Optimal. Leaf size=27 \[ \frac {\text {Shi}\left (\cosh ^{-1}(a x)\right )}{4 a^3}+\frac {\text {Shi}\left (3 \cosh ^{-1}(a x)\right )}{4 a^3} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.06, antiderivative size = 27, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 3, integrand size = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.300, Rules used = {5670, 5448, 3298} \[ \frac {\text {Shi}\left (\cosh ^{-1}(a x)\right )}{4 a^3}+\frac {\text {Shi}\left (3 \cosh ^{-1}(a x)\right )}{4 a^3} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 3298
Rule 5448
Rule 5670
Rubi steps
\begin {align*} \int \frac {x^2}{\cosh ^{-1}(a x)} \, dx &=\frac {\operatorname {Subst}\left (\int \frac {\cosh ^2(x) \sinh (x)}{x} \, dx,x,\cosh ^{-1}(a x)\right )}{a^3}\\ &=\frac {\operatorname {Subst}\left (\int \left (\frac {\sinh (x)}{4 x}+\frac {\sinh (3 x)}{4 x}\right ) \, dx,x,\cosh ^{-1}(a x)\right )}{a^3}\\ &=\frac {\operatorname {Subst}\left (\int \frac {\sinh (x)}{x} \, dx,x,\cosh ^{-1}(a x)\right )}{4 a^3}+\frac {\operatorname {Subst}\left (\int \frac {\sinh (3 x)}{x} \, dx,x,\cosh ^{-1}(a x)\right )}{4 a^3}\\ &=\frac {\text {Shi}\left (\cosh ^{-1}(a x)\right )}{4 a^3}+\frac {\text {Shi}\left (3 \cosh ^{-1}(a x)\right )}{4 a^3}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A] time = 0.06, size = 20, normalized size = 0.74 \[ \frac {\text {Shi}\left (\cosh ^{-1}(a x)\right )+\text {Shi}\left (3 \cosh ^{-1}(a x)\right )}{4 a^3} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
fricas [F] time = 0.56, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {x^{2}}{\operatorname {arcosh}\left (a x\right )}, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {x^{2}}{\operatorname {arcosh}\left (a x\right )}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [A] time = 0.03, size = 22, normalized size = 0.81 \[ \frac {\frac {\Shi \left (\mathrm {arccosh}\left (a x \right )\right )}{4}+\frac {\Shi \left (3 \,\mathrm {arccosh}\left (a x \right )\right )}{4}}{a^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {x^{2}}{\operatorname {arcosh}\left (a x\right )}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [F] time = 0.00, size = -1, normalized size = -0.04 \[ \int \frac {x^2}{\mathrm {acosh}\left (a\,x\right )} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {x^{2}}{\operatorname {acosh}{\left (a x \right )}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________