\(\int \frac {\text {arctanh}(a x)^3}{x \sqrt {1-a^2 x^2}} \, dx\) [384]

Optimal result

Integrand size = 24, antiderivative size = 102 \[ \int \frac {\text {arctanh}(a x)^3}{x \sqrt {1-a^2 x^2}} \, dx=-2 \text {arctanh}\left (e^{\text {arctanh}(a x)}\right ) \text {arctanh}(a x)^3-3 \text {arctanh}(a x)^2 \operatorname {PolyLog}\left (2,-e^{\text {arctanh}(a x)}\right )+3 \text {arctanh}(a x)^2 \operatorname {PolyLog}\left (2,e^{\text {arctanh}(a x)}\right )+6 \text {arctanh}(a x) \operatorname {PolyLog}\left (3,-e^{\text {arctanh}(a x)}\right )-6 \text {arctanh}(a x) \operatorname {PolyLog}\left (3,e^{\text {arctanh}(a x)}\right )-6 \operatorname {PolyLog}\left (4,-e^{\text {arctanh}(a x)}\right )+6 \operatorname {PolyLog}\left (4,e^{\text {arctanh}(a x)}\right ) \] Output:

-2*arctanh((a*x+1)/(-a^2*x^2+1)^(1/2))*arctanh(a*x)^3-3*arctanh(a*x)^2*pol 
ylog(2,-(a*x+1)/(-a^2*x^2+1)^(1/2))+3*arctanh(a*x)^2*polylog(2,(a*x+1)/(-a 
^2*x^2+1)^(1/2))+6*arctanh(a*x)*polylog(3,-(a*x+1)/(-a^2*x^2+1)^(1/2))-6*a 
rctanh(a*x)*polylog(3,(a*x+1)/(-a^2*x^2+1)^(1/2))-6*polylog(4,-(a*x+1)/(-a 
^2*x^2+1)^(1/2))+6*polylog(4,(a*x+1)/(-a^2*x^2+1)^(1/2))
 

Mathematica [A] (verified)

Time = 0.13 (sec) , antiderivative size = 146, normalized size of antiderivative = 1.43 \[ \int \frac {\text {arctanh}(a x)^3}{x \sqrt {1-a^2 x^2}} \, dx=\frac {1}{8} \left (\pi ^4-2 \text {arctanh}(a x)^4-8 \text {arctanh}(a x)^3 \log \left (1+e^{-\text {arctanh}(a x)}\right )+8 \text {arctanh}(a x)^3 \log \left (1-e^{\text {arctanh}(a x)}\right )+24 \text {arctanh}(a x)^2 \operatorname {PolyLog}\left (2,-e^{-\text {arctanh}(a x)}\right )+24 \text {arctanh}(a x)^2 \operatorname {PolyLog}\left (2,e^{\text {arctanh}(a x)}\right )+48 \text {arctanh}(a x) \operatorname {PolyLog}\left (3,-e^{-\text {arctanh}(a x)}\right )-48 \text {arctanh}(a x) \operatorname {PolyLog}\left (3,e^{\text {arctanh}(a x)}\right )+48 \operatorname {PolyLog}\left (4,-e^{-\text {arctanh}(a x)}\right )+48 \operatorname {PolyLog}\left (4,e^{\text {arctanh}(a x)}\right )\right ) \] Input:

Integrate[ArcTanh[a*x]^3/(x*Sqrt[1 - a^2*x^2]),x]
 

Output:

(Pi^4 - 2*ArcTanh[a*x]^4 - 8*ArcTanh[a*x]^3*Log[1 + E^(-ArcTanh[a*x])] + 8 
*ArcTanh[a*x]^3*Log[1 - E^ArcTanh[a*x]] + 24*ArcTanh[a*x]^2*PolyLog[2, -E^ 
(-ArcTanh[a*x])] + 24*ArcTanh[a*x]^2*PolyLog[2, E^ArcTanh[a*x]] + 48*ArcTa 
nh[a*x]*PolyLog[3, -E^(-ArcTanh[a*x])] - 48*ArcTanh[a*x]*PolyLog[3, E^ArcT 
anh[a*x]] + 48*PolyLog[4, -E^(-ArcTanh[a*x])] + 48*PolyLog[4, E^ArcTanh[a* 
x]])/8
 

Rubi [C] (verified)

Result contains complex when optimal does not.

Time = 0.64 (sec) , antiderivative size = 122, normalized size of antiderivative = 1.20, number of steps used = 9, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.333, Rules used = {6582, 3042, 26, 4670, 3011, 7163, 2720, 7143}

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.

\(\displaystyle \int \frac {\text {arctanh}(a x)^3}{x \sqrt {1-a^2 x^2}} \, dx\)

\(\Big \downarrow \) 6582

\(\displaystyle \int \frac {\sqrt {1-a^2 x^2} \text {arctanh}(a x)^3}{a x}d\text {arctanh}(a x)\)

\(\Big \downarrow \) 3042

\(\displaystyle \int i \text {arctanh}(a x)^3 \csc (i \text {arctanh}(a x))d\text {arctanh}(a x)\)

\(\Big \downarrow \) 26

\(\displaystyle i \int \text {arctanh}(a x)^3 \csc (i \text {arctanh}(a x))d\text {arctanh}(a x)\)

\(\Big \downarrow \) 4670

\(\displaystyle i \left (3 i \int \text {arctanh}(a x)^2 \log \left (1-e^{\text {arctanh}(a x)}\right )d\text {arctanh}(a x)-3 i \int \text {arctanh}(a x)^2 \log \left (1+e^{\text {arctanh}(a x)}\right )d\text {arctanh}(a x)+2 i \text {arctanh}\left (e^{\text {arctanh}(a x)}\right ) \text {arctanh}(a x)^3\right )\)

\(\Big \downarrow \) 3011

\(\displaystyle i \left (-3 i \left (2 \int \text {arctanh}(a x) \operatorname {PolyLog}\left (2,-e^{\text {arctanh}(a x)}\right )d\text {arctanh}(a x)-\text {arctanh}(a x)^2 \operatorname {PolyLog}\left (2,-e^{\text {arctanh}(a x)}\right )\right )+3 i \left (2 \int \text {arctanh}(a x) \operatorname {PolyLog}\left (2,e^{\text {arctanh}(a x)}\right )d\text {arctanh}(a x)-\text {arctanh}(a x)^2 \operatorname {PolyLog}\left (2,e^{\text {arctanh}(a x)}\right )\right )+2 i \text {arctanh}\left (e^{\text {arctanh}(a x)}\right ) \text {arctanh}(a x)^3\right )\)

\(\Big \downarrow \) 7163

\(\displaystyle i \left (-3 i \left (2 \left (\text {arctanh}(a x) \operatorname {PolyLog}\left (3,-e^{\text {arctanh}(a x)}\right )-\int \operatorname {PolyLog}\left (3,-e^{\text {arctanh}(a x)}\right )d\text {arctanh}(a x)\right )-\text {arctanh}(a x)^2 \operatorname {PolyLog}\left (2,-e^{\text {arctanh}(a x)}\right )\right )+3 i \left (2 \left (\text {arctanh}(a x) \operatorname {PolyLog}\left (3,e^{\text {arctanh}(a x)}\right )-\int \operatorname {PolyLog}\left (3,e^{\text {arctanh}(a x)}\right )d\text {arctanh}(a x)\right )-\text {arctanh}(a x)^2 \operatorname {PolyLog}\left (2,e^{\text {arctanh}(a x)}\right )\right )+2 i \text {arctanh}\left (e^{\text {arctanh}(a x)}\right ) \text {arctanh}(a x)^3\right )\)

\(\Big \downarrow \) 2720

\(\displaystyle i \left (-3 i \left (2 \left (\text {arctanh}(a x) \operatorname {PolyLog}\left (3,-e^{\text {arctanh}(a x)}\right )-\int e^{-\text {arctanh}(a x)} \operatorname {PolyLog}\left (3,-e^{\text {arctanh}(a x)}\right )de^{\text {arctanh}(a x)}\right )-\text {arctanh}(a x)^2 \operatorname {PolyLog}\left (2,-e^{\text {arctanh}(a x)}\right )\right )+3 i \left (2 \left (\text {arctanh}(a x) \operatorname {PolyLog}\left (3,e^{\text {arctanh}(a x)}\right )-\int e^{-\text {arctanh}(a x)} \operatorname {PolyLog}\left (3,e^{\text {arctanh}(a x)}\right )de^{\text {arctanh}(a x)}\right )-\text {arctanh}(a x)^2 \operatorname {PolyLog}\left (2,e^{\text {arctanh}(a x)}\right )\right )+2 i \text {arctanh}\left (e^{\text {arctanh}(a x)}\right ) \text {arctanh}(a x)^3\right )\)

\(\Big \downarrow \) 7143

\(\displaystyle i \left (-3 i \left (2 \left (\text {arctanh}(a x) \operatorname {PolyLog}\left (3,-e^{\text {arctanh}(a x)}\right )-\operatorname {PolyLog}\left (4,-e^{\text {arctanh}(a x)}\right )\right )-\text {arctanh}(a x)^2 \operatorname {PolyLog}\left (2,-e^{\text {arctanh}(a x)}\right )\right )+3 i \left (2 \left (\text {arctanh}(a x) \operatorname {PolyLog}\left (3,e^{\text {arctanh}(a x)}\right )-\operatorname {PolyLog}\left (4,e^{\text {arctanh}(a x)}\right )\right )-\text {arctanh}(a x)^2 \operatorname {PolyLog}\left (2,e^{\text {arctanh}(a x)}\right )\right )+2 i \text {arctanh}\left (e^{\text {arctanh}(a x)}\right ) \text {arctanh}(a x)^3\right )\)

Input:

Int[ArcTanh[a*x]^3/(x*Sqrt[1 - a^2*x^2]),x]
 

Output:

I*((2*I)*ArcTanh[E^ArcTanh[a*x]]*ArcTanh[a*x]^3 - (3*I)*(-(ArcTanh[a*x]^2* 
PolyLog[2, -E^ArcTanh[a*x]]) + 2*(ArcTanh[a*x]*PolyLog[3, -E^ArcTanh[a*x]] 
 - PolyLog[4, -E^ArcTanh[a*x]])) + (3*I)*(-(ArcTanh[a*x]^2*PolyLog[2, E^Ar 
cTanh[a*x]]) + 2*(ArcTanh[a*x]*PolyLog[3, E^ArcTanh[a*x]] - PolyLog[4, E^A 
rcTanh[a*x]])))
 

Defintions of rubi rules used

rule 26

Int[(Complex[0, a_])*(Fx_), x_Symbol] :> Simp[(Complex[Identity[0], a])   I 
nt[Fx, x], x] /; FreeQ[a, x] && EqQ[a^2, 1]
 

rule 2720

Int[u_, x_Symbol] :> With[{v = FunctionOfExponential[u, x]}, Simp[v/D[v, x] 
   Subst[Int[FunctionOfExponentialFunction[u, x]/x, x], x, v], x]] /; Funct 
ionOfExponentialQ[u, x] &&  !MatchQ[u, (w_)*((a_.)*(v_)^(n_))^(m_) /; FreeQ 
[{a, m, n}, x] && IntegerQ[m*n]] &&  !MatchQ[u, E^((c_.)*((a_.) + (b_.)*x)) 
*(F_)[v_] /; FreeQ[{a, b, c}, x] && InverseFunctionQ[F[x]]]
 

rule 3011

Int[Log[1 + (e_.)*((F_)^((c_.)*((a_.) + (b_.)*(x_))))^(n_.)]*((f_.) + (g_.) 
*(x_))^(m_.), x_Symbol] :> Simp[(-(f + g*x)^m)*(PolyLog[2, (-e)*(F^(c*(a + 
b*x)))^n]/(b*c*n*Log[F])), x] + Simp[g*(m/(b*c*n*Log[F]))   Int[(f + g*x)^( 
m - 1)*PolyLog[2, (-e)*(F^(c*(a + b*x)))^n], x], x] /; FreeQ[{F, a, b, c, e 
, f, g, n}, x] && GtQ[m, 0]
 

rule 3042

Int[u_, x_Symbol] :> Int[DeactivateTrig[u, x], x] /; FunctionOfTrigOfLinear 
Q[u, x]
 

rule 4670

Int[csc[(e_.) + (Complex[0, fz_])*(f_.)*(x_)]*((c_.) + (d_.)*(x_))^(m_.), x 
_Symbol] :> Simp[-2*(c + d*x)^m*(ArcTanh[E^((-I)*e + f*fz*x)]/(f*fz*I)), x] 
 + (-Simp[d*(m/(f*fz*I))   Int[(c + d*x)^(m - 1)*Log[1 - E^((-I)*e + f*fz*x 
)], x], x] + Simp[d*(m/(f*fz*I))   Int[(c + d*x)^(m - 1)*Log[1 + E^((-I)*e 
+ f*fz*x)], x], x]) /; FreeQ[{c, d, e, f, fz}, x] && IGtQ[m, 0]
 

rule 6582

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

rule 7143

Int[PolyLog[n_, (c_.)*((a_.) + (b_.)*(x_))^(p_.)]/((d_.) + (e_.)*(x_)), x_S 
ymbol] :> Simp[PolyLog[n + 1, c*(a + b*x)^p]/(e*p), x] /; FreeQ[{a, b, c, d 
, e, n, p}, x] && EqQ[b*d, a*e]
 

rule 7163

Int[((e_.) + (f_.)*(x_))^(m_.)*PolyLog[n_, (d_.)*((F_)^((c_.)*((a_.) + (b_. 
)*(x_))))^(p_.)], x_Symbol] :> Simp[(e + f*x)^m*(PolyLog[n + 1, d*(F^(c*(a 
+ b*x)))^p]/(b*c*p*Log[F])), x] - Simp[f*(m/(b*c*p*Log[F]))   Int[(e + f*x) 
^(m - 1)*PolyLog[n + 1, d*(F^(c*(a + b*x)))^p], x], x] /; FreeQ[{F, a, b, c 
, d, e, f, n, p}, x] && GtQ[m, 0]
 

Maple [A] (verified)

Time = 0.54 (sec) , antiderivative size = 215, normalized size of antiderivative = 2.11

method	result	size
default	\(\operatorname {arctanh}\left (a x \right )^{3} \ln \left (1-\frac {a x +1}{\sqrt {-a^{2} x^{2}+1}}\right )+3 \operatorname {arctanh}\left (a x \right )^{2} \operatorname {polylog}\left (2, \frac {a x +1}{\sqrt {-a^{2} x^{2}+1}}\right )-6 \,\operatorname {arctanh}\left (a x \right ) \operatorname {polylog}\left (3, \frac {a x +1}{\sqrt {-a^{2} x^{2}+1}}\right )+6 \operatorname {polylog}\left (4, \frac {a x +1}{\sqrt {-a^{2} x^{2}+1}}\right )-\operatorname {arctanh}\left (a x \right )^{3} \ln \left (1+\frac {a x +1}{\sqrt {-a^{2} x^{2}+1}}\right )-3 \operatorname {arctanh}\left (a x \right )^{2} \operatorname {polylog}\left (2, -\frac {a x +1}{\sqrt {-a^{2} x^{2}+1}}\right )+6 \,\operatorname {arctanh}\left (a x \right ) \operatorname {polylog}\left (3, -\frac {a x +1}{\sqrt {-a^{2} x^{2}+1}}\right )-6 \operatorname {polylog}\left (4, -\frac {a x +1}{\sqrt {-a^{2} x^{2}+1}}\right )\)	\(215\)

Input:

int(arctanh(a*x)^3/x/(-a^2*x^2+1)^(1/2),x,method=_RETURNVERBOSE)
 

Output:

arctanh(a*x)^3*ln(1-(a*x+1)/(-a^2*x^2+1)^(1/2))+3*arctanh(a*x)^2*polylog(2 
,(a*x+1)/(-a^2*x^2+1)^(1/2))-6*arctanh(a*x)*polylog(3,(a*x+1)/(-a^2*x^2+1) 
^(1/2))+6*polylog(4,(a*x+1)/(-a^2*x^2+1)^(1/2))-arctanh(a*x)^3*ln(1+(a*x+1 
)/(-a^2*x^2+1)^(1/2))-3*arctanh(a*x)^2*polylog(2,-(a*x+1)/(-a^2*x^2+1)^(1/ 
2))+6*arctanh(a*x)*polylog(3,-(a*x+1)/(-a^2*x^2+1)^(1/2))-6*polylog(4,-(a* 
x+1)/(-a^2*x^2+1)^(1/2))
 

Fricas [F]

\[ \int \frac {\text {arctanh}(a x)^3}{x \sqrt {1-a^2 x^2}} \, dx=\int { \frac {\operatorname {artanh}\left (a x\right )^{3}}{\sqrt {-a^{2} x^{2} + 1} x} \,d x } \] Input:

integrate(arctanh(a*x)^3/x/(-a^2*x^2+1)^(1/2),x, algorithm="fricas")
 

Output:

integral(-sqrt(-a^2*x^2 + 1)*arctanh(a*x)^3/(a^2*x^3 - x), x)
 

Sympy [F]

\[ \int \frac {\text {arctanh}(a x)^3}{x \sqrt {1-a^2 x^2}} \, dx=\int \frac {\operatorname {atanh}^{3}{\left (a x \right )}}{x \sqrt {- \left (a x - 1\right ) \left (a x + 1\right )}}\, dx \] Input:

integrate(atanh(a*x)**3/x/(-a**2*x**2+1)**(1/2),x)
 

Output:

Integral(atanh(a*x)**3/(x*sqrt(-(a*x - 1)*(a*x + 1))), x)
 

Maxima [F]

\[ \int \frac {\text {arctanh}(a x)^3}{x \sqrt {1-a^2 x^2}} \, dx=\int { \frac {\operatorname {artanh}\left (a x\right )^{3}}{\sqrt {-a^{2} x^{2} + 1} x} \,d x } \] Input:

integrate(arctanh(a*x)^3/x/(-a^2*x^2+1)^(1/2),x, algorithm="maxima")
 

Output:

integrate(arctanh(a*x)^3/(sqrt(-a^2*x^2 + 1)*x), x)
 

Giac [F]

\[ \int \frac {\text {arctanh}(a x)^3}{x \sqrt {1-a^2 x^2}} \, dx=\int { \frac {\operatorname {artanh}\left (a x\right )^{3}}{\sqrt {-a^{2} x^{2} + 1} x} \,d x } \] Input:

integrate(arctanh(a*x)^3/x/(-a^2*x^2+1)^(1/2),x, algorithm="giac")
 

Output:

integrate(arctanh(a*x)^3/(sqrt(-a^2*x^2 + 1)*x), x)
 

Mupad [F(-1)]

Timed out. \[ \int \frac {\text {arctanh}(a x)^3}{x \sqrt {1-a^2 x^2}} \, dx=\int \frac {{\mathrm {atanh}\left (a\,x\right )}^3}{x\,\sqrt {1-a^2\,x^2}} \,d x \] Input:

int(atanh(a*x)^3/(x*(1 - a^2*x^2)^(1/2)),x)
                                                                                    
                                                                                    
 

Output:

int(atanh(a*x)^3/(x*(1 - a^2*x^2)^(1/2)), x)
 

Reduce [F]

\[ \int \frac {\text {arctanh}(a x)^3}{x \sqrt {1-a^2 x^2}} \, dx=\int \frac {\mathit {atanh} \left (a x \right )^{3}}{\sqrt {-a^{2} x^{2}+1}\, x}d x \] Input:

int(atanh(a*x)^3/x/(-a^2*x^2+1)^(1/2),x)
 

Output:

int(atanh(a*x)**3/(sqrt( - a**2*x**2 + 1)*x),x)