3.1.77 \(\int \frac {1}{\sqrt [3]{1-3 x^2} (3-x^2)} \, dx\) [77]

3.1.77.1 Optimal result

Integrand size = 21, antiderivative size = 81 \[ \int \frac {1}{\sqrt [3]{1-3 x^2} \left (3-x^2\right )} \, dx=\frac {1}{4} \arctan \left (\frac {1-\sqrt [3]{1-3 x^2}}{x}\right )+\frac {\text {arctanh}\left (\frac {x}{\sqrt {3}}\right )}{4 \sqrt {3}}-\frac {\text {arctanh}\left (\frac {\left (1-\sqrt [3]{1-3 x^2}\right )^2}{3 \sqrt {3} x}\right )}{4 \sqrt {3}} \]

output

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

3.1.77.2 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 = 126, normalized size of antiderivative = 1.56 \[ \int \frac {1}{\sqrt [3]{1-3 x^2} \left (3-x^2\right )} \, dx=-\frac {9 x \operatorname {AppellF1}\left (\frac {1}{2},\frac {1}{3},1,\frac {3}{2},3 x^2,\frac {x^2}{3}\right )}{\sqrt [3]{1-3 x^2} \left (-3+x^2\right ) \left (9 \operatorname {AppellF1}\left (\frac {1}{2},\frac {1}{3},1,\frac {3}{2},3 x^2,\frac {x^2}{3}\right )+2 x^2 \left (\operatorname {AppellF1}\left (\frac {3}{2},\frac {1}{3},2,\frac {5}{2},3 x^2,\frac {x^2}{3}\right )+3 \operatorname {AppellF1}\left (\frac {3}{2},\frac {4}{3},1,\frac {5}{2},3 x^2,\frac {x^2}{3}\right )\right )\right )} \]

input

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

output

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

3.1.77.3 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, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.048, Rules used = {307}

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.

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

\(\Big \downarrow \) 307

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

input

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

output

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

3.1.77.3.1 Defintions of rubi rules used

rule 307

Int[1/(((a_) + (b_.)*(x_)^2)^(1/3)*((c_) + (d_.)*(x_)^2)), x_Symbol] :> Wit 
h[{q = Rt[-b/a, 2]}, Simp[(-q)*(ArcTanh[q*(x/3)]/(12*Rt[a, 3]*d)), x] + (Si 
mp[q*(ArcTanh[(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*(ArcTan[(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] && NegQ[b/a]
 

3.1.77.4 Maple [C] (warning: unable to verify)

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

Time = 7.80 (sec) , antiderivative size = 538, normalized size of antiderivative = 6.64

method	result	size
trager	\(-\frac {\operatorname {RootOf}\left (\textit {\_Z}^{2}-3\right ) \ln \left (\frac {8 \left (-3 x^{2}+1\right )^{\frac {1}{3}} \operatorname {RootOf}\left (\textit {\_Z}^{2}-3\right )^{2} \operatorname {RootOf}\left (4 \textit {\_Z} \operatorname {RootOf}\left (\textit {\_Z}^{2}-3\right )+48 \textit {\_Z}^{2}+1\right ) x +192 \left (-3 x^{2}+1\right )^{\frac {1}{3}} \operatorname {RootOf}\left (\textit {\_Z}^{2}-3\right ) {\operatorname {RootOf}\left (4 \textit {\_Z} \operatorname {RootOf}\left (\textit {\_Z}^{2}-3\right )+48 \textit {\_Z}^{2}+1\right )}^{2} x -16 \operatorname {RootOf}\left (\textit {\_Z}^{2}-3\right )^{2} \operatorname {RootOf}\left (4 \textit {\_Z} \operatorname {RootOf}\left (\textit {\_Z}^{2}-3\right )+48 \textit {\_Z}^{2}+1\right ) x -384 \operatorname {RootOf}\left (\textit {\_Z}^{2}-3\right ) {\operatorname {RootOf}\left (4 \textit {\_Z} \operatorname {RootOf}\left (\textit {\_Z}^{2}-3\right )+48 \textit {\_Z}^{2}+1\right )}^{2} x +12 \operatorname {RootOf}\left (\textit {\_Z}^{2}-3\right ) \operatorname {RootOf}\left (4 \textit {\_Z} \operatorname {RootOf}\left (\textit {\_Z}^{2}-3\right )+48 \textit {\_Z}^{2}+1\right ) x^{2}+24 \left (-3 x^{2}+1\right )^{\frac {1}{3}} \operatorname {RootOf}\left (4 \textit {\_Z} \operatorname {RootOf}\left (\textit {\_Z}^{2}-3\right )+48 \textit {\_Z}^{2}+1\right ) \operatorname {RootOf}\left (\textit {\_Z}^{2}-3\right )+6 \left (-3 x^{2}+1\right )^{\frac {2}{3}}+12 \operatorname {RootOf}\left (4 \textit {\_Z} \operatorname {RootOf}\left (\textit {\_Z}^{2}-3\right )+48 \textit {\_Z}^{2}+1\right ) \operatorname {RootOf}\left (\textit {\_Z}^{2}-3\right )-4 \operatorname {RootOf}\left (\textit {\_Z}^{2}-3\right ) x -96 \operatorname {RootOf}\left (4 \textit {\_Z} \operatorname {RootOf}\left (\textit {\_Z}^{2}-3\right )+48 \textit {\_Z}^{2}+1\right ) x +3 x^{2}+3}{x^{2}-3}\right )}{12}-\frac {\ln \left (\frac {2 \left (-3 x^{2}+1\right )^{\frac {1}{3}} \operatorname {RootOf}\left (\textit {\_Z}^{2}-3\right ) x +48 \left (-3 x^{2}+1\right )^{\frac {1}{3}} \operatorname {RootOf}\left (4 \textit {\_Z} \operatorname {RootOf}\left (\textit {\_Z}^{2}-3\right )+48 \textit {\_Z}^{2}+1\right ) x +6 \left (-3 x^{2}+1\right )^{\frac {2}{3}}+4 \operatorname {RootOf}\left (\textit {\_Z}^{2}-3\right ) x +96 \operatorname {RootOf}\left (4 \textit {\_Z} \operatorname {RootOf}\left (\textit {\_Z}^{2}-3\right )+48 \textit {\_Z}^{2}+1\right ) x -3 x^{2}+6 \left (-3 x^{2}+1\right )^{\frac {1}{3}}-3}{x^{2}-3}\right ) \operatorname {RootOf}\left (\textit {\_Z}^{2}-3\right )}{12}-\ln \left (\frac {2 \left (-3 x^{2}+1\right )^{\frac {1}{3}} \operatorname {RootOf}\left (\textit {\_Z}^{2}-3\right ) x +48 \left (-3 x^{2}+1\right )^{\frac {1}{3}} \operatorname {RootOf}\left (4 \textit {\_Z} \operatorname {RootOf}\left (\textit {\_Z}^{2}-3\right )+48 \textit {\_Z}^{2}+1\right ) x +6 \left (-3 x^{2}+1\right )^{\frac {2}{3}}+4 \operatorname {RootOf}\left (\textit {\_Z}^{2}-3\right ) x +96 \operatorname {RootOf}\left (4 \textit {\_Z} \operatorname {RootOf}\left (\textit {\_Z}^{2}-3\right )+48 \textit {\_Z}^{2}+1\right ) x -3 x^{2}+6 \left (-3 x^{2}+1\right )^{\frac {1}{3}}-3}{x^{2}-3}\right ) \operatorname {RootOf}\left (4 \textit {\_Z} \operatorname {RootOf}\left (\textit {\_Z}^{2}-3\right )+48 \textit {\_Z}^{2}+1\right )\)	\(538\)

input

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

output

-1/12*RootOf(_Z^2-3)*ln((8*(-3*x^2+1)^(1/3)*RootOf(_Z^2-3)^2*RootOf(4*_Z*R 
ootOf(_Z^2-3)+48*_Z^2+1)*x+192*(-3*x^2+1)^(1/3)*RootOf(_Z^2-3)*RootOf(4*_Z 
*RootOf(_Z^2-3)+48*_Z^2+1)^2*x-16*RootOf(_Z^2-3)^2*RootOf(4*_Z*RootOf(_Z^2 
-3)+48*_Z^2+1)*x-384*RootOf(_Z^2-3)*RootOf(4*_Z*RootOf(_Z^2-3)+48*_Z^2+1)^ 
2*x+12*RootOf(_Z^2-3)*RootOf(4*_Z*RootOf(_Z^2-3)+48*_Z^2+1)*x^2+24*(-3*x^2 
+1)^(1/3)*RootOf(4*_Z*RootOf(_Z^2-3)+48*_Z^2+1)*RootOf(_Z^2-3)+6*(-3*x^2+1 
)^(2/3)+12*RootOf(4*_Z*RootOf(_Z^2-3)+48*_Z^2+1)*RootOf(_Z^2-3)-4*RootOf(_ 
Z^2-3)*x-96*RootOf(4*_Z*RootOf(_Z^2-3)+48*_Z^2+1)*x+3*x^2+3)/(x^2-3))-1/12 
*ln((2*(-3*x^2+1)^(1/3)*RootOf(_Z^2-3)*x+48*(-3*x^2+1)^(1/3)*RootOf(4*_Z*R 
ootOf(_Z^2-3)+48*_Z^2+1)*x+6*(-3*x^2+1)^(2/3)+4*RootOf(_Z^2-3)*x+96*RootOf 
(4*_Z*RootOf(_Z^2-3)+48*_Z^2+1)*x-3*x^2+6*(-3*x^2+1)^(1/3)-3)/(x^2-3))*Roo 
tOf(_Z^2-3)-ln((2*(-3*x^2+1)^(1/3)*RootOf(_Z^2-3)*x+48*(-3*x^2+1)^(1/3)*Ro 
otOf(4*_Z*RootOf(_Z^2-3)+48*_Z^2+1)*x+6*(-3*x^2+1)^(2/3)+4*RootOf(_Z^2-3)* 
x+96*RootOf(4*_Z*RootOf(_Z^2-3)+48*_Z^2+1)*x-3*x^2+6*(-3*x^2+1)^(1/3)-3)/( 
x^2-3))*RootOf(4*_Z*RootOf(_Z^2-3)+48*_Z^2+1)
 

3.1.77.5 Fricas [C] (verification not implemented)

Result contains complex when optimal does not.

Time = 1.30 (sec) , antiderivative size = 1210, normalized size of antiderivative = 14.94 \[ \int \frac {1}{\sqrt [3]{1-3 x^2} \left (3-x^2\right )} \, dx=\text {Too large to display} \]

input

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

output

1/144*sqrt(6)*sqrt(-I*sqrt(3) - 1)*log(-(42*x^5 + 276*x^3 + sqrt(6)*(x^6 + 
 105*x^4 + 63*x^2 - 9)*sqrt(-I*sqrt(3) - 1) - 3*(40*x^3 + sqrt(6)*(x^4 + 1 
2*x^2 - sqrt(3)*(-I*x^4 - 12*I*x^2 - 3*I) + 3)*sqrt(-I*sqrt(3) - 1) + 72*x 
)*(-3*x^2 + 1)^(2/3) + 6*sqrt(3)*(-7*I*x^5 - 46*I*x^3 + 9*I*x) - 3*(2*x^5 
+ 52*x^3 - sqrt(6)*(5*x^4 + 18*x^2 + sqrt(3)*(-5*I*x^4 - 18*I*x^2 + 3*I) - 
 3)*sqrt(-I*sqrt(3) - 1) - 2*sqrt(3)*(-I*x^5 - 26*I*x^3 - 9*I*x) + 18*x)*( 
-3*x^2 + 1)^(1/3) - 54*x)/(x^6 - 9*x^4 + 27*x^2 - 27)) - 1/144*sqrt(6)*sqr 
t(-I*sqrt(3) - 1)*log(-(42*x^5 + 276*x^3 - sqrt(6)*(x^6 + 105*x^4 + 63*x^2 
 - 9)*sqrt(-I*sqrt(3) - 1) - 3*(40*x^3 - sqrt(6)*(x^4 + 12*x^2 + sqrt(3)*( 
I*x^4 + 12*I*x^2 + 3*I) + 3)*sqrt(-I*sqrt(3) - 1) + 72*x)*(-3*x^2 + 1)^(2/ 
3) + 6*sqrt(3)*(-7*I*x^5 - 46*I*x^3 + 9*I*x) - 3*(2*x^5 + 52*x^3 + sqrt(6) 
*(5*x^4 + 18*x^2 - sqrt(3)*(5*I*x^4 + 18*I*x^2 - 3*I) - 3)*sqrt(-I*sqrt(3) 
 - 1) - 2*sqrt(3)*(-I*x^5 - 26*I*x^3 - 9*I*x) + 18*x)*(-3*x^2 + 1)^(1/3) - 
 54*x)/(x^6 - 9*x^4 + 27*x^2 - 27)) + 1/144*sqrt(6)*sqrt(I*sqrt(3) - 1)*lo 
g(-(42*x^5 + 276*x^3 - 24*(5*x^3 + 9*x)*(-3*x^2 + 1)^(2/3) + 6*sqrt(3)*(7* 
I*x^5 + 46*I*x^3 - 9*I*x) - (3*sqrt(6)*(x^4 + 12*x^2 - sqrt(3)*(I*x^4 + 12 
*I*x^2 + 3*I) + 3)*(-3*x^2 + 1)^(2/3) - 3*sqrt(6)*(5*x^4 + 18*x^2 + sqrt(3 
)*(5*I*x^4 + 18*I*x^2 - 3*I) - 3)*(-3*x^2 + 1)^(1/3) - sqrt(6)*(x^6 + 105* 
x^4 + 63*x^2 - 9))*sqrt(I*sqrt(3) - 1) - 6*(x^5 + 26*x^3 - sqrt(3)*(I*x^5 
+ 26*I*x^3 + 9*I*x) + 9*x)*(-3*x^2 + 1)^(1/3) - 54*x)/(x^6 - 9*x^4 + 27...
 

3.1.77.6 Sympy [F]

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

input

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

output

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

3.1.77.7 Maxima [F]

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

input

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

output

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

3.1.77.8 Giac [F]

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

input

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

output

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

3.1.77.9 Mupad [F(-1)]

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

input

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

output

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

3.1.77.10 Reduce [F]

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

input

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

output

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