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]

Shi(arccosh(a*x))/a

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Rubi [A]  time = 0.02, antiderivative size = 9, normalized size of antiderivative = 1.00, 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 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 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]

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*}

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Mathematica [A]  time = 0.02, size = 9, normalized size = 1.00 \[ \frac {\text {Shi}\left (\cosh ^{-1}(a x)\right )}{a} \]

Antiderivative was successfully verified.

[In]

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

[Out]

SinhIntegral[ArcCosh[a*x]]/a

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fricas [F]  time = 0.64, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {1}{\operatorname {arcosh}\left (a x\right )}, x\right ) \]

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)

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giac [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {1}{\operatorname {arcosh}\left (a x\right )}\,{d x} \]

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)

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maple [A]  time = 0.02, size = 10, normalized size = 1.11 \[ \frac {\Shi \left (\mathrm {arccosh}\left (a x \right )\right )}{a} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Shi(arccosh(a*x))/a

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maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {1}{\operatorname {arcosh}\left (a x\right )}\,{d x} \]

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)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

int(1/acosh(a*x), x)

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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {1}{\operatorname {acosh}{\left (a x \right )}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Integral(1/acosh(a*x), x)

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