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

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 = 0.0129699, 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, Rules used = {751} \[ \frac{\log \left (-\sqrt [3]{2} \sqrt{3} \sqrt [3]{x^2+1}-x+\sqrt{3}\right )}{2\ 2^{2/3}}-\frac{\tan ^{-1}\left (\frac{2^{2/3} \left (\sqrt{3}-x\right )}{3 \sqrt [3]{x^2+1}}+\frac{1}{\sqrt{3}}\right )}{2^{2/3} \sqrt{3}}-\frac{\log \left (x+\sqrt{3}\right )}{2\ 2^{2/3}} \]

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))

Rule 751

Int[1/(((d_) + (e_.)*(x_))*((a_) + (c_.)*(x_)^2)^(1/3)), x_Symbol] :> With[{q = Rt[(6*c^2*e^2)/d^2, 3]}, -Simp
[(Sqrt[3]*c*e*ArcTan[1/Sqrt[3] + (2*c*(d - e*x))/(Sqrt[3]*d*q*(a + c*x^2)^(1/3))])/(d^2*q^2), x] + (-Simp[(3*c
*e*Log[d + e*x])/(2*d^2*q^2), x] + Simp[(3*c*e*Log[c*d - c*e*x - d*q*(a + c*x^2)^(1/3)])/(2*d^2*q^2), x])] /;
FreeQ[{a, c, d, e}, x] && EqQ[c*d^2 - 3*a*e^2, 0]

Rubi steps

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

Mathematica [C]  time = 0.0800111, size = 102, normalized size = 0.98 \[ -\frac{3 \sqrt [3]{\frac{x-i}{x+\sqrt{3}}} \sqrt [3]{\frac{x+i}{x+\sqrt{3}}} F_1\left (\frac{2}{3};\frac{1}{3},\frac{1}{3};\frac{5}{3};\frac{-i+\sqrt{3}}{x+\sqrt{3}},\frac{i+\sqrt{3}}{x+\sqrt{3}}\right )}{2 \sqrt [3]{x^2+1}} \]

Warning: Unable to verify antiderivative.

[In]

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

[Out]

(-3*((-I + x)/(Sqrt[3] + x))^(1/3)*((I + x)/(Sqrt[3] + x))^(1/3)*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))

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Maple [F]  time = 0.377, size = 0, normalized size = 0. \begin{align*} \int{\frac{1}{x+\sqrt{3}}{\frac{1}{\sqrt [3]{{x}^{2}+1}}}}\, dx \end{align*}

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. \begin{align*} \int \frac{1}{{\left (x^{2} + 1\right )}^{\frac{1}{3}}{\left (x + \sqrt{3}\right )}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [B]  time = 43.6501, size = 799, normalized size = 7.68 \begin{align*} -\frac{1}{6} \cdot 4^{\frac{1}{6}} \sqrt{3} \arctan \left (\frac{4^{\frac{1}{6}} \sqrt{3}{\left (6 \cdot 4^{\frac{2}{3}}{\left (x^{4} + 8 \, \sqrt{3} x^{3} - 18 \, x^{2} - 27\right )}{\left (x^{2} + 1\right )}^{\frac{2}{3}} + 4^{\frac{1}{3}}{\left (x^{6} + 99 \, x^{4} + 243 \, x^{2} + 12 \, \sqrt{3}{\left (x^{5} + 10 \, x^{3} + 9 \, x\right )} + 81\right )} + 4 \,{\left (21 \, x^{4} + 54 \, x^{2} + \sqrt{3}{\left (x^{5} - 42 \, x^{3} - 27 \, x\right )} + 81\right )}{\left (x^{2} + 1\right )}^{\frac{1}{3}}\right )}}{6 \,{\left (x^{6} - 225 \, x^{4} - 405 \, x^{2} - 243\right )}}\right ) - \frac{1}{24} \cdot 4^{\frac{2}{3}} \log \left (\frac{3 \cdot 4^{\frac{2}{3}}{\left (x^{2} - 2 \, \sqrt{3} x + 3\right )}{\left (x^{2} + 1\right )}^{\frac{2}{3}} + 4^{\frac{1}{3}}{\left (x^{4} + 18 \, x^{2} - 4 \, \sqrt{3}{\left (x^{3} + 3 \, x\right )} + 9\right )} + 2 \,{\left (9 \, x^{2} - \sqrt{3}{\left (x^{3} + 9 \, x\right )} + 9\right )}{\left (x^{2} + 1\right )}^{\frac{1}{3}}}{x^{4} - 6 \, x^{2} + 9}\right ) + \frac{1}{12} \cdot 4^{\frac{2}{3}} \log \left (\frac{4^{\frac{1}{3}}{\left (x^{2} - 2 \, \sqrt{3} x + 3\right )} + 2 \,{\left (x^{2} + 1\right )}^{\frac{1}{3}}{\left (\sqrt{3} x - 3\right )}}{x^{2} - 3}\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/6*4^(1/6)*sqrt(3)*arctan(1/6*4^(1/6)*sqrt(3)*(6*4^(2/3)*(x^4 + 8*sqrt(3)*x^3 - 18*x^2 - 27)*(x^2 + 1)^(2/3)
+ 4^(1/3)*(x^6 + 99*x^4 + 243*x^2 + 12*sqrt(3)*(x^5 + 10*x^3 + 9*x) + 81) + 4*(21*x^4 + 54*x^2 + sqrt(3)*(x^5
- 42*x^3 - 27*x) + 81)*(x^2 + 1)^(1/3))/(x^6 - 225*x^4 - 405*x^2 - 243)) - 1/24*4^(2/3)*log((3*4^(2/3)*(x^2 -
2*sqrt(3)*x + 3)*(x^2 + 1)^(2/3) + 4^(1/3)*(x^4 + 18*x^2 - 4*sqrt(3)*(x^3 + 3*x) + 9) + 2*(9*x^2 - sqrt(3)*(x
^3 + 9*x) + 9)*(x^2 + 1)^(1/3))/(x^4 - 6*x^2 + 9)) + 1/12*4^(2/3)*log((4^(1/3)*(x^2 - 2*sqrt(3)*x + 3) + 2*(x^
2 + 1)^(1/3)*(sqrt(3)*x - 3))/(x^2 - 3))

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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