3.57 \(\int \frac{1}{\sinh ^{-1}(a x)^2} \, dx\)

Optimal. Leaf size=34 \[ \frac{\text{Shi}\left (\sinh ^{-1}(a x)\right )}{a}-\frac{\sqrt{a^2 x^2+1}}{a \sinh ^{-1}(a x)} \]

[Out]

-(Sqrt[1 + a^2*x^2]/(a*ArcSinh[a*x])) + SinhIntegral[ArcSinh[a*x]]/a

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Rubi [A]  time = 0.0833157, antiderivative size = 34, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 6, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.5, Rules used = {5655, 5779, 3298} \[ \frac{\text{Shi}\left (\sinh ^{-1}(a x)\right )}{a}-\frac{\sqrt{a^2 x^2+1}}{a \sinh ^{-1}(a x)} \]

Antiderivative was successfully verified.

[In]

Int[ArcSinh[a*x]^(-2),x]

[Out]

-(Sqrt[1 + a^2*x^2]/(a*ArcSinh[a*x])) + SinhIntegral[ArcSinh[a*x]]/a

Rule 5655

Int[((a_.) + ArcSinh[(c_.)*(x_)]*(b_.))^(n_), x_Symbol] :> Simp[(Sqrt[1 + c^2*x^2]*(a + b*ArcSinh[c*x])^(n + 1
))/(b*c*(n + 1)), x] - Dist[c/(b*(n + 1)), Int[(x*(a + b*ArcSinh[c*x])^(n + 1))/Sqrt[1 + c^2*x^2], x], x] /; F
reeQ[{a, b, c}, x] && LtQ[n, -1]

Rule 5779

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

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]

Rubi steps

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

Mathematica [A]  time = 0.0478953, size = 31, normalized size = 0.91 \[ \frac{\text{Shi}\left (\sinh ^{-1}(a x)\right )-\frac{\sqrt{a^2 x^2+1}}{\sinh ^{-1}(a x)}}{a} \]

Antiderivative was successfully verified.

[In]

Integrate[ArcSinh[a*x]^(-2),x]

[Out]

(-(Sqrt[1 + a^2*x^2]/ArcSinh[a*x]) + SinhIntegral[ArcSinh[a*x]])/a

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Maple [A]  time = 0.023, size = 30, normalized size = 0.9 \begin{align*}{\frac{1}{a} \left ( -{\frac{1}{{\it Arcsinh} \left ( ax \right ) }\sqrt{{a}^{2}{x}^{2}+1}}+{\it Shi} \left ({\it Arcsinh} \left ( ax \right ) \right ) \right ) } \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/arcsinh(a*x)^2,x)

[Out]

1/a*(-1/arcsinh(a*x)*(a^2*x^2+1)^(1/2)+Shi(arcsinh(a*x)))

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} -\frac{a^{3} x^{3} + a x +{\left (a^{2} x^{2} + 1\right )}^{\frac{3}{2}}}{{\left (a^{3} x^{2} + \sqrt{a^{2} x^{2} + 1} a^{2} x + a\right )} \log \left (a x + \sqrt{a^{2} x^{2} + 1}\right )} + \int \frac{a^{4} x^{4} + 2 \, a^{2} x^{2} +{\left (a^{2} x^{2} + 1\right )}{\left (a^{2} x^{2} - 1\right )} +{\left (2 \, a^{3} x^{3} + a x\right )} \sqrt{a^{2} x^{2} + 1} + 1}{{\left (a^{4} x^{4} +{\left (a^{2} x^{2} + 1\right )} a^{2} x^{2} + 2 \, a^{2} x^{2} + 2 \,{\left (a^{3} x^{3} + a x\right )} \sqrt{a^{2} x^{2} + 1} + 1\right )} \log \left (a x + \sqrt{a^{2} x^{2} + 1}\right )}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/arcsinh(a*x)^2,x, algorithm="maxima")

[Out]

-(a^3*x^3 + a*x + (a^2*x^2 + 1)^(3/2))/((a^3*x^2 + sqrt(a^2*x^2 + 1)*a^2*x + a)*log(a*x + sqrt(a^2*x^2 + 1)))
+ integrate((a^4*x^4 + 2*a^2*x^2 + (a^2*x^2 + 1)*(a^2*x^2 - 1) + (2*a^3*x^3 + a*x)*sqrt(a^2*x^2 + 1) + 1)/((a^
4*x^4 + (a^2*x^2 + 1)*a^2*x^2 + 2*a^2*x^2 + 2*(a^3*x^3 + a*x)*sqrt(a^2*x^2 + 1) + 1)*log(a*x + sqrt(a^2*x^2 +
1))), x)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/arcsinh(a*x)^2,x, algorithm="fricas")

[Out]

integral(arcsinh(a*x)^(-2), x)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/asinh(a*x)**2,x)

[Out]

Integral(asinh(a*x)**(-2), x)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/arcsinh(a*x)^2,x, algorithm="giac")

[Out]

integrate(arcsinh(a*x)^(-2), x)