\(\int \frac {x^2 \text {arcsinh}(a x)^n}{\sqrt {1+a^2 x^2}} \, dx\) [478]

Optimal result

Integrand size = 23, antiderivative size = 80 \[ \int \frac {x^2 \text {arcsinh}(a x)^n}{\sqrt {1+a^2 x^2}} \, dx=-\frac {\text {arcsinh}(a x)^{1+n}}{2 a^3 (1+n)}+\frac {2^{-3-n} (-\text {arcsinh}(a x))^{-n} \text {arcsinh}(a x)^n \Gamma (1+n,-2 \text {arcsinh}(a x))}{a^3}-\frac {2^{-3-n} \Gamma (1+n,2 \text {arcsinh}(a x))}{a^3} \] Output:

-1/2*arcsinh(a*x)^(1+n)/a^3/(1+n)+2^(-3-n)*arcsinh(a*x)^n*GAMMA(1+n,-2*arc 
sinh(a*x))/a^3/((-arcsinh(a*x))^n)-2^(-3-n)*GAMMA(1+n,2*arcsinh(a*x))/a^3
 

Mathematica [A] (verified)

Time = 0.17 (sec) , antiderivative size = 86, normalized size of antiderivative = 1.08 \[ \int \frac {x^2 \text {arcsinh}(a x)^n}{\sqrt {1+a^2 x^2}} \, dx=\frac {2^{-3-n} (-\text {arcsinh}(a x))^{-n} \left ((1+n) \text {arcsinh}(a x)^n \Gamma (1+n,-2 \text {arcsinh}(a x))-(-\text {arcsinh}(a x))^n \left (2^{2+n} \text {arcsinh}(a x)^{1+n}+(1+n) \Gamma (1+n,2 \text {arcsinh}(a x))\right )\right )}{a^3 (1+n)} \] Input:

Integrate[(x^2*ArcSinh[a*x]^n)/Sqrt[1 + a^2*x^2],x]
 

Output:

(2^(-3 - n)*((1 + n)*ArcSinh[a*x]^n*Gamma[1 + n, -2*ArcSinh[a*x]] - (-ArcS 
inh[a*x])^n*(2^(2 + n)*ArcSinh[a*x]^(1 + n) + (1 + n)*Gamma[1 + n, 2*ArcSi 
nh[a*x]])))/(a^3*(1 + n)*(-ArcSinh[a*x])^n)
 

Rubi [A] (verified)

Time = 0.53 (sec) , antiderivative size = 75, normalized size of antiderivative = 0.94, number of steps used = 6, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.217, Rules used = {6234, 3042, 25, 3793, 2009}

Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.

\(\displaystyle \int \frac {x^2 \text {arcsinh}(a x)^n}{\sqrt {a^2 x^2+1}} \, dx\)

\(\Big \downarrow \) 6234

\(\displaystyle \frac {\int a^2 x^2 \text {arcsinh}(a x)^nd\text {arcsinh}(a x)}{a^3}\)

\(\Big \downarrow \) 3042

\(\displaystyle \frac {\int -\text {arcsinh}(a x)^n \sin (i \text {arcsinh}(a x))^2d\text {arcsinh}(a x)}{a^3}\)

\(\Big \downarrow \) 25

\(\displaystyle -\frac {\int \text {arcsinh}(a x)^n \sin (i \text {arcsinh}(a x))^2d\text {arcsinh}(a x)}{a^3}\)

\(\Big \downarrow \) 3793

\(\displaystyle -\frac {\int \left (\frac {1}{2} \text {arcsinh}(a x)^n-\frac {1}{2} \text {arcsinh}(a x)^n \cosh (2 \text {arcsinh}(a x))\right )d\text {arcsinh}(a x)}{a^3}\)

\(\Big \downarrow \) 2009

\(\displaystyle \frac {-\frac {\text {arcsinh}(a x)^{n+1}}{2 (n+1)}+2^{-n-3} \text {arcsinh}(a x)^n (-\text {arcsinh}(a x))^{-n} \Gamma (n+1,-2 \text {arcsinh}(a x))-2^{-n-3} \Gamma (n+1,2 \text {arcsinh}(a x))}{a^3}\)

Input:

Int[(x^2*ArcSinh[a*x]^n)/Sqrt[1 + a^2*x^2],x]
 

Output:

(-1/2*ArcSinh[a*x]^(1 + n)/(1 + n) + (2^(-3 - n)*ArcSinh[a*x]^n*Gamma[1 + 
n, -2*ArcSinh[a*x]])/(-ArcSinh[a*x])^n - 2^(-3 - n)*Gamma[1 + n, 2*ArcSinh 
[a*x]])/a^3
 

Defintions of rubi rules used

rule 25

Int[-(Fx_), x_Symbol] :> Simp[Identity[-1]   Int[Fx, x], x]
 

rule 2009

Int[u_, x_Symbol] :> Simp[IntSum[u, x], x] /; SumQ[u]
 

rule 3042

Int[u_, x_Symbol] :> Int[DeactivateTrig[u, x], x] /; FunctionOfTrigOfLinear 
Q[u, x]
 

rule 3793

Int[((c_.) + (d_.)*(x_))^(m_)*sin[(e_.) + (f_.)*(x_)]^(n_), x_Symbol] :> In 
t[ExpandTrigReduce[(c + d*x)^m, Sin[e + f*x]^n, x], x] /; FreeQ[{c, d, e, f 
, m}, x] && IGtQ[n, 1] && ( !RationalQ[m] || (GeQ[m, -1] && LtQ[m, 1]))
 

rule 6234

Int[((a_.) + ArcSinh[(c_.)*(x_)]*(b_.))^(n_.)*(x_)^(m_.)*((d_) + (e_.)*(x_) 
^2)^(p_.), x_Symbol] :> Simp[(1/(b*c^(m + 1)))*Simp[(d + e*x^2)^p/(1 + c^2* 
x^2)^p]   Subst[Int[x^n*Sinh[-a/b + x/b]^m*Cosh[-a/b + x/b]^(2*p + 1), x], 
x, a + b*ArcSinh[c*x]], x] /; FreeQ[{a, b, c, d, e, n}, x] && EqQ[e, c^2*d] 
 && IGtQ[2*p + 2, 0] && IGtQ[m, 0]
 

Maple [F]

\[\int \frac {x^{2} \operatorname {arcsinh}\left (x a \right )^{n}}{\sqrt {a^{2} x^{2}+1}}d x\]

Input:

int(x^2*arcsinh(x*a)^n/(a^2*x^2+1)^(1/2),x)
 

Output:

int(x^2*arcsinh(x*a)^n/(a^2*x^2+1)^(1/2),x)
 

Fricas [F]

\[ \int \frac {x^2 \text {arcsinh}(a x)^n}{\sqrt {1+a^2 x^2}} \, dx=\int { \frac {x^{2} \operatorname {arsinh}\left (a x\right )^{n}}{\sqrt {a^{2} x^{2} + 1}} \,d x } \] Input:

integrate(x^2*arcsinh(a*x)^n/(a^2*x^2+1)^(1/2),x, algorithm="fricas")
 

Output:

integral(x^2*arcsinh(a*x)^n/sqrt(a^2*x^2 + 1), x)
 

Sympy [F]

\[ \int \frac {x^2 \text {arcsinh}(a x)^n}{\sqrt {1+a^2 x^2}} \, dx=\int \frac {x^{2} \operatorname {asinh}^{n}{\left (a x \right )}}{\sqrt {a^{2} x^{2} + 1}}\, dx \] Input:

integrate(x**2*asinh(a*x)**n/(a**2*x**2+1)**(1/2),x)
 

Output:

Integral(x**2*asinh(a*x)**n/sqrt(a**2*x**2 + 1), x)
 

Maxima [F]

\[ \int \frac {x^2 \text {arcsinh}(a x)^n}{\sqrt {1+a^2 x^2}} \, dx=\int { \frac {x^{2} \operatorname {arsinh}\left (a x\right )^{n}}{\sqrt {a^{2} x^{2} + 1}} \,d x } \] Input:

integrate(x^2*arcsinh(a*x)^n/(a^2*x^2+1)^(1/2),x, algorithm="maxima")
 

Output:

integrate(x^2*arcsinh(a*x)^n/sqrt(a^2*x^2 + 1), x)
 

Giac [F]

\[ \int \frac {x^2 \text {arcsinh}(a x)^n}{\sqrt {1+a^2 x^2}} \, dx=\int { \frac {x^{2} \operatorname {arsinh}\left (a x\right )^{n}}{\sqrt {a^{2} x^{2} + 1}} \,d x } \] Input:

integrate(x^2*arcsinh(a*x)^n/(a^2*x^2+1)^(1/2),x, algorithm="giac")
 

Output:

integrate(x^2*arcsinh(a*x)^n/sqrt(a^2*x^2 + 1), x)
 

Mupad [F(-1)]

Timed out. \[ \int \frac {x^2 \text {arcsinh}(a x)^n}{\sqrt {1+a^2 x^2}} \, dx=\int \frac {x^2\,{\mathrm {asinh}\left (a\,x\right )}^n}{\sqrt {a^2\,x^2+1}} \,d x \] Input:

int((x^2*asinh(a*x)^n)/(a^2*x^2 + 1)^(1/2),x)
 

Output:

int((x^2*asinh(a*x)^n)/(a^2*x^2 + 1)^(1/2), x)
 

Reduce [F]

\[ \int \frac {x^2 \text {arcsinh}(a x)^n}{\sqrt {1+a^2 x^2}} \, dx=\int \frac {\mathit {asinh} \left (a x \right )^{n} x^{2}}{\sqrt {a^{2} x^{2}+1}}d x \] Input:

int(x^2*asinh(a*x)^n/(a^2*x^2+1)^(1/2),x)
 

Output:

int((asinh(a*x)**n*x**2)/sqrt(a**2*x**2 + 1),x)