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

Optimal. Leaf size=68 \[ -\frac{\sqrt [4]{4 x^4+3}}{x}-\frac{\tan ^{-1}\left (\frac{\sqrt{2} x}{\sqrt [4]{4 x^4+3}}\right )}{\sqrt{2}}+\frac{\tanh ^{-1}\left (\frac{\sqrt{2} x}{\sqrt [4]{4 x^4+3}}\right )}{\sqrt{2}} \]

[Out]

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

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Rubi [A]  time = 0.0208885, antiderivative size = 68, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 5, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.333, Rules used = {277, 331, 298, 203, 206} \[ -\frac{\sqrt [4]{4 x^4+3}}{x}-\frac{\tan ^{-1}\left (\frac{\sqrt{2} x}{\sqrt [4]{4 x^4+3}}\right )}{\sqrt{2}}+\frac{\tanh ^{-1}\left (\frac{\sqrt{2} x}{\sqrt [4]{4 x^4+3}}\right )}{\sqrt{2}} \]

Antiderivative was successfully verified.

[In]

Int[(3 + 4*x^4)^(1/4)/x^2,x]

[Out]

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

Rule 277

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

Rule 331

Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Dist[a^(p + (m + 1)/n), Subst[Int[x^m/(1 - b*x^n)^(
p + (m + 1)/n + 1), x], x, x/(a + b*x^n)^(1/n)], x] /; FreeQ[{a, b}, x] && IGtQ[n, 0] && LtQ[-1, p, 0] && NeQ[
p, -2^(-1)] && IntegersQ[m, p + (m + 1)/n]

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])

Rubi steps

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

Mathematica [C]  time = 0.0040984, size = 27, normalized size = 0.4 \[ -\frac{\sqrt [4]{3} \, _2F_1\left (-\frac{1}{4},-\frac{1}{4};\frac{3}{4};-\frac{4 x^4}{3}\right )}{x} \]

Antiderivative was successfully verified.

[In]

Integrate[(3 + 4*x^4)^(1/4)/x^2,x]

[Out]

-((3^(1/4)*Hypergeometric2F1[-1/4, -1/4, 3/4, (-4*x^4)/3])/x)

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Maple [C]  time = 0.04, size = 35, normalized size = 0.5 \begin{align*} -{\frac{1}{x}\sqrt [4]{4\,{x}^{4}+3}}+{\frac{4\,\sqrt [4]{3}{x}^{3}}{9}{\mbox{$_2$F$_1$}({\frac{3}{4}},{\frac{3}{4}};\,{\frac{7}{4}};\,-{\frac{4\,{x}^{4}}{3}})}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]  time = 1.46827, size = 112, normalized size = 1.65 \begin{align*} \frac{1}{2} \, \sqrt{2} \arctan \left (\frac{\sqrt{2}{\left (4 \, x^{4} + 3\right )}^{\frac{1}{4}}}{2 \, x}\right ) - \frac{1}{4} \, \sqrt{2} \log \left (-\frac{\sqrt{2} - \frac{{\left (4 \, x^{4} + 3\right )}^{\frac{1}{4}}}{x}}{\sqrt{2} + \frac{{\left (4 \, x^{4} + 3\right )}^{\frac{1}{4}}}{x}}\right ) - \frac{{\left (4 \, x^{4} + 3\right )}^{\frac{1}{4}}}{x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [B]  time = 26.1558, size = 390, normalized size = 5.74 \begin{align*} -\frac{2 \, \sqrt{2} x \arctan \left (\frac{4}{3} \, \sqrt{2}{\left (4 \, x^{4} + 3\right )}^{\frac{1}{4}} x^{3} + \frac{2}{3} \, \sqrt{2}{\left (4 \, x^{4} + 3\right )}^{\frac{3}{4}} x\right ) - \sqrt{2} x \log \left (-256 \, x^{8} - 192 \, x^{4} - 4 \, \sqrt{2}{\left (16 \, x^{5} + 3 \, x\right )}{\left (4 \, x^{4} + 3\right )}^{\frac{3}{4}} - 8 \, \sqrt{2}{\left (16 \, x^{7} + 9 \, x^{3}\right )}{\left (4 \, x^{4} + 3\right )}^{\frac{1}{4}} - 16 \,{\left (8 \, x^{6} + 3 \, x^{2}\right )} \sqrt{4 \, x^{4} + 3} - 9\right ) + 8 \,{\left (4 \, x^{4} + 3\right )}^{\frac{1}{4}}}{8 \, x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/8*(2*sqrt(2)*x*arctan(4/3*sqrt(2)*(4*x^4 + 3)^(1/4)*x^3 + 2/3*sqrt(2)*(4*x^4 + 3)^(3/4)*x) - sqrt(2)*x*log(
-256*x^8 - 192*x^4 - 4*sqrt(2)*(16*x^5 + 3*x)*(4*x^4 + 3)^(3/4) - 8*sqrt(2)*(16*x^7 + 9*x^3)*(4*x^4 + 3)^(1/4)
- 16*(8*x^6 + 3*x^2)*sqrt(4*x^4 + 3) - 9) + 8*(4*x^4 + 3)^(1/4))/x

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Sympy [C]  time = 1.05336, size = 41, normalized size = 0.6 \begin{align*} \frac{\sqrt [4]{3} \Gamma \left (- \frac{1}{4}\right ){{}_{2}F_{1}\left (\begin{matrix} - \frac{1}{4}, - \frac{1}{4} \\ \frac{3}{4} \end{matrix}\middle |{\frac{4 x^{4} e^{i \pi }}{3}} \right )}}{4 x \Gamma \left (\frac{3}{4}\right )} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

3**(1/4)*gamma(-1/4)*hyper((-1/4, -1/4), (3/4,), 4*x**4*exp_polar(I*pi)/3)/(4*x*gamma(3/4))

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Giac [A]  time = 1.08321, size = 112, normalized size = 1.65 \begin{align*} \frac{1}{2} \, \sqrt{2} \arctan \left (\frac{\sqrt{2}{\left (4 \, x^{4} + 3\right )}^{\frac{1}{4}}}{2 \, x}\right ) - \frac{1}{4} \, \sqrt{2} \log \left (-\frac{\sqrt{2} - \frac{{\left (4 \, x^{4} + 3\right )}^{\frac{1}{4}}}{x}}{\sqrt{2} + \frac{{\left (4 \, x^{4} + 3\right )}^{\frac{1}{4}}}{x}}\right ) - \frac{{\left (4 \, x^{4} + 3\right )}^{\frac{1}{4}}}{x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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