\(\int \frac {1}{7+3 x^4} \, dx\) [38]

Optimal result

Integrand size = 9, antiderivative size = 171 \[ \int \frac {1}{7+3 x^4} \, dx=-\frac {\arctan \left (1-\sqrt [4]{\frac {3}{7}} \sqrt {2} x\right )}{2 \sqrt {2} \sqrt [4]{3} 7^{3/4}}+\frac {\arctan \left (1+\sqrt [4]{\frac {3}{7}} \sqrt {2} x\right )}{2 \sqrt {2} \sqrt [4]{3} 7^{3/4}}-\frac {\log \left (\sqrt {21}-\sqrt {2} 3^{3/4} \sqrt [4]{7} x+3 x^2\right )}{4 \sqrt {2} \sqrt [4]{3} 7^{3/4}}+\frac {\log \left (\sqrt {21}+\sqrt {2} 3^{3/4} \sqrt [4]{7} x+3 x^2\right )}{4 \sqrt {2} \sqrt [4]{3} 7^{3/4}} \] Output:

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)
 

Mathematica [A] (verified)

Time = 0.03 (sec) , antiderivative size = 120, normalized size of antiderivative = 0.70 \[ \int \frac {1}{7+3 x^4} \, dx=\frac {-2 \arctan \left (1-\sqrt [4]{\frac {3}{7}} \sqrt {2} x\right )+2 \arctan \left (1+\sqrt [4]{\frac {3}{7}} \sqrt {2} x\right )-\log \left (7-\sqrt {2} \sqrt [4]{3} 7^{3/4} x+\sqrt {21} x^2\right )+\log \left (7+\sqrt {2} \sqrt [4]{3} 7^{3/4} x+\sqrt {21} x^2\right )}{4 \sqrt {2} \sqrt [4]{3} 7^{3/4}} \] Input:

Integrate[(7 + 3*x^4)^(-1),x]
 

Output:

(-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))
 

Rubi [A] (verified)

Time = 0.31 (sec) , antiderivative size = 166, normalized size of antiderivative = 0.97, number of steps used = 8, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.778, Rules used = {755, 1476, 1082, 217, 1479, 25, 1103}

Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.

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

\(\Big \downarrow \) 755

\(\displaystyle \frac {\int \frac {\sqrt {7}-\sqrt {3} x^2}{3 x^4+7}dx}{2 \sqrt {7}}+\frac {\int \frac {\sqrt {3} x^2+\sqrt {7}}{3 x^4+7}dx}{2 \sqrt {7}}\)

\(\Big \downarrow \) 1476

\(\displaystyle \frac {\frac {\int \frac {1}{x^2-\sqrt {2} \sqrt [4]{\frac {7}{3}} x+\sqrt {\frac {7}{3}}}dx}{2 \sqrt {3}}+\frac {\int \frac {1}{x^2+\sqrt {2} \sqrt [4]{\frac {7}{3}} x+\sqrt {\frac {7}{3}}}dx}{2 \sqrt {3}}}{2 \sqrt {7}}+\frac {\int \frac {\sqrt {7}-\sqrt {3} x^2}{3 x^4+7}dx}{2 \sqrt {7}}\)

\(\Big \downarrow \) 1082

\(\displaystyle \frac {\int \frac {\sqrt {7}-\sqrt {3} x^2}{3 x^4+7}dx}{2 \sqrt {7}}+\frac {\frac {\int \frac {1}{-\left (1-\sqrt [4]{\frac {3}{7}} \sqrt {2} x\right )^2-1}d\left (1-\sqrt [4]{\frac {3}{7}} \sqrt {2} x\right )}{\sqrt {2} \sqrt [4]{21}}-\frac {\int \frac {1}{-\left (\sqrt [4]{\frac {3}{7}} \sqrt {2} x+1\right )^2-1}d\left (\sqrt [4]{\frac {3}{7}} \sqrt {2} x+1\right )}{\sqrt {2} \sqrt [4]{21}}}{2 \sqrt {7}}\)

\(\Big \downarrow \) 217

\(\displaystyle \frac {\int \frac {\sqrt {7}-\sqrt {3} x^2}{3 x^4+7}dx}{2 \sqrt {7}}+\frac {\frac {\arctan \left (\sqrt [4]{\frac {3}{7}} \sqrt {2} x+1\right )}{\sqrt {2} \sqrt [4]{21}}-\frac {\arctan \left (1-\sqrt [4]{\frac {3}{7}} \sqrt {2} x\right )}{\sqrt {2} \sqrt [4]{21}}}{2 \sqrt {7}}\)

\(\Big \downarrow \) 1479

\(\displaystyle \frac {-\frac {\int -\frac {\sqrt {2} 3^{3/4} \sqrt [4]{7}-6 x}{3 x^2-\sqrt {2} 3^{3/4} \sqrt [4]{7} x+\sqrt {21}}dx}{2 \sqrt {2} \sqrt [4]{21}}-\frac {\int -\frac {6 x+\sqrt {2} 3^{3/4} \sqrt [4]{7}}{3 x^2+\sqrt {2} 3^{3/4} \sqrt [4]{7} x+\sqrt {21}}dx}{2 \sqrt {2} \sqrt [4]{21}}}{2 \sqrt {7}}+\frac {\frac {\arctan \left (\sqrt [4]{\frac {3}{7}} \sqrt {2} x+1\right )}{\sqrt {2} \sqrt [4]{21}}-\frac {\arctan \left (1-\sqrt [4]{\frac {3}{7}} \sqrt {2} x\right )}{\sqrt {2} \sqrt [4]{21}}}{2 \sqrt {7}}\)

\(\Big \downarrow \) 25

\(\displaystyle \frac {\frac {\int \frac {\sqrt {2} 3^{3/4} \sqrt [4]{7}-6 x}{3 x^2-\sqrt {2} 3^{3/4} \sqrt [4]{7} x+\sqrt {21}}dx}{2 \sqrt {2} \sqrt [4]{21}}+\frac {\int \frac {6 x+\sqrt {2} 3^{3/4} \sqrt [4]{7}}{3 x^2+\sqrt {2} 3^{3/4} \sqrt [4]{7} x+\sqrt {21}}dx}{2 \sqrt {2} \sqrt [4]{21}}}{2 \sqrt {7}}+\frac {\frac {\arctan \left (\sqrt [4]{\frac {3}{7}} \sqrt {2} x+1\right )}{\sqrt {2} \sqrt [4]{21}}-\frac {\arctan \left (1-\sqrt [4]{\frac {3}{7}} \sqrt {2} x\right )}{\sqrt {2} \sqrt [4]{21}}}{2 \sqrt {7}}\)

\(\Big \downarrow \) 1103

\(\displaystyle \frac {\frac {\arctan \left (\sqrt [4]{\frac {3}{7}} \sqrt {2} x+1\right )}{\sqrt {2} \sqrt [4]{21}}-\frac {\arctan \left (1-\sqrt [4]{\frac {3}{7}} \sqrt {2} x\right )}{\sqrt {2} \sqrt [4]{21}}}{2 \sqrt {7}}+\frac {\frac {\log \left (3 x^2+\sqrt {2} 3^{3/4} \sqrt [4]{7} x+\sqrt {21}\right )}{2 \sqrt {2} \sqrt [4]{21}}-\frac {\log \left (3 x^2-\sqrt {2} 3^{3/4} \sqrt [4]{7} x+\sqrt {21}\right )}{2 \sqrt {2} \sqrt [4]{21}}}{2 \sqrt {7}}\)

Input:

Int[(7 + 3*x^4)^(-1),x]
 

Output:

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

Defintions of rubi rules used

rule 25

Int[-(Fx_), x_Symbol] :> Simp[Identity[-1]   Int[Fx, x], x]
 

rule 217

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 755

Int[((a_) + (b_.)*(x_)^4)^(-1), x_Symbol] :> With[{r = Numerator[Rt[a/b, 2] 
], s = Denominator[Rt[a/b, 2]]}, Simp[1/(2*r)   Int[(r - s*x^2)/(a + b*x^4) 
, x], x] + Simp[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 1082

Int[((a_) + (b_.)*(x_) + (c_.)*(x_)^2)^(-1), x_Symbol] :> With[{q = 1 - 4*S 
implify[a*(c/b^2)]}, Simp[-2/b   Subst[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])] /; Fre 
eQ[{a, b, c}, x]
 

rule 1103

Int[((d_) + (e_.)*(x_))/((a_.) + (b_.)*(x_) + (c_.)*(x_)^2), x_Symbol] :> S 
imp[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 1476

Int[((d_) + (e_.)*(x_)^2)/((a_) + (c_.)*(x_)^4), x_Symbol] :> With[{q = Rt[ 
2*(d/e), 2]}, Simp[e/(2*c)   Int[1/Simp[d/e + q*x + x^2, x], x], x] + Simp[ 
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 1479

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

Maple [C] (verified)

Result contains higher order function than in optimal. Order 9 vs. order 3.

Time = 0.07 (sec) , antiderivative size = 24, normalized size of antiderivative = 0.14

method	result	size
risch	\(\frac {\left (\munderset {\textit {\_R} =\operatorname {RootOf}\left (3 \textit {\_Z}^{4}+7\right )}{\sum }\frac {\ln \left (x -\textit {\_R} \right )}{\textit {\_R}^{3}}\right )}{12}\)	\(24\)
default	\(\frac {\sqrt {3}\, 21^{\frac {1}{4}} \sqrt {2}\, \left (\ln \left (\frac {x^{2}+\frac {\sqrt {3}\, 21^{\frac {1}{4}} x \sqrt {2}}{3}+\frac {\sqrt {21}}{3}}{x^{2}-\frac {\sqrt {3}\, 21^{\frac {1}{4}} x \sqrt {2}}{3}+\frac {\sqrt {21}}{3}}\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {3}\, 21^{\frac {3}{4}} x}{21}+1\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {3}\, 21^{\frac {3}{4}} x}{21}-1\right )\right )}{168}\)	\(93\)
meijerg	\(\frac {1029^{\frac {3}{4}} \left (-\frac {x \sqrt {2}\, \ln \left (1-\frac {\sqrt {2}\, 3^{\frac {1}{4}} 7^{\frac {3}{4}} \left (x^{4}\right )^{\frac {1}{4}}}{7}+\frac {\sqrt {3}\, \sqrt {7}\, \sqrt {x^{4}}}{7}\right )}{2 \left (x^{4}\right )^{\frac {1}{4}}}+\frac {x \sqrt {2}\, \arctan \left (\frac {\sqrt {2}\, 3^{\frac {1}{4}} 7^{\frac {3}{4}} \left (x^{4}\right )^{\frac {1}{4}}}{14-\sqrt {2}\, 3^{\frac {1}{4}} 7^{\frac {3}{4}} \left (x^{4}\right )^{\frac {1}{4}}}\right )}{\left (x^{4}\right )^{\frac {1}{4}}}+\frac {x \sqrt {2}\, \ln \left (1+\frac {\sqrt {2}\, 3^{\frac {1}{4}} 7^{\frac {3}{4}} \left (x^{4}\right )^{\frac {1}{4}}}{7}+\frac {\sqrt {3}\, \sqrt {7}\, \sqrt {x^{4}}}{7}\right )}{2 \left (x^{4}\right )^{\frac {1}{4}}}+\frac {x \sqrt {2}\, \arctan \left (\frac {\sqrt {2}\, 3^{\frac {1}{4}} 7^{\frac {3}{4}} \left (x^{4}\right )^{\frac {1}{4}}}{14+\sqrt {2}\, 3^{\frac {1}{4}} 7^{\frac {3}{4}} \left (x^{4}\right )^{\frac {1}{4}}}\right )}{\left (x^{4}\right )^{\frac {1}{4}}}\right )}{4116}\)	\(187\)

Input:

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

Output:

1/12*sum(1/_R^3*ln(x-_R),_R=RootOf(3*_Z^4+7))
 

Fricas [A] (verification not implemented)

Time = 0.07 (sec) , antiderivative size = 74, normalized size of antiderivative = 0.43 \[ \int \frac {1}{7+3 x^4} \, dx=\frac {1}{8232} \cdot 4116^{\frac {3}{4}} \arctan \left (\frac {1}{7} \cdot 4116^{\frac {1}{4}} x + 1\right ) + \frac {1}{8232} \cdot 4116^{\frac {3}{4}} \arctan \left (\frac {1}{7} \cdot 4116^{\frac {1}{4}} x - 1\right ) + \frac {1}{16464} \cdot 4116^{\frac {3}{4}} \log \left (294 \, x^{2} + 4116^{\frac {3}{4}} x + 98 \, \sqrt {21}\right ) - \frac {1}{16464} \cdot 4116^{\frac {3}{4}} \log \left (294 \, x^{2} - 4116^{\frac {3}{4}} x + 98 \, \sqrt {21}\right ) \] Input:

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

Output:

1/8232*4116^(3/4)*arctan(1/7*4116^(1/4)*x + 1) + 1/8232*4116^(3/4)*arctan( 
1/7*4116^(1/4)*x - 1) + 1/16464*4116^(3/4)*log(294*x^2 + 4116^(3/4)*x + 98 
*sqrt(21)) - 1/16464*4116^(3/4)*log(294*x^2 - 4116^(3/4)*x + 98*sqrt(21))
 

Sympy [A] (verification not implemented)

Time = 0.18 (sec) , antiderivative size = 151, normalized size of antiderivative = 0.88 \[ \int \frac {1}{7+3 x^4} \, dx=- \frac {\sqrt [4]{189} \sqrt {2} \log {\left (x^{2} - \frac {\sqrt [4]{189} \sqrt {2} x}{3} + \frac {\sqrt {21}}{3} \right )}}{168} + \frac {\sqrt [4]{189} \sqrt {2} \log {\left (x^{2} + \frac {\sqrt [4]{189} \sqrt {2} x}{3} + \frac {\sqrt {21}}{3} \right )}}{168} + \frac {\sqrt {2} \cdot 3^{\frac {3}{4}} \cdot \sqrt [4]{7} \operatorname {atan}{\left (\frac {\sqrt {2} \cdot \sqrt [4]{3} \cdot 7^{\frac {3}{4}} x}{7} - 1 \right )}}{84} + \frac {\sqrt {2} \cdot 3^{\frac {3}{4}} \cdot \sqrt [4]{7} \operatorname {atan}{\left (\frac {\sqrt {2} \cdot \sqrt [4]{3} \cdot 7^{\frac {3}{4}} x}{7} + 1 \right )}}{84} \] Input:

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

Output:

-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 + s 
qrt(2)*3**(3/4)*7**(1/4)*atan(sqrt(2)*3**(1/4)*7**(3/4)*x/7 - 1)/84 + sqrt 
(2)*3**(3/4)*7**(1/4)*atan(sqrt(2)*3**(1/4)*7**(3/4)*x/7 + 1)/84
 

Maxima [A] (verification not implemented)

Time = 0.11 (sec) , antiderivative size = 151, normalized size of antiderivative = 0.88 \[ \int \frac {1}{7+3 x^4} \, dx=\frac {1}{84} \cdot 7^{\frac {1}{4}} 3^{\frac {3}{4}} \sqrt {2} \arctan \left (\frac {1}{42} \cdot 7^{\frac {3}{4}} 3^{\frac {3}{4}} \sqrt {2} {\left (2 \, \sqrt {3} x + 7^{\frac {1}{4}} 3^{\frac {1}{4}} \sqrt {2}\right )}\right ) + \frac {1}{84} \cdot 7^{\frac {1}{4}} 3^{\frac {3}{4}} \sqrt {2} \arctan \left (\frac {1}{42} \cdot 7^{\frac {3}{4}} 3^{\frac {3}{4}} \sqrt {2} {\left (2 \, \sqrt {3} x - 7^{\frac {1}{4}} 3^{\frac {1}{4}} \sqrt {2}\right )}\right ) + \frac {1}{168} \cdot 7^{\frac {1}{4}} 3^{\frac {3}{4}} \sqrt {2} \log \left (\sqrt {3} x^{2} + 7^{\frac {1}{4}} 3^{\frac {1}{4}} \sqrt {2} x + \sqrt {7}\right ) - \frac {1}{168} \cdot 7^{\frac {1}{4}} 3^{\frac {3}{4}} \sqrt {2} \log \left (\sqrt {3} x^{2} - 7^{\frac {1}{4}} 3^{\frac {1}{4}} \sqrt {2} x + \sqrt {7}\right ) \] Input:

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

Output:

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))) + 1/168* 
7^(1/4)*3^(3/4)*sqrt(2)*log(sqrt(3)*x^2 + 7^(1/4)*3^(1/4)*sqrt(2)*x + sqrt 
(7)) - 1/168*7^(1/4)*3^(3/4)*sqrt(2)*log(sqrt(3)*x^2 - 7^(1/4)*3^(1/4)*sqr 
t(2)*x + sqrt(7))
 

Giac [A] (verification not implemented)

Time = 0.13 (sec) , antiderivative size = 95, normalized size of antiderivative = 0.56 \[ \int \frac {1}{7+3 x^4} \, dx=\frac {1}{84} \cdot 756^{\frac {1}{4}} \arctan \left (\frac {3}{14} \, \left (\frac {7}{3}\right )^{\frac {3}{4}} \sqrt {2} {\left (2 \, x + \left (\frac {7}{3}\right )^{\frac {1}{4}} \sqrt {2}\right )}\right ) + \frac {1}{84} \cdot 756^{\frac {1}{4}} \arctan \left (\frac {3}{14} \, \left (\frac {7}{3}\right )^{\frac {3}{4}} \sqrt {2} {\left (2 \, x - \left (\frac {7}{3}\right )^{\frac {1}{4}} \sqrt {2}\right )}\right ) + \frac {1}{168} \cdot 756^{\frac {1}{4}} \log \left (x^{2} + \left (\frac {7}{3}\right )^{\frac {1}{4}} \sqrt {2} x + \sqrt {\frac {7}{3}}\right ) - \frac {1}{168} \cdot 756^{\frac {1}{4}} \log \left (x^{2} - \left (\frac {7}{3}\right )^{\frac {1}{4}} \sqrt {2} x + \sqrt {\frac {7}{3}}\right ) \] Input:

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

Output:

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)) - 1/1 
68*756^(1/4)*log(x^2 - (7/3)^(1/4)*sqrt(2)*x + sqrt(7/3))
 

Mupad [B] (verification not implemented)

Time = 0.06 (sec) , antiderivative size = 45, normalized size of antiderivative = 0.26 \[ \int \frac {1}{7+3 x^4} \, dx=\sqrt {2}\,{189}^{1/4}\,\mathrm {atan}\left (\sqrt {2}\,{189}^{3/4}\,x\,\left (\frac {1}{126}-\frac {1}{126}{}\mathrm {i}\right )\right )\,\left (\frac {1}{84}+\frac {1}{84}{}\mathrm {i}\right )+\sqrt {2}\,{189}^{1/4}\,\mathrm {atan}\left (\sqrt {2}\,{189}^{3/4}\,x\,\left (\frac {1}{126}+\frac {1}{126}{}\mathrm {i}\right )\right )\,\left (\frac {1}{84}-\frac {1}{84}{}\mathrm {i}\right ) \] Input:

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

Output:

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

Reduce [B] (verification not implemented)

Time = 0.15 (sec) , antiderivative size = 93, normalized size of antiderivative = 0.54 \[ \int \frac {1}{7+3 x^4} \, dx=\frac {\sqrt {6}\, 21^{\frac {1}{4}} \left (-2 \mathit {atan} \left (\frac {\left (\sqrt {2}\, 21^{\frac {1}{4}}-2 \sqrt {3}\, x \right ) 21^{\frac {3}{4}}}{21 \sqrt {2}}\right )+2 \mathit {atan} \left (\frac {\left (\sqrt {2}\, 21^{\frac {1}{4}}+2 \sqrt {3}\, x \right ) 21^{\frac {3}{4}}}{21 \sqrt {2}}\right )-\mathrm {log}\left (-\sqrt {2}\, 21^{\frac {1}{4}} x +\sqrt {7}+\sqrt {3}\, x^{2}\right )+\mathrm {log}\left (\sqrt {2}\, 21^{\frac {1}{4}} x +\sqrt {7}+\sqrt {3}\, x^{2}\right )\right )}{168} \] Input:

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

Output:

(sqrt(6)*21**(1/4)*( - 2*atan((sqrt(2)*21**(1/4) - 2*sqrt(3)*x)/(sqrt(2)*2 
1**(1/4))) + 2*atan((sqrt(2)*21**(1/4) + 2*sqrt(3)*x)/(sqrt(2)*21**(1/4))) 
 - log( - sqrt(2)*21**(1/4)*x + sqrt(7) + sqrt(3)*x**2) + log(sqrt(2)*21** 
(1/4)*x + sqrt(7) + sqrt(3)*x**2)))/168