Integrand size = 24, antiderivative size = 102 \[ \int \frac {\text {arctanh}(a x)^3}{x \sqrt {1-a^2 x^2}} \, dx=-2 \text {arctanh}\left (e^{\text {arctanh}(a x)}\right ) \text {arctanh}(a x)^3-3 \text {arctanh}(a x)^2 \operatorname {PolyLog}\left (2,-e^{\text {arctanh}(a x)}\right )+3 \text {arctanh}(a x)^2 \operatorname {PolyLog}\left (2,e^{\text {arctanh}(a x)}\right )+6 \text {arctanh}(a x) \operatorname {PolyLog}\left (3,-e^{\text {arctanh}(a x)}\right )-6 \text {arctanh}(a x) \operatorname {PolyLog}\left (3,e^{\text {arctanh}(a x)}\right )-6 \operatorname {PolyLog}\left (4,-e^{\text {arctanh}(a x)}\right )+6 \operatorname {PolyLog}\left (4,e^{\text {arctanh}(a x)}\right ) \]
-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))
Time = 0.13 (sec) , antiderivative size = 146, normalized size of antiderivative = 1.43 \[ \int \frac {\text {arctanh}(a x)^3}{x \sqrt {1-a^2 x^2}} \, dx=\frac {1}{8} \left (\pi ^4-2 \text {arctanh}(a x)^4-8 \text {arctanh}(a x)^3 \log \left (1+e^{-\text {arctanh}(a x)}\right )+8 \text {arctanh}(a x)^3 \log \left (1-e^{\text {arctanh}(a x)}\right )+24 \text {arctanh}(a x)^2 \operatorname {PolyLog}\left (2,-e^{-\text {arctanh}(a x)}\right )+24 \text {arctanh}(a x)^2 \operatorname {PolyLog}\left (2,e^{\text {arctanh}(a x)}\right )+48 \text {arctanh}(a x) \operatorname {PolyLog}\left (3,-e^{-\text {arctanh}(a x)}\right )-48 \text {arctanh}(a x) \operatorname {PolyLog}\left (3,e^{\text {arctanh}(a x)}\right )+48 \operatorname {PolyLog}\left (4,-e^{-\text {arctanh}(a x)}\right )+48 \operatorname {PolyLog}\left (4,e^{\text {arctanh}(a x)}\right )\right ) \]
(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
Result contains complex when optimal does not.
Time = 0.62 (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 |
\(\displaystyle i \left (-3 i \left (2 \left (\text {arctanh}(a x) \operatorname {PolyLog}\left (3,-e^{\text {arctanh}(a x)}\right )-\operatorname {PolyLog}\left (4,-e^{\text {arctanh}(a x)}\right )\right )-\text {arctanh}(a x)^2 \operatorname {PolyLog}\left (2,-e^{\text {arctanh}(a x)}\right )\right )+3 i \left (2 \left (\text {arctanh}(a x) \operatorname {PolyLog}\left (3,e^{\text {arctanh}(a x)}\right )-\operatorname {PolyLog}\left (4,e^{\text {arctanh}(a x)}\right )\right )-\text {arctanh}(a x)^2 \operatorname {PolyLog}\left (2,e^{\text {arctanh}(a x)}\right )\right )+2 i \text {arctanh}\left (e^{\text {arctanh}(a x)}\right ) \text {arctanh}(a x)^3\right )\) |
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]])))
3.4.84.3.1 Defintions of rubi rules used
Int[(Complex[0, a_])*(Fx_), x_Symbol] :> Simp[(Complex[Identity[0], a]) I nt[Fx, x], x] /; FreeQ[a, x] && EqQ[a^2, 1]
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]]]
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]
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]
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]
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]
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]
Time = 0.16 (sec) , antiderivative size = 215, normalized size of antiderivative = 2.11
method | result | size |
default | \(\operatorname {arctanh}\left (a x \right )^{3} \ln \left (1-\frac {a x +1}{\sqrt {-a^{2} x^{2}+1}}\right )+3 \operatorname {arctanh}\left (a x \right )^{2} \operatorname {polylog}\left (2, \frac {a x +1}{\sqrt {-a^{2} x^{2}+1}}\right )-6 \,\operatorname {arctanh}\left (a x \right ) \operatorname {polylog}\left (3, \frac {a x +1}{\sqrt {-a^{2} x^{2}+1}}\right )+6 \operatorname {polylog}\left (4, \frac {a x +1}{\sqrt {-a^{2} x^{2}+1}}\right )-\operatorname {arctanh}\left (a x \right )^{3} \ln \left (1+\frac {a x +1}{\sqrt {-a^{2} x^{2}+1}}\right )-3 \operatorname {arctanh}\left (a x \right )^{2} \operatorname {polylog}\left (2, -\frac {a x +1}{\sqrt {-a^{2} x^{2}+1}}\right )+6 \,\operatorname {arctanh}\left (a x \right ) \operatorname {polylog}\left (3, -\frac {a x +1}{\sqrt {-a^{2} x^{2}+1}}\right )-6 \operatorname {polylog}\left (4, -\frac {a x +1}{\sqrt {-a^{2} x^{2}+1}}\right )\) | \(215\) |
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))
\[ \int \frac {\text {arctanh}(a x)^3}{x \sqrt {1-a^2 x^2}} \, dx=\int { \frac {\operatorname {artanh}\left (a x\right )^{3}}{\sqrt {-a^{2} x^{2} + 1} x} \,d x } \]
\[ \int \frac {\text {arctanh}(a x)^3}{x \sqrt {1-a^2 x^2}} \, dx=\int \frac {\operatorname {atanh}^{3}{\left (a x \right )}}{x \sqrt {- \left (a x - 1\right ) \left (a x + 1\right )}}\, dx \]
\[ \int \frac {\text {arctanh}(a x)^3}{x \sqrt {1-a^2 x^2}} \, dx=\int { \frac {\operatorname {artanh}\left (a x\right )^{3}}{\sqrt {-a^{2} x^{2} + 1} x} \,d x } \]
\[ \int \frac {\text {arctanh}(a x)^3}{x \sqrt {1-a^2 x^2}} \, dx=\int { \frac {\operatorname {artanh}\left (a x\right )^{3}}{\sqrt {-a^{2} x^{2} + 1} x} \,d x } \]
Timed out. \[ \int \frac {\text {arctanh}(a x)^3}{x \sqrt {1-a^2 x^2}} \, dx=\int \frac {{\mathrm {atanh}\left (a\,x\right )}^3}{x\,\sqrt {1-a^2\,x^2}} \,d x \]