3.1041 \(\int \frac{1}{x^4 \sqrt [4]{2-3 x^2} (4-3 x^2)} \, dx\)

Optimal. Leaf size=184 \[ -\frac{3 \left (2-3 x^2\right )^{3/4}}{16 x}-\frac{\left (2-3 x^2\right )^{3/4}}{24 x^3}+\frac{3 \sqrt{3} \tan ^{-1}\left (\frac{2^{3/4}-\sqrt [4]{2} \sqrt{2-3 x^2}}{\sqrt{3} x \sqrt [4]{2-3 x^2}}\right )}{32\ 2^{3/4}}+\frac{3 \sqrt{3} \tanh ^{-1}\left (\frac{\sqrt [4]{2} \sqrt{2-3 x^2}+2^{3/4}}{\sqrt{3} x \sqrt [4]{2-3 x^2}}\right )}{32\ 2^{3/4}}-\frac{3 \sqrt{3} E\left (\left .\frac{1}{2} \sin ^{-1}\left (\sqrt{\frac{3}{2}} x\right )\right |2\right )}{8\ 2^{3/4}} \]

[Out]

-(2 - 3*x^2)^(3/4)/(24*x^3) - (3*(2 - 3*x^2)^(3/4))/(16*x) + (3*Sqrt[3]*ArcTan[(2^(3/4) - 2^(1/4)*Sqrt[2 - 3*x
^2])/(Sqrt[3]*x*(2 - 3*x^2)^(1/4))])/(32*2^(3/4)) + (3*Sqrt[3]*ArcTanh[(2^(3/4) + 2^(1/4)*Sqrt[2 - 3*x^2])/(Sq
rt[3]*x*(2 - 3*x^2)^(1/4))])/(32*2^(3/4)) - (3*Sqrt[3]*EllipticE[ArcSin[Sqrt[3/2]*x]/2, 2])/(8*2^(3/4))

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Rubi [A]  time = 0.0870461, antiderivative size = 184, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 4, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.167, Rules used = {440, 325, 228, 397} \[ -\frac{3 \left (2-3 x^2\right )^{3/4}}{16 x}-\frac{\left (2-3 x^2\right )^{3/4}}{24 x^3}+\frac{3 \sqrt{3} \tan ^{-1}\left (\frac{2^{3/4}-\sqrt [4]{2} \sqrt{2-3 x^2}}{\sqrt{3} x \sqrt [4]{2-3 x^2}}\right )}{32\ 2^{3/4}}+\frac{3 \sqrt{3} \tanh ^{-1}\left (\frac{\sqrt [4]{2} \sqrt{2-3 x^2}+2^{3/4}}{\sqrt{3} x \sqrt [4]{2-3 x^2}}\right )}{32\ 2^{3/4}}-\frac{3 \sqrt{3} E\left (\left .\frac{1}{2} \sin ^{-1}\left (\sqrt{\frac{3}{2}} x\right )\right |2\right )}{8\ 2^{3/4}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-(2 - 3*x^2)^(3/4)/(24*x^3) - (3*(2 - 3*x^2)^(3/4))/(16*x) + (3*Sqrt[3]*ArcTan[(2^(3/4) - 2^(1/4)*Sqrt[2 - 3*x
^2])/(Sqrt[3]*x*(2 - 3*x^2)^(1/4))])/(32*2^(3/4)) + (3*Sqrt[3]*ArcTanh[(2^(3/4) + 2^(1/4)*Sqrt[2 - 3*x^2])/(Sq
rt[3]*x*(2 - 3*x^2)^(1/4))])/(32*2^(3/4)) - (3*Sqrt[3]*EllipticE[ArcSin[Sqrt[3/2]*x]/2, 2])/(8*2^(3/4))

Rule 440

Int[(x_)^(m_)/(((a_) + (b_.)*(x_)^2)^(1/4)*((c_) + (d_.)*(x_)^2)), x_Symbol] :> Int[ExpandIntegrand[x^m/((a +
b*x^2)^(1/4)*(c + d*x^2)), x], x] /; FreeQ[{a, b, c, d}, x] && EqQ[b*c - 2*a*d, 0] && IntegerQ[m] && (PosQ[a]
|| IntegerQ[m/2])

Rule 325

Int[((c_.)*(x_))^(m_)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[((c*x)^(m + 1)*(a + b*x^n)^(p + 1))/(a*
c*(m + 1)), x] - Dist[(b*(m + n*(p + 1) + 1))/(a*c^n*(m + 1)), Int[(c*x)^(m + n)*(a + b*x^n)^p, x], x] /; Free
Q[{a, b, c, p}, x] && IGtQ[n, 0] && LtQ[m, -1] && IntBinomialQ[a, b, c, n, m, p, x]

Rule 228

Int[((a_) + (b_.)*(x_)^2)^(-1/4), x_Symbol] :> Simp[(2*EllipticE[(1*ArcSin[Rt[-(b/a), 2]*x])/2, 2])/(a^(1/4)*R
t[-(b/a), 2]), x] /; FreeQ[{a, b}, x] && GtQ[a, 0] && NegQ[b/a]

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}{x^4 \sqrt [4]{2-3 x^2} \left (4-3 x^2\right )} \, dx &=\int \left (\frac{1}{4 x^4 \sqrt [4]{2-3 x^2}}+\frac{3}{16 x^2 \sqrt [4]{2-3 x^2}}-\frac{9}{16 \sqrt [4]{2-3 x^2} \left (-4+3 x^2\right )}\right ) \, dx\\ &=\frac{3}{16} \int \frac{1}{x^2 \sqrt [4]{2-3 x^2}} \, dx+\frac{1}{4} \int \frac{1}{x^4 \sqrt [4]{2-3 x^2}} \, dx-\frac{9}{16} \int \frac{1}{\sqrt [4]{2-3 x^2} \left (-4+3 x^2\right )} \, dx\\ &=-\frac{\left (2-3 x^2\right )^{3/4}}{24 x^3}-\frac{3 \left (2-3 x^2\right )^{3/4}}{32 x}+\frac{3 \sqrt{3} \tan ^{-1}\left (\frac{2^{3/4}-\sqrt [4]{2} \sqrt{2-3 x^2}}{\sqrt{3} x \sqrt [4]{2-3 x^2}}\right )}{32\ 2^{3/4}}+\frac{3 \sqrt{3} \tanh ^{-1}\left (\frac{2^{3/4}+\sqrt [4]{2} \sqrt{2-3 x^2}}{\sqrt{3} x \sqrt [4]{2-3 x^2}}\right )}{32\ 2^{3/4}}-\frac{9}{64} \int \frac{1}{\sqrt [4]{2-3 x^2}} \, dx+\frac{3}{16} \int \frac{1}{x^2 \sqrt [4]{2-3 x^2}} \, dx\\ &=-\frac{\left (2-3 x^2\right )^{3/4}}{24 x^3}-\frac{3 \left (2-3 x^2\right )^{3/4}}{16 x}+\frac{3 \sqrt{3} \tan ^{-1}\left (\frac{2^{3/4}-\sqrt [4]{2} \sqrt{2-3 x^2}}{\sqrt{3} x \sqrt [4]{2-3 x^2}}\right )}{32\ 2^{3/4}}+\frac{3 \sqrt{3} \tanh ^{-1}\left (\frac{2^{3/4}+\sqrt [4]{2} \sqrt{2-3 x^2}}{\sqrt{3} x \sqrt [4]{2-3 x^2}}\right )}{32\ 2^{3/4}}-\frac{3 \sqrt{3} E\left (\left .\frac{1}{2} \sin ^{-1}\left (\sqrt{\frac{3}{2}} x\right )\right |2\right )}{16\ 2^{3/4}}-\frac{9}{64} \int \frac{1}{\sqrt [4]{2-3 x^2}} \, dx\\ &=-\frac{\left (2-3 x^2\right )^{3/4}}{24 x^3}-\frac{3 \left (2-3 x^2\right )^{3/4}}{16 x}+\frac{3 \sqrt{3} \tan ^{-1}\left (\frac{2^{3/4}-\sqrt [4]{2} \sqrt{2-3 x^2}}{\sqrt{3} x \sqrt [4]{2-3 x^2}}\right )}{32\ 2^{3/4}}+\frac{3 \sqrt{3} \tanh ^{-1}\left (\frac{2^{3/4}+\sqrt [4]{2} \sqrt{2-3 x^2}}{\sqrt{3} x \sqrt [4]{2-3 x^2}}\right )}{32\ 2^{3/4}}-\frac{3 \sqrt{3} E\left (\left .\frac{1}{2} \sin ^{-1}\left (\sqrt{\frac{3}{2}} x\right )\right |2\right )}{8\ 2^{3/4}}\\ \end{align*}

Mathematica [C]  time = 0.154913, size = 156, normalized size = 0.85 \[ \frac{1}{8} \left (2-3 x^2\right )^{3/4} \left (\frac{9 x F_1\left (\frac{1}{2};-\frac{3}{4},1;\frac{3}{2};\frac{3 x^2}{2},\frac{3 x^2}{4}\right )}{\left (3 x^2-4\right ) \left (x^2 \left (2 F_1\left (\frac{3}{2};-\frac{3}{4},2;\frac{5}{2};\frac{3 x^2}{2},\frac{3 x^2}{4}\right )-3 F_1\left (\frac{3}{2};\frac{1}{4},1;\frac{5}{2};\frac{3 x^2}{2},\frac{3 x^2}{4}\right )\right )+4 F_1\left (\frac{1}{2};-\frac{3}{4},1;\frac{3}{2};\frac{3 x^2}{2},\frac{3 x^2}{4}\right )\right )}-\frac{9 x^2+2}{6 x^3}\right ) \]

Warning: Unable to verify antiderivative.

[In]

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

[Out]

((2 - 3*x^2)^(3/4)*(-(2 + 9*x^2)/(6*x^3) + (9*x*AppellF1[1/2, -3/4, 1, 3/2, (3*x^2)/2, (3*x^2)/4])/((-4 + 3*x^
2)*(4*AppellF1[1/2, -3/4, 1, 3/2, (3*x^2)/2, (3*x^2)/4] + x^2*(2*AppellF1[3/2, -3/4, 2, 5/2, (3*x^2)/2, (3*x^2
)/4] - 3*AppellF1[3/2, 1/4, 1, 5/2, (3*x^2)/2, (3*x^2)/4])))))/8

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Maple [F]  time = 0.076, size = 0, normalized size = 0. \begin{align*} \int{\frac{1}{{x}^{4} \left ( -3\,{x}^{2}+4 \right ) }{\frac{1}{\sqrt [4]{-3\,{x}^{2}+2}}}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x^4/(-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^4), x)

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Fricas [F]  time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (\frac{{\left (-3 \, x^{2} + 2\right )}^{\frac{3}{4}}}{9 \, x^{8} - 18 \, x^{6} + 8 \, x^{4}}, x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integral((-3*x^2 + 2)^(3/4)/(9*x^8 - 18*x^6 + 8*x^4), x)

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} - \int \frac{1}{3 x^{6} \sqrt [4]{2 - 3 x^{2}} - 4 x^{4} \sqrt [4]{2 - 3 x^{2}}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-Integral(1/(3*x**6*(2 - 3*x**2)**(1/4) - 4*x**4*(2 - 3*x**2)**(1/4)), x)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x^4/(-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^4), x)