3.4.3 \(\int \frac {\sqrt [4]{3+4 x^4}}{x^2} \, dx\) [303]

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

[Out]

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

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Rubi [A]
time = 0.02, antiderivative size = 68, normalized size of antiderivative = 1.00, 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 = {283, 338, 304, 209, 212} \begin {gather*} -\frac {\text {ArcTan}\left (\frac {\sqrt {2} x}{\sqrt [4]{4 x^4+3}}\right )}{\sqrt {2}}-\frac {\sqrt [4]{4 x^4+3}}{x}+\frac {\tanh ^{-1}\left (\frac {\sqrt {2} x}{\sqrt [4]{4 x^4+3}}\right )}{\sqrt {2}} \end {gather*}

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 209

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

Rule 212

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

Rule 283

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 304

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 338

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]

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 \text {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}+\text {Subst}\left (\int \frac {1}{1-2 x^2} \, dx,x,\frac {x}{\sqrt [4]{3+4 x^4}}\right )-\text {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*}

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Mathematica [A]
time = 0.14, size = 68, normalized size = 1.00 \begin {gather*} -\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 {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(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]

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Maple [C] Result contains higher order function than in optimal. Order 5 vs. order 3.
time = 2.44, size = 20, normalized size = 0.29

method	result	size
meijerg	\(-\frac {3^{\frac {1}{4}} \hypergeom \left (\left [-\frac {1}{4}, -\frac {1}{4}\right ], \left [\frac {3}{4}\right ], -\frac {4 x^{4}}{3}\right )}{x}\)	\(20\)
risch	\(-\frac {\left (4 x^{4}+3\right )^{\frac {1}{4}}}{x}+\frac {4 \,3^{\frac {1}{4}} x^{3} \hypergeom \left (\left [\frac {3}{4}, \frac {3}{4}\right ], \left [\frac {7}{4}\right ], -\frac {4 x^{4}}{3}\right )}{9}\)	\(35\)
trager	\(-\frac {\left (4 x^{4}+3\right )^{\frac {1}{4}}}{x}+\frac {\RootOf \left (\textit {\_Z}^{2}+2\right ) \ln \left (-4 \sqrt {4 x^{4}+3}\, \RootOf \left (\textit {\_Z}^{2}+2\right ) x^{2}+8 \RootOf \left (\textit {\_Z}^{2}+2\right ) x^{4}-4 \left (4 x^{4}+3\right )^{\frac {3}{4}} x +8 x^{3} \left (4 x^{4}+3\right )^{\frac {1}{4}}+3 \RootOf \left (\textit {\_Z}^{2}+2\right )\right )}{4}-\frac {\RootOf \left (\textit {\_Z}^{2}-2\right ) \ln \left (4 \RootOf \left (\textit {\_Z}^{2}-2\right ) \sqrt {4 x^{4}+3}\, x^{2}+8 \RootOf \left (\textit {\_Z}^{2}-2\right ) x^{4}-4 \left (4 x^{4}+3\right )^{\frac {3}{4}} x -8 x^{3} \left (4 x^{4}+3\right )^{\frac {1}{4}}+3 \RootOf \left (\textit {\_Z}^{2}-2\right )\right )}{4}\)	\(166\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]
time = 4.01, size = 83, normalized size = 1.22 \begin {gather*} \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 {gather*}

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] Leaf count of result is larger than twice the leaf count of optimal. 146 vs. \(2 (55) = 110\).
time = 3.33, size = 146, normalized size = 2.15 \begin {gather*} -\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 {gather*}

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] Result contains complex when optimal does not.
time = 0.51, size = 41, normalized size = 0.60 \begin {gather*} \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 {gather*}

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 = 0.70, size = 83, normalized size = 1.22 \begin {gather*} \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 {gather*}

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

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Mupad [B]
time = 0.42, size = 18, normalized size = 0.26 \begin {gather*} -\frac {3^{1/4}\,{{}}_2{\mathrm {F}}_1\left (-\frac {1}{4},-\frac {1}{4};\ \frac {3}{4};\ -\frac {4\,x^4}{3}\right )}{x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((4*x^4 + 3)^(1/4)/x^2,x)

[Out]

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

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