\(\int \frac {\arctan (a x)}{\sqrt {c+a^2 c x^2}} \, dx\) [227]

Optimal result

Integrand size = 19, antiderivative size = 193 \[ \int \frac {\arctan (a x)}{\sqrt {c+a^2 c x^2}} \, dx=-\frac {2 i \sqrt {1+a^2 x^2} \arctan (a x) \arctan \left (\frac {\sqrt {1+i a x}}{\sqrt {1-i a x}}\right )}{a \sqrt {c+a^2 c x^2}}+\frac {i \sqrt {1+a^2 x^2} \operatorname {PolyLog}\left (2,-\frac {i \sqrt {1+i a x}}{\sqrt {1-i a x}}\right )}{a \sqrt {c+a^2 c x^2}}-\frac {i \sqrt {1+a^2 x^2} \operatorname {PolyLog}\left (2,\frac {i \sqrt {1+i a x}}{\sqrt {1-i a x}}\right )}{a \sqrt {c+a^2 c x^2}} \]

[Out]

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

Rubi [A] (verified)

Time = 0.04 (sec) , antiderivative size = 193, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.105, Rules used = {5010, 5006} \[ \int \frac {\arctan (a x)}{\sqrt {c+a^2 c x^2}} \, dx=-\frac {2 i \sqrt {a^2 x^2+1} \arctan (a x) \arctan \left (\frac {\sqrt {1+i a x}}{\sqrt {1-i a x}}\right )}{a \sqrt {a^2 c x^2+c}}+\frac {i \sqrt {a^2 x^2+1} \operatorname {PolyLog}\left (2,-\frac {i \sqrt {i a x+1}}{\sqrt {1-i a x}}\right )}{a \sqrt {a^2 c x^2+c}}-\frac {i \sqrt {a^2 x^2+1} \operatorname {PolyLog}\left (2,\frac {i \sqrt {i a x+1}}{\sqrt {1-i a x}}\right )}{a \sqrt {a^2 c x^2+c}} \]

[In]

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

[Out]

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

Rule 5006

Int[((a_.) + ArcTan[(c_.)*(x_)]*(b_.))/Sqrt[(d_) + (e_.)*(x_)^2], x_Symbol] :> Simp[-2*I*(a + b*ArcTan[c*x])*(
ArcTan[Sqrt[1 + I*c*x]/Sqrt[1 - I*c*x]]/(c*Sqrt[d])), x] + (Simp[I*b*(PolyLog[2, (-I)*(Sqrt[1 + I*c*x]/Sqrt[1
- I*c*x])]/(c*Sqrt[d])), x] - Simp[I*b*(PolyLog[2, I*(Sqrt[1 + I*c*x]/Sqrt[1 - I*c*x])]/(c*Sqrt[d])), x]) /; F
reeQ[{a, b, c, d, e}, x] && EqQ[e, c^2*d] && GtQ[d, 0]

Rule 5010

Int[((a_.) + ArcTan[(c_.)*(x_)]*(b_.))^(p_.)/Sqrt[(d_) + (e_.)*(x_)^2], x_Symbol] :> Dist[Sqrt[1 + c^2*x^2]/Sq
rt[d + e*x^2], Int[(a + b*ArcTan[c*x])^p/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)}{\sqrt {1+a^2 x^2}} \, dx}{\sqrt {c+a^2 c x^2}} \\ & = -\frac {2 i \sqrt {1+a^2 x^2} \arctan (a x) \arctan \left (\frac {\sqrt {1+i a x}}{\sqrt {1-i a x}}\right )}{a \sqrt {c+a^2 c x^2}}+\frac {i \sqrt {1+a^2 x^2} \operatorname {PolyLog}\left (2,-\frac {i \sqrt {1+i a x}}{\sqrt {1-i a x}}\right )}{a \sqrt {c+a^2 c x^2}}-\frac {i \sqrt {1+a^2 x^2} \operatorname {PolyLog}\left (2,\frac {i \sqrt {1+i a x}}{\sqrt {1-i a x}}\right )}{a \sqrt {c+a^2 c x^2}} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.10 (sec) , antiderivative size = 90, normalized size of antiderivative = 0.47 \[ \int \frac {\arctan (a x)}{\sqrt {c+a^2 c x^2}} \, dx=-\frac {i \sqrt {c \left (1+a^2 x^2\right )} \left (2 \arctan \left (e^{i \arctan (a x)}\right ) \arctan (a x)-\operatorname {PolyLog}\left (2,-i e^{i \arctan (a x)}\right )+\operatorname {PolyLog}\left (2,i e^{i \arctan (a x)}\right )\right )}{a c \sqrt {1+a^2 x^2}} \]

[In]

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

[Out]

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

Maple [A] (verified)

Time = 0.39 (sec) , antiderivative size = 150, normalized size of antiderivative = 0.78

method	result	size
default	\(-\frac {\left (\arctan \left (a x \right ) \ln \left (1+\frac {i \left (i a x +1\right )}{\sqrt {a^{2} x^{2}+1}}\right )-\arctan \left (a x \right ) \ln \left (1-\frac {i \left (i a x +1\right )}{\sqrt {a^{2} x^{2}+1}}\right )-i \operatorname {dilog}\left (1+\frac {i \left (i a x +1\right )}{\sqrt {a^{2} x^{2}+1}}\right )+i \operatorname {dilog}\left (1-\frac {i \left (i a x +1\right )}{\sqrt {a^{2} x^{2}+1}}\right )\right ) \sqrt {c \left (a x -i\right ) \left (a x +i\right )}}{\sqrt {a^{2} x^{2}+1}\, c a}\)	\(150\)

[In]

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

[Out]

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

Fricas [F]

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

[In]

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

[Out]

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

Sympy [F]

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

[In]

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

[Out]

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

Maxima [F]

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

[In]

integrate(arctan(a*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)

Giac [F]

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

[In]

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

[Out]

sage0*x

Mupad [F(-1)]

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

[In]

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

[Out]

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