Integrand size = 24, antiderivative size = 197 \[ \int \frac {\sqrt {1-a^2 x^2} \text {arctanh}(a x)^2}{x^2} \, dx=-\frac {\sqrt {1-a^2 x^2} \text {arctanh}(a x)^2}{x}-2 a \arctan \left (e^{\text {arctanh}(a x)}\right ) \text {arctanh}(a x)^2-4 a \text {arctanh}(a x) \text {arctanh}\left (\frac {\sqrt {1-a x}}{\sqrt {1+a x}}\right )+2 i a \text {arctanh}(a x) \operatorname {PolyLog}\left (2,-i e^{\text {arctanh}(a x)}\right )-2 i a \text {arctanh}(a x) \operatorname {PolyLog}\left (2,i e^{\text {arctanh}(a x)}\right )+2 a \operatorname {PolyLog}\left (2,-\frac {\sqrt {1-a x}}{\sqrt {1+a x}}\right )-2 a \operatorname {PolyLog}\left (2,\frac {\sqrt {1-a x}}{\sqrt {1+a x}}\right )-2 i a \operatorname {PolyLog}\left (3,-i e^{\text {arctanh}(a x)}\right )+2 i a \operatorname {PolyLog}\left (3,i e^{\text {arctanh}(a x)}\right ) \]
-2*a*arctan((a*x+1)/(-a^2*x^2+1)^(1/2))*arctanh(a*x)^2-4*a*arctanh(a*x)*ar ctanh((-a*x+1)^(1/2)/(a*x+1)^(1/2))+2*I*a*arctanh(a*x)*polylog(2,-I*(a*x+1 )/(-a^2*x^2+1)^(1/2))-2*I*a*arctanh(a*x)*polylog(2,I*(a*x+1)/(-a^2*x^2+1)^ (1/2))+2*a*polylog(2,-(-a*x+1)^(1/2)/(a*x+1)^(1/2))-2*a*polylog(2,(-a*x+1) ^(1/2)/(a*x+1)^(1/2))-2*I*a*polylog(3,-I*(a*x+1)/(-a^2*x^2+1)^(1/2))+2*I*a *polylog(3,I*(a*x+1)/(-a^2*x^2+1)^(1/2))-arctanh(a*x)^2*(-a^2*x^2+1)^(1/2) /x
Time = 0.58 (sec) , antiderivative size = 223, normalized size of antiderivative = 1.13 \[ \int \frac {\sqrt {1-a^2 x^2} \text {arctanh}(a x)^2}{x^2} \, dx=a \left (-\frac {\sqrt {1-a^2 x^2} \text {arctanh}(a x)^2}{a x}+2 \text {arctanh}(a x) \log \left (1-e^{-\text {arctanh}(a x)}\right )+i \text {arctanh}(a x)^2 \log \left (1-i e^{-\text {arctanh}(a x)}\right )-i \text {arctanh}(a x)^2 \log \left (1+i e^{-\text {arctanh}(a x)}\right )-2 \text {arctanh}(a x) \log \left (1+e^{-\text {arctanh}(a x)}\right )+2 \operatorname {PolyLog}\left (2,-e^{-\text {arctanh}(a x)}\right )+2 i \text {arctanh}(a x) \operatorname {PolyLog}\left (2,-i e^{-\text {arctanh}(a x)}\right )-2 i \text {arctanh}(a x) \operatorname {PolyLog}\left (2,i e^{-\text {arctanh}(a x)}\right )-2 \operatorname {PolyLog}\left (2,e^{-\text {arctanh}(a x)}\right )+2 i \operatorname {PolyLog}\left (3,-i e^{-\text {arctanh}(a x)}\right )-2 i \operatorname {PolyLog}\left (3,i e^{-\text {arctanh}(a x)}\right )\right ) \]
a*(-((Sqrt[1 - a^2*x^2]*ArcTanh[a*x]^2)/(a*x)) + 2*ArcTanh[a*x]*Log[1 - E^ (-ArcTanh[a*x])] + I*ArcTanh[a*x]^2*Log[1 - I/E^ArcTanh[a*x]] - I*ArcTanh[ a*x]^2*Log[1 + I/E^ArcTanh[a*x]] - 2*ArcTanh[a*x]*Log[1 + E^(-ArcTanh[a*x] )] + 2*PolyLog[2, -E^(-ArcTanh[a*x])] + (2*I)*ArcTanh[a*x]*PolyLog[2, (-I) /E^ArcTanh[a*x]] - (2*I)*ArcTanh[a*x]*PolyLog[2, I/E^ArcTanh[a*x]] - 2*Pol yLog[2, E^(-ArcTanh[a*x])] + (2*I)*PolyLog[3, (-I)/E^ArcTanh[a*x]] - (2*I) *PolyLog[3, I/E^ArcTanh[a*x]])
Time = 1.43 (sec) , antiderivative size = 193, normalized size of antiderivative = 0.98, number of steps used = 10, number of rules used = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.375, Rules used = {6576, 6514, 3042, 4668, 3011, 2720, 6570, 6580, 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.
\(\displaystyle \int \frac {\sqrt {1-a^2 x^2} \text {arctanh}(a x)^2}{x^2} \, dx\) |
\(\Big \downarrow \) 6576 |
\(\displaystyle \int \frac {\text {arctanh}(a x)^2}{x^2 \sqrt {1-a^2 x^2}}dx-a^2 \int \frac {\text {arctanh}(a x)^2}{\sqrt {1-a^2 x^2}}dx\) |
\(\Big \downarrow \) 6514 |
\(\displaystyle \int \frac {\text {arctanh}(a x)^2}{x^2 \sqrt {1-a^2 x^2}}dx-a \int \sqrt {1-a^2 x^2} \text {arctanh}(a x)^2d\text {arctanh}(a x)\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \int \frac {\text {arctanh}(a x)^2}{x^2 \sqrt {1-a^2 x^2}}dx-a \int \text {arctanh}(a x)^2 \csc \left (i \text {arctanh}(a x)+\frac {\pi }{2}\right )d\text {arctanh}(a x)\) |
\(\Big \downarrow \) 4668 |
\(\displaystyle \int \frac {\text {arctanh}(a x)^2}{x^2 \sqrt {1-a^2 x^2}}dx-a \left (-2 i \int \text {arctanh}(a x) \log \left (1-i e^{\text {arctanh}(a x)}\right )d\text {arctanh}(a x)+2 i \int \text {arctanh}(a x) \log \left (1+i e^{\text {arctanh}(a x)}\right )d\text {arctanh}(a x)+2 \text {arctanh}(a x)^2 \arctan \left (e^{\text {arctanh}(a x)}\right )\right )\) |
\(\Big \downarrow \) 3011 |
\(\displaystyle \int \frac {\text {arctanh}(a x)^2}{x^2 \sqrt {1-a^2 x^2}}dx-a \left (2 i \left (\int \operatorname {PolyLog}\left (2,-i e^{\text {arctanh}(a x)}\right )d\text {arctanh}(a x)-\text {arctanh}(a x) \operatorname {PolyLog}\left (2,-i e^{\text {arctanh}(a x)}\right )\right )-2 i \left (\int \operatorname {PolyLog}\left (2,i e^{\text {arctanh}(a x)}\right )d\text {arctanh}(a x)-\text {arctanh}(a x) \operatorname {PolyLog}\left (2,i e^{\text {arctanh}(a x)}\right )\right )+2 \text {arctanh}(a x)^2 \arctan \left (e^{\text {arctanh}(a x)}\right )\right )\) |
\(\Big \downarrow \) 2720 |
\(\displaystyle \int \frac {\text {arctanh}(a x)^2}{x^2 \sqrt {1-a^2 x^2}}dx-a \left (2 i \left (\int e^{-\text {arctanh}(a x)} \operatorname {PolyLog}\left (2,-i e^{\text {arctanh}(a x)}\right )de^{\text {arctanh}(a x)}-\text {arctanh}(a x) \operatorname {PolyLog}\left (2,-i e^{\text {arctanh}(a x)}\right )\right )-2 i \left (\int e^{-\text {arctanh}(a x)} \operatorname {PolyLog}\left (2,i e^{\text {arctanh}(a x)}\right )de^{\text {arctanh}(a x)}-\text {arctanh}(a x) \operatorname {PolyLog}\left (2,i e^{\text {arctanh}(a x)}\right )\right )+2 \text {arctanh}(a x)^2 \arctan \left (e^{\text {arctanh}(a x)}\right )\right )\) |
\(\Big \downarrow \) 6570 |
\(\displaystyle 2 a \int \frac {\text {arctanh}(a x)}{x \sqrt {1-a^2 x^2}}dx-a \left (2 i \left (\int e^{-\text {arctanh}(a x)} \operatorname {PolyLog}\left (2,-i e^{\text {arctanh}(a x)}\right )de^{\text {arctanh}(a x)}-\text {arctanh}(a x) \operatorname {PolyLog}\left (2,-i e^{\text {arctanh}(a x)}\right )\right )-2 i \left (\int e^{-\text {arctanh}(a x)} \operatorname {PolyLog}\left (2,i e^{\text {arctanh}(a x)}\right )de^{\text {arctanh}(a x)}-\text {arctanh}(a x) \operatorname {PolyLog}\left (2,i e^{\text {arctanh}(a x)}\right )\right )+2 \text {arctanh}(a x)^2 \arctan \left (e^{\text {arctanh}(a x)}\right )\right )-\frac {\sqrt {1-a^2 x^2} \text {arctanh}(a x)^2}{x}\) |
\(\Big \downarrow \) 6580 |
\(\displaystyle -a \left (2 i \left (\int e^{-\text {arctanh}(a x)} \operatorname {PolyLog}\left (2,-i e^{\text {arctanh}(a x)}\right )de^{\text {arctanh}(a x)}-\text {arctanh}(a x) \operatorname {PolyLog}\left (2,-i e^{\text {arctanh}(a x)}\right )\right )-2 i \left (\int e^{-\text {arctanh}(a x)} \operatorname {PolyLog}\left (2,i e^{\text {arctanh}(a x)}\right )de^{\text {arctanh}(a x)}-\text {arctanh}(a x) \operatorname {PolyLog}\left (2,i e^{\text {arctanh}(a x)}\right )\right )+2 \text {arctanh}(a x)^2 \arctan \left (e^{\text {arctanh}(a x)}\right )\right )-\frac {\sqrt {1-a^2 x^2} \text {arctanh}(a x)^2}{x}+2 a \left (-2 \text {arctanh}(a x) \text {arctanh}\left (\frac {\sqrt {1-a x}}{\sqrt {a x+1}}\right )+\operatorname {PolyLog}\left (2,-\frac {\sqrt {1-a x}}{\sqrt {a x+1}}\right )-\operatorname {PolyLog}\left (2,\frac {\sqrt {1-a x}}{\sqrt {a x+1}}\right )\right )\) |
\(\Big \downarrow \) 7143 |
\(\displaystyle -\frac {\sqrt {1-a^2 x^2} \text {arctanh}(a x)^2}{x}-a \left (2 \text {arctanh}(a x)^2 \arctan \left (e^{\text {arctanh}(a x)}\right )+2 i \left (\operatorname {PolyLog}\left (3,-i e^{\text {arctanh}(a x)}\right )-\text {arctanh}(a x) \operatorname {PolyLog}\left (2,-i e^{\text {arctanh}(a x)}\right )\right )-2 i \left (\operatorname {PolyLog}\left (3,i e^{\text {arctanh}(a x)}\right )-\text {arctanh}(a x) \operatorname {PolyLog}\left (2,i e^{\text {arctanh}(a x)}\right )\right )\right )+2 a \left (-2 \text {arctanh}(a x) \text {arctanh}\left (\frac {\sqrt {1-a x}}{\sqrt {a x+1}}\right )+\operatorname {PolyLog}\left (2,-\frac {\sqrt {1-a x}}{\sqrt {a x+1}}\right )-\operatorname {PolyLog}\left (2,\frac {\sqrt {1-a x}}{\sqrt {a x+1}}\right )\right )\) |
-((Sqrt[1 - a^2*x^2]*ArcTanh[a*x]^2)/x) + 2*a*(-2*ArcTanh[a*x]*ArcTanh[Sqr t[1 - a*x]/Sqrt[1 + a*x]] + PolyLog[2, -(Sqrt[1 - a*x]/Sqrt[1 + a*x])] - P olyLog[2, Sqrt[1 - a*x]/Sqrt[1 + a*x]]) - a*(2*ArcTan[E^ArcTanh[a*x]]*ArcT anh[a*x]^2 + (2*I)*(-(ArcTanh[a*x]*PolyLog[2, (-I)*E^ArcTanh[a*x]]) + Poly Log[3, (-I)*E^ArcTanh[a*x]]) - (2*I)*(-(ArcTanh[a*x]*PolyLog[2, I*E^ArcTan h[a*x]]) + PolyLog[3, I*E^ArcTanh[a*x]]))
3.5.44.3.1 Defintions of rubi rules used
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_.) + 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]
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]
Int[((a_.) + ArcTanh[(c_.)*(x_)]*(b_.))^(p_.)*((f_.)*(x_))^(m_.)*((d_) + (e _.)*(x_)^2)^(q_.), x_Symbol] :> Simp[(f*x)^(m + 1)*(d + e*x^2)^(q + 1)*((a + b*ArcTanh[c*x])^p/(d*(m + 1))), x] - Simp[b*c*(p/(m + 1)) Int[(f*x)^(m + 1)*(d + e*x^2)^q*(a + b*ArcTanh[c*x])^(p - 1), x], x] /; FreeQ[{a, b, c, d, e, f, m, q}, x] && EqQ[c^2*d + e, 0] && EqQ[m + 2*q + 3, 0] && GtQ[p, 0] && NeQ[m, -1]
Int[((a_.) + ArcTanh[(c_.)*(x_)]*(b_.))^(p_.)*((f_.)*(x_))^(m_)*((d_) + (e_ .)*(x_)^2)^(q_.), x_Symbol] :> Simp[d Int[(f*x)^m*(d + e*x^2)^(q - 1)*(a + b*ArcTanh[c*x])^p, x], x] - Simp[c^2*(d/f^2) Int[(f*x)^(m + 2)*(d + e*x ^2)^(q - 1)*(a + b*ArcTanh[c*x])^p, x], x] /; FreeQ[{a, b, c, d, e, f, m}, x] && EqQ[c^2*d + e, 0] && GtQ[q, 0] && IGtQ[p, 0] && (RationalQ[m] || (EqQ [p, 1] && IntegerQ[q]))
Int[((a_.) + ArcTanh[(c_.)*(x_)]*(b_.))/((x_)*Sqrt[(d_) + (e_.)*(x_)^2]), x _Symbol] :> Simp[(-2/Sqrt[d])*(a + b*ArcTanh[c*x])*ArcTanh[Sqrt[1 - c*x]/Sq rt[1 + c*x]], x] + (Simp[(b/Sqrt[d])*PolyLog[2, -Sqrt[1 - c*x]/Sqrt[1 + c*x ]], x] - Simp[(b/Sqrt[d])*PolyLog[2, Sqrt[1 - c*x]/Sqrt[1 + c*x]], x]) /; F reeQ[{a, b, c, d, e}, x] && EqQ[c^2*d + e, 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]
\[\int \frac {\operatorname {arctanh}\left (a x \right )^{2} \sqrt {-a^{2} x^{2}+1}}{x^{2}}d x\]
\[ \int \frac {\sqrt {1-a^2 x^2} \text {arctanh}(a x)^2}{x^2} \, dx=\int { \frac {\sqrt {-a^{2} x^{2} + 1} \operatorname {artanh}\left (a x\right )^{2}}{x^{2}} \,d x } \]
\[ \int \frac {\sqrt {1-a^2 x^2} \text {arctanh}(a x)^2}{x^2} \, dx=\int \frac {\sqrt {- \left (a x - 1\right ) \left (a x + 1\right )} \operatorname {atanh}^{2}{\left (a x \right )}}{x^{2}}\, dx \]
\[ \int \frac {\sqrt {1-a^2 x^2} \text {arctanh}(a x)^2}{x^2} \, dx=\int { \frac {\sqrt {-a^{2} x^{2} + 1} \operatorname {artanh}\left (a x\right )^{2}}{x^{2}} \,d x } \]
Exception generated. \[ \int \frac {\sqrt {1-a^2 x^2} \text {arctanh}(a x)^2}{x^2} \, dx=\text {Exception raised: TypeError} \]
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
Timed out. \[ \int \frac {\sqrt {1-a^2 x^2} \text {arctanh}(a x)^2}{x^2} \, dx=\int \frac {{\mathrm {atanh}\left (a\,x\right )}^2\,\sqrt {1-a^2\,x^2}}{x^2} \,d x \]