3.152 \(\int \frac {a}{2+3 x^4} \, dx\)

Optimal. Leaf size=101 \[ -\frac {a \log \left (3 x^2-6^{3/4} x+\sqrt {6}\right )}{8 \sqrt [4]{6}}+\frac {a \log \left (3 x^2+6^{3/4} x+\sqrt {6}\right )}{8 \sqrt [4]{6}}-\frac {a \tan ^{-1}\left (1-\sqrt [4]{6} x\right )}{4 \sqrt [4]{6}}+\frac {a \tan ^{-1}\left (\sqrt [4]{6} x+1\right )}{4 \sqrt [4]{6}} \]

[Out]

1/24*a*arctan(-1+6^(1/4)*x)*6^(3/4)+1/24*a*arctan(1+6^(1/4)*x)*6^(3/4)-1/48*a*ln(-6^(3/4)*x+3*x^2+6^(1/2))*6^(
3/4)+1/48*a*ln(6^(3/4)*x+3*x^2+6^(1/2))*6^(3/4)

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Rubi [A]  time = 0.10, antiderivative size = 101, normalized size of antiderivative = 1.00, number of steps used = 10, number of rules used = 7, integrand size = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.636, Rules used = {12, 211, 1165, 628, 1162, 617, 204} \[ -\frac {a \log \left (3 x^2-6^{3/4} x+\sqrt {6}\right )}{8 \sqrt [4]{6}}+\frac {a \log \left (3 x^2+6^{3/4} x+\sqrt {6}\right )}{8 \sqrt [4]{6}}-\frac {a \tan ^{-1}\left (1-\sqrt [4]{6} x\right )}{4 \sqrt [4]{6}}+\frac {a \tan ^{-1}\left (\sqrt [4]{6} x+1\right )}{4 \sqrt [4]{6}} \]

Antiderivative was successfully verified.

[In]

Int[a/(2 + 3*x^4),x]

[Out]

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

Rule 12

Int[(a_)*(u_), x_Symbol] :> Dist[a, Int[u, x], x] /; FreeQ[a, x] &&  !MatchQ[u, (b_)*(v_) /; FreeQ[b, x]]

Rule 204

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

Rule 211

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 617

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 628

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 1162

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 1165

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

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Mathematica [A]  time = 0.03, size = 78, normalized size = 0.77 \[ \frac {a \left (-\log \left (\sqrt {6} x^2-2 \sqrt [4]{6} x+2\right )+\log \left (\sqrt {6} x^2+2 \sqrt [4]{6} x+2\right )-2 \tan ^{-1}\left (1-\sqrt [4]{6} x\right )+2 \tan ^{-1}\left (\sqrt [4]{6} x+1\right )\right )}{8 \sqrt [4]{6}} \]

Antiderivative was successfully verified.

[In]

Integrate[a/(2 + 3*x^4),x]

[Out]

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

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fricas [B]  time = 0.82, size = 284, normalized size = 2.81 \[ -\frac {1}{48} \cdot 24^{\frac {3}{4}} \sqrt {2} {\left (a^{4}\right )}^{\frac {1}{4}} \arctan \left (-\frac {4 \, a^{3} + 2 \cdot 24^{\frac {1}{4}} \sqrt {2} {\left (a^{4}\right )}^{\frac {3}{4}} x - 24^{\frac {1}{4}} \sqrt {2} \sqrt {\frac {1}{3}} {\left (a^{4}\right )}^{\frac {3}{4}} \sqrt {\frac {12 \, a^{2} x^{2} + 24^{\frac {3}{4}} \sqrt {2} {\left (a^{4}\right )}^{\frac {1}{4}} a x + 4 \, \sqrt {6} \sqrt {a^{4}}}{a^{2}}}}{4 \, a^{3}}\right ) - \frac {1}{48} \cdot 24^{\frac {3}{4}} \sqrt {2} {\left (a^{4}\right )}^{\frac {1}{4}} \arctan \left (\frac {4 \, a^{3} - 2 \cdot 24^{\frac {1}{4}} \sqrt {2} {\left (a^{4}\right )}^{\frac {3}{4}} x + 24^{\frac {1}{4}} \sqrt {2} \sqrt {\frac {1}{3}} {\left (a^{4}\right )}^{\frac {3}{4}} \sqrt {\frac {12 \, a^{2} x^{2} - 24^{\frac {3}{4}} \sqrt {2} {\left (a^{4}\right )}^{\frac {1}{4}} a x + 4 \, \sqrt {6} \sqrt {a^{4}}}{a^{2}}}}{4 \, a^{3}}\right ) + \frac {1}{192} \cdot 24^{\frac {3}{4}} \sqrt {2} {\left (a^{4}\right )}^{\frac {1}{4}} \log \left (12 \, a^{2} x^{2} + 24^{\frac {3}{4}} \sqrt {2} {\left (a^{4}\right )}^{\frac {1}{4}} a x + 4 \, \sqrt {6} \sqrt {a^{4}}\right ) - \frac {1}{192} \cdot 24^{\frac {3}{4}} \sqrt {2} {\left (a^{4}\right )}^{\frac {1}{4}} \log \left (12 \, a^{2} x^{2} - 24^{\frac {3}{4}} \sqrt {2} {\left (a^{4}\right )}^{\frac {1}{4}} a x + 4 \, \sqrt {6} \sqrt {a^{4}}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/48*24^(3/4)*sqrt(2)*(a^4)^(1/4)*arctan(-1/4*(4*a^3 + 2*24^(1/4)*sqrt(2)*(a^4)^(3/4)*x - 24^(1/4)*sqrt(2)*sq
rt(1/3)*(a^4)^(3/4)*sqrt((12*a^2*x^2 + 24^(3/4)*sqrt(2)*(a^4)^(1/4)*a*x + 4*sqrt(6)*sqrt(a^4))/a^2))/a^3) - 1/
48*24^(3/4)*sqrt(2)*(a^4)^(1/4)*arctan(1/4*(4*a^3 - 2*24^(1/4)*sqrt(2)*(a^4)^(3/4)*x + 24^(1/4)*sqrt(2)*sqrt(1
/3)*(a^4)^(3/4)*sqrt((12*a^2*x^2 - 24^(3/4)*sqrt(2)*(a^4)^(1/4)*a*x + 4*sqrt(6)*sqrt(a^4))/a^2))/a^3) + 1/192*
24^(3/4)*sqrt(2)*(a^4)^(1/4)*log(12*a^2*x^2 + 24^(3/4)*sqrt(2)*(a^4)^(1/4)*a*x + 4*sqrt(6)*sqrt(a^4)) - 1/192*
24^(3/4)*sqrt(2)*(a^4)^(1/4)*log(12*a^2*x^2 - 24^(3/4)*sqrt(2)*(a^4)^(1/4)*a*x + 4*sqrt(6)*sqrt(a^4))

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giac [A]  time = 0.20, size = 97, normalized size = 0.96 \[ \frac {1}{48} \, {\left (2 \cdot 6^{\frac {3}{4}} \arctan \left (\frac {3}{4} \, \sqrt {2} \left (\frac {2}{3}\right )^{\frac {3}{4}} {\left (2 \, x + \sqrt {2} \left (\frac {2}{3}\right )^{\frac {1}{4}}\right )}\right ) + 2 \cdot 6^{\frac {3}{4}} \arctan \left (\frac {3}{4} \, \sqrt {2} \left (\frac {2}{3}\right )^{\frac {3}{4}} {\left (2 \, x - \sqrt {2} \left (\frac {2}{3}\right )^{\frac {1}{4}}\right )}\right ) + 6^{\frac {3}{4}} \log \left (x^{2} + \sqrt {2} \left (\frac {2}{3}\right )^{\frac {1}{4}} x + \sqrt {\frac {2}{3}}\right ) - 6^{\frac {3}{4}} \log \left (x^{2} - \sqrt {2} \left (\frac {2}{3}\right )^{\frac {1}{4}} x + \sqrt {\frac {2}{3}}\right )\right )} a \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/48*(2*6^(3/4)*arctan(3/4*sqrt(2)*(2/3)^(3/4)*(2*x + sqrt(2)*(2/3)^(1/4))) + 2*6^(3/4)*arctan(3/4*sqrt(2)*(2/
3)^(3/4)*(2*x - sqrt(2)*(2/3)^(1/4))) + 6^(3/4)*log(x^2 + sqrt(2)*(2/3)^(1/4)*x + sqrt(2/3)) - 6^(3/4)*log(x^2
- sqrt(2)*(2/3)^(1/4)*x + sqrt(2/3)))*a

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maple [A]  time = 0.04, size = 114, normalized size = 1.13 \[ \frac {\sqrt {3}\, 6^{\frac {1}{4}} \sqrt {2}\, a \arctan \left (\frac {\sqrt {2}\, \sqrt {3}\, 6^{\frac {3}{4}} x}{6}-1\right )}{24}+\frac {\sqrt {3}\, 6^{\frac {1}{4}} \sqrt {2}\, a \arctan \left (\frac {\sqrt {2}\, \sqrt {3}\, 6^{\frac {3}{4}} x}{6}+1\right )}{24}+\frac {\sqrt {3}\, 6^{\frac {1}{4}} \sqrt {2}\, a \ln \left (\frac {x^{2}+\frac {\sqrt {3}\, 6^{\frac {1}{4}} \sqrt {2}\, x}{3}+\frac {\sqrt {6}}{3}}{x^{2}-\frac {\sqrt {3}\, 6^{\frac {1}{4}} \sqrt {2}\, x}{3}+\frac {\sqrt {6}}{3}}\right )}{48} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/24*a*3^(1/2)*6^(1/4)*2^(1/2)*arctan(1/6*2^(1/2)*3^(1/2)*6^(3/4)*x+1)+1/24*a*3^(1/2)*6^(1/4)*2^(1/2)*arctan(1
/6*2^(1/2)*3^(1/2)*6^(3/4)*x-1)+1/48*a*3^(1/2)*6^(1/4)*2^(1/2)*ln((x^2+1/3*3^(1/2)*6^(1/4)*2^(1/2)*x+1/3*6^(1/
2))/(x^2-1/3*3^(1/2)*6^(1/4)*2^(1/2)*x+1/3*6^(1/2)))

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maxima [A]  time = 2.94, size = 123, normalized size = 1.22 \[ \frac {1}{48} \, {\left (2 \cdot 3^{\frac {3}{4}} 2^{\frac {3}{4}} \arctan \left (\frac {1}{6} \cdot 3^{\frac {3}{4}} 2^{\frac {1}{4}} {\left (2 \, \sqrt {3} x + 3^{\frac {1}{4}} 2^{\frac {3}{4}}\right )}\right ) + 2 \cdot 3^{\frac {3}{4}} 2^{\frac {3}{4}} \arctan \left (\frac {1}{6} \cdot 3^{\frac {3}{4}} 2^{\frac {1}{4}} {\left (2 \, \sqrt {3} x - 3^{\frac {1}{4}} 2^{\frac {3}{4}}\right )}\right ) + 3^{\frac {3}{4}} 2^{\frac {3}{4}} \log \left (\sqrt {3} x^{2} + 3^{\frac {1}{4}} 2^{\frac {3}{4}} x + \sqrt {2}\right ) - 3^{\frac {3}{4}} 2^{\frac {3}{4}} \log \left (\sqrt {3} x^{2} - 3^{\frac {1}{4}} 2^{\frac {3}{4}} x + \sqrt {2}\right )\right )} a \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/48*(2*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))) + 2*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))) + 3^(3/4)*2^(3/4)*log(sqrt(3)*x^2 + 3^(1/4)*2^(3/4)*x +
sqrt(2)) - 3^(3/4)*2^(3/4)*log(sqrt(3)*x^2 - 3^(1/4)*2^(3/4)*x + sqrt(2)))*a

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mupad [B]  time = 0.12, size = 36, normalized size = 0.36 \[ -\frac {{\left (-1\right )}^{1/4}\,{6144}^{3/4}\,a\,\left (\mathrm {atan}\left (\frac {{\left (-1\right )}^{1/4}\,{6144}^{1/4}\,x}{8}\right )\,1{}\mathrm {i}+\mathrm {atanh}\left (\frac {{\left (-1\right )}^{1/4}\,{6144}^{1/4}\,x}{8}\right )\,1{}\mathrm {i}\right )}{3072} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(a/(3*x^4 + 2),x)

[Out]

-((-1)^(1/4)*6144^(3/4)*a*(atan(((-1)^(1/4)*6144^(1/4)*x)/8)*1i + atanh(((-1)^(1/4)*6144^(1/4)*x)/8)*1i))/3072

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sympy [A]  time = 0.44, size = 88, normalized size = 0.87 \[ a \left (- \frac {6^{\frac {3}{4}} \log {\left (x^{2} - \frac {6^{\frac {3}{4}} x}{3} + \frac {\sqrt {6}}{3} \right )}}{48} + \frac {6^{\frac {3}{4}} \log {\left (x^{2} + \frac {6^{\frac {3}{4}} x}{3} + \frac {\sqrt {6}}{3} \right )}}{48} + \frac {6^{\frac {3}{4}} \operatorname {atan}{\left (\sqrt [4]{6} x - 1 \right )}}{24} + \frac {6^{\frac {3}{4}} \operatorname {atan}{\left (\sqrt [4]{6} x + 1 \right )}}{24}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

a*(-6**(3/4)*log(x**2 - 6**(3/4)*x/3 + sqrt(6)/3)/48 + 6**(3/4)*log(x**2 + 6**(3/4)*x/3 + sqrt(6)/3)/48 + 6**(
3/4)*atan(6**(1/4)*x - 1)/24 + 6**(3/4)*atan(6**(1/4)*x + 1)/24)

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