Optimal. Leaf size=9 \[ \frac{\text{Shi}\left (\cosh ^{-1}(a x)\right )}{a} \]
[Out]
________________________________________________________________________________________
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]
[Out]
Rule 5658
Rule 3298
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]
[Out]
________________________________________________________________________________________
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]
[Out]
________________________________________________________________________________________
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]
[Out]
________________________________________________________________________________________
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]
[Out]
________________________________________________________________________________________
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]
[Out]
________________________________________________________________________________________
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]
[Out]