Integrand size = 24, antiderivative size = 68 \[ \int \frac {\text {arctanh}(a x)^2}{x \sqrt {1-a^2 x^2}} \, dx=-2 \text {arctanh}\left (e^{\text {arctanh}(a x)}\right ) \text {arctanh}(a x)^2-2 \text {arctanh}(a x) \operatorname {PolyLog}\left (2,-e^{\text {arctanh}(a x)}\right )+2 \text {arctanh}(a x) \operatorname {PolyLog}\left (2,e^{\text {arctanh}(a x)}\right )+2 \operatorname {PolyLog}\left (3,-e^{\text {arctanh}(a x)}\right )-2 \operatorname {PolyLog}\left (3,e^{\text {arctanh}(a x)}\right ) \]
-2*arctanh((a*x+1)/(-a^2*x^2+1)^(1/2))*arctanh(a*x)^2-2*arctanh(a*x)*polyl og(2,-(a*x+1)/(-a^2*x^2+1)^(1/2))+2*arctanh(a*x)*polylog(2,(a*x+1)/(-a^2*x ^2+1)^(1/2))+2*polylog(3,-(a*x+1)/(-a^2*x^2+1)^(1/2))-2*polylog(3,(a*x+1)/ (-a^2*x^2+1)^(1/2))
Time = 0.10 (sec) , antiderivative size = 100, normalized size of antiderivative = 1.47 \[ \int \frac {\text {arctanh}(a x)^2}{x \sqrt {1-a^2 x^2}} \, dx=\text {arctanh}(a x)^2 \log \left (1-e^{-\text {arctanh}(a x)}\right )-\text {arctanh}(a x)^2 \log \left (1+e^{-\text {arctanh}(a x)}\right )+2 \text {arctanh}(a x) \operatorname {PolyLog}\left (2,-e^{-\text {arctanh}(a x)}\right )-2 \text {arctanh}(a x) \operatorname {PolyLog}\left (2,e^{-\text {arctanh}(a x)}\right )+2 \operatorname {PolyLog}\left (3,-e^{-\text {arctanh}(a x)}\right )-2 \operatorname {PolyLog}\left (3,e^{-\text {arctanh}(a x)}\right ) \]
ArcTanh[a*x]^2*Log[1 - E^(-ArcTanh[a*x])] - ArcTanh[a*x]^2*Log[1 + E^(-Arc Tanh[a*x])] + 2*ArcTanh[a*x]*PolyLog[2, -E^(-ArcTanh[a*x])] - 2*ArcTanh[a* x]*PolyLog[2, E^(-ArcTanh[a*x])] + 2*PolyLog[3, -E^(-ArcTanh[a*x])] - 2*Po lyLog[3, E^(-ArcTanh[a*x])]
Result contains complex when optimal does not.
Time = 0.49 (sec) , antiderivative size = 80, normalized size of antiderivative = 1.18, number of steps used = 8, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.292, Rules used = {6582, 3042, 26, 4670, 3011, 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.
\(\displaystyle \int \frac {\text {arctanh}(a x)^2}{x \sqrt {1-a^2 x^2}} \, dx\) |
\(\Big \downarrow \) 6582 |
\(\displaystyle \int \frac {\sqrt {1-a^2 x^2} \text {arctanh}(a x)^2}{a x}d\text {arctanh}(a x)\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \int i \text {arctanh}(a x)^2 \csc (i \text {arctanh}(a x))d\text {arctanh}(a x)\) |
\(\Big \downarrow \) 26 |
\(\displaystyle i \int \text {arctanh}(a x)^2 \csc (i \text {arctanh}(a x))d\text {arctanh}(a x)\) |
\(\Big \downarrow \) 4670 |
\(\displaystyle i \left (2 i \int \text {arctanh}(a x) \log \left (1-e^{\text {arctanh}(a x)}\right )d\text {arctanh}(a x)-2 i \int \text {arctanh}(a x) \log \left (1+e^{\text {arctanh}(a x)}\right )d\text {arctanh}(a x)+2 i \text {arctanh}\left (e^{\text {arctanh}(a x)}\right ) \text {arctanh}(a x)^2\right )\) |
\(\Big \downarrow \) 3011 |
\(\displaystyle i \left (-2 i \left (\int \operatorname {PolyLog}\left (2,-e^{\text {arctanh}(a x)}\right )d\text {arctanh}(a x)-\text {arctanh}(a x) \operatorname {PolyLog}\left (2,-e^{\text {arctanh}(a x)}\right )\right )+2 i \left (\int \operatorname {PolyLog}\left (2,e^{\text {arctanh}(a x)}\right )d\text {arctanh}(a x)-\text {arctanh}(a x) \operatorname {PolyLog}\left (2,e^{\text {arctanh}(a x)}\right )\right )+2 i \text {arctanh}\left (e^{\text {arctanh}(a x)}\right ) \text {arctanh}(a x)^2\right )\) |
\(\Big \downarrow \) 2720 |
\(\displaystyle i \left (-2 i \left (\int e^{-\text {arctanh}(a x)} \operatorname {PolyLog}\left (2,-e^{\text {arctanh}(a x)}\right )de^{\text {arctanh}(a x)}-\text {arctanh}(a x) \operatorname {PolyLog}\left (2,-e^{\text {arctanh}(a x)}\right )\right )+2 i \left (\int e^{-\text {arctanh}(a x)} \operatorname {PolyLog}\left (2,e^{\text {arctanh}(a x)}\right )de^{\text {arctanh}(a x)}-\text {arctanh}(a x) \operatorname {PolyLog}\left (2,e^{\text {arctanh}(a x)}\right )\right )+2 i \text {arctanh}\left (e^{\text {arctanh}(a x)}\right ) \text {arctanh}(a x)^2\right )\) |
\(\Big \downarrow \) 7143 |
\(\displaystyle i \left (-2 i \left (\operatorname {PolyLog}\left (3,-e^{\text {arctanh}(a x)}\right )-\text {arctanh}(a x) \operatorname {PolyLog}\left (2,-e^{\text {arctanh}(a x)}\right )\right )+2 i \left (\operatorname {PolyLog}\left (3,e^{\text {arctanh}(a x)}\right )-\text {arctanh}(a x) \operatorname {PolyLog}\left (2,e^{\text {arctanh}(a x)}\right )\right )+2 i \text {arctanh}\left (e^{\text {arctanh}(a x)}\right ) \text {arctanh}(a x)^2\right )\) |
I*((2*I)*ArcTanh[E^ArcTanh[a*x]]*ArcTanh[a*x]^2 - (2*I)*(-(ArcTanh[a*x]*Po lyLog[2, -E^ArcTanh[a*x]]) + PolyLog[3, -E^ArcTanh[a*x]]) + (2*I)*(-(ArcTa nh[a*x]*PolyLog[2, E^ArcTanh[a*x]]) + PolyLog[3, E^ArcTanh[a*x]]))
3.4.77.3.1 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]
Time = 0.17 (sec) , antiderivative size = 158, normalized size of antiderivative = 2.32
method | result | size |
default | \(\operatorname {arctanh}\left (a x \right )^{2} \ln \left (1-\frac {a x +1}{\sqrt {-a^{2} x^{2}+1}}\right )+2 \,\operatorname {arctanh}\left (a x \right ) \operatorname {polylog}\left (2, \frac {a x +1}{\sqrt {-a^{2} x^{2}+1}}\right )-2 \operatorname {polylog}\left (3, \frac {a x +1}{\sqrt {-a^{2} x^{2}+1}}\right )-\operatorname {arctanh}\left (a x \right )^{2} \ln \left (1+\frac {a x +1}{\sqrt {-a^{2} x^{2}+1}}\right )-2 \,\operatorname {arctanh}\left (a x \right ) \operatorname {polylog}\left (2, -\frac {a x +1}{\sqrt {-a^{2} x^{2}+1}}\right )+2 \operatorname {polylog}\left (3, -\frac {a x +1}{\sqrt {-a^{2} x^{2}+1}}\right )\) | \(158\) |
arctanh(a*x)^2*ln(1-(a*x+1)/(-a^2*x^2+1)^(1/2))+2*arctanh(a*x)*polylog(2,( a*x+1)/(-a^2*x^2+1)^(1/2))-2*polylog(3,(a*x+1)/(-a^2*x^2+1)^(1/2))-arctanh (a*x)^2*ln(1+(a*x+1)/(-a^2*x^2+1)^(1/2))-2*arctanh(a*x)*polylog(2,-(a*x+1) /(-a^2*x^2+1)^(1/2))+2*polylog(3,-(a*x+1)/(-a^2*x^2+1)^(1/2))
\[ \int \frac {\text {arctanh}(a x)^2}{x \sqrt {1-a^2 x^2}} \, dx=\int { \frac {\operatorname {artanh}\left (a x\right )^{2}}{\sqrt {-a^{2} x^{2} + 1} x} \,d x } \]
\[ \int \frac {\text {arctanh}(a x)^2}{x \sqrt {1-a^2 x^2}} \, dx=\int \frac {\operatorname {atanh}^{2}{\left (a x \right )}}{x \sqrt {- \left (a x - 1\right ) \left (a x + 1\right )}}\, dx \]
\[ \int \frac {\text {arctanh}(a x)^2}{x \sqrt {1-a^2 x^2}} \, dx=\int { \frac {\operatorname {artanh}\left (a x\right )^{2}}{\sqrt {-a^{2} x^{2} + 1} x} \,d x } \]
\[ \int \frac {\text {arctanh}(a x)^2}{x \sqrt {1-a^2 x^2}} \, dx=\int { \frac {\operatorname {artanh}\left (a x\right )^{2}}{\sqrt {-a^{2} x^{2} + 1} x} \,d x } \]
Timed out. \[ \int \frac {\text {arctanh}(a x)^2}{x \sqrt {1-a^2 x^2}} \, dx=\int \frac {{\mathrm {atanh}\left (a\,x\right )}^2}{x\,\sqrt {1-a^2\,x^2}} \,d x \]