3.83 \(\int \frac{1}{\sinh ^{-1}(a+b x)} \, dx\)

Optimal. Leaf size=11 \[ \frac{\text{Chi}\left (\sinh ^{-1}(a+b x)\right )}{b} \]

[Out]

CoshIntegral[ArcSinh[a + b*x]]/b

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Rubi [A]  time = 0.0233665, antiderivative size = 11, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 8, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.375, Rules used = {5863, 5657, 3301} \[ \frac{\text{Chi}\left (\sinh ^{-1}(a+b x)\right )}{b} \]

Antiderivative was successfully verified.

[In]

Int[ArcSinh[a + b*x]^(-1),x]

[Out]

CoshIntegral[ArcSinh[a + b*x]]/b

Rule 5863

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

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

Mathematica [A]  time = 0.0069485, size = 11, normalized size = 1. \[ \frac{\text{Chi}\left (\sinh ^{-1}(a+b x)\right )}{b} \]

Antiderivative was successfully verified.

[In]

Integrate[ArcSinh[a + b*x]^(-1),x]

[Out]

CoshIntegral[ArcSinh[a + b*x]]/b

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Maple [A]  time = 0.023, size = 12, normalized size = 1.1 \begin{align*}{\frac{{\it Chi} \left ({\it Arcsinh} \left ( bx+a \right ) \right ) }{b}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Chi(arcsinh(b*x+a))/b

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate(1/arcsinh(b*x + a), x)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integral(1/arcsinh(b*x + a), x)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Integral(1/asinh(a + b*x), x)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate(1/arcsinh(b*x + a), x)