\(\int \frac {\arctan (a x)^3}{x^2 (c+a^2 c x^2)^{3/2}} \, dx\) [448]

Optimal result

Integrand size = 24, antiderivative size = 377 \[ \int \frac {\arctan (a x)^3}{x^2 \left (c+a^2 c x^2\right )^{3/2}} \, dx=\frac {6 a}{c \sqrt {c+a^2 c x^2}}+\frac {6 a^2 x \arctan (a x)}{c \sqrt {c+a^2 c x^2}}-\frac {3 a \arctan (a x)^2}{c \sqrt {c+a^2 c x^2}}-\frac {a^2 x \arctan (a x)^3}{c \sqrt {c+a^2 c x^2}}-\frac {\sqrt {c+a^2 c x^2} \arctan (a x)^3}{c^2 x}-\frac {6 a \sqrt {1+a^2 x^2} \arctan (a x)^2 \text {arctanh}\left (e^{i \arctan (a x)}\right )}{c \sqrt {c+a^2 c x^2}}+\frac {6 i a \sqrt {1+a^2 x^2} \arctan (a x) \operatorname {PolyLog}\left (2,-e^{i \arctan (a x)}\right )}{c \sqrt {c+a^2 c x^2}}-\frac {6 i a \sqrt {1+a^2 x^2} \arctan (a x) \operatorname {PolyLog}\left (2,e^{i \arctan (a x)}\right )}{c \sqrt {c+a^2 c x^2}}-\frac {6 a \sqrt {1+a^2 x^2} \operatorname {PolyLog}\left (3,-e^{i \arctan (a x)}\right )}{c \sqrt {c+a^2 c x^2}}+\frac {6 a \sqrt {1+a^2 x^2} \operatorname {PolyLog}\left (3,e^{i \arctan (a x)}\right )}{c \sqrt {c+a^2 c x^2}} \] Output:

6*a/c/(a^2*c*x^2+c)^(1/2)+6*a^2*x*arctan(a*x)/c/(a^2*c*x^2+c)^(1/2)-3*a*ar 
ctan(a*x)^2/c/(a^2*c*x^2+c)^(1/2)-a^2*x*arctan(a*x)^3/c/(a^2*c*x^2+c)^(1/2 
)-(a^2*c*x^2+c)^(1/2)*arctan(a*x)^3/c^2/x-6*a*(a^2*x^2+1)^(1/2)*arctan(a*x 
)^2*arctanh((1+I*a*x)/(a^2*x^2+1)^(1/2))/c/(a^2*c*x^2+c)^(1/2)+6*I*a*(a^2* 
x^2+1)^(1/2)*arctan(a*x)*polylog(2,-(1+I*a*x)/(a^2*x^2+1)^(1/2))/c/(a^2*c* 
x^2+c)^(1/2)-6*I*a*(a^2*x^2+1)^(1/2)*arctan(a*x)*polylog(2,(1+I*a*x)/(a^2* 
x^2+1)^(1/2))/c/(a^2*c*x^2+c)^(1/2)-6*a*(a^2*x^2+1)^(1/2)*polylog(3,-(1+I* 
a*x)/(a^2*x^2+1)^(1/2))/c/(a^2*c*x^2+c)^(1/2)+6*a*(a^2*x^2+1)^(1/2)*polylo 
g(3,(1+I*a*x)/(a^2*x^2+1)^(1/2))/c/(a^2*c*x^2+c)^(1/2)
 

Mathematica [A] (warning: unable to verify)

Time = 1.08 (sec) , antiderivative size = 301, normalized size of antiderivative = 0.80 \[ \int \frac {\arctan (a x)^3}{x^2 \left (c+a^2 c x^2\right )^{3/2}} \, dx=\frac {a \left (12+12 a x \arctan (a x)-6 \arctan (a x)^2-2 a x \arctan (a x)^3-\frac {1}{2} a x \arctan (a x)^3 \csc ^2\left (\frac {1}{2} \arctan (a x)\right )+6 \sqrt {1+a^2 x^2} \arctan (a x)^2 \log \left (1-e^{i \arctan (a x)}\right )-6 \sqrt {1+a^2 x^2} \arctan (a x)^2 \log \left (1+e^{i \arctan (a x)}\right )+12 i \sqrt {1+a^2 x^2} \arctan (a x) \operatorname {PolyLog}\left (2,-e^{i \arctan (a x)}\right )-12 i \sqrt {1+a^2 x^2} \arctan (a x) \operatorname {PolyLog}\left (2,e^{i \arctan (a x)}\right )-12 \sqrt {1+a^2 x^2} \operatorname {PolyLog}\left (3,-e^{i \arctan (a x)}\right )+12 \sqrt {1+a^2 x^2} \operatorname {PolyLog}\left (3,e^{i \arctan (a x)}\right )-\frac {2 \left (1+a^2 x^2\right ) \arctan (a x)^3 \sin ^2\left (\frac {1}{2} \arctan (a x)\right )}{a x}\right )}{2 c \sqrt {c+a^2 c x^2}} \] Input:

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

Output:

(a*(12 + 12*a*x*ArcTan[a*x] - 6*ArcTan[a*x]^2 - 2*a*x*ArcTan[a*x]^3 - (a*x 
*ArcTan[a*x]^3*Csc[ArcTan[a*x]/2]^2)/2 + 6*Sqrt[1 + a^2*x^2]*ArcTan[a*x]^2 
*Log[1 - E^(I*ArcTan[a*x])] - 6*Sqrt[1 + a^2*x^2]*ArcTan[a*x]^2*Log[1 + E^ 
(I*ArcTan[a*x])] + (12*I)*Sqrt[1 + a^2*x^2]*ArcTan[a*x]*PolyLog[2, -E^(I*A 
rcTan[a*x])] - (12*I)*Sqrt[1 + a^2*x^2]*ArcTan[a*x]*PolyLog[2, E^(I*ArcTan 
[a*x])] - 12*Sqrt[1 + a^2*x^2]*PolyLog[3, -E^(I*ArcTan[a*x])] + 12*Sqrt[1 
+ a^2*x^2]*PolyLog[3, E^(I*ArcTan[a*x])] - (2*(1 + a^2*x^2)*ArcTan[a*x]^3* 
Sin[ArcTan[a*x]/2]^2)/(a*x)))/(2*c*Sqrt[c + a^2*c*x^2])
 

Rubi [A] (verified)

Time = 1.82 (sec) , antiderivative size = 268, normalized size of antiderivative = 0.71, number of steps used = 12, number of rules used = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.458, Rules used = {5501, 5433, 5429, 5479, 5493, 5491, 3042, 4671, 3011, 2720, 7143}

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.

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

\(\Big \downarrow \) 5501

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

\(\Big \downarrow \) 5433

\(\displaystyle \frac {\int \frac {\arctan (a x)^3}{x^2 \sqrt {a^2 c x^2+c}}dx}{c}-a^2 \left (-6 \int \frac {\arctan (a x)}{\left (a^2 c x^2+c\right )^{3/2}}dx+\frac {x \arctan (a x)^3}{c \sqrt {a^2 c x^2+c}}+\frac {3 \arctan (a x)^2}{a c \sqrt {a^2 c x^2+c}}\right )\)

\(\Big \downarrow \) 5429

\(\displaystyle \frac {\int \frac {\arctan (a x)^3}{x^2 \sqrt {a^2 c x^2+c}}dx}{c}-a^2 \left (\frac {x \arctan (a x)^3}{c \sqrt {a^2 c x^2+c}}+\frac {3 \arctan (a x)^2}{a c \sqrt {a^2 c x^2+c}}-6 \left (\frac {x \arctan (a x)}{c \sqrt {a^2 c x^2+c}}+\frac {1}{a c \sqrt {a^2 c x^2+c}}\right )\right )\)

\(\Big \downarrow \) 5479

\(\displaystyle \frac {3 a \int \frac {\arctan (a x)^2}{x \sqrt {a^2 c x^2+c}}dx-\frac {\arctan (a x)^3 \sqrt {a^2 c x^2+c}}{c x}}{c}-a^2 \left (\frac {x \arctan (a x)^3}{c \sqrt {a^2 c x^2+c}}+\frac {3 \arctan (a x)^2}{a c \sqrt {a^2 c x^2+c}}-6 \left (\frac {x \arctan (a x)}{c \sqrt {a^2 c x^2+c}}+\frac {1}{a c \sqrt {a^2 c x^2+c}}\right )\right )\)

\(\Big \downarrow \) 5493

\(\displaystyle \frac {\frac {3 a \sqrt {a^2 x^2+1} \int \frac {\arctan (a x)^2}{x \sqrt {a^2 x^2+1}}dx}{\sqrt {a^2 c x^2+c}}-\frac {\arctan (a x)^3 \sqrt {a^2 c x^2+c}}{c x}}{c}-a^2 \left (\frac {x \arctan (a x)^3}{c \sqrt {a^2 c x^2+c}}+\frac {3 \arctan (a x)^2}{a c \sqrt {a^2 c x^2+c}}-6 \left (\frac {x \arctan (a x)}{c \sqrt {a^2 c x^2+c}}+\frac {1}{a c \sqrt {a^2 c x^2+c}}\right )\right )\)

\(\Big \downarrow \) 5491

\(\displaystyle \frac {\frac {3 a \sqrt {a^2 x^2+1} \int \frac {\sqrt {a^2 x^2+1} \arctan (a x)^2}{a x}d\arctan (a x)}{\sqrt {a^2 c x^2+c}}-\frac {\arctan (a x)^3 \sqrt {a^2 c x^2+c}}{c x}}{c}-a^2 \left (\frac {x \arctan (a x)^3}{c \sqrt {a^2 c x^2+c}}+\frac {3 \arctan (a x)^2}{a c \sqrt {a^2 c x^2+c}}-6 \left (\frac {x \arctan (a x)}{c \sqrt {a^2 c x^2+c}}+\frac {1}{a c \sqrt {a^2 c x^2+c}}\right )\right )\)

\(\Big \downarrow \) 3042

\(\displaystyle \frac {\frac {3 a \sqrt {a^2 x^2+1} \int \arctan (a x)^2 \csc (\arctan (a x))d\arctan (a x)}{\sqrt {a^2 c x^2+c}}-\frac {\arctan (a x)^3 \sqrt {a^2 c x^2+c}}{c x}}{c}-a^2 \left (\frac {x \arctan (a x)^3}{c \sqrt {a^2 c x^2+c}}+\frac {3 \arctan (a x)^2}{a c \sqrt {a^2 c x^2+c}}-6 \left (\frac {x \arctan (a x)}{c \sqrt {a^2 c x^2+c}}+\frac {1}{a c \sqrt {a^2 c x^2+c}}\right )\right )\)

\(\Big \downarrow \) 4671

\(\displaystyle -a^2 \left (\frac {x \arctan (a x)^3}{c \sqrt {a^2 c x^2+c}}+\frac {3 \arctan (a x)^2}{a c \sqrt {a^2 c x^2+c}}-6 \left (\frac {x \arctan (a x)}{c \sqrt {a^2 c x^2+c}}+\frac {1}{a c \sqrt {a^2 c x^2+c}}\right )\right )+\frac {-\frac {\arctan (a x)^3 \sqrt {a^2 c x^2+c}}{c x}+\frac {3 a \sqrt {a^2 x^2+1} \left (-2 \int \arctan (a x) \log \left (1-e^{i \arctan (a x)}\right )d\arctan (a x)+2 \int \arctan (a x) \log \left (1+e^{i \arctan (a x)}\right )d\arctan (a x)-2 \arctan (a x)^2 \text {arctanh}\left (e^{i \arctan (a x)}\right )\right )}{\sqrt {a^2 c x^2+c}}}{c}\)

\(\Big \downarrow \) 3011

\(\displaystyle -a^2 \left (\frac {x \arctan (a x)^3}{c \sqrt {a^2 c x^2+c}}+\frac {3 \arctan (a x)^2}{a c \sqrt {a^2 c x^2+c}}-6 \left (\frac {x \arctan (a x)}{c \sqrt {a^2 c x^2+c}}+\frac {1}{a c \sqrt {a^2 c x^2+c}}\right )\right )+\frac {-\frac {\arctan (a x)^3 \sqrt {a^2 c x^2+c}}{c x}+\frac {3 a \sqrt {a^2 x^2+1} \left (2 \left (i \arctan (a x) \operatorname {PolyLog}\left (2,-e^{i \arctan (a x)}\right )-i \int \operatorname {PolyLog}\left (2,-e^{i \arctan (a x)}\right )d\arctan (a x)\right )-2 \left (i \arctan (a x) \operatorname {PolyLog}\left (2,e^{i \arctan (a x)}\right )-i \int \operatorname {PolyLog}\left (2,e^{i \arctan (a x)}\right )d\arctan (a x)\right )-2 \arctan (a x)^2 \text {arctanh}\left (e^{i \arctan (a x)}\right )\right )}{\sqrt {a^2 c x^2+c}}}{c}\)

\(\Big \downarrow \) 2720

\(\displaystyle -a^2 \left (\frac {x \arctan (a x)^3}{c \sqrt {a^2 c x^2+c}}+\frac {3 \arctan (a x)^2}{a c \sqrt {a^2 c x^2+c}}-6 \left (\frac {x \arctan (a x)}{c \sqrt {a^2 c x^2+c}}+\frac {1}{a c \sqrt {a^2 c x^2+c}}\right )\right )+\frac {-\frac {\arctan (a x)^3 \sqrt {a^2 c x^2+c}}{c x}+\frac {3 a \sqrt {a^2 x^2+1} \left (2 \left (i \arctan (a x) \operatorname {PolyLog}\left (2,-e^{i \arctan (a x)}\right )-\int e^{-i \arctan (a x)} \operatorname {PolyLog}\left (2,-e^{i \arctan (a x)}\right )de^{i \arctan (a x)}\right )-2 \left (i \arctan (a x) \operatorname {PolyLog}\left (2,e^{i \arctan (a x)}\right )-\int e^{-i \arctan (a x)} \operatorname {PolyLog}\left (2,e^{i \arctan (a x)}\right )de^{i \arctan (a x)}\right )-2 \arctan (a x)^2 \text {arctanh}\left (e^{i \arctan (a x)}\right )\right )}{\sqrt {a^2 c x^2+c}}}{c}\)

\(\Big \downarrow \) 7143

\(\displaystyle -a^2 \left (\frac {x \arctan (a x)^3}{c \sqrt {a^2 c x^2+c}}+\frac {3 \arctan (a x)^2}{a c \sqrt {a^2 c x^2+c}}-6 \left (\frac {x \arctan (a x)}{c \sqrt {a^2 c x^2+c}}+\frac {1}{a c \sqrt {a^2 c x^2+c}}\right )\right )+\frac {-\frac {\arctan (a x)^3 \sqrt {a^2 c x^2+c}}{c x}+\frac {3 a \sqrt {a^2 x^2+1} \left (-2 \arctan (a x)^2 \text {arctanh}\left (e^{i \arctan (a x)}\right )+2 \left (i \arctan (a x) \operatorname {PolyLog}\left (2,-e^{i \arctan (a x)}\right )-\operatorname {PolyLog}\left (3,-e^{i \arctan (a x)}\right )\right )-2 \left (i \arctan (a x) \operatorname {PolyLog}\left (2,e^{i \arctan (a x)}\right )-\operatorname {PolyLog}\left (3,e^{i \arctan (a x)}\right )\right )\right )}{\sqrt {a^2 c x^2+c}}}{c}\)

Input:

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

Output:

-(a^2*((3*ArcTan[a*x]^2)/(a*c*Sqrt[c + a^2*c*x^2]) + (x*ArcTan[a*x]^3)/(c* 
Sqrt[c + a^2*c*x^2]) - 6*(1/(a*c*Sqrt[c + a^2*c*x^2]) + (x*ArcTan[a*x])/(c 
*Sqrt[c + a^2*c*x^2])))) + (-((Sqrt[c + a^2*c*x^2]*ArcTan[a*x]^3)/(c*x)) + 
 (3*a*Sqrt[1 + a^2*x^2]*(-2*ArcTan[a*x]^2*ArcTanh[E^(I*ArcTan[a*x])] + 2*( 
I*ArcTan[a*x]*PolyLog[2, -E^(I*ArcTan[a*x])] - PolyLog[3, -E^(I*ArcTan[a*x 
])]) - 2*(I*ArcTan[a*x]*PolyLog[2, E^(I*ArcTan[a*x])] - PolyLog[3, E^(I*Ar 
cTan[a*x])])))/Sqrt[c + a^2*c*x^2])/c
 

Defintions of rubi rules used

rule 2720

Int[u_, x_Symbol] :> With[{v = FunctionOfExponential[u, x]}, Simp[v/D[v, x] 
   Subst[Int[FunctionOfExponentialFunction[u, x]/x, x], x, v], x]] /; Funct 
ionOfExponentialQ[u, x] &&  !MatchQ[u, (w_)*((a_.)*(v_)^(n_))^(m_) /; FreeQ 
[{a, m, n}, x] && IntegerQ[m*n]] &&  !MatchQ[u, E^((c_.)*((a_.) + (b_.)*x)) 
*(F_)[v_] /; FreeQ[{a, b, c}, x] && InverseFunctionQ[F[x]]]
 

rule 3011

Int[Log[1 + (e_.)*((F_)^((c_.)*((a_.) + (b_.)*(x_))))^(n_.)]*((f_.) + (g_.) 
*(x_))^(m_.), x_Symbol] :> Simp[(-(f + g*x)^m)*(PolyLog[2, (-e)*(F^(c*(a + 
b*x)))^n]/(b*c*n*Log[F])), x] + Simp[g*(m/(b*c*n*Log[F]))   Int[(f + g*x)^( 
m - 1)*PolyLog[2, (-e)*(F^(c*(a + b*x)))^n], x], x] /; FreeQ[{F, a, b, c, e 
, f, g, n}, x] && GtQ[m, 0]
 

rule 3042

Int[u_, x_Symbol] :> Int[DeactivateTrig[u, x], x] /; FunctionOfTrigOfLinear 
Q[u, x]
 

rule 4671

Int[csc[(e_.) + (f_.)*(x_)]*((c_.) + (d_.)*(x_))^(m_.), x_Symbol] :> Simp[- 
2*(c + d*x)^m*(ArcTanh[E^(I*(e + f*x))]/f), x] + (-Simp[d*(m/f)   Int[(c + 
d*x)^(m - 1)*Log[1 - E^(I*(e + f*x))], x], x] + Simp[d*(m/f)   Int[(c + d*x 
)^(m - 1)*Log[1 + E^(I*(e + f*x))], x], x]) /; FreeQ[{c, d, e, f}, x] && IG 
tQ[m, 0]
 

rule 5429

Int[((a_.) + ArcTan[(c_.)*(x_)]*(b_.))/((d_) + (e_.)*(x_)^2)^(3/2), x_Symbo 
l] :> Simp[b/(c*d*Sqrt[d + e*x^2]), x] + Simp[x*((a + b*ArcTan[c*x])/(d*Sqr 
t[d + e*x^2])), x] /; FreeQ[{a, b, c, d, e}, x] && EqQ[e, c^2*d]
 

rule 5433

Int[((a_.) + ArcTan[(c_.)*(x_)]*(b_.))^(p_)/((d_) + (e_.)*(x_)^2)^(3/2), x_ 
Symbol] :> Simp[b*p*((a + b*ArcTan[c*x])^(p - 1)/(c*d*Sqrt[d + e*x^2])), x] 
 + (Simp[x*((a + b*ArcTan[c*x])^p/(d*Sqrt[d + e*x^2])), x] - Simp[b^2*p*(p 
- 1)   Int[(a + b*ArcTan[c*x])^(p - 2)/(d + e*x^2)^(3/2), x], x]) /; FreeQ[ 
{a, b, c, d, e}, x] && EqQ[e, c^2*d] && GtQ[p, 1]
 

rule 5479

Int[((a_.) + ArcTan[(c_.)*(x_)]*(b_.))^(p_.)*((f_.)*(x_))^(m_.)*((d_) + (e_ 
.)*(x_)^2)^(q_.), x_Symbol] :> Simp[(f*x)^(m + 1)*(d + e*x^2)^(q + 1)*((a + 
 b*ArcTan[c*x])^p/(d*f*(m + 1))), x] - Simp[b*c*(p/(f*(m + 1)))   Int[(f*x) 
^(m + 1)*(d + e*x^2)^q*(a + b*ArcTan[c*x])^(p - 1), x], x] /; FreeQ[{a, b, 
c, d, e, f, m, q}, x] && EqQ[e, c^2*d] && EqQ[m + 2*q + 3, 0] && GtQ[p, 0] 
&& NeQ[m, -1]
 

rule 5491

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

rule 5493

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

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 7143

Int[PolyLog[n_, (c_.)*((a_.) + (b_.)*(x_))^(p_.)]/((d_.) + (e_.)*(x_)), x_S 
ymbol] :> Simp[PolyLog[n + 1, c*(a + b*x)^p]/(e*p), x] /; FreeQ[{a, b, c, d 
, e, n, p}, x] && EqQ[b*d, a*e]
 

Maple [A] (verified)

Time = 4.65 (sec) , antiderivative size = 356, normalized size of antiderivative = 0.94

method	result	size
default	\(-\frac {a \left (\arctan \left (a x \right )^{3}-6 \arctan \left (a x \right )+3 i \arctan \left (a x \right )^{2}-6 i\right ) \left (a x -i\right ) \sqrt {c \left (a x -i\right ) \left (a x +i\right )}}{2 \left (a^{2} x^{2}+1\right ) c^{2}}-\frac {\sqrt {c \left (a x -i\right ) \left (a x +i\right )}\, \left (a x +i\right ) \left (\arctan \left (a x \right )^{3}-6 \arctan \left (a x \right )-3 i \arctan \left (a x \right )^{2}+6 i\right ) a}{2 \left (a^{2} x^{2}+1\right ) c^{2}}-\frac {\arctan \left (a x \right )^{3} \sqrt {c \left (a x -i\right ) \left (a x +i\right )}}{c^{2} x}-\frac {3 a \left (\arctan \left (a x \right )^{2} \ln \left (1+\frac {i a x +1}{\sqrt {a^{2} x^{2}+1}}\right )-2 i \arctan \left (a x \right ) \operatorname {polylog}\left (2, -\frac {i a x +1}{\sqrt {a^{2} x^{2}+1}}\right )-\arctan \left (a x \right )^{2} \ln \left (1-\frac {i a x +1}{\sqrt {a^{2} x^{2}+1}}\right )+2 i \arctan \left (a x \right ) \operatorname {polylog}\left (2, \frac {i a x +1}{\sqrt {a^{2} x^{2}+1}}\right )+2 \operatorname {polylog}\left (3, -\frac {i a x +1}{\sqrt {a^{2} x^{2}+1}}\right )-2 \operatorname {polylog}\left (3, \frac {i a x +1}{\sqrt {a^{2} x^{2}+1}}\right )\right ) \sqrt {c \left (a x -i\right ) \left (a x +i\right )}}{\sqrt {a^{2} x^{2}+1}\, c^{2}}\)	\(356\)

Input:

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

Output:

-1/2*a*(arctan(a*x)^3-6*arctan(a*x)+3*I*arctan(a*x)^2-6*I)*(a*x-I)*(c*(a*x 
-I)*(a*x+I))^(1/2)/(a^2*x^2+1)/c^2-1/2*(c*(a*x-I)*(a*x+I))^(1/2)*(a*x+I)*( 
arctan(a*x)^3-6*arctan(a*x)-3*I*arctan(a*x)^2+6*I)*a/(a^2*x^2+1)/c^2-arcta 
n(a*x)^3*(c*(a*x-I)*(a*x+I))^(1/2)/c^2/x-3*a*(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) 
)-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))-2 
*polylog(3,(1+I*a*x)/(a^2*x^2+1)^(1/2)))*(c*(a*x-I)*(a*x+I))^(1/2)/(a^2*x^ 
2+1)^(1/2)/c^2
 

Fricas [F]

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

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

Output:

integral(sqrt(a^2*c*x^2 + c)*arctan(a*x)^3/(a^4*c^2*x^6 + 2*a^2*c^2*x^4 + 
c^2*x^2), x)
 

Sympy [F]

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

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

Output:

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

Maxima [F]

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

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

Output:

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

Giac [F(-2)]

Exception generated. \[ \int \frac {\arctan (a x)^3}{x^2 \left (c+a^2 c x^2\right )^{3/2}} \, dx=\text {Exception raised: TypeError} \] Input:

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

Output:

Exception raised: TypeError >> an error occurred running a Giac command:IN 
PUT:sage2:=int(sage0,sageVARx):;OUTPUT:sym2poly/r2sym(const gen & e,const 
index_m & i,const vecteur & l) Error: Bad Argument Value
                                                                                    
                                                                                    
 

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

\[ \int \frac {\arctan (a x)^3}{x^2 \left (c+a^2 c x^2\right )^{3/2}} \, dx=\frac {\int \frac {\mathit {atan} \left (a x \right )^{3}}{\sqrt {a^{2} x^{2}+1}\, a^{2} x^{4}+\sqrt {a^{2} x^{2}+1}\, x^{2}}d x}{\sqrt {c}\, c} \] Input:

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

Output:

int(atan(a*x)**3/(sqrt(a**2*x**2 + 1)*a**2*x**4 + sqrt(a**2*x**2 + 1)*x**2 
),x)/(sqrt(c)*c)