Integrand size = 20, antiderivative size = 107 \[ \int \frac {x}{\sqrt [3]{1-x^2} \left (3+x^2\right )^2} \, dx=-\frac {\left (1-x^2\right )^{2/3}}{8 \left (3+x^2\right )}+\frac {\arctan \left (\frac {1+\sqrt [3]{2} \sqrt [3]{1-x^2}}{\sqrt {3}}\right )}{8\ 2^{2/3} \sqrt {3}}-\frac {\log \left (3+x^2\right )}{48\ 2^{2/3}}+\frac {\log \left (2^{2/3}-\sqrt [3]{1-x^2}\right )}{16\ 2^{2/3}} \] Output:
-1/8*(-x^2+1)^(2/3)/(x^2+3)+1/48*3^(1/2)*arctan(1/3*(1+2^(1/3)*(-x^2+1)^(1 /3))*3^(1/2))*2^(1/3)-1/96*ln(x^2+3)*2^(1/3)+1/32*ln(2^(2/3)-(-x^2+1)^(1/3 ))*2^(1/3)
Time = 0.13 (sec) , antiderivative size = 112, normalized size of antiderivative = 1.05 \[ \int \frac {x}{\sqrt [3]{1-x^2} \left (3+x^2\right )^2} \, dx=\frac {1}{96} \left (-\frac {12 \left (1-x^2\right )^{2/3}}{3+x^2}+2 \sqrt [3]{2} \sqrt {3} \arctan \left (\frac {1+\sqrt [3]{2-2 x^2}}{\sqrt {3}}\right )+2 \sqrt [3]{2} \log \left (-2+\sqrt [3]{2-2 x^2}\right )-\sqrt [3]{2} \log \left (4+2 \sqrt [3]{2-2 x^2}+\left (2-2 x^2\right )^{2/3}\right )\right ) \] Input:
Integrate[x/((1 - x^2)^(1/3)*(3 + x^2)^2),x]
Output:
((-12*(1 - x^2)^(2/3))/(3 + x^2) + 2*2^(1/3)*Sqrt[3]*ArcTan[(1 + (2 - 2*x^ 2)^(1/3))/Sqrt[3]] + 2*2^(1/3)*Log[-2 + (2 - 2*x^2)^(1/3)] - 2^(1/3)*Log[4 + 2*(2 - 2*x^2)^(1/3) + (2 - 2*x^2)^(2/3)])/96
Time = 0.22 (sec) , antiderivative size = 113, normalized size of antiderivative = 1.06, number of steps used = 7, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.300, Rules used = {353, 52, 67, 16, 1082, 217}
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.
| \(\displaystyle \int \frac {x}{\sqrt [3]{1-x^2} \left (x^2+3\right )^2} \, dx\) |
| \(\Big \downarrow \) 353 |
| \(\displaystyle \frac {1}{2} \int \frac {1}{\sqrt [3]{1-x^2} \left (x^2+3\right )^2}dx^2\) |
| \(\Big \downarrow \) 52 |
| \(\displaystyle \frac {1}{2} \left (\frac {1}{12} \int \frac {1}{\sqrt [3]{1-x^2} \left (x^2+3\right )}dx^2-\frac {\left (1-x^2\right )^{2/3}}{4 \left (x^2+3\right )}\right )\) |
| \(\Big \downarrow \) 67 |
| \(\displaystyle \frac {1}{2} \left (\frac {1}{12} \left (-\frac {3 \int \frac {1}{2^{2/3}-\sqrt [3]{1-x^2}}d\sqrt [3]{1-x^2}}{2\ 2^{2/3}}+\frac {3}{2} \int \frac {1}{x^4+2^{2/3} \sqrt [3]{1-x^2}+2 \sqrt [3]{2}}d\sqrt [3]{1-x^2}-\frac {\log \left (x^2+3\right )}{2\ 2^{2/3}}\right )-\frac {\left (1-x^2\right )^{2/3}}{4 \left (x^2+3\right )}\right )\) |
| \(\Big \downarrow \) 16 |
| \(\displaystyle \frac {1}{2} \left (\frac {1}{12} \left (\frac {3}{2} \int \frac {1}{x^4+2^{2/3} \sqrt [3]{1-x^2}+2 \sqrt [3]{2}}d\sqrt [3]{1-x^2}-\frac {\log \left (x^2+3\right )}{2\ 2^{2/3}}+\frac {3 \log \left (2^{2/3}-\sqrt [3]{1-x^2}\right )}{2\ 2^{2/3}}\right )-\frac {\left (1-x^2\right )^{2/3}}{4 \left (x^2+3\right )}\right )\) |
| \(\Big \downarrow \) 1082 |
| \(\displaystyle \frac {1}{2} \left (\frac {1}{12} \left (-\frac {3 \int \frac {1}{-x^4-3}d\left (\sqrt [3]{2} \sqrt [3]{1-x^2}+1\right )}{2^{2/3}}-\frac {\log \left (x^2+3\right )}{2\ 2^{2/3}}+\frac {3 \log \left (2^{2/3}-\sqrt [3]{1-x^2}\right )}{2\ 2^{2/3}}\right )-\frac {\left (1-x^2\right )^{2/3}}{4 \left (x^2+3\right )}\right )\) |
| \(\Big \downarrow \) 217 |
| \(\displaystyle \frac {1}{2} \left (\frac {1}{12} \left (\frac {\sqrt {3} \arctan \left (\frac {\sqrt [3]{2} \sqrt [3]{1-x^2}+1}{\sqrt {3}}\right )}{2^{2/3}}-\frac {\log \left (x^2+3\right )}{2\ 2^{2/3}}+\frac {3 \log \left (2^{2/3}-\sqrt [3]{1-x^2}\right )}{2\ 2^{2/3}}\right )-\frac {\left (1-x^2\right )^{2/3}}{4 \left (x^2+3\right )}\right )\) |
Input:
Int[x/((1 - x^2)^(1/3)*(3 + x^2)^2),x]
Output:
(-1/4*(1 - x^2)^(2/3)/(3 + x^2) + ((Sqrt[3]*ArcTan[(1 + 2^(1/3)*(1 - x^2)^ (1/3))/Sqrt[3]])/2^(2/3) - Log[3 + x^2]/(2*2^(2/3)) + (3*Log[2^(2/3) - (1 - x^2)^(1/3)])/(2*2^(2/3)))/12)/2
Int[(c_.)/((a_.) + (b_.)*(x_)), x_Symbol] :> Simp[c*(Log[RemoveContent[a + b*x, x]]/b), x] /; FreeQ[{a, b, c}, x]
Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[ (a + b*x)^(m + 1)*((c + d*x)^(n + 1)/((b*c - a*d)*(m + 1))), x] - Simp[d*(( m + n + 2)/((b*c - a*d)*(m + 1))) Int[(a + b*x)^(m + 1)*(c + d*x)^n, x], x] /; FreeQ[{a, b, c, d, n}, x] && ILtQ[m, -1] && FractionQ[n] && LtQ[n, 0]
Int[1/(((a_.) + (b_.)*(x_))*((c_.) + (d_.)*(x_))^(1/3)), x_Symbol] :> With[ {q = Rt[(b*c - a*d)/b, 3]}, Simp[-Log[RemoveContent[a + b*x, x]]/(2*b*q), x ] + (Simp[3/(2*b) Subst[Int[1/(q^2 + q*x + x^2), x], x, (c + d*x)^(1/3)], x] - Simp[3/(2*b*q) Subst[Int[1/(q - x), x], x, (c + d*x)^(1/3)], x])] / ; FreeQ[{a, b, c, d}, x] && PosQ[(b*c - a*d)/b]
Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(-(Rt[-a, 2]*Rt[-b, 2])^( -1))*ArcTan[Rt[-b, 2]*(x/Rt[-a, 2])], x] /; FreeQ[{a, b}, x] && PosQ[a/b] & & (LtQ[a, 0] || LtQ[b, 0])
Int[(x_)*((a_) + (b_.)*(x_)^2)^(p_.)*((c_) + (d_.)*(x_)^2)^(q_.), x_Symbol] :> Simp[1/2 Subst[Int[(a + b*x)^p*(c + d*x)^q, x], x, x^2], x] /; FreeQ[ {a, b, c, d, p, q}, x] && NeQ[b*c - a*d, 0]
Int[((a_) + (b_.)*(x_) + (c_.)*(x_)^2)^(-1), x_Symbol] :> With[{q = 1 - 4*S implify[a*(c/b^2)]}, Simp[-2/b Subst[Int[1/(q - x^2), x], x, 1 + 2*c*(x/b )], x] /; RationalQ[q] && (EqQ[q^2, 1] || !RationalQ[b^2 - 4*a*c])] /; Fre eQ[{a, b, c}, x]
Time = 5.81 (sec) , antiderivative size = 108, normalized size of antiderivative = 1.01
| method | result | size |
| pseudoelliptic | \(\frac {\left (x^{2}+3\right ) \left (2 \sqrt {3}\, \arctan \left (\frac {\left (1+2^{\frac {1}{3}} \left (-x^{2}+1\right )^{\frac {1}{3}}\right ) \sqrt {3}}{3}\right )+2 \ln \left (\left (-x^{2}+1\right )^{\frac {1}{3}}-2^{\frac {2}{3}}\right )-\ln \left (\left (-x^{2}+1\right )^{\frac {2}{3}}+2^{\frac {2}{3}} \left (-x^{2}+1\right )^{\frac {1}{3}}+2 \,2^{\frac {1}{3}}\right )\right ) 2^{\frac {1}{3}}-12 \left (-x^{2}+1\right )^{\frac {2}{3}}}{96 x^{2}+288}\) | \(108\) |
| trager | \(-\frac {\left (-x^{2}+1\right )^{\frac {2}{3}}}{8 \left (x^{2}+3\right )}+\frac {3 \operatorname {RootOf}\left (\operatorname {RootOf}\left (\textit {\_Z}^{3}-2\right )^{2}+36 \textit {\_Z} \operatorname {RootOf}\left (\textit {\_Z}^{3}-2\right )+1296 \textit {\_Z}^{2}\right ) \ln \left (\frac {72 \operatorname {RootOf}\left (\operatorname {RootOf}\left (\textit {\_Z}^{3}-2\right )^{2}+36 \textit {\_Z} \operatorname {RootOf}\left (\textit {\_Z}^{3}-2\right )+1296 \textit {\_Z}^{2}\right ) \operatorname {RootOf}\left (\textit {\_Z}^{3}-2\right )^{3} x^{2}+1728 {\operatorname {RootOf}\left (\operatorname {RootOf}\left (\textit {\_Z}^{3}-2\right )^{2}+36 \textit {\_Z} \operatorname {RootOf}\left (\textit {\_Z}^{3}-2\right )+1296 \textit {\_Z}^{2}\right )}^{2} \operatorname {RootOf}\left (\textit {\_Z}^{3}-2\right )^{2} x^{2}+504 \left (-x^{2}+1\right )^{\frac {1}{3}} \operatorname {RootOf}\left (\operatorname {RootOf}\left (\textit {\_Z}^{3}-2\right )^{2}+36 \textit {\_Z} \operatorname {RootOf}\left (\textit {\_Z}^{3}-2\right )+1296 \textit {\_Z}^{2}\right ) \operatorname {RootOf}\left (\textit {\_Z}^{3}-2\right )+5 \operatorname {RootOf}\left (\textit {\_Z}^{3}-2\right ) x^{2}+120 \operatorname {RootOf}\left (\operatorname {RootOf}\left (\textit {\_Z}^{3}-2\right )^{2}+36 \textit {\_Z} \operatorname {RootOf}\left (\textit {\_Z}^{3}-2\right )+1296 \textit {\_Z}^{2}\right ) x^{2}+14 \left (-x^{2}+1\right )^{\frac {2}{3}}-21 \operatorname {RootOf}\left (\textit {\_Z}^{3}-2\right )-504 \operatorname {RootOf}\left (\operatorname {RootOf}\left (\textit {\_Z}^{3}-2\right )^{2}+36 \textit {\_Z} \operatorname {RootOf}\left (\textit {\_Z}^{3}-2\right )+1296 \textit {\_Z}^{2}\right )}{x^{2}+3}\right )}{4}+\frac {\operatorname {RootOf}\left (\textit {\_Z}^{3}-2\right ) \ln \left (-\frac {72 \operatorname {RootOf}\left (\operatorname {RootOf}\left (\textit {\_Z}^{3}-2\right )^{2}+36 \textit {\_Z} \operatorname {RootOf}\left (\textit {\_Z}^{3}-2\right )+1296 \textit {\_Z}^{2}\right ) \operatorname {RootOf}\left (\textit {\_Z}^{3}-2\right )^{3} x^{2}+3888 {\operatorname {RootOf}\left (\operatorname {RootOf}\left (\textit {\_Z}^{3}-2\right )^{2}+36 \textit {\_Z} \operatorname {RootOf}\left (\textit {\_Z}^{3}-2\right )+1296 \textit {\_Z}^{2}\right )}^{2} \operatorname {RootOf}\left (\textit {\_Z}^{3}-2\right )^{2} x^{2}-756 \left (-x^{2}+1\right )^{\frac {1}{3}} \operatorname {RootOf}\left (\operatorname {RootOf}\left (\textit {\_Z}^{3}-2\right )^{2}+36 \textit {\_Z} \operatorname {RootOf}\left (\textit {\_Z}^{3}-2\right )+1296 \textit {\_Z}^{2}\right ) \operatorname {RootOf}\left (\textit {\_Z}^{3}-2\right )-\operatorname {RootOf}\left (\textit {\_Z}^{3}-2\right ) x^{2}-54 \operatorname {RootOf}\left (\operatorname {RootOf}\left (\textit {\_Z}^{3}-2\right )^{2}+36 \textit {\_Z} \operatorname {RootOf}\left (\textit {\_Z}^{3}-2\right )+1296 \textit {\_Z}^{2}\right ) x^{2}-21 \left (-x^{2}+1\right )^{\frac {2}{3}}+21 \operatorname {RootOf}\left (\textit {\_Z}^{3}-2\right )+1134 \operatorname {RootOf}\left (\operatorname {RootOf}\left (\textit {\_Z}^{3}-2\right )^{2}+36 \textit {\_Z} \operatorname {RootOf}\left (\textit {\_Z}^{3}-2\right )+1296 \textit {\_Z}^{2}\right )}{x^{2}+3}\right )}{48}\) | \(479\) |
| risch | \(\text {Expression too large to display}\) | \(657\) |
Input:
int(1/(-x^2+1)^(1/3)/(x^2+3)^2*x,x,method=_RETURNVERBOSE)
Output:
((x^2+3)*(2*3^(1/2)*arctan(1/3*(1+2^(1/3)*(-x^2+1)^(1/3))*3^(1/2))+2*ln((- x^2+1)^(1/3)-2^(2/3))-ln((-x^2+1)^(2/3)+2^(2/3)*(-x^2+1)^(1/3)+2*2^(1/3))) *2^(1/3)-12*(-x^2+1)^(2/3))/(96*x^2+288)
Time = 0.08 (sec) , antiderivative size = 121, normalized size of antiderivative = 1.13 \[ \int \frac {x}{\sqrt [3]{1-x^2} \left (3+x^2\right )^2} \, dx=\frac {12 \cdot 4^{\frac {1}{6}} \sqrt {\frac {1}{3}} {\left (x^{2} + 3\right )} \arctan \left (\frac {1}{2} \cdot 4^{\frac {1}{6}} \sqrt {\frac {1}{3}} {\left (4^{\frac {1}{3}} + 2 \, {\left (-x^{2} + 1\right )}^{\frac {1}{3}}\right )}\right ) - 4^{\frac {2}{3}} {\left (x^{2} + 3\right )} \log \left (4^{\frac {2}{3}} + 4^{\frac {1}{3}} {\left (-x^{2} + 1\right )}^{\frac {1}{3}} + {\left (-x^{2} + 1\right )}^{\frac {2}{3}}\right ) + 2 \cdot 4^{\frac {2}{3}} {\left (x^{2} + 3\right )} \log \left (-4^{\frac {1}{3}} + {\left (-x^{2} + 1\right )}^{\frac {1}{3}}\right ) - 24 \, {\left (-x^{2} + 1\right )}^{\frac {2}{3}}}{192 \, {\left (x^{2} + 3\right )}} \] Input:
integrate(x/(-x^2+1)^(1/3)/(x^2+3)^2,x, algorithm="fricas")
Output:
1/192*(12*4^(1/6)*sqrt(1/3)*(x^2 + 3)*arctan(1/2*4^(1/6)*sqrt(1/3)*(4^(1/3 ) + 2*(-x^2 + 1)^(1/3))) - 4^(2/3)*(x^2 + 3)*log(4^(2/3) + 4^(1/3)*(-x^2 + 1)^(1/3) + (-x^2 + 1)^(2/3)) + 2*4^(2/3)*(x^2 + 3)*log(-4^(1/3) + (-x^2 + 1)^(1/3)) - 24*(-x^2 + 1)^(2/3))/(x^2 + 3)
\[ \int \frac {x}{\sqrt [3]{1-x^2} \left (3+x^2\right )^2} \, dx=\int \frac {x}{\sqrt [3]{- \left (x - 1\right ) \left (x + 1\right )} \left (x^{2} + 3\right )^{2}}\, dx \] Input:
integrate(x/(-x**2+1)**(1/3)/(x**2+3)**2,x)
Output:
Integral(x/((-(x - 1)*(x + 1))**(1/3)*(x**2 + 3)**2), x)
Time = 0.14 (sec) , antiderivative size = 104, normalized size of antiderivative = 0.97 \[ \int \frac {x}{\sqrt [3]{1-x^2} \left (3+x^2\right )^2} \, dx=\frac {1}{96} \cdot 4^{\frac {2}{3}} \sqrt {3} \arctan \left (\frac {1}{12} \cdot 4^{\frac {2}{3}} \sqrt {3} {\left (4^{\frac {1}{3}} + 2 \, {\left (-x^{2} + 1\right )}^{\frac {1}{3}}\right )}\right ) - \frac {1}{192} \cdot 4^{\frac {2}{3}} \log \left (4^{\frac {2}{3}} + 4^{\frac {1}{3}} {\left (-x^{2} + 1\right )}^{\frac {1}{3}} + {\left (-x^{2} + 1\right )}^{\frac {2}{3}}\right ) + \frac {1}{96} \cdot 4^{\frac {2}{3}} \log \left (-4^{\frac {1}{3}} + {\left (-x^{2} + 1\right )}^{\frac {1}{3}}\right ) - \frac {{\left (-x^{2} + 1\right )}^{\frac {2}{3}}}{8 \, {\left (x^{2} + 3\right )}} \] Input:
integrate(x/(-x^2+1)^(1/3)/(x^2+3)^2,x, algorithm="maxima")
Output:
1/96*4^(2/3)*sqrt(3)*arctan(1/12*4^(2/3)*sqrt(3)*(4^(1/3) + 2*(-x^2 + 1)^( 1/3))) - 1/192*4^(2/3)*log(4^(2/3) + 4^(1/3)*(-x^2 + 1)^(1/3) + (-x^2 + 1) ^(2/3)) + 1/96*4^(2/3)*log(-4^(1/3) + (-x^2 + 1)^(1/3)) - 1/8*(-x^2 + 1)^( 2/3)/(x^2 + 3)
Time = 0.14 (sec) , antiderivative size = 104, normalized size of antiderivative = 0.97 \[ \int \frac {x}{\sqrt [3]{1-x^2} \left (3+x^2\right )^2} \, dx=\frac {1}{96} \cdot 4^{\frac {2}{3}} \sqrt {3} \arctan \left (\frac {1}{12} \cdot 4^{\frac {2}{3}} \sqrt {3} {\left (4^{\frac {1}{3}} + 2 \, {\left (-x^{2} + 1\right )}^{\frac {1}{3}}\right )}\right ) - \frac {1}{192} \cdot 4^{\frac {2}{3}} \log \left (4^{\frac {2}{3}} + 4^{\frac {1}{3}} {\left (-x^{2} + 1\right )}^{\frac {1}{3}} + {\left (-x^{2} + 1\right )}^{\frac {2}{3}}\right ) + \frac {1}{96} \cdot 4^{\frac {2}{3}} \log \left (4^{\frac {1}{3}} - {\left (-x^{2} + 1\right )}^{\frac {1}{3}}\right ) - \frac {{\left (-x^{2} + 1\right )}^{\frac {2}{3}}}{8 \, {\left (x^{2} + 3\right )}} \] Input:
integrate(x/(-x^2+1)^(1/3)/(x^2+3)^2,x, algorithm="giac")
Output:
1/96*4^(2/3)*sqrt(3)*arctan(1/12*4^(2/3)*sqrt(3)*(4^(1/3) + 2*(-x^2 + 1)^( 1/3))) - 1/192*4^(2/3)*log(4^(2/3) + 4^(1/3)*(-x^2 + 1)^(1/3) + (-x^2 + 1) ^(2/3)) + 1/96*4^(2/3)*log(4^(1/3) - (-x^2 + 1)^(1/3)) - 1/8*(-x^2 + 1)^(2 /3)/(x^2 + 3)
Time = 1.54 (sec) , antiderivative size = 126, normalized size of antiderivative = 1.18 \[ \int \frac {x}{\sqrt [3]{1-x^2} \left (3+x^2\right )^2} \, dx=\frac {2^{1/3}\,\ln \left (\frac {{\left (1-x^2\right )}^{1/3}}{64}-\frac {2^{2/3}}{64}\right )}{48}-\frac {{\left (1-x^2\right )}^{2/3}}{8\,\left (x^2+3\right )}+\frac {2^{1/3}\,\ln \left (\frac {{\left (1-x^2\right )}^{1/3}}{64}-\frac {2^{2/3}\,{\left (-1+\sqrt {3}\,1{}\mathrm {i}\right )}^2}{256}\right )\,\left (-1+\sqrt {3}\,1{}\mathrm {i}\right )}{96}-\frac {2^{1/3}\,\ln \left (\frac {{\left (1-x^2\right )}^{1/3}}{64}-\frac {2^{2/3}\,{\left (1+\sqrt {3}\,1{}\mathrm {i}\right )}^2}{256}\right )\,\left (1+\sqrt {3}\,1{}\mathrm {i}\right )}{96} \] Input:
int(x/((1 - x^2)^(1/3)*(x^2 + 3)^2),x)
Output:
(2^(1/3)*log((1 - x^2)^(1/3)/64 - 2^(2/3)/64))/48 - (1 - x^2)^(2/3)/(8*(x^ 2 + 3)) + (2^(1/3)*log((1 - x^2)^(1/3)/64 - (2^(2/3)*(3^(1/2)*1i - 1)^2)/2 56)*(3^(1/2)*1i - 1))/96 - (2^(1/3)*log((1 - x^2)^(1/3)/64 - (2^(2/3)*(3^( 1/2)*1i + 1)^2)/256)*(3^(1/2)*1i + 1))/96
\[ \int \frac {x}{\sqrt [3]{1-x^2} \left (3+x^2\right )^2} \, dx=\int \frac {x}{\left (-x^{2}+1\right )^{\frac {1}{3}} x^{4}+6 \left (-x^{2}+1\right )^{\frac {1}{3}} x^{2}+9 \left (-x^{2}+1\right )^{\frac {1}{3}}}d x \] Input:
int(x/(-x^2+1)^(1/3)/(x^2+3)^2,x)
Output:
int(x/(( - x**2 + 1)**(1/3)*x**4 + 6*( - x**2 + 1)**(1/3)*x**2 + 9*( - x** 2 + 1)**(1/3)),x)