3.442 \(\int \sqrt{1-a^2 x^2} \tanh ^{-1}(a x)^2 \, dx\)

Optimal. Leaf size=158 \[ -\frac{i \tanh ^{-1}(a x) \text{PolyLog}\left (2,-i e^{\tanh ^{-1}(a x)}\right )}{a}+\frac{i \tanh ^{-1}(a x) \text{PolyLog}\left (2,i e^{\tanh ^{-1}(a x)}\right )}{a}+\frac{i \text{PolyLog}\left (3,-i e^{\tanh ^{-1}(a x)}\right )}{a}-\frac{i \text{PolyLog}\left (3,i e^{\tanh ^{-1}(a x)}\right )}{a}+\frac{1}{2} x \sqrt{1-a^2 x^2} \tanh ^{-1}(a x)^2+\frac{\sqrt{1-a^2 x^2} \tanh ^{-1}(a x)}{a}-\frac{\sin ^{-1}(a x)}{a}+\frac{\tanh ^{-1}(a x)^2 \tan ^{-1}\left (e^{\tanh ^{-1}(a x)}\right )}{a} \]

[Out]

-(ArcSin[a*x]/a) + (Sqrt[1 - a^2*x^2]*ArcTanh[a*x])/a + (x*Sqrt[1 - a^2*x^2]*ArcTanh[a*x]^2)/2 + (ArcTan[E^Arc
Tanh[a*x]]*ArcTanh[a*x]^2)/a - (I*ArcTanh[a*x]*PolyLog[2, (-I)*E^ArcTanh[a*x]])/a + (I*ArcTanh[a*x]*PolyLog[2,
I*E^ArcTanh[a*x]])/a + (I*PolyLog[3, (-I)*E^ArcTanh[a*x]])/a - (I*PolyLog[3, I*E^ArcTanh[a*x]])/a

________________________________________________________________________________________

Rubi [A]  time = 0.1462, antiderivative size = 158, normalized size of antiderivative = 1., number of steps used = 10, number of rules used = 7, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.333, Rules used = {5944, 5952, 4180, 2531, 2282, 6589, 216} \[ -\frac{i \tanh ^{-1}(a x) \text{PolyLog}\left (2,-i e^{\tanh ^{-1}(a x)}\right )}{a}+\frac{i \tanh ^{-1}(a x) \text{PolyLog}\left (2,i e^{\tanh ^{-1}(a x)}\right )}{a}+\frac{i \text{PolyLog}\left (3,-i e^{\tanh ^{-1}(a x)}\right )}{a}-\frac{i \text{PolyLog}\left (3,i e^{\tanh ^{-1}(a x)}\right )}{a}+\frac{1}{2} x \sqrt{1-a^2 x^2} \tanh ^{-1}(a x)^2+\frac{\sqrt{1-a^2 x^2} \tanh ^{-1}(a x)}{a}-\frac{\sin ^{-1}(a x)}{a}+\frac{\tanh ^{-1}(a x)^2 \tan ^{-1}\left (e^{\tanh ^{-1}(a x)}\right )}{a} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-(ArcSin[a*x]/a) + (Sqrt[1 - a^2*x^2]*ArcTanh[a*x])/a + (x*Sqrt[1 - a^2*x^2]*ArcTanh[a*x]^2)/2 + (ArcTan[E^Arc
Tanh[a*x]]*ArcTanh[a*x]^2)/a - (I*ArcTanh[a*x]*PolyLog[2, (-I)*E^ArcTanh[a*x]])/a + (I*ArcTanh[a*x]*PolyLog[2,
I*E^ArcTanh[a*x]])/a + (I*PolyLog[3, (-I)*E^ArcTanh[a*x]])/a - (I*PolyLog[3, I*E^ArcTanh[a*x]])/a

Rule 5944

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

Rule 5952

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

Rule 4180

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] + (-Dist[(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] + Dist[(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 2531

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] + Dist[(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 2282

Int[u_, x_Symbol] :> With[{v = FunctionOfExponential[u, x]}, Dist[v/D[v, x], Subst[Int[FunctionOfExponentialFu
nction[u, x]/x, x], x, v], x]] /; FunctionOfExponentialQ[u, x] &&  !MatchQ[u, (w_)*((a_.)*(v_)^(n_))^(m_) /; F
reeQ[{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 6589

Int[PolyLog[n_, (c_.)*((a_.) + (b_.)*(x_))^(p_.)]/((d_.) + (e_.)*(x_)), x_Symbol] :> 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 216

Int[1/Sqrt[(a_) + (b_.)*(x_)^2], x_Symbol] :> Simp[ArcSin[(Rt[-b, 2]*x)/Sqrt[a]]/Rt[-b, 2], x] /; FreeQ[{a, b}
, x] && GtQ[a, 0] && NegQ[b]

Rubi steps

\begin{align*} \int \sqrt{1-a^2 x^2} \tanh ^{-1}(a x)^2 \, dx &=\frac{\sqrt{1-a^2 x^2} \tanh ^{-1}(a x)}{a}+\frac{1}{2} x \sqrt{1-a^2 x^2} \tanh ^{-1}(a x)^2+\frac{1}{2} \int \frac{\tanh ^{-1}(a x)^2}{\sqrt{1-a^2 x^2}} \, dx-\int \frac{1}{\sqrt{1-a^2 x^2}} \, dx\\ &=-\frac{\sin ^{-1}(a x)}{a}+\frac{\sqrt{1-a^2 x^2} \tanh ^{-1}(a x)}{a}+\frac{1}{2} x \sqrt{1-a^2 x^2} \tanh ^{-1}(a x)^2+\frac{\operatorname{Subst}\left (\int x^2 \text{sech}(x) \, dx,x,\tanh ^{-1}(a x)\right )}{2 a}\\ &=-\frac{\sin ^{-1}(a x)}{a}+\frac{\sqrt{1-a^2 x^2} \tanh ^{-1}(a x)}{a}+\frac{1}{2} x \sqrt{1-a^2 x^2} \tanh ^{-1}(a x)^2+\frac{\tan ^{-1}\left (e^{\tanh ^{-1}(a x)}\right ) \tanh ^{-1}(a x)^2}{a}-\frac{i \operatorname{Subst}\left (\int x \log \left (1-i e^x\right ) \, dx,x,\tanh ^{-1}(a x)\right )}{a}+\frac{i \operatorname{Subst}\left (\int x \log \left (1+i e^x\right ) \, dx,x,\tanh ^{-1}(a x)\right )}{a}\\ &=-\frac{\sin ^{-1}(a x)}{a}+\frac{\sqrt{1-a^2 x^2} \tanh ^{-1}(a x)}{a}+\frac{1}{2} x \sqrt{1-a^2 x^2} \tanh ^{-1}(a x)^2+\frac{\tan ^{-1}\left (e^{\tanh ^{-1}(a x)}\right ) \tanh ^{-1}(a x)^2}{a}-\frac{i \tanh ^{-1}(a x) \text{Li}_2\left (-i e^{\tanh ^{-1}(a x)}\right )}{a}+\frac{i \tanh ^{-1}(a x) \text{Li}_2\left (i e^{\tanh ^{-1}(a x)}\right )}{a}+\frac{i \operatorname{Subst}\left (\int \text{Li}_2\left (-i e^x\right ) \, dx,x,\tanh ^{-1}(a x)\right )}{a}-\frac{i \operatorname{Subst}\left (\int \text{Li}_2\left (i e^x\right ) \, dx,x,\tanh ^{-1}(a x)\right )}{a}\\ &=-\frac{\sin ^{-1}(a x)}{a}+\frac{\sqrt{1-a^2 x^2} \tanh ^{-1}(a x)}{a}+\frac{1}{2} x \sqrt{1-a^2 x^2} \tanh ^{-1}(a x)^2+\frac{\tan ^{-1}\left (e^{\tanh ^{-1}(a x)}\right ) \tanh ^{-1}(a x)^2}{a}-\frac{i \tanh ^{-1}(a x) \text{Li}_2\left (-i e^{\tanh ^{-1}(a x)}\right )}{a}+\frac{i \tanh ^{-1}(a x) \text{Li}_2\left (i e^{\tanh ^{-1}(a x)}\right )}{a}+\frac{i \operatorname{Subst}\left (\int \frac{\text{Li}_2(-i x)}{x} \, dx,x,e^{\tanh ^{-1}(a x)}\right )}{a}-\frac{i \operatorname{Subst}\left (\int \frac{\text{Li}_2(i x)}{x} \, dx,x,e^{\tanh ^{-1}(a x)}\right )}{a}\\ &=-\frac{\sin ^{-1}(a x)}{a}+\frac{\sqrt{1-a^2 x^2} \tanh ^{-1}(a x)}{a}+\frac{1}{2} x \sqrt{1-a^2 x^2} \tanh ^{-1}(a x)^2+\frac{\tan ^{-1}\left (e^{\tanh ^{-1}(a x)}\right ) \tanh ^{-1}(a x)^2}{a}-\frac{i \tanh ^{-1}(a x) \text{Li}_2\left (-i e^{\tanh ^{-1}(a x)}\right )}{a}+\frac{i \tanh ^{-1}(a x) \text{Li}_2\left (i e^{\tanh ^{-1}(a x)}\right )}{a}+\frac{i \text{Li}_3\left (-i e^{\tanh ^{-1}(a x)}\right )}{a}-\frac{i \text{Li}_3\left (i e^{\tanh ^{-1}(a x)}\right )}{a}\\ \end{align*}

Mathematica [A]  time = 0.714139, size = 187, normalized size = 1.18 \[ \frac{\sqrt{1-a^2 x^2} \left (-\frac{i \left (2 \tanh ^{-1}(a x) \text{PolyLog}\left (2,-i e^{-\tanh ^{-1}(a x)}\right )-2 \tanh ^{-1}(a x) \text{PolyLog}\left (2,i e^{-\tanh ^{-1}(a x)}\right )+2 \text{PolyLog}\left (3,-i e^{-\tanh ^{-1}(a x)}\right )-2 \text{PolyLog}\left (3,i e^{-\tanh ^{-1}(a x)}\right )+\tanh ^{-1}(a x)^2 \log \left (1-i e^{-\tanh ^{-1}(a x)}\right )-\tanh ^{-1}(a x)^2 \log \left (1+i e^{-\tanh ^{-1}(a x)}\right )-4 i \tan ^{-1}\left (\tanh \left (\frac{1}{2} \tanh ^{-1}(a x)\right )\right )\right )}{\sqrt{1-a^2 x^2}}+a x \tanh ^{-1}(a x)^2+2 \tanh ^{-1}(a x)\right )}{2 a} \]

Warning: Unable to verify antiderivative.

[In]

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

[Out]

(Sqrt[1 - a^2*x^2]*(2*ArcTanh[a*x] + a*x*ArcTanh[a*x]^2 - (I*((-4*I)*ArcTan[Tanh[ArcTanh[a*x]/2]] + ArcTanh[a*
x]^2*Log[1 - I/E^ArcTanh[a*x]] - ArcTanh[a*x]^2*Log[1 + I/E^ArcTanh[a*x]] + 2*ArcTanh[a*x]*PolyLog[2, (-I)/E^A
rcTanh[a*x]] - 2*ArcTanh[a*x]*PolyLog[2, I/E^ArcTanh[a*x]] + 2*PolyLog[3, (-I)/E^ArcTanh[a*x]] - 2*PolyLog[3,
I/E^ArcTanh[a*x]]))/Sqrt[1 - a^2*x^2]))/(2*a)

________________________________________________________________________________________

Maple [F]  time = 0.206, size = 0, normalized size = 0. \begin{align*} \int \sqrt{-{a}^{2}{x}^{2}+1} \left ({\it Artanh} \left ( ax \right ) \right ) ^{2}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \sqrt{-a^{2} x^{2} + 1} \operatorname{artanh}\left (a x\right )^{2}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

Fricas [F]  time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (\sqrt{-a^{2} x^{2} + 1} \operatorname{artanh}\left (a x\right )^{2}, x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \sqrt{- \left (a x - 1\right ) \left (a x + 1\right )} \operatorname{atanh}^{2}{\left (a x \right )}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Integral(sqrt(-(a*x - 1)*(a*x + 1))*atanh(a*x)**2, x)

________________________________________________________________________________________

Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \sqrt{-a^{2} x^{2} + 1} \operatorname{artanh}\left (a x\right )^{2}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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