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

Optimal. Leaf size=120 \[ \frac {\tan ^{-1}\left (\frac {2-\sqrt {2} \sqrt {2-3 x^2}}{\sqrt [4]{2} \sqrt {3} x \sqrt [4]{2-3 x^2}}\right )}{2\ 2^{3/4} \sqrt {3}}+\frac {\tanh ^{-1}\left (\frac {\sqrt {2} \sqrt {2-3 x^2}+2}{\sqrt [4]{2} \sqrt {3} x \sqrt [4]{2-3 x^2}}\right )}{2\ 2^{3/4} \sqrt {3}} \]

________________________________________________________________________________________

Rubi [A]  time = 0.02, antiderivative size = 120, 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 = {397} \begin {gather*} \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 {\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}} \end {gather*}

Antiderivative was successfully verified.

[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 397

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

Rubi steps

\begin {align*} \int \frac {1}{\sqrt [4]{2-3 x^2} \left (4-3 x^2\right )} \, dx &=\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 [C]  time = 0.13, size = 135, normalized size = 1.12 \begin {gather*} -\frac {4 x F_1\left (\frac {1}{2};\frac {1}{4},1;\frac {3}{2};\frac {3 x^2}{2},\frac {3 x^2}{4}\right )}{\sqrt [4]{2-3 x^2} \left (3 x^2-4\right ) \left (x^2 \left (2 F_1\left (\frac {3}{2};\frac {1}{4},2;\frac {5}{2};\frac {3 x^2}{2},\frac {3 x^2}{4}\right )+F_1\left (\frac {3}{2};\frac {5}{4},1;\frac {5}{2};\frac {3 x^2}{2},\frac {3 x^2}{4}\right )\right )+4 F_1\left (\frac {1}{2};\frac {1}{4},1;\frac {3}{2};\frac {3 x^2}{2},\frac {3 x^2}{4}\right )\right )} \end {gather*}

Warning: Unable to verify antiderivative.

[In]

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

[Out]

(-4*x*AppellF1[1/2, 1/4, 1, 3/2, (3*x^2)/2, (3*x^2)/4])/((2 - 3*x^2)^(1/4)*(-4 + 3*x^2)*(4*AppellF1[1/2, 1/4,
1, 3/2, (3*x^2)/2, (3*x^2)/4] + x^2*(2*AppellF1[3/2, 1/4, 2, 5/2, (3*x^2)/2, (3*x^2)/4] + AppellF1[3/2, 5/4, 1
, 5/2, (3*x^2)/2, (3*x^2)/4])))

________________________________________________________________________________________

IntegrateAlgebraic [A]  time = 0.32, size = 137, normalized size = 1.14 \begin {gather*} \frac {\tan ^{-1}\left (\frac {\frac {\sqrt {3} x^2}{2 \sqrt [4]{2}}-\frac {\sqrt [4]{2} \sqrt {2-3 x^2}}{\sqrt {3}}}{x \sqrt [4]{2-3 x^2}}\right )}{4\ 2^{3/4} \sqrt {3}}+\frac {\tanh ^{-1}\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}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

________________________________________________________________________________________

fricas [B]  time = 9.05, size = 553, normalized size = 4.61 \begin {gather*} \frac {1}{72} \cdot 18^{\frac {3}{4}} \sqrt {2} \arctan \left (-\frac {6 \cdot 18^{\frac {3}{4}} \sqrt {2} {\left (-3 \, x^{2} + 2\right )}^{\frac {1}{4}} x^{3} + 54 \, x^{4} + 24 \cdot 18^{\frac {1}{4}} \sqrt {2} {\left (-3 \, x^{2} + 2\right )}^{\frac {3}{4}} x + 12 \, \sqrt {2} {\left (3 \, x^{2} - 4\right )} \sqrt {-3 \, x^{2} + 2} - 72 \, x^{2} + {\left (18^{\frac {3}{4}} \sqrt {2} {\left (3 \, x^{3} + 4 \, x\right )} \sqrt {-3 \, x^{2} + 2} - 72 \, {\left (-3 \, x^{2} + 2\right )}^{\frac {1}{4}} x^{2} - 6 \cdot 18^{\frac {1}{4}} \sqrt {2} {\left (3 \, x^{3} - 4 \, x\right )} - 48 \, \sqrt {2} {\left (-3 \, x^{2} + 2\right )}^{\frac {3}{4}}\right )} \sqrt {-\frac {3 \, \sqrt {2} x^{2} + 2 \cdot 18^{\frac {1}{4}} \sqrt {2} {\left (-3 \, x^{2} + 2\right )}^{\frac {1}{4}} x + 4 \, \sqrt {-3 \, x^{2} + 2}}{3 \, x^{2} - 4}}}{6 \, {\left (9 \, x^{4} + 24 \, x^{2} - 16\right )}}\right ) - \frac {1}{72} \cdot 18^{\frac {3}{4}} \sqrt {2} \arctan \left (\frac {6 \cdot 18^{\frac {3}{4}} \sqrt {2} {\left (-3 \, x^{2} + 2\right )}^{\frac {1}{4}} x^{3} - 54 \, x^{4} + 24 \cdot 18^{\frac {1}{4}} \sqrt {2} {\left (-3 \, x^{2} + 2\right )}^{\frac {3}{4}} x - 12 \, \sqrt {2} {\left (3 \, x^{2} - 4\right )} \sqrt {-3 \, x^{2} + 2} + 72 \, x^{2} + {\left (18^{\frac {3}{4}} \sqrt {2} {\left (3 \, x^{3} + 4 \, x\right )} \sqrt {-3 \, x^{2} + 2} + 72 \, {\left (-3 \, x^{2} + 2\right )}^{\frac {1}{4}} x^{2} - 6 \cdot 18^{\frac {1}{4}} \sqrt {2} {\left (3 \, x^{3} - 4 \, x\right )} + 48 \, \sqrt {2} {\left (-3 \, x^{2} + 2\right )}^{\frac {3}{4}}\right )} \sqrt {-\frac {3 \, \sqrt {2} x^{2} - 2 \cdot 18^{\frac {1}{4}} \sqrt {2} {\left (-3 \, x^{2} + 2\right )}^{\frac {1}{4}} x + 4 \, \sqrt {-3 \, x^{2} + 2}}{3 \, x^{2} - 4}}}{6 \, {\left (9 \, x^{4} + 24 \, x^{2} - 16\right )}}\right ) + \frac {1}{288} \cdot 18^{\frac {3}{4}} \sqrt {2} \log \left (-\frac {36 \, {\left (3 \, \sqrt {2} x^{2} + 2 \cdot 18^{\frac {1}{4}} \sqrt {2} {\left (-3 \, x^{2} + 2\right )}^{\frac {1}{4}} x + 4 \, \sqrt {-3 \, x^{2} + 2}\right )}}{3 \, x^{2} - 4}\right ) - \frac {1}{288} \cdot 18^{\frac {3}{4}} \sqrt {2} \log \left (-\frac {36 \, {\left (3 \, \sqrt {2} x^{2} - 2 \cdot 18^{\frac {1}{4}} \sqrt {2} {\left (-3 \, x^{2} + 2\right )}^{\frac {1}{4}} x + 4 \, \sqrt {-3 \, x^{2} + 2}\right )}}{3 \, x^{2} - 4}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/72*18^(3/4)*sqrt(2)*arctan(-1/6*(6*18^(3/4)*sqrt(2)*(-3*x^2 + 2)^(1/4)*x^3 + 54*x^4 + 24*18^(1/4)*sqrt(2)*(-
3*x^2 + 2)^(3/4)*x + 12*sqrt(2)*(3*x^2 - 4)*sqrt(-3*x^2 + 2) - 72*x^2 + (18^(3/4)*sqrt(2)*(3*x^3 + 4*x)*sqrt(-
3*x^2 + 2) - 72*(-3*x^2 + 2)^(1/4)*x^2 - 6*18^(1/4)*sqrt(2)*(3*x^3 - 4*x) - 48*sqrt(2)*(-3*x^2 + 2)^(3/4))*sqr
t(-(3*sqrt(2)*x^2 + 2*18^(1/4)*sqrt(2)*(-3*x^2 + 2)^(1/4)*x + 4*sqrt(-3*x^2 + 2))/(3*x^2 - 4)))/(9*x^4 + 24*x^
2 - 16)) - 1/72*18^(3/4)*sqrt(2)*arctan(1/6*(6*18^(3/4)*sqrt(2)*(-3*x^2 + 2)^(1/4)*x^3 - 54*x^4 + 24*18^(1/4)*
sqrt(2)*(-3*x^2 + 2)^(3/4)*x - 12*sqrt(2)*(3*x^2 - 4)*sqrt(-3*x^2 + 2) + 72*x^2 + (18^(3/4)*sqrt(2)*(3*x^3 + 4
*x)*sqrt(-3*x^2 + 2) + 72*(-3*x^2 + 2)^(1/4)*x^2 - 6*18^(1/4)*sqrt(2)*(3*x^3 - 4*x) + 48*sqrt(2)*(-3*x^2 + 2)^
(3/4))*sqrt(-(3*sqrt(2)*x^2 - 2*18^(1/4)*sqrt(2)*(-3*x^2 + 2)^(1/4)*x + 4*sqrt(-3*x^2 + 2))/(3*x^2 - 4)))/(9*x
^4 + 24*x^2 - 16)) + 1/288*18^(3/4)*sqrt(2)*log(-36*(3*sqrt(2)*x^2 + 2*18^(1/4)*sqrt(2)*(-3*x^2 + 2)^(1/4)*x +
4*sqrt(-3*x^2 + 2))/(3*x^2 - 4)) - 1/288*18^(3/4)*sqrt(2)*log(-36*(3*sqrt(2)*x^2 - 2*18^(1/4)*sqrt(2)*(-3*x^2
+ 2)^(1/4)*x + 4*sqrt(-3*x^2 + 2))/(3*x^2 - 4))

________________________________________________________________________________________

giac [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int -\frac {1}{{\left (3 \, x^{2} - 4\right )} {\left (-3 \, x^{2} + 2\right )}^{\frac {1}{4}}}\,{d x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[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)

________________________________________________________________________________________

maple [C]  time = 1.60, size = 187, normalized size = 1.56 \begin {gather*} -\frac {\RootOf \left (\textit {\_Z}^{2}+\RootOf \left (\textit {\_Z}^{4}+72\right )^{2}\right ) \ln \left (-\frac {-3 x \RootOf \left (\textit {\_Z}^{4}+72\right )^{2}+\left (-3 x^{2}+2\right )^{\frac {1}{4}} \RootOf \left (\textit {\_Z}^{2}+\RootOf \left (\textit {\_Z}^{4}+72\right )^{2}\right ) \RootOf \left (\textit {\_Z}^{4}+72\right )^{2}-18 \sqrt {-3 x^{2}+2}\, x +6 \left (-3 x^{2}+2\right )^{\frac {3}{4}} \RootOf \left (\textit {\_Z}^{2}+\RootOf \left (\textit {\_Z}^{4}+72\right )^{2}\right )}{3 x^{2}-4}\right )}{24}+\frac {\RootOf \left (\textit {\_Z}^{4}+72\right ) \ln \left (-\frac {3 x \RootOf \left (\textit {\_Z}^{4}+72\right )^{2}+\left (-3 x^{2}+2\right )^{\frac {1}{4}} \RootOf \left (\textit {\_Z}^{4}+72\right )^{3}-18 \sqrt {-3 x^{2}+2}\, x -6 \left (-3 x^{2}+2\right )^{\frac {3}{4}} \RootOf \left (\textit {\_Z}^{4}+72\right )}{3 x^{2}-4}\right )}{24} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} -\int \frac {1}{{\left (3 \, x^{2} - 4\right )} {\left (-3 \, x^{2} + 2\right )}^{\frac {1}{4}}}\,{d x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[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)

________________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

[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)

________________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

[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)

________________________________________________________________________________________