3.313 \(\int \frac{1}{(-2-3 x^2) \sqrt [4]{-1-3 x^2}} \, dx\)

Optimal. Leaf size=61 \[ -\frac{\tan ^{-1}\left (\frac{\sqrt{\frac{3}{2}} x}{\sqrt [4]{-3 x^2-1}}\right )}{2 \sqrt{6}}-\frac{\tanh ^{-1}\left (\frac{\sqrt{\frac{3}{2}} x}{\sqrt [4]{-3 x^2-1}}\right )}{2 \sqrt{6}} \]

[Out]

-ArcTan[(Sqrt[3/2]*x)/(-1 - 3*x^2)^(1/4)]/(2*Sqrt[6]) - ArcTanh[(Sqrt[3/2]*x)/(-1 - 3*x^2)^(1/4)]/(2*Sqrt[6])

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Rubi [A]  time = 0.0098623, antiderivative size = 61, 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 = {398} \[ -\frac{\tan ^{-1}\left (\frac{\sqrt{\frac{3}{2}} x}{\sqrt [4]{-3 x^2-1}}\right )}{2 \sqrt{6}}-\frac{\tanh ^{-1}\left (\frac{\sqrt{\frac{3}{2}} x}{\sqrt [4]{-3 x^2-1}}\right )}{2 \sqrt{6}} \]

Antiderivative was successfully verified.

[In]

Int[1/((-2 - 3*x^2)*(-1 - 3*x^2)^(1/4)),x]

[Out]

-ArcTan[(Sqrt[3/2]*x)/(-1 - 3*x^2)^(1/4)]/(2*Sqrt[6]) - ArcTanh[(Sqrt[3/2]*x)/(-1 - 3*x^2)^(1/4)]/(2*Sqrt[6])

Rule 398

Int[1/(((a_) + (b_.)*(x_)^2)^(1/4)*((c_) + (d_.)*(x_)^2)), x_Symbol] :> With[{q = Rt[-(b^2/a), 4]}, Simp[(b*Ar
cTan[(q*x)/(Sqrt[2]*(a + b*x^2)^(1/4))])/(2*Sqrt[2]*a*d*q), x] + Simp[(b*ArcTanh[(q*x)/(Sqrt[2]*(a + b*x^2)^(1
/4))])/(2*Sqrt[2]*a*d*q), x]] /; FreeQ[{a, b, c, d}, x] && EqQ[b*c - 2*a*d, 0] && NegQ[b^2/a]

Rubi steps

\begin{align*} \int \frac{1}{\left (-2-3 x^2\right ) \sqrt [4]{-1-3 x^2}} \, dx &=-\frac{\tan ^{-1}\left (\frac{\sqrt{\frac{3}{2}} x}{\sqrt [4]{-1-3 x^2}}\right )}{2 \sqrt{6}}-\frac{\tanh ^{-1}\left (\frac{\sqrt{\frac{3}{2}} x}{\sqrt [4]{-1-3 x^2}}\right )}{2 \sqrt{6}}\\ \end{align*}

Mathematica [C]  time = 0.124174, size = 127, normalized size = 2.08 \[ \frac{2 x F_1\left (\frac{1}{2};\frac{1}{4},1;\frac{3}{2};-3 x^2,-\frac{3 x^2}{2}\right )}{\sqrt [4]{-3 x^2-1} \left (3 x^2+2\right ) \left (x^2 \left (2 F_1\left (\frac{3}{2};\frac{1}{4},2;\frac{5}{2};-3 x^2,-\frac{3 x^2}{2}\right )+F_1\left (\frac{3}{2};\frac{5}{4},1;\frac{5}{2};-3 x^2,-\frac{3 x^2}{2}\right )\right )-2 F_1\left (\frac{1}{2};\frac{1}{4},1;\frac{3}{2};-3 x^2,-\frac{3 x^2}{2}\right )\right )} \]

Warning: Unable to verify antiderivative.

[In]

Integrate[1/((-2 - 3*x^2)*(-1 - 3*x^2)^(1/4)),x]

[Out]

(2*x*AppellF1[1/2, 1/4, 1, 3/2, -3*x^2, (-3*x^2)/2])/((-1 - 3*x^2)^(1/4)*(2 + 3*x^2)*(-2*AppellF1[1/2, 1/4, 1,
3/2, -3*x^2, (-3*x^2)/2] + x^2*(2*AppellF1[3/2, 1/4, 2, 5/2, -3*x^2, (-3*x^2)/2] + AppellF1[3/2, 5/4, 1, 5/2,
-3*x^2, (-3*x^2)/2])))

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Maple [F]  time = 0.027, size = 0, normalized size = 0. \begin{align*} \int{\frac{1}{-3\,{x}^{2}-2}{\frac{1}{\sqrt [4]{-3\,{x}^{2}-1}}}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(-3*x^2-2)/(-3*x^2-1)^(1/4),x)

[Out]

int(1/(-3*x^2-2)/(-3*x^2-1)^(1/4),x)

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} -\int \frac{1}{{\left (3 \, x^{2} + 2\right )}{\left (-3 \, x^{2} - 1\right )}^{\frac{1}{4}}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(-3*x^2-2)/(-3*x^2-1)^(1/4),x, algorithm="maxima")

[Out]

-integrate(1/((3*x^2 + 2)*(-3*x^2 - 1)^(1/4)), x)

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Fricas [C]  time = 25.7431, size = 683, normalized size = 11.2 \begin{align*} -\frac{1}{24} \, \sqrt{6} \log \left (\frac{\sqrt{6} \sqrt{-3 \, x^{2} - 1} x - \sqrt{6} x + 2 \,{\left (-3 \, x^{2} - 1\right )}^{\frac{3}{4}} - 2 \,{\left (-3 \, x^{2} - 1\right )}^{\frac{1}{4}}}{3 \,{\left (3 \, x^{2} + 2\right )}}\right ) + \frac{1}{24} \, \sqrt{6} \log \left (-\frac{\sqrt{6} \sqrt{-3 \, x^{2} - 1} x - \sqrt{6} x - 2 \,{\left (-3 \, x^{2} - 1\right )}^{\frac{3}{4}} + 2 \,{\left (-3 \, x^{2} - 1\right )}^{\frac{1}{4}}}{3 \,{\left (3 \, x^{2} + 2\right )}}\right ) + \frac{1}{24} i \, \sqrt{6} \log \left (\frac{i \, \sqrt{6} \sqrt{-3 \, x^{2} - 1} x + i \, \sqrt{6} x + 2 \,{\left (-3 \, x^{2} - 1\right )}^{\frac{3}{4}} + 2 \,{\left (-3 \, x^{2} - 1\right )}^{\frac{1}{4}}}{3 \,{\left (3 \, x^{2} + 2\right )}}\right ) - \frac{1}{24} i \, \sqrt{6} \log \left (\frac{-i \, \sqrt{6} \sqrt{-3 \, x^{2} - 1} x - i \, \sqrt{6} x + 2 \,{\left (-3 \, x^{2} - 1\right )}^{\frac{3}{4}} + 2 \,{\left (-3 \, x^{2} - 1\right )}^{\frac{1}{4}}}{3 \,{\left (3 \, x^{2} + 2\right )}}\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(-3*x^2-2)/(-3*x^2-1)^(1/4),x, algorithm="fricas")

[Out]

-1/24*sqrt(6)*log(1/3*(sqrt(6)*sqrt(-3*x^2 - 1)*x - sqrt(6)*x + 2*(-3*x^2 - 1)^(3/4) - 2*(-3*x^2 - 1)^(1/4))/(
3*x^2 + 2)) + 1/24*sqrt(6)*log(-1/3*(sqrt(6)*sqrt(-3*x^2 - 1)*x - sqrt(6)*x - 2*(-3*x^2 - 1)^(3/4) + 2*(-3*x^2
- 1)^(1/4))/(3*x^2 + 2)) + 1/24*I*sqrt(6)*log(1/3*(I*sqrt(6)*sqrt(-3*x^2 - 1)*x + I*sqrt(6)*x + 2*(-3*x^2 - 1
)^(3/4) + 2*(-3*x^2 - 1)^(1/4))/(3*x^2 + 2)) - 1/24*I*sqrt(6)*log(1/3*(-I*sqrt(6)*sqrt(-3*x^2 - 1)*x - I*sqrt(
6)*x + 2*(-3*x^2 - 1)^(3/4) + 2*(-3*x^2 - 1)^(1/4))/(3*x^2 + 2))

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} - \int \frac{1}{3 x^{2} \sqrt [4]{- 3 x^{2} - 1} + 2 \sqrt [4]{- 3 x^{2} - 1}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(-3*x**2-2)/(-3*x**2-1)**(1/4),x)

[Out]

-Integral(1/(3*x**2*(-3*x**2 - 1)**(1/4) + 2*(-3*x**2 - 1)**(1/4)), x)

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int -\frac{1}{{\left (3 \, x^{2} + 2\right )}{\left (-3 \, x^{2} - 1\right )}^{\frac{1}{4}}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(-3*x^2-2)/(-3*x^2-1)^(1/4),x, algorithm="giac")

[Out]

integrate(-1/((3*x^2 + 2)*(-3*x^2 - 1)^(1/4)), x)