\(\int \frac {x}{\sqrt [4]{2-3 x^2} (4-3 x^2)} \, dx\) [1034]

Optimal result

Integrand size = 22, antiderivative size = 91 \[ \int \frac {x}{\sqrt [4]{2-3 x^2} \left (4-3 x^2\right )} \, dx=\frac {\arctan \left (\frac {\sqrt {2}-\sqrt {2-3 x^2}}{2^{3/4} \sqrt [4]{2-3 x^2}}\right )}{3\ 2^{3/4}}+\frac {\text {arctanh}\left (\frac {\sqrt {2}+\sqrt {2-3 x^2}}{2^{3/4} \sqrt [4]{2-3 x^2}}\right )}{3\ 2^{3/4}} \]

[Out]

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

Rubi [A] (verified)

Time = 0.01 (sec) , antiderivative size = 91, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.045, Rules used = {450} \[ \int \frac {x}{\sqrt [4]{2-3 x^2} \left (4-3 x^2\right )} \, dx=\frac {\arctan \left (\frac {\sqrt {2}-\sqrt {2-3 x^2}}{2^{3/4} \sqrt [4]{2-3 x^2}}\right )}{3\ 2^{3/4}}+\frac {\text {arctanh}\left (\frac {\sqrt {2-3 x^2}+\sqrt {2}}{2^{3/4} \sqrt [4]{2-3 x^2}}\right )}{3\ 2^{3/4}} \]

[In]

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

[Out]

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

Rule 450

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

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

Mathematica [A] (verified)

Time = 0.09 (sec) , antiderivative size = 76, normalized size of antiderivative = 0.84 \[ \int \frac {x}{\sqrt [4]{2-3 x^2} \left (4-3 x^2\right )} \, dx=\frac {\arctan \left (\frac {\sqrt {2}-\sqrt {2-3 x^2}}{2^{3/4} \sqrt [4]{2-3 x^2}}\right )+\text {arctanh}\left (\frac {2 \sqrt [4]{4-6 x^2}}{2+\sqrt {4-6 x^2}}\right )}{3\ 2^{3/4}} \]

[In]

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

[Out]

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

Maple [A] (verified)

Time = 3.22 (sec) , antiderivative size = 100, normalized size of antiderivative = 1.10

method	result	size
pseudoelliptic	\(-\frac {2^{\frac {1}{4}} \left (\ln \left (\frac {-2^{\frac {3}{4}} \left (-3 x^{2}+2\right )^{\frac {1}{4}}+\sqrt {2}+\sqrt {-3 x^{2}+2}}{2^{\frac {3}{4}} \left (-3 x^{2}+2\right )^{\frac {1}{4}}+\sqrt {2}+\sqrt {-3 x^{2}+2}}\right )+2 \arctan \left (2^{\frac {1}{4}} \left (-3 x^{2}+2\right )^{\frac {1}{4}}+1\right )+2 \arctan \left (-1+2^{\frac {1}{4}} \left (-3 x^{2}+2\right )^{\frac {1}{4}}\right )\right )}{12}\)	\(100\)
trager	\(\frac {\operatorname {RootOf}\left (\textit {\_Z}^{2}+\operatorname {RootOf}\left (\textit {\_Z}^{4}+8\right )^{2}\right ) \ln \left (\frac {\operatorname {RootOf}\left (\textit {\_Z}^{2}+\operatorname {RootOf}\left (\textit {\_Z}^{4}+8\right )^{2}\right ) \operatorname {RootOf}\left (\textit {\_Z}^{4}+8\right )^{2} \left (-3 x^{2}+2\right )^{\frac {3}{4}}+2 \operatorname {RootOf}\left (\textit {\_Z}^{4}+8\right )^{2} \sqrt {-3 x^{2}+2}-4 \operatorname {RootOf}\left (\textit {\_Z}^{2}+\operatorname {RootOf}\left (\textit {\_Z}^{4}+8\right )^{2}\right ) \left (-3 x^{2}+2\right )^{\frac {1}{4}}-6 x^{2}}{3 x^{2}-4}\right )}{12}-\frac {\operatorname {RootOf}\left (\textit {\_Z}^{4}+8\right ) \ln \left (\frac {\operatorname {RootOf}\left (\textit {\_Z}^{4}+8\right )^{3} \left (-3 x^{2}+2\right )^{\frac {3}{4}}-2 \operatorname {RootOf}\left (\textit {\_Z}^{4}+8\right )^{2} \sqrt {-3 x^{2}+2}+4 \operatorname {RootOf}\left (\textit {\_Z}^{4}+8\right ) \left (-3 x^{2}+2\right )^{\frac {1}{4}}-6 x^{2}}{3 x^{2}-4}\right )}{12}\)	\(187\)

[In]

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

[Out]

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

Fricas [C] (verification not implemented)

Result contains complex when optimal does not.

Time = 0.27 (sec) , antiderivative size = 93, normalized size of antiderivative = 1.02 \[ \int \frac {x}{\sqrt [4]{2-3 x^2} \left (4-3 x^2\right )} \, dx=-\left (\frac {1}{12} i - \frac {1}{12}\right ) \cdot 2^{\frac {1}{4}} \log \left (\left (i + 1\right ) \cdot 2^{\frac {3}{4}} + 2 \, {\left (-3 \, x^{2} + 2\right )}^{\frac {1}{4}}\right ) + \left (\frac {1}{12} i + \frac {1}{12}\right ) \cdot 2^{\frac {1}{4}} \log \left (-\left (i - 1\right ) \cdot 2^{\frac {3}{4}} + 2 \, {\left (-3 \, x^{2} + 2\right )}^{\frac {1}{4}}\right ) - \left (\frac {1}{12} i + \frac {1}{12}\right ) \cdot 2^{\frac {1}{4}} \log \left (\left (i - 1\right ) \cdot 2^{\frac {3}{4}} + 2 \, {\left (-3 \, x^{2} + 2\right )}^{\frac {1}{4}}\right ) + \left (\frac {1}{12} i - \frac {1}{12}\right ) \cdot 2^{\frac {1}{4}} \log \left (-\left (i + 1\right ) \cdot 2^{\frac {3}{4}} + 2 \, {\left (-3 \, x^{2} + 2\right )}^{\frac {1}{4}}\right ) \]

[In]

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

[Out]

-(1/12*I - 1/12)*2^(1/4)*log((I + 1)*2^(3/4) + 2*(-3*x^2 + 2)^(1/4)) + (1/12*I + 1/12)*2^(1/4)*log(-(I - 1)*2^
(3/4) + 2*(-3*x^2 + 2)^(1/4)) - (1/12*I + 1/12)*2^(1/4)*log((I - 1)*2^(3/4) + 2*(-3*x^2 + 2)^(1/4)) + (1/12*I
- 1/12)*2^(1/4)*log(-(I + 1)*2^(3/4) + 2*(-3*x^2 + 2)^(1/4))

Sympy [F]

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

[In]

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

[Out]

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

Maxima [A] (verification not implemented)

none

Time = 0.30 (sec) , antiderivative size = 118, normalized size of antiderivative = 1.30 \[ \int \frac {x}{\sqrt [4]{2-3 x^2} \left (4-3 x^2\right )} \, dx=-\frac {1}{6} \cdot 2^{\frac {1}{4}} \arctan \left (\frac {1}{2} \cdot 2^{\frac {1}{4}} {\left (2^{\frac {3}{4}} + 2 \, {\left (-3 \, x^{2} + 2\right )}^{\frac {1}{4}}\right )}\right ) - \frac {1}{6} \cdot 2^{\frac {1}{4}} \arctan \left (-\frac {1}{2} \cdot 2^{\frac {1}{4}} {\left (2^{\frac {3}{4}} - 2 \, {\left (-3 \, x^{2} + 2\right )}^{\frac {1}{4}}\right )}\right ) + \frac {1}{12} \cdot 2^{\frac {1}{4}} \log \left (2^{\frac {3}{4}} {\left (-3 \, x^{2} + 2\right )}^{\frac {1}{4}} + \sqrt {2} + \sqrt {-3 \, x^{2} + 2}\right ) - \frac {1}{12} \cdot 2^{\frac {1}{4}} \log \left (-2^{\frac {3}{4}} {\left (-3 \, x^{2} + 2\right )}^{\frac {1}{4}} + \sqrt {2} + \sqrt {-3 \, x^{2} + 2}\right ) \]

[In]

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

[Out]

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

Giac [A] (verification not implemented)

none

Time = 0.29 (sec) , antiderivative size = 118, normalized size of antiderivative = 1.30 \[ \int \frac {x}{\sqrt [4]{2-3 x^2} \left (4-3 x^2\right )} \, dx=-\frac {1}{6} \cdot 2^{\frac {1}{4}} \arctan \left (\frac {1}{2} \cdot 2^{\frac {1}{4}} {\left (2^{\frac {3}{4}} + 2 \, {\left (-3 \, x^{2} + 2\right )}^{\frac {1}{4}}\right )}\right ) - \frac {1}{6} \cdot 2^{\frac {1}{4}} \arctan \left (-\frac {1}{2} \cdot 2^{\frac {1}{4}} {\left (2^{\frac {3}{4}} - 2 \, {\left (-3 \, x^{2} + 2\right )}^{\frac {1}{4}}\right )}\right ) + \frac {1}{12} \cdot 2^{\frac {1}{4}} \log \left (2^{\frac {3}{4}} {\left (-3 \, x^{2} + 2\right )}^{\frac {1}{4}} + \sqrt {2} + \sqrt {-3 \, x^{2} + 2}\right ) - \frac {1}{12} \cdot 2^{\frac {1}{4}} \log \left (-2^{\frac {3}{4}} {\left (-3 \, x^{2} + 2\right )}^{\frac {1}{4}} + \sqrt {2} + \sqrt {-3 \, x^{2} + 2}\right ) \]

[In]

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

[Out]

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

Mupad [B] (verification not implemented)

Time = 0.15 (sec) , antiderivative size = 49, normalized size of antiderivative = 0.54 \[ \int \frac {x}{\sqrt [4]{2-3 x^2} \left (4-3 x^2\right )} \, dx=2^{1/4}\,\mathrm {atan}\left (2^{1/4}\,{\left (2-3\,x^2\right )}^{1/4}\,\left (\frac {1}{2}-\frac {1}{2}{}\mathrm {i}\right )\right )\,\left (-\frac {1}{6}+\frac {1}{6}{}\mathrm {i}\right )+2^{1/4}\,\mathrm {atan}\left (2^{1/4}\,{\left (2-3\,x^2\right )}^{1/4}\,\left (\frac {1}{2}+\frac {1}{2}{}\mathrm {i}\right )\right )\,\left (-\frac {1}{6}-\frac {1}{6}{}\mathrm {i}\right ) \]

[In]

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

[Out]

- 2^(1/4)*atan(2^(1/4)*(2 - 3*x^2)^(1/4)*(1/2 - 1i/2))*(1/6 - 1i/6) - 2^(1/4)*atan(2^(1/4)*(2 - 3*x^2)^(1/4)*(
1/2 + 1i/2))*(1/6 + 1i/6)