

Warning: Unable to verify antiderivative.

[In]

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

[Out]

(-9*x*AppellF1[1/2, 1/3, 1, 3/2, -3*x^2, -x^2/3])/((3 + x^2)*(1 + 3*x^2)^(1/3)*(-9*AppellF1[1/2, 1/3, 1, 3/2,
-3*x^2, -x^2/3] + 2*x^2*(AppellF1[3/2, 1/3, 2, 5/2, -3*x^2, -x^2/3] + 3*AppellF1[3/2, 4/3, 1, 5/2, -3*x^2, -x^
2/3])))

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Maple [F]  time = 0.026, size = 0, normalized size = 0. \begin{align*} \int{\frac{1}{{x}^{2}+3}{\frac{1}{\sqrt [3]{3\,{x}^{2}+1}}}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{{\left (3 \, x^{2} + 1\right )}^{\frac{1}{3}}{\left (x^{2} + 3\right )}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [B]  time = 15.5078, size = 921, normalized size = 11.37 \begin{align*} \frac{1}{36} \, \sqrt{3} \arctan \left (\frac{4 \, \sqrt{3}{\left (3 \, x^{4} - 10 \, x^{3} - 36 \, x^{2} + 18 \, x + 9\right )}{\left (3 \, x^{2} + 1\right )}^{\frac{2}{3}} - 4 \, \sqrt{3}{\left (x^{5} + 15 \, x^{4} - 26 \, x^{3} - 54 \, x^{2} + 9 \, x - 9\right )}{\left (3 \, x^{2} + 1\right )}^{\frac{1}{3}} + \sqrt{3}{\left (x^{6} - 2 \, x^{5} - 105 \, x^{4} - 28 \, x^{3} + 63 \, x^{2} + 126 \, x + 9\right )}}{x^{6} + 126 \, x^{5} - 225 \, x^{4} - 828 \, x^{3} - 81 \, x^{2} - 162 \, x + 81}\right ) - \frac{1}{36} \, \sqrt{3} \arctan \left (\frac{2 \,{\left (2 \, \sqrt{3}{\left (23 \, x^{3} + 9 \, x\right )}{\left (3 \, x^{2} + 1\right )}^{\frac{2}{3}} + \sqrt{3}{\left (x^{5} - 80 \, x^{3} - 9 \, x\right )}{\left (3 \, x^{2} + 1\right )}^{\frac{1}{3}} + \sqrt{3}{\left (11 \, x^{5} + 10 \, x^{3} - 9 \, x\right )}\right )}}{x^{6} - 657 \, x^{4} - 189 \, x^{2} - 27}\right ) + \frac{1}{24} \, \log \left (\frac{x^{6} + 108 \, x^{5} + 549 \, x^{4} + 360 \, x^{3} + 99 \, x^{2} + 6 \,{\left (3 \, x^{4} + 32 \, x^{3} + 42 \, x^{2} + 3\right )}{\left (3 \, x^{2} + 1\right )}^{\frac{2}{3}} + 6 \,{\left (x^{5} + 27 \, x^{4} + 70 \, x^{3} + 18 \, x^{2} + 9 \, x + 3\right )}{\left (3 \, x^{2} + 1\right )}^{\frac{1}{3}} + 108 \, x - 9}{x^{6} + 9 \, x^{4} + 27 \, x^{2} + 27}\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/36*sqrt(3)*arctan((4*sqrt(3)*(3*x^4 - 10*x^3 - 36*x^2 + 18*x + 9)*(3*x^2 + 1)^(2/3) - 4*sqrt(3)*(x^5 + 15*x^
4 - 26*x^3 - 54*x^2 + 9*x - 9)*(3*x^2 + 1)^(1/3) + sqrt(3)*(x^6 - 2*x^5 - 105*x^4 - 28*x^3 + 63*x^2 + 126*x +
9))/(x^6 + 126*x^5 - 225*x^4 - 828*x^3 - 81*x^2 - 162*x + 81)) - 1/36*sqrt(3)*arctan(2*(2*sqrt(3)*(23*x^3 + 9*
x)*(3*x^2 + 1)^(2/3) + sqrt(3)*(x^5 - 80*x^3 - 9*x)*(3*x^2 + 1)^(1/3) + sqrt(3)*(11*x^5 + 10*x^3 - 9*x))/(x^6
- 657*x^4 - 189*x^2 - 27)) + 1/24*log((x^6 + 108*x^5 + 549*x^4 + 360*x^3 + 99*x^2 + 6*(3*x^4 + 32*x^3 + 42*x^2
+ 3)*(3*x^2 + 1)^(2/3) + 6*(x^5 + 27*x^4 + 70*x^3 + 18*x^2 + 9*x + 3)*(3*x^2 + 1)^(1/3) + 108*x - 9)/(x^6 + 9
*x^4 + 27*x^2 + 27))

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{\left (x^{2} + 3\right ) \sqrt [3]{3 x^{2} + 1}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{{\left (3 \, x^{2} + 1\right )}^{\frac{1}{3}}{\left (x^{2} + 3\right )}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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