\(\int \frac {\text {arctanh}(a x)^3}{\sqrt {1-a^2 x^2}} \, dx\) [383]

Optimal result

Integrand size = 21, antiderivative size = 153 \[ \int \frac {\text {arctanh}(a x)^3}{\sqrt {1-a^2 x^2}} \, dx=\frac {2 \arctan \left (e^{\text {arctanh}(a x)}\right ) \text {arctanh}(a x)^3}{a}-\frac {3 i \text {arctanh}(a x)^2 \operatorname {PolyLog}\left (2,-i e^{\text {arctanh}(a x)}\right )}{a}+\frac {3 i \text {arctanh}(a x)^2 \operatorname {PolyLog}\left (2,i e^{\text {arctanh}(a x)}\right )}{a}+\frac {6 i \text {arctanh}(a x) \operatorname {PolyLog}\left (3,-i e^{\text {arctanh}(a x)}\right )}{a}-\frac {6 i \text {arctanh}(a x) \operatorname {PolyLog}\left (3,i e^{\text {arctanh}(a x)}\right )}{a}-\frac {6 i \operatorname {PolyLog}\left (4,-i e^{\text {arctanh}(a x)}\right )}{a}+\frac {6 i \operatorname {PolyLog}\left (4,i e^{\text {arctanh}(a x)}\right )}{a} \] Output:

2*arctan((a*x+1)/(-a^2*x^2+1)^(1/2))*arctanh(a*x)^3/a-3*I*arctanh(a*x)^2*p 
olylog(2,-I*(a*x+1)/(-a^2*x^2+1)^(1/2))/a+3*I*arctanh(a*x)^2*polylog(2,I*( 
a*x+1)/(-a^2*x^2+1)^(1/2))/a+6*I*arctanh(a*x)*polylog(3,-I*(a*x+1)/(-a^2*x 
^2+1)^(1/2))/a-6*I*arctanh(a*x)*polylog(3,I*(a*x+1)/(-a^2*x^2+1)^(1/2))/a- 
6*I*polylog(4,-I*(a*x+1)/(-a^2*x^2+1)^(1/2))/a+6*I*polylog(4,I*(a*x+1)/(-a 
^2*x^2+1)^(1/2))/a
 

Mathematica [B] (verified)

Both result and optimal contain complex but leaf count is larger than twice the leaf count of optimal. \(451\) vs. \(2(153)=306\).

Time = 0.30 (sec) , antiderivative size = 451, normalized size of antiderivative = 2.95 \[ \int \frac {\text {arctanh}(a x)^3}{\sqrt {1-a^2 x^2}} \, dx=-\frac {i \left (7 \pi ^4+8 i \pi ^3 \text {arctanh}(a x)+24 \pi ^2 \text {arctanh}(a x)^2-32 i \pi \text {arctanh}(a x)^3-16 \text {arctanh}(a x)^4+8 i \pi ^3 \log \left (1+i e^{-\text {arctanh}(a x)}\right )+48 \pi ^2 \text {arctanh}(a x) \log \left (1+i e^{-\text {arctanh}(a x)}\right )-96 i \pi \text {arctanh}(a x)^2 \log \left (1+i e^{-\text {arctanh}(a x)}\right )-64 \text {arctanh}(a x)^3 \log \left (1+i e^{-\text {arctanh}(a x)}\right )-48 \pi ^2 \text {arctanh}(a x) \log \left (1-i e^{\text {arctanh}(a x)}\right )+96 i \pi \text {arctanh}(a x)^2 \log \left (1-i e^{\text {arctanh}(a x)}\right )-8 i \pi ^3 \log \left (1+i e^{\text {arctanh}(a x)}\right )+64 \text {arctanh}(a x)^3 \log \left (1+i e^{\text {arctanh}(a x)}\right )+8 i \pi ^3 \log \left (\tan \left (\frac {1}{4} (\pi +2 i \text {arctanh}(a x))\right )\right )-48 (\pi -2 i \text {arctanh}(a x))^2 \operatorname {PolyLog}\left (2,-i e^{-\text {arctanh}(a x)}\right )+192 \text {arctanh}(a x)^2 \operatorname {PolyLog}\left (2,-i e^{\text {arctanh}(a x)}\right )-48 \pi ^2 \operatorname {PolyLog}\left (2,i e^{\text {arctanh}(a x)}\right )+192 i \pi \text {arctanh}(a x) \operatorname {PolyLog}\left (2,i e^{\text {arctanh}(a x)}\right )+192 i \pi \operatorname {PolyLog}\left (3,-i e^{-\text {arctanh}(a x)}\right )+384 \text {arctanh}(a x) \operatorname {PolyLog}\left (3,-i e^{-\text {arctanh}(a x)}\right )-384 \text {arctanh}(a x) \operatorname {PolyLog}\left (3,-i e^{\text {arctanh}(a x)}\right )-192 i \pi \operatorname {PolyLog}\left (3,i e^{\text {arctanh}(a x)}\right )+384 \operatorname {PolyLog}\left (4,-i e^{-\text {arctanh}(a x)}\right )+384 \operatorname {PolyLog}\left (4,-i e^{\text {arctanh}(a x)}\right )\right )}{64 a} \] Input:

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

Output:

((-1/64*I)*(7*Pi^4 + (8*I)*Pi^3*ArcTanh[a*x] + 24*Pi^2*ArcTanh[a*x]^2 - (3 
2*I)*Pi*ArcTanh[a*x]^3 - 16*ArcTanh[a*x]^4 + (8*I)*Pi^3*Log[1 + I/E^ArcTan 
h[a*x]] + 48*Pi^2*ArcTanh[a*x]*Log[1 + I/E^ArcTanh[a*x]] - (96*I)*Pi*ArcTa 
nh[a*x]^2*Log[1 + I/E^ArcTanh[a*x]] - 64*ArcTanh[a*x]^3*Log[1 + I/E^ArcTan 
h[a*x]] - 48*Pi^2*ArcTanh[a*x]*Log[1 - I*E^ArcTanh[a*x]] + (96*I)*Pi*ArcTa 
nh[a*x]^2*Log[1 - I*E^ArcTanh[a*x]] - (8*I)*Pi^3*Log[1 + I*E^ArcTanh[a*x]] 
 + 64*ArcTanh[a*x]^3*Log[1 + I*E^ArcTanh[a*x]] + (8*I)*Pi^3*Log[Tan[(Pi + 
(2*I)*ArcTanh[a*x])/4]] - 48*(Pi - (2*I)*ArcTanh[a*x])^2*PolyLog[2, (-I)/E 
^ArcTanh[a*x]] + 192*ArcTanh[a*x]^2*PolyLog[2, (-I)*E^ArcTanh[a*x]] - 48*P 
i^2*PolyLog[2, I*E^ArcTanh[a*x]] + (192*I)*Pi*ArcTanh[a*x]*PolyLog[2, I*E^ 
ArcTanh[a*x]] + (192*I)*Pi*PolyLog[3, (-I)/E^ArcTanh[a*x]] + 384*ArcTanh[a 
*x]*PolyLog[3, (-I)/E^ArcTanh[a*x]] - 384*ArcTanh[a*x]*PolyLog[3, (-I)*E^A 
rcTanh[a*x]] - (192*I)*Pi*PolyLog[3, I*E^ArcTanh[a*x]] + 384*PolyLog[4, (- 
I)/E^ArcTanh[a*x]] + 384*PolyLog[4, (-I)*E^ArcTanh[a*x]]))/a
 

Rubi [A] (verified)

Time = 0.60 (sec) , antiderivative size = 138, normalized size of antiderivative = 0.90, number of steps used = 8, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.333, Rules used = {6514, 3042, 4668, 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.

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

\(\Big \downarrow \) 6514

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

\(\Big \downarrow \) 3042

\(\displaystyle \frac {\int \text {arctanh}(a x)^3 \csc \left (i \text {arctanh}(a x)+\frac {\pi }{2}\right )d\text {arctanh}(a x)}{a}\)

\(\Big \downarrow \) 4668

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

\(\Big \downarrow \) 3011

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

\(\Big \downarrow \) 7163

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

\(\Big \downarrow \) 2720

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

\(\Big \downarrow \) 7143

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

Input:

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

Output:

(2*ArcTan[E^ArcTanh[a*x]]*ArcTanh[a*x]^3 + (3*I)*(-(ArcTanh[a*x]^2*PolyLog 
[2, (-I)*E^ArcTanh[a*x]]) + 2*(ArcTanh[a*x]*PolyLog[3, (-I)*E^ArcTanh[a*x] 
] - PolyLog[4, (-I)*E^ArcTanh[a*x]])) - (3*I)*(-(ArcTanh[a*x]^2*PolyLog[2, 
 I*E^ArcTanh[a*x]]) + 2*(ArcTanh[a*x]*PolyLog[3, I*E^ArcTanh[a*x]] - PolyL 
og[4, I*E^ArcTanh[a*x]])))/a
 

Defintions of rubi rules used

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 4668

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

rule 6514

Int[((a_.) + ArcTanh[(c_.)*(x_)]*(b_.))^(p_.)/Sqrt[(d_) + (e_.)*(x_)^2], x_ 
Symbol] :> Simp[1/(c*Sqrt[d])   Subst[Int[(a + b*x)^p*Sech[x], x], x, ArcTa 
nh[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 [F]

\[\int \frac {\operatorname {arctanh}\left (a x \right )^{3}}{\sqrt {-a^{2} x^{2}+1}}d x\]

Input:

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

Output:

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

Fricas [F]

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

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

Sympy [F]

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

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

Output:

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

Maxima [F]

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

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

Output:

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

Giac [F]

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

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

Output:

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

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

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

Output:

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