\(\int \frac {\arctan (a x)^2}{x^2 (c+a^2 c x^2)^{3/2}} \, dx\) [344]

Optimal result

Integrand size = 24, antiderivative size = 293 \[ \int \frac {\arctan (a x)^2}{x^2 \left (c+a^2 c x^2\right )^{3/2}} \, dx=\frac {2 a^2 x}{c \sqrt {c+a^2 c x^2}}-\frac {2 a \arctan (a x)}{c \sqrt {c+a^2 c x^2}}-\frac {a^2 x \arctan (a x)^2}{c \sqrt {c+a^2 c x^2}}-\frac {\sqrt {c+a^2 c x^2} \arctan (a x)^2}{c^2 x}-\frac {4 a \sqrt {1+a^2 x^2} \arctan (a x) \text {arctanh}\left (\frac {\sqrt {1+i a x}}{\sqrt {1-i a x}}\right )}{c \sqrt {c+a^2 c x^2}}+\frac {2 i a \sqrt {1+a^2 x^2} \operatorname {PolyLog}\left (2,-\frac {\sqrt {1+i a x}}{\sqrt {1-i a x}}\right )}{c \sqrt {c+a^2 c x^2}}-\frac {2 i a \sqrt {1+a^2 x^2} \operatorname {PolyLog}\left (2,\frac {\sqrt {1+i a x}}{\sqrt {1-i a x}}\right )}{c \sqrt {c+a^2 c x^2}} \] Output:

2*a^2*x/c/(a^2*c*x^2+c)^(1/2)-2*a*arctan(a*x)/c/(a^2*c*x^2+c)^(1/2)-a^2*x* 
arctan(a*x)^2/c/(a^2*c*x^2+c)^(1/2)-(a^2*c*x^2+c)^(1/2)*arctan(a*x)^2/c^2/ 
x-4*a*(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)+2*I*a*(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)-2*I*a*(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] (verified)

Time = 0.81 (sec) , antiderivative size = 226, normalized size of antiderivative = 0.77 \[ \int \frac {\arctan (a x)^2}{x^2 \left (c+a^2 c x^2\right )^{3/2}} \, dx=\frac {a \left (4 a x-4 \arctan (a x)-2 a x \arctan (a x)^2-\frac {1}{2} a x \arctan (a x)^2 \csc ^2\left (\frac {1}{2} \arctan (a x)\right )+4 \sqrt {1+a^2 x^2} \arctan (a x) \log \left (1-e^{i \arctan (a x)}\right )-4 \sqrt {1+a^2 x^2} \arctan (a x) \log \left (1+e^{i \arctan (a x)}\right )+4 i \sqrt {1+a^2 x^2} \operatorname {PolyLog}\left (2,-e^{i \arctan (a x)}\right )-4 i \sqrt {1+a^2 x^2} \operatorname {PolyLog}\left (2,e^{i \arctan (a x)}\right )-\frac {2 \left (1+a^2 x^2\right ) \arctan (a x)^2 \sin ^2\left (\frac {1}{2} \arctan (a x)\right )}{a x}\right )}{2 c \sqrt {c+a^2 c x^2}} \] Input:

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

Output:

(a*(4*a*x - 4*ArcTan[a*x] - 2*a*x*ArcTan[a*x]^2 - (a*x*ArcTan[a*x]^2*Csc[A 
rcTan[a*x]/2]^2)/2 + 4*Sqrt[1 + a^2*x^2]*ArcTan[a*x]*Log[1 - E^(I*ArcTan[a 
*x])] - 4*Sqrt[1 + a^2*x^2]*ArcTan[a*x]*Log[1 + E^(I*ArcTan[a*x])] + (4*I) 
*Sqrt[1 + a^2*x^2]*PolyLog[2, -E^(I*ArcTan[a*x])] - (4*I)*Sqrt[1 + a^2*x^2 
]*PolyLog[2, E^(I*ArcTan[a*x])] - (2*(1 + a^2*x^2)*ArcTan[a*x]^2*Sin[ArcTa 
n[a*x]/2]^2)/(a*x)))/(2*c*Sqrt[c + a^2*c*x^2])
 

Rubi [A] (verified)

Time = 1.02 (sec) , antiderivative size = 237, normalized size of antiderivative = 0.81, number of steps used = 6, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {5501, 5433, 208, 5479, 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.

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

\(\Big \downarrow \) 5501

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

\(\Big \downarrow \) 5433

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

\(\Big \downarrow \) 208

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

\(\Big \downarrow \) 5479

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

\(\Big \downarrow \) 5493

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

\(\Big \downarrow \) 5489

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

Input:

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

Output:

-(a^2*((-2*x)/(c*Sqrt[c + a^2*c*x^2]) + (2*ArcTan[a*x])/(a*c*Sqrt[c + a^2* 
c*x^2]) + (x*ArcTan[a*x]^2)/(c*Sqrt[c + a^2*c*x^2]))) + (-((Sqrt[c + a^2*c 
*x^2]*ArcTan[a*x]^2)/(c*x)) + (2*a*Sqrt[1 + a^2*x^2]*(-2*ArcTan[a*x]*ArcTa 
nh[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]]))/Sqrt[c + a^ 
2*c*x^2])/c
 

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 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 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 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 = 2.18 (sec) , antiderivative size = 279, normalized size of antiderivative = 0.95

method	result	size
default	\(-\frac {a \left (\arctan \left (a x \right )^{2}-2+2 i \arctan \left (a x \right )\right ) \left (a x -i\right ) \sqrt {c \left (a x -i\right ) \left (a x +i\right )}}{2 \left (a^{2} x^{2}+1\right ) c^{2}}-\frac {\sqrt {c \left (a x -i\right ) \left (a x +i\right )}\, \left (a x +i\right ) \left (\arctan \left (a x \right )^{2}-2-2 i \arctan \left (a x \right )\right ) a}{2 \left (a^{2} x^{2}+1\right ) c^{2}}-\frac {\arctan \left (a x \right )^{2} \sqrt {c \left (a x -i\right ) \left (a x +i\right )}}{c^{2} x}-\frac {2 i a \left (i \arctan \left (a x \right ) \ln \left (1-\frac {i a x +1}{\sqrt {a^{2} x^{2}+1}}\right )-i \arctan \left (a x \right ) \ln \left (1+\frac {i a x +1}{\sqrt {a^{2} x^{2}+1}}\right )+\operatorname {polylog}\left (2, \frac {i a x +1}{\sqrt {a^{2} x^{2}+1}}\right )-\operatorname {polylog}\left (2, -\frac {i a x +1}{\sqrt {a^{2} x^{2}+1}}\right )\right ) \sqrt {c \left (a x -i\right ) \left (a x +i\right )}}{\sqrt {a^{2} x^{2}+1}\, c^{2}}\)	\(279\)

Input:

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

Output:

-1/2*a*(arctan(a*x)^2-2+2*I*arctan(a*x))*(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)^2-2-2* 
I*arctan(a*x))*a/(a^2*x^2+1)/c^2-arctan(a*x)^2*(c*(a*x-I)*(a*x+I))^(1/2)/c 
^2/x-2*I*a*(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))-p 
olylog(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]

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

integrate(arctan(a*x)^2/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)^2/(a^4*c^2*x^6 + 2*a^2*c^2*x^4 + 
c^2*x^2), x)
 

Sympy [F]

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

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

Output:

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

Maxima [F]

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

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

Output:

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

Giac [F(-2)]

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

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

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

Output:

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

Reduce [F]

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

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

Output:

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