\(\int \frac {x^3 \arctan (a x)^2}{c+a^2 c x^2} \, dx\) [283]

Optimal result

Integrand size = 22, antiderivative size = 169 \[ \int \frac {x^3 \arctan (a x)^2}{c+a^2 c x^2} \, dx=-\frac {x \arctan (a x)}{a^3 c}+\frac {\arctan (a x)^2}{2 a^4 c}+\frac {x^2 \arctan (a x)^2}{2 a^2 c}+\frac {i \arctan (a x)^3}{3 a^4 c}+\frac {\arctan (a x)^2 \log \left (\frac {2}{1+i a x}\right )}{a^4 c}+\frac {\log \left (1+a^2 x^2\right )}{2 a^4 c}+\frac {i \arctan (a x) \operatorname {PolyLog}\left (2,1-\frac {2}{1+i a x}\right )}{a^4 c}+\frac {\operatorname {PolyLog}\left (3,1-\frac {2}{1+i a x}\right )}{2 a^4 c} \] Output:

-x*arctan(a*x)/a^3/c+1/2*arctan(a*x)^2/a^4/c+1/2*x^2*arctan(a*x)^2/a^2/c+1 
/3*I*arctan(a*x)^3/a^4/c+arctan(a*x)^2*ln(2/(1+I*a*x))/a^4/c+1/2*ln(a^2*x^ 
2+1)/a^4/c+I*arctan(a*x)*polylog(2,1-2/(1+I*a*x))/a^4/c+1/2*polylog(3,1-2/ 
(1+I*a*x))/a^4/c
 

Mathematica [A] (verified)

Time = 0.10 (sec) , antiderivative size = 123, normalized size of antiderivative = 0.73 \[ \int \frac {x^3 \arctan (a x)^2}{c+a^2 c x^2} \, dx=\frac {-a x \arctan (a x)+\frac {1}{2} \left (1+a^2 x^2\right ) \arctan (a x)^2-\frac {1}{3} i \arctan (a x)^3+\arctan (a x)^2 \log \left (1+e^{2 i \arctan (a x)}\right )-\log \left (\frac {1}{\sqrt {1+a^2 x^2}}\right )-i \arctan (a x) \operatorname {PolyLog}\left (2,-e^{2 i \arctan (a x)}\right )+\frac {1}{2} \operatorname {PolyLog}\left (3,-e^{2 i \arctan (a x)}\right )}{a^4 c} \] Input:

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

Output:

(-(a*x*ArcTan[a*x]) + ((1 + a^2*x^2)*ArcTan[a*x]^2)/2 - (I/3)*ArcTan[a*x]^ 
3 + ArcTan[a*x]^2*Log[1 + E^((2*I)*ArcTan[a*x])] - Log[1/Sqrt[1 + a^2*x^2] 
] - I*ArcTan[a*x]*PolyLog[2, -E^((2*I)*ArcTan[a*x])] + PolyLog[3, -E^((2*I 
)*ArcTan[a*x])]/2)/(a^4*c)
 

Rubi [A] (verified)

Time = 1.24 (sec) , antiderivative size = 175, normalized size of antiderivative = 1.04, number of steps used = 11, number of rules used = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.500, Rules used = {5451, 27, 5361, 5451, 5345, 240, 5419, 5455, 5379, 5529, 7164}

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.

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

\(\Big \downarrow \) 5451

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 5361

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

\(\Big \downarrow \) 5451

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

\(\Big \downarrow \) 5345

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

\(\Big \downarrow \) 240

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

\(\Big \downarrow \) 5419

\(\displaystyle \frac {\frac {1}{2} x^2 \arctan (a x)^2-a \left (\frac {x \arctan (a x)-\frac {\log \left (a^2 x^2+1\right )}{2 a}}{a^2}-\frac {\arctan (a x)^2}{2 a^3}\right )}{a^2 c}-\frac {\int \frac {x \arctan (a x)^2}{a^2 x^2+1}dx}{a^2 c}\)

\(\Big \downarrow \) 5455

\(\displaystyle \frac {\frac {1}{2} x^2 \arctan (a x)^2-a \left (\frac {x \arctan (a x)-\frac {\log \left (a^2 x^2+1\right )}{2 a}}{a^2}-\frac {\arctan (a x)^2}{2 a^3}\right )}{a^2 c}-\frac {-\frac {\int \frac {\arctan (a x)^2}{i-a x}dx}{a}-\frac {i \arctan (a x)^3}{3 a^2}}{a^2 c}\)

\(\Big \downarrow \) 5379

\(\displaystyle \frac {\frac {1}{2} x^2 \arctan (a x)^2-a \left (\frac {x \arctan (a x)-\frac {\log \left (a^2 x^2+1\right )}{2 a}}{a^2}-\frac {\arctan (a x)^2}{2 a^3}\right )}{a^2 c}-\frac {-\frac {\frac {\arctan (a x)^2 \log \left (\frac {2}{1+i a x}\right )}{a}-2 \int \frac {\arctan (a x) \log \left (\frac {2}{i a x+1}\right )}{a^2 x^2+1}dx}{a}-\frac {i \arctan (a x)^3}{3 a^2}}{a^2 c}\)

\(\Big \downarrow \) 5529

\(\displaystyle \frac {\frac {1}{2} x^2 \arctan (a x)^2-a \left (\frac {x \arctan (a x)-\frac {\log \left (a^2 x^2+1\right )}{2 a}}{a^2}-\frac {\arctan (a x)^2}{2 a^3}\right )}{a^2 c}-\frac {-\frac {\frac {\arctan (a x)^2 \log \left (\frac {2}{1+i a x}\right )}{a}-2 \left (\frac {1}{2} i \int \frac {\operatorname {PolyLog}\left (2,1-\frac {2}{i a x+1}\right )}{a^2 x^2+1}dx-\frac {i \arctan (a x) \operatorname {PolyLog}\left (2,1-\frac {2}{i a x+1}\right )}{2 a}\right )}{a}-\frac {i \arctan (a x)^3}{3 a^2}}{a^2 c}\)

\(\Big \downarrow \) 7164

\(\displaystyle \frac {\frac {1}{2} x^2 \arctan (a x)^2-a \left (\frac {x \arctan (a x)-\frac {\log \left (a^2 x^2+1\right )}{2 a}}{a^2}-\frac {\arctan (a x)^2}{2 a^3}\right )}{a^2 c}-\frac {-\frac {i \arctan (a x)^3}{3 a^2}-\frac {\frac {\arctan (a x)^2 \log \left (\frac {2}{1+i a x}\right )}{a}-2 \left (-\frac {i \arctan (a x) \operatorname {PolyLog}\left (2,1-\frac {2}{i a x+1}\right )}{2 a}-\frac {\operatorname {PolyLog}\left (3,1-\frac {2}{i a x+1}\right )}{4 a}\right )}{a}}{a^2 c}\)

Input:

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

Output:

((x^2*ArcTan[a*x]^2)/2 - a*(-1/2*ArcTan[a*x]^2/a^3 + (x*ArcTan[a*x] - Log[ 
1 + a^2*x^2]/(2*a))/a^2))/(a^2*c) - (((-1/3*I)*ArcTan[a*x]^3)/a^2 - ((ArcT 
an[a*x]^2*Log[2/(1 + I*a*x)])/a - 2*(((-1/2*I)*ArcTan[a*x]*PolyLog[2, 1 - 
2/(1 + I*a*x)])/a - PolyLog[3, 1 - 2/(1 + I*a*x)]/(4*a)))/a)/(a^2*c)
 

Defintions of rubi rules used

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 240

Int[(x_)/((a_) + (b_.)*(x_)^2), x_Symbol] :> Simp[Log[RemoveContent[a + b*x 
^2, x]]/(2*b), x] /; FreeQ[{a, b}, x]
 

rule 5345

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

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 5419

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

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]
 

rule 5529

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

rule 7164

Int[(u_)*PolyLog[n_, v_], x_Symbol] :> With[{w = DerivativeDivides[v, u*v, 
x]}, Simp[w*PolyLog[n + 1, v], x] /;  !FalseQ[w]] /; FreeQ[n, x]
 

Maple [C] (warning: unable to verify)

Result contains higher order function than in optimal. Order 9 vs. order 4.

Time = 19.53 (sec) , antiderivative size = 874, normalized size of antiderivative = 5.17

method	result	size
derivativedivides	\(\text {Expression too large to display}\)	\(874\)
default	\(\text {Expression too large to display}\)	\(874\)
parts	\(\text {Expression too large to display}\)	\(929\)

Input:

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

Output:

1/a^4*(1/2/c*arctan(a*x)^2*a^2*x^2-1/2/c*arctan(a*x)^2*ln(a^2*x^2+1)-1/c*( 
I*arctan(a*x)*polylog(2,-(1+I*a*x)^2/(a^2*x^2+1))-1/2*polylog(3,-(1+I*a*x) 
^2/(a^2*x^2+1))-arctan(a*x)^2*ln((1+I*a*x)/(a^2*x^2+1)^(1/2))+1/4*I*Pi*csg 
n(I/((1+I*a*x)^2/(a^2*x^2+1)+1)^2)*csgn(I*(1+I*a*x)^2/(a^2*x^2+1))*csgn(I* 
(1+I*a*x)^2/(a^2*x^2+1)/((1+I*a*x)^2/(a^2*x^2+1)+1)^2)*arctan(a*x)^2+1/2*I 
*Pi*csgn(I*((1+I*a*x)^2/(a^2*x^2+1)+1))*csgn(I*((1+I*a*x)^2/(a^2*x^2+1)+1) 
^2)^2*arctan(a*x)^2+ln((1+I*a*x)^2/(a^2*x^2+1)+1)+1/4*I*Pi*csgn(I*(1+I*a*x 
)^2/(a^2*x^2+1)/((1+I*a*x)^2/(a^2*x^2+1)+1)^2)^3*arctan(a*x)^2-1/2*arctan( 
a*x)^2-1/4*I*Pi*csgn(I*((1+I*a*x)^2/(a^2*x^2+1)+1))^2*csgn(I*((1+I*a*x)^2/ 
(a^2*x^2+1)+1)^2)*arctan(a*x)^2-1/2*I*Pi*csgn(I*(1+I*a*x)/(a^2*x^2+1)^(1/2 
))*csgn(I*(1+I*a*x)^2/(a^2*x^2+1))^2*arctan(a*x)^2-1/4*I*Pi*csgn(I*(1+I*a* 
x)^2/(a^2*x^2+1))*csgn(I*(1+I*a*x)^2/(a^2*x^2+1)/((1+I*a*x)^2/(a^2*x^2+1)+ 
1)^2)^2*arctan(a*x)^2-1/4*I*Pi*csgn(I/((1+I*a*x)^2/(a^2*x^2+1)+1)^2)*csgn( 
I*(1+I*a*x)^2/(a^2*x^2+1)/((1+I*a*x)^2/(a^2*x^2+1)+1)^2)^2*arctan(a*x)^2+1 
/4*I*Pi*csgn(I*(1+I*a*x)^2/(a^2*x^2+1))^3*arctan(a*x)^2+1/3*I*arctan(a*x)^ 
3-1/4*I*Pi*csgn(I*((1+I*a*x)^2/(a^2*x^2+1)+1)^2)^3*arctan(a*x)^2+arctan(a* 
x)*(a*x-I)+1/4*I*Pi*csgn(I*(1+I*a*x)/(a^2*x^2+1)^(1/2))^2*csgn(I*(1+I*a*x) 
^2/(a^2*x^2+1))*arctan(a*x)^2-arctan(a*x)^2*ln(2)))
 

Fricas [F]

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

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

Output:

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

Sympy [F]

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

integrate(x**3*atan(a*x)**2/(a**2*c*x**2+c),x)
 

Output:

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

Maxima [F]

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

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

Output:

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

Giac [F]

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

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

Output:

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

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

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

Output:

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