\(\int x \sqrt {a+b \text {arcsinh}(c x)} \, dx\) [148]

Optimal result

Integrand size = 14, antiderivative size = 145 \[ \int x \sqrt {a+b \text {arcsinh}(c x)} \, dx=\frac {\sqrt {a+b \text {arcsinh}(c x)}}{4 c^2}+\frac {1}{2} x^2 \sqrt {a+b \text {arcsinh}(c x)}-\frac {\sqrt {b} e^{\frac {2 a}{b}} \sqrt {\frac {\pi }{2}} \text {erf}\left (\frac {\sqrt {2} \sqrt {a+b \text {arcsinh}(c x)}}{\sqrt {b}}\right )}{16 c^2}-\frac {\sqrt {b} e^{-\frac {2 a}{b}} \sqrt {\frac {\pi }{2}} \text {erfi}\left (\frac {\sqrt {2} \sqrt {a+b \text {arcsinh}(c x)}}{\sqrt {b}}\right )}{16 c^2} \] Output:

1/4*(a+b*arcsinh(c*x))^(1/2)/c^2+1/2*x^2*(a+b*arcsinh(c*x))^(1/2)-1/32*b^( 
1/2)*exp(2*a/b)*2^(1/2)*Pi^(1/2)*erf(2^(1/2)*(a+b*arcsinh(c*x))^(1/2)/b^(1 
/2))/c^2-1/32*b^(1/2)*2^(1/2)*Pi^(1/2)*erfi(2^(1/2)*(a+b*arcsinh(c*x))^(1/ 
2)/b^(1/2))/c^2/exp(2*a/b)
 

Mathematica [A] (verified)

Time = 0.06 (sec) , antiderivative size = 111, normalized size of antiderivative = 0.77 \[ \int x \sqrt {a+b \text {arcsinh}(c x)} \, dx=\frac {e^{-\frac {2 a}{b}} \left (-b \sqrt {-\frac {a+b \text {arcsinh}(c x)}{b}} \Gamma \left (\frac {3}{2},-\frac {2 (a+b \text {arcsinh}(c x))}{b}\right )+b e^{\frac {4 a}{b}} \sqrt {\frac {a}{b}+\text {arcsinh}(c x)} \Gamma \left (\frac {3}{2},\frac {2 (a+b \text {arcsinh}(c x))}{b}\right )\right )}{8 \sqrt {2} c^2 \sqrt {a+b \text {arcsinh}(c x)}} \] Input:

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

Output:

(-(b*Sqrt[-((a + b*ArcSinh[c*x])/b)]*Gamma[3/2, (-2*(a + b*ArcSinh[c*x]))/ 
b]) + b*E^((4*a)/b)*Sqrt[a/b + ArcSinh[c*x]]*Gamma[3/2, (2*(a + b*ArcSinh[ 
c*x]))/b])/(8*Sqrt[2]*c^2*E^((2*a)/b)*Sqrt[a + b*ArcSinh[c*x]])
 

Rubi [A] (verified)

Time = 0.97 (sec) , antiderivative size = 142, normalized size of antiderivative = 0.98, number of steps used = 7, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.429, Rules used = {6192, 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.

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

\(\Big \downarrow \) 6192

\(\displaystyle \frac {1}{2} x^2 \sqrt {a+b \text {arcsinh}(c x)}-\frac {1}{4} b c \int \frac {x^2}{\sqrt {c^2 x^2+1} \sqrt {a+b \text {arcsinh}(c x)}}dx\)

\(\Big \downarrow \) 6234

\(\displaystyle \frac {1}{2} x^2 \sqrt {a+b \text {arcsinh}(c x)}-\frac {\int \frac {\sinh ^2\left (\frac {a}{b}-\frac {a+b \text {arcsinh}(c x)}{b}\right )}{\sqrt {a+b \text {arcsinh}(c x)}}d(a+b \text {arcsinh}(c x))}{4 c^2}\)

\(\Big \downarrow \) 3042

\(\displaystyle \frac {1}{2} x^2 \sqrt {a+b \text {arcsinh}(c x)}-\frac {\int -\frac {\sin \left (\frac {i a}{b}-\frac {i (a+b \text {arcsinh}(c x))}{b}\right )^2}{\sqrt {a+b \text {arcsinh}(c x)}}d(a+b \text {arcsinh}(c x))}{4 c^2}\)

\(\Big \downarrow \) 25

\(\displaystyle \frac {1}{2} x^2 \sqrt {a+b \text {arcsinh}(c x)}+\frac {\int \frac {\sin \left (\frac {i a}{b}-\frac {i (a+b \text {arcsinh}(c x))}{b}\right )^2}{\sqrt {a+b \text {arcsinh}(c x)}}d(a+b \text {arcsinh}(c x))}{4 c^2}\)

\(\Big \downarrow \) 3793

\(\displaystyle \frac {\int \left (\frac {1}{2 \sqrt {a+b \text {arcsinh}(c x)}}-\frac {\cosh \left (\frac {2 a}{b}-\frac {2 (a+b \text {arcsinh}(c x))}{b}\right )}{2 \sqrt {a+b \text {arcsinh}(c x)}}\right )d(a+b \text {arcsinh}(c x))}{4 c^2}+\frac {1}{2} x^2 \sqrt {a+b \text {arcsinh}(c x)}\)

\(\Big \downarrow \) 2009

\(\displaystyle \frac {1}{2} x^2 \sqrt {a+b \text {arcsinh}(c x)}-\frac {\frac {1}{4} \sqrt {\frac {\pi }{2}} \sqrt {b} e^{\frac {2 a}{b}} \text {erf}\left (\frac {\sqrt {2} \sqrt {a+b \text {arcsinh}(c x)}}{\sqrt {b}}\right )+\frac {1}{4} \sqrt {\frac {\pi }{2}} \sqrt {b} e^{-\frac {2 a}{b}} \text {erfi}\left (\frac {\sqrt {2} \sqrt {a+b \text {arcsinh}(c x)}}{\sqrt {b}}\right )-\sqrt {a+b \text {arcsinh}(c x)}}{4 c^2}\)

Input:

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

Output:

(x^2*Sqrt[a + b*ArcSinh[c*x]])/2 - (-Sqrt[a + b*ArcSinh[c*x]] + (Sqrt[b]*E 
^((2*a)/b)*Sqrt[Pi/2]*Erf[(Sqrt[2]*Sqrt[a + b*ArcSinh[c*x]])/Sqrt[b]])/4 + 
 (Sqrt[b]*Sqrt[Pi/2]*Erfi[(Sqrt[2]*Sqrt[a + b*ArcSinh[c*x]])/Sqrt[b]])/(4* 
E^((2*a)/b)))/(4*c^2)
 

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 6192

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

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]

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

Input:

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

Output:

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

Fricas [F(-2)]

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

integrate(x*(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]

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

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

Output:

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

Maxima [F]

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

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

Output:

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

Giac [F]

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

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

Output:

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

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

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

Output:

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