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

Optimal. Leaf size=104 \[ \frac{\log \left (-\sqrt [3]{2} \sqrt{3} \sqrt [3]{x^2+1}-x+\sqrt{3}\right )}{2\ 2^{2/3}}-\frac{\tan ^{-1}\left (\frac{2^{2/3} \left (\sqrt{3}-x\right )}{3 \sqrt [3]{x^2+1}}+\frac{1}{\sqrt{3}}\right )}{2^{2/3} \sqrt{3}}-\frac{\log \left (x+\sqrt{3}\right )}{2\ 2^{2/3}} \]

[Out]

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

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Rubi [A]  time = 0.0602989, antiderivative size = 104, 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 \[ \frac{\log \left (-\sqrt [3]{2} \sqrt{3} \sqrt [3]{x^2+1}-x+\sqrt{3}\right )}{2\ 2^{2/3}}-\frac{\tan ^{-1}\left (\frac{2^{2/3} \left (\sqrt{3}-x\right )}{3 \sqrt [3]{x^2+1}}+\frac{1}{\sqrt{3}}\right )}{2^{2/3} \sqrt{3}}-\frac{\log \left (x+\sqrt{3}\right )}{2\ 2^{2/3}} \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [A]  time = 3.77697, size = 90, normalized size = 0.87 \[ - \frac{\sqrt [3]{2} \log{\left (x + \sqrt{3} \right )}}{4} + \frac{\sqrt [3]{2} \log{\left (- x - \sqrt [3]{2} \sqrt{3} \sqrt [3]{x^{2} + 1} + \sqrt{3} \right )}}{4} - \frac{\sqrt [3]{2} \sqrt{3} \operatorname{atan}{\left (\frac{2^{\frac{2}{3}} \left (- x + \sqrt{3}\right )}{3 \sqrt [3]{x^{2} + 1}} + \frac{\sqrt{3}}{3} \right )}}{6} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-2**(1/3)*log(x + sqrt(3))/4 + 2**(1/3)*log(-x - 2**(1/3)*sqrt(3)*(x**2 + 1)**(1
/3) + sqrt(3))/4 - 2**(1/3)*sqrt(3)*atan(2**(2/3)*(-x + sqrt(3))/(3*(x**2 + 1)**
(1/3)) + sqrt(3)/3)/6

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Mathematica [C]  time = 0.843254, size = 256, normalized size = 2.46 \[ -\frac{15 \left (x+\sqrt{3}\right ) F_1\left (\frac{2}{3};\frac{1}{3},\frac{1}{3};\frac{5}{3};\frac{-i+\sqrt{3}}{x+\sqrt{3}},\frac{i+\sqrt{3}}{x+\sqrt{3}}\right )}{2 \sqrt [3]{x^2+1} \left (5 \left (x+\sqrt{3}\right ) F_1\left (\frac{2}{3};\frac{1}{3},\frac{1}{3};\frac{5}{3};\frac{-i+\sqrt{3}}{x+\sqrt{3}},\frac{i+\sqrt{3}}{x+\sqrt{3}}\right )+\left (\sqrt{3}+i\right ) F_1\left (\frac{5}{3};\frac{1}{3},\frac{4}{3};\frac{8}{3};\frac{-i+\sqrt{3}}{x+\sqrt{3}},\frac{i+\sqrt{3}}{x+\sqrt{3}}\right )+\left (\sqrt{3}-i\right ) F_1\left (\frac{5}{3};\frac{4}{3},\frac{1}{3};\frac{8}{3};\frac{-i+\sqrt{3}}{x+\sqrt{3}},\frac{i+\sqrt{3}}{x+\sqrt{3}}\right )\right )} \]

Warning: Unable to verify antiderivative.

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

[Out]

(-15*(Sqrt[3] + x)*AppellF1[2/3, 1/3, 1/3, 5/3, (-I + Sqrt[3])/(Sqrt[3] + x), (I
+ Sqrt[3])/(Sqrt[3] + x)])/(2*(1 + x^2)^(1/3)*(5*(Sqrt[3] + x)*AppellF1[2/3, 1/
3, 1/3, 5/3, (-I + Sqrt[3])/(Sqrt[3] + x), (I + Sqrt[3])/(Sqrt[3] + x)] + (I + S
qrt[3])*AppellF1[5/3, 1/3, 4/3, 8/3, (-I + Sqrt[3])/(Sqrt[3] + x), (I + Sqrt[3])
/(Sqrt[3] + x)] + (-I + Sqrt[3])*AppellF1[5/3, 4/3, 1/3, 8/3, (-I + Sqrt[3])/(Sq
rt[3] + x), (I + Sqrt[3])/(Sqrt[3] + x)]))

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Maple [F]  time = 0.059, size = 0, normalized size = 0. \[ \int{\frac{1}{x+\sqrt{3}}{\frac{1}{\sqrt [3]{{x}^{2}+1}}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(1/(x^2+1)^(1/3)/(x+3^(1/2)),x)

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

[Out]

Timed out

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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