3.4.13 \(\int \frac {1}{(-2-3 x^2) \sqrt [4]{-1-3 x^2}} \, dx\) [313]

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

[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 = {407} \begin {gather*} -\frac {\text {ArcTan}\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]

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

Rule 407

Int[1/(((a_) + (b_.)*(x_)^2)^(1/4)*((c_) + (d_.)*(x_)^2)), x_Symbol] :> With[{q = Rt[-b^2/a, 4]}, Simp[(b/(2*S
qrt[2]*a*d*q))*ArcTan[q*(x/(Sqrt[2]*(a + b*x^2)^(1/4)))], x] + Simp[(b/(2*Sqrt[2]*a*d*q))*ArcTanh[q*(x/(Sqrt[2
]*(a + b*x^2)^(1/4)))], 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 [A]
time = 0.10, size = 56, normalized size = 0.92 \begin {gather*} -\frac {-\tan ^{-1}\left (\frac {\sqrt {\frac {2}{3}} \sqrt [4]{-1-3 x^2}}{x}\right )+\tanh ^{-1}\left (\frac {\sqrt {\frac {3}{2}} x}{\sqrt [4]{-1-3 x^2}}\right )}{2 \sqrt {6}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [C] Result contains higher order function than in optimal. Order 9 vs. order 3.
time = 0.96, size = 137, normalized size = 2.25

method	result	size
trager	\(-\frac {\RootOf \left (\textit {\_Z}^{2}-6\right ) \ln \left (\frac {\RootOf \left (\textit {\_Z}^{2}-6\right ) \left (-3 x^{2}-1\right )^{\frac {3}{4}}+3 \sqrt {-3 x^{2}-1}\, x -\RootOf \left (\textit {\_Z}^{2}-6\right ) \left (-3 x^{2}-1\right )^{\frac {1}{4}}-3 x}{3 x^{2}+2}\right )}{12}-\frac {\RootOf \left (\textit {\_Z}^{2}+6\right ) \ln \left (\frac {\RootOf \left (\textit {\_Z}^{2}+6\right ) \left (-3 x^{2}-1\right )^{\frac {3}{4}}+3 \sqrt {-3 x^{2}-1}\, x +\RootOf \left (\textit {\_Z}^{2}+6\right ) \left (-3 x^{2}-1\right )^{\frac {1}{4}}+3 x}{3 x^{2}+2}\right )}{12}\)	\(137\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[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*} \text {Failed to integrate} \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 + 2)*(-3*x^2 - 1)^(1/4)), x)

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Fricas [C] Result contains complex when optimal does not.
time = 3.49, size = 243, normalized size = 3.98 \begin {gather*} -\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 {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]

-1/24*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 = 0.00, size = 0, normalized size = 0.00 \begin {gather*} - \int \frac {1}{3 x^{2} \sqrt [4]{- 3 x^{2} - 1} + 2 \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*(-3*x**2 - 1)**(1/4) + 2*(-3*x**2 - 1)**(1/4)), x)

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Giac [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \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 + 2)*(-3*x^2 - 1)^(1/4)), 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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