\(\int \frac {\arctan (a x)}{x^3 (c+a^2 c x^2)^{3/2}} \, dx\) [238]

Optimal result

Integrand size = 22, antiderivative size = 300 \[ \int \frac {\arctan (a x)}{x^3 \left (c+a^2 c x^2\right )^{3/2}} \, dx=\frac {a^3 x}{c \sqrt {c+a^2 c x^2}}-\frac {a \sqrt {c+a^2 c x^2}}{2 c^2 x}-\frac {a^2 \arctan (a x)}{c \sqrt {c+a^2 c x^2}}-\frac {\sqrt {c+a^2 c x^2} \arctan (a x)}{2 c^2 x^2}+\frac {3 a^2 \sqrt {1+a^2 x^2} \arctan (a x) \text {arctanh}\left (\frac {\sqrt {1+i a x}}{\sqrt {1-i a x}}\right )}{c \sqrt {c+a^2 c x^2}}-\frac {3 i a^2 \sqrt {1+a^2 x^2} \operatorname {PolyLog}\left (2,-\frac {\sqrt {1+i a x}}{\sqrt {1-i a x}}\right )}{2 c \sqrt {c+a^2 c x^2}}+\frac {3 i a^2 \sqrt {1+a^2 x^2} \operatorname {PolyLog}\left (2,\frac {\sqrt {1+i a x}}{\sqrt {1-i a x}}\right )}{2 c \sqrt {c+a^2 c x^2}} \] Output:

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

Mathematica [A] (warning: unable to verify)

Time = 0.95 (sec) , antiderivative size = 258, normalized size of antiderivative = 0.86 \[ \int \frac {\arctan (a x)}{x^3 \left (c+a^2 c x^2\right )^{3/2}} \, dx=-\frac {a^2 \left (-8 a x+8 \arctan (a x)+a x \csc ^2\left (\frac {1}{2} \arctan (a x)\right )+\sqrt {1+a^2 x^2} \arctan (a x) \csc ^2\left (\frac {1}{2} \arctan (a x)\right )+12 \sqrt {1+a^2 x^2} \arctan (a x) \log \left (1-e^{i \arctan (a x)}\right )-12 \sqrt {1+a^2 x^2} \arctan (a x) \log \left (1+e^{i \arctan (a x)}\right )+12 i \sqrt {1+a^2 x^2} \operatorname {PolyLog}\left (2,-e^{i \arctan (a x)}\right )-12 i \sqrt {1+a^2 x^2} \operatorname {PolyLog}\left (2,e^{i \arctan (a x)}\right )-\sqrt {1+a^2 x^2} \arctan (a x) \sec ^2\left (\frac {1}{2} \arctan (a x)\right )+2 \sqrt {1+a^2 x^2} \tan \left (\frac {1}{2} \arctan (a x)\right )\right )}{8 c \sqrt {c+a^2 c x^2}} \] Input:

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

Output:

-1/8*(a^2*(-8*a*x + 8*ArcTan[a*x] + a*x*Csc[ArcTan[a*x]/2]^2 + Sqrt[1 + a^ 
2*x^2]*ArcTan[a*x]*Csc[ArcTan[a*x]/2]^2 + 12*Sqrt[1 + a^2*x^2]*ArcTan[a*x] 
*Log[1 - E^(I*ArcTan[a*x])] - 12*Sqrt[1 + a^2*x^2]*ArcTan[a*x]*Log[1 + E^( 
I*ArcTan[a*x])] + (12*I)*Sqrt[1 + a^2*x^2]*PolyLog[2, -E^(I*ArcTan[a*x])] 
- (12*I)*Sqrt[1 + a^2*x^2]*PolyLog[2, E^(I*ArcTan[a*x])] - Sqrt[1 + a^2*x^ 
2]*ArcTan[a*x]*Sec[ArcTan[a*x]/2]^2 + 2*Sqrt[1 + a^2*x^2]*Tan[ArcTan[a*x]/ 
2]))/(c*Sqrt[c + a^2*c*x^2])
 

Rubi [A] (verified)

Time = 1.79 (sec) , antiderivative size = 376, normalized size of antiderivative = 1.25, number of steps used = 10, number of rules used = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.455, Rules used = {5501, 5497, 242, 5493, 5489, 5501, 5465, 208, 5493, 5489}

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.

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

\(\Big \downarrow \) 5501

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

\(\Big \downarrow \) 5497

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

\(\Big \downarrow \) 242

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

\(\Big \downarrow \) 5493

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

\(\Big \downarrow \) 5489

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

\(\Big \downarrow \) 5501

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

\(\Big \downarrow \) 5465

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

\(\Big \downarrow \) 208

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

\(\Big \downarrow \) 5493

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

\(\Big \downarrow \) 5489

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

Input:

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

Output:

(-1/2*(a*Sqrt[c + a^2*c*x^2])/(c*x) - (Sqrt[c + a^2*c*x^2]*ArcTan[a*x])/(2 
*c*x^2) - (a^2*Sqrt[1 + a^2*x^2]*(-2*ArcTan[a*x]*ArcTanh[Sqrt[1 + I*a*x]/S 
qrt[1 - I*a*x]] + I*PolyLog[2, -(Sqrt[1 + I*a*x]/Sqrt[1 - I*a*x])] - I*Pol 
yLog[2, Sqrt[1 + I*a*x]/Sqrt[1 - I*a*x]]))/(2*Sqrt[c + a^2*c*x^2]))/c - a^ 
2*(-(a^2*(x/(a*c*Sqrt[c + a^2*c*x^2]) - ArcTan[a*x]/(a^2*c*Sqrt[c + a^2*c* 
x^2]))) + (Sqrt[1 + a^2*x^2]*(-2*ArcTan[a*x]*ArcTanh[Sqrt[1 + I*a*x]/Sqrt[ 
1 - I*a*x]] + I*PolyLog[2, -(Sqrt[1 + I*a*x]/Sqrt[1 - I*a*x])] - I*PolyLog 
[2, Sqrt[1 + I*a*x]/Sqrt[1 - I*a*x]]))/(c*Sqrt[c + a^2*c*x^2]))
 

Defintions of rubi rules used

rule 208

Int[((a_) + (b_.)*(x_)^2)^(-3/2), x_Symbol] :> Simp[x/(a*Sqrt[a + b*x^2]), 
x] /; FreeQ[{a, b}, x]
 

rule 242

Int[((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> Simp[(c*x)^ 
(m + 1)*((a + b*x^2)^(p + 1)/(a*c*(m + 1))), x] /; FreeQ[{a, b, c, m, p}, x 
] && EqQ[m + 2*p + 3, 0] && NeQ[m, -1]
 

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 5489

Int[((a_.) + ArcTan[(c_.)*(x_)]*(b_.))/((x_)*Sqrt[(d_) + (e_.)*(x_)^2]), x_ 
Symbol] :> Simp[(-2/Sqrt[d])*(a + b*ArcTan[c*x])*ArcTanh[Sqrt[1 + I*c*x]/Sq 
rt[1 - I*c*x]], x] + (Simp[I*(b/Sqrt[d])*PolyLog[2, -Sqrt[1 + I*c*x]/Sqrt[1 
 - I*c*x]], x] - Simp[I*(b/Sqrt[d])*PolyLog[2, Sqrt[1 + I*c*x]/Sqrt[1 - I*c 
*x]], x]) /; FreeQ[{a, b, c, d, e}, x] && EqQ[e, c^2*d] && 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 5497

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

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]
 

Maple [A] (verified)

Time = 1.39 (sec) , antiderivative size = 273, normalized size of antiderivative = 0.91

method	result	size
default	\(-\frac {a^{2} \left (\arctan \left (a x \right )+i\right ) \left (i a x +1\right ) \sqrt {c \left (a x -i\right ) \left (a x +i\right )}}{2 \left (a^{2} x^{2}+1\right ) c^{2}}+\frac {\sqrt {c \left (a x -i\right ) \left (a x +i\right )}\, \left (i a x -1\right ) \left (\arctan \left (a x \right )-i\right ) a^{2}}{2 \left (a^{2} x^{2}+1\right ) c^{2}}-\frac {\left (a x +\arctan \left (a x \right )\right ) \sqrt {c \left (a x -i\right ) \left (a x +i\right )}}{2 c^{2} x^{2}}+\frac {3 i a^{2} \left (i \arctan \left (a x \right ) \ln \left (1-\frac {i a x +1}{\sqrt {a^{2} x^{2}+1}}\right )-i \arctan \left (a x \right ) \ln \left (1+\frac {i a x +1}{\sqrt {a^{2} x^{2}+1}}\right )+\operatorname {polylog}\left (2, \frac {i a x +1}{\sqrt {a^{2} x^{2}+1}}\right )-\operatorname {polylog}\left (2, -\frac {i a x +1}{\sqrt {a^{2} x^{2}+1}}\right )\right ) \sqrt {c \left (a x -i\right ) \left (a x +i\right )}}{2 \sqrt {a^{2} x^{2}+1}\, c^{2}}\)	\(273\)

Input:

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

Output:

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

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

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

Output:

integral(sqrt(a^2*c*x^2 + c)*arctan(a*x)/(a^4*c^2*x^7 + 2*a^2*c^2*x^5 + c^ 
2*x^3), x)
 

Sympy [F]

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

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

Output:

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

Maxima [F]

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

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

Output:

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

Giac [F]

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

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

Output:

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

\[ \int \frac {\arctan (a x)}{x^3 \left (c+a^2 c x^2\right )^{3/2}} \, dx=\frac {\int \frac {\mathit {atan} \left (a x \right )}{\sqrt {a^{2} x^{2}+1}\, a^{2} x^{5}+\sqrt {a^{2} x^{2}+1}\, x^{3}}d x}{\sqrt {c}\, c} \] Input:

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

Output:

int(atan(a*x)/(sqrt(a**2*x**2 + 1)*a**2*x**5 + sqrt(a**2*x**2 + 1)*x**3),x 
)/(sqrt(c)*c)