Optimal. Leaf size=49 \[ \frac{\sinh ^{-1}(a x)^n \left (-\sinh ^{-1}(a x)\right )^{-n} \text{Gamma}\left (n+1,-\sinh ^{-1}(a x)\right )}{2 a^2}+\frac{\text{Gamma}\left (n+1,\sinh ^{-1}(a x)\right )}{2 a^2} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.117022, antiderivative size = 49, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.143, Rules used = {5779, 3308, 2181} \[ \frac{\sinh ^{-1}(a x)^n \left (-\sinh ^{-1}(a x)\right )^{-n} \text{Gamma}\left (n+1,-\sinh ^{-1}(a x)\right )}{2 a^2}+\frac{\text{Gamma}\left (n+1,\sinh ^{-1}(a x)\right )}{2 a^2} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 5779
Rule 3308
Rule 2181
Rubi steps
\begin{align*} \int \frac{x \sinh ^{-1}(a x)^n}{\sqrt{1+a^2 x^2}} \, dx &=\frac{\operatorname{Subst}\left (\int x^n \sinh (x) \, dx,x,\sinh ^{-1}(a x)\right )}{a^2}\\ &=-\frac{\operatorname{Subst}\left (\int e^{-x} x^n \, dx,x,\sinh ^{-1}(a x)\right )}{2 a^2}+\frac{\operatorname{Subst}\left (\int e^x x^n \, dx,x,\sinh ^{-1}(a x)\right )}{2 a^2}\\ &=\frac{\left (-\sinh ^{-1}(a x)\right )^{-n} \sinh ^{-1}(a x)^n \Gamma \left (1+n,-\sinh ^{-1}(a x)\right )}{2 a^2}+\frac{\Gamma \left (1+n,\sinh ^{-1}(a x)\right )}{2 a^2}\\ \end{align*}
Mathematica [A] time = 0.0716987, size = 43, normalized size = 0.88 \[ \frac{\sinh ^{-1}(a x)^n \left (-\sinh ^{-1}(a x)\right )^{-n} \text{Gamma}\left (n+1,-\sinh ^{-1}(a x)\right )+\text{Gamma}\left (n+1,\sinh ^{-1}(a x)\right )}{2 a^2} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [F] time = 0.087, size = 0, normalized size = 0. \begin{align*} \int{x \left ({\it Arcsinh} \left ( ax \right ) \right ) ^{n}{\frac{1}{\sqrt{{a}^{2}{x}^{2}+1}}}}\, dx \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 \operatorname{arsinh}\left (a x\right )^{n}}{\sqrt{a^{2} x^{2} + 1}}\,{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 \operatorname{arsinh}\left (a x\right )^{n}}{\sqrt{a^{2} x^{2} + 1}}, 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 \operatorname{asinh}^{n}{\left (a x \right )}}{\sqrt{a^{2} x^{2} + 1}}\, 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 \operatorname{arsinh}\left (a x\right )^{n}}{\sqrt{a^{2} x^{2} + 1}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]