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

Optimal. Leaf size=9 \[ \frac {\text {Shi}\left (\tanh ^{-1}(a x)\right )}{a^2} \]

[Out]

Shi(arctanh(a*x))/a^2

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Rubi [A]  time = 0.10, antiderivative size = 9, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 22, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.091, Rules used = {6034, 3298} \[ \frac {\text {Shi}\left (\tanh ^{-1}(a x)\right )}{a^2} \]

Antiderivative was successfully verified.

[In]

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

[Out]

SinhIntegral[ArcTanh[a*x]]/a^2

Rule 3298

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

Rule 6034

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

Rubi steps

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

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Mathematica [A]  time = 0.07, size = 9, normalized size = 1.00 \[ \frac {\text {Shi}\left (\tanh ^{-1}(a x)\right )}{a^2} \]

Antiderivative was successfully verified.

[In]

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

[Out]

SinhIntegral[ArcTanh[a*x]]/a^2

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fricas [F]  time = 0.45, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {\sqrt {-a^{2} x^{2} + 1} x}{{\left (a^{4} x^{4} - 2 \, a^{2} x^{2} + 1\right )} \operatorname {artanh}\left (a x\right )}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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giac [F(-2)]  time = 0.00, size = 0, normalized size = 0.00 \[ \text {Exception raised: TypeError} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Exception raised: TypeError >> An error occurred running a Giac command:INPUT:sage2:=int(sage0,x):;OUTPUT:sym2
poly/r2sym(const gen & e,const index_m & i,const vecteur & l) Error: Bad Argument Value

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maple [B]  time = 0.32, size = 26, normalized size = 2.89 \[ -\frac {\Ei \left (1, -\arctanh \left (a x \right )\right )}{2 a^{2}}+\frac {\Ei \left (1, \arctanh \left (a x \right )\right )}{2 a^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {x}{{\left (-a^{2} x^{2} + 1\right )}^{\frac {3}{2}} \operatorname {artanh}\left (a x\right )}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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mupad [F]  time = 0.00, size = -1, normalized size = -0.11 \[ \int \frac {x}{\mathrm {atanh}\left (a\,x\right )\,{\left (1-a^2\,x^2\right )}^{3/2}} \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {x}{\left (- \left (a x - 1\right ) \left (a x + 1\right )\right )^{\frac {3}{2}} \operatorname {atanh}{\left (a x \right )}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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