\(\int \frac {1}{\sqrt [4]{2-3 x^2} (4-3 x^2)} \, dx\) [305]

Optimal result

Integrand size = 21, antiderivative size = 120 \[ \int \frac {1}{\sqrt [4]{2-3 x^2} \left (4-3 x^2\right )} \, dx=\frac {\arctan \left (\frac {2-\sqrt {2} \sqrt {2-3 x^2}}{\sqrt [4]{2} \sqrt {3} x \sqrt [4]{2-3 x^2}}\right )}{2\ 2^{3/4} \sqrt {3}}+\frac {\text {arctanh}\left (\frac {2+\sqrt {2} \sqrt {2-3 x^2}}{\sqrt [4]{2} \sqrt {3} x \sqrt [4]{2-3 x^2}}\right )}{2\ 2^{3/4} \sqrt {3}} \]

[Out]

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

Rubi [A] (verified)

Time = 0.01 (sec) , antiderivative size = 120, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.048, Rules used = {406} \[ \int \frac {1}{\sqrt [4]{2-3 x^2} \left (4-3 x^2\right )} \, dx=\frac {\arctan \left (\frac {2-\sqrt {2} \sqrt {2-3 x^2}}{\sqrt [4]{2} \sqrt {3} x \sqrt [4]{2-3 x^2}}\right )}{2\ 2^{3/4} \sqrt {3}}+\frac {\text {arctanh}\left (\frac {\sqrt {2} \sqrt {2-3 x^2}+2}{\sqrt [4]{2} \sqrt {3} x \sqrt [4]{2-3 x^2}}\right )}{2\ 2^{3/4} \sqrt {3}} \]

[In]

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

[Out]

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

Rule 406

Int[1/(((a_) + (b_.)*(x_)^2)^(1/4)*((c_) + (d_.)*(x_)^2)), x_Symbol] :> With[{q = Rt[b^2/a, 4]}, Simp[(-b/(2*a
*d*q))*ArcTan[(b + q^2*Sqrt[a + b*x^2])/(q^3*x*(a + b*x^2)^(1/4))], x] - Simp[(b/(2*a*d*q))*ArcTanh[(b - q^2*S
qrt[a + b*x^2])/(q^3*x*(a + b*x^2)^(1/4))], x]] /; FreeQ[{a, b, c, d}, x] && EqQ[b*c - 2*a*d, 0] && PosQ[b^2/a
]

Rubi steps \begin{align*} \text {integral}& = \frac {\tan ^{-1}\left (\frac {2-\sqrt {2} \sqrt {2-3 x^2}}{\sqrt [4]{2} \sqrt {3} x \sqrt [4]{2-3 x^2}}\right )}{2\ 2^{3/4} \sqrt {3}}+\frac {\tanh ^{-1}\left (\frac {2+\sqrt {2} \sqrt {2-3 x^2}}{\sqrt [4]{2} \sqrt {3} x \sqrt [4]{2-3 x^2}}\right )}{2\ 2^{3/4} \sqrt {3}} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.24 (sec) , antiderivative size = 119, normalized size of antiderivative = 0.99 \[ \int \frac {1}{\sqrt [4]{2-3 x^2} \left (4-3 x^2\right )} \, dx=\frac {\arctan \left (\frac {3 \sqrt {2} x^2-4 \sqrt {2-3 x^2}}{2\ 2^{3/4} \sqrt {3} x \sqrt [4]{2-3 x^2}}\right )+\text {arctanh}\left (\frac {2\ 2^{3/4} \sqrt {3} x \sqrt [4]{2-3 x^2}}{3 \sqrt {2} x^2+4 \sqrt {2-3 x^2}}\right )}{4\ 2^{3/4} \sqrt {3}} \]

[In]

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

[Out]

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

Maple [C] (warning: unable to verify)

Result contains higher order function than in optimal. Order 9 vs. order 3.

Time = 2.47 (sec) , antiderivative size = 188, normalized size of antiderivative = 1.57

method	result	size
trager	\(-\frac {\operatorname {RootOf}\left (\textit {\_Z}^{2}+\operatorname {RootOf}\left (\textit {\_Z}^{4}+72\right )^{2}\right ) \ln \left (-\frac {6 \left (-3 x^{2}+2\right )^{\frac {3}{4}} \operatorname {RootOf}\left (\textit {\_Z}^{2}+\operatorname {RootOf}\left (\textit {\_Z}^{4}+72\right )^{2}\right )+\left (-3 x^{2}+2\right )^{\frac {1}{4}} \operatorname {RootOf}\left (\textit {\_Z}^{2}+\operatorname {RootOf}\left (\textit {\_Z}^{4}+72\right )^{2}\right ) \operatorname {RootOf}\left (\textit {\_Z}^{4}+72\right )^{2}-18 \sqrt {-3 x^{2}+2}\, x -3 \operatorname {RootOf}\left (\textit {\_Z}^{4}+72\right )^{2} x}{3 x^{2}-4}\right )}{24}-\frac {\operatorname {RootOf}\left (\textit {\_Z}^{4}+72\right ) \ln \left (-\frac {6 \left (-3 x^{2}+2\right )^{\frac {3}{4}} \operatorname {RootOf}\left (\textit {\_Z}^{4}+72\right )-\left (-3 x^{2}+2\right )^{\frac {1}{4}} \operatorname {RootOf}\left (\textit {\_Z}^{4}+72\right )^{3}-18 \sqrt {-3 x^{2}+2}\, x +3 \operatorname {RootOf}\left (\textit {\_Z}^{4}+72\right )^{2} x}{3 x^{2}-4}\right )}{24}\)	\(188\)

[In]

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

[Out]

-1/24*RootOf(_Z^2+RootOf(_Z^4+72)^2)*ln(-(6*(-3*x^2+2)^(3/4)*RootOf(_Z^2+RootOf(_Z^4+72)^2)+(-3*x^2+2)^(1/4)*R
ootOf(_Z^2+RootOf(_Z^4+72)^2)*RootOf(_Z^4+72)^2-18*(-3*x^2+2)^(1/2)*x-3*RootOf(_Z^4+72)^2*x)/(3*x^2-4))-1/24*R
ootOf(_Z^4+72)*ln(-(6*(-3*x^2+2)^(3/4)*RootOf(_Z^4+72)-(-3*x^2+2)^(1/4)*RootOf(_Z^4+72)^3-18*(-3*x^2+2)^(1/2)*
x+3*RootOf(_Z^4+72)^2*x)/(3*x^2-4))

Fricas [C] (verification not implemented)

Result contains complex when optimal does not.

Time = 1.93 (sec) , antiderivative size = 289, normalized size of antiderivative = 2.41 \[ \int \frac {1}{\sqrt [4]{2-3 x^2} \left (4-3 x^2\right )} \, dx=-\left (\frac {1}{288} i + \frac {1}{288}\right ) \cdot 18^{\frac {3}{4}} \sqrt {2} \log \left (\frac {\left (i + 1\right ) \cdot 18^{\frac {3}{4}} \sqrt {2} x + \left (3 i - 3\right ) \cdot 18^{\frac {1}{4}} \sqrt {2} \sqrt {-3 \, x^{2} + 2} x - 12 i \, \sqrt {2} {\left (-3 \, x^{2} + 2\right )}^{\frac {1}{4}} + 12 \, {\left (-3 \, x^{2} + 2\right )}^{\frac {3}{4}}}{3 \, x^{2} - 4}\right ) + \left (\frac {1}{288} i - \frac {1}{288}\right ) \cdot 18^{\frac {3}{4}} \sqrt {2} \log \left (\frac {-\left (i - 1\right ) \cdot 18^{\frac {3}{4}} \sqrt {2} x - \left (3 i + 3\right ) \cdot 18^{\frac {1}{4}} \sqrt {2} \sqrt {-3 \, x^{2} + 2} x + 12 i \, \sqrt {2} {\left (-3 \, x^{2} + 2\right )}^{\frac {1}{4}} + 12 \, {\left (-3 \, x^{2} + 2\right )}^{\frac {3}{4}}}{3 \, x^{2} - 4}\right ) - \left (\frac {1}{288} i - \frac {1}{288}\right ) \cdot 18^{\frac {3}{4}} \sqrt {2} \log \left (\frac {\left (i - 1\right ) \cdot 18^{\frac {3}{4}} \sqrt {2} x + \left (3 i + 3\right ) \cdot 18^{\frac {1}{4}} \sqrt {2} \sqrt {-3 \, x^{2} + 2} x + 12 i \, \sqrt {2} {\left (-3 \, x^{2} + 2\right )}^{\frac {1}{4}} + 12 \, {\left (-3 \, x^{2} + 2\right )}^{\frac {3}{4}}}{3 \, x^{2} - 4}\right ) + \left (\frac {1}{288} i + \frac {1}{288}\right ) \cdot 18^{\frac {3}{4}} \sqrt {2} \log \left (\frac {-\left (i + 1\right ) \cdot 18^{\frac {3}{4}} \sqrt {2} x - \left (3 i - 3\right ) \cdot 18^{\frac {1}{4}} \sqrt {2} \sqrt {-3 \, x^{2} + 2} x - 12 i \, \sqrt {2} {\left (-3 \, x^{2} + 2\right )}^{\frac {1}{4}} + 12 \, {\left (-3 \, x^{2} + 2\right )}^{\frac {3}{4}}}{3 \, x^{2} - 4}\right ) \]

[In]

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

[Out]

-(1/288*I + 1/288)*18^(3/4)*sqrt(2)*log(((I + 1)*18^(3/4)*sqrt(2)*x + (3*I - 3)*18^(1/4)*sqrt(2)*sqrt(-3*x^2 +
2)*x - 12*I*sqrt(2)*(-3*x^2 + 2)^(1/4) + 12*(-3*x^2 + 2)^(3/4))/(3*x^2 - 4)) + (1/288*I - 1/288)*18^(3/4)*sqr
t(2)*log((-(I - 1)*18^(3/4)*sqrt(2)*x - (3*I + 3)*18^(1/4)*sqrt(2)*sqrt(-3*x^2 + 2)*x + 12*I*sqrt(2)*(-3*x^2 +
2)^(1/4) + 12*(-3*x^2 + 2)^(3/4))/(3*x^2 - 4)) - (1/288*I - 1/288)*18^(3/4)*sqrt(2)*log(((I - 1)*18^(3/4)*sqr
t(2)*x + (3*I + 3)*18^(1/4)*sqrt(2)*sqrt(-3*x^2 + 2)*x + 12*I*sqrt(2)*(-3*x^2 + 2)^(1/4) + 12*(-3*x^2 + 2)^(3/
4))/(3*x^2 - 4)) + (1/288*I + 1/288)*18^(3/4)*sqrt(2)*log((-(I + 1)*18^(3/4)*sqrt(2)*x - (3*I - 3)*18^(1/4)*sq
rt(2)*sqrt(-3*x^2 + 2)*x - 12*I*sqrt(2)*(-3*x^2 + 2)^(1/4) + 12*(-3*x^2 + 2)^(3/4))/(3*x^2 - 4))

Sympy [F]

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

[In]

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

[Out]

-Integral(1/(3*x**2*(2 - 3*x**2)**(1/4) - 4*(2 - 3*x**2)**(1/4)), x)

Maxima [F]

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

[In]

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

[Out]

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

Giac [F]

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

[In]

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

[Out]

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

Mupad [F(-1)]

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

[In]

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

[Out]

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