\(\int \frac {x^3 \text {arcsinh}(a x)^n}{\sqrt {1+a^2 x^2}} \, dx\) [477]

Optimal result

Integrand size = 23, antiderivative size = 113 \[ \int \frac {x^3 \text {arcsinh}(a x)^n}{\sqrt {1+a^2 x^2}} \, dx=\frac {3^{-1-n} (-\text {arcsinh}(a x))^{-n} \text {arcsinh}(a x)^n \Gamma (1+n,-3 \text {arcsinh}(a x))}{8 a^4}-\frac {3 (-\text {arcsinh}(a x))^{-n} \text {arcsinh}(a x)^n \Gamma (1+n,-\text {arcsinh}(a x))}{8 a^4}-\frac {3 \Gamma (1+n,\text {arcsinh}(a x))}{8 a^4}+\frac {3^{-1-n} \Gamma (1+n,3 \text {arcsinh}(a x))}{8 a^4} \] Output:

1/8*3^(-1-n)*arcsinh(a*x)^n*GAMMA(1+n,-3*arcsinh(a*x))/a^4/((-arcsinh(a*x) 
)^n)-3/8*arcsinh(a*x)^n*GAMMA(1+n,-arcsinh(a*x))/a^4/((-arcsinh(a*x))^n)-3 
/8*GAMMA(1+n,arcsinh(a*x))/a^4+1/8*3^(-1-n)*GAMMA(1+n,3*arcsinh(a*x))/a^4
 

Mathematica [A] (verified)

Time = 0.14 (sec) , antiderivative size = 100, normalized size of antiderivative = 0.88 \[ \int \frac {x^3 \text {arcsinh}(a x)^n}{\sqrt {1+a^2 x^2}} \, dx=\frac {3^{-1-n} (-\text {arcsinh}(a x))^{-n} \left (\text {arcsinh}(a x)^n \Gamma (1+n,-3 \text {arcsinh}(a x))-3^{2+n} \text {arcsinh}(a x)^n \Gamma (1+n,-\text {arcsinh}(a x))+(-\text {arcsinh}(a x))^n \left (-3^{2+n} \Gamma (1+n,\text {arcsinh}(a x))+\Gamma (1+n,3 \text {arcsinh}(a x))\right )\right )}{8 a^4} \] Input:

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

Output:

(3^(-1 - n)*(ArcSinh[a*x]^n*Gamma[1 + n, -3*ArcSinh[a*x]] - 3^(2 + n)*ArcS 
inh[a*x]^n*Gamma[1 + n, -ArcSinh[a*x]] + (-ArcSinh[a*x])^n*(-(3^(2 + n)*Ga 
mma[1 + n, ArcSinh[a*x]]) + Gamma[1 + n, 3*ArcSinh[a*x]])))/(8*a^4*(-ArcSi 
nh[a*x])^n)
 

Rubi [C] (verified)

Result contains complex when optimal does not.

Time = 0.79 (sec) , antiderivative size = 116, normalized size of antiderivative = 1.03, number of steps used = 6, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.217, Rules used = {6234, 3042, 26, 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.

\(\displaystyle \int \frac {x^3 \text {arcsinh}(a x)^n}{\sqrt {a^2 x^2+1}} \, dx\)

\(\Big \downarrow \) 6234

\(\displaystyle \frac {\int a^3 x^3 \text {arcsinh}(a x)^nd\text {arcsinh}(a x)}{a^4}\)

\(\Big \downarrow \) 3042

\(\displaystyle \frac {\int i \text {arcsinh}(a x)^n \sin (i \text {arcsinh}(a x))^3d\text {arcsinh}(a x)}{a^4}\)

\(\Big \downarrow \) 26

\(\displaystyle \frac {i \int \text {arcsinh}(a x)^n \sin (i \text {arcsinh}(a x))^3d\text {arcsinh}(a x)}{a^4}\)

\(\Big \downarrow \) 3793

\(\displaystyle \frac {i \int \left (\frac {3}{4} i a x \text {arcsinh}(a x)^n-\frac {1}{4} i \text {arcsinh}(a x)^n \sinh (3 \text {arcsinh}(a x))\right )d\text {arcsinh}(a x)}{a^4}\)

\(\Big \downarrow \) 2009

\(\displaystyle \frac {i \left (-\frac {1}{8} i 3^{-n-1} \text {arcsinh}(a x)^n (-\text {arcsinh}(a x))^{-n} \Gamma (n+1,-3 \text {arcsinh}(a x))+\frac {3}{8} i \text {arcsinh}(a x)^n (-\text {arcsinh}(a x))^{-n} \Gamma (n+1,-\text {arcsinh}(a x))+\frac {3}{8} i \Gamma (n+1,\text {arcsinh}(a x))-\frac {1}{8} i 3^{-n-1} \Gamma (n+1,3 \text {arcsinh}(a x))\right )}{a^4}\)

Input:

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

Output:

(I*(((-1/8*I)*3^(-1 - n)*ArcSinh[a*x]^n*Gamma[1 + n, -3*ArcSinh[a*x]])/(-A 
rcSinh[a*x])^n + (((3*I)/8)*ArcSinh[a*x]^n*Gamma[1 + n, -ArcSinh[a*x]])/(- 
ArcSinh[a*x])^n + ((3*I)/8)*Gamma[1 + n, ArcSinh[a*x]] - (I/8)*3^(-1 - n)* 
Gamma[1 + n, 3*ArcSinh[a*x]]))/a^4
 

Defintions of rubi rules used

rule 26

Int[(Complex[0, a_])*(Fx_), x_Symbol] :> Simp[(Complex[Identity[0], a])   I 
nt[Fx, x], x] /; FreeQ[a, x] && EqQ[a^2, 1]
 

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]

\[\int \frac {x^{3} \operatorname {arcsinh}\left (x a \right )^{n}}{\sqrt {a^{2} x^{2}+1}}d x\]

Input:

int(x^3*arcsinh(x*a)^n/(a^2*x^2+1)^(1/2),x)
 

Output:

int(x^3*arcsinh(x*a)^n/(a^2*x^2+1)^(1/2),x)
 

Fricas [F]

\[ \int \frac {x^3 \text {arcsinh}(a x)^n}{\sqrt {1+a^2 x^2}} \, dx=\int { \frac {x^{3} \operatorname {arsinh}\left (a x\right )^{n}}{\sqrt {a^{2} x^{2} + 1}} \,d x } \] Input:

integrate(x^3*arcsinh(a*x)^n/(a^2*x^2+1)^(1/2),x, algorithm="fricas")
 

Output:

integral(x^3*arcsinh(a*x)^n/sqrt(a^2*x^2 + 1), x)
 

Sympy [F]

\[ \int \frac {x^3 \text {arcsinh}(a x)^n}{\sqrt {1+a^2 x^2}} \, dx=\int \frac {x^{3} \operatorname {asinh}^{n}{\left (a x \right )}}{\sqrt {a^{2} x^{2} + 1}}\, dx \] Input:

integrate(x**3*asinh(a*x)**n/(a**2*x**2+1)**(1/2),x)
 

Output:

Integral(x**3*asinh(a*x)**n/sqrt(a**2*x**2 + 1), x)
 

Maxima [F]

\[ \int \frac {x^3 \text {arcsinh}(a x)^n}{\sqrt {1+a^2 x^2}} \, dx=\int { \frac {x^{3} \operatorname {arsinh}\left (a x\right )^{n}}{\sqrt {a^{2} x^{2} + 1}} \,d x } \] Input:

integrate(x^3*arcsinh(a*x)^n/(a^2*x^2+1)^(1/2),x, algorithm="maxima")
 

Output:

integrate(x^3*arcsinh(a*x)^n/sqrt(a^2*x^2 + 1), x)
 

Giac [F(-2)]

Exception generated. \[ \int \frac {x^3 \text {arcsinh}(a x)^n}{\sqrt {1+a^2 x^2}} \, dx=\text {Exception raised: TypeError} \] Input:

integrate(x^3*arcsinh(a*x)^n/(a^2*x^2+1)^(1/2),x, algorithm="giac")
 

Output:

Exception raised: TypeError >> an error occurred running a Giac command:IN 
PUT:sage2:=int(sage0,sageVARx):;OUTPUT:sym2poly/r2sym(const gen & e,const 
index_m & i,const vecteur & l) Error: Bad Argument Value
 

Mupad [F(-1)]

Timed out. \[ \int \frac {x^3 \text {arcsinh}(a x)^n}{\sqrt {1+a^2 x^2}} \, dx=\int \frac {x^3\,{\mathrm {asinh}\left (a\,x\right )}^n}{\sqrt {a^2\,x^2+1}} \,d x \] Input:

int((x^3*asinh(a*x)^n)/(a^2*x^2 + 1)^(1/2),x)
 

Output:

int((x^3*asinh(a*x)^n)/(a^2*x^2 + 1)^(1/2), x)
 

Reduce [F]

\[ \int \frac {x^3 \text {arcsinh}(a x)^n}{\sqrt {1+a^2 x^2}} \, dx=\int \frac {\mathit {asinh} \left (a x \right )^{n} x^{3}}{\sqrt {a^{2} x^{2}+1}}d x \] Input:

int(x^3*asinh(a*x)^n/(a^2*x^2+1)^(1/2),x)
 

Output:

int((asinh(a*x)**n*x**3)/sqrt(a**2*x**2 + 1),x)