3.3.87 \(\int \frac {\arctan (a x)^2}{x (c+a^2 c x^2)} \, dx\) [287]

3.3.87.1 Optimal result

Integrand size = 22, antiderivative size = 91 \[ \int \frac {\arctan (a x)^2}{x \left (c+a^2 c x^2\right )} \, dx=-\frac {i \arctan (a x)^3}{3 c}+\frac {\arctan (a x)^2 \log \left (2-\frac {2}{1-i a x}\right )}{c}-\frac {i \arctan (a x) \operatorname {PolyLog}\left (2,-1+\frac {2}{1-i a x}\right )}{c}+\frac {\operatorname {PolyLog}\left (3,-1+\frac {2}{1-i a x}\right )}{2 c} \]

output

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

3.3.87.2 Mathematica [A] (verified)

Time = 0.13 (sec) , antiderivative size = 83, normalized size of antiderivative = 0.91 \[ \int \frac {\arctan (a x)^2}{x \left (c+a^2 c x^2\right )} \, dx=\frac {i \arctan (a x)^3}{3 c}+\frac {\arctan (a x)^2 \log \left (1-e^{-2 i \arctan (a x)}\right )}{c}+\frac {i \arctan (a x) \operatorname {PolyLog}\left (2,e^{-2 i \arctan (a x)}\right )}{c}+\frac {\operatorname {PolyLog}\left (3,e^{-2 i \arctan (a x)}\right )}{2 c} \]

input

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

output

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

3.3.87.3 Rubi [A] (verified)

Time = 0.53 (sec) , antiderivative size = 107, normalized size of antiderivative = 1.18, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.182, Rules used = {5459, 5403, 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.

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

\(\Big \downarrow \) 5459

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

\(\Big \downarrow \) 5403

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

\(\Big \downarrow \) 5527

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

\(\Big \downarrow \) 7164

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

input

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

output

((-1/3*I)*ArcTan[a*x]^3)/c + (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
 

3.3.87.3.1 Defintions of rubi rules used

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 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 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]
 

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

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

Time = 11.53 (sec) , antiderivative size = 1578, normalized size of antiderivative = 17.34

method	result	size
derivativedivides	\(\text {Expression too large to display}\)	\(1578\)
default	\(\text {Expression too large to display}\)	\(1578\)
parts	\(\text {Expression too large to display}\)	\(1989\)

input

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

output

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

3.3.87.5 Fricas [F]

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

input

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

output

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

3.3.87.6 Sympy [F]

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

input

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

output

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

3.3.87.7 Maxima [F]

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

input

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

output

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

3.3.87.8 Giac [F]

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

input

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

output

sage0*x
 

3.3.87.9 Mupad [F(-1)]

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

input

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

output

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