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]

-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))

________________________________________________________________________________________

Rubi [A]  time = 0.108253, antiderivative size = 171, normalized size of antiderivative = 1., 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 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 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]

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 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 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])

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 = 0.0481946, size = 120, normalized size = 0.7 \[ \frac{-\log \left (\sqrt{21} x^2-\sqrt{2} \sqrt [4]{3} 7^{3/4} x+7\right )+\log \left (\sqrt{21} x^2+\sqrt{2} \sqrt [4]{3} 7^{3/4} x+7\right )-2 \tan ^{-1}\left (1-\sqrt [4]{\frac{3}{7}} \sqrt{2} x\right )+2 \tan ^{-1}\left (\sqrt [4]{\frac{3}{7}} \sqrt{2} x+1\right )}{4 \sqrt{2} \sqrt [4]{3} 7^{3/4}} \]

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))

________________________________________________________________________________________

Maple [A]  time = 0.005, size = 111, normalized size = 0.7 \begin{align*}{\frac{\sqrt{3}\sqrt [4]{21}\sqrt{2}}{84}\arctan \left ({\frac{\sqrt{2}\sqrt{3}{21}^{{\frac{3}{4}}}x}{21}}+1 \right ) }+{\frac{\sqrt{3}\sqrt [4]{21}\sqrt{2}}{84}\arctan \left ({\frac{\sqrt{2}\sqrt{3}{21}^{{\frac{3}{4}}}x}{21}}-1 \right ) }+{\frac{\sqrt{3}\sqrt [4]{21}\sqrt{2}}{168}\ln \left ({ \left ({x}^{2}+{\frac{\sqrt{3}\sqrt [4]{21}x\sqrt{2}}{3}}+{\frac{\sqrt{21}}{3}} \right ) \left ({x}^{2}-{\frac{\sqrt{3}\sqrt [4]{21}x\sqrt{2}}{3}}+{\frac{\sqrt{21}}{3}} \right ) ^{-1}} \right ) } \end{align*}

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)+1/84*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)))

________________________________________________________________________________________

Maxima [A]  time = 1.46404, size = 204, normalized size = 1.19 \begin{align*} \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 ) \end{align*}

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))

________________________________________________________________________________________

Fricas [A]  time = 1.98422, size = 621, normalized size = 3.63 \begin{align*} -\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 ) \end{align*}

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))

________________________________________________________________________________________

Sympy [A]  time = 0.397105, size = 151, normalized size = 0.88 \begin{align*} - \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} \end{align*}

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

________________________________________________________________________________________

Giac [A]  time = 1.11397, size = 128, normalized size = 0.75 \begin{align*} \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 ) \end{align*}

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))