\(\int \frac {\arctan (a x)^2}{x (c+a^2 c x^2)} \, dx\) [287]
Optimal result
Integrand size = 22, antiderivative size = 91 \[
\int \frac {\arctan (a x)^2}{x \left (c+a^2 c x^2\right )} \, dx=-\frac {i \arctan (a x)^3}{3 c}+\frac {\arctan (a x)^2 \log \left (2-\frac {2}{1-i a x}\right )}{c}-\frac {i \arctan (a x) \operatorname {PolyLog}\left (2,-1+\frac {2}{1-i a x}\right )}{c}+\frac {\operatorname {PolyLog}\left (3,-1+\frac {2}{1-i a x}\right )}{2 c}
\]
[Out]
-1/3*I*arctan(a*x)^3/c+arctan(a*x)^2*ln(2-2/(1-I*a*x))/c-I*arctan(a*x)*polylog(2,-1+2/(1-I*a*x))/c+1/2*polylog
(3,-1+2/(1-I*a*x))/c
Rubi [A] (verified)
Time = 0.13 (sec) , antiderivative size = 91, normalized size of antiderivative = 1.00, number of
steps used = 4, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.227, Rules used = {5044, 4988, 5004, 5112, 6745}
\[
\int \frac {\arctan (a x)^2}{x \left (c+a^2 c x^2\right )} \, dx=-\frac {i \arctan (a x) \operatorname {PolyLog}\left (2,\frac {2}{1-i a x}-1\right )}{c}-\frac {i \arctan (a x)^3}{3 c}+\frac {\arctan (a x)^2 \log \left (2-\frac {2}{1-i a x}\right )}{c}+\frac {\operatorname {PolyLog}\left (3,\frac {2}{1-i a x}-1\right )}{2 c}
\]
[In]
Int[ArcTan[a*x]^2/(x*(c + a^2*c*x^2)),x]
[Out]
((-1/3*I)*ArcTan[a*x]^3)/c + (ArcTan[a*x]^2*Log[2 - 2/(1 - I*a*x)])/c - (I*ArcTan[a*x]*PolyLog[2, -1 + 2/(1 -
I*a*x)])/c + PolyLog[3, -1 + 2/(1 - I*a*x)]/(2*c)
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 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 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}& = -\frac {i \arctan (a x)^3}{3 c}+\frac {i \int \frac {\arctan (a x)^2}{x (i+a x)} \, dx}{c} \\ & = -\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 {(2 a) \int \frac {\arctan (a x) \log \left (2-\frac {2}{1-i a x}\right )}{1+a^2 x^2} \, dx}{c} \\ & = -\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 {(i a) \int \frac {\operatorname {PolyLog}\left (2,-1+\frac {2}{1-i a x}\right )}{1+a^2 x^2} \, dx}{c} \\ & = -\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} \\
\end{align*}
Mathematica [A] (verified)
Time = 0.13 (sec) , antiderivative size = 83, normalized size of antiderivative = 0.91
\[
\int \frac {\arctan (a x)^2}{x \left (c+a^2 c x^2\right )} \, dx=\frac {i \arctan (a x)^3}{3 c}+\frac {\arctan (a x)^2 \log \left (1-e^{-2 i \arctan (a x)}\right )}{c}+\frac {i \arctan (a x) \operatorname {PolyLog}\left (2,e^{-2 i \arctan (a x)}\right )}{c}+\frac {\operatorname {PolyLog}\left (3,e^{-2 i \arctan (a x)}\right )}{2 c}
\]
[In]
Integrate[ArcTan[a*x]^2/(x*(c + a^2*c*x^2)),x]
[Out]
((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)
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 |
\(\text {Expression too large to display}\) |
\(1989\) |
| | |
|
|
|
|
[In]
int(arctan(a*x)^2/x/(a^2*c*x^2+c),x,method=_RETURNVERBOSE)
[Out]
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*P
i*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)+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)/((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))*csgn(I*(1+I*a*x)^2/(a^2*x^2+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))^3+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+4*ln(2))*arctan(a*x)^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*polyl
og(3,(1+I*a*x)/(a^2*x^2+1)^(1/2))-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)))
Fricas [F]
\[
\int \frac {\arctan (a x)^2}{x \left (c+a^2 c x^2\right )} \, dx=\int { \frac {\arctan \left (a x\right )^{2}}{{\left (a^{2} c x^{2} + c\right )} x} \,d x }
\]
[In]
integrate(arctan(a*x)^2/x/(a^2*c*x^2+c),x, algorithm="fricas")
[Out]
integral(arctan(a*x)^2/(a^2*c*x^3 + c*x), x)
Sympy [F]
\[
\int \frac {\arctan (a x)^2}{x \left (c+a^2 c x^2\right )} \, dx=\frac {\int \frac {\operatorname {atan}^{2}{\left (a x \right )}}{a^{2} x^{3} + x}\, dx}{c}
\]
[In]
integrate(atan(a*x)**2/x/(a**2*c*x**2+c),x)
[Out]
Integral(atan(a*x)**2/(a**2*x**3 + x), x)/c
Maxima [F]
\[
\int \frac {\arctan (a x)^2}{x \left (c+a^2 c x^2\right )} \, dx=\int { \frac {\arctan \left (a x\right )^{2}}{{\left (a^{2} c x^{2} + c\right )} x} \,d x }
\]
[In]
integrate(arctan(a*x)^2/x/(a^2*c*x^2+c),x, algorithm="maxima")
[Out]
integrate(arctan(a*x)^2/((a^2*c*x^2 + c)*x), x)
Giac [F]
\[
\int \frac {\arctan (a x)^2}{x \left (c+a^2 c x^2\right )} \, dx=\int { \frac {\arctan \left (a x\right )^{2}}{{\left (a^{2} c x^{2} + c\right )} x} \,d x }
\]
[In]
integrate(arctan(a*x)^2/x/(a^2*c*x^2+c),x, algorithm="giac")
[Out]
sage0*x
Mupad [F(-1)]
Timed out. \[
\int \frac {\arctan (a x)^2}{x \left (c+a^2 c x^2\right )} \, dx=\int \frac {{\mathrm {atan}\left (a\,x\right )}^2}{x\,\left (c\,a^2\,x^2+c\right )} \,d x
\]
[In]
int(atan(a*x)^2/(x*(c + a^2*c*x^2)),x)
[Out]
int(atan(a*x)^2/(x*(c + a^2*c*x^2)), x)