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

Optimal result

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

[Out]

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)))

Rubi [A] (verified)

Time = 0.01 (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 = {403} \[ \int \frac {1}{\sqrt [3]{1+x^2} \left (9+x^2\right )} \, dx=\frac {1}{12} \arctan \left (\frac {\left (1-\sqrt [3]{x^2+1}\right )^2}{3 x}\right )+\frac {1}{12} \arctan \left (\frac {x}{3}\right )-\frac {\text {arctanh}\left (\frac {\sqrt {3} \left (1-\sqrt [3]{x^2+1}\right )}{x}\right )}{4 \sqrt {3}} \]

[In]

Int[1/((1 + x^2)^(1/3)*(9 + x^2)),x]

[Out]

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])

Rule 403

Int[1/(((a_) + (b_.)*(x_)^2)^(1/3)*((c_) + (d_.)*(x_)^2)), x_Symbol] :> With[{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]

Rubi steps \begin{align*} \text {integral}& = \frac {1}{12} \arctan \left (\frac {x}{3}\right )+\frac {1}{12} \arctan \left (\frac {\left (1-\sqrt [3]{1+x^2}\right )^2}{3 x}\right )-\frac {\text {arctanh}\left (\frac {\sqrt {3} \left (1-\sqrt [3]{1+x^2}\right )}{x}\right )}{4 \sqrt {3}} \\ \end{align*}

Mathematica [C] (warning: unable to verify)

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

Time = 4.13 (sec) , antiderivative size = 124, normalized size of antiderivative = 0.65 \[ \int \frac {1}{\sqrt [3]{1+x^2} \left (9+x^2\right )} \, dx=-\frac {27 x \operatorname {AppellF1}\left (\frac {1}{2},\frac {1}{3},1,\frac {3}{2},-x^2,-\frac {x^2}{9}\right )}{\sqrt [3]{1+x^2} \left (9+x^2\right ) \left (-27 \operatorname {AppellF1}\left (\frac {1}{2},\frac {1}{3},1,\frac {3}{2},-x^2,-\frac {x^2}{9}\right )+2 x^2 \left (\operatorname {AppellF1}\left (\frac {3}{2},\frac {1}{3},2,\frac {5}{2},-x^2,-\frac {x^2}{9}\right )+3 \operatorname {AppellF1}\left (\frac {3}{2},\frac {4}{3},1,\frac {5}{2},-x^2,-\frac {x^2}{9}\right )\right )\right )} \]

[In]

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

[Out]

(-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])))

Maple [C] (verified)

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

Time = 3.76 (sec) , antiderivative size = 616, normalized size of antiderivative = 3.24

method	result	size
trager	\(-144 \operatorname {RootOf}\left (20736 \textit {\_Z}^{4}-144 \textit {\_Z}^{2}+1\right )^{3} \ln \left (\frac {497664 \left (x^{2}+1\right )^{\frac {1}{3}} \operatorname {RootOf}\left (20736 \textit {\_Z}^{4}-144 \textit {\_Z}^{2}+1\right )^{5} x -995328 \operatorname {RootOf}\left (20736 \textit {\_Z}^{4}-144 \textit {\_Z}^{2}+1\right )^{5} x -6912 \left (x^{2}+1\right )^{\frac {1}{3}} \operatorname {RootOf}\left (20736 \textit {\_Z}^{4}-144 \textit {\_Z}^{2}+1\right )^{3} x +20736 \operatorname {RootOf}\left (20736 \textit {\_Z}^{4}-144 \textit {\_Z}^{2}+1\right )^{3} x -144 \operatorname {RootOf}\left (20736 \textit {\_Z}^{4}-144 \textit {\_Z}^{2}+1\right )^{2} x^{2}+864 \operatorname {RootOf}\left (20736 \textit {\_Z}^{4}-144 \textit {\_Z}^{2}+1\right )^{2} \left (x^{2}+1\right )^{\frac {1}{3}}+432 \operatorname {RootOf}\left (20736 \textit {\_Z}^{4}-144 \textit {\_Z}^{2}+1\right )^{2}-96 \operatorname {RootOf}\left (20736 \textit {\_Z}^{4}-144 \textit {\_Z}^{2}+1\right ) x -6 \left (x^{2}+1\right )^{\frac {2}{3}}+x^{2}-3}{x^{2}+9}\right )+\operatorname {RootOf}\left (20736 \textit {\_Z}^{4}-144 \textit {\_Z}^{2}+1\right ) \ln \left (\frac {497664 \left (x^{2}+1\right )^{\frac {1}{3}} \operatorname {RootOf}\left (20736 \textit {\_Z}^{4}-144 \textit {\_Z}^{2}+1\right )^{5} x -995328 \operatorname {RootOf}\left (20736 \textit {\_Z}^{4}-144 \textit {\_Z}^{2}+1\right )^{5} x -6912 \left (x^{2}+1\right )^{\frac {1}{3}} \operatorname {RootOf}\left (20736 \textit {\_Z}^{4}-144 \textit {\_Z}^{2}+1\right )^{3} x +20736 \operatorname {RootOf}\left (20736 \textit {\_Z}^{4}-144 \textit {\_Z}^{2}+1\right )^{3} x -144 \operatorname {RootOf}\left (20736 \textit {\_Z}^{4}-144 \textit {\_Z}^{2}+1\right )^{2} x^{2}+864 \operatorname {RootOf}\left (20736 \textit {\_Z}^{4}-144 \textit {\_Z}^{2}+1\right )^{2} \left (x^{2}+1\right )^{\frac {1}{3}}+432 \operatorname {RootOf}\left (20736 \textit {\_Z}^{4}-144 \textit {\_Z}^{2}+1\right )^{2}-96 \operatorname {RootOf}\left (20736 \textit {\_Z}^{4}-144 \textit {\_Z}^{2}+1\right ) x -6 \left (x^{2}+1\right )^{\frac {2}{3}}+x^{2}-3}{x^{2}+9}\right )-\operatorname {RootOf}\left (20736 \textit {\_Z}^{4}-144 \textit {\_Z}^{2}+1\right ) \ln \left (\frac {82944 \left (x^{2}+1\right )^{\frac {1}{3}} \operatorname {RootOf}\left (20736 \textit {\_Z}^{4}-144 \textit {\_Z}^{2}+1\right )^{5} x -165888 \operatorname {RootOf}\left (20736 \textit {\_Z}^{4}-144 \textit {\_Z}^{2}+1\right )^{5} x -1728 \left (x^{2}+1\right )^{\frac {1}{3}} \operatorname {RootOf}\left (20736 \textit {\_Z}^{4}-144 \textit {\_Z}^{2}+1\right )^{3} x +2304 \operatorname {RootOf}\left (20736 \textit {\_Z}^{4}-144 \textit {\_Z}^{2}+1\right )^{3} x +24 \operatorname {RootOf}\left (20736 \textit {\_Z}^{4}-144 \textit {\_Z}^{2}+1\right )^{2} x^{2}-144 \operatorname {RootOf}\left (20736 \textit {\_Z}^{4}-144 \textit {\_Z}^{2}+1\right )^{2} \left (x^{2}+1\right )^{\frac {1}{3}}+8 \left (x^{2}+1\right )^{\frac {1}{3}} \operatorname {RootOf}\left (20736 \textit {\_Z}^{4}-144 \textit {\_Z}^{2}+1\right ) x -72 \operatorname {RootOf}\left (20736 \textit {\_Z}^{4}-144 \textit {\_Z}^{2}+1\right )^{2}-\left (x^{2}+1\right )^{\frac {2}{3}}+\left (x^{2}+1\right )^{\frac {1}{3}}}{x^{2}+9}\right )\)	\(616\)

[In]

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

[Out]

-144*RootOf(20736*_Z^4-144*_Z^2+1)^3*ln((497664*(x^2+1)^(1/3)*RootOf(20736*_Z^4-144*_Z^2+1)^5*x-995328*RootOf(
20736*_Z^4-144*_Z^2+1)^5*x-6912*(x^2+1)^(1/3)*RootOf(20736*_Z^4-144*_Z^2+1)^3*x+20736*RootOf(20736*_Z^4-144*_Z
^2+1)^3*x-144*RootOf(20736*_Z^4-144*_Z^2+1)^2*x^2+864*RootOf(20736*_Z^4-144*_Z^2+1)^2*(x^2+1)^(1/3)+432*RootOf
(20736*_Z^4-144*_Z^2+1)^2-96*RootOf(20736*_Z^4-144*_Z^2+1)*x-6*(x^2+1)^(2/3)+x^2-3)/(x^2+9))+RootOf(20736*_Z^4
-144*_Z^2+1)*ln((497664*(x^2+1)^(1/3)*RootOf(20736*_Z^4-144*_Z^2+1)^5*x-995328*RootOf(20736*_Z^4-144*_Z^2+1)^5
*x-6912*(x^2+1)^(1/3)*RootOf(20736*_Z^4-144*_Z^2+1)^3*x+20736*RootOf(20736*_Z^4-144*_Z^2+1)^3*x-144*RootOf(207
36*_Z^4-144*_Z^2+1)^2*x^2+864*RootOf(20736*_Z^4-144*_Z^2+1)^2*(x^2+1)^(1/3)+432*RootOf(20736*_Z^4-144*_Z^2+1)^
2-96*RootOf(20736*_Z^4-144*_Z^2+1)*x-6*(x^2+1)^(2/3)+x^2-3)/(x^2+9))-RootOf(20736*_Z^4-144*_Z^2+1)*ln((82944*(
x^2+1)^(1/3)*RootOf(20736*_Z^4-144*_Z^2+1)^5*x-165888*RootOf(20736*_Z^4-144*_Z^2+1)^5*x-1728*(x^2+1)^(1/3)*Roo
tOf(20736*_Z^4-144*_Z^2+1)^3*x+2304*RootOf(20736*_Z^4-144*_Z^2+1)^3*x+24*RootOf(20736*_Z^4-144*_Z^2+1)^2*x^2-1
44*RootOf(20736*_Z^4-144*_Z^2+1)^2*(x^2+1)^(1/3)+8*(x^2+1)^(1/3)*RootOf(20736*_Z^4-144*_Z^2+1)*x-72*RootOf(207
36*_Z^4-144*_Z^2+1)^2-(x^2+1)^(2/3)+(x^2+1)^(1/3))/(x^2+9))

Fricas [B] (verification not implemented)

Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 1059 vs. \(2 (137) = 274\).

Time = 1.14 (sec) , antiderivative size = 1059, normalized size of antiderivative = 5.57 \[ \int \frac {1}{\sqrt [3]{1+x^2} \left (9+x^2\right )} \, dx=\text {Too large to display} \]

[In]

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

[Out]

-1/144*sqrt(2)*sqrt(-I*sqrt(3) + 1)*log(-(42*x^5 - 828*x^3 + sqrt(2)*(x^6 - 315*x^4 + 567*x^2 + 243)*sqrt(-I*s
qrt(3) + 1) + 9*(40*x^3 + (sqrt(3)*sqrt(2)*(-I*x^4 + 36*I*x^2 - 27*I) + sqrt(2)*(x^4 - 36*x^2 + 27))*sqrt(-I*s
qrt(3) + 1) - 216*x)*(x^2 + 1)^(2/3) + 6*sqrt(3)*(7*I*x^5 - 138*I*x^3 - 81*I*x) - 3*(2*x^5 - 156*x^3 - 2*sqrt(
3)*(I*x^5 - 78*I*x^3 + 81*I*x) - 3*(sqrt(3)*sqrt(2)*(-5*I*x^4 + 54*I*x^2 + 27*I) - sqrt(2)*(5*x^4 - 54*x^2 - 2
7))*sqrt(-I*sqrt(3) + 1) + 162*x)*(x^2 + 1)^(1/3) - 486*x)/(x^6 + 27*x^4 + 243*x^2 + 729)) + 1/144*sqrt(2)*sqr
t(-I*sqrt(3) + 1)*log(-(42*x^5 - 828*x^3 - sqrt(2)*(x^6 - 315*x^4 + 567*x^2 + 243)*sqrt(-I*sqrt(3) + 1) + 9*(4
0*x^3 + (sqrt(3)*sqrt(2)*(I*x^4 - 36*I*x^2 + 27*I) - sqrt(2)*(x^4 - 36*x^2 + 27))*sqrt(-I*sqrt(3) + 1) - 216*x
)*(x^2 + 1)^(2/3) + 6*sqrt(3)*(7*I*x^5 - 138*I*x^3 - 81*I*x) - 3*(2*x^5 - 156*x^3 - 2*sqrt(3)*(I*x^5 - 78*I*x^
3 + 81*I*x) - 3*(sqrt(3)*sqrt(2)*(5*I*x^4 - 54*I*x^2 - 27*I) + sqrt(2)*(5*x^4 - 54*x^2 - 27))*sqrt(-I*sqrt(3)
+ 1) + 162*x)*(x^2 + 1)^(1/3) - 486*x)/(x^6 + 27*x^4 + 243*x^2 + 729)) - 1/144*sqrt(2)*sqrt(I*sqrt(3) + 1)*log
(-(42*x^5 - 828*x^3 + 72*(5*x^3 - 27*x)*(x^2 + 1)^(2/3) + 6*sqrt(3)*(-7*I*x^5 + 138*I*x^3 + 81*I*x) + (9*(sqrt
(3)*sqrt(2)*(I*x^4 - 36*I*x^2 + 27*I) + sqrt(2)*(x^4 - 36*x^2 + 27))*(x^2 + 1)^(2/3) + sqrt(2)*(x^6 - 315*x^4
+ 567*x^2 + 243) + 9*(sqrt(3)*sqrt(2)*(5*I*x^4 - 54*I*x^2 - 27*I) - sqrt(2)*(5*x^4 - 54*x^2 - 27))*(x^2 + 1)^(
1/3))*sqrt(I*sqrt(3) + 1) - 6*(x^5 - 78*x^3 - sqrt(3)*(-I*x^5 + 78*I*x^3 - 81*I*x) + 81*x)*(x^2 + 1)^(1/3) - 4
86*x)/(x^6 + 27*x^4 + 243*x^2 + 729)) + 1/144*sqrt(2)*sqrt(I*sqrt(3) + 1)*log(-(42*x^5 - 828*x^3 + 72*(5*x^3 -
27*x)*(x^2 + 1)^(2/3) + 6*sqrt(3)*(-7*I*x^5 + 138*I*x^3 + 81*I*x) + (9*(sqrt(3)*sqrt(2)*(-I*x^4 + 36*I*x^2 -
27*I) - sqrt(2)*(x^4 - 36*x^2 + 27))*(x^2 + 1)^(2/3) - sqrt(2)*(x^6 - 315*x^4 + 567*x^2 + 243) + 9*(sqrt(3)*sq
rt(2)*(-5*I*x^4 + 54*I*x^2 + 27*I) + sqrt(2)*(5*x^4 - 54*x^2 - 27))*(x^2 + 1)^(1/3))*sqrt(I*sqrt(3) + 1) - 6*(
x^5 - 78*x^3 - sqrt(3)*(-I*x^5 + 78*I*x^3 - 81*I*x) + 81*x)*(x^2 + 1)^(1/3) - 486*x)/(x^6 + 27*x^4 + 243*x^2 +
729)) - 1/36*arctan(6*(11*x^5 + 30*x^3 + 6*(23*x^3 + 27*x)*(x^2 + 1)^(2/3) + (x^5 - 240*x^3 - 81*x)*(x^2 + 1)
^(1/3) - 81*x)/(x^6 - 1971*x^4 - 1701*x^2 - 729))

Sympy [F]

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

[In]

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

[Out]

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

Maxima [F]

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

[In]

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

[Out]

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

Giac [F]

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

[In]

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

[Out]

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

Mupad [F(-1)]

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

[In]

int(1/((x^2 + 1)^(1/3)*(x^2 + 9)),x)

[Out]

int(1/((x^2 + 1)^(1/3)*(x^2 + 9)), x)