\(\int \frac {\arctan (a x)}{x \sqrt {c+a^2 c x^2}} \, dx\) [228]

Optimal result

Integrand size = 22, antiderivative size = 177 \[ \int \frac {\arctan (a x)}{x \sqrt {c+a^2 c x^2}} \, dx=-\frac {2 \sqrt {1+a^2 x^2} \arctan (a x) \text {arctanh}\left (\frac {\sqrt {1+i a x}}{\sqrt {1-i a x}}\right )}{\sqrt {c+a^2 c x^2}}+\frac {i \sqrt {1+a^2 x^2} \operatorname {PolyLog}\left (2,-\frac {\sqrt {1+i a x}}{\sqrt {1-i a x}}\right )}{\sqrt {c+a^2 c x^2}}-\frac {i \sqrt {1+a^2 x^2} \operatorname {PolyLog}\left (2,\frac {\sqrt {1+i a x}}{\sqrt {1-i a x}}\right )}{\sqrt {c+a^2 c x^2}} \]

[Out]

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

Rubi [A] (verified)

Time = 0.10 (sec) , antiderivative size = 177, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.091, Rules used = {5078, 5074} \[ \int \frac {\arctan (a x)}{x \sqrt {c+a^2 c x^2}} \, dx=-\frac {2 \sqrt {a^2 x^2+1} \arctan (a x) \text {arctanh}\left (\frac {\sqrt {1+i a x}}{\sqrt {1-i a x}}\right )}{\sqrt {a^2 c x^2+c}}+\frac {i \sqrt {a^2 x^2+1} \operatorname {PolyLog}\left (2,-\frac {\sqrt {i a x+1}}{\sqrt {1-i a x}}\right )}{\sqrt {a^2 c x^2+c}}-\frac {i \sqrt {a^2 x^2+1} \operatorname {PolyLog}\left (2,\frac {\sqrt {i a x+1}}{\sqrt {1-i a x}}\right )}{\sqrt {a^2 c x^2+c}} \]

[In]

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

[Out]

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

Rule 5074

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]/Sqrt[1 - I*c*x]], x] + (Simp[I*(b/Sqrt[d])*PolyLog[2, -Sqrt[1 + I*c*x]/S
qrt[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 5078

Int[((a_.) + ArcTan[(c_.)*(x_)]*(b_.))^(p_.)/((x_)*Sqrt[(d_) + (e_.)*(x_)^2]), x_Symbol] :> Dist[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]

Rubi steps \begin{align*} \text {integral}& = \frac {\sqrt {1+a^2 x^2} \int \frac {\arctan (a x)}{x \sqrt {1+a^2 x^2}} \, dx}{\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 )}{\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 )}{\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 )}{\sqrt {c+a^2 c x^2}} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.23 (sec) , antiderivative size = 100, normalized size of antiderivative = 0.56 \[ \int \frac {\arctan (a x)}{x \sqrt {c+a^2 c x^2}} \, dx=\frac {\sqrt {1+a^2 x^2} \left (\arctan (a x) \left (\log \left (1-e^{i \arctan (a x)}\right )-\log \left (1+e^{i \arctan (a x)}\right )\right )+i \operatorname {PolyLog}\left (2,-e^{i \arctan (a x)}\right )-i \operatorname {PolyLog}\left (2,e^{i \arctan (a x)}\right )\right )}{\sqrt {c \left (1+a^2 x^2\right )}} \]

[In]

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

[Out]

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

Maple [A] (verified)

Time = 0.43 (sec) , antiderivative size = 139, normalized size of antiderivative = 0.79

method	result	size
default	\(\frac {i \left (i \arctan \left (a x \right ) \ln \left (\frac {i a x +1}{\sqrt {a^{2} x^{2}+1}}+1\right )-i \arctan \left (a x \right ) \ln \left (1-\frac {i a x +1}{\sqrt {a^{2} x^{2}+1}}\right )+\operatorname {polylog}\left (2, -\frac {i a x +1}{\sqrt {a^{2} x^{2}+1}}\right )-\operatorname {polylog}\left (2, \frac {i a x +1}{\sqrt {a^{2} x^{2}+1}}\right )\right ) \sqrt {c \left (a x -i\right ) \left (a x +i\right )}}{\sqrt {a^{2} x^{2}+1}\, c}\)	\(139\)

[In]

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

[Out]

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

Fricas [F]

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

[In]

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

[Out]

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

Sympy [F]

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

[In]

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

[Out]

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

Maxima [F]

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

[In]

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

[Out]

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

Giac [F]

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

[In]

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

[Out]

sage0*x

Mupad [F(-1)]

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

[In]

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

[Out]

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