\(\int \frac {x^{2/3}}{1+x^2} \, dx\) [348]

Optimal result

Integrand size = 13, antiderivative size = 73 \[ \int \frac {x^{2/3}}{1+x^2} \, dx=-\frac {1}{2} \arctan \left (\sqrt {3}-2 \sqrt [3]{x}\right )+\frac {1}{2} \arctan \left (\sqrt {3}+2 \sqrt [3]{x}\right )+\arctan \left (\sqrt [3]{x}\right )-\frac {1}{2} \sqrt {3} \text {arctanh}\left (\frac {\sqrt {3} \sqrt [3]{x}}{1+x^{2/3}}\right ) \] Output:

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

Mathematica [C] (verified)

Result contains complex when optimal does not.

Time = 0.09 (sec) , antiderivative size = 79, normalized size of antiderivative = 1.08 \[ \int \frac {x^{2/3}}{1+x^2} \, dx=\arctan \left (\sqrt [3]{x}\right )+\frac {1}{2} \left (1-i \sqrt {3}\right ) \arctan \left (\frac {1}{2} \left (1-i \sqrt {3}\right ) \sqrt [3]{x}\right )+\frac {1}{2} \left (1+i \sqrt {3}\right ) \arctan \left (\frac {1}{2} \left (1+i \sqrt {3}\right ) \sqrt [3]{x}\right ) \] Input:

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

Output:

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
 

Rubi [A] (verified)

Time = 0.26 (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

\(\displaystyle 3 \left (\frac {1}{6} \left (\frac {1}{2} \sqrt {3} \log \left (x^{2/3}-\sqrt {3} \sqrt [3]{x}+1\right )-\arctan \left (\sqrt {3}-2 \sqrt [3]{x}\right )\right )+\frac {1}{6} \left (\arctan \left (2 \sqrt [3]{x}+\sqrt {3}\right )-\frac {1}{2} \sqrt {3} \log \left (x^{2/3}+\sqrt {3} \sqrt [3]{x}+1\right )\right )+\frac {1}{3} \arctan \left (\sqrt [3]{x}\right )\right )\)

Input:

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

Output:

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)
 

Defintions of rubi rules used

rule 25

Int[-(Fx_), x_Symbol] :> Simp[Identity[-1]   Int[Fx, x], x]
 

rule 27

Int[(a_)*(Fx_), x_Symbol] :> Simp[a   Int[Fx, x], x] /; FreeQ[a, x] &&  !Ma 
tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
 

rule 216

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])
 

rule 217

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])
 

rule 266

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]
 

rule 824

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]
 

rule 1083

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]
 

rule 1103

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]
 

rule 1142

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]
 

Maple [A] (verified)

Time = 5.48 (sec) , antiderivative size = 69, normalized size of antiderivative = 0.95

method	result	size
derivativedivides	\(\frac {\sqrt {3}\, \ln \left (x^{\frac {2}{3}}-\sqrt {3}\, x^{\frac {1}{3}}+1\right )}{4}+\frac {\arctan \left (-\sqrt {3}+2 x^{\frac {1}{3}}\right )}{2}-\frac {\sqrt {3}\, \ln \left (x^{\frac {2}{3}}+\sqrt {3}\, x^{\frac {1}{3}}+1\right )}{4}+\frac {\arctan \left (\sqrt {3}+2 x^{\frac {1}{3}}\right )}{2}+\arctan \left (x^{\frac {1}{3}}\right )\)	\(69\)
default	\(\frac {\sqrt {3}\, \ln \left (x^{\frac {2}{3}}-\sqrt {3}\, x^{\frac {1}{3}}+1\right )}{4}+\frac {\arctan \left (-\sqrt {3}+2 x^{\frac {1}{3}}\right )}{2}-\frac {\sqrt {3}\, \ln \left (x^{\frac {2}{3}}+\sqrt {3}\, x^{\frac {1}{3}}+1\right )}{4}+\frac {\arctan \left (\sqrt {3}+2 x^{\frac {1}{3}}\right )}{2}+\arctan \left (x^{\frac {1}{3}}\right )\)	\(69\)
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	\(\text {Expression too large to display}\)	\(1341\)

Input:

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

Output:

1/4*3^(1/2)*ln(x^(2/3)-3^(1/2)*x^(1/3)+1)+1/2*arctan(-3^(1/2)+2*x^(1/3))-1 
/4*3^(1/2)*ln(x^(2/3)+3^(1/2)*x^(1/3)+1)+1/2*arctan(3^(1/2)+2*x^(1/3))+arc 
tan(x^(1/3))
 

Fricas [A] (verification not implemented)

Time = 0.07 (sec) , antiderivative size = 68, normalized size of antiderivative = 0.93 \[ \int \frac {x^{2/3}}{1+x^2} \, dx=-\frac {1}{4} \, \sqrt {3} \log \left (\sqrt {3} x^{\frac {1}{3}} + x^{\frac {2}{3}} + 1\right ) + \frac {1}{4} \, \sqrt {3} \log \left (-\sqrt {3} x^{\frac {1}{3}} + x^{\frac {2}{3}} + 1\right ) + \frac {1}{2} \, \arctan \left (\sqrt {3} + 2 \, x^{\frac {1}{3}}\right ) + \frac {1}{2} \, \arctan \left (-\sqrt {3} + 2 \, x^{\frac {1}{3}}\right ) + \arctan \left (x^{\frac {1}{3}}\right ) \] Input:

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

Output:

-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))
 

Sympy [A] (verification not implemented)

Time = 0.35 (sec) , antiderivative size = 94, normalized size of antiderivative = 1.29 \[ \int \frac {x^{2/3}}{1+x^2} \, dx=\frac {\sqrt {3} \log {\left (4 x^{\frac {2}{3}} - 4 \sqrt {3} \sqrt [3]{x} + 4 \right )}}{4} - \frac {\sqrt {3} \log {\left (4 x^{\frac {2}{3}} + 4 \sqrt {3} \sqrt [3]{x} + 4 \right )}}{4} + \operatorname {atan}{\left (\sqrt [3]{x} \right )} + \frac {\operatorname {atan}{\left (2 \sqrt [3]{x} - \sqrt {3} \right )}}{2} + \frac {\operatorname {atan}{\left (2 \sqrt [3]{x} + \sqrt {3} \right )}}{2} \] Input:

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

Output:

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
                                                                                    
                                                                                    
 

Maxima [A] (verification not implemented)

Time = 0.11 (sec) , antiderivative size = 68, normalized size of antiderivative = 0.93 \[ \int \frac {x^{2/3}}{1+x^2} \, dx=-\frac {1}{4} \, \sqrt {3} \log \left (\sqrt {3} x^{\frac {1}{3}} + x^{\frac {2}{3}} + 1\right ) + \frac {1}{4} \, \sqrt {3} \log \left (-\sqrt {3} x^{\frac {1}{3}} + x^{\frac {2}{3}} + 1\right ) + \frac {1}{2} \, \arctan \left (\sqrt {3} + 2 \, x^{\frac {1}{3}}\right ) + \frac {1}{2} \, \arctan \left (-\sqrt {3} + 2 \, x^{\frac {1}{3}}\right ) + \arctan \left (x^{\frac {1}{3}}\right ) \] Input:

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

Output:

-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))
 

Giac [A] (verification not implemented)

Time = 0.12 (sec) , antiderivative size = 68, normalized size of antiderivative = 0.93 \[ \int \frac {x^{2/3}}{1+x^2} \, dx=-\frac {1}{4} \, \sqrt {3} \log \left (\sqrt {3} x^{\frac {1}{3}} + x^{\frac {2}{3}} + 1\right ) + \frac {1}{4} \, \sqrt {3} \log \left (-\sqrt {3} x^{\frac {1}{3}} + x^{\frac {2}{3}} + 1\right ) + \frac {1}{2} \, \arctan \left (\sqrt {3} + 2 \, x^{\frac {1}{3}}\right ) + \frac {1}{2} \, \arctan \left (-\sqrt {3} + 2 \, x^{\frac {1}{3}}\right ) + \arctan \left (x^{\frac {1}{3}}\right ) \] Input:

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

Output:

-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))
 

Mupad [B] (verification not implemented)

Time = 0.11 (sec) , antiderivative size = 57, normalized size of antiderivative = 0.78 \[ \int \frac {x^{2/3}}{1+x^2} \, dx=\mathrm {atan}\left (x^{1/3}\right )-\mathrm {atan}\left (\frac {486\,x^{1/3}}{-243+\sqrt {3}\,243{}\mathrm {i}}\right )\,\left (\frac {1}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}\right )-\mathrm {atan}\left (\frac {486\,x^{1/3}}{243+\sqrt {3}\,243{}\mathrm {i}}\right )\,\left (-\frac {1}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}\right ) \] Input:

int(x^(2/3)/(x^2 + 1),x)
 

Output:

atan(x^(1/3)) - atan((486*x^(1/3))/(3^(1/2)*243i - 243))*((3^(1/2)*1i)/2 + 
 1/2) - atan((486*x^(1/3))/(3^(1/2)*243i + 243))*((3^(1/2)*1i)/2 - 1/2)
 

Reduce [B] (verification not implemented)

Time = 0.21 (sec) , antiderivative size = 62, normalized size of antiderivative = 0.85 \[ \int \frac {x^{2/3}}{1+x^2} \, dx=\mathit {atan} \left (x^{\frac {1}{3}}\right )+\frac {\mathit {atan} \left (2 x^{\frac {1}{3}}-\sqrt {3}\right )}{2}+\frac {\mathit {atan} \left (2 x^{\frac {1}{3}}+\sqrt {3}\right )}{2}+\frac {\sqrt {3}\, \mathrm {log}\left (x^{\frac {2}{3}}-x^{\frac {1}{3}} \sqrt {3}+1\right )}{4}-\frac {\sqrt {3}\, \mathrm {log}\left (x^{\frac {2}{3}}+x^{\frac {1}{3}} \sqrt {3}+1\right )}{4} \] Input:

int(x^(2/3)/(x^2+1),x)
 

Output:

(4*atan(x**(1/3)) + 2*atan(2*x**(1/3) - sqrt(3)) + 2*atan(2*x**(1/3) + sqr 
t(3)) + sqrt(3)*log(x**(2/3) - x**(1/3)*sqrt(3) + 1) - sqrt(3)*log(x**(2/3 
) + x**(1/3)*sqrt(3) + 1))/4