\(\int \frac {1}{(a+b \text {arcsinh}(c x))^{3/2}} \, dx\) [169]

Optimal result

Integrand size = 12, antiderivative size = 116 \[ \int \frac {1}{(a+b \text {arcsinh}(c x))^{3/2}} \, dx=-\frac {2 \sqrt {1+c^2 x^2}}{b c \sqrt {a+b \text {arcsinh}(c x)}}-\frac {e^{a/b} \sqrt {\pi } \text {erf}\left (\frac {\sqrt {a+b \text {arcsinh}(c x)}}{\sqrt {b}}\right )}{b^{3/2} c}+\frac {e^{-\frac {a}{b}} \sqrt {\pi } \text {erfi}\left (\frac {\sqrt {a+b \text {arcsinh}(c x)}}{\sqrt {b}}\right )}{b^{3/2} c} \] Output:

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

Mathematica [A] (verified)

Time = 0.17 (sec) , antiderivative size = 137, normalized size of antiderivative = 1.18 \[ \int \frac {1}{(a+b \text {arcsinh}(c x))^{3/2}} \, dx=\frac {e^{-\frac {a+b \text {arcsinh}(c x)}{b}} \left (-e^{a/b} \left (1+e^{2 \text {arcsinh}(c x)}\right )+e^{\frac {2 a}{b}+\text {arcsinh}(c x)} \sqrt {\frac {a}{b}+\text {arcsinh}(c x)} \Gamma \left (\frac {1}{2},\frac {a}{b}+\text {arcsinh}(c x)\right )+e^{\text {arcsinh}(c x)} \sqrt {-\frac {a+b \text {arcsinh}(c x)}{b}} \Gamma \left (\frac {1}{2},-\frac {a+b \text {arcsinh}(c x)}{b}\right )\right )}{b c \sqrt {a+b \text {arcsinh}(c x)}} \] Input:

Integrate[(a + b*ArcSinh[c*x])^(-3/2),x]
 

Output:

(-(E^(a/b)*(1 + E^(2*ArcSinh[c*x]))) + E^((2*a)/b + ArcSinh[c*x])*Sqrt[a/b 
 + ArcSinh[c*x]]*Gamma[1/2, a/b + ArcSinh[c*x]] + E^ArcSinh[c*x]*Sqrt[-((a 
 + b*ArcSinh[c*x])/b)]*Gamma[1/2, -((a + b*ArcSinh[c*x])/b)])/(b*c*E^((a + 
 b*ArcSinh[c*x])/b)*Sqrt[a + b*ArcSinh[c*x]])
 

Rubi [C] (verified)

Result contains complex when optimal does not.

Time = 0.64 (sec) , antiderivative size = 130, normalized size of antiderivative = 1.12, number of steps used = 10, number of rules used = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.750, Rules used = {6188, 6234, 25, 3042, 26, 3789, 2611, 2633, 2634}

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.

\(\displaystyle \int \frac {1}{(a+b \text {arcsinh}(c x))^{3/2}} \, dx\)

\(\Big \downarrow \) 6188

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

\(\Big \downarrow \) 6234

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

\(\Big \downarrow \) 25

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 26

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

\(\Big \downarrow \) 3789

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

\(\Big \downarrow \) 2611

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

\(\Big \downarrow \) 2633

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

\(\Big \downarrow \) 2634

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

Input:

Int[(a + b*ArcSinh[c*x])^(-3/2),x]
 

Output:

(-2*Sqrt[1 + c^2*x^2])/(b*c*Sqrt[a + b*ArcSinh[c*x]]) + ((2*I)*((I/2)*Sqrt 
[b]*E^(a/b)*Sqrt[Pi]*Erf[Sqrt[a + b*ArcSinh[c*x]]/Sqrt[b]] - ((I/2)*Sqrt[b 
]*Sqrt[Pi]*Erfi[Sqrt[a + b*ArcSinh[c*x]]/Sqrt[b]])/E^(a/b)))/(b^2*c)
 

Defintions of rubi rules used

rule 25

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

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 2611

Int[(F_)^((g_.)*((e_.) + (f_.)*(x_)))/Sqrt[(c_.) + (d_.)*(x_)], x_Symbol] : 
> Simp[2/d   Subst[Int[F^(g*(e - c*(f/d)) + f*g*(x^2/d)), x], x, Sqrt[c + d 
*x]], x] /; FreeQ[{F, c, d, e, f, g}, x] &&  !TrueQ[$UseGamma]
 

rule 2633

Int[(F_)^((a_.) + (b_.)*((c_.) + (d_.)*(x_))^2), x_Symbol] :> Simp[F^a*Sqrt 
[Pi]*(Erfi[(c + d*x)*Rt[b*Log[F], 2]]/(2*d*Rt[b*Log[F], 2])), x] /; FreeQ[{ 
F, a, b, c, d}, x] && PosQ[b]
 

rule 2634

Int[(F_)^((a_.) + (b_.)*((c_.) + (d_.)*(x_))^2), x_Symbol] :> Simp[F^a*Sqrt 
[Pi]*(Erf[(c + d*x)*Rt[(-b)*Log[F], 2]]/(2*d*Rt[(-b)*Log[F], 2])), x] /; Fr 
eeQ[{F, a, b, c, d}, x] && NegQ[b]
 

rule 3042

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

rule 3789

Int[((c_.) + (d_.)*(x_))^(m_.)*sin[(e_.) + (f_.)*(x_)], x_Symbol] :> Simp[I 
/2   Int[(c + d*x)^m/E^(I*(e + f*x)), x], x] - Simp[I/2   Int[(c + d*x)^m*E 
^(I*(e + f*x)), x], x] /; FreeQ[{c, d, e, f, m}, x]
 

rule 6188

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

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

Input:

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

Output:

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

Fricas [F(-2)]

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

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

Output:

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

Sympy [F]

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

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

Output:

Integral((a + b*asinh(c*x))**(-3/2), x)
 

Maxima [F]

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

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

Output:

integrate((b*arcsinh(c*x) + a)^(-3/2), x)
 

Giac [F]

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

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

Output:

integrate((b*arcsinh(c*x) + a)^(-3/2), x)
 

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

\[ \int \frac {1}{(a+b \text {arcsinh}(c x))^{3/2}} \, dx =\text {Too large to display} \] Input:

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

Output:

(2*sqrt(c**2*x**2 + 1)*sqrt(asinh(c*x)*b + a)*asinh(c*x) - asinh(c*x)*int( 
(sqrt(asinh(c*x)*b + a)*asinh(c*x)*x**2)/(asinh(c*x)**2*b**2*c**2*x**2 + a 
sinh(c*x)**2*b**2 + 2*asinh(c*x)*a*b*c**2*x**2 + 2*asinh(c*x)*a*b + a**2*c 
**2*x**2 + a**2),x)*b**2*c**3 - asinh(c*x)*int((sqrt(asinh(c*x)*b + a)*asi 
nh(c*x))/(asinh(c*x)**2*b**2*c**2*x**2 + asinh(c*x)**2*b**2 + 2*asinh(c*x) 
*a*b*c**2*x**2 + 2*asinh(c*x)*a*b + a**2*c**2*x**2 + a**2),x)*b**2*c - 2*a 
sinh(c*x)*int((sqrt(c**2*x**2 + 1)*sqrt(asinh(c*x)*b + a)*asinh(c*x)*x)/(a 
sinh(c*x)**2*b**2*c**2*x**2 + asinh(c*x)**2*b**2 + 2*asinh(c*x)*a*b*c**2*x 
**2 + 2*asinh(c*x)*a*b + a**2*c**2*x**2 + a**2),x)*a*b*c**2 - 2*asinh(c*x) 
*int((sqrt(c**2*x**2 + 1)*sqrt(asinh(c*x)*b + a)*asinh(c*x)**2*x)/(asinh(c 
*x)**2*b**2*c**2*x**2 + asinh(c*x)**2*b**2 + 2*asinh(c*x)*a*b*c**2*x**2 + 
2*asinh(c*x)*a*b + a**2*c**2*x**2 + a**2),x)*b**2*c**2 - int((sqrt(asinh(c 
*x)*b + a)*asinh(c*x)*x**2)/(asinh(c*x)**2*b**2*c**2*x**2 + asinh(c*x)**2* 
b**2 + 2*asinh(c*x)*a*b*c**2*x**2 + 2*asinh(c*x)*a*b + a**2*c**2*x**2 + a* 
*2),x)*a*b*c**3 - int((sqrt(asinh(c*x)*b + a)*asinh(c*x))/(asinh(c*x)**2*b 
**2*c**2*x**2 + asinh(c*x)**2*b**2 + 2*asinh(c*x)*a*b*c**2*x**2 + 2*asinh( 
c*x)*a*b + a**2*c**2*x**2 + a**2),x)*a*b*c - 2*int((sqrt(c**2*x**2 + 1)*sq 
rt(asinh(c*x)*b + a)*asinh(c*x)*x)/(asinh(c*x)**2*b**2*c**2*x**2 + asinh(c 
*x)**2*b**2 + 2*asinh(c*x)*a*b*c**2*x**2 + 2*asinh(c*x)*a*b + a**2*c**2*x* 
*2 + a**2),x)*a**2*c**2 - 2*int((sqrt(c**2*x**2 + 1)*sqrt(asinh(c*x)*b ...