\(\int \frac {x^3 \arctan (x)}{1+x^2} \, dx\) [672]

Optimal result

Integrand size = 13, antiderivative size = 67 \[ \int \frac {x^3 \arctan (x)}{1+x^2} \, dx=-\frac {x}{2}+\frac {\arctan (x)}{2}+\frac {1}{2} x^2 \arctan (x)+\frac {1}{2} i \arctan (x)^2+\arctan (x) \log \left (\frac {2}{1+i x}\right )+\frac {1}{2} i \operatorname {PolyLog}\left (2,1-\frac {2}{1+i x}\right ) \] Output:

-1/2*x+1/2*arctan(x)+1/2*x^2*arctan(x)+1/2*I*arctan(x)^2+arctan(x)*ln(2/(1 
+I*x))+1/2*I*polylog(2,1-2/(1+I*x))
 

Mathematica [A] (verified)

Time = 0.02 (sec) , antiderivative size = 57, normalized size of antiderivative = 0.85 \[ \int \frac {x^3 \arctan (x)}{1+x^2} \, dx=\frac {1}{2} \left (-x+i \arctan (x)^2+\arctan (x) \left (1+x^2+2 \log \left (-\frac {2 i}{-i+x}\right )\right )+i \operatorname {PolyLog}\left (2,\frac {i+x}{-i+x}\right )\right ) \] Input:

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

Output:

(-x + I*ArcTan[x]^2 + ArcTan[x]*(1 + x^2 + 2*Log[(-2*I)/(-I + x)]) + I*Pol 
yLog[2, (I + x)/(-I + x)])/2
 

Rubi [A] (verified)

Time = 0.43 (sec) , antiderivative size = 66, normalized size of antiderivative = 0.99, number of steps used = 9, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.615, Rules used = {5451, 5361, 262, 216, 5455, 5379, 2849, 2752}

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.

\(\displaystyle \int \frac {x^3 \arctan (x)}{x^2+1} \, dx\)

\(\Big \downarrow \) 5451

\(\displaystyle \int x \arctan (x)dx-\int \frac {x \arctan (x)}{x^2+1}dx\)

\(\Big \downarrow \) 5361

\(\displaystyle -\int \frac {x \arctan (x)}{x^2+1}dx-\frac {1}{2} \int \frac {x^2}{x^2+1}dx+\frac {1}{2} x^2 \arctan (x)\)

\(\Big \downarrow \) 262

\(\displaystyle -\int \frac {x \arctan (x)}{x^2+1}dx+\frac {1}{2} \left (\int \frac {1}{x^2+1}dx-x\right )+\frac {1}{2} x^2 \arctan (x)\)

\(\Big \downarrow \) 216

\(\displaystyle -\int \frac {x \arctan (x)}{x^2+1}dx+\frac {1}{2} x^2 \arctan (x)+\frac {1}{2} (\arctan (x)-x)\)

\(\Big \downarrow \) 5455

\(\displaystyle \int \frac {\arctan (x)}{i-x}dx+\frac {1}{2} x^2 \arctan (x)+\frac {1}{2} i \arctan (x)^2+\frac {1}{2} (\arctan (x)-x)\)

\(\Big \downarrow \) 5379

\(\displaystyle -\int \frac {\log \left (\frac {2}{i x+1}\right )}{x^2+1}dx+\frac {1}{2} x^2 \arctan (x)+\frac {1}{2} i \arctan (x)^2+\frac {1}{2} (\arctan (x)-x)+\arctan (x) \log \left (\frac {2}{1+i x}\right )\)

\(\Big \downarrow \) 2849

\(\displaystyle i \int \frac {\log \left (\frac {2}{i x+1}\right )}{1-\frac {2}{i x+1}}d\frac {1}{i x+1}+\frac {1}{2} x^2 \arctan (x)+\frac {1}{2} i \arctan (x)^2+\frac {1}{2} (\arctan (x)-x)+\arctan (x) \log \left (\frac {2}{1+i x}\right )\)

\(\Big \downarrow \) 2752

\(\displaystyle \frac {1}{2} x^2 \arctan (x)+\frac {1}{2} i \arctan (x)^2+\frac {1}{2} (\arctan (x)-x)+\arctan (x) \log \left (\frac {2}{1+i x}\right )+\frac {1}{2} i \operatorname {PolyLog}\left (2,1-\frac {2}{i x+1}\right )\)

Input:

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

Output:

(x^2*ArcTan[x])/2 + (I/2)*ArcTan[x]^2 + (-x + ArcTan[x])/2 + ArcTan[x]*Log 
[2/(1 + I*x)] + (I/2)*PolyLog[2, 1 - 2/(1 + I*x)]
 

Defintions of rubi rules used

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 262

Int[((c_.)*(x_))^(m_)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> Simp[c*(c*x) 
^(m - 1)*((a + b*x^2)^(p + 1)/(b*(m + 2*p + 1))), x] - Simp[a*c^2*((m - 1)/ 
(b*(m + 2*p + 1)))   Int[(c*x)^(m - 2)*(a + b*x^2)^p, x], x] /; FreeQ[{a, b 
, c, p}, x] && GtQ[m, 2 - 1] && NeQ[m + 2*p + 1, 0] && IntBinomialQ[a, b, c 
, 2, m, p, x]
 

rule 2752

Int[Log[(c_.)*(x_)]/((d_) + (e_.)*(x_)), x_Symbol] :> Simp[(-e^(-1))*PolyLo 
g[2, 1 - c*x], x] /; FreeQ[{c, d, e}, x] && EqQ[e + c*d, 0]
 

rule 2849

Int[Log[(c_.)/((d_) + (e_.)*(x_))]/((f_) + (g_.)*(x_)^2), x_Symbol] :> Simp 
[-e/g   Subst[Int[Log[2*d*x]/(1 - 2*d*x), x], x, 1/(d + e*x)], x] /; FreeQ[ 
{c, d, e, f, g}, x] && EqQ[c, 2*d] && EqQ[e^2*f + d^2*g, 0]
 

rule 5361

Int[((a_.) + ArcTan[(c_.)*(x_)^(n_.)]*(b_.))^(p_.)*(x_)^(m_.), x_Symbol] :> 
 Simp[x^(m + 1)*((a + b*ArcTan[c*x^n])^p/(m + 1)), x] - Simp[b*c*n*(p/(m + 
1))   Int[x^(m + n)*((a + b*ArcTan[c*x^n])^(p - 1)/(1 + c^2*x^(2*n))), x], 
x] /; FreeQ[{a, b, c, m, n}, x] && IGtQ[p, 0] && (EqQ[p, 1] || (EqQ[n, 1] & 
& IntegerQ[m])) && NeQ[m, -1]
 

rule 5379

Int[((a_.) + ArcTan[(c_.)*(x_)]*(b_.))^(p_.)/((d_) + (e_.)*(x_)), x_Symbol] 
 :> Simp[(-(a + b*ArcTan[c*x])^p)*(Log[2/(1 + e*(x/d))]/e), x] + Simp[b*c*( 
p/e)   Int[(a + b*ArcTan[c*x])^(p - 1)*(Log[2/(1 + e*(x/d))]/(1 + c^2*x^2)) 
, x], x] /; FreeQ[{a, b, c, d, e}, x] && IGtQ[p, 0] && EqQ[c^2*d^2 + e^2, 0 
]
 

rule 5451

Int[(((a_.) + ArcTan[(c_.)*(x_)]*(b_.))^(p_.)*((f_.)*(x_))^(m_))/((d_) + (e 
_.)*(x_)^2), x_Symbol] :> Simp[f^2/e   Int[(f*x)^(m - 2)*(a + b*ArcTan[c*x] 
)^p, x], x] - Simp[d*(f^2/e)   Int[(f*x)^(m - 2)*((a + b*ArcTan[c*x])^p/(d 
+ e*x^2)), x], x] /; FreeQ[{a, b, c, d, e, f}, x] && GtQ[p, 0] && GtQ[m, 1]
 

rule 5455

Int[(((a_.) + ArcTan[(c_.)*(x_)]*(b_.))^(p_.)*(x_))/((d_) + (e_.)*(x_)^2), 
x_Symbol] :> Simp[(-I)*((a + b*ArcTan[c*x])^(p + 1)/(b*e*(p + 1))), x] - Si 
mp[1/(c*d)   Int[(a + b*ArcTan[c*x])^p/(I - c*x), x], x] /; FreeQ[{a, b, c, 
 d, e}, x] && EqQ[e, c^2*d] && IGtQ[p, 0]
 

Maple [B] (verified)

Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 125 vs. \(2 (53 ) = 106\).

Time = 0.30 (sec) , antiderivative size = 126, normalized size of antiderivative = 1.88

method	result	size
default	\(\frac {x^{2} \arctan \left (x \right )}{2}-\frac {\arctan \left (x \right ) \ln \left (x^{2}+1\right )}{2}-\frac {x}{2}+\frac {\arctan \left (x \right )}{2}-\frac {i \left (\ln \left (x -i\right ) \ln \left (x^{2}+1\right )-\frac {\ln \left (x -i\right )^{2}}{2}-\operatorname {dilog}\left (-\frac {i \left (x +i\right )}{2}\right )-\ln \left (x -i\right ) \ln \left (-\frac {i \left (x +i\right )}{2}\right )\right )}{4}+\frac {i \left (\ln \left (x +i\right ) \ln \left (x^{2}+1\right )-\frac {\ln \left (x +i\right )^{2}}{2}-\operatorname {dilog}\left (\frac {i \left (x -i\right )}{2}\right )-\ln \left (x +i\right ) \ln \left (\frac {i \left (x -i\right )}{2}\right )\right )}{4}\)	\(126\)
parts	\(\frac {x^{2} \arctan \left (x \right )}{2}-\frac {\arctan \left (x \right ) \ln \left (x^{2}+1\right )}{2}-\frac {x}{2}+\frac {\arctan \left (x \right )}{2}-\frac {i \left (\ln \left (x -i\right ) \ln \left (x^{2}+1\right )-\frac {\ln \left (x -i\right )^{2}}{2}-\operatorname {dilog}\left (-\frac {i \left (x +i\right )}{2}\right )-\ln \left (x -i\right ) \ln \left (-\frac {i \left (x +i\right )}{2}\right )\right )}{4}+\frac {i \left (\ln \left (x +i\right ) \ln \left (x^{2}+1\right )-\frac {\ln \left (x +i\right )^{2}}{2}-\operatorname {dilog}\left (\frac {i \left (x -i\right )}{2}\right )-\ln \left (x +i\right ) \ln \left (\frac {i \left (x -i\right )}{2}\right )\right )}{4}\)	\(126\)
risch	\(-\frac {i \ln \left (\frac {1}{2}+\frac {i x}{2}\right ) \ln \left (-i x +1\right )}{4}-\frac {i \ln \left (i x +1\right )}{4}+\frac {i x^{2} \ln \left (-i x +1\right )}{4}-\frac {x}{2}+\frac {i \ln \left (i x +1\right )^{2}}{8}-\frac {i x^{2} \ln \left (i x +1\right )}{4}-\frac {i \operatorname {dilog}\left (\frac {1}{2}+\frac {i x}{2}\right )}{4}+\frac {i \ln \left (-i x +1\right )}{4}+\frac {i \operatorname {dilog}\left (\frac {1}{2}-\frac {i x}{2}\right )}{4}-\frac {i \ln \left (-i x +1\right )^{2}}{8}+\frac {i \ln \left (\frac {1}{2}-\frac {i x}{2}\right ) \ln \left (i x +1\right )}{4}\)	\(129\)

Input:

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

Output:

1/2*x^2*arctan(x)-1/2*arctan(x)*ln(x^2+1)-1/2*x+1/2*arctan(x)-1/4*I*(ln(x- 
I)*ln(x^2+1)-1/2*ln(x-I)^2-dilog(-1/2*I*(x+I))-ln(x-I)*ln(-1/2*I*(x+I)))+1 
/4*I*(ln(x+I)*ln(x^2+1)-1/2*ln(x+I)^2-dilog(1/2*I*(x-I))-ln(x+I)*ln(1/2*I* 
(x-I)))
                                                                                    
                                                                                    
 

Fricas [F]

\[ \int \frac {x^3 \arctan (x)}{1+x^2} \, dx=\int { \frac {x^{3} \arctan \left (x\right )}{x^{2} + 1} \,d x } \] Input:

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

Output:

integral(x^3*arctan(x)/(x^2 + 1), x)
 

Sympy [F]

\[ \int \frac {x^3 \arctan (x)}{1+x^2} \, dx=\int \frac {x^{3} \operatorname {atan}{\left (x \right )}}{x^{2} + 1}\, dx \] Input:

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

Output:

Integral(x**3*atan(x)/(x**2 + 1), x)
 

Maxima [F]

\[ \int \frac {x^3 \arctan (x)}{1+x^2} \, dx=\int { \frac {x^{3} \arctan \left (x\right )}{x^{2} + 1} \,d x } \] Input:

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

Output:

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

Giac [F]

\[ \int \frac {x^3 \arctan (x)}{1+x^2} \, dx=\int { \frac {x^{3} \arctan \left (x\right )}{x^{2} + 1} \,d x } \] Input:

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

Output:

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

Mupad [F(-1)]

Timed out. \[ \int \frac {x^3 \arctan (x)}{1+x^2} \, dx=\int \frac {x^3\,\mathrm {atan}\left (x\right )}{x^2+1} \,d x \] Input:

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

Output:

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

Reduce [F]

\[ \int \frac {x^3 \arctan (x)}{1+x^2} \, dx=\int \frac {\mathit {atan} \left (x \right ) x^{3}}{x^{2}+1}d x \] Input:

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

Output:

int((atan(x)*x**3)/(x**2 + 1),x)