3.1050 \(\int \frac{1}{(-2+3 x^2) \sqrt [4]{-1+3 x^2}} \, dx\)

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 = 0.0089125, 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 = 0.0260957, size = 127, normalized size = 2.08 \[ \frac{2 x F_1\left (\frac{1}{2};\frac{1}{4},1;\frac{3}{2};3 x^2,\frac{3 x^2}{2}\right )}{\left (3 x^2-2\right ) \sqrt [4]{3 x^2-1} \left (x^2 \left (2 F_1\left (\frac{3}{2};\frac{1}{4},2;\frac{5}{2};3 x^2,\frac{3 x^2}{2}\right )+F_1\left (\frac{3}{2};\frac{5}{4},1;\frac{5}{2};3 x^2,\frac{3 x^2}{2}\right )\right )+2 F_1\left (\frac{1}{2};\frac{1}{4},1;\frac{3}{2};3 x^2,\frac{3 x^2}{2}\right )\right )} \]

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])/((-2 + 3*x^2)*(-1 + 3*x^2)^(1/4)*(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., 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} - 1\right )}^{\frac{1}{4}}{\left (3 \, x^{2} - 2\right )}}\,{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 - 1)^(1/4)*(3*x^2 - 2)), x)

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Fricas [B]  time = 24.0225, size = 281, normalized size = 4.61 \begin{align*} \frac{1}{12} \, \sqrt{6} \arctan \left (\frac{\sqrt{6}{\left (3 \, x^{2} - 1\right )}^{\frac{1}{4}}}{3 \, x}\right ) + \frac{1}{24} \, \sqrt{6} \log \left (-\frac{9 \, x^{4} - 6 \, \sqrt{6}{\left (3 \, x^{2} - 1\right )}^{\frac{1}{4}} x^{3} + 12 \, \sqrt{3 \, x^{2} - 1} x^{2} - 4 \, \sqrt{6}{\left (3 \, x^{2} - 1\right )}^{\frac{3}{4}} x + 12 \, x^{2} - 4}{9 \, x^{4} - 12 \, x^{2} + 4}\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]

1/12*sqrt(6)*arctan(1/3*sqrt(6)*(3*x^2 - 1)^(1/4)/x) + 1/24*sqrt(6)*log(-(9*x^4 - 6*sqrt(6)*(3*x^2 - 1)^(1/4)*
x^3 + 12*sqrt(3*x^2 - 1)*x^2 - 4*sqrt(6)*(3*x^2 - 1)^(3/4)*x + 12*x^2 - 4)/(9*x^4 - 12*x^2 + 4))

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