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

Optimal. Leaf size=91 \[ \frac {\tan ^{-1}\left (\frac {\sqrt {2}-\sqrt {2-3 x^2}}{2^{3/4} \sqrt [4]{2-3 x^2}}\right )}{3\ 2^{3/4}}+\frac {\tanh ^{-1}\left (\frac {\sqrt {2-3 x^2}+\sqrt {2}}{2^{3/4} \sqrt [4]{2-3 x^2}}\right )}{3\ 2^{3/4}} \]

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Rubi [A]  time = 0.01, antiderivative size = 91, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, integrand size = 22, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.045, Rules used = {439} \begin {gather*} \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 {2-3 x^2}+\sqrt {2}}{2^{3/4} \sqrt [4]{2-3 x^2}}\right )}{3\ 2^{3/4}} \end {gather*}

Antiderivative was successfully verified.

[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 439

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

Rubi steps

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

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Mathematica [C]  time = 0.01, size = 34, normalized size = 0.37 \begin {gather*} -\frac {1}{9} \left (2-3 x^2\right )^{3/4} \, _2F_1\left (\frac {3}{4},1;\frac {7}{4};\frac {1}{2} \left (3 x^2-2\right )\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

-1/9*((2 - 3*x^2)^(3/4)*Hypergeometric2F1[3/4, 1, 7/4, (-2 + 3*x^2)/2])

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IntegrateAlgebraic [A]  time = 0.12, size = 97, normalized size = 1.07 \begin {gather*} \frac {\tanh ^{-1}\left (\frac {2 \sqrt [4]{2} \sqrt [4]{2-3 x^2}}{\sqrt {2} \sqrt {2-3 x^2}+2}\right )}{3\ 2^{3/4}}-\frac {\tan ^{-1}\left (\frac {\frac {\sqrt {2-3 x^2}}{2^{3/4}}-\frac {1}{\sqrt [4]{2}}}{\sqrt [4]{2-3 x^2}}\right )}{3\ 2^{3/4}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

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fricas [B]  time = 1.40, size = 189, normalized size = 2.08 \begin {gather*} \frac {1}{3} \cdot 2^{\frac {1}{4}} \arctan \left (2^{\frac {1}{4}} \sqrt {2^{\frac {3}{4}} {\left (-3 \, x^{2} + 2\right )}^{\frac {1}{4}} + \sqrt {2} + \sqrt {-3 \, x^{2} + 2}} - 2^{\frac {1}{4}} {\left (-3 \, x^{2} + 2\right )}^{\frac {1}{4}} - 1\right ) + \frac {1}{3} \cdot 2^{\frac {1}{4}} \arctan \left (\frac {1}{2} \cdot 2^{\frac {1}{4}} \sqrt {-4 \cdot 2^{\frac {3}{4}} {\left (-3 \, x^{2} + 2\right )}^{\frac {1}{4}} + 4 \, \sqrt {2} + 4 \, \sqrt {-3 \, x^{2} + 2}} - 2^{\frac {1}{4}} {\left (-3 \, x^{2} + 2\right )}^{\frac {1}{4}} + 1\right ) + \frac {1}{12} \cdot 2^{\frac {1}{4}} \log \left (4 \cdot 2^{\frac {3}{4}} {\left (-3 \, x^{2} + 2\right )}^{\frac {1}{4}} + 4 \, \sqrt {2} + 4 \, \sqrt {-3 \, x^{2} + 2}\right ) - \frac {1}{12} \cdot 2^{\frac {1}{4}} \log \left (-4 \cdot 2^{\frac {3}{4}} {\left (-3 \, x^{2} + 2\right )}^{\frac {1}{4}} + 4 \, \sqrt {2} + 4 \, \sqrt {-3 \, x^{2} + 2}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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giac [A]  time = 0.37, size = 118, normalized size = 1.30 \begin {gather*} -\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 ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[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))

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maple [C]  time = 3.96, size = 189, normalized size = 2.08 \begin {gather*} -\frac {\RootOf \left (\textit {\_Z}^{2}+\RootOf \left (\textit {\_Z}^{4}+8\right )^{2}\right ) \ln \left (-\frac {\left (-3 x^{2}+2\right )^{\frac {3}{4}} \RootOf \left (\textit {\_Z}^{2}+\RootOf \left (\textit {\_Z}^{4}+8\right )^{2}\right ) \RootOf \left (\textit {\_Z}^{4}+8\right )^{2}+6 x^{2}-2 \sqrt {-3 x^{2}+2}\, \RootOf \left (\textit {\_Z}^{4}+8\right )^{2}-4 \left (-3 x^{2}+2\right )^{\frac {1}{4}} \RootOf \left (\textit {\_Z}^{2}+\RootOf \left (\textit {\_Z}^{4}+8\right )^{2}\right )}{3 x^{2}-4}\right )}{12}+\frac {\RootOf \left (\textit {\_Z}^{4}+8\right ) \ln \left (-\frac {\left (-3 x^{2}+2\right )^{\frac {3}{4}} \RootOf \left (\textit {\_Z}^{4}+8\right )^{3}+6 x^{2}+2 \sqrt {-3 x^{2}+2}\, \RootOf \left (\textit {\_Z}^{4}+8\right )^{2}+4 \left (-3 x^{2}+2\right )^{\frac {1}{4}} \RootOf \left (\textit {\_Z}^{4}+8\right )}{3 x^{2}-4}\right )}{12} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/12*RootOf(_Z^4+8)*ln(-(RootOf(_Z^4+8)^3*(-3*x^2+2)^(3/4)+2*RootOf(_Z^4+8)^2*(-3*x^2+2)^(1/2)+4*RootOf(_Z^4+8
)*(-3*x^2+2)^(1/4)+6*x^2)/(3*x^2-4))-1/12*RootOf(_Z^2+RootOf(_Z^4+8)^2)*ln(-(RootOf(_Z^2+RootOf(_Z^4+8)^2)*Roo
tOf(_Z^4+8)^2*(-3*x^2+2)^(3/4)-2*RootOf(_Z^4+8)^2*(-3*x^2+2)^(1/2)-4*RootOf(_Z^2+RootOf(_Z^4+8)^2)*(-3*x^2+2)^
(1/4)+6*x^2)/(3*x^2-4))

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maxima [A]  time = 1.90, size = 118, normalized size = 1.30 \begin {gather*} -\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 ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[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))

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mupad [B]  time = 0.15, size = 49, normalized size = 0.54 \begin {gather*} 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 ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[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)

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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} - \int \frac {x}{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(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)

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