\(\int \frac {\sqrt {c+a^2 c x^2} \arctan (a x)}{x} \, dx\) [204]

Optimal result

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

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

Mathematica [A] (verified)

Time = 0.20 (sec) , antiderivative size = 164, normalized size of antiderivative = 0.72 \[ \int \frac {\sqrt {c+a^2 c x^2} \arctan (a x)}{x} \, dx=\frac {\sqrt {c+a^2 c x^2} \left (\sqrt {1+a^2 x^2} \arctan (a x)+\arctan (a x) \log \left (1-e^{i \arctan (a x)}\right )-\arctan (a x) \log \left (1+e^{i \arctan (a x)}\right )+\log \left (\cos \left (\frac {1}{2} \arctan (a x)\right )-\sin \left (\frac {1}{2} \arctan (a x)\right )\right )-\log \left (\cos \left (\frac {1}{2} \arctan (a x)\right )+\sin \left (\frac {1}{2} \arctan (a x)\right )\right )+i \operatorname {PolyLog}\left (2,-e^{i \arctan (a x)}\right )-i \operatorname {PolyLog}\left (2,e^{i \arctan (a x)}\right )\right )}{\sqrt {1+a^2 x^2}} \] Input:

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

Output:

(Sqrt[c + a^2*c*x^2]*(Sqrt[1 + a^2*x^2]*ArcTan[a*x] + ArcTan[a*x]*Log[1 - 
E^(I*ArcTan[a*x])] - ArcTan[a*x]*Log[1 + E^(I*ArcTan[a*x])] + Log[Cos[ArcT 
an[a*x]/2] - Sin[ArcTan[a*x]/2]] - Log[Cos[ArcTan[a*x]/2] + Sin[ArcTan[a*x 
]/2]] + I*PolyLog[2, -E^(I*ArcTan[a*x])] - I*PolyLog[2, E^(I*ArcTan[a*x])] 
))/Sqrt[1 + a^2*x^2]
 

Rubi [A] (verified)

Time = 0.59 (sec) , antiderivative size = 175, normalized size of antiderivative = 0.76, number of steps used = 6, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.227, Rules used = {5481, 224, 219, 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.

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

\(\Big \downarrow \) 5481

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

\(\Big \downarrow \) 224

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

\(\Big \downarrow \) 219

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

\(\Big \downarrow \) 5493

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

\(\Big \downarrow \) 5489

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

Input:

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

Output:

Sqrt[c + a^2*c*x^2]*ArcTan[a*x] - Sqrt[c]*ArcTanh[(a*Sqrt[c]*x)/Sqrt[c + a 
^2*c*x^2]] + (c*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*Po 
lyLog[2, Sqrt[1 + I*a*x]/Sqrt[1 - I*a*x]]))/Sqrt[c + a^2*c*x^2]
 

Defintions of rubi rules used

rule 219

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1/(Rt[a, 2]*Rt[-b, 2]))* 
ArcTanh[Rt[-b, 2]*(x/Rt[a, 2])], x] /; FreeQ[{a, b}, x] && NegQ[a/b] && (Gt 
Q[a, 0] || LtQ[b, 0])
 

rule 224

Int[1/Sqrt[(a_) + (b_.)*(x_)^2], x_Symbol] :> Subst[Int[1/(1 - b*x^2), x], 
x, x/Sqrt[a + b*x^2]] /; FreeQ[{a, b}, x] &&  !GtQ[a, 0]
 

rule 5481

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

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]
 

Maple [A] (verified)

Time = 1.49 (sec) , antiderivative size = 151, normalized size of antiderivative = 0.66

method	result	size
default	\(\sqrt {c \left (a x -i\right ) \left (a x +i\right )}\, \arctan \left (a x \right )-\frac {\sqrt {c \left (a x -i\right ) \left (a x +i\right )}\, \left (\arctan \left (a x \right ) \ln \left (1+\frac {i a x +1}{\sqrt {a^{2} x^{2}+1}}\right )-i \operatorname {dilog}\left (\frac {i a x +1}{\sqrt {a^{2} x^{2}+1}}\right )-i \operatorname {dilog}\left (1+\frac {i a x +1}{\sqrt {a^{2} x^{2}+1}}\right )-2 i \arctan \left (\frac {i a x +1}{\sqrt {a^{2} x^{2}+1}}\right )\right )}{\sqrt {a^{2} x^{2}+1}}\)	\(151\)

Input:

int((a^2*c*x^2+c)^(1/2)*arctan(a*x)/x,x,method=_RETURNVERBOSE)
 

Output:

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

Fricas [F]

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

integrate((a^2*c*x^2+c)^(1/2)*arctan(a*x)/x,x, algorithm="fricas")
 

Output:

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

Sympy [F]

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

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

Output:

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

Maxima [F]

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

integrate((a^2*c*x^2+c)^(1/2)*arctan(a*x)/x,x, algorithm="maxima")
 

Output:

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

Giac [F(-2)]

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

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

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

Output:

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

Reduce [F]

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

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

Output:

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