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