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

Optimal. Leaf size=129 \[ -\frac{\tan ^{-1}\left (\frac{2 \sqrt [4]{2} \sqrt{3 x^2+2}+2\ 2^{3/4}}{2 \sqrt{3} x \sqrt [4]{3 x^2+2}}\right )}{2\ 2^{3/4} \sqrt{3}}-\frac{\tanh ^{-1}\left (\frac{2\ 2^{3/4}-2 \sqrt [4]{2} \sqrt{3 x^2+2}}{2 \sqrt{3} x \sqrt [4]{3 x^2+2}}\right )}{2\ 2^{3/4} \sqrt{3}} \]

[Out]

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

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Rubi [A]  time = 0.0191751, antiderivative size = 129, normalized size of antiderivative = 1., 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} \[ -\frac{\tan ^{-1}\left (\frac{2 \sqrt [4]{2} \sqrt{3 x^2+2}+2\ 2^{3/4}}{2 \sqrt{3} x \sqrt [4]{3 x^2+2}}\right )}{2\ 2^{3/4} \sqrt{3}}-\frac{\tanh ^{-1}\left (\frac{2\ 2^{3/4}-2 \sqrt [4]{2} \sqrt{3 x^2+2}}{2 \sqrt{3} x \sqrt [4]{3 x^2+2}}\right )}{2\ 2^{3/4} \sqrt{3}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-ArcTan[(2*2^(3/4) + 2*2^(1/4)*Sqrt[2 + 3*x^2])/(2*Sqrt[3]*x*(2 + 3*x^2)^(1/4))]/(2*2^(3/4)*Sqrt[3]) - ArcTanh
[(2*2^(3/4) - 2*2^(1/4)*Sqrt[2 + 3*x^2])/(2*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\ 2^{3/4}+2 \sqrt [4]{2} \sqrt{2+3 x^2}}{2 \sqrt{3} x \sqrt [4]{2+3 x^2}}\right )}{2\ 2^{3/4} \sqrt{3}}-\frac{\tanh ^{-1}\left (\frac{2\ 2^{3/4}-2 \sqrt [4]{2} \sqrt{2+3 x^2}}{2 \sqrt{3} x \sqrt [4]{2+3 x^2}}\right )}{2\ 2^{3/4} \sqrt{3}}\\ \end{align*}

Mathematica [C]  time = 0.106849, size = 135, normalized size = 1.05 \[ -\frac{4 x F_1\left (\frac{1}{2};\frac{1}{4},1;\frac{3}{2};-\frac{3 x^2}{2},-\frac{3 x^2}{4}\right )}{\sqrt [4]{3 x^2+2} \left (3 x^2+4\right ) \left (x^2 \left (2 F_1\left (\frac{3}{2};\frac{1}{4},2;\frac{5}{2};-\frac{3 x^2}{2},-\frac{3 x^2}{4}\right )+F_1\left (\frac{3}{2};\frac{5}{4},1;\frac{5}{2};-\frac{3 x^2}{2},-\frac{3 x^2}{4}\right )\right )-4 F_1\left (\frac{1}{2};\frac{1}{4},1;\frac{3}{2};-\frac{3 x^2}{2},-\frac{3 x^2}{4}\right )\right )} \]

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])))

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Maple [F]  time = 0.026, size = 0, normalized size = 0. \begin{align*} \int{\frac{1}{3\,{x}^{2}+4}{\frac{1}{\sqrt [4]{3\,{x}^{2}+2}}}}\, dx \end{align*}

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]

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

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)

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Fricas [B]  time = 27.7936, size = 1520, normalized size = 11.78 \begin{align*} \text{result too large to display} \end{align*}

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))*sqrt((3*s
qrt(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))

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

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)**(1/4)*(3*x**2 + 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}}}\,{d x} \end{align*}

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)