\(\int \sqrt {c-a^2 c x^2} \text {arctanh}(a x) \, dx\) [466]

Optimal result

Integrand size = 20, antiderivative size = 235 \[ \int \sqrt {c-a^2 c x^2} \text {arctanh}(a x) \, dx=\frac {\sqrt {c-a^2 c x^2}}{2 a}+\frac {1}{2} x \sqrt {c-a^2 c x^2} \text {arctanh}(a x)-\frac {c \sqrt {1-a^2 x^2} \arctan \left (\frac {\sqrt {1-a x}}{\sqrt {1+a x}}\right ) \text {arctanh}(a x)}{a \sqrt {c-a^2 c x^2}}-\frac {i c \sqrt {1-a^2 x^2} \operatorname {PolyLog}\left (2,-\frac {i \sqrt {1-a x}}{\sqrt {1+a x}}\right )}{2 a \sqrt {c-a^2 c x^2}}+\frac {i c \sqrt {1-a^2 x^2} \operatorname {PolyLog}\left (2,\frac {i \sqrt {1-a x}}{\sqrt {1+a x}}\right )}{2 a \sqrt {c-a^2 c x^2}} \] Output:

1/2*(-a^2*c*x^2+c)^(1/2)/a+1/2*x*(-a^2*c*x^2+c)^(1/2)*arctanh(a*x)-c*(-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)-1/2*I*c*(-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)+1/2*I*c*(-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 = 0.27 (sec) , antiderivative size = 119, normalized size of antiderivative = 0.51 \[ \int \sqrt {c-a^2 c x^2} \text {arctanh}(a x) \, dx=\frac {\sqrt {c \left (1-a^2 x^2\right )} \left (1+a x \text {arctanh}(a x)-\frac {i \left (\text {arctanh}(a x) \left (\log \left (1-i e^{-\text {arctanh}(a x)}\right )-\log \left (1+i e^{-\text {arctanh}(a x)}\right )\right )+\operatorname {PolyLog}\left (2,-i e^{-\text {arctanh}(a x)}\right )-\operatorname {PolyLog}\left (2,i e^{-\text {arctanh}(a x)}\right )\right )}{\sqrt {1-a^2 x^2}}\right )}{2 a} \] Input:

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

Output:

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

Rubi [A] (verified)

Time = 0.47 (sec) , antiderivative size = 176, normalized size of antiderivative = 0.75, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.150, Rules used = {6504, 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.

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

\(\Big \downarrow \) 6504

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

\(\Big \downarrow \) 6516

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

\(\Big \downarrow \) 6512

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

Input:

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

Output:

Sqrt[c - a^2*c*x^2]/(2*a) + (x*Sqrt[c - a^2*c*x^2]*ArcTanh[a*x])/2 + (c*Sq 
rt[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*S 
qrt[1 - a*x])/Sqrt[1 + a*x]])/a))/(2*Sqrt[c - a^2*c*x^2])
 

Defintions of rubi rules used

rule 6504

Int[((a_.) + ArcTanh[(c_.)*(x_)]*(b_.))*((d_) + (e_.)*(x_)^2)^(q_.), x_Symb 
ol] :> Simp[b*((d + e*x^2)^q/(2*c*q*(2*q + 1))), x] + (Simp[x*(d + e*x^2)^q 
*((a + b*ArcTanh[c*x])/(2*q + 1)), x] + Simp[2*d*(q/(2*q + 1))   Int[(d + e 
*x^2)^(q - 1)*(a + b*ArcTanh[c*x]), x], x]) /; FreeQ[{a, b, c, d, e}, x] && 
 EqQ[c^2*d + e, 0] && GtQ[q, 0]
 

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 = 0.58 (sec) , antiderivative size = 319, normalized size of antiderivative = 1.36

method	result	size
default	\(\frac {\left (a x \,\operatorname {arctanh}\left (a x \right )+1\right ) \sqrt {-\left (a x -1\right ) c \left (a x +1\right )}}{2 a}+\frac {i \sqrt {-\left (a x -1\right ) c \left (a x +1\right )}\, \sqrt {-a^{2} x^{2}+1}\, \operatorname {arctanh}\left (a x \right ) \ln \left (1+\frac {i \left (a x +1\right )}{\sqrt {-a^{2} x^{2}+1}}\right )}{2 a \left (a x +1\right ) \left (a x -1\right )}-\frac {i \sqrt {-\left (a x -1\right ) c \left (a x +1\right )}\, \sqrt {-a^{2} x^{2}+1}\, \operatorname {arctanh}\left (a x \right ) \ln \left (1-\frac {i \left (a x +1\right )}{\sqrt {-a^{2} x^{2}+1}}\right )}{2 a \left (a x +1\right ) \left (a x -1\right )}+\frac {i \sqrt {-\left (a x -1\right ) c \left (a x +1\right )}\, \sqrt {-a^{2} x^{2}+1}\, \operatorname {dilog}\left (1+\frac {i \left (a x +1\right )}{\sqrt {-a^{2} x^{2}+1}}\right )}{2 a \left (a x +1\right ) \left (a x -1\right )}-\frac {i \sqrt {-\left (a x -1\right ) c \left (a x +1\right )}\, \sqrt {-a^{2} x^{2}+1}\, \operatorname {dilog}\left (1-\frac {i \left (a x +1\right )}{\sqrt {-a^{2} x^{2}+1}}\right )}{2 a \left (a x +1\right ) \left (a x -1\right )}\)	\(319\)

Input:

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

Output:

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

Fricas [F]

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

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

Output:

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

Sympy [F]

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

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

Output:

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

Maxima [F]

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

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

Output:

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

Giac [F(-2)]

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

integrate((-a^2*c*x^2+c)^(1/2)*arctanh(a*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. \[ \int \sqrt {c-a^2 c x^2} \text {arctanh}(a x) \, dx=\int \mathrm {atanh}\left (a\,x\right )\,\sqrt {c-a^2\,c\,x^2} \,d x \] 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]

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

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

Output:

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