Optimal. Leaf size=27 \[ \frac{\text{Shi}\left (3 \sinh ^{-1}(a x)\right )}{4 a^4}-\frac{3 \text{Shi}\left (\sinh ^{-1}(a x)\right )}{4 a^4} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.155289, antiderivative size = 27, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 3, integrand size = 23, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.13, Rules used = {5779, 3312, 3298} \[ \frac{\text{Shi}\left (3 \sinh ^{-1}(a x)\right )}{4 a^4}-\frac{3 \text{Shi}\left (\sinh ^{-1}(a x)\right )}{4 a^4} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 5779
Rule 3312
Rule 3298
Rubi steps
\begin{align*} \int \frac{x^3}{\sqrt{1+a^2 x^2} \sinh ^{-1}(a x)} \, dx &=\frac{\operatorname{Subst}\left (\int \frac{\sinh ^3(x)}{x} \, dx,x,\sinh ^{-1}(a x)\right )}{a^4}\\ &=\frac{i \operatorname{Subst}\left (\int \left (\frac{3 i \sinh (x)}{4 x}-\frac{i \sinh (3 x)}{4 x}\right ) \, dx,x,\sinh ^{-1}(a x)\right )}{a^4}\\ &=\frac{\operatorname{Subst}\left (\int \frac{\sinh (3 x)}{x} \, dx,x,\sinh ^{-1}(a x)\right )}{4 a^4}-\frac{3 \operatorname{Subst}\left (\int \frac{\sinh (x)}{x} \, dx,x,\sinh ^{-1}(a x)\right )}{4 a^4}\\ &=-\frac{3 \text{Shi}\left (\sinh ^{-1}(a x)\right )}{4 a^4}+\frac{\text{Shi}\left (3 \sinh ^{-1}(a x)\right )}{4 a^4}\\ \end{align*}
Mathematica [A] time = 0.0774442, size = 22, normalized size = 0.81 \[ \frac{\text{Shi}\left (3 \sinh ^{-1}(a x)\right )-3 \text{Shi}\left (\sinh ^{-1}(a x)\right )}{4 a^4} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [A] time = 0.043, size = 23, normalized size = 0.9 \begin{align*} -{\frac{3\,{\it Shi} \left ({\it Arcsinh} \left ( ax \right ) \right ) -{\it Shi} \left ( 3\,{\it Arcsinh} \left ( ax \right ) \right ) }{4\,{a}^{4}}} \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{x^{3}}{\sqrt{a^{2} x^{2} + 1} \operatorname{arsinh}\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{x^{3}}{\sqrt{a^{2} x^{2} + 1} \operatorname{arsinh}\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{x^{3}}{\sqrt{a^{2} x^{2} + 1} \operatorname{asinh}{\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{x^{3}}{\sqrt{a^{2} x^{2} + 1} \operatorname{arsinh}\left (a x\right )}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]