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

Optimal result

Integrand size = 25, antiderivative size = 85 \[ \int \frac {1}{\left (-2 a-3 x^2\right ) \sqrt [4]{-a-3 x^2}} \, dx=-\frac {\arctan \left (\frac {\sqrt {\frac {3}{2}} x}{\sqrt [4]{a} \sqrt [4]{-a-3 x^2}}\right )}{2 \sqrt {6} a^{3/4}}-\frac {\text {arctanh}\left (\frac {\sqrt {\frac {3}{2}} x}{\sqrt [4]{a} \sqrt [4]{-a-3 x^2}}\right )}{2 \sqrt {6} a^{3/4}} \]

[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)

Rubi [A] (verified)

Time = 0.01 (sec) , antiderivative size = 85, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.040, Rules used = {407} \[ \int \frac {1}{\left (-2 a-3 x^2\right ) \sqrt [4]{-a-3 x^2}} \, dx=-\frac {\arctan \left (\frac {\sqrt {\frac {3}{2}} x}{\sqrt [4]{a} \sqrt [4]{-a-3 x^2}}\right )}{2 \sqrt {6} a^{3/4}}-\frac {\text {arctanh}\left (\frac {\sqrt {\frac {3}{2}} x}{\sqrt [4]{a} \sqrt [4]{-a-3 x^2}}\right )}{2 \sqrt {6} a^{3/4}} \]

[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*} \text {integral}& = -\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*}

Mathematica [A] (verified)

Time = 0.21 (sec) , antiderivative size = 77, normalized size of antiderivative = 0.91 \[ \int \frac {1}{\left (-2 a-3 x^2\right ) \sqrt [4]{-a-3 x^2}} \, dx=-\frac {-\arctan \left (\frac {\sqrt {\frac {2}{3}} \sqrt [4]{a} \sqrt [4]{-a-3 x^2}}{x}\right )+\text {arctanh}\left (\frac {\sqrt {\frac {2}{3}} \sqrt [4]{a} \sqrt [4]{-a-3 x^2}}{x}\right )}{2 \sqrt {6} a^{3/4}} \]

[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))

Maple [F]

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

[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)

Fricas [C] (verification not implemented)

Result contains complex when optimal does not.

Time = 4.89 (sec) , antiderivative size = 373, normalized size of antiderivative = 4.39 \[ \int \frac {1}{\left (-2 a-3 x^2\right ) \sqrt [4]{-a-3 x^2}} \, dx=\frac {1}{4} \, \left (\frac {1}{36}\right )^{\frac {1}{4}} \frac {1}{a^{3}}^{\frac {1}{4}} \log \left (-\frac {18 \, \left (\frac {1}{36}\right )^{\frac {3}{4}} \sqrt {-3 \, x^{2} - a} a^{2} \frac {1}{a^{3}}^{\frac {3}{4}} x + {\left (-3 \, x^{2} - a\right )}^{\frac {1}{4}} a^{2} \sqrt {\frac {1}{a^{3}}} - 3 \, \left (\frac {1}{36}\right )^{\frac {1}{4}} a \frac {1}{a^{3}}^{\frac {1}{4}} x - {\left (-3 \, x^{2} - a\right )}^{\frac {3}{4}}}{3 \, x^{2} + 2 \, a}\right ) - \frac {1}{4} \, \left (\frac {1}{36}\right )^{\frac {1}{4}} \frac {1}{a^{3}}^{\frac {1}{4}} \log \left (\frac {18 \, \left (\frac {1}{36}\right )^{\frac {3}{4}} \sqrt {-3 \, x^{2} - a} a^{2} \frac {1}{a^{3}}^{\frac {3}{4}} x - {\left (-3 \, x^{2} - a\right )}^{\frac {1}{4}} a^{2} \sqrt {\frac {1}{a^{3}}} - 3 \, \left (\frac {1}{36}\right )^{\frac {1}{4}} a \frac {1}{a^{3}}^{\frac {1}{4}} x + {\left (-3 \, x^{2} - a\right )}^{\frac {3}{4}}}{3 \, x^{2} + 2 \, a}\right ) + \frac {1}{4} i \, \left (\frac {1}{36}\right )^{\frac {1}{4}} \frac {1}{a^{3}}^{\frac {1}{4}} \log \left (\frac {18 i \, \left (\frac {1}{36}\right )^{\frac {3}{4}} \sqrt {-3 \, x^{2} - a} a^{2} \frac {1}{a^{3}}^{\frac {3}{4}} x + {\left (-3 \, x^{2} - a\right )}^{\frac {1}{4}} a^{2} \sqrt {\frac {1}{a^{3}}} + 3 i \, \left (\frac {1}{36}\right )^{\frac {1}{4}} a \frac {1}{a^{3}}^{\frac {1}{4}} x + {\left (-3 \, x^{2} - a\right )}^{\frac {3}{4}}}{3 \, x^{2} + 2 \, a}\right ) - \frac {1}{4} i \, \left (\frac {1}{36}\right )^{\frac {1}{4}} \frac {1}{a^{3}}^{\frac {1}{4}} \log \left (\frac {-18 i \, \left (\frac {1}{36}\right )^{\frac {3}{4}} \sqrt {-3 \, x^{2} - a} a^{2} \frac {1}{a^{3}}^{\frac {3}{4}} x + {\left (-3 \, x^{2} - a\right )}^{\frac {1}{4}} a^{2} \sqrt {\frac {1}{a^{3}}} - 3 i \, \left (\frac {1}{36}\right )^{\frac {1}{4}} a \frac {1}{a^{3}}^{\frac {1}{4}} x + {\left (-3 \, x^{2} - a\right )}^{\frac {3}{4}}}{3 \, x^{2} + 2 \, a}\right ) \]

[In]

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

[Out]

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)) + 1/4*I*(1/36)^(1/4)*(a^(-3))^(1/
4)*log((18*I*(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*I*(1
/36)^(1/4)*a*(a^(-3))^(1/4)*x + (-3*x^2 - a)^(3/4))/(3*x^2 + 2*a)) - 1/4*I*(1/36)^(1/4)*(a^(-3))^(1/4)*log((-1
8*I*(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*I*(1/36)^(1/4
)*a*(a^(-3))^(1/4)*x + (-3*x^2 - a)^(3/4))/(3*x^2 + 2*a))

Sympy [F]

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

[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)

Maxima [F]

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

[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)

Giac [F]

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

[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)

Mupad [F(-1)]

Timed out. \[ \int \frac {1}{\left (-2 a-3 x^2\right ) \sqrt [4]{-a-3 x^2}} \, dx=-\int \frac {1}{\left (3\,x^2+2\,a\right )\,{\left (-3\,x^2-a\right )}^{1/4}} \,d x \]

[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)