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

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

[Out]

CoshIntegral[ArcSinh[a*x]]/a

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Rubi [A]  time = 0.017505, antiderivative size = 9, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 6, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.333, Rules used = {5657, 3301} \[ \frac{\text{Chi}\left (\sinh ^{-1}(a x)\right )}{a} \]

Antiderivative was successfully verified.

[In]

Int[ArcSinh[a*x]^(-1),x]

[Out]

CoshIntegral[ArcSinh[a*x]]/a

Rule 5657

Int[((a_.) + ArcSinh[(c_.)*(x_)]*(b_.))^(n_), x_Symbol] :> Dist[1/(b*c), Subst[Int[x^n*Cosh[a/b - x/b], x], x,
a + b*ArcSinh[c*x]], x] /; FreeQ[{a, b, c, n}, x]

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

Mathematica [A]  time = 0.0089621, size = 9, normalized size = 1. \[ \frac{\text{Chi}\left (\sinh ^{-1}(a x)\right )}{a} \]

Antiderivative was successfully verified.

[In]

Integrate[ArcSinh[a*x]^(-1),x]

[Out]

CoshIntegral[ArcSinh[a*x]]/a

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Maple [A]  time = 0.019, size = 10, normalized size = 1.1 \begin{align*}{\frac{{\it Chi} \left ({\it Arcsinh} \left ( ax \right ) \right ) }{a}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Chi(arcsinh(a*x))/a

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate(1/arcsinh(a*x), 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 )}, x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integral(1/arcsinh(a*x), x)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Integral(1/asinh(a*x), 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 )}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate(1/arcsinh(a*x), x)