3.1035 \(\int \frac{1}{x \sqrt [4]{2-3 x^2} (4-3 x^2)} \, dx\)

Optimal. Leaf size=145 \[ \frac{\tan ^{-1}\left (\frac{\sqrt [4]{2-3 x^2}}{\sqrt [4]{2}}\right )}{4 \sqrt [4]{2}}+\frac{\tan ^{-1}\left (\frac{\sqrt{2}-\sqrt{2-3 x^2}}{2^{3/4} \sqrt [4]{2-3 x^2}}\right )}{4\ 2^{3/4}}-\frac{\tanh ^{-1}\left (\frac{\sqrt [4]{2-3 x^2}}{\sqrt [4]{2}}\right )}{4 \sqrt [4]{2}}+\frac{\tanh ^{-1}\left (\frac{\sqrt{2-3 x^2}+\sqrt{2}}{2^{3/4} \sqrt [4]{2-3 x^2}}\right )}{4\ 2^{3/4}} \]

[Out]

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

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Rubi [A]  time = 0.0833905, antiderivative size = 145, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 7, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.292, Rules used = {440, 266, 63, 298, 203, 206, 439} \[ \frac{\tan ^{-1}\left (\frac{\sqrt [4]{2-3 x^2}}{\sqrt [4]{2}}\right )}{4 \sqrt [4]{2}}+\frac{\tan ^{-1}\left (\frac{\sqrt{2}-\sqrt{2-3 x^2}}{2^{3/4} \sqrt [4]{2-3 x^2}}\right )}{4\ 2^{3/4}}-\frac{\tanh ^{-1}\left (\frac{\sqrt [4]{2-3 x^2}}{\sqrt [4]{2}}\right )}{4 \sqrt [4]{2}}+\frac{\tanh ^{-1}\left (\frac{\sqrt{2-3 x^2}+\sqrt{2}}{2^{3/4} \sqrt [4]{2-3 x^2}}\right )}{4\ 2^{3/4}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

ArcTan[(2 - 3*x^2)^(1/4)/2^(1/4)]/(4*2^(1/4)) + ArcTan[(Sqrt[2] - Sqrt[2 - 3*x^2])/(2^(3/4)*(2 - 3*x^2)^(1/4))
]/(4*2^(3/4)) - ArcTanh[(2 - 3*x^2)^(1/4)/2^(1/4)]/(4*2^(1/4)) + ArcTanh[(Sqrt[2] + Sqrt[2 - 3*x^2])/(2^(3/4)*
(2 - 3*x^2)^(1/4))]/(4*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 266

Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Dist[1/n, Subst[Int[x^(Simplify[(m + 1)/n] - 1)*(a
+ b*x)^p, x], x, x^n], x] /; FreeQ[{a, b, m, n, p}, x] && IntegerQ[Simplify[(m + 1)/n]]

Rule 63

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> With[{p = Denominator[m]}, Dist[p/b, Sub
st[Int[x^(p*(m + 1) - 1)*(c - (a*d)/b + (d*x^p)/b)^n, x], x, (a + b*x)^(1/p)], x]] /; FreeQ[{a, b, c, d}, x] &
& NeQ[b*c - a*d, 0] && LtQ[-1, m, 0] && LeQ[-1, n, 0] && LeQ[Denominator[n], Denominator[m]] && IntLinearQ[a,
b, c, d, m, n, x]

Rule 298

Int[(x_)^2/((a_) + (b_.)*(x_)^4), x_Symbol] :> With[{r = Numerator[Rt[-(a/b), 2]], s = Denominator[Rt[-(a/b),
2]]}, Dist[s/(2*b), Int[1/(r + s*x^2), x], x] - Dist[s/(2*b), Int[1/(r - s*x^2), x], x]] /; FreeQ[{a, b}, x] &
&  !GtQ[a/b, 0]

Rule 203

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1*ArcTan[(Rt[b, 2]*x)/Rt[a, 2]])/(Rt[a, 2]*Rt[b, 2]), x] /;
FreeQ[{a, b}, x] && PosQ[a/b] && (GtQ[a, 0] || GtQ[b, 0])

Rule 206

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

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{1}{x \sqrt [4]{2-3 x^2} \left (4-3 x^2\right )} \, dx &=\int \left (\frac{1}{4 x \sqrt [4]{2-3 x^2}}-\frac{3 x}{4 \sqrt [4]{2-3 x^2} \left (-4+3 x^2\right )}\right ) \, dx\\ &=\frac{1}{4} \int \frac{1}{x \sqrt [4]{2-3 x^2}} \, dx-\frac{3}{4} \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 )}{4\ 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 )}{4\ 2^{3/4}}+\frac{1}{8} \operatorname{Subst}\left (\int \frac{1}{\sqrt [4]{2-3 x} x} \, dx,x,x^2\right )\\ &=\frac{\tan ^{-1}\left (\frac{\sqrt{2}-\sqrt{2-3 x^2}}{2^{3/4} \sqrt [4]{2-3 x^2}}\right )}{4\ 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 )}{4\ 2^{3/4}}-\frac{1}{6} \operatorname{Subst}\left (\int \frac{x^2}{\frac{2}{3}-\frac{x^4}{3}} \, dx,x,\sqrt [4]{2-3 x^2}\right )\\ &=\frac{\tan ^{-1}\left (\frac{\sqrt{2}-\sqrt{2-3 x^2}}{2^{3/4} \sqrt [4]{2-3 x^2}}\right )}{4\ 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 )}{4\ 2^{3/4}}-\frac{1}{4} \operatorname{Subst}\left (\int \frac{1}{\sqrt{2}-x^2} \, dx,x,\sqrt [4]{2-3 x^2}\right )+\frac{1}{4} \operatorname{Subst}\left (\int \frac{1}{\sqrt{2}+x^2} \, dx,x,\sqrt [4]{2-3 x^2}\right )\\ &=\frac{\tan ^{-1}\left (\frac{\sqrt [4]{2-3 x^2}}{\sqrt [4]{2}}\right )}{4 \sqrt [4]{2}}+\frac{\tan ^{-1}\left (\frac{\sqrt{2}-\sqrt{2-3 x^2}}{2^{3/4} \sqrt [4]{2-3 x^2}}\right )}{4\ 2^{3/4}}-\frac{\tanh ^{-1}\left (\frac{\sqrt [4]{2-3 x^2}}{\sqrt [4]{2}}\right )}{4 \sqrt [4]{2}}+\frac{\tanh ^{-1}\left (\frac{\sqrt{2}+\sqrt{2-3 x^2}}{2^{3/4} \sqrt [4]{2-3 x^2}}\right )}{4\ 2^{3/4}}\\ \end{align*}

Mathematica [C]  time = 0.0407167, size = 73, normalized size = 0.5 \[ \frac{1}{24} \left (3\ 2^{3/4} \left (\tan ^{-1}\left (\sqrt [4]{1-\frac{3 x^2}{2}}\right )-\tanh ^{-1}\left (\sqrt [4]{1-\frac{3 x^2}{2}}\right )\right )-2 \left (2-3 x^2\right )^{3/4} \, _2F_1\left (\frac{3}{4},1;\frac{7}{4};\frac{3 x^2}{2}-1\right )\right ) \]

Antiderivative was successfully verified.

[In]

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

[Out]

(3*2^(3/4)*(ArcTan[(1 - (3*x^2)/2)^(1/4)] - ArcTanh[(1 - (3*x^2)/2)^(1/4)]) - 2*(2 - 3*x^2)^(3/4)*Hypergeometr
ic2F1[3/4, 1, 7/4, -1 + (3*x^2)/2])/24

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Maple [F]  time = 0.05, size = 0, normalized size = 0. \begin{align*} \int{\frac{1}{x \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/(-3*x^2+2)^(1/4)/(-3*x^2+4),x)

[Out]

int(1/x/(-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}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x/(-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), x)

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Fricas [B]  time = 1.54202, size = 844, normalized size = 5.82 \begin{align*} -\frac{1}{4} \cdot 2^{\frac{3}{4}} \arctan \left (\frac{1}{2} \cdot 2^{\frac{3}{4}} \sqrt{\sqrt{2} + \sqrt{-3 \, x^{2} + 2}} - \frac{1}{2} \cdot 2^{\frac{3}{4}}{\left (-3 \, x^{2} + 2\right )}^{\frac{1}{4}}\right ) - \frac{1}{16} \cdot 2^{\frac{3}{4}} \log \left (2^{\frac{1}{4}} +{\left (-3 \, x^{2} + 2\right )}^{\frac{1}{4}}\right ) + \frac{1}{16} \cdot 2^{\frac{3}{4}} \log \left (-2^{\frac{1}{4}} +{\left (-3 \, x^{2} + 2\right )}^{\frac{1}{4}}\right ) + \frac{1}{4} \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}{4} \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}{16} \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}{16} \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{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/4*2^(3/4)*arctan(1/2*2^(3/4)*sqrt(sqrt(2) + sqrt(-3*x^2 + 2)) - 1/2*2^(3/4)*(-3*x^2 + 2)^(1/4)) - 1/16*2^(3
/4)*log(2^(1/4) + (-3*x^2 + 2)^(1/4)) + 1/16*2^(3/4)*log(-2^(1/4) + (-3*x^2 + 2)^(1/4)) + 1/4*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) + 1/4*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) + 1/16*2^(1/4)*log(4*2^(3/4)*(-3*x^2 + 2)^(1/4) + 4*sqrt(2) + 4*sqrt(-3*x^2 + 2)) - 1/16*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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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} - \int \frac{1}{3 x^{3} \sqrt [4]{2 - 3 x^{2}} - 4 x \sqrt [4]{2 - 3 x^{2}}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Giac [A]  time = 1.26835, size = 292, normalized size = 2.01 \begin{align*} -\frac{1}{16} \cdot 4^{\frac{3}{8}} \sqrt{2} \arctan \left (\frac{1}{8} \cdot 4^{\frac{7}{8}} \sqrt{2}{\left (4^{\frac{1}{8}} \sqrt{2} + 2 \,{\left (-3 \, x^{2} + 2\right )}^{\frac{1}{4}}\right )}\right ) - \frac{1}{16} \cdot 4^{\frac{3}{8}} \sqrt{2} \arctan \left (-\frac{1}{8} \cdot 4^{\frac{7}{8}} \sqrt{2}{\left (4^{\frac{1}{8}} \sqrt{2} - 2 \,{\left (-3 \, x^{2} + 2\right )}^{\frac{1}{4}}\right )}\right ) + \frac{1}{32} \cdot 4^{\frac{3}{8}} \sqrt{2} \log \left (4^{\frac{1}{8}} \sqrt{2}{\left (-3 \, x^{2} + 2\right )}^{\frac{1}{4}} + \sqrt{-3 \, x^{2} + 2} + 4^{\frac{1}{4}}\right ) - \frac{1}{32} \cdot 4^{\frac{3}{8}} \sqrt{2} \log \left (-4^{\frac{1}{8}} \sqrt{2}{\left (-3 \, x^{2} + 2\right )}^{\frac{1}{4}} + \sqrt{-3 \, x^{2} + 2} + 4^{\frac{1}{4}}\right ) + \frac{1}{8} \cdot 4^{\frac{1}{8}} \sqrt{2} \arctan \left (\frac{1}{4} \cdot 4^{\frac{7}{8}}{\left (-3 \, x^{2} + 2\right )}^{\frac{1}{4}}\right ) + \frac{1}{16} \cdot 4^{\frac{1}{8}} \sqrt{2} \log \left (-{\left (-3 \, x^{2} + 2\right )}^{\frac{1}{4}} + 4^{\frac{1}{8}}\right ) - \frac{1}{16} \cdot 4^{\frac{3}{8}} \log \left ({\left (-3 \, x^{2} + 2\right )}^{\frac{1}{4}} + 4^{\frac{1}{8}}\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/16*4^(3/8)*sqrt(2)*arctan(1/8*4^(7/8)*sqrt(2)*(4^(1/8)*sqrt(2) + 2*(-3*x^2 + 2)^(1/4))) - 1/16*4^(3/8)*sqrt
(2)*arctan(-1/8*4^(7/8)*sqrt(2)*(4^(1/8)*sqrt(2) - 2*(-3*x^2 + 2)^(1/4))) + 1/32*4^(3/8)*sqrt(2)*log(4^(1/8)*s
qrt(2)*(-3*x^2 + 2)^(1/4) + sqrt(-3*x^2 + 2) + 4^(1/4)) - 1/32*4^(3/8)*sqrt(2)*log(-4^(1/8)*sqrt(2)*(-3*x^2 +
2)^(1/4) + sqrt(-3*x^2 + 2) + 4^(1/4)) + 1/8*4^(1/8)*sqrt(2)*arctan(1/4*4^(7/8)*(-3*x^2 + 2)^(1/4)) + 1/16*4^(
1/8)*sqrt(2)*log(-(-3*x^2 + 2)^(1/4) + 4^(1/8)) - 1/16*4^(3/8)*log((-3*x^2 + 2)^(1/4) + 4^(1/8))