\(\int \frac {\text {arctanh}(a x)}{\sqrt {c-a^2 c x^2}} \, dx\) [467]

Optimal result

Integrand size = 20, antiderivative size = 182 \[ \int \frac {\text {arctanh}(a x)}{\sqrt {c-a^2 c x^2}} \, dx=-\frac {2 \sqrt {1-a^2 x^2} \arctan \left (\frac {\sqrt {1-a x}}{\sqrt {1+a x}}\right ) \text {arctanh}(a x)}{a \sqrt {c-a^2 c x^2}}-\frac {i \sqrt {1-a^2 x^2} \operatorname {PolyLog}\left (2,-\frac {i \sqrt {1-a x}}{\sqrt {1+a x}}\right )}{a \sqrt {c-a^2 c x^2}}+\frac {i \sqrt {1-a^2 x^2} \operatorname {PolyLog}\left (2,\frac {i \sqrt {1-a x}}{\sqrt {1+a x}}\right )}{a \sqrt {c-a^2 c x^2}} \] Output:

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

Mathematica [A] (verified)

Time = 0.15 (sec) , antiderivative size = 109, normalized size of antiderivative = 0.60 \[ \int \frac {\text {arctanh}(a x)}{\sqrt {c-a^2 c x^2}} \, dx=-\frac {i \sqrt {c \left (1-a^2 x^2\right )} \left (\text {arctanh}(a x) \left (\log \left (1-i e^{-\text {arctanh}(a x)}\right )-\log \left (1+i e^{-\text {arctanh}(a x)}\right )\right )+\operatorname {PolyLog}\left (2,-i e^{-\text {arctanh}(a x)}\right )-\operatorname {PolyLog}\left (2,i e^{-\text {arctanh}(a x)}\right )\right )}{a c \sqrt {1-a^2 x^2}} \] Input:

Integrate[ArcTanh[a*x]/Sqrt[c - a^2*c*x^2],x]
 

Output:

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

Rubi [A] (verified)

Time = 0.34 (sec) , antiderivative size = 125, normalized size of antiderivative = 0.69, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.100, Rules used = {6516, 6512}

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.

\(\displaystyle \int \frac {\text {arctanh}(a x)}{\sqrt {c-a^2 c x^2}} \, dx\)

\(\Big \downarrow \) 6516

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

\(\Big \downarrow \) 6512

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

Input:

Int[ArcTanh[a*x]/Sqrt[c - a^2*c*x^2],x]
 

Output:

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

Defintions of rubi rules used

rule 6512

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

rule 6516

Int[((a_.) + ArcTanh[(c_.)*(x_)]*(b_.))^(p_.)/Sqrt[(d_) + (e_.)*(x_)^2], x_ 
Symbol] :> Simp[Sqrt[1 - c^2*x^2]/Sqrt[d + e*x^2]   Int[(a + b*ArcTanh[c*x] 
)^p/Sqrt[1 - c^2*x^2], x], x] /; FreeQ[{a, b, c, d, e}, x] && EqQ[c^2*d + e 
, 0] && IGtQ[p, 0] &&  !GtQ[d, 0]
 

Maple [A] (verified)

Time = 0.55 (sec) , antiderivative size = 302, normalized size of antiderivative = 1.66

method	result	size
default	\(\frac {i \sqrt {-\left (a x -1\right ) c \left (a x +1\right )}\, \sqrt {-a^{2} x^{2}+1}\, \operatorname {arctanh}\left (a x \right ) \ln \left (1+\frac {i \left (a x +1\right )}{\sqrt {-a^{2} x^{2}+1}}\right )}{\left (a x -1\right ) \left (a x +1\right ) c a}-\frac {i \sqrt {-\left (a x -1\right ) c \left (a x +1\right )}\, \sqrt {-a^{2} x^{2}+1}\, \operatorname {arctanh}\left (a x \right ) \ln \left (1-\frac {i \left (a x +1\right )}{\sqrt {-a^{2} x^{2}+1}}\right )}{\left (a x -1\right ) \left (a x +1\right ) c a}+\frac {i \sqrt {-\left (a x -1\right ) c \left (a x +1\right )}\, \sqrt {-a^{2} x^{2}+1}\, \operatorname {dilog}\left (1+\frac {i \left (a x +1\right )}{\sqrt {-a^{2} x^{2}+1}}\right )}{\left (a x -1\right ) \left (a x +1\right ) c a}-\frac {i \sqrt {-\left (a x -1\right ) c \left (a x +1\right )}\, \sqrt {-a^{2} x^{2}+1}\, \operatorname {dilog}\left (1-\frac {i \left (a x +1\right )}{\sqrt {-a^{2} x^{2}+1}}\right )}{\left (a x -1\right ) \left (a x +1\right ) c a}\)	\(302\)

Input:

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

Output:

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

Fricas [F]

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

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

Output:

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

Sympy [F]

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

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

Output:

Integral(atanh(a*x)/sqrt(-c*(a*x - 1)*(a*x + 1)), x)
 

Maxima [F]

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

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

Output:

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

Giac [F]

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

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

Output:

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

Mupad [F(-1)]

Timed out. \[ \int \frac {\text {arctanh}(a x)}{\sqrt {c-a^2 c x^2}} \, dx=\int \frac {\mathrm {atanh}\left (a\,x\right )}{\sqrt {c-a^2\,c\,x^2}} \,d x \] Input:

int(atanh(a*x)/(c - a^2*c*x^2)^(1/2),x)
 

Output:

int(atanh(a*x)/(c - a^2*c*x^2)^(1/2), x)
 

Reduce [F]

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

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

Output:

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