3.38 \(\int \frac {1}{7+3 x^4} \, dx\)

Optimal. Leaf size=171 \[ -\frac {\log \left (3 x^2-\sqrt {2} 3^{3/4} \sqrt [4]{7} x+\sqrt {21}\right )}{4 \sqrt {2} \sqrt [4]{3} 7^{3/4}}+\frac {\log \left (3 x^2+\sqrt {2} 3^{3/4} \sqrt [4]{7} x+\sqrt {21}\right )}{4 \sqrt {2} \sqrt [4]{3} 7^{3/4}}-\frac {\tan ^{-1}\left (1-\sqrt [4]{\frac {3}{7}} \sqrt {2} x\right )}{2 \sqrt {2} \sqrt [4]{3} 7^{3/4}}+\frac {\tan ^{-1}\left (\sqrt [4]{\frac {3}{7}} \sqrt {2} x+1\right )}{2 \sqrt {2} \sqrt [4]{3} 7^{3/4}} \]

[Out]

1/84*arctan(-1+1/7*3^(1/4)*7^(3/4)*x*2^(1/2))*3^(3/4)*7^(1/4)*2^(1/2)+1/84*arctan(1+1/7*3^(1/4)*7^(3/4)*x*2^(1
/2))*3^(3/4)*7^(1/4)*2^(1/2)-1/168*ln(3*x^2-3^(3/4)*7^(1/4)*x*2^(1/2)+21^(1/2))*3^(3/4)*7^(1/4)*2^(1/2)+1/168*
ln(3*x^2+3^(3/4)*7^(1/4)*x*2^(1/2)+21^(1/2))*3^(3/4)*7^(1/4)*2^(1/2)

________________________________________________________________________________________

Rubi [A]  time = 0.11, antiderivative size = 171, normalized size of antiderivative = 1.00, number of steps used = 9, number of rules used = 6, integrand size = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.667, Rules used = {211, 1165, 628, 1162, 617, 204} \[ -\frac {\log \left (3 x^2-\sqrt {2} 3^{3/4} \sqrt [4]{7} x+\sqrt {21}\right )}{4 \sqrt {2} \sqrt [4]{3} 7^{3/4}}+\frac {\log \left (3 x^2+\sqrt {2} 3^{3/4} \sqrt [4]{7} x+\sqrt {21}\right )}{4 \sqrt {2} \sqrt [4]{3} 7^{3/4}}-\frac {\tan ^{-1}\left (1-\sqrt [4]{\frac {3}{7}} \sqrt {2} x\right )}{2 \sqrt {2} \sqrt [4]{3} 7^{3/4}}+\frac {\tan ^{-1}\left (\sqrt [4]{\frac {3}{7}} \sqrt {2} x+1\right )}{2 \sqrt {2} \sqrt [4]{3} 7^{3/4}} \]

Antiderivative was successfully verified.

[In]

Int[(7 + 3*x^4)^(-1),x]

[Out]

-ArcTan[1 - (3/7)^(1/4)*Sqrt[2]*x]/(2*Sqrt[2]*3^(1/4)*7^(3/4)) + ArcTan[1 + (3/7)^(1/4)*Sqrt[2]*x]/(2*Sqrt[2]*
3^(1/4)*7^(3/4)) - Log[Sqrt[21] - Sqrt[2]*3^(3/4)*7^(1/4)*x + 3*x^2]/(4*Sqrt[2]*3^(1/4)*7^(3/4)) + Log[Sqrt[21
] + Sqrt[2]*3^(3/4)*7^(1/4)*x + 3*x^2]/(4*Sqrt[2]*3^(1/4)*7^(3/4))

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 {1}{7+3 x^4} \, dx &=\frac {\int \frac {\sqrt {7}-\sqrt {3} x^2}{7+3 x^4} \, dx}{2 \sqrt {7}}+\frac {\int \frac {\sqrt {7}+\sqrt {3} x^2}{7+3 x^4} \, dx}{2 \sqrt {7}}\\ &=-\frac {\int \frac {\sqrt {2} \sqrt [4]{\frac {7}{3}}+2 x}{-\sqrt {\frac {7}{3}}-\sqrt {2} \sqrt [4]{\frac {7}{3}} x-x^2} \, dx}{4 \sqrt {2} \sqrt [4]{3} 7^{3/4}}-\frac {\int \frac {\sqrt {2} \sqrt [4]{\frac {7}{3}}-2 x}{-\sqrt {\frac {7}{3}}+\sqrt {2} \sqrt [4]{\frac {7}{3}} x-x^2} \, dx}{4 \sqrt {2} \sqrt [4]{3} 7^{3/4}}+\frac {\int \frac {1}{\sqrt {\frac {7}{3}}-\sqrt {2} \sqrt [4]{\frac {7}{3}} x+x^2} \, dx}{4 \sqrt {21}}+\frac {\int \frac {1}{\sqrt {\frac {7}{3}}+\sqrt {2} \sqrt [4]{\frac {7}{3}} x+x^2} \, dx}{4 \sqrt {21}}\\ &=-\frac {\log \left (\sqrt {21}-\sqrt {2} 3^{3/4} \sqrt [4]{7} x+3 x^2\right )}{4 \sqrt {2} \sqrt [4]{3} 7^{3/4}}+\frac {\log \left (\sqrt {21}+\sqrt {2} 3^{3/4} \sqrt [4]{7} x+3 x^2\right )}{4 \sqrt {2} \sqrt [4]{3} 7^{3/4}}+\frac {\operatorname {Subst}\left (\int \frac {1}{-1-x^2} \, dx,x,1-\sqrt [4]{\frac {3}{7}} \sqrt {2} x\right )}{2 \sqrt {2} \sqrt [4]{3} 7^{3/4}}-\frac {\operatorname {Subst}\left (\int \frac {1}{-1-x^2} \, dx,x,1+\sqrt [4]{\frac {3}{7}} \sqrt {2} x\right )}{2 \sqrt {2} \sqrt [4]{3} 7^{3/4}}\\ &=-\frac {\tan ^{-1}\left (1-\sqrt [4]{\frac {3}{7}} \sqrt {2} x\right )}{2 \sqrt {2} \sqrt [4]{3} 7^{3/4}}+\frac {\tan ^{-1}\left (1+\sqrt [4]{\frac {3}{7}} \sqrt {2} x\right )}{2 \sqrt {2} \sqrt [4]{3} 7^{3/4}}-\frac {\log \left (\sqrt {21}-\sqrt {2} 3^{3/4} \sqrt [4]{7} x+3 x^2\right )}{4 \sqrt {2} \sqrt [4]{3} 7^{3/4}}+\frac {\log \left (\sqrt {21}+\sqrt {2} 3^{3/4} \sqrt [4]{7} x+3 x^2\right )}{4 \sqrt {2} \sqrt [4]{3} 7^{3/4}}\\ \end {align*}

________________________________________________________________________________________

Mathematica [A]  time = 0.05, size = 120, normalized size = 0.70 \[ \frac {-\log \left (\sqrt {21} x^2-\sqrt {2} \sqrt [4]{3} 7^{3/4} x+7\right )+\log \left (\sqrt {21} x^2+\sqrt {2} \sqrt [4]{3} 7^{3/4} x+7\right )-2 \tan ^{-1}\left (1-\sqrt [4]{\frac {3}{7}} \sqrt {2} x\right )+2 \tan ^{-1}\left (\sqrt [4]{\frac {3}{7}} \sqrt {2} x+1\right )}{4 \sqrt {2} \sqrt [4]{3} 7^{3/4}} \]

Antiderivative was successfully verified.

[In]

Integrate[(7 + 3*x^4)^(-1),x]

[Out]

(-2*ArcTan[1 - (3/7)^(1/4)*Sqrt[2]*x] + 2*ArcTan[1 + (3/7)^(1/4)*Sqrt[2]*x] - Log[7 - Sqrt[2]*3^(1/4)*7^(3/4)*
x + Sqrt[21]*x^2] + Log[7 + Sqrt[2]*3^(1/4)*7^(3/4)*x + Sqrt[21]*x^2])/(4*Sqrt[2]*3^(1/4)*7^(3/4))

________________________________________________________________________________________

fricas [A]  time = 0.43, size = 163, normalized size = 0.95 \[ -\frac {1}{2058} \cdot 1029^{\frac {3}{4}} \sqrt {2} \arctan \left (\frac {1}{147} \cdot 1029^{\frac {1}{4}} \sqrt {3} \sqrt {2} \sqrt {1029^{\frac {3}{4}} \sqrt {2} x + 147 \, x^{2} + 49 \, \sqrt {21}} - \frac {1}{7} \cdot 1029^{\frac {1}{4}} \sqrt {2} x - 1\right ) - \frac {1}{2058} \cdot 1029^{\frac {3}{4}} \sqrt {2} \arctan \left (\frac {1}{147} \cdot 1029^{\frac {1}{4}} \sqrt {3} \sqrt {2} \sqrt {-1029^{\frac {3}{4}} \sqrt {2} x + 147 \, x^{2} + 49 \, \sqrt {21}} - \frac {1}{7} \cdot 1029^{\frac {1}{4}} \sqrt {2} x + 1\right ) + \frac {1}{8232} \cdot 1029^{\frac {3}{4}} \sqrt {2} \log \left (1029^{\frac {3}{4}} \sqrt {2} x + 147 \, x^{2} + 49 \, \sqrt {21}\right ) - \frac {1}{8232} \cdot 1029^{\frac {3}{4}} \sqrt {2} \log \left (-1029^{\frac {3}{4}} \sqrt {2} x + 147 \, x^{2} + 49 \, \sqrt {21}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(3*x^4+7),x, algorithm="fricas")

[Out]

-1/2058*1029^(3/4)*sqrt(2)*arctan(1/147*1029^(1/4)*sqrt(3)*sqrt(2)*sqrt(1029^(3/4)*sqrt(2)*x + 147*x^2 + 49*sq
rt(21)) - 1/7*1029^(1/4)*sqrt(2)*x - 1) - 1/2058*1029^(3/4)*sqrt(2)*arctan(1/147*1029^(1/4)*sqrt(3)*sqrt(2)*sq
rt(-1029^(3/4)*sqrt(2)*x + 147*x^2 + 49*sqrt(21)) - 1/7*1029^(1/4)*sqrt(2)*x + 1) + 1/8232*1029^(3/4)*sqrt(2)*
log(1029^(3/4)*sqrt(2)*x + 147*x^2 + 49*sqrt(21)) - 1/8232*1029^(3/4)*sqrt(2)*log(-1029^(3/4)*sqrt(2)*x + 147*
x^2 + 49*sqrt(21))

________________________________________________________________________________________

giac [A]  time = 1.17, size = 95, normalized size = 0.56 \[ \frac {1}{84} \cdot 756^{\frac {1}{4}} \arctan \left (\frac {3}{14} \, \left (\frac {7}{3}\right )^{\frac {3}{4}} \sqrt {2} {\left (2 \, x + \left (\frac {7}{3}\right )^{\frac {1}{4}} \sqrt {2}\right )}\right ) + \frac {1}{84} \cdot 756^{\frac {1}{4}} \arctan \left (\frac {3}{14} \, \left (\frac {7}{3}\right )^{\frac {3}{4}} \sqrt {2} {\left (2 \, x - \left (\frac {7}{3}\right )^{\frac {1}{4}} \sqrt {2}\right )}\right ) + \frac {1}{168} \cdot 756^{\frac {1}{4}} \log \left (x^{2} + \left (\frac {7}{3}\right )^{\frac {1}{4}} \sqrt {2} x + \sqrt {\frac {7}{3}}\right ) - \frac {1}{168} \cdot 756^{\frac {1}{4}} \log \left (x^{2} - \left (\frac {7}{3}\right )^{\frac {1}{4}} \sqrt {2} x + \sqrt {\frac {7}{3}}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(3*x^4+7),x, algorithm="giac")

[Out]

1/84*756^(1/4)*arctan(3/14*(7/3)^(3/4)*sqrt(2)*(2*x + (7/3)^(1/4)*sqrt(2))) + 1/84*756^(1/4)*arctan(3/14*(7/3)
^(3/4)*sqrt(2)*(2*x - (7/3)^(1/4)*sqrt(2))) + 1/168*756^(1/4)*log(x^2 + (7/3)^(1/4)*sqrt(2)*x + sqrt(7/3)) - 1
/168*756^(1/4)*log(x^2 - (7/3)^(1/4)*sqrt(2)*x + sqrt(7/3))

________________________________________________________________________________________

maple [A]  time = 0.01, size = 111, normalized size = 0.65 \[ \frac {\sqrt {3}\, 21^{\frac {1}{4}} \sqrt {2}\, \arctan \left (\frac {\sqrt {2}\, \sqrt {3}\, 21^{\frac {3}{4}} x}{21}-1\right )}{84}+\frac {\sqrt {3}\, 21^{\frac {1}{4}} \sqrt {2}\, \arctan \left (\frac {\sqrt {2}\, \sqrt {3}\, 21^{\frac {3}{4}} x}{21}+1\right )}{84}+\frac {\sqrt {3}\, 21^{\frac {1}{4}} \sqrt {2}\, \ln \left (\frac {x^{2}+\frac {\sqrt {3}\, 21^{\frac {1}{4}} \sqrt {2}\, x}{3}+\frac {\sqrt {21}}{3}}{x^{2}-\frac {\sqrt {3}\, 21^{\frac {1}{4}} \sqrt {2}\, x}{3}+\frac {\sqrt {21}}{3}}\right )}{168} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(3*x^4+7),x)

[Out]

1/84*3^(1/2)*21^(1/4)*2^(1/2)*arctan(1/21*2^(1/2)*3^(1/2)*21^(3/4)*x-1)+1/168*3^(1/2)*21^(1/4)*2^(1/2)*ln((x^2
+1/3*3^(1/2)*21^(1/4)*x*2^(1/2)+1/3*21^(1/2))/(x^2-1/3*3^(1/2)*21^(1/4)*x*2^(1/2)+1/3*21^(1/2)))+1/84*3^(1/2)*
21^(1/4)*2^(1/2)*arctan(1/21*2^(1/2)*3^(1/2)*21^(3/4)*x+1)

________________________________________________________________________________________

maxima [A]  time = 0.97, size = 151, normalized size = 0.88 \[ \frac {1}{84} \cdot 7^{\frac {1}{4}} 3^{\frac {3}{4}} \sqrt {2} \arctan \left (\frac {1}{42} \cdot 7^{\frac {3}{4}} 3^{\frac {3}{4}} \sqrt {2} {\left (2 \, \sqrt {3} x + 7^{\frac {1}{4}} 3^{\frac {1}{4}} \sqrt {2}\right )}\right ) + \frac {1}{84} \cdot 7^{\frac {1}{4}} 3^{\frac {3}{4}} \sqrt {2} \arctan \left (\frac {1}{42} \cdot 7^{\frac {3}{4}} 3^{\frac {3}{4}} \sqrt {2} {\left (2 \, \sqrt {3} x - 7^{\frac {1}{4}} 3^{\frac {1}{4}} \sqrt {2}\right )}\right ) + \frac {1}{168} \cdot 7^{\frac {1}{4}} 3^{\frac {3}{4}} \sqrt {2} \log \left (\sqrt {3} x^{2} + 7^{\frac {1}{4}} 3^{\frac {1}{4}} \sqrt {2} x + \sqrt {7}\right ) - \frac {1}{168} \cdot 7^{\frac {1}{4}} 3^{\frac {3}{4}} \sqrt {2} \log \left (\sqrt {3} x^{2} - 7^{\frac {1}{4}} 3^{\frac {1}{4}} \sqrt {2} x + \sqrt {7}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(3*x^4+7),x, algorithm="maxima")

[Out]

1/84*7^(1/4)*3^(3/4)*sqrt(2)*arctan(1/42*7^(3/4)*3^(3/4)*sqrt(2)*(2*sqrt(3)*x + 7^(1/4)*3^(1/4)*sqrt(2))) + 1/
84*7^(1/4)*3^(3/4)*sqrt(2)*arctan(1/42*7^(3/4)*3^(3/4)*sqrt(2)*(2*sqrt(3)*x - 7^(1/4)*3^(1/4)*sqrt(2))) + 1/16
8*7^(1/4)*3^(3/4)*sqrt(2)*log(sqrt(3)*x^2 + 7^(1/4)*3^(1/4)*sqrt(2)*x + sqrt(7)) - 1/168*7^(1/4)*3^(3/4)*sqrt(
2)*log(sqrt(3)*x^2 - 7^(1/4)*3^(1/4)*sqrt(2)*x + sqrt(7))

________________________________________________________________________________________

mupad [B]  time = 0.11, size = 45, normalized size = 0.26 \[ \sqrt {2}\,{189}^{1/4}\,\mathrm {atan}\left (\sqrt {2}\,{189}^{3/4}\,x\,\left (\frac {1}{126}-\frac {1}{126}{}\mathrm {i}\right )\right )\,\left (\frac {1}{84}+\frac {1}{84}{}\mathrm {i}\right )+\sqrt {2}\,{189}^{1/4}\,\mathrm {atan}\left (\sqrt {2}\,{189}^{3/4}\,x\,\left (\frac {1}{126}+\frac {1}{126}{}\mathrm {i}\right )\right )\,\left (\frac {1}{84}-\frac {1}{84}{}\mathrm {i}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(3*x^4 + 7),x)

[Out]

2^(1/2)*189^(1/4)*atan(2^(1/2)*189^(3/4)*x*(1/126 - 1i/126))*(1/84 + 1i/84) + 2^(1/2)*189^(1/4)*atan(2^(1/2)*1
89^(3/4)*x*(1/126 + 1i/126))*(1/84 - 1i/84)

________________________________________________________________________________________

sympy [A]  time = 0.40, size = 151, normalized size = 0.88 \[ - \frac {\sqrt [4]{189} \sqrt {2} \log {\left (x^{2} - \frac {\sqrt [4]{189} \sqrt {2} x}{3} + \frac {\sqrt {21}}{3} \right )}}{168} + \frac {\sqrt [4]{189} \sqrt {2} \log {\left (x^{2} + \frac {\sqrt [4]{189} \sqrt {2} x}{3} + \frac {\sqrt {21}}{3} \right )}}{168} + \frac {\sqrt {2} \cdot 3^{\frac {3}{4}} \sqrt [4]{7} \operatorname {atan}{\left (\frac {\sqrt {2} \sqrt [4]{3} \cdot 7^{\frac {3}{4}} x}{7} - 1 \right )}}{84} + \frac {\sqrt {2} \cdot 3^{\frac {3}{4}} \sqrt [4]{7} \operatorname {atan}{\left (\frac {\sqrt {2} \sqrt [4]{3} \cdot 7^{\frac {3}{4}} x}{7} + 1 \right )}}{84} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(3*x**4+7),x)

[Out]

-189**(1/4)*sqrt(2)*log(x**2 - 189**(1/4)*sqrt(2)*x/3 + sqrt(21)/3)/168 + 189**(1/4)*sqrt(2)*log(x**2 + 189**(
1/4)*sqrt(2)*x/3 + sqrt(21)/3)/168 + sqrt(2)*3**(3/4)*7**(1/4)*atan(sqrt(2)*3**(1/4)*7**(3/4)*x/7 - 1)/84 + sq
rt(2)*3**(3/4)*7**(1/4)*atan(sqrt(2)*3**(1/4)*7**(3/4)*x/7 + 1)/84

________________________________________________________________________________________