\(\int x^m \text {arcsinh}(a x)^2 \, dx\) [102]

Optimal result

Integrand size = 10, antiderivative size = 137 \[ \int x^m \text {arcsinh}(a x)^2 \, dx=\frac {x^{1+m} \text {arcsinh}(a x)^2}{1+m}-\frac {2 a x^{2+m} \text {arcsinh}(a x) \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {2+m}{2},\frac {4+m}{2},-a^2 x^2\right )}{2+3 m+m^2}+\frac {2 a^2 x^{3+m} \, _3F_2\left (1,\frac {3}{2}+\frac {m}{2},\frac {3}{2}+\frac {m}{2};2+\frac {m}{2},\frac {5}{2}+\frac {m}{2};-a^2 x^2\right )}{6+11 m+6 m^2+m^3} \] Output:

x^(1+m)*arcsinh(a*x)^2/(1+m)-2*a*x^(2+m)*arcsinh(a*x)*hypergeom([1/2, 1+1/ 
2*m],[2+1/2*m],-a^2*x^2)/(m^2+3*m+2)+2*a^2*x^(3+m)*hypergeom([1, 3/2+1/2*m 
, 3/2+1/2*m],[2+1/2*m, 5/2+1/2*m],-a^2*x^2)/(m^3+6*m^2+11*m+6)
 

Mathematica [A] (verified)

Time = 0.03 (sec) , antiderivative size = 123, normalized size of antiderivative = 0.90 \[ \int x^m \text {arcsinh}(a x)^2 \, dx=\frac {x^{1+m} \left ((3+m) \text {arcsinh}(a x) \left ((2+m) \text {arcsinh}(a x)-2 a x \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {2+m}{2},\frac {4+m}{2},-a^2 x^2\right )\right )+2 a^2 x^2 \, _3F_2\left (1,\frac {3}{2}+\frac {m}{2},\frac {3}{2}+\frac {m}{2};2+\frac {m}{2},\frac {5}{2}+\frac {m}{2};-a^2 x^2\right )\right )}{(1+m) (2+m) (3+m)} \] Input:

Integrate[x^m*ArcSinh[a*x]^2,x]
 

Output:

(x^(1 + m)*((3 + m)*ArcSinh[a*x]*((2 + m)*ArcSinh[a*x] - 2*a*x*Hypergeomet 
ric2F1[1/2, (2 + m)/2, (4 + m)/2, -(a^2*x^2)]) + 2*a^2*x^2*HypergeometricP 
FQ[{1, 3/2 + m/2, 3/2 + m/2}, {2 + m/2, 5/2 + m/2}, -(a^2*x^2)]))/((1 + m) 
*(2 + m)*(3 + m))
 

Rubi [A] (verified)

Time = 0.34 (sec) , antiderivative size = 132, normalized size of antiderivative = 0.96, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.200, Rules used = {6191, 6232}

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.

\(\displaystyle \int x^m \text {arcsinh}(a x)^2 \, dx\)

\(\Big \downarrow \) 6191

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

\(\Big \downarrow \) 6232

\(\displaystyle \frac {x^{m+1} \text {arcsinh}(a x)^2}{m+1}-\frac {2 a \left (\frac {x^{m+2} \text {arcsinh}(a x) \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {m+2}{2},\frac {m+4}{2},-a^2 x^2\right )}{m+2}-\frac {a x^{m+3} \, _3F_2\left (1,\frac {m}{2}+\frac {3}{2},\frac {m}{2}+\frac {3}{2};\frac {m}{2}+2,\frac {m}{2}+\frac {5}{2};-a^2 x^2\right )}{m^2+5 m+6}\right )}{m+1}\)

Input:

Int[x^m*ArcSinh[a*x]^2,x]
 

Output:

(x^(1 + m)*ArcSinh[a*x]^2)/(1 + m) - (2*a*((x^(2 + m)*ArcSinh[a*x]*Hyperge 
ometric2F1[1/2, (2 + m)/2, (4 + m)/2, -(a^2*x^2)])/(2 + m) - (a*x^(3 + m)* 
HypergeometricPFQ[{1, 3/2 + m/2, 3/2 + m/2}, {2 + m/2, 5/2 + m/2}, -(a^2*x 
^2)])/(6 + 5*m + m^2)))/(1 + m)
 

Defintions of rubi rules used

rule 6191

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

rule 6232

Int[(((a_.) + ArcSinh[(c_.)*(x_)]*(b_.))*((f_.)*(x_))^(m_))/Sqrt[(d_) + (e_ 
.)*(x_)^2], x_Symbol] :> Simp[((f*x)^(m + 1)/(f*(m + 1)))*Simp[Sqrt[1 + c^2 
*x^2]/Sqrt[d + e*x^2]]*(a + b*ArcSinh[c*x])*Hypergeometric2F1[1/2, (1 + m)/ 
2, (3 + m)/2, (-c^2)*x^2], x] - Simp[b*c*((f*x)^(m + 2)/(f^2*(m + 1)*(m + 2 
)))*Simp[Sqrt[1 + c^2*x^2]/Sqrt[d + e*x^2]]*HypergeometricPFQ[{1, 1 + m/2, 
1 + m/2}, {3/2 + m/2, 2 + m/2}, (-c^2)*x^2], x] /; FreeQ[{a, b, c, d, e, f, 
 m}, x] && EqQ[e, c^2*d] &&  !IntegerQ[m]
 

Maple [F]

\[\int x^{m} \operatorname {arcsinh}\left (x a \right )^{2}d x\]

Input:

int(x^m*arcsinh(x*a)^2,x)
 

Output:

int(x^m*arcsinh(x*a)^2,x)
 

Fricas [F]

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

integrate(x^m*arcsinh(a*x)^2,x, algorithm="fricas")
 

Output:

integral(x^m*arcsinh(a*x)^2, x)
 

Sympy [F]

\[ \int x^m \text {arcsinh}(a x)^2 \, dx=\int x^{m} \operatorname {asinh}^{2}{\left (a x \right )}\, dx \] Input:

integrate(x**m*asinh(a*x)**2,x)
 

Output:

Integral(x**m*asinh(a*x)**2, x)
 

Maxima [F]

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

integrate(x^m*arcsinh(a*x)^2,x, algorithm="maxima")
 

Output:

x*x^m*log(a*x + sqrt(a^2*x^2 + 1))^2/(m + 1) - integrate(2*(sqrt(a^2*x^2 + 
 1)*a^2*x^2*x^m + (a^3*x^3 + a*x)*x^m)*log(a*x + sqrt(a^2*x^2 + 1))/(a^3*( 
m + 1)*x^3 + a*(m + 1)*x + (a^2*(m + 1)*x^2 + m + 1)*sqrt(a^2*x^2 + 1)), x 
)
 

Giac [F]

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

integrate(x^m*arcsinh(a*x)^2,x, algorithm="giac")
 

Output:

integrate(x^m*arcsinh(a*x)^2, x)
 

Mupad [F(-1)]

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

int(x^m*asinh(a*x)^2,x)
 

Output:

int(x^m*asinh(a*x)^2, x)
 

Reduce [F]

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

int(x^m*asinh(a*x)^2,x)
 

Output:

int(x**m*asinh(a*x)**2,x)