3.8.5 \(\int \frac {1}{(2+3 x^4)^2} \, dx\) [705]

Optimal. Leaf size=131 \[ \frac {x}{8 \left (2+3 x^4\right )}-\frac {3^{3/4} \tan ^{-1}\left (1-\sqrt [4]{6} x\right )}{32 \sqrt [4]{2}}+\frac {3^{3/4} \tan ^{-1}\left (1+\sqrt [4]{6} x\right )}{32 \sqrt [4]{2}}-\frac {3^{3/4} \log \left (\sqrt {6}-6^{3/4} x+3 x^2\right )}{64 \sqrt [4]{2}}+\frac {3^{3/4} \log \left (\sqrt {6}+6^{3/4} x+3 x^2\right )}{64 \sqrt [4]{2}} \]

[Out]

1/8*x/(3*x^4+2)+1/64*3^(3/4)*arctan(-1+6^(1/4)*x)*2^(3/4)+1/64*3^(3/4)*arctan(1+6^(1/4)*x)*2^(3/4)-1/128*3^(3/
4)*ln(-6^(3/4)*x+3*x^2+6^(1/2))*2^(3/4)+1/128*3^(3/4)*ln(6^(3/4)*x+3*x^2+6^(1/2))*2^(3/4)

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Rubi [A]
time = 0.05, antiderivative size = 131, normalized size of antiderivative = 1.00, number of steps used = 10, number of rules used = 7, integrand size = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.778, Rules used = {205, 217, 1179, 642, 1176, 631, 210} \begin {gather*} -\frac {3^{3/4} \text {ArcTan}\left (1-\sqrt [4]{6} x\right )}{32 \sqrt [4]{2}}+\frac {3^{3/4} \text {ArcTan}\left (\sqrt [4]{6} x+1\right )}{32 \sqrt [4]{2}}+\frac {x}{8 \left (3 x^4+2\right )}-\frac {3^{3/4} \log \left (3 x^2-6^{3/4} x+\sqrt {6}\right )}{64 \sqrt [4]{2}}+\frac {3^{3/4} \log \left (3 x^2+6^{3/4} x+\sqrt {6}\right )}{64 \sqrt [4]{2}} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(2 + 3*x^4)^(-2),x]

[Out]

x/(8*(2 + 3*x^4)) - (3^(3/4)*ArcTan[1 - 6^(1/4)*x])/(32*2^(1/4)) + (3^(3/4)*ArcTan[1 + 6^(1/4)*x])/(32*2^(1/4)
) - (3^(3/4)*Log[Sqrt[6] - 6^(3/4)*x + 3*x^2])/(64*2^(1/4)) + (3^(3/4)*Log[Sqrt[6] + 6^(3/4)*x + 3*x^2])/(64*2
^(1/4))

Rule 205

Int[((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[(-x)*((a + b*x^n)^(p + 1)/(a*n*(p + 1))), x] + Dist[(n*(p
+ 1) + 1)/(a*n*(p + 1)), Int[(a + b*x^n)^(p + 1), x], x] /; FreeQ[{a, b}, x] && IGtQ[n, 0] && LtQ[p, -1] && (
IntegerQ[2*p] || (n == 2 && IntegerQ[4*p]) || (n == 2 && IntegerQ[3*p]) || Denominator[p + 1/n] < Denominator[
p])

Rule 210

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

Rule 217

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

Rule 631

Int[((a_) + (b_.)*(x_) + (c_.)*(x_)^2)^(-1), x_Symbol] :> With[{q = 1 - 4*Simplify[a*(c/b^2)]}, Dist[-2/b, Sub
st[Int[1/(q - x^2), x], x, 1 + 2*c*(x/b)], x] /; RationalQ[q] && (EqQ[q^2, 1] ||  !RationalQ[b^2 - 4*a*c])] /;
FreeQ[{a, b, c}, x] && NeQ[b^2 - 4*a*c, 0]

Rule 642

Int[((d_) + (e_.)*(x_))/((a_.) + (b_.)*(x_) + (c_.)*(x_)^2), x_Symbol] :> Simp[d*(Log[RemoveContent[a + b*x +
c*x^2, x]]/b), x] /; FreeQ[{a, b, c, d, e}, x] && EqQ[2*c*d - b*e, 0]

Rule 1176

Int[((d_) + (e_.)*(x_)^2)/((a_) + (c_.)*(x_)^4), x_Symbol] :> With[{q = Rt[2*(d/e), 2]}, Dist[e/(2*c), Int[1/S
imp[d/e + q*x + x^2, x], x], x] + Dist[e/(2*c), Int[1/Simp[d/e - q*x + x^2, x], x], x]] /; FreeQ[{a, c, d, e},
x] && EqQ[c*d^2 - a*e^2, 0] && PosQ[d*e]

Rule 1179

Int[((d_) + (e_.)*(x_)^2)/((a_) + (c_.)*(x_)^4), x_Symbol] :> With[{q = Rt[-2*(d/e), 2]}, Dist[e/(2*c*q), Int[
(q - 2*x)/Simp[d/e + q*x - x^2, x], x], x] + Dist[e/(2*c*q), Int[(q + 2*x)/Simp[d/e - q*x - x^2, x], x], x]] /
; FreeQ[{a, c, d, e}, x] && EqQ[c*d^2 - a*e^2, 0] && NegQ[d*e]

Rubi steps

\begin {align*} \int \frac {1}{\left (2+3 x^4\right )^2} \, dx &=\frac {x}{8 \left (2+3 x^4\right )}+\frac {3}{8} \int \frac {1}{2+3 x^4} \, dx\\ &=\frac {x}{8 \left (2+3 x^4\right )}+\frac {3 \int \frac {\sqrt {2}-\sqrt {3} x^2}{2+3 x^4} \, dx}{16 \sqrt {2}}+\frac {3 \int \frac {\sqrt {2}+\sqrt {3} x^2}{2+3 x^4} \, dx}{16 \sqrt {2}}\\ &=\frac {x}{8 \left (2+3 x^4\right )}+\frac {1}{32} \sqrt {\frac {3}{2}} \int \frac {1}{\sqrt {\frac {2}{3}}-\frac {2^{3/4} x}{\sqrt [4]{3}}+x^2} \, dx+\frac {1}{32} \sqrt {\frac {3}{2}} \int \frac {1}{\sqrt {\frac {2}{3}}+\frac {2^{3/4} x}{\sqrt [4]{3}}+x^2} \, dx-\frac {3^{3/4} \int \frac {\frac {2^{3/4}}{\sqrt [4]{3}}+2 x}{-\sqrt {\frac {2}{3}}-\frac {2^{3/4} x}{\sqrt [4]{3}}-x^2} \, dx}{64 \sqrt [4]{2}}-\frac {3^{3/4} \int \frac {\frac {2^{3/4}}{\sqrt [4]{3}}-2 x}{-\sqrt {\frac {2}{3}}+\frac {2^{3/4} x}{\sqrt [4]{3}}-x^2} \, dx}{64 \sqrt [4]{2}}\\ &=\frac {x}{8 \left (2+3 x^4\right )}-\frac {3^{3/4} \log \left (\sqrt {6}-6^{3/4} x+3 x^2\right )}{64 \sqrt [4]{2}}+\frac {3^{3/4} \log \left (\sqrt {6}+6^{3/4} x+3 x^2\right )}{64 \sqrt [4]{2}}+\frac {3^{3/4} \text {Subst}\left (\int \frac {1}{-1-x^2} \, dx,x,1-\sqrt [4]{6} x\right )}{32 \sqrt [4]{2}}-\frac {3^{3/4} \text {Subst}\left (\int \frac {1}{-1-x^2} \, dx,x,1+\sqrt [4]{6} x\right )}{32 \sqrt [4]{2}}\\ &=\frac {x}{8 \left (2+3 x^4\right )}-\frac {3^{3/4} \tan ^{-1}\left (1-\sqrt [4]{6} x\right )}{32 \sqrt [4]{2}}+\frac {3^{3/4} \tan ^{-1}\left (1+\sqrt [4]{6} x\right )}{32 \sqrt [4]{2}}-\frac {3^{3/4} \log \left (\sqrt {6}-6^{3/4} x+3 x^2\right )}{64 \sqrt [4]{2}}+\frac {3^{3/4} \log \left (\sqrt {6}+6^{3/4} x+3 x^2\right )}{64 \sqrt [4]{2}}\\ \end {align*}

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Mathematica [A]
time = 0.04, size = 105, normalized size = 0.80 \begin {gather*} \frac {1}{128} \left (\frac {16 x}{2+3 x^4}-2\ 6^{3/4} \tan ^{-1}\left (1-\sqrt [4]{6} x\right )+2\ 6^{3/4} \tan ^{-1}\left (1+\sqrt [4]{6} x\right )-6^{3/4} \log \left (2-2 \sqrt [4]{6} x+\sqrt {6} x^2\right )+6^{3/4} \log \left (2+2 \sqrt [4]{6} x+\sqrt {6} x^2\right )\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(2 + 3*x^4)^(-2),x]

[Out]

((16*x)/(2 + 3*x^4) - 2*6^(3/4)*ArcTan[1 - 6^(1/4)*x] + 2*6^(3/4)*ArcTan[1 + 6^(1/4)*x] - 6^(3/4)*Log[2 - 2*6^
(1/4)*x + Sqrt[6]*x^2] + 6^(3/4)*Log[2 + 2*6^(1/4)*x + Sqrt[6]*x^2])/128

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Maple [A]
time = 0.14, size = 106, normalized size = 0.81

method	result	size
risch	\(\frac {x}{24 x^{4}+16}+\frac {\left (\munderset {\textit {\_R} =\RootOf \left (3 \textit {\_Z}^{4}+2\right )}{\sum }\frac {\ln \left (x -\textit {\_R} \right )}{\textit {\_R}^{3}}\right )}{32}\)	\(35\)
default	\(\frac {x}{24 x^{4}+16}+\frac {\sqrt {3}\, 6^{\frac {1}{4}} \sqrt {2}\, \left (\ln \left (\frac {x^{2}+\frac {\sqrt {3}\, 6^{\frac {1}{4}} x \sqrt {2}}{3}+\frac {\sqrt {6}}{3}}{x^{2}-\frac {\sqrt {3}\, 6^{\frac {1}{4}} x \sqrt {2}}{3}+\frac {\sqrt {6}}{3}}\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {3}\, 6^{\frac {3}{4}} x}{6}+1\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {3}\, 6^{\frac {3}{4}} x}{6}-1\right )\right )}{128}\)	\(106\)
meijerg	\(\frac {24^{\frac {3}{4}} \left (\frac {2 \,24^{\frac {1}{4}} x}{6 x^{4}+4}-\frac {3 x \sqrt {2}\, \ln \left (1-6^{\frac {1}{4}} \left (x^{4}\right )^{\frac {1}{4}}+\frac {\sqrt {3}\, \sqrt {2}\, \sqrt {x^{4}}}{2}\right )}{8 \left (x^{4}\right )^{\frac {1}{4}}}+\frac {3 x \sqrt {2}\, \arctan \left (\frac {3^{\frac {1}{4}} 8^{\frac {3}{4}} \left (x^{4}\right )^{\frac {1}{4}}}{8-3^{\frac {1}{4}} 8^{\frac {3}{4}} \left (x^{4}\right )^{\frac {1}{4}}}\right )}{4 \left (x^{4}\right )^{\frac {1}{4}}}+\frac {3 x \sqrt {2}\, \ln \left (1+6^{\frac {1}{4}} \left (x^{4}\right )^{\frac {1}{4}}+\frac {\sqrt {3}\, \sqrt {2}\, \sqrt {x^{4}}}{2}\right )}{8 \left (x^{4}\right )^{\frac {1}{4}}}+\frac {3 x \sqrt {2}\, \arctan \left (\frac {3^{\frac {1}{4}} 8^{\frac {3}{4}} \left (x^{4}\right )^{\frac {1}{4}}}{8+3^{\frac {1}{4}} 8^{\frac {3}{4}} \left (x^{4}\right )^{\frac {1}{4}}}\right )}{4 \left (x^{4}\right )^{\frac {1}{4}}}\right )}{192}\)	\(179\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/8*x/(3*x^4+2)+1/128*3^(1/2)*6^(1/4)*2^(1/2)*(ln((x^2+1/3*3^(1/2)*6^(1/4)*x*2^(1/2)+1/3*6^(1/2))/(x^2-1/3*3^(
1/2)*6^(1/4)*x*2^(1/2)+1/3*6^(1/2)))+2*arctan(1/6*2^(1/2)*3^(1/2)*6^(3/4)*x+1)+2*arctan(1/6*2^(1/2)*3^(1/2)*6^
(3/4)*x-1))

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Maxima [A]
time = 0.50, size = 133, normalized size = 1.02 \begin {gather*} \frac {1}{64} \cdot 3^{\frac {3}{4}} 2^{\frac {3}{4}} \arctan \left (\frac {1}{6} \cdot 3^{\frac {3}{4}} 2^{\frac {1}{4}} {\left (2 \, \sqrt {3} x + 3^{\frac {1}{4}} 2^{\frac {3}{4}}\right )}\right ) + \frac {1}{64} \cdot 3^{\frac {3}{4}} 2^{\frac {3}{4}} \arctan \left (\frac {1}{6} \cdot 3^{\frac {3}{4}} 2^{\frac {1}{4}} {\left (2 \, \sqrt {3} x - 3^{\frac {1}{4}} 2^{\frac {3}{4}}\right )}\right ) + \frac {1}{128} \cdot 3^{\frac {3}{4}} 2^{\frac {3}{4}} \log \left (\sqrt {3} x^{2} + 3^{\frac {1}{4}} 2^{\frac {3}{4}} x + \sqrt {2}\right ) - \frac {1}{128} \cdot 3^{\frac {3}{4}} 2^{\frac {3}{4}} \log \left (\sqrt {3} x^{2} - 3^{\frac {1}{4}} 2^{\frac {3}{4}} x + \sqrt {2}\right ) + \frac {x}{8 \, {\left (3 \, x^{4} + 2\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/64*3^(3/4)*2^(3/4)*arctan(1/6*3^(3/4)*2^(1/4)*(2*sqrt(3)*x + 3^(1/4)*2^(3/4))) + 1/64*3^(3/4)*2^(3/4)*arctan
(1/6*3^(3/4)*2^(1/4)*(2*sqrt(3)*x - 3^(1/4)*2^(3/4))) + 1/128*3^(3/4)*2^(3/4)*log(sqrt(3)*x^2 + 3^(1/4)*2^(3/4
)*x + sqrt(2)) - 1/128*3^(3/4)*2^(3/4)*log(sqrt(3)*x^2 - 3^(1/4)*2^(3/4)*x + sqrt(2)) + 1/8*x/(3*x^4 + 2)

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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 250 vs. \(2 (92) = 184\).
time = 0.38, size = 250, normalized size = 1.91 \begin {gather*} -\frac {4 \cdot 27^{\frac {1}{4}} 8^{\frac {3}{4}} \sqrt {2} {\left (3 \, x^{4} + 2\right )} \arctan \left (\frac {1}{108} \cdot 27^{\frac {3}{4}} 8^{\frac {1}{4}} \sqrt {3} \sqrt {2} \sqrt {27^{\frac {1}{4}} 8^{\frac {3}{4}} \sqrt {2} x + 12 \, x^{2} + 4 \, \sqrt {3} \sqrt {2}} - \frac {1}{18} \cdot 27^{\frac {3}{4}} 8^{\frac {1}{4}} \sqrt {2} x - 1\right ) + 4 \cdot 27^{\frac {1}{4}} 8^{\frac {3}{4}} \sqrt {2} {\left (3 \, x^{4} + 2\right )} \arctan \left (-\frac {1}{18} \cdot 27^{\frac {3}{4}} 8^{\frac {1}{4}} \sqrt {2} x + \frac {1}{216} \cdot 27^{\frac {3}{4}} 8^{\frac {1}{4}} \sqrt {2} \sqrt {-12 \cdot 27^{\frac {1}{4}} 8^{\frac {3}{4}} \sqrt {2} x + 144 \, x^{2} + 48 \, \sqrt {3} \sqrt {2}} + 1\right ) - 27^{\frac {1}{4}} 8^{\frac {3}{4}} \sqrt {2} {\left (3 \, x^{4} + 2\right )} \log \left (12 \cdot 27^{\frac {1}{4}} 8^{\frac {3}{4}} \sqrt {2} x + 144 \, x^{2} + 48 \, \sqrt {3} \sqrt {2}\right ) + 27^{\frac {1}{4}} 8^{\frac {3}{4}} \sqrt {2} {\left (3 \, x^{4} + 2\right )} \log \left (-12 \cdot 27^{\frac {1}{4}} 8^{\frac {3}{4}} \sqrt {2} x + 144 \, x^{2} + 48 \, \sqrt {3} \sqrt {2}\right ) - 64 \, x}{512 \, {\left (3 \, x^{4} + 2\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/512*(4*27^(1/4)*8^(3/4)*sqrt(2)*(3*x^4 + 2)*arctan(1/108*27^(3/4)*8^(1/4)*sqrt(3)*sqrt(2)*sqrt(27^(1/4)*8^(
3/4)*sqrt(2)*x + 12*x^2 + 4*sqrt(3)*sqrt(2)) - 1/18*27^(3/4)*8^(1/4)*sqrt(2)*x - 1) + 4*27^(1/4)*8^(3/4)*sqrt(
2)*(3*x^4 + 2)*arctan(-1/18*27^(3/4)*8^(1/4)*sqrt(2)*x + 1/216*27^(3/4)*8^(1/4)*sqrt(2)*sqrt(-12*27^(1/4)*8^(3
/4)*sqrt(2)*x + 144*x^2 + 48*sqrt(3)*sqrt(2)) + 1) - 27^(1/4)*8^(3/4)*sqrt(2)*(3*x^4 + 2)*log(12*27^(1/4)*8^(3
/4)*sqrt(2)*x + 144*x^2 + 48*sqrt(3)*sqrt(2)) + 27^(1/4)*8^(3/4)*sqrt(2)*(3*x^4 + 2)*log(-12*27^(1/4)*8^(3/4)*
sqrt(2)*x + 144*x^2 + 48*sqrt(3)*sqrt(2)) - 64*x)/(3*x^4 + 2)

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Sympy [A]
time = 0.19, size = 95, normalized size = 0.73 \begin {gather*} \frac {x}{24 x^{4} + 16} - \frac {6^{\frac {3}{4}} \log {\left (x^{2} - \frac {6^{\frac {3}{4}} x}{3} + \frac {\sqrt {6}}{3} \right )}}{128} + \frac {6^{\frac {3}{4}} \log {\left (x^{2} + \frac {6^{\frac {3}{4}} x}{3} + \frac {\sqrt {6}}{3} \right )}}{128} + \frac {6^{\frac {3}{4}} \operatorname {atan}{\left (\sqrt [4]{6} x - 1 \right )}}{64} + \frac {6^{\frac {3}{4}} \operatorname {atan}{\left (\sqrt [4]{6} x + 1 \right )}}{64} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

x/(24*x**4 + 16) - 6**(3/4)*log(x**2 - 6**(3/4)*x/3 + sqrt(6)/3)/128 + 6**(3/4)*log(x**2 + 6**(3/4)*x/3 + sqrt
(6)/3)/128 + 6**(3/4)*atan(6**(1/4)*x - 1)/64 + 6**(3/4)*atan(6**(1/4)*x + 1)/64

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Giac [A]
time = 0.45, size = 107, normalized size = 0.82 \begin {gather*} \frac {1}{64} \cdot 6^{\frac {3}{4}} \arctan \left (\frac {3}{4} \, \sqrt {2} \left (\frac {2}{3}\right )^{\frac {3}{4}} {\left (2 \, x + \sqrt {2} \left (\frac {2}{3}\right )^{\frac {1}{4}}\right )}\right ) + \frac {1}{64} \cdot 6^{\frac {3}{4}} \arctan \left (\frac {3}{4} \, \sqrt {2} \left (\frac {2}{3}\right )^{\frac {3}{4}} {\left (2 \, x - \sqrt {2} \left (\frac {2}{3}\right )^{\frac {1}{4}}\right )}\right ) + \frac {1}{128} \cdot 6^{\frac {3}{4}} \log \left (x^{2} + \sqrt {2} \left (\frac {2}{3}\right )^{\frac {1}{4}} x + \sqrt {\frac {2}{3}}\right ) - \frac {1}{128} \cdot 6^{\frac {3}{4}} \log \left (x^{2} - \sqrt {2} \left (\frac {2}{3}\right )^{\frac {1}{4}} x + \sqrt {\frac {2}{3}}\right ) + \frac {x}{8 \, {\left (3 \, x^{4} + 2\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/64*6^(3/4)*arctan(3/4*sqrt(2)*(2/3)^(3/4)*(2*x + sqrt(2)*(2/3)^(1/4))) + 1/64*6^(3/4)*arctan(3/4*sqrt(2)*(2/
3)^(3/4)*(2*x - sqrt(2)*(2/3)^(1/4))) + 1/128*6^(3/4)*log(x^2 + sqrt(2)*(2/3)^(1/4)*x + sqrt(2/3)) - 1/128*6^(
3/4)*log(x^2 - sqrt(2)*(2/3)^(1/4)*x + sqrt(2/3)) + 1/8*x/(3*x^4 + 2)

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Mupad [B]
time = 1.06, size = 44, normalized size = 0.34 \begin {gather*} \frac {x}{24\,\left (x^4+\frac {2}{3}\right )}+6^{3/4}\,\mathrm {atan}\left (6^{1/4}\,x\,\left (\frac {1}{2}-\frac {1}{2}{}\mathrm {i}\right )\right )\,\left (\frac {1}{64}+\frac {1}{64}{}\mathrm {i}\right )+6^{3/4}\,\mathrm {atan}\left (6^{1/4}\,x\,\left (\frac {1}{2}+\frac {1}{2}{}\mathrm {i}\right )\right )\,\left (\frac {1}{64}-\frac {1}{64}{}\mathrm {i}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(3*x^4 + 2)^2,x)

[Out]

6^(3/4)*atan(6^(1/4)*x*(1/2 - 1i/2))*(1/64 + 1i/64) + 6^(3/4)*atan(6^(1/4)*x*(1/2 + 1i/2))*(1/64 - 1i/64) + x/
(24*(x^4 + 2/3))

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