3.3.36 \(\int \frac {\arctan (a x)}{x (c+a^2 c x^2)^{3/2}} \, dx\) [236]

3.3.36.1 Optimal result

Integrand size = 22, antiderivative size = 229 \[ \int \frac {\arctan (a x)}{x \left (c+a^2 c x^2\right )^{3/2}} \, dx=-\frac {a x}{c \sqrt {c+a^2 c x^2}}+\frac {\arctan (a x)}{c \sqrt {c+a^2 c x^2}}-\frac {2 \sqrt {1+a^2 x^2} \arctan (a x) \text {arctanh}\left (\frac {\sqrt {1+i a x}}{\sqrt {1-i a x}}\right )}{c \sqrt {c+a^2 c x^2}}+\frac {i \sqrt {1+a^2 x^2} \operatorname {PolyLog}\left (2,-\frac {\sqrt {1+i a x}}{\sqrt {1-i a x}}\right )}{c \sqrt {c+a^2 c x^2}}-\frac {i \sqrt {1+a^2 x^2} \operatorname {PolyLog}\left (2,\frac {\sqrt {1+i a x}}{\sqrt {1-i a x}}\right )}{c \sqrt {c+a^2 c x^2}} \]

output

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

3.3.36.2 Mathematica [A] (verified)

Time = 0.26 (sec) , antiderivative size = 141, normalized size of antiderivative = 0.62 \[ \int \frac {\arctan (a x)}{x \left (c+a^2 c x^2\right )^{3/2}} \, dx=\frac {\sqrt {1+a^2 x^2} \left (-\frac {a x}{\sqrt {1+a^2 x^2}}+\frac {\arctan (a x)}{\sqrt {1+a^2 x^2}}+\arctan (a x) \log \left (1-e^{i \arctan (a x)}\right )-\arctan (a x) \log \left (1+e^{i \arctan (a x)}\right )+i \operatorname {PolyLog}\left (2,-e^{i \arctan (a x)}\right )-i \operatorname {PolyLog}\left (2,e^{i \arctan (a x)}\right )\right )}{c \sqrt {c \left (1+a^2 x^2\right )}} \]

input

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

output

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

3.3.36.3 Rubi [A] (verified)

Time = 0.68 (sec) , antiderivative size = 182, normalized size of antiderivative = 0.79, number of steps used = 5, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.227, Rules used = {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.

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

\(\Big \downarrow \) 5501

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

\(\Big \downarrow \) 5465

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

\(\Big \downarrow \) 208

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

\(\Big \downarrow \) 5493

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

\(\Big \downarrow \) 5489

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

input

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

output

-(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])
 

3.3.36.3.1 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 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 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]
 

3.3.36.4 Maple [A] (verified)

Time = 0.42 (sec) , antiderivative size = 232, normalized size of antiderivative = 1.01

method	result	size
default	\(\frac {\left (\arctan \left (a x \right )+i\right ) \left (i a x +1\right ) \sqrt {c \left (a x -i\right ) \left (a x +i\right )}}{2 \left (a^{2} x^{2}+1\right ) c^{2}}-\frac {\sqrt {c \left (a x -i\right ) \left (a x +i\right )}\, \left (i a x -1\right ) \left (\arctan \left (a x \right )-i\right )}{2 \left (a^{2} x^{2}+1\right ) c^{2}}+\frac {i \left (i \arctan \left (a x \right ) \ln \left (\frac {i a x +1}{\sqrt {a^{2} x^{2}+1}}+1\right )-i \arctan \left (a x \right ) \ln \left (1-\frac {i a x +1}{\sqrt {a^{2} x^{2}+1}}\right )+\operatorname {polylog}\left (2, -\frac {i a x +1}{\sqrt {a^{2} x^{2}+1}}\right )-\operatorname {polylog}\left (2, \frac {i a x +1}{\sqrt {a^{2} x^{2}+1}}\right )\right ) \sqrt {c \left (a x -i\right ) \left (a x +i\right )}}{\sqrt {a^{2} x^{2}+1}\, c^{2}}\)	\(232\)

input

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

output

1/2*(arctan(a*x)+I)*(1+I*a*x)*(c*(a*x-I)*(I+a*x))^(1/2)/(a^2*x^2+1)/c^2-1/ 
2*(c*(a*x-I)*(I+a*x))^(1/2)*(I*a*x-1)*(arctan(a*x)-I)/(a^2*x^2+1)/c^2+I*(I 
*arctan(a*x)*ln((1+I*a*x)/(a^2*x^2+1)^(1/2)+1)-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)*(I+a*x))^(1/2)/(a^2*x^2+1)^(1/2)/c^2
 

3.3.36.5 Fricas [F]

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

input

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

3.3.36.6 Sympy [F(-2)]

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

input

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

output

Exception raised: TypeError >> Invalid comparison of non-real zoo
 

3.3.36.7 Maxima [F]

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

input

integrate(arctan(a*x)/x/(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), x)
 

3.3.36.8 Giac [F]

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

input

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

output

sage0*x
 

3.3.36.9 Mupad [F(-1)]

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

input

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

output

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