\(\int \frac {1}{(3+x^2) \sqrt [3]{1+3 x^2}} \, dx\) [78]

Optimal result

Integrand size = 19, antiderivative size = 81 \[ \int \frac {1}{\left (3+x^2\right ) \sqrt [3]{1+3 x^2}} \, dx=\frac {\arctan \left (\frac {x}{\sqrt {3}}\right )}{4 \sqrt {3}}+\frac {\arctan \left (\frac {\left (1-\sqrt [3]{1+3 x^2}\right )^2}{3 \sqrt {3} x}\right )}{4 \sqrt {3}}-\frac {1}{4} \text {arctanh}\left (\frac {1-\sqrt [3]{1+3 x^2}}{x}\right ) \] Output:

-1/4*arctanh((1-(3*x^2+1)^(1/3))/x)+1/12*arctan(1/3*x*3^(1/2))*3^(1/2)+1/1 
2*arctan(1/9*(1-(3*x^2+1)^(1/3))^2/x*3^(1/2))*3^(1/2)
                                                                                    
                                                                                    
 

Mathematica [C] (warning: unable to verify)

Result contains higher order function than in optimal. Order 6 vs. order 3 in optimal.

Time = 3.92 (sec) , antiderivative size = 126, normalized size of antiderivative = 1.56 \[ \int \frac {1}{\left (3+x^2\right ) \sqrt [3]{1+3 x^2}} \, dx=-\frac {9 x \operatorname {AppellF1}\left (\frac {1}{2},\frac {1}{3},1,\frac {3}{2},-3 x^2,-\frac {x^2}{3}\right )}{\left (3+x^2\right ) \sqrt [3]{1+3 x^2} \left (-9 \operatorname {AppellF1}\left (\frac {1}{2},\frac {1}{3},1,\frac {3}{2},-3 x^2,-\frac {x^2}{3}\right )+2 x^2 \left (\operatorname {AppellF1}\left (\frac {3}{2},\frac {1}{3},2,\frac {5}{2},-3 x^2,-\frac {x^2}{3}\right )+3 \operatorname {AppellF1}\left (\frac {3}{2},\frac {4}{3},1,\frac {5}{2},-3 x^2,-\frac {x^2}{3}\right )\right )\right )} \] Input:

Integrate[1/((3 + x^2)*(1 + 3*x^2)^(1/3)),x]
 

Output:

(-9*x*AppellF1[1/2, 1/3, 1, 3/2, -3*x^2, -1/3*x^2])/((3 + x^2)*(1 + 3*x^2) 
^(1/3)*(-9*AppellF1[1/2, 1/3, 1, 3/2, -3*x^2, -1/3*x^2] + 2*x^2*(AppellF1[ 
3/2, 1/3, 2, 5/2, -3*x^2, -1/3*x^2] + 3*AppellF1[3/2, 4/3, 1, 5/2, -3*x^2, 
 -1/3*x^2])))
 

Rubi [A] (verified)

Time = 0.16 (sec) , antiderivative size = 81, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.053, Rules used = {306}

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.

\(\displaystyle \int \frac {1}{\left (x^2+3\right ) \sqrt [3]{3 x^2+1}} \, dx\)

\(\Big \downarrow \) 306

\(\displaystyle \frac {\arctan \left (\frac {\left (1-\sqrt [3]{3 x^2+1}\right )^2}{3 \sqrt {3} x}\right )}{4 \sqrt {3}}+\frac {\arctan \left (\frac {x}{\sqrt {3}}\right )}{4 \sqrt {3}}-\frac {1}{4} \text {arctanh}\left (\frac {1-\sqrt [3]{3 x^2+1}}{x}\right )\)

Input:

Int[1/((3 + x^2)*(1 + 3*x^2)^(1/3)),x]
 

Output:

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

Defintions of rubi rules used

rule 306

Int[1/(((a_) + (b_.)*(x_)^2)^(1/3)*((c_) + (d_.)*(x_)^2)), x_Symbol] :> Wit 
h[{q = Rt[b/a, 2]}, Simp[q*(ArcTan[q*(x/3)]/(12*Rt[a, 3]*d)), x] + (Simp[q* 
(ArcTan[(Rt[a, 3] - (a + b*x^2)^(1/3))^2/(3*Rt[a, 3]^2*q*x)]/(12*Rt[a, 3]*d 
)), x] - Simp[q*(ArcTanh[(Sqrt[3]*(Rt[a, 3] - (a + b*x^2)^(1/3)))/(Rt[a, 3] 
*q*x)]/(4*Sqrt[3]*Rt[a, 3]*d)), x])] /; FreeQ[{a, b, c, d}, x] && NeQ[b*c - 
 a*d, 0] && EqQ[b*c - 9*a*d, 0] && PosQ[b/a]
 

Maple [C] (verified)

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

Time = 2.07 (sec) , antiderivative size = 443, normalized size of antiderivative = 5.47

method	result	size
trager	\(\operatorname {RootOf}\left (48 \textit {\_Z}^{2}+12 \textit {\_Z} +1\right ) \ln \left (-\frac {-24 \operatorname {RootOf}\left (48 \textit {\_Z}^{2}+12 \textit {\_Z} +1\right ) \left (3 x^{2}+1\right )^{\frac {1}{3}} x +12 \operatorname {RootOf}\left (48 \textit {\_Z}^{2}+12 \textit {\_Z} +1\right ) x^{2}+2 \left (3 x^{2}+1\right )^{\frac {2}{3}}-24 \operatorname {RootOf}\left (48 \textit {\_Z}^{2}+12 \textit {\_Z} +1\right ) \left (3 x^{2}+1\right )^{\frac {1}{3}}-4 \left (3 x^{2}+1\right )^{\frac {1}{3}} x +48 \operatorname {RootOf}\left (48 \textit {\_Z}^{2}+12 \textit {\_Z} +1\right ) x +x^{2}-4 \left (3 x^{2}+1\right )^{\frac {1}{3}}-12 \operatorname {RootOf}\left (48 \textit {\_Z}^{2}+12 \textit {\_Z} +1\right )+4 x -1}{x^{2}+3}\right )-\frac {\ln \left (-\frac {12 \operatorname {RootOf}\left (48 \textit {\_Z}^{2}+12 \textit {\_Z} +1\right ) \left (3 x^{2}+1\right )^{\frac {1}{3}} x -6 \operatorname {RootOf}\left (48 \textit {\_Z}^{2}+12 \textit {\_Z} +1\right ) x^{2}+\left (3 x^{2}+1\right )^{\frac {2}{3}}+12 \operatorname {RootOf}\left (48 \textit {\_Z}^{2}+12 \textit {\_Z} +1\right ) \left (3 x^{2}+1\right )^{\frac {1}{3}}+\left (3 x^{2}+1\right )^{\frac {1}{3}} x -24 \operatorname {RootOf}\left (48 \textit {\_Z}^{2}+12 \textit {\_Z} +1\right ) x -x^{2}+\left (3 x^{2}+1\right )^{\frac {1}{3}}+6 \operatorname {RootOf}\left (48 \textit {\_Z}^{2}+12 \textit {\_Z} +1\right )-4 x +1}{x^{2}+3}\right )}{4}-\ln \left (-\frac {12 \operatorname {RootOf}\left (48 \textit {\_Z}^{2}+12 \textit {\_Z} +1\right ) \left (3 x^{2}+1\right )^{\frac {1}{3}} x -6 \operatorname {RootOf}\left (48 \textit {\_Z}^{2}+12 \textit {\_Z} +1\right ) x^{2}+\left (3 x^{2}+1\right )^{\frac {2}{3}}+12 \operatorname {RootOf}\left (48 \textit {\_Z}^{2}+12 \textit {\_Z} +1\right ) \left (3 x^{2}+1\right )^{\frac {1}{3}}+\left (3 x^{2}+1\right )^{\frac {1}{3}} x -24 \operatorname {RootOf}\left (48 \textit {\_Z}^{2}+12 \textit {\_Z} +1\right ) x -x^{2}+\left (3 x^{2}+1\right )^{\frac {1}{3}}+6 \operatorname {RootOf}\left (48 \textit {\_Z}^{2}+12 \textit {\_Z} +1\right )-4 x +1}{x^{2}+3}\right ) \operatorname {RootOf}\left (48 \textit {\_Z}^{2}+12 \textit {\_Z} +1\right )\)	\(443\)

Input:

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

Output:

RootOf(48*_Z^2+12*_Z+1)*ln(-(-24*RootOf(48*_Z^2+12*_Z+1)*(3*x^2+1)^(1/3)*x 
+12*RootOf(48*_Z^2+12*_Z+1)*x^2+2*(3*x^2+1)^(2/3)-24*RootOf(48*_Z^2+12*_Z+ 
1)*(3*x^2+1)^(1/3)-4*(3*x^2+1)^(1/3)*x+48*RootOf(48*_Z^2+12*_Z+1)*x+x^2-4* 
(3*x^2+1)^(1/3)-12*RootOf(48*_Z^2+12*_Z+1)+4*x-1)/(x^2+3))-1/4*ln(-(12*Roo 
tOf(48*_Z^2+12*_Z+1)*(3*x^2+1)^(1/3)*x-6*RootOf(48*_Z^2+12*_Z+1)*x^2+(3*x^ 
2+1)^(2/3)+12*RootOf(48*_Z^2+12*_Z+1)*(3*x^2+1)^(1/3)+(3*x^2+1)^(1/3)*x-24 
*RootOf(48*_Z^2+12*_Z+1)*x-x^2+(3*x^2+1)^(1/3)+6*RootOf(48*_Z^2+12*_Z+1)-4 
*x+1)/(x^2+3))-ln(-(12*RootOf(48*_Z^2+12*_Z+1)*(3*x^2+1)^(1/3)*x-6*RootOf( 
48*_Z^2+12*_Z+1)*x^2+(3*x^2+1)^(2/3)+12*RootOf(48*_Z^2+12*_Z+1)*(3*x^2+1)^ 
(1/3)+(3*x^2+1)^(1/3)*x-24*RootOf(48*_Z^2+12*_Z+1)*x-x^2+(3*x^2+1)^(1/3)+6 
*RootOf(48*_Z^2+12*_Z+1)-4*x+1)/(x^2+3))*RootOf(48*_Z^2+12*_Z+1)
 

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 345 vs. \(2 (59) = 118\).

Time = 0.86 (sec) , antiderivative size = 345, normalized size of antiderivative = 4.26 \[ \int \frac {1}{\left (3+x^2\right ) \sqrt [3]{1+3 x^2}} \, dx=\frac {1}{36} \, \sqrt {3} \arctan \left (\frac {4 \, \sqrt {3} {\left (3 \, x^{4} - 10 \, x^{3} - 36 \, x^{2} + 18 \, x + 9\right )} {\left (3 \, x^{2} + 1\right )}^{\frac {2}{3}} - 4 \, \sqrt {3} {\left (x^{5} + 15 \, x^{4} - 26 \, x^{3} - 54 \, x^{2} + 9 \, x - 9\right )} {\left (3 \, x^{2} + 1\right )}^{\frac {1}{3}} + \sqrt {3} {\left (x^{6} - 2 \, x^{5} - 105 \, x^{4} - 28 \, x^{3} + 63 \, x^{2} + 126 \, x + 9\right )}}{x^{6} + 126 \, x^{5} - 225 \, x^{4} - 828 \, x^{3} - 81 \, x^{2} - 162 \, x + 81}\right ) - \frac {1}{36} \, \sqrt {3} \arctan \left (\frac {2 \, {\left (2 \, \sqrt {3} {\left (23 \, x^{3} + 9 \, x\right )} {\left (3 \, x^{2} + 1\right )}^{\frac {2}{3}} + \sqrt {3} {\left (x^{5} - 80 \, x^{3} - 9 \, x\right )} {\left (3 \, x^{2} + 1\right )}^{\frac {1}{3}} + \sqrt {3} {\left (11 \, x^{5} + 10 \, x^{3} - 9 \, x\right )}\right )}}{x^{6} - 657 \, x^{4} - 189 \, x^{2} - 27}\right ) + \frac {1}{24} \, \log \left (\frac {x^{6} + 108 \, x^{5} + 549 \, x^{4} + 360 \, x^{3} + 99 \, x^{2} + 6 \, {\left (3 \, x^{4} + 32 \, x^{3} + 42 \, x^{2} + 3\right )} {\left (3 \, x^{2} + 1\right )}^{\frac {2}{3}} + 6 \, {\left (x^{5} + 27 \, x^{4} + 70 \, x^{3} + 18 \, x^{2} + 9 \, x + 3\right )} {\left (3 \, x^{2} + 1\right )}^{\frac {1}{3}} + 108 \, x - 9}{x^{6} + 9 \, x^{4} + 27 \, x^{2} + 27}\right ) \] Input:

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

Output:

1/36*sqrt(3)*arctan((4*sqrt(3)*(3*x^4 - 10*x^3 - 36*x^2 + 18*x + 9)*(3*x^2 
 + 1)^(2/3) - 4*sqrt(3)*(x^5 + 15*x^4 - 26*x^3 - 54*x^2 + 9*x - 9)*(3*x^2 
+ 1)^(1/3) + sqrt(3)*(x^6 - 2*x^5 - 105*x^4 - 28*x^3 + 63*x^2 + 126*x + 9) 
)/(x^6 + 126*x^5 - 225*x^4 - 828*x^3 - 81*x^2 - 162*x + 81)) - 1/36*sqrt(3 
)*arctan(2*(2*sqrt(3)*(23*x^3 + 9*x)*(3*x^2 + 1)^(2/3) + sqrt(3)*(x^5 - 80 
*x^3 - 9*x)*(3*x^2 + 1)^(1/3) + sqrt(3)*(11*x^5 + 10*x^3 - 9*x))/(x^6 - 65 
7*x^4 - 189*x^2 - 27)) + 1/24*log((x^6 + 108*x^5 + 549*x^4 + 360*x^3 + 99* 
x^2 + 6*(3*x^4 + 32*x^3 + 42*x^2 + 3)*(3*x^2 + 1)^(2/3) + 6*(x^5 + 27*x^4 
+ 70*x^3 + 18*x^2 + 9*x + 3)*(3*x^2 + 1)^(1/3) + 108*x - 9)/(x^6 + 9*x^4 + 
 27*x^2 + 27))
 

Sympy [F]

\[ \int \frac {1}{\left (3+x^2\right ) \sqrt [3]{1+3 x^2}} \, dx=\int \frac {1}{\left (x^{2} + 3\right ) \sqrt [3]{3 x^{2} + 1}}\, dx \] Input:

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

Output:

Integral(1/((x**2 + 3)*(3*x**2 + 1)**(1/3)), x)
 

Maxima [F]

\[ \int \frac {1}{\left (3+x^2\right ) \sqrt [3]{1+3 x^2}} \, dx=\int { \frac {1}{{\left (3 \, x^{2} + 1\right )}^{\frac {1}{3}} {\left (x^{2} + 3\right )}} \,d x } \] Input:

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

Output:

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

Giac [F]

\[ \int \frac {1}{\left (3+x^2\right ) \sqrt [3]{1+3 x^2}} \, dx=\int { \frac {1}{{\left (3 \, x^{2} + 1\right )}^{\frac {1}{3}} {\left (x^{2} + 3\right )}} \,d x } \] Input:

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

Output:

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

Mupad [F(-1)]

Timed out. \[ \int \frac {1}{\left (3+x^2\right ) \sqrt [3]{1+3 x^2}} \, dx=\int \frac {1}{\left (x^2+3\right )\,{\left (3\,x^2+1\right )}^{1/3}} \,d x \] Input:

int(1/((x^2 + 3)*(3*x^2 + 1)^(1/3)),x)
 

Output:

int(1/((x^2 + 3)*(3*x^2 + 1)^(1/3)), x)
 

Reduce [F]

\[ \int \frac {1}{\left (3+x^2\right ) \sqrt [3]{1+3 x^2}} \, dx=\int \frac {1}{\left (3 x^{2}+1\right )^{\frac {1}{3}} x^{2}+3 \left (3 x^{2}+1\right )^{\frac {1}{3}}}d x \] Input:

int(1/(x^2+3)/(3*x^2+1)^(1/3),x)
 

Output:

int(1/((3*x**2 + 1)**(1/3)*x**2 + 3*(3*x**2 + 1)**(1/3)),x)