\(\int \frac {1}{\sqrt [3]{1+x^2} (9+x^2)} \, dx\) [2378]

Optimal result

Integrand size = 17, antiderivative size = 190 \[ \int \frac {1}{\sqrt [3]{1+x^2} \left (9+x^2\right )} \, dx=\frac {1}{12} \arctan \left (\frac {x}{1+2 \sqrt [3]{1+x^2}}\right )+\frac {i \arctan \left (\frac {-\frac {1}{\sqrt {3}}-\frac {i x}{\sqrt {3}}+\frac {\sqrt [3]{1+x^2}}{\sqrt {3}}}{\sqrt [3]{1+x^2}}\right )}{8 \sqrt {3}}-\frac {i \arctan \left (\frac {-\frac {1}{\sqrt {3}}+\frac {i x}{\sqrt {3}}+\frac {\sqrt [3]{1+x^2}}{\sqrt {3}}}{\sqrt [3]{1+x^2}}\right )}{8 \sqrt {3}}-\frac {1}{24} i \text {arctanh}\left (\frac {2 i x-2 i x \sqrt [3]{1+x^2}}{-1+x^2+2 \sqrt [3]{1+x^2}-4 \left (1+x^2\right )^{2/3}}\right ) \] Output:

1/12*arctan(x/(1+2*(x^2+1)^(1/3)))+1/24*I*arctan((-1/3*3^(1/2)-1/3*I*x*3^( 
1/2)+1/3*(x^2+1)^(1/3)*3^(1/2))/(x^2+1)^(1/3))*3^(1/2)-1/24*I*arctan((-1/3 
*3^(1/2)+1/3*I*x*3^(1/2)+1/3*(x^2+1)^(1/3)*3^(1/2))/(x^2+1)^(1/3))*3^(1/2) 
-1/24*I*arctanh((2*I*x-2*I*x*(x^2+1)^(1/3))/(-1+x^2+2*(x^2+1)^(1/3)-4*(x^2 
+1)^(2/3)))
                                                                                    
                                                                                    
 

Mathematica [C] (warning: unable to verify)

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

Time = 4.47 (sec) , antiderivative size = 124, normalized size of antiderivative = 0.65 \[ \int \frac {1}{\sqrt [3]{1+x^2} \left (9+x^2\right )} \, dx=-\frac {27 x \operatorname {AppellF1}\left (\frac {1}{2},\frac {1}{3},1,\frac {3}{2},-x^2,-\frac {x^2}{9}\right )}{\sqrt [3]{1+x^2} \left (9+x^2\right ) \left (-27 \operatorname {AppellF1}\left (\frac {1}{2},\frac {1}{3},1,\frac {3}{2},-x^2,-\frac {x^2}{9}\right )+2 x^2 \left (\operatorname {AppellF1}\left (\frac {3}{2},\frac {1}{3},2,\frac {5}{2},-x^2,-\frac {x^2}{9}\right )+3 \operatorname {AppellF1}\left (\frac {3}{2},\frac {4}{3},1,\frac {5}{2},-x^2,-\frac {x^2}{9}\right )\right )\right )} \] Input:

Integrate[1/((1 + x^2)^(1/3)*(9 + x^2)),x]
 

Output:

(-27*x*AppellF1[1/2, 1/3, 1, 3/2, -x^2, -1/9*x^2])/((1 + x^2)^(1/3)*(9 + x 
^2)*(-27*AppellF1[1/2, 1/3, 1, 3/2, -x^2, -1/9*x^2] + 2*x^2*(AppellF1[3/2, 
 1/3, 2, 5/2, -x^2, -1/9*x^2] + 3*AppellF1[3/2, 4/3, 1, 5/2, -x^2, -1/9*x^ 
2])))
 

Rubi [A] (verified)

Time = 0.16 (sec) , antiderivative size = 70, normalized size of antiderivative = 0.37, number of steps used = 1, number of rules used = 1, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.059, 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.

\(\displaystyle \int \frac {1}{\sqrt [3]{x^2+1} \left (x^2+9\right )} \, dx\)

\(\Big \downarrow \) 306

\(\displaystyle \frac {1}{12} \arctan \left (\frac {\left (1-\sqrt [3]{x^2+1}\right )^2}{3 x}\right )+\frac {1}{12} \arctan \left (\frac {x}{3}\right )-\frac {\text {arctanh}\left (\frac {\sqrt {3} \left (1-\sqrt [3]{x^2+1}\right )}{x}\right )}{4 \sqrt {3}}\)

Input:

Int[1/((1 + x^2)^(1/3)*(9 + x^2)),x]
 

Output:

ArcTan[x/3]/12 + ArcTan[(1 - (1 + x^2)^(1/3))^2/(3*x)]/12 - ArcTanh[(Sqrt[ 
3]*(1 - (1 + x^2)^(1/3)))/x]/(4*Sqrt[3])
 

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] (warning: unable to verify)

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

Time = 8.06 (sec) , antiderivative size = 416, normalized size of antiderivative = 2.19

method	result	size
trager	\(-\frac {\operatorname {RootOf}\left (\textit {\_Z}^{2}+1\right ) \ln \left (-\frac {-12 \operatorname {RootOf}\left (\textit {\_Z}^{2}+1\right )^{2} \left (x^{2}+1\right )^{\frac {1}{3}} \operatorname {RootOf}\left (-12 \textit {\_Z} \operatorname {RootOf}\left (\textit {\_Z}^{2}+1\right )+144 \textit {\_Z}^{2}-1\right ) x +288 \operatorname {RootOf}\left (\textit {\_Z}^{2}+1\right ) \left (x^{2}+1\right )^{\frac {1}{3}} {\operatorname {RootOf}\left (-12 \textit {\_Z} \operatorname {RootOf}\left (\textit {\_Z}^{2}+1\right )+144 \textit {\_Z}^{2}-1\right )}^{2} x +24 \operatorname {RootOf}\left (\textit {\_Z}^{2}+1\right )^{2} \operatorname {RootOf}\left (-12 \textit {\_Z} \operatorname {RootOf}\left (\textit {\_Z}^{2}+1\right )+144 \textit {\_Z}^{2}-1\right ) x -576 \operatorname {RootOf}\left (\textit {\_Z}^{2}+1\right ) {\operatorname {RootOf}\left (-12 \textit {\_Z} \operatorname {RootOf}\left (\textit {\_Z}^{2}+1\right )+144 \textit {\_Z}^{2}-1\right )}^{2} x +6 \operatorname {RootOf}\left (\textit {\_Z}^{2}+1\right ) \operatorname {RootOf}\left (-12 \textit {\_Z} \operatorname {RootOf}\left (\textit {\_Z}^{2}+1\right )+144 \textit {\_Z}^{2}-1\right ) x^{2}-36 \left (x^{2}+1\right )^{\frac {1}{3}} \operatorname {RootOf}\left (-12 \textit {\_Z} \operatorname {RootOf}\left (\textit {\_Z}^{2}+1\right )+144 \textit {\_Z}^{2}-1\right ) \operatorname {RootOf}\left (\textit {\_Z}^{2}+1\right )-\operatorname {RootOf}\left (\textit {\_Z}^{2}+1\right ) \left (x^{2}+1\right )^{\frac {1}{3}} x +24 \left (x^{2}+1\right )^{\frac {1}{3}} \operatorname {RootOf}\left (-12 \textit {\_Z} \operatorname {RootOf}\left (\textit {\_Z}^{2}+1\right )+144 \textit {\_Z}^{2}-1\right ) x +3 \left (x^{2}+1\right )^{\frac {2}{3}}-18 \operatorname {RootOf}\left (-12 \textit {\_Z} \operatorname {RootOf}\left (\textit {\_Z}^{2}+1\right )+144 \textit {\_Z}^{2}-1\right ) \operatorname {RootOf}\left (\textit {\_Z}^{2}+1\right )-3 \left (x^{2}+1\right )^{\frac {1}{3}}}{x^{2}+9}\right )}{12}+\operatorname {RootOf}\left (-12 \textit {\_Z} \operatorname {RootOf}\left (\textit {\_Z}^{2}+1\right )+144 \textit {\_Z}^{2}-1\right ) \ln \left (-\frac {-2 \operatorname {RootOf}\left (\textit {\_Z}^{2}+1\right ) \left (x^{2}+1\right )^{\frac {1}{3}} x +48 \left (x^{2}+1\right )^{\frac {1}{3}} \operatorname {RootOf}\left (-12 \textit {\_Z} \operatorname {RootOf}\left (\textit {\_Z}^{2}+1\right )+144 \textit {\_Z}^{2}-1\right ) x +6 \left (x^{2}+1\right )^{\frac {2}{3}}-4 \operatorname {RootOf}\left (\textit {\_Z}^{2}+1\right ) x +96 \operatorname {RootOf}\left (-12 \textit {\_Z} \operatorname {RootOf}\left (\textit {\_Z}^{2}+1\right )+144 \textit {\_Z}^{2}-1\right ) x +x^{2}+6 \left (x^{2}+1\right )^{\frac {1}{3}}-3}{x^{2}+9}\right )\)	\(416\)

Input:

int(1/(x^2+1)^(1/3)/(x^2+9),x,method=_RETURNVERBOSE)
 

Output:

-1/12*RootOf(_Z^2+1)*ln(-(-12*RootOf(_Z^2+1)^2*(x^2+1)^(1/3)*RootOf(-12*_Z 
*RootOf(_Z^2+1)+144*_Z^2-1)*x+288*RootOf(_Z^2+1)*(x^2+1)^(1/3)*RootOf(-12* 
_Z*RootOf(_Z^2+1)+144*_Z^2-1)^2*x+24*RootOf(_Z^2+1)^2*RootOf(-12*_Z*RootOf 
(_Z^2+1)+144*_Z^2-1)*x-576*RootOf(_Z^2+1)*RootOf(-12*_Z*RootOf(_Z^2+1)+144 
*_Z^2-1)^2*x+6*RootOf(_Z^2+1)*RootOf(-12*_Z*RootOf(_Z^2+1)+144*_Z^2-1)*x^2 
-36*(x^2+1)^(1/3)*RootOf(-12*_Z*RootOf(_Z^2+1)+144*_Z^2-1)*RootOf(_Z^2+1)- 
RootOf(_Z^2+1)*(x^2+1)^(1/3)*x+24*(x^2+1)^(1/3)*RootOf(-12*_Z*RootOf(_Z^2+ 
1)+144*_Z^2-1)*x+3*(x^2+1)^(2/3)-18*RootOf(-12*_Z*RootOf(_Z^2+1)+144*_Z^2- 
1)*RootOf(_Z^2+1)-3*(x^2+1)^(1/3))/(x^2+9))+RootOf(-12*_Z*RootOf(_Z^2+1)+1 
44*_Z^2-1)*ln(-(-2*RootOf(_Z^2+1)*(x^2+1)^(1/3)*x+48*(x^2+1)^(1/3)*RootOf( 
-12*_Z*RootOf(_Z^2+1)+144*_Z^2-1)*x+6*(x^2+1)^(2/3)-4*RootOf(_Z^2+1)*x+96* 
RootOf(-12*_Z*RootOf(_Z^2+1)+144*_Z^2-1)*x+x^2+6*(x^2+1)^(1/3)-3)/(x^2+9))
 

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 781 vs. \(2 (137) = 274\).

Time = 0.89 (sec) , antiderivative size = 781, normalized size of antiderivative = 4.11 \[ \int \frac {1}{\sqrt [3]{1+x^2} \left (9+x^2\right )} \, dx=\text {Too large to display} \] Input:

integrate(1/(x^2+1)^(1/3)/(x^2+9),x, algorithm="fricas")
 

Output:

1/144*sqrt(3)*log((x^6 + 1647*x^4 + 891*x^2 + 18*(3*x^4 + 32*sqrt(3)*x^3 + 
 126*x^2 + 27)*(x^2 + 1)^(2/3) + 108*sqrt(3)*(x^5 + 10*x^3 + 9*x) + 6*(81* 
x^4 + 162*x^2 + sqrt(3)*(x^5 + 210*x^3 + 81*x) + 81)*(x^2 + 1)^(1/3) - 243 
)/(x^6 + 27*x^4 + 243*x^2 + 729)) - 1/144*sqrt(3)*log((x^6 + 1647*x^4 + 89 
1*x^2 + 18*(3*x^4 - 32*sqrt(3)*x^3 + 126*x^2 + 27)*(x^2 + 1)^(2/3) - 108*s 
qrt(3)*(x^5 + 10*x^3 + 9*x) + 6*(81*x^4 + 162*x^2 - sqrt(3)*(x^5 + 210*x^3 
 + 81*x) + 81)*(x^2 + 1)^(1/3) - 243)/(x^6 + 27*x^4 + 243*x^2 + 729)) - 1/ 
72*arctan((384*x^11 - 130320*x^9 + 2379456*x^7 - 629856*x^5 - 1259712*x^3 
+ 36*(388*x^9 - 27864*x^7 + 303264*x^5 + 17496*x^3 + sqrt(3)*(x^10 + 549*x 
^8 - 8046*x^6 + 129762*x^4 - 19683*x^2 + 59049) - 236196*x)*(x^2 + 1)^(2/3 
) + sqrt(3)*(x^12 - 234*x^10 + 229311*x^8 - 1214028*x^6 + 6816879*x^4 + 60 
22998*x^2 + 531441) + 12*(x^11 - 6423*x^9 + 225018*x^7 - 1106622*x^5 - 154 
1835*x^3 + 3*sqrt(3)*(37*x^10 - 675*x^8 + 34722*x^6 - 97686*x^4 + 59049*x^ 
2 + 59049) - 177147*x)*(x^2 + 1)^(1/3) - 8503056*x)/(x^12 - 48978*x^10 + 2 
332071*x^8 - 16419996*x^6 - 24151041*x^4 - 9565938*x^2 + 4782969)) + 1/72* 
arctan(-(384*x^11 - 130320*x^9 + 2379456*x^7 - 629856*x^5 - 1259712*x^3 + 
36*(388*x^9 - 27864*x^7 + 303264*x^5 + 17496*x^3 - sqrt(3)*(x^10 + 549*x^8 
 - 8046*x^6 + 129762*x^4 - 19683*x^2 + 59049) - 236196*x)*(x^2 + 1)^(2/3) 
- sqrt(3)*(x^12 - 234*x^10 + 229311*x^8 - 1214028*x^6 + 6816879*x^4 + 6022 
998*x^2 + 531441) + 12*(x^11 - 6423*x^9 + 225018*x^7 - 1106622*x^5 - 15...
 

Sympy [F]

\[ \int \frac {1}{\sqrt [3]{1+x^2} \left (9+x^2\right )} \, dx=\int \frac {1}{\sqrt [3]{x^{2} + 1} \left (x^{2} + 9\right )}\, dx \] Input:

integrate(1/(x**2+1)**(1/3)/(x**2+9),x)
 

Output:

Integral(1/((x**2 + 1)**(1/3)*(x**2 + 9)), x)
 

Maxima [F]

\[ \int \frac {1}{\sqrt [3]{1+x^2} \left (9+x^2\right )} \, dx=\int { \frac {1}{{\left (x^{2} + 9\right )} {\left (x^{2} + 1\right )}^{\frac {1}{3}}} \,d x } \] Input:

integrate(1/(x^2+1)^(1/3)/(x^2+9),x, algorithm="maxima")
 

Output:

integrate(1/((x^2 + 9)*(x^2 + 1)^(1/3)), x)
 

Giac [F]

\[ \int \frac {1}{\sqrt [3]{1+x^2} \left (9+x^2\right )} \, dx=\int { \frac {1}{{\left (x^{2} + 9\right )} {\left (x^{2} + 1\right )}^{\frac {1}{3}}} \,d x } \] Input:

integrate(1/(x^2+1)^(1/3)/(x^2+9),x, algorithm="giac")
 

Output:

integrate(1/((x^2 + 9)*(x^2 + 1)^(1/3)), x)
 

Mupad [F(-1)]

Timed out. \[ \int \frac {1}{\sqrt [3]{1+x^2} \left (9+x^2\right )} \, dx=\int \frac {1}{{\left (x^2+1\right )}^{1/3}\,\left (x^2+9\right )} \,d x \] Input:

int(1/((x^2 + 1)^(1/3)*(x^2 + 9)),x)
 

Output:

int(1/((x^2 + 1)^(1/3)*(x^2 + 9)), x)
 

Reduce [F]

\[ \int \frac {1}{\sqrt [3]{1+x^2} \left (9+x^2\right )} \, dx=\int \frac {1}{\left (x^{2}+1\right )^{\frac {1}{3}} x^{2}+9 \left (x^{2}+1\right )^{\frac {1}{3}}}d x \] Input:

int(1/(x^2+1)^(1/3)/(x^2+9),x)
 

Output:

int(1/((x**2 + 1)**(1/3)*x**2 + 9*(x**2 + 1)**(1/3)),x)