3.3.95 \(\int \frac {\arctan (a x)^2}{x (c+a^2 c x^2)^2} \, dx\) [295]

3.3.95.1 Optimal result

Integrand size = 22, antiderivative size = 170 \[ \int \frac {\arctan (a x)^2}{x \left (c+a^2 c x^2\right )^2} \, dx=-\frac {1}{4 c^2 \left (1+a^2 x^2\right )}-\frac {a x \arctan (a x)}{2 c^2 \left (1+a^2 x^2\right )}-\frac {\arctan (a x)^2}{4 c^2}+\frac {\arctan (a x)^2}{2 c^2 \left (1+a^2 x^2\right )}-\frac {i \arctan (a x)^3}{3 c^2}+\frac {\arctan (a x)^2 \log \left (2-\frac {2}{1-i a x}\right )}{c^2}-\frac {i \arctan (a x) \operatorname {PolyLog}\left (2,-1+\frac {2}{1-i a x}\right )}{c^2}+\frac {\operatorname {PolyLog}\left (3,-1+\frac {2}{1-i a x}\right )}{2 c^2} \]

output

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

3.3.95.2 Mathematica [A] (verified)

Time = 0.17 (sec) , antiderivative size = 119, normalized size of antiderivative = 0.70 \[ \int \frac {\arctan (a x)^2}{x \left (c+a^2 c x^2\right )^2} \, dx=\frac {-i \pi ^3+8 i \arctan (a x)^3-3 \cos (2 \arctan (a x))+6 \arctan (a x)^2 \cos (2 \arctan (a x))+24 \arctan (a x)^2 \log \left (1-e^{-2 i \arctan (a x)}\right )+24 i \arctan (a x) \operatorname {PolyLog}\left (2,e^{-2 i \arctan (a x)}\right )+12 \operatorname {PolyLog}\left (3,e^{-2 i \arctan (a x)}\right )-6 \arctan (a x) \sin (2 \arctan (a x))}{24 c^2} \]

input

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

output

((-I)*Pi^3 + (8*I)*ArcTan[a*x]^3 - 3*Cos[2*ArcTan[a*x]] + 6*ArcTan[a*x]^2* 
Cos[2*ArcTan[a*x]] + 24*ArcTan[a*x]^2*Log[1 - E^((-2*I)*ArcTan[a*x])] + (2 
4*I)*ArcTan[a*x]*PolyLog[2, E^((-2*I)*ArcTan[a*x])] + 12*PolyLog[3, E^((-2 
*I)*ArcTan[a*x])] - 6*ArcTan[a*x]*Sin[2*ArcTan[a*x]])/(24*c^2)
 

3.3.95.3 Rubi [A] (verified)

Time = 1.05 (sec) , antiderivative size = 195, normalized size of antiderivative = 1.15, number of steps used = 9, number of rules used = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.409, Rules used = {5501, 27, 5459, 5403, 5465, 5427, 241, 5527, 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.

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

\(\Big \downarrow \) 5501

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 5459

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

\(\Big \downarrow \) 5403

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

\(\Big \downarrow \) 5465

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

\(\Big \downarrow \) 5427

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

\(\Big \downarrow \) 241

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

\(\Big \downarrow \) 5527

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

\(\Big \downarrow \) 7164

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

input

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

output

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

3.3.95.3.1 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 241

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

rule 5403

Int[((a_.) + ArcTan[(c_.)*(x_)]*(b_.))^(p_.)/((x_)*((d_) + (e_.)*(x_))), x_ 
Symbol] :> Simp[(a + b*ArcTan[c*x])^p*(Log[2 - 2/(1 + e*(x/d))]/d), x] - Si 
mp[b*c*(p/d)   Int[(a + b*ArcTan[c*x])^(p - 1)*(Log[2 - 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 5427

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

rule 5459

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

rule 5465

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

rule 5501

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

rule 5527

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]
 

3.3.95.4 Maple [C] (warning: unable to verify)

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

Time = 16.09 (sec) , antiderivative size = 1677, normalized size of antiderivative = 9.86

method	result	size
derivativedivides	\(\text {Expression too large to display}\)	\(1677\)
default	\(\text {Expression too large to display}\)	\(1677\)
parts	\(\text {Expression too large to display}\)	\(2098\)

input

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

output

1/c^2*arctan(a*x)^2*ln(a*x)-1/2/c^2*arctan(a*x)^2*ln(a^2*x^2+1)+1/2*arctan 
(a*x)^2/c^2/(a^2*x^2+1)-1/c^2*(-arctan(a*x)^2*ln((1+I*a*x)/(a^2*x^2+1)^(1/ 
2))+1/3*I*arctan(a*x)^3-I*arctan(a*x)*(I+a*x)/(8*a*x-8*I)-1/16*(I+a*x)/(a* 
x-I)+I*arctan(a*x)*(a*x-I)/(8*a*x+8*I)-1/16*(a*x-I)/(I+a*x)+arctan(a*x)^2* 
ln((1+I*a*x)^2/(a^2*x^2+1)-1)-arctan(a*x)^2*ln((1+I*a*x)/(a^2*x^2+1)^(1/2) 
+1)+2*I*arctan(a*x)*polylog(2,-(1+I*a*x)/(a^2*x^2+1)^(1/2))-2*polylog(3,-( 
1+I*a*x)/(a^2*x^2+1)^(1/2))-arctan(a*x)^2*ln(1-(1+I*a*x)/(a^2*x^2+1)^(1/2) 
)+2*I*arctan(a*x)*polylog(2,(1+I*a*x)/(a^2*x^2+1)^(1/2))-2*polylog(3,(1+I* 
a*x)/(a^2*x^2+1)^(1/2))+1/4*(I*Pi*csgn(I*(1+I*a*x)^2/(a^2*x^2+1))^3+2*I*Pi 
*csgn(I*((1+I*a*x)^2/(a^2*x^2+1)-1)/((1+I*a*x)^2/(a^2*x^2+1)+1))*csgn(((1+ 
I*a*x)^2/(a^2*x^2+1)-1)/((1+I*a*x)^2/(a^2*x^2+1)+1))^2+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)/((1+I*a*x)^2/(a 
^2*x^2+1)+1))^2-I*Pi*csgn(I*((1+I*a*x)^2/(a^2*x^2+1)+1)^2)^3-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)-I*Pi*c 
sgn(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+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))*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*I*Pi*csgn(((1+I*a*x)^2/(a^2*x^2+1)-1)/((1+I*a*x)^2/(a^2* 
x^2+1)+1))^2-2*I*Pi*csgn(((1+I*a*x)^2/(a^2*x^2+1)-1)/((1+I*a*x)^2/(a^2*x^2 
+1)+1))^3+I*Pi*csgn(I*(1+I*a*x)/(a^2*x^2+1)^(1/2))^2*csgn(I*(1+I*a*x)^2...
 

3.3.95.5 Fricas [F]

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

input

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

output

integral(arctan(a*x)^2/(a^4*c^2*x^5 + 2*a^2*c^2*x^3 + c^2*x), x)
 

3.3.95.6 Sympy [F]

\[ \int \frac {\arctan (a x)^2}{x \left (c+a^2 c x^2\right )^2} \, dx=\frac {\int \frac {\operatorname {atan}^{2}{\left (a x \right )}}{a^{4} x^{5} + 2 a^{2} x^{3} + x}\, dx}{c^{2}} \]

input

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

output

Integral(atan(a*x)**2/(a**4*x**5 + 2*a**2*x**3 + x), x)/c**2
 

3.3.95.7 Maxima [F]

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

input

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

output

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

3.3.95.8 Giac [F]

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

input

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

output

sage0*x
 

3.3.95.9 Mupad [F(-1)]

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

input

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

output

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