3.414 \(\int \frac{1}{(1-a^2 x^2)^{3/2} \tanh ^{-1}(a x)} \, dx\)

Optimal. Leaf size=9 \[ \frac{\text{Chi}\left (\tanh ^{-1}(a x)\right )}{a} \]

[Out]

CoshIntegral[ArcTanh[a*x]]/a

________________________________________________________________________________________

Rubi [A]  time = 0.0622549, antiderivative size = 9, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.095, Rules used = {5968, 3301} \[ \frac{\text{Chi}\left (\tanh ^{-1}(a x)\right )}{a} \]

Antiderivative was successfully verified.

[In]

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

[Out]

CoshIntegral[ArcTanh[a*x]]/a

Rule 5968

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

Rule 3301

Int[sin[(e_.) + (Complex[0, fz_])*(f_.)*(x_)]/((c_.) + (d_.)*(x_)), x_Symbol] :> Simp[CoshIntegral[(c*f*fz)/d
+ f*fz*x]/d, x] /; FreeQ[{c, d, e, f, fz}, x] && EqQ[d*(e - Pi/2) - c*f*fz*I, 0]

Rubi steps

\begin{align*} \int \frac{1}{\left (1-a^2 x^2\right )^{3/2} \tanh ^{-1}(a x)} \, dx &=\frac{\operatorname{Subst}\left (\int \frac{\cosh (x)}{x} \, dx,x,\tanh ^{-1}(a x)\right )}{a}\\ &=\frac{\text{Chi}\left (\tanh ^{-1}(a x)\right )}{a}\\ \end{align*}

Mathematica [A]  time = 0.0804722, size = 9, normalized size = 1. \[ \frac{\text{Chi}\left (\tanh ^{-1}(a x)\right )}{a} \]

Antiderivative was successfully verified.

[In]

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

[Out]

CoshIntegral[ArcTanh[a*x]]/a

________________________________________________________________________________________

Maple [A]  time = 0.219, size = 10, normalized size = 1.1 \begin{align*}{\frac{{\it Chi} \left ({\it Artanh} \left ( ax \right ) \right ) }{a}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Chi(arctanh(a*x))/a

________________________________________________________________________________________

Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{{\left (-a^{2} x^{2} + 1\right )}^{\frac{3}{2}} \operatorname{artanh}\left (a x\right )}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

Fricas [F]  time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (\frac{\sqrt{-a^{2} x^{2} + 1}}{{\left (a^{4} x^{4} - 2 \, a^{2} x^{2} + 1\right )} \operatorname{artanh}\left (a x\right )}, x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integral(sqrt(-a^2*x^2 + 1)/((a^4*x^4 - 2*a^2*x^2 + 1)*arctanh(a*x)), x)

________________________________________________________________________________________

Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{\left (- \left (a x - 1\right ) \left (a x + 1\right )\right )^{\frac{3}{2}} \operatorname{atanh}{\left (a x \right )}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{{\left (-a^{2} x^{2} + 1\right )}^{\frac{3}{2}} \operatorname{artanh}\left (a x\right )}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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