Integrand size = 19, antiderivative size = 113 \[ \int \frac {1}{\sqrt [3]{1-x^2} \left (3+x^2\right )} \, dx=\frac {\arctan \left (\frac {\sqrt {3}}{x}\right )}{2\ 2^{2/3} \sqrt {3}}+\frac {\arctan \left (\frac {\sqrt {3} \left (1-\sqrt [3]{2} \sqrt [3]{1-x^2}\right )}{x}\right )}{2\ 2^{2/3} \sqrt {3}}-\frac {\text {arctanh}(x)}{6\ 2^{2/3}}+\frac {\text {arctanh}\left (\frac {x}{1+\sqrt [3]{2} \sqrt [3]{1-x^2}}\right )}{2\ 2^{2/3}} \] Output:
-1/12*arctanh(x)*2^(1/3)+1/4*arctanh(x/(1+2^(1/3)*(-x^2+1)^(1/3)))*2^(1/3) +1/12*arctan(3^(1/2)/x)*2^(1/3)*3^(1/2)+1/12*arctan((1-2^(1/3)*(-x^2+1)^(1 /3))*3^(1/2)/x)*2^(1/3)*3^(1/2)
Result contains higher order function than in optimal. Order 6 vs. order 3 in optimal.
Time = 3.93 (sec) , antiderivative size = 118, normalized size of antiderivative = 1.04 \[ \int \frac {1}{\sqrt [3]{1-x^2} \left (3+x^2\right )} \, dx=-\frac {9 x \operatorname {AppellF1}\left (\frac {1}{2},\frac {1}{3},1,\frac {3}{2},x^2,-\frac {x^2}{3}\right )}{\sqrt [3]{1-x^2} \left (3+x^2\right ) \left (-9 \operatorname {AppellF1}\left (\frac {1}{2},\frac {1}{3},1,\frac {3}{2},x^2,-\frac {x^2}{3}\right )+2 x^2 \left (\operatorname {AppellF1}\left (\frac {3}{2},\frac {1}{3},2,\frac {5}{2},x^2,-\frac {x^2}{3}\right )-\operatorname {AppellF1}\left (\frac {3}{2},\frac {4}{3},1,\frac {5}{2},x^2,-\frac {x^2}{3}\right )\right )\right )} \] Input:
Integrate[1/((1 - x^2)^(1/3)*(3 + x^2)),x]
Output:
(-9*x*AppellF1[1/2, 1/3, 1, 3/2, x^2, -1/3*x^2])/((1 - x^2)^(1/3)*(3 + x^2 )*(-9*AppellF1[1/2, 1/3, 1, 3/2, x^2, -1/3*x^2] + 2*x^2*(AppellF1[3/2, 1/3 , 2, 5/2, x^2, -1/3*x^2] - AppellF1[3/2, 4/3, 1, 5/2, x^2, -1/3*x^2])))
Time = 0.17 (sec) , antiderivative size = 113, 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 = {305}
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.
\(\displaystyle \int \frac {1}{\sqrt [3]{1-x^2} \left (x^2+3\right )} \, dx\) |
\(\Big \downarrow \) 305 |
\(\displaystyle \frac {\arctan \left (\frac {\sqrt {3} \left (1-\sqrt [3]{2} \sqrt [3]{1-x^2}\right )}{x}\right )}{2\ 2^{2/3} \sqrt {3}}+\frac {\arctan \left (\frac {\sqrt {3}}{x}\right )}{2\ 2^{2/3} \sqrt {3}}+\frac {\text {arctanh}\left (\frac {x}{\sqrt [3]{2} \sqrt [3]{1-x^2}+1}\right )}{2\ 2^{2/3}}-\frac {\text {arctanh}(x)}{6\ 2^{2/3}}\) |
Input:
Int[1/((1 - x^2)^(1/3)*(3 + x^2)),x]
Output:
ArcTan[Sqrt[3]/x]/(2*2^(2/3)*Sqrt[3]) + ArcTan[(Sqrt[3]*(1 - 2^(1/3)*(1 - x^2)^(1/3)))/x]/(2*2^(2/3)*Sqrt[3]) - ArcTanh[x]/(6*2^(2/3)) + ArcTanh[x/( 1 + 2^(1/3)*(1 - x^2)^(1/3))]/(2*2^(2/3))
Int[1/(((a_) + (b_.)*(x_)^2)^(1/3)*((c_) + (d_.)*(x_)^2)), x_Symbol] :> Wit h[{q = Rt[-b/a, 2]}, Simp[q*(ArcTan[Sqrt[3]/(q*x)]/(2*2^(2/3)*Sqrt[3]*a^(1/ 3)*d)), x] + (Simp[q*(ArcTanh[(a^(1/3)*q*x)/(a^(1/3) + 2^(1/3)*(a + b*x^2)^ (1/3))]/(2*2^(2/3)*a^(1/3)*d)), x] - Simp[q*(ArcTanh[q*x]/(6*2^(2/3)*a^(1/3 )*d)), x] + Simp[q*(ArcTan[Sqrt[3]*((a^(1/3) - 2^(1/3)*(a + b*x^2)^(1/3))/( a^(1/3)*q*x))]/(2*2^(2/3)*Sqrt[3]*a^(1/3)*d)), x])] /; FreeQ[{a, b, c, d}, x] && NeQ[b*c - a*d, 0] && EqQ[b*c + 3*a*d, 0] && NegQ[b/a]
Result contains higher order function than in optimal. Order 9 vs. order 3.
Time = 14.20 (sec) , antiderivative size = 938, normalized size of antiderivative = 8.30
Input:
int(1/(-x^2+1)^(1/3)/(x^2+3),x,method=_RETURNVERBOSE)
Output:
1/36*RootOf(_Z^6+108)*ln((-486*RootOf(_Z^6+108)+189*x^2*RootOf(_Z^6+108)^4 -RootOf(_Z^6+108)^4*x^6-18*RootOf(_Z^6+108)*x^6-72*RootOf(_Z^6+108)^4*x^5- 225*RootOf(_Z^6+108)^4*x^4+72*RootOf(_Z^6+108)^4*x^3-1296*RootOf(_Z^6+108) *x^5-4050*RootOf(_Z^6+108)*x^4+1296*RootOf(_Z^6+108)*x^3+3402*RootOf(_Z^6+ 108)*x^2-27*RootOf(_Z^6+108)^4+1296*(-x^2+1)^(2/3)*x^4+9072*(-x^2+1)^(2/3) *x^3+3888*(-x^2+1)^(2/3)*x^2-3888*(-x^2+1)^(2/3)*x-6*(-x^2+1)^(1/3)*RootOf (_Z^6+108)^5*x^5-108*(-x^2+1)^(1/3)*RootOf(_Z^6+108)^5*x^4-144*(-x^2+1)^(1 /3)*RootOf(_Z^6+108)^5*x^3+108*(-x^2+1)^(1/3)*RootOf(_Z^6+108)^5*x^2+54*(- x^2+1)^(1/3)*RootOf(_Z^6+108)^5*x-36*RootOf(_Z^6+108)^2*(-x^2+1)^(1/3)*x^5 -648*RootOf(_Z^6+108)^2*(-x^2+1)^(1/3)*x^4-864*RootOf(_Z^6+108)^2*(-x^2+1) ^(1/3)*x^3+648*RootOf(_Z^6+108)^2*(-x^2+1)^(1/3)*x^2+324*RootOf(_Z^6+108)^ 2*(-x^2+1)^(1/3)*x)/(x^2+3)^3)-1/432*ln((RootOf(_Z^6+108)^4*x^6+72*RootOf( _Z^6+108)^4*x^5+36*RootOf(_Z^6+108)^2*(-x^2+1)^(1/3)*x^5+225*RootOf(_Z^6+1 08)^4*x^4+648*RootOf(_Z^6+108)^2*(-x^2+1)^(1/3)*x^4-72*RootOf(_Z^6+108)^4* x^3+648*(-x^2+1)^(2/3)*x^4+864*RootOf(_Z^6+108)^2*(-x^2+1)^(1/3)*x^3-189*x ^2*RootOf(_Z^6+108)^4+4536*(-x^2+1)^(2/3)*x^3-648*RootOf(_Z^6+108)^2*(-x^2 +1)^(1/3)*x^2+1944*(-x^2+1)^(2/3)*x^2-324*RootOf(_Z^6+108)^2*(-x^2+1)^(1/3 )*x+27*RootOf(_Z^6+108)^4-1944*(-x^2+1)^(2/3)*x)/(x^2+3)^3)*RootOf(_Z^6+10 8)^4+1/72*ln((RootOf(_Z^6+108)^4*x^6+72*RootOf(_Z^6+108)^4*x^5+36*RootOf(_ Z^6+108)^2*(-x^2+1)^(1/3)*x^5+225*RootOf(_Z^6+108)^4*x^4+648*RootOf(_Z^...
Leaf count of result is larger than twice the leaf count of optimal. 1132 vs. \(2 (81) = 162\).
Time = 0.50 (sec) , antiderivative size = 1132, normalized size of antiderivative = 10.02 \[ \int \frac {1}{\sqrt [3]{1-x^2} \left (3+x^2\right )} \, dx=\text {Too large to display} \] Input:
integrate(1/(-x^2+1)^(1/3)/(x^2+3),x, algorithm="fricas")
Output:
-1/24*(-1/432)^(1/6)*(sqrt(-3) + 1)*log((2*(-1/2)^(2/3)*(5*x^5 - 30*x^3 + sqrt(-3)*(5*x^5 - 30*x^3 + 9*x) + 9*x) + 12*(3*x^3 + sqrt(-1/3)*(x^4 - 9*x ^2) - 3*x)*(-x^2 + 1)^(2/3) + 2*(-x^2 + 1)^(1/3)*(864*(-1/432)^(5/6)*(x^4 - 3*x^2 - sqrt(-3)*(x^4 - 3*x^2)) - (-1/2)^(1/3)*(x^5 - 18*x^3 - sqrt(-3)* (x^5 - 18*x^3 + 9*x) + 9*x)) - (-1/432)^(1/6)*(x^6 - 69*x^4 + 63*x^2 + sqr t(-3)*(x^6 - 69*x^4 + 63*x^2 - 27) - 27))/(x^6 + 9*x^4 + 27*x^2 + 27)) + 1 /24*(-1/432)^(1/6)*(sqrt(-3) + 1)*log((2*(-1/2)^(2/3)*(5*x^5 - 30*x^3 + sq rt(-3)*(5*x^5 - 30*x^3 + 9*x) + 9*x) + 12*(3*x^3 - sqrt(-1/3)*(x^4 - 9*x^2 ) - 3*x)*(-x^2 + 1)^(2/3) - 2*(-x^2 + 1)^(1/3)*(864*(-1/432)^(5/6)*(x^4 - 3*x^2 - sqrt(-3)*(x^4 - 3*x^2)) + (-1/2)^(1/3)*(x^5 - 18*x^3 - sqrt(-3)*(x ^5 - 18*x^3 + 9*x) + 9*x)) + (-1/432)^(1/6)*(x^6 - 69*x^4 + 63*x^2 + sqrt( -3)*(x^6 - 69*x^4 + 63*x^2 - 27) - 27))/(x^6 + 9*x^4 + 27*x^2 + 27)) + 1/2 4*(-1/432)^(1/6)*(sqrt(-3) - 1)*log((2*(-1/2)^(2/3)*(5*x^5 - 30*x^3 - sqrt (-3)*(5*x^5 - 30*x^3 + 9*x) + 9*x) + 12*(3*x^3 + sqrt(-1/3)*(x^4 - 9*x^2) - 3*x)*(-x^2 + 1)^(2/3) + 2*(-x^2 + 1)^(1/3)*(864*(-1/432)^(5/6)*(x^4 - 3* x^2 + sqrt(-3)*(x^4 - 3*x^2)) - (-1/2)^(1/3)*(x^5 - 18*x^3 + sqrt(-3)*(x^5 - 18*x^3 + 9*x) + 9*x)) - (-1/432)^(1/6)*(x^6 - 69*x^4 + 63*x^2 - sqrt(-3 )*(x^6 - 69*x^4 + 63*x^2 - 27) - 27))/(x^6 + 9*x^4 + 27*x^2 + 27)) - 1/24* (-1/432)^(1/6)*(sqrt(-3) - 1)*log((2*(-1/2)^(2/3)*(5*x^5 - 30*x^3 - sqrt(- 3)*(5*x^5 - 30*x^3 + 9*x) + 9*x) + 12*(3*x^3 - sqrt(-1/3)*(x^4 - 9*x^2)...
\[ \int \frac {1}{\sqrt [3]{1-x^2} \left (3+x^2\right )} \, dx=\int \frac {1}{\sqrt [3]{- \left (x - 1\right ) \left (x + 1\right )} \left (x^{2} + 3\right )}\, dx \] Input:
integrate(1/(-x**2+1)**(1/3)/(x**2+3),x)
Output:
Integral(1/((-(x - 1)*(x + 1))**(1/3)*(x**2 + 3)), x)
\[ \int \frac {1}{\sqrt [3]{1-x^2} \left (3+x^2\right )} \, dx=\int { \frac {1}{{\left (x^{2} + 3\right )} {\left (-x^{2} + 1\right )}^{\frac {1}{3}}} \,d x } \] Input:
integrate(1/(-x^2+1)^(1/3)/(x^2+3),x, algorithm="maxima")
Output:
integrate(1/((x^2 + 3)*(-x^2 + 1)^(1/3)), x)
\[ \int \frac {1}{\sqrt [3]{1-x^2} \left (3+x^2\right )} \, dx=\int { \frac {1}{{\left (x^{2} + 3\right )} {\left (-x^{2} + 1\right )}^{\frac {1}{3}}} \,d x } \] Input:
integrate(1/(-x^2+1)^(1/3)/(x^2+3),x, algorithm="giac")
Output:
integrate(1/((x^2 + 3)*(-x^2 + 1)^(1/3)), x)
Timed out. \[ \int \frac {1}{\sqrt [3]{1-x^2} \left (3+x^2\right )} \, dx=\int \frac {1}{{\left (1-x^2\right )}^{1/3}\,\left (x^2+3\right )} \,d x \] Input:
int(1/((1 - x^2)^(1/3)*(x^2 + 3)),x)
Output:
int(1/((1 - x^2)^(1/3)*(x^2 + 3)), x)
\[ \int \frac {1}{\sqrt [3]{1-x^2} \left (3+x^2\right )} \, dx=\int \frac {1}{\left (-x^{2}+1\right )^{\frac {1}{3}} x^{2}+3 \left (-x^{2}+1\right )^{\frac {1}{3}}}d x \] Input:
int(1/(-x^2+1)^(1/3)/(x^2+3),x)
Output:
int(1/(( - x**2 + 1)**(1/3)*x**2 + 3*( - x**2 + 1)**(1/3)),x)