\(\int \frac {x^2}{\sqrt {a+b \text {arcsinh}(c x)}} \, dx\) [162]

Optimal result

Integrand size = 16, antiderivative size = 194 \[ \int \frac {x^2}{\sqrt {a+b \text {arcsinh}(c x)}} \, dx=-\frac {e^{a/b} \sqrt {\pi } \text {erf}\left (\frac {\sqrt {a+b \text {arcsinh}(c x)}}{\sqrt {b}}\right )}{8 \sqrt {b} c^3}+\frac {e^{\frac {3 a}{b}} \sqrt {\frac {\pi }{3}} \text {erf}\left (\frac {\sqrt {3} \sqrt {a+b \text {arcsinh}(c x)}}{\sqrt {b}}\right )}{8 \sqrt {b} c^3}-\frac {e^{-\frac {a}{b}} \sqrt {\pi } \text {erfi}\left (\frac {\sqrt {a+b \text {arcsinh}(c x)}}{\sqrt {b}}\right )}{8 \sqrt {b} c^3}+\frac {e^{-\frac {3 a}{b}} \sqrt {\frac {\pi }{3}} \text {erfi}\left (\frac {\sqrt {3} \sqrt {a+b \text {arcsinh}(c x)}}{\sqrt {b}}\right )}{8 \sqrt {b} c^3} \] Output:

-1/8*exp(a/b)*Pi^(1/2)*erf((a+b*arcsinh(c*x))^(1/2)/b^(1/2))/b^(1/2)/c^3+1 
/24*exp(3*a/b)*3^(1/2)*Pi^(1/2)*erf(3^(1/2)*(a+b*arcsinh(c*x))^(1/2)/b^(1/ 
2))/b^(1/2)/c^3-1/8*Pi^(1/2)*erfi((a+b*arcsinh(c*x))^(1/2)/b^(1/2))/b^(1/2 
)/c^3/exp(a/b)+1/24*3^(1/2)*Pi^(1/2)*erfi(3^(1/2)*(a+b*arcsinh(c*x))^(1/2) 
/b^(1/2))/b^(1/2)/c^3/exp(3*a/b)
                                                                                    
                                                                                    
 

Mathematica [A] (verified)

Time = 0.17 (sec) , antiderivative size = 196, normalized size of antiderivative = 1.01 \[ \int \frac {x^2}{\sqrt {a+b \text {arcsinh}(c x)}} \, dx=\frac {e^{-\frac {3 a}{b}} \left (3 e^{\frac {4 a}{b}} \sqrt {\frac {a}{b}+\text {arcsinh}(c x)} \Gamma \left (\frac {1}{2},\frac {a}{b}+\text {arcsinh}(c x)\right )+\sqrt {3} \sqrt {-\frac {a+b \text {arcsinh}(c x)}{b}} \Gamma \left (\frac {1}{2},-\frac {3 (a+b \text {arcsinh}(c x))}{b}\right )-3 e^{\frac {2 a}{b}} \sqrt {-\frac {a+b \text {arcsinh}(c x)}{b}} \Gamma \left (\frac {1}{2},-\frac {a+b \text {arcsinh}(c x)}{b}\right )-\sqrt {3} e^{\frac {6 a}{b}} \sqrt {\frac {a}{b}+\text {arcsinh}(c x)} \Gamma \left (\frac {1}{2},\frac {3 (a+b \text {arcsinh}(c x))}{b}\right )\right )}{24 c^3 \sqrt {a+b \text {arcsinh}(c x)}} \] Input:

Integrate[x^2/Sqrt[a + b*ArcSinh[c*x]],x]
 

Output:

(3*E^((4*a)/b)*Sqrt[a/b + ArcSinh[c*x]]*Gamma[1/2, a/b + ArcSinh[c*x]] + S 
qrt[3]*Sqrt[-((a + b*ArcSinh[c*x])/b)]*Gamma[1/2, (-3*(a + b*ArcSinh[c*x]) 
)/b] - 3*E^((2*a)/b)*Sqrt[-((a + b*ArcSinh[c*x])/b)]*Gamma[1/2, -((a + b*A 
rcSinh[c*x])/b)] - Sqrt[3]*E^((6*a)/b)*Sqrt[a/b + ArcSinh[c*x]]*Gamma[1/2, 
 (3*(a + b*ArcSinh[c*x]))/b])/(24*c^3*E^((3*a)/b)*Sqrt[a + b*ArcSinh[c*x]] 
)
 

Rubi [A] (verified)

Time = 0.55 (sec) , antiderivative size = 189, normalized size of antiderivative = 0.97, number of steps used = 4, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.188, Rules used = {6195, 5971, 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.

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

\(\Big \downarrow \) 6195

\(\displaystyle \frac {\int \frac {\cosh \left (\frac {a}{b}-\frac {a+b \text {arcsinh}(c x)}{b}\right ) \sinh ^2\left (\frac {a}{b}-\frac {a+b \text {arcsinh}(c x)}{b}\right )}{\sqrt {a+b \text {arcsinh}(c x)}}d(a+b \text {arcsinh}(c x))}{b c^3}\)

\(\Big \downarrow \) 5971

\(\displaystyle \frac {\int \left (\frac {\cosh \left (\frac {3 a}{b}-\frac {3 (a+b \text {arcsinh}(c x))}{b}\right )}{4 \sqrt {a+b \text {arcsinh}(c x)}}-\frac {\cosh \left (\frac {a}{b}-\frac {a+b \text {arcsinh}(c x)}{b}\right )}{4 \sqrt {a+b \text {arcsinh}(c x)}}\right )d(a+b \text {arcsinh}(c x))}{b c^3}\)

\(\Big \downarrow \) 2009

\(\displaystyle \frac {-\frac {1}{8} \sqrt {\pi } \sqrt {b} e^{a/b} \text {erf}\left (\frac {\sqrt {a+b \text {arcsinh}(c x)}}{\sqrt {b}}\right )+\frac {1}{8} \sqrt {\frac {\pi }{3}} \sqrt {b} e^{\frac {3 a}{b}} \text {erf}\left (\frac {\sqrt {3} \sqrt {a+b \text {arcsinh}(c x)}}{\sqrt {b}}\right )-\frac {1}{8} \sqrt {\pi } \sqrt {b} e^{-\frac {a}{b}} \text {erfi}\left (\frac {\sqrt {a+b \text {arcsinh}(c x)}}{\sqrt {b}}\right )+\frac {1}{8} \sqrt {\frac {\pi }{3}} \sqrt {b} e^{-\frac {3 a}{b}} \text {erfi}\left (\frac {\sqrt {3} \sqrt {a+b \text {arcsinh}(c x)}}{\sqrt {b}}\right )}{b c^3}\)

Input:

Int[x^2/Sqrt[a + b*ArcSinh[c*x]],x]
 

Output:

(-1/8*(Sqrt[b]*E^(a/b)*Sqrt[Pi]*Erf[Sqrt[a + b*ArcSinh[c*x]]/Sqrt[b]]) + ( 
Sqrt[b]*E^((3*a)/b)*Sqrt[Pi/3]*Erf[(Sqrt[3]*Sqrt[a + b*ArcSinh[c*x]])/Sqrt 
[b]])/8 - (Sqrt[b]*Sqrt[Pi]*Erfi[Sqrt[a + b*ArcSinh[c*x]]/Sqrt[b]])/(8*E^( 
a/b)) + (Sqrt[b]*Sqrt[Pi/3]*Erfi[(Sqrt[3]*Sqrt[a + b*ArcSinh[c*x]])/Sqrt[b 
]])/(8*E^((3*a)/b)))/(b*c^3)
 

Defintions of rubi rules used

rule 2009

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

rule 5971

Int[Cosh[(a_.) + (b_.)*(x_)]^(p_.)*((c_.) + (d_.)*(x_))^(m_.)*Sinh[(a_.) + 
(b_.)*(x_)]^(n_.), x_Symbol] :> Int[ExpandTrigReduce[(c + d*x)^m, Sinh[a + 
b*x]^n*Cosh[a + b*x]^p, x], x] /; FreeQ[{a, b, c, d, m}, x] && IGtQ[n, 0] & 
& IGtQ[p, 0]
 

rule 6195

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

Maple [F]

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

Input:

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

Output:

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

Fricas [F(-2)]

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

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

Output:

Exception raised: TypeError >>  Error detected within library code:   inte 
grate: implementation incomplete (constant residues)
 

Sympy [F]

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

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

Output:

Integral(x**2/sqrt(a + b*asinh(c*x)), x)
 

Maxima [F]

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

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

Output:

integrate(x^2/sqrt(b*arcsinh(c*x) + a), x)
 

Giac [F]

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

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

Output:

integrate(x^2/sqrt(b*arcsinh(c*x) + a), x)
 

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

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

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

Output:

int((sqrt(asinh(c*x)*b + a)*x**2)/(asinh(c*x)*b + a),x)