Integrand size = 13, antiderivative size = 73 \[ \int \frac {x^{2/3}}{1+x^2} \, dx=-\frac {1}{2} \arctan \left (\sqrt {3}-2 \sqrt [3]{x}\right )+\frac {1}{2} \arctan \left (\sqrt {3}+2 \sqrt [3]{x}\right )+\arctan \left (\sqrt [3]{x}\right )-\frac {1}{2} \sqrt {3} \text {arctanh}\left (\frac {\sqrt {3} \sqrt [3]{x}}{1+x^{2/3}}\right ) \]
arctan(x^(1/3))+1/2*arctan(2*x^(1/3)-3^(1/2))+1/2*arctan(2*x^(1/3)+3^(1/2) )-1/2*arctanh(x^(1/3)*3^(1/2)/(1+x^(2/3)))*3^(1/2)
Result contains complex when optimal does not.
Time = 0.13 (sec) , antiderivative size = 79, normalized size of antiderivative = 1.08 \[ \int \frac {x^{2/3}}{1+x^2} \, dx=\arctan \left (\sqrt [3]{x}\right )+\frac {1}{2} \left (1-i \sqrt {3}\right ) \arctan \left (\frac {1}{2} \left (1-i \sqrt {3}\right ) \sqrt [3]{x}\right )+\frac {1}{2} \left (1+i \sqrt {3}\right ) \arctan \left (\frac {1}{2} \left (1+i \sqrt {3}\right ) \sqrt [3]{x}\right ) \]
ArcTan[x^(1/3)] + ((1 - I*Sqrt[3])*ArcTan[((1 - I*Sqrt[3])*x^(1/3))/2])/2 + ((1 + I*Sqrt[3])*ArcTan[((1 + I*Sqrt[3])*x^(1/3))/2])/2
Time = 0.28 (sec) , antiderivative size = 110, normalized size of antiderivative = 1.51, number of steps used = 10, number of rules used = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.692, Rules used = {266, 824, 27, 216, 1142, 25, 1083, 217, 1103}
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.
\(\displaystyle \int \frac {x^{2/3}}{x^2+1} \, dx\) |
\(\Big \downarrow \) 266 |
\(\displaystyle 3 \int \frac {x^{4/3}}{x^2+1}d\sqrt [3]{x}\) |
\(\Big \downarrow \) 824 |
\(\displaystyle 3 \left (\frac {1}{3} \int \frac {1}{x^{2/3}+1}d\sqrt [3]{x}+\frac {1}{3} \int -\frac {1-\sqrt {3} \sqrt [3]{x}}{2 \left (x^{2/3}-\sqrt {3} \sqrt [3]{x}+1\right )}d\sqrt [3]{x}+\frac {1}{3} \int -\frac {\sqrt {3} \sqrt [3]{x}+1}{2 \left (x^{2/3}+\sqrt {3} \sqrt [3]{x}+1\right )}d\sqrt [3]{x}\right )\) |
\(\Big \downarrow \) 27 |
\(\displaystyle 3 \left (\frac {1}{3} \int \frac {1}{x^{2/3}+1}d\sqrt [3]{x}-\frac {1}{6} \int \frac {1-\sqrt {3} \sqrt [3]{x}}{x^{2/3}-\sqrt {3} \sqrt [3]{x}+1}d\sqrt [3]{x}-\frac {1}{6} \int \frac {\sqrt {3} \sqrt [3]{x}+1}{x^{2/3}+\sqrt {3} \sqrt [3]{x}+1}d\sqrt [3]{x}\right )\) |
\(\Big \downarrow \) 216 |
\(\displaystyle 3 \left (-\frac {1}{6} \int \frac {1-\sqrt {3} \sqrt [3]{x}}{x^{2/3}-\sqrt {3} \sqrt [3]{x}+1}d\sqrt [3]{x}-\frac {1}{6} \int \frac {\sqrt {3} \sqrt [3]{x}+1}{x^{2/3}+\sqrt {3} \sqrt [3]{x}+1}d\sqrt [3]{x}+\frac {1}{3} \arctan \left (\sqrt [3]{x}\right )\right )\) |
\(\Big \downarrow \) 1142 |
\(\displaystyle 3 \left (\frac {1}{6} \left (\frac {1}{2} \int \frac {1}{x^{2/3}-\sqrt {3} \sqrt [3]{x}+1}d\sqrt [3]{x}+\frac {1}{2} \sqrt {3} \int -\frac {\sqrt {3}-2 \sqrt [3]{x}}{x^{2/3}-\sqrt {3} \sqrt [3]{x}+1}d\sqrt [3]{x}\right )+\frac {1}{6} \left (\frac {1}{2} \int \frac {1}{x^{2/3}+\sqrt {3} \sqrt [3]{x}+1}d\sqrt [3]{x}-\frac {1}{2} \sqrt {3} \int \frac {2 \sqrt [3]{x}+\sqrt {3}}{x^{2/3}+\sqrt {3} \sqrt [3]{x}+1}d\sqrt [3]{x}\right )+\frac {1}{3} \arctan \left (\sqrt [3]{x}\right )\right )\) |
\(\Big \downarrow \) 25 |
\(\displaystyle 3 \left (\frac {1}{6} \left (\frac {1}{2} \int \frac {1}{x^{2/3}-\sqrt {3} \sqrt [3]{x}+1}d\sqrt [3]{x}-\frac {1}{2} \sqrt {3} \int \frac {\sqrt {3}-2 \sqrt [3]{x}}{x^{2/3}-\sqrt {3} \sqrt [3]{x}+1}d\sqrt [3]{x}\right )+\frac {1}{6} \left (\frac {1}{2} \int \frac {1}{x^{2/3}+\sqrt {3} \sqrt [3]{x}+1}d\sqrt [3]{x}-\frac {1}{2} \sqrt {3} \int \frac {2 \sqrt [3]{x}+\sqrt {3}}{x^{2/3}+\sqrt {3} \sqrt [3]{x}+1}d\sqrt [3]{x}\right )+\frac {1}{3} \arctan \left (\sqrt [3]{x}\right )\right )\) |
\(\Big \downarrow \) 1083 |
\(\displaystyle 3 \left (\frac {1}{6} \left (-\int \frac {1}{-x^{2/3}-1}d\left (2 \sqrt [3]{x}-\sqrt {3}\right )-\frac {1}{2} \sqrt {3} \int \frac {\sqrt {3}-2 \sqrt [3]{x}}{x^{2/3}-\sqrt {3} \sqrt [3]{x}+1}d\sqrt [3]{x}\right )+\frac {1}{6} \left (-\int \frac {1}{-x^{2/3}-1}d\left (2 \sqrt [3]{x}+\sqrt {3}\right )-\frac {1}{2} \sqrt {3} \int \frac {2 \sqrt [3]{x}+\sqrt {3}}{x^{2/3}+\sqrt {3} \sqrt [3]{x}+1}d\sqrt [3]{x}\right )+\frac {1}{3} \arctan \left (\sqrt [3]{x}\right )\right )\) |
\(\Big \downarrow \) 217 |
\(\displaystyle 3 \left (\frac {1}{6} \left (-\frac {1}{2} \sqrt {3} \int \frac {\sqrt {3}-2 \sqrt [3]{x}}{x^{2/3}-\sqrt {3} \sqrt [3]{x}+1}d\sqrt [3]{x}-\arctan \left (\sqrt {3}-2 \sqrt [3]{x}\right )\right )+\frac {1}{6} \left (\arctan \left (2 \sqrt [3]{x}+\sqrt {3}\right )-\frac {1}{2} \sqrt {3} \int \frac {2 \sqrt [3]{x}+\sqrt {3}}{x^{2/3}+\sqrt {3} \sqrt [3]{x}+1}d\sqrt [3]{x}\right )+\frac {1}{3} \arctan \left (\sqrt [3]{x}\right )\right )\) |
\(\Big \downarrow \) 1103 |
\(\displaystyle 3 \left (\frac {1}{6} \left (\frac {1}{2} \sqrt {3} \log \left (x^{2/3}-\sqrt {3} \sqrt [3]{x}+1\right )-\arctan \left (\sqrt {3}-2 \sqrt [3]{x}\right )\right )+\frac {1}{6} \left (\arctan \left (2 \sqrt [3]{x}+\sqrt {3}\right )-\frac {1}{2} \sqrt {3} \log \left (x^{2/3}+\sqrt {3} \sqrt [3]{x}+1\right )\right )+\frac {1}{3} \arctan \left (\sqrt [3]{x}\right )\right )\) |
3*(ArcTan[x^(1/3)]/3 + (-ArcTan[Sqrt[3] - 2*x^(1/3)] + (Sqrt[3]*Log[1 - Sq rt[3]*x^(1/3) + x^(2/3)])/2)/6 + (ArcTan[Sqrt[3] + 2*x^(1/3)] - (Sqrt[3]*L og[1 + Sqrt[3]*x^(1/3) + x^(2/3)])/2)/6)
3.4.38.3.1 Defintions of rubi rules used
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1/(Rt[a, 2]*Rt[b, 2]))*A rcTan[Rt[b, 2]*(x/Rt[a, 2])], x] /; FreeQ[{a, b}, x] && PosQ[a/b] && (GtQ[a , 0] || GtQ[b, 0])
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[((c_.)*(x_))^(m_)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> With[{k = De nominator[m]}, Simp[k/c Subst[Int[x^(k*(m + 1) - 1)*(a + b*(x^(2*k)/c^2)) ^p, x], x, (c*x)^(1/k)], x]] /; FreeQ[{a, b, c, p}, x] && FractionQ[m] && I ntBinomialQ[a, b, c, 2, m, p, x]
Int[(x_)^(m_.)/((a_) + (b_.)*(x_)^(n_)), x_Symbol] :> Module[{r = Numerator [Rt[a/b, n]], s = Denominator[Rt[a/b, n]], k, u}, Simp[u = Int[(r*Cos[(2*k - 1)*m*(Pi/n)] - s*Cos[(2*k - 1)*(m + 1)*(Pi/n)]*x)/(r^2 - 2*r*s*Cos[(2*k - 1)*(Pi/n)]*x + s^2*x^2), x] + Int[(r*Cos[(2*k - 1)*m*(Pi/n)] + s*Cos[(2*k - 1)*(m + 1)*(Pi/n)]*x)/(r^2 + 2*r*s*Cos[(2*k - 1)*(Pi/n)]*x + s^2*x^2), x] ; 2*(-1)^(m/2)*(r^(m + 2)/(a*n*s^m)) Int[1/(r^2 + s^2*x^2), x] + 2*(r^(m + 1)/(a*n*s^m)) Sum[u, {k, 1, (n - 2)/4}], x]] /; FreeQ[{a, b}, x] && IGt Q[(n - 2)/4, 0] && IGtQ[m, 0] && LtQ[m, n - 1] && PosQ[a/b]
Int[((a_) + (b_.)*(x_) + (c_.)*(x_)^2)^(-1), x_Symbol] :> Simp[-2 Subst[I nt[1/Simp[b^2 - 4*a*c - x^2, x], x], x, b + 2*c*x], x] /; FreeQ[{a, b, c}, x]
Int[((d_) + (e_.)*(x_))/((a_.) + (b_.)*(x_) + (c_.)*(x_)^2), x_Symbol] :> S imp[d*(Log[RemoveContent[a + b*x + c*x^2, x]]/b), x] /; FreeQ[{a, b, c, d, e}, x] && EqQ[2*c*d - b*e, 0]
Int[((d_.) + (e_.)*(x_))/((a_) + (b_.)*(x_) + (c_.)*(x_)^2), x_Symbol] :> S imp[(2*c*d - b*e)/(2*c) Int[1/(a + b*x + c*x^2), x], x] + Simp[e/(2*c) Int[(b + 2*c*x)/(a + b*x + c*x^2), x], x] /; FreeQ[{a, b, c, d, e}, x]
Time = 6.45 (sec) , antiderivative size = 69, normalized size of antiderivative = 0.95
method | result | size |
derivativedivides | \(\arctan \left (x^{\frac {1}{3}}\right )+\frac {\sqrt {3}\, \ln \left (x^{\frac {2}{3}}-\sqrt {3}\, x^{\frac {1}{3}}+1\right )}{4}+\frac {\arctan \left (2 x^{\frac {1}{3}}-\sqrt {3}\right )}{2}-\frac {\sqrt {3}\, \ln \left (x^{\frac {2}{3}}+\sqrt {3}\, x^{\frac {1}{3}}+1\right )}{4}+\frac {\arctan \left (2 x^{\frac {1}{3}}+\sqrt {3}\right )}{2}\) | \(69\) |
default | \(\arctan \left (x^{\frac {1}{3}}\right )+\frac {\sqrt {3}\, \ln \left (x^{\frac {2}{3}}-\sqrt {3}\, x^{\frac {1}{3}}+1\right )}{4}+\frac {\arctan \left (2 x^{\frac {1}{3}}-\sqrt {3}\right )}{2}-\frac {\sqrt {3}\, \ln \left (x^{\frac {2}{3}}+\sqrt {3}\, x^{\frac {1}{3}}+1\right )}{4}+\frac {\arctan \left (2 x^{\frac {1}{3}}+\sqrt {3}\right )}{2}\) | \(69\) |
meijerg | \(\frac {x^{\frac {5}{3}} \sqrt {3}\, \ln \left (1-\sqrt {3}\, \left (x^{2}\right )^{\frac {1}{6}}+\left (x^{2}\right )^{\frac {1}{3}}\right )}{4 \left (x^{2}\right )^{\frac {5}{6}}}+\frac {x^{\frac {5}{3}} \arctan \left (\frac {\left (x^{2}\right )^{\frac {1}{6}}}{2-\sqrt {3}\, \left (x^{2}\right )^{\frac {1}{6}}}\right )}{2 \left (x^{2}\right )^{\frac {5}{6}}}+\frac {x^{\frac {5}{3}} \arctan \left (\left (x^{2}\right )^{\frac {1}{6}}\right )}{\left (x^{2}\right )^{\frac {5}{6}}}-\frac {x^{\frac {5}{3}} \sqrt {3}\, \ln \left (1+\sqrt {3}\, \left (x^{2}\right )^{\frac {1}{6}}+\left (x^{2}\right )^{\frac {1}{3}}\right )}{4 \left (x^{2}\right )^{\frac {5}{6}}}+\frac {x^{\frac {5}{3}} \arctan \left (\frac {\left (x^{2}\right )^{\frac {1}{6}}}{2+\sqrt {3}\, \left (x^{2}\right )^{\frac {1}{6}}}\right )}{2 \left (x^{2}\right )^{\frac {5}{6}}}\) | \(142\) |
trager | \(\text {Expression too large to display}\) | \(2024\) |
arctan(x^(1/3))+1/4*3^(1/2)*ln(x^(2/3)-3^(1/2)*x^(1/3)+1)+1/2*arctan(2*x^( 1/3)-3^(1/2))-1/4*3^(1/2)*ln(x^(2/3)+3^(1/2)*x^(1/3)+1)+1/2*arctan(2*x^(1/ 3)+3^(1/2))
Leaf count of result is larger than twice the leaf count of optimal. 155 vs. \(2 (51) = 102\).
Time = 0.26 (sec) , antiderivative size = 155, normalized size of antiderivative = 2.12 \[ \int \frac {x^{2/3}}{1+x^2} \, dx=\frac {1}{4} \, \sqrt {2} \sqrt {\sqrt {-3} + 1} \log \left (\sqrt {2} \sqrt {\sqrt {-3} + 1} {\left (\sqrt {-3} - 1\right )} + 4 \, x^{\frac {1}{3}}\right ) - \frac {1}{4} \, \sqrt {2} \sqrt {\sqrt {-3} + 1} \log \left (-\sqrt {2} \sqrt {\sqrt {-3} + 1} {\left (\sqrt {-3} - 1\right )} + 4 \, x^{\frac {1}{3}}\right ) - \frac {1}{4} \, \sqrt {2} \sqrt {-\sqrt {-3} + 1} \log \left (\sqrt {2} {\left (\sqrt {-3} + 1\right )} \sqrt {-\sqrt {-3} + 1} + 4 \, x^{\frac {1}{3}}\right ) + \frac {1}{4} \, \sqrt {2} \sqrt {-\sqrt {-3} + 1} \log \left (-\sqrt {2} {\left (\sqrt {-3} + 1\right )} \sqrt {-\sqrt {-3} + 1} + 4 \, x^{\frac {1}{3}}\right ) + \arctan \left (x^{\frac {1}{3}}\right ) \]
1/4*sqrt(2)*sqrt(sqrt(-3) + 1)*log(sqrt(2)*sqrt(sqrt(-3) + 1)*(sqrt(-3) - 1) + 4*x^(1/3)) - 1/4*sqrt(2)*sqrt(sqrt(-3) + 1)*log(-sqrt(2)*sqrt(sqrt(-3 ) + 1)*(sqrt(-3) - 1) + 4*x^(1/3)) - 1/4*sqrt(2)*sqrt(-sqrt(-3) + 1)*log(s qrt(2)*(sqrt(-3) + 1)*sqrt(-sqrt(-3) + 1) + 4*x^(1/3)) + 1/4*sqrt(2)*sqrt( -sqrt(-3) + 1)*log(-sqrt(2)*(sqrt(-3) + 1)*sqrt(-sqrt(-3) + 1) + 4*x^(1/3) ) + arctan(x^(1/3))
Time = 0.38 (sec) , antiderivative size = 94, normalized size of antiderivative = 1.29 \[ \int \frac {x^{2/3}}{1+x^2} \, dx=\frac {\sqrt {3} \log {\left (4 x^{\frac {2}{3}} - 4 \sqrt {3} \sqrt [3]{x} + 4 \right )}}{4} - \frac {\sqrt {3} \log {\left (4 x^{\frac {2}{3}} + 4 \sqrt {3} \sqrt [3]{x} + 4 \right )}}{4} + \operatorname {atan}{\left (\sqrt [3]{x} \right )} + \frac {\operatorname {atan}{\left (2 \sqrt [3]{x} - \sqrt {3} \right )}}{2} + \frac {\operatorname {atan}{\left (2 \sqrt [3]{x} + \sqrt {3} \right )}}{2} \]
sqrt(3)*log(4*x**(2/3) - 4*sqrt(3)*x**(1/3) + 4)/4 - sqrt(3)*log(4*x**(2/3 ) + 4*sqrt(3)*x**(1/3) + 4)/4 + atan(x**(1/3)) + atan(2*x**(1/3) - sqrt(3) )/2 + atan(2*x**(1/3) + sqrt(3))/2
Time = 0.28 (sec) , antiderivative size = 68, normalized size of antiderivative = 0.93 \[ \int \frac {x^{2/3}}{1+x^2} \, dx=-\frac {1}{4} \, \sqrt {3} \log \left (\sqrt {3} x^{\frac {1}{3}} + x^{\frac {2}{3}} + 1\right ) + \frac {1}{4} \, \sqrt {3} \log \left (-\sqrt {3} x^{\frac {1}{3}} + x^{\frac {2}{3}} + 1\right ) + \frac {1}{2} \, \arctan \left (\sqrt {3} + 2 \, x^{\frac {1}{3}}\right ) + \frac {1}{2} \, \arctan \left (-\sqrt {3} + 2 \, x^{\frac {1}{3}}\right ) + \arctan \left (x^{\frac {1}{3}}\right ) \]
-1/4*sqrt(3)*log(sqrt(3)*x^(1/3) + x^(2/3) + 1) + 1/4*sqrt(3)*log(-sqrt(3) *x^(1/3) + x^(2/3) + 1) + 1/2*arctan(sqrt(3) + 2*x^(1/3)) + 1/2*arctan(-sq rt(3) + 2*x^(1/3)) + arctan(x^(1/3))
Time = 0.28 (sec) , antiderivative size = 68, normalized size of antiderivative = 0.93 \[ \int \frac {x^{2/3}}{1+x^2} \, dx=-\frac {1}{4} \, \sqrt {3} \log \left (\sqrt {3} x^{\frac {1}{3}} + x^{\frac {2}{3}} + 1\right ) + \frac {1}{4} \, \sqrt {3} \log \left (-\sqrt {3} x^{\frac {1}{3}} + x^{\frac {2}{3}} + 1\right ) + \frac {1}{2} \, \arctan \left (\sqrt {3} + 2 \, x^{\frac {1}{3}}\right ) + \frac {1}{2} \, \arctan \left (-\sqrt {3} + 2 \, x^{\frac {1}{3}}\right ) + \arctan \left (x^{\frac {1}{3}}\right ) \]
-1/4*sqrt(3)*log(sqrt(3)*x^(1/3) + x^(2/3) + 1) + 1/4*sqrt(3)*log(-sqrt(3) *x^(1/3) + x^(2/3) + 1) + 1/2*arctan(sqrt(3) + 2*x^(1/3)) + 1/2*arctan(-sq rt(3) + 2*x^(1/3)) + arctan(x^(1/3))
Time = 4.80 (sec) , antiderivative size = 57, normalized size of antiderivative = 0.78 \[ \int \frac {x^{2/3}}{1+x^2} \, dx=\mathrm {atan}\left (x^{1/3}\right )-\mathrm {atan}\left (\frac {486\,x^{1/3}}{-243+\sqrt {3}\,243{}\mathrm {i}}\right )\,\left (\frac {1}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}\right )-\mathrm {atan}\left (\frac {486\,x^{1/3}}{243+\sqrt {3}\,243{}\mathrm {i}}\right )\,\left (-\frac {1}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}\right ) \]