3.4.17 \(\int \frac {1}{(-2 a-3 x^2) \sqrt [4]{-a-3 x^2}} \, dx\) [317]

Optimal. Leaf size=85 \[ -\frac {\tan ^{-1}\left (\frac {\sqrt {\frac {3}{2}} x}{\sqrt [4]{a} \sqrt [4]{-a-3 x^2}}\right )}{2 \sqrt {6} a^{3/4}}-\frac {\tanh ^{-1}\left (\frac {\sqrt {\frac {3}{2}} x}{\sqrt [4]{a} \sqrt [4]{-a-3 x^2}}\right )}{2 \sqrt {6} a^{3/4}} \]

[Out]

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

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Rubi [A]
time = 0.01, antiderivative size = 85, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, integrand size = 25, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.040, Rules used = {407} \begin {gather*} -\frac {\text {ArcTan}\left (\frac {\sqrt {\frac {3}{2}} x}{\sqrt [4]{a} \sqrt [4]{-a-3 x^2}}\right )}{2 \sqrt {6} a^{3/4}}-\frac {\tanh ^{-1}\left (\frac {\sqrt {\frac {3}{2}} x}{\sqrt [4]{a} \sqrt [4]{-a-3 x^2}}\right )}{2 \sqrt {6} a^{3/4}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

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

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Mathematica [A]
time = 0.15, size = 77, normalized size = 0.91 \begin {gather*} -\frac {-\tan ^{-1}\left (\frac {\sqrt {\frac {2}{3}} \sqrt [4]{a} \sqrt [4]{-a-3 x^2}}{x}\right )+\tanh ^{-1}\left (\frac {\sqrt {\frac {2}{3}} \sqrt [4]{a} \sqrt [4]{-a-3 x^2}}{x}\right )}{2 \sqrt {6} a^{3/4}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [F]
time = 0.03, size = 0, normalized size = 0.00 \[\int \frac {1}{\left (-3 x^{2}-2 a \right ) \left (-3 x^{2}-a \right )^{\frac {1}{4}}}\, dx\]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

[Out]

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

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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 278 vs. \(2 (59) = 118\).
time = 6.42, size = 278, normalized size = 3.27 \begin {gather*} -\left (\frac {1}{36}\right )^{\frac {1}{4}} \frac {1}{a^{3}}^{\frac {1}{4}} \arctan \left (\frac {2 \, {\left (\sqrt {\frac {1}{2}} {\left (6 \, \left (\frac {1}{36}\right )^{\frac {3}{4}} a^{3} \frac {1}{a^{3}}^{\frac {3}{4}} - \left (\frac {1}{36}\right )^{\frac {1}{4}} \sqrt {-3 \, x^{2} - a} a \frac {1}{a^{3}}^{\frac {1}{4}}\right )} \sqrt {-a \sqrt {\frac {1}{a^{3}}}} - \left (\frac {1}{36}\right )^{\frac {1}{4}} {\left (-3 \, x^{2} - a\right )}^{\frac {1}{4}} a \frac {1}{a^{3}}^{\frac {1}{4}}\right )}}{x}\right ) + \frac {1}{4} \, \left (\frac {1}{36}\right )^{\frac {1}{4}} \frac {1}{a^{3}}^{\frac {1}{4}} \log \left (-\frac {18 \, \left (\frac {1}{36}\right )^{\frac {3}{4}} \sqrt {-3 \, x^{2} - a} a^{2} \frac {1}{a^{3}}^{\frac {3}{4}} x + {\left (-3 \, x^{2} - a\right )}^{\frac {1}{4}} a^{2} \sqrt {\frac {1}{a^{3}}} - 3 \, \left (\frac {1}{36}\right )^{\frac {1}{4}} a \frac {1}{a^{3}}^{\frac {1}{4}} x - {\left (-3 \, x^{2} - a\right )}^{\frac {3}{4}}}{3 \, x^{2} + 2 \, a}\right ) - \frac {1}{4} \, \left (\frac {1}{36}\right )^{\frac {1}{4}} \frac {1}{a^{3}}^{\frac {1}{4}} \log \left (\frac {18 \, \left (\frac {1}{36}\right )^{\frac {3}{4}} \sqrt {-3 \, x^{2} - a} a^{2} \frac {1}{a^{3}}^{\frac {3}{4}} x - {\left (-3 \, x^{2} - a\right )}^{\frac {1}{4}} a^{2} \sqrt {\frac {1}{a^{3}}} - 3 \, \left (\frac {1}{36}\right )^{\frac {1}{4}} a \frac {1}{a^{3}}^{\frac {1}{4}} x + {\left (-3 \, x^{2} - a\right )}^{\frac {3}{4}}}{3 \, x^{2} + 2 \, a}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-(1/36)^(1/4)*(a^(-3))^(1/4)*arctan(2*(sqrt(1/2)*(6*(1/36)^(3/4)*a^3*(a^(-3))^(3/4) - (1/36)^(1/4)*sqrt(-3*x^2
- a)*a*(a^(-3))^(1/4))*sqrt(-a*sqrt(a^(-3))) - (1/36)^(1/4)*(-3*x^2 - a)^(1/4)*a*(a^(-3))^(1/4))/x) + 1/4*(1/
36)^(1/4)*(a^(-3))^(1/4)*log(-(18*(1/36)^(3/4)*sqrt(-3*x^2 - a)*a^2*(a^(-3))^(3/4)*x + (-3*x^2 - a)^(1/4)*a^2*
sqrt(a^(-3)) - 3*(1/36)^(1/4)*a*(a^(-3))^(1/4)*x - (-3*x^2 - a)^(3/4))/(3*x^2 + 2*a)) - 1/4*(1/36)^(1/4)*(a^(-
3))^(1/4)*log((18*(1/36)^(3/4)*sqrt(-3*x^2 - a)*a^2*(a^(-3))^(3/4)*x - (-3*x^2 - a)^(1/4)*a^2*sqrt(a^(-3)) - 3
*(1/36)^(1/4)*a*(a^(-3))^(1/4)*x + (-3*x^2 - a)^(3/4))/(3*x^2 + 2*a))

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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} - \int \frac {1}{2 a \sqrt [4]{- a - 3 x^{2}} + 3 x^{2} \sqrt [4]{- a - 3 x^{2}}}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

[Out]

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

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Mupad [F]
time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} -\int \frac {1}{\left (3\,x^2+2\,a\right )\,{\left (-3\,x^2-a\right )}^{1/4}} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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