Optimal. Leaf size=29 \[ \frac{3 c \text{Shi}\left (\cosh ^{-1}(a x)\right )}{4 a}-\frac{c \text{Shi}\left (3 \cosh ^{-1}(a x)\right )}{4 a} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.080804, antiderivative size = 29, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 3, integrand size = 18, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.167, Rules used = {5700, 3312, 3298} \[ \frac{3 c \text{Shi}\left (\cosh ^{-1}(a x)\right )}{4 a}-\frac{c \text{Shi}\left (3 \cosh ^{-1}(a x)\right )}{4 a} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 5700
Rule 3312
Rule 3298
Rubi steps
\begin{align*} \int \frac{c-a^2 c x^2}{\cosh ^{-1}(a x)} \, dx &=-\frac{c \operatorname{Subst}\left (\int \frac{\sinh ^3(x)}{x} \, dx,x,\cosh ^{-1}(a x)\right )}{a}\\ &=-\frac{(i c) \operatorname{Subst}\left (\int \left (\frac{3 i \sinh (x)}{4 x}-\frac{i \sinh (3 x)}{4 x}\right ) \, dx,x,\cosh ^{-1}(a x)\right )}{a}\\ &=-\frac{c \operatorname{Subst}\left (\int \frac{\sinh (3 x)}{x} \, dx,x,\cosh ^{-1}(a x)\right )}{4 a}+\frac{(3 c) \operatorname{Subst}\left (\int \frac{\sinh (x)}{x} \, dx,x,\cosh ^{-1}(a x)\right )}{4 a}\\ &=\frac{3 c \text{Shi}\left (\cosh ^{-1}(a x)\right )}{4 a}-\frac{c \text{Shi}\left (3 \cosh ^{-1}(a x)\right )}{4 a}\\ \end{align*}
Mathematica [A] time = 0.115985, size = 25, normalized size = 0.86 \[ \frac{c \left (3 \text{Shi}\left (\cosh ^{-1}(a x)\right )-\text{Shi}\left (3 \cosh ^{-1}(a x)\right )\right )}{4 a} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [A] time = 0.03, size = 24, normalized size = 0.8 \begin{align*}{\frac{c \left ( 3\,{\it Shi} \left ({\rm arccosh} \left (ax\right ) \right ) -{\it Shi} \left ( 3\,{\rm arccosh} \left (ax\right ) \right ) \right ) }{4\,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{a^{2} c x^{2} - c}{\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{a^{2} c x^{2} - c}{\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*} - c \left (\int \frac{a^{2} x^{2}}{\operatorname{acosh}{\left (a x \right )}}\, dx + \int - \frac{1}{\operatorname{acosh}{\left (a x \right )}}\, dx\right ) \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{a^{2} c x^{2} - c}{\operatorname{arcosh}\left (a x\right )}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]