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

Optimal result

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

[Out]

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

Rubi [A] (verified)

Time = 0.24 (sec) , antiderivative size = 170, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.409, Rules used = {5086, 5044, 4988, 5004, 5112, 6745, 5050, 5012, 267} \[ \int \frac {\arctan (a x)^2}{x \left (c+a^2 c x^2\right )^2} \, dx=\frac {\arctan (a x)^2}{2 c^2 \left (a^2 x^2+1\right )}-\frac {a x \arctan (a x)}{2 c^2 \left (a^2 x^2+1\right )}-\frac {1}{4 c^2 \left (a^2 x^2+1\right )}-\frac {i \arctan (a x) \operatorname {PolyLog}\left (2,\frac {2}{1-i a x}-1\right )}{c^2}-\frac {i \arctan (a x)^3}{3 c^2}-\frac {\arctan (a x)^2}{4 c^2}+\frac {\arctan (a x)^2 \log \left (2-\frac {2}{1-i a x}\right )}{c^2}+\frac {\operatorname {PolyLog}\left (3,\frac {2}{1-i a x}-1\right )}{2 c^2} \]

[In]

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

[Out]

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

Rule 267

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

Rule 4988

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] - Dist[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 5004

Int[((a_.) + ArcTan[(c_.)*(x_)]*(b_.))^(p_.)/((d_) + (e_.)*(x_)^2), x_Symbol] :> 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 5012

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

Rule 5044

Int[((a_.) + ArcTan[(c_.)*(x_)]*(b_.))^(p_.)/((x_)*((d_) + (e_.)*(x_)^2)), x_Symbol] :> Simp[(-I)*((a + b*ArcT
an[c*x])^(p + 1)/(b*d*(p + 1))), x] + Dist[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 5050

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] - Dist[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 5086

Int[((a_.) + ArcTan[(c_.)*(x_)]*(b_.))^(p_.)*(x_)^(m_)*((d_) + (e_.)*(x_)^2)^(q_), x_Symbol] :> Dist[1/d, Int[
x^m*(d + e*x^2)^(q + 1)*(a + b*ArcTan[c*x])^p, x], x] - Dist[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 5112

Int[(Log[u_]*((a_.) + ArcTan[(c_.)*(x_)]*(b_.))^(p_.))/((d_) + (e_.)*(x_)^2), x_Symbol] :> Simp[I*(a + b*ArcTa
n[c*x])^p*(PolyLog[2, 1 - u]/(2*c*d)), x] - Dist[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 6745

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]

Rubi steps \begin{align*} \text {integral}& = -\left (a^2 \int \frac {x \arctan (a x)^2}{\left (c+a^2 c x^2\right )^2} \, dx\right )+\frac {\int \frac {\arctan (a x)^2}{x \left (c+a^2 c x^2\right )} \, dx}{c} \\ & = \frac {\arctan (a x)^2}{2 c^2 \left (1+a^2 x^2\right )}-\frac {i \arctan (a x)^3}{3 c^2}-a \int \frac {\arctan (a x)}{\left (c+a^2 c x^2\right )^2} \, dx+\frac {i \int \frac {\arctan (a x)^2}{x (i+a x)} \, dx}{c^2} \\ & = -\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 {1}{2} a^2 \int \frac {x}{\left (c+a^2 c x^2\right )^2} \, dx-\frac {(2 a) \int \frac {\arctan (a x) \log \left (2-\frac {2}{1-i a x}\right )}{1+a^2 x^2} \, dx}{c^2} \\ & = -\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 {(i a) \int \frac {\operatorname {PolyLog}\left (2,-1+\frac {2}{1-i a x}\right )}{1+a^2 x^2} \, dx}{c^2} \\ & = -\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} \\ \end{align*}

Mathematica [A] (verified)

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

[In]

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

[Out]

((-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])] + (24*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)

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	\(\text {Expression too large to display}\)	\(2098\)

[In]

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

[Out]

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*(-arct
an(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)-arct
an(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*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+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/(a^2*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)/((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)/(
(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)^2)
^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))*csgn(I*((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+I*a*x)^2/(a^2*x^2+1)+1)^2)^3-2*I*P
i*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))^2-2*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))*csgn(I*(1+I*a*x)^2/(a^2*x^2+1)/((1+I*a*x)^2/(a^2*x^2+1)+1)^2)^2-4*ln(2)+1)*arctan(a*x)^2
)

Fricas [F]

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

[In]

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

[Out]

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

Sympy [F]

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

[In]

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

[Out]

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

Maxima [F]

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

[In]

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

[Out]

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

Giac [F]

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

[In]

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

[Out]

sage0*x

Mupad [F(-1)]

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

[In]

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

[Out]

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