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]

-1/12*arctan(1/2*x*6^(1/2)/(-3*x^2-1)^(1/4))*6^(1/2)-1/12*arctanh(1/2*x*6^(1/2)/(-3*x^2-1)^(1/4))*6^(1/2)

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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} \[ -\frac {\tan ^{-1}\left (\frac {\sqrt {\frac {3}{2}} x}{\sqrt [4]{-3 x^2-1}}\right )}{2 \sqrt {6}}-\frac {\tanh ^{-1}\left (\frac {\sqrt {\frac {3}{2}} x}{\sqrt [4]{-3 x^2-1}}\right )}{2 \sqrt {6}} \]

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*}

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Mathematica [C]  time = 0.13, 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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fricas [C]  time = 4.43, size = 243, normalized size = 3.98 \[ -\frac {1}{24} \, \sqrt {6} \log \left (\frac {\sqrt {6} \sqrt {-3 \, x^{2} - 1} x - \sqrt {6} x + 2 \, {\left (-3 \, x^{2} - 1\right )}^{\frac {3}{4}} - 2 \, {\left (-3 \, x^{2} - 1\right )}^{\frac {1}{4}}}{3 \, {\left (3 \, x^{2} + 2\right )}}\right ) + \frac {1}{24} \, \sqrt {6} \log \left (-\frac {\sqrt {6} \sqrt {-3 \, x^{2} - 1} x - \sqrt {6} x - 2 \, {\left (-3 \, x^{2} - 1\right )}^{\frac {3}{4}} + 2 \, {\left (-3 \, x^{2} - 1\right )}^{\frac {1}{4}}}{3 \, {\left (3 \, x^{2} + 2\right )}}\right ) + \frac {1}{24} i \, \sqrt {6} \log \left (\frac {i \, \sqrt {6} \sqrt {-3 \, x^{2} - 1} x + i \, \sqrt {6} x + 2 \, {\left (-3 \, x^{2} - 1\right )}^{\frac {3}{4}} + 2 \, {\left (-3 \, x^{2} - 1\right )}^{\frac {1}{4}}}{3 \, {\left (3 \, x^{2} + 2\right )}}\right ) - \frac {1}{24} i \, \sqrt {6} \log \left (\frac {-i \, \sqrt {6} \sqrt {-3 \, x^{2} - 1} x - i \, \sqrt {6} x + 2 \, {\left (-3 \, x^{2} - 1\right )}^{\frac {3}{4}} + 2 \, {\left (-3 \, x^{2} - 1\right )}^{\frac {1}{4}}}{3 \, {\left (3 \, x^{2} + 2\right )}}\right ) \]

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

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)

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maple [C]  time = 1.11, size = 138, normalized size = 2.26 \[ -\frac {\RootOf \left (\textit {\_Z}^{2}-6\right ) \ln \left (\frac {3 \sqrt {-3 x^{2}-1}\, x -3 x +\left (-3 x^{2}-1\right )^{\frac {3}{4}} \RootOf \left (\textit {\_Z}^{2}-6\right )-\left (-3 x^{2}-1\right )^{\frac {1}{4}} \RootOf \left (\textit {\_Z}^{2}-6\right )}{3 x^{2}+2}\right )}{12}+\frac {\RootOf \left (\textit {\_Z}^{2}+6\right ) \ln \left (-\frac {-3 \sqrt {-3 x^{2}-1}\, x -3 x +\left (-3 x^{2}-1\right )^{\frac {3}{4}} \RootOf \left (\textit {\_Z}^{2}+6\right )+\left (-3 x^{2}-1\right )^{\frac {1}{4}} \RootOf \left (\textit {\_Z}^{2}+6\right )}{3 x^{2}+2}\right )}{12} \]

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 \[ -\int \frac {1}{{\left (3 \, x^{2} + 2\right )} {\left (-3 \, x^{2} - 1\right )}^{\frac {1}{4}}}\,{d x} \]

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

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

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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