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

Optimal. Leaf size=101 \[ -\frac{\log \left (-\sqrt [3]{2} \sqrt{3} \sqrt [3]{x^2+1}+x+\sqrt{3}\right )}{2\ 2^{2/3}}+\frac{\tan ^{-1}\left (\frac{2^{2/3} \left (x+\sqrt{3}\right )}{3 \sqrt [3]{x^2+1}}+\frac{1}{\sqrt{3}}\right )}{2^{2/3} \sqrt{3}}+\frac{\log \left (\sqrt{3}-x\right )}{2\ 2^{2/3}} \]

[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 = 0.0144118, antiderivative size = 101, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.048, Rules used = {751} \[ -\frac{\log \left (-\sqrt [3]{2} \sqrt{3} \sqrt [3]{x^2+1}+x+\sqrt{3}\right )}{2\ 2^{2/3}}+\frac{\tan ^{-1}\left (\frac{2^{2/3} \left (x+\sqrt{3}\right )}{3 \sqrt [3]{x^2+1}}+\frac{1}{\sqrt{3}}\right )}{2^{2/3} \sqrt{3}}+\frac{\log \left (\sqrt{3}-x\right )}{2\ 2^{2/3}} \]

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 = 0.0828518, size = 110, normalized size = 1.09 \[ \frac{3 \sqrt [3]{\frac{x-i}{x-\sqrt{3}}} \sqrt [3]{\frac{x+i}{x-\sqrt{3}}} F_1\left (\frac{2}{3};\frac{1}{3},\frac{1}{3};\frac{5}{3};\frac{-i+\sqrt{3}}{\sqrt{3}-x},\frac{i+\sqrt{3}}{\sqrt{3}-x}\right )}{2 \sqrt [3]{x^2+1}} \]

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 = 0.413, 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 = 40.936, size = 1006, normalized size = 9.96 \begin{align*} \frac{1}{6} \cdot 4^{\frac{1}{6}} \sqrt{3} \left (-1\right )^{\frac{1}{3}} \arctan \left (\frac{4^{\frac{1}{6}} \sqrt{3}{\left (6 \cdot 4^{\frac{2}{3}}{\left (8 \, \sqrt{3} \left (-1\right )^{\frac{2}{3}} x^{3} - \left (-1\right )^{\frac{2}{3}}{\left (x^{4} - 18 \, x^{2} - 27\right )}\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 (x^{2} + 1\right )}^{\frac{1}{3}}{\left (\sqrt{3} \left (-1\right )^{\frac{1}{3}}{\left (x^{5} - 42 \, x^{3} - 27 \, x\right )} - 3 \, \left (-1\right )^{\frac{1}{3}}{\left (7 \, x^{4} + 18 \, x^{2} + 27\right )}\right )}\right )}}{6 \,{\left (x^{6} - 225 \, x^{4} - 405 \, x^{2} - 243\right )}}\right ) - \frac{1}{24} \cdot 4^{\frac{2}{3}} \left (-1\right )^{\frac{1}{3}} \log \left (-\frac{3 \cdot 4^{\frac{2}{3}}{\left (x^{2} + 1\right )}^{\frac{2}{3}}{\left (2 \, \sqrt{3} \left (-1\right )^{\frac{1}{3}} x + \left (-1\right )^{\frac{1}{3}}{\left (x^{2} + 3\right )}\right )} - 4^{\frac{1}{3}}{\left (4 \, \sqrt{3} \left (-1\right )^{\frac{2}{3}}{\left (x^{3} + 3 \, x\right )} + \left (-1\right )^{\frac{2}{3}}{\left (x^{4} + 18 \, x^{2} + 9\right )}\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}} \left (-1\right )^{\frac{1}{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} \left (-1\right )^{\frac{1}{3}} x + 3 \, \left (-1\right )^{\frac{1}{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)*(-1)^(1/3)*arctan(1/6*4^(1/6)*sqrt(3)*(6*4^(2/3)*(8*sqrt(3)*(-1)^(2/3)*x^3 - (-1)^(2/3)*(x
^4 - 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*(x^2 + 1)^(1/3)*(sqrt(3)*(-1)^(1/3)*(x^5 - 42*x^3 - 27*x) - 3*(-1)^(1/3)*(7*x^4 + 18*x^2 + 27)))/(x^6 - 225
*x^4 - 405*x^2 - 243)) - 1/24*4^(2/3)*(-1)^(1/3)*log(-(3*4^(2/3)*(x^2 + 1)^(2/3)*(2*sqrt(3)*(-1)^(1/3)*x + (-1
)^(1/3)*(x^2 + 3)) - 4^(1/3)*(4*sqrt(3)*(-1)^(2/3)*(x^3 + 3*x) + (-1)^(2/3)*(x^4 + 18*x^2 + 9)) - 2*(9*x^2 + s
qrt(3)*(x^3 + 9*x) + 9)*(x^2 + 1)^(1/3))/(x^4 - 6*x^2 + 9)) + 1/12*4^(2/3)*(-1)^(1/3)*log(-(4^(1/3)*(x^2 + 2*s
qrt(3)*x + 3) + 2*(x^2 + 1)^(1/3)*(sqrt(3)*(-1)^(1/3)*x + 3*(-1)^(1/3)))/(x^2 - 3))

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