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

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

[Out]

SinhIntegral[ArcCosh[a*x]]/a

________________________________________________________________________________________

Rubi [A]  time = 0.0171132, 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 = {5658, 3298} \[ \frac{\text{Shi}\left (\cosh ^{-1}(a x)\right )}{a} \]

Antiderivative was successfully verified.

[In]

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

[Out]

SinhIntegral[ArcCosh[a*x]]/a

Rule 5658

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

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

Mathematica [A]  time = 0.0224391, size = 9, normalized size = 1. \[ \frac{\text{Shi}\left (\cosh ^{-1}(a x)\right )}{a} \]

Antiderivative was successfully verified.

[In]

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

[Out]

SinhIntegral[ArcCosh[a*x]]/a

________________________________________________________________________________________

Maple [A]  time = 0.021, size = 10, normalized size = 1.1 \begin{align*}{\frac{{\it Shi} \left ({\rm arccosh} \left (ax\right ) \right ) }{a}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Shi(arccosh(a*x))/a

________________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate(1/arccosh(a*x), x)

________________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integral(1/arccosh(a*x), x)

________________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Integral(1/acosh(a*x), x)

________________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate(1/arccosh(a*x), x)