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

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

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Rubi [A]  time = 0.01, antiderivative size = 61, normalized size of antiderivative = 1.00, 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} \begin {gather*} -\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}} \end {gather*}

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*}

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Mathematica [C]  time = 0.15, size = 127, normalized size = 2.08 \begin {gather*} \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 )} \end {gather*}

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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IntegrateAlgebraic [A]  time = 0.12, size = 63, normalized size = 1.03 \begin {gather*} \frac {\tan ^{-1}\left (\frac {\sqrt {\frac {2}{3}} \sqrt [4]{3 x^2-1}}{x}\right )}{2 \sqrt {6}}-\frac {\tanh ^{-1}\left (\frac {\sqrt {\frac {3}{2}} x}{\sqrt [4]{3 x^2-1}}\right )}{2 \sqrt {6}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

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fricas [B]  time = 9.42, size = 104, normalized size = 1.70 \begin {gather*} \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 {gather*}

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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giac [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {1}{{\left (3 \, x^{2} - 1\right )}^{\frac {1}{4}} {\left (3 \, x^{2} - 2\right )}}\,{d x} \end {gather*}

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)

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maple [C]  time = 1.14, size = 138, normalized size = 2.26 \begin {gather*} \frac {\RootOf \left (\textit {\_Z}^{2}-6\right ) \ln \left (-\frac {-3 \sqrt {3 x^{2}-1}\, x -3 x +\left (3 x^{2}-1\right )^{\frac {3}{4}} \RootOf \left (\textit {\_Z}^{2}-6\right )+\left (3 x^{2}-1\right )^{\frac {1}{4}} \RootOf \left (\textit {\_Z}^{2}-6\right )}{3 x^{2}-2}\right )}{12}-\frac {\RootOf \left (\textit {\_Z}^{2}+6\right ) \ln \left (\frac {3 \sqrt {3 x^{2}-1}\, x -3 x +\left (3 x^{2}-1\right )^{\frac {3}{4}} \RootOf \left (\textit {\_Z}^{2}+6\right )-\left (3 x^{2}-1\right )^{\frac {1}{4}} \RootOf \left (\textit {\_Z}^{2}+6\right )}{3 x^{2}-2}\right )}{12} \end {gather*}

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]

-1/12*RootOf(_Z^2+6)*ln((RootOf(_Z^2+6)*(3*x^2-1)^(3/4)+3*(3*x^2-1)^(1/2)*x-RootOf(_Z^2+6)*(3*x^2-1)^(1/4)-3*x
)/(3*x^2-2))+1/12*RootOf(_Z^2-6)*ln(-(RootOf(_Z^2-6)*(3*x^2-1)^(3/4)-3*(3*x^2-1)^(1/2)*x+RootOf(_Z^2-6)*(3*x^2
-1)^(1/4)-3*x)/(3*x^2-2))

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maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {1}{{\left (3 \, x^{2} - 1\right )}^{\frac {1}{4}} {\left (3 \, x^{2} - 2\right )}}\,{d x} \end {gather*}

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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mupad [F]  time = 0.00, size = -1, normalized size = -0.02 \begin {gather*} \int \frac {1}{{\left (3\,x^2-1\right )}^{1/4}\,\left (3\,x^2-2\right )} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {1}{\left (3 x^{2} - 2\right ) \sqrt [4]{3 x^{2} - 1}}\, dx \end {gather*}

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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