Integrand size = 19, antiderivative size = 78 \[ \int \frac {1}{(3-x) \sqrt [3]{1-x^2}} \, dx=-\frac {1}{4} \sqrt {3} \arctan \left (\frac {1}{\sqrt {3}}-\frac {(1+x)^{2/3}}{\sqrt {3} \sqrt [3]{1-x}}\right )-\frac {1}{4} \log (3-x)+\frac {3}{8} \log \left (-\sqrt [3]{1-x}-\frac {1}{2} (1+x)^{2/3}\right ) \]
-1/4*ln(3-x)+3/8*ln(-(1-x)^(1/3)-1/2*(1+x)^(2/3))+1/4*arctan(-1/3*3^(1/2)+ 1/3*(1+x)^(2/3)/(1-x)^(1/3)*3^(1/2))*3^(1/2)
Time = 0.19 (sec) , antiderivative size = 107, normalized size of antiderivative = 1.37 \[ \int \frac {1}{(3-x) \sqrt [3]{1-x^2}} \, dx=\frac {1}{8} \left (-2 \sqrt {3} \arctan \left (\frac {\sqrt {3} \sqrt [3]{1-x^2}}{1+x-\sqrt [3]{1-x^2}}\right )+2 \log \left (1+x+2 \sqrt [3]{1-x^2}\right )-\log \left (1+2 x+x^2-2 (1+x) \sqrt [3]{1-x^2}+4 \left (1-x^2\right )^{2/3}\right )\right ) \]
(-2*Sqrt[3]*ArcTan[(Sqrt[3]*(1 - x^2)^(1/3))/(1 + x - (1 - x^2)^(1/3))] + 2*Log[1 + x + 2*(1 - x^2)^(1/3)] - Log[1 + 2*x + x^2 - 2*(1 + x)*(1 - x^2) ^(1/3) + 4*(1 - x^2)^(2/3)])/8
Time = 0.18 (sec) , antiderivative size = 78, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.105, Rules used = {502, 133}
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.
| \(\displaystyle \int \frac {1}{(3-x) \sqrt [3]{1-x^2}} \, dx\) |
| \(\Big \downarrow \) 502 |
| \(\displaystyle \int \frac {1}{\sqrt [3]{1-x} (3-x) \sqrt [3]{x+1}}dx\) |
| \(\Big \downarrow \) 133 |
| \(\displaystyle -\frac {1}{4} \sqrt {3} \arctan \left (\frac {1}{\sqrt {3}}-\frac {(x+1)^{2/3}}{\sqrt {3} \sqrt [3]{1-x}}\right )-\frac {1}{4} \log (3-x)+\frac {3}{8} \log \left (-\frac {1}{2} (x+1)^{2/3}-\sqrt [3]{1-x}\right )\) |
-1/4*(Sqrt[3]*ArcTan[1/Sqrt[3] - (1 + x)^(2/3)/(Sqrt[3]*(1 - x)^(1/3))]) - Log[3 - x]/4 + (3*Log[-(1 - x)^(1/3) - (1 + x)^(2/3)/2])/8
3.8.14.3.1 Defintions of rubi rules used
Int[1/(((a_.) + (b_.)*(x_))*((c_.) + (d_.)*(x_))^(1/3)*((e_.) + (f_.)*(x_)) ^(1/3)), x_] :> With[{q = Rt[b*((b*e - a*f)/(b*c - a*d)^2), 3]}, Simp[-Log[ a + b*x]/(2*q*(b*c - a*d)), x] + (-Simp[Sqrt[3]*(ArcTan[1/Sqrt[3] + 2*q*((c + d*x)^(2/3)/(Sqrt[3]*(e + f*x)^(1/3)))]/(2*q*(b*c - a*d))), x] + Simp[3*( Log[q*(c + d*x)^(2/3) - (e + f*x)^(1/3)]/(4*q*(b*c - a*d))), x])] /; FreeQ[ {a, b, c, d, e, f}, x] && EqQ[2*b*d*e - b*c*f - a*d*f, 0]
Int[1/(((c_) + (d_.)*(x_))*((a_) + (b_.)*(x_)^2)^(1/3)), x_Symbol] :> Simp[ a^(1/3) Int[1/((c + d*x)*(1 - 3*d*(x/c))^(1/3)*(1 + 3*d*(x/c))^(1/3)), x] , x] /; FreeQ[{a, b, c, d}, x] && EqQ[b*c^2 + 9*a*d^2, 0] && GtQ[a, 0]
Result contains higher order function than in optimal. Order 9 vs. order 3.
Time = 2.19 (sec) , antiderivative size = 605, normalized size of antiderivative = 7.76
| method | result | size |
| trager | \(\frac {\ln \left (-\frac {448 \operatorname {RootOf}\left (4 \textit {\_Z}^{2}+2 \textit {\_Z} +1\right )^{2} x^{2}+864 \operatorname {RootOf}\left (4 \textit {\_Z}^{2}+2 \textit {\_Z} +1\right ) \left (-x^{2}+1\right )^{\frac {2}{3}}-516 \operatorname {RootOf}\left (4 \textit {\_Z}^{2}+2 \textit {\_Z} +1\right ) \left (-x^{2}+1\right )^{\frac {1}{3}} x +1344 \operatorname {RootOf}\left (4 \textit {\_Z}^{2}+2 \textit {\_Z} +1\right )^{2} x -20 \operatorname {RootOf}\left (4 \textit {\_Z}^{2}+2 \textit {\_Z} +1\right ) x^{2}-516 \left (-x^{2}+1\right )^{\frac {2}{3}}-516 \operatorname {RootOf}\left (4 \textit {\_Z}^{2}+2 \textit {\_Z} +1\right ) \left (-x^{2}+1\right )^{\frac {1}{3}}-474 x \left (-x^{2}+1\right )^{\frac {1}{3}}-1656 \operatorname {RootOf}\left (4 \textit {\_Z}^{2}+2 \textit {\_Z} +1\right ) x -3 x^{2}-474 \left (-x^{2}+1\right )^{\frac {1}{3}}+1596 \operatorname {RootOf}\left (4 \textit {\_Z}^{2}+2 \textit {\_Z} +1\right )+162 x -171}{\left (-3+x \right )^{2}}\right )}{4}-\frac {\ln \left (\frac {544 \operatorname {RootOf}\left (4 \textit {\_Z}^{2}+2 \textit {\_Z} +1\right )^{2} x^{2}+864 \operatorname {RootOf}\left (4 \textit {\_Z}^{2}+2 \textit {\_Z} +1\right ) \left (-x^{2}+1\right )^{\frac {2}{3}}+948 \operatorname {RootOf}\left (4 \textit {\_Z}^{2}+2 \textit {\_Z} +1\right ) \left (-x^{2}+1\right )^{\frac {1}{3}} x +1632 \operatorname {RootOf}\left (4 \textit {\_Z}^{2}+2 \textit {\_Z} +1\right )^{2} x +286 \operatorname {RootOf}\left (4 \textit {\_Z}^{2}+2 \textit {\_Z} +1\right ) x^{2}+948 \left (-x^{2}+1\right )^{\frac {2}{3}}+948 \operatorname {RootOf}\left (4 \textit {\_Z}^{2}+2 \textit {\_Z} +1\right ) \left (-x^{2}+1\right )^{\frac {1}{3}}+258 x \left (-x^{2}+1\right )^{\frac {1}{3}}+2796 \operatorname {RootOf}\left (4 \textit {\_Z}^{2}+2 \textit {\_Z} +1\right ) x +21 x^{2}+258 \left (-x^{2}+1\right )^{\frac {1}{3}}-1938 \operatorname {RootOf}\left (4 \textit {\_Z}^{2}+2 \textit {\_Z} +1\right )+234 x -171}{\left (-3+x \right )^{2}}\right )}{4}-\frac {\ln \left (\frac {544 \operatorname {RootOf}\left (4 \textit {\_Z}^{2}+2 \textit {\_Z} +1\right )^{2} x^{2}+864 \operatorname {RootOf}\left (4 \textit {\_Z}^{2}+2 \textit {\_Z} +1\right ) \left (-x^{2}+1\right )^{\frac {2}{3}}+948 \operatorname {RootOf}\left (4 \textit {\_Z}^{2}+2 \textit {\_Z} +1\right ) \left (-x^{2}+1\right )^{\frac {1}{3}} x +1632 \operatorname {RootOf}\left (4 \textit {\_Z}^{2}+2 \textit {\_Z} +1\right )^{2} x +286 \operatorname {RootOf}\left (4 \textit {\_Z}^{2}+2 \textit {\_Z} +1\right ) x^{2}+948 \left (-x^{2}+1\right )^{\frac {2}{3}}+948 \operatorname {RootOf}\left (4 \textit {\_Z}^{2}+2 \textit {\_Z} +1\right ) \left (-x^{2}+1\right )^{\frac {1}{3}}+258 x \left (-x^{2}+1\right )^{\frac {1}{3}}+2796 \operatorname {RootOf}\left (4 \textit {\_Z}^{2}+2 \textit {\_Z} +1\right ) x +21 x^{2}+258 \left (-x^{2}+1\right )^{\frac {1}{3}}-1938 \operatorname {RootOf}\left (4 \textit {\_Z}^{2}+2 \textit {\_Z} +1\right )+234 x -171}{\left (-3+x \right )^{2}}\right ) \operatorname {RootOf}\left (4 \textit {\_Z}^{2}+2 \textit {\_Z} +1\right )}{2}\) | \(605\) |
1/4*ln(-(448*RootOf(4*_Z^2+2*_Z+1)^2*x^2+864*RootOf(4*_Z^2+2*_Z+1)*(-x^2+1 )^(2/3)-516*RootOf(4*_Z^2+2*_Z+1)*(-x^2+1)^(1/3)*x+1344*RootOf(4*_Z^2+2*_Z +1)^2*x-20*RootOf(4*_Z^2+2*_Z+1)*x^2-516*(-x^2+1)^(2/3)-516*RootOf(4*_Z^2+ 2*_Z+1)*(-x^2+1)^(1/3)-474*x*(-x^2+1)^(1/3)-1656*RootOf(4*_Z^2+2*_Z+1)*x-3 *x^2-474*(-x^2+1)^(1/3)+1596*RootOf(4*_Z^2+2*_Z+1)+162*x-171)/(-3+x)^2)-1/ 4*ln((544*RootOf(4*_Z^2+2*_Z+1)^2*x^2+864*RootOf(4*_Z^2+2*_Z+1)*(-x^2+1)^( 2/3)+948*RootOf(4*_Z^2+2*_Z+1)*(-x^2+1)^(1/3)*x+1632*RootOf(4*_Z^2+2*_Z+1) ^2*x+286*RootOf(4*_Z^2+2*_Z+1)*x^2+948*(-x^2+1)^(2/3)+948*RootOf(4*_Z^2+2* _Z+1)*(-x^2+1)^(1/3)+258*x*(-x^2+1)^(1/3)+2796*RootOf(4*_Z^2+2*_Z+1)*x+21* x^2+258*(-x^2+1)^(1/3)-1938*RootOf(4*_Z^2+2*_Z+1)+234*x-171)/(-3+x)^2)-1/2 *ln((544*RootOf(4*_Z^2+2*_Z+1)^2*x^2+864*RootOf(4*_Z^2+2*_Z+1)*(-x^2+1)^(2 /3)+948*RootOf(4*_Z^2+2*_Z+1)*(-x^2+1)^(1/3)*x+1632*RootOf(4*_Z^2+2*_Z+1)^ 2*x+286*RootOf(4*_Z^2+2*_Z+1)*x^2+948*(-x^2+1)^(2/3)+948*RootOf(4*_Z^2+2*_ Z+1)*(-x^2+1)^(1/3)+258*x*(-x^2+1)^(1/3)+2796*RootOf(4*_Z^2+2*_Z+1)*x+21*x ^2+258*(-x^2+1)^(1/3)-1938*RootOf(4*_Z^2+2*_Z+1)+234*x-171)/(-3+x)^2)*Root Of(4*_Z^2+2*_Z+1)
Time = 0.45 (sec) , antiderivative size = 113, normalized size of antiderivative = 1.45 \[ \int \frac {1}{(3-x) \sqrt [3]{1-x^2}} \, dx=-\frac {1}{4} \, \sqrt {3} \arctan \left (\frac {18031 \, \sqrt {3} {\left (-x^{2} + 1\right )}^{\frac {1}{3}} {\left (x + 1\right )} + \sqrt {3} {\left (5054 \, x^{2} - 8497 \, x + 23659\right )} + 57889 \, \sqrt {3} {\left (-x^{2} + 1\right )}^{\frac {2}{3}}}{6859 \, x^{2} + 240699 \, x - 220122}\right ) + \frac {1}{8} \, \log \left (\frac {x^{2} + 6 \, {\left (-x^{2} + 1\right )}^{\frac {1}{3}} {\left (x + 1\right )} - 6 \, x + 12 \, {\left (-x^{2} + 1\right )}^{\frac {2}{3}} + 9}{x^{2} - 6 \, x + 9}\right ) \]
-1/4*sqrt(3)*arctan((18031*sqrt(3)*(-x^2 + 1)^(1/3)*(x + 1) + sqrt(3)*(505 4*x^2 - 8497*x + 23659) + 57889*sqrt(3)*(-x^2 + 1)^(2/3))/(6859*x^2 + 2406 99*x - 220122)) + 1/8*log((x^2 + 6*(-x^2 + 1)^(1/3)*(x + 1) - 6*x + 12*(-x ^2 + 1)^(2/3) + 9)/(x^2 - 6*x + 9))
\[ \int \frac {1}{(3-x) \sqrt [3]{1-x^2}} \, dx=- \int \frac {1}{x \sqrt [3]{1 - x^{2}} - 3 \sqrt [3]{1 - x^{2}}}\, dx \]
\[ \int \frac {1}{(3-x) \sqrt [3]{1-x^2}} \, dx=\int { -\frac {1}{{\left (-x^{2} + 1\right )}^{\frac {1}{3}} {\left (x - 3\right )}} \,d x } \]
\[ \int \frac {1}{(3-x) \sqrt [3]{1-x^2}} \, dx=\int { -\frac {1}{{\left (-x^{2} + 1\right )}^{\frac {1}{3}} {\left (x - 3\right )}} \,d x } \]
Timed out. \[ \int \frac {1}{(3-x) \sqrt [3]{1-x^2}} \, dx=-\int \frac {1}{{\left (1-x^2\right )}^{1/3}\,\left (x-3\right )} \,d x \]