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

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 = 0.0369219, 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 \[ \frac{\tan ^{-1}\left (\frac{\left (1-\sqrt [3]{3 x^2+1}\right )^2}{3 \sqrt{3} x}\right )}{4 \sqrt{3}}-\frac{1}{4} \tanh ^{-1}\left (\frac{1-\sqrt [3]{3 x^2+1}}{x}\right )+\frac{\tan ^{-1}\left (\frac{x}{\sqrt{3}}\right )}{4 \sqrt{3}} \]

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

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Rubi in Sympy [A]  time = 3.07163, size = 20, normalized size = 0.25 \[ \frac{x \operatorname{appellf_{1}}{\left (\frac{1}{2},\frac{1}{3},1,\frac{3}{2},- 3 x^{2},- \frac{x^{2}}{3} \right )}}{3} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(1/(x**2+3)/(3*x**2+1)**(1/3),x)

[Out]

x*appellf1(1/2, 1/3, 1, 3/2, -3*x**2, -x**2/3)/3

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Mathematica [C]  time = 0.140462, 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 = 0.04, size = 0, normalized size = 0. \[ \int{\frac{1}{{x}^{2}+3}{\frac{1}{\sqrt [3]{3\,{x}^{2}+1}}}}\, dx \]

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. \[ \int \frac{1}{{\left (3 \, x^{2} + 1\right )}^{\frac{1}{3}}{\left (x^{2} + 3\right )}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Fricas [F(-1)]  time = 0., size = 0, normalized size = 0. \[ \text{Timed out} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Timed out

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

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/XCAS [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{{\left (3 \, x^{2} + 1\right )}^{\frac{1}{3}}{\left (x^{2} + 3\right )}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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