\(\int \frac {1}{(a+b \text {arcsinh}(c x))^2} \, dx\) [27]

Optimal result

Integrand size = 10, antiderivative size = 85 \[ \int \frac {1}{(a+b \text {arcsinh}(c x))^2} \, dx=-\frac {\sqrt {1+c^2 x^2}}{b c (a+b \text {arcsinh}(c x))}-\frac {\text {Chi}\left (\frac {a+b \text {arcsinh}(c x)}{b}\right ) \sinh \left (\frac {a}{b}\right )}{b^2 c}+\frac {\cosh \left (\frac {a}{b}\right ) \text {Shi}\left (\frac {a+b \text {arcsinh}(c x)}{b}\right )}{b^2 c} \] Output:

-(c^2*x^2+1)^(1/2)/b/c/(a+b*arcsinh(c*x))-Chi((a+b*arcsinh(c*x))/b)*sinh(a 
/b)/b^2/c+cosh(a/b)*Shi((a+b*arcsinh(c*x))/b)/b^2/c
 

Mathematica [A] (verified)

Time = 0.14 (sec) , antiderivative size = 71, normalized size of antiderivative = 0.84 \[ \int \frac {1}{(a+b \text {arcsinh}(c x))^2} \, dx=\frac {-\frac {b \sqrt {1+c^2 x^2}}{a+b \text {arcsinh}(c x)}-\text {Chi}\left (\frac {a}{b}+\text {arcsinh}(c x)\right ) \sinh \left (\frac {a}{b}\right )+\cosh \left (\frac {a}{b}\right ) \text {Shi}\left (\frac {a}{b}+\text {arcsinh}(c x)\right )}{b^2 c} \] Input:

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

Output:

(-((b*Sqrt[1 + c^2*x^2])/(a + b*ArcSinh[c*x])) - CoshIntegral[a/b + ArcSin 
h[c*x]]*Sinh[a/b] + Cosh[a/b]*SinhIntegral[a/b + ArcSinh[c*x]])/(b^2*c)
 

Rubi [C] (verified)

Result contains complex when optimal does not.

Time = 0.65 (sec) , antiderivative size = 89, normalized size of antiderivative = 1.05, number of steps used = 12, number of rules used = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 1.100, Rules used = {6188, 6234, 25, 3042, 26, 3784, 26, 3042, 26, 3779, 3782}

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.

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

\(\Big \downarrow \) 6188

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

\(\Big \downarrow \) 6234

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

\(\Big \downarrow \) 25

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 26

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

\(\Big \downarrow \) 3784

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

\(\Big \downarrow \) 26

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 26

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

\(\Big \downarrow \) 3779

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

\(\Big \downarrow \) 3782

\(\displaystyle -\frac {\sqrt {c^2 x^2+1}}{b c (a+b \text {arcsinh}(c x))}+\frac {i \left (i \sinh \left (\frac {a}{b}\right ) \text {Chi}\left (\frac {a+b \text {arcsinh}(c x)}{b}\right )-i \cosh \left (\frac {a}{b}\right ) \text {Shi}\left (\frac {a+b \text {arcsinh}(c x)}{b}\right )\right )}{b^2 c}\)

Input:

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

Output:

-(Sqrt[1 + c^2*x^2]/(b*c*(a + b*ArcSinh[c*x]))) + (I*(I*CoshIntegral[(a + 
b*ArcSinh[c*x])/b]*Sinh[a/b] - I*Cosh[a/b]*SinhIntegral[(a + b*ArcSinh[c*x 
])/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 3042

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

rule 3779

Int[sin[(e_.) + (Complex[0, fz_])*(f_.)*(x_)]/((c_.) + (d_.)*(x_)), x_Symbo 
l] :> Simp[I*(SinhIntegral[c*f*(fz/d) + f*fz*x]/d), x] /; FreeQ[{c, d, e, f 
, fz}, x] && EqQ[d*e - c*f*fz*I, 0]
 

rule 3782

Int[sin[(e_.) + (Complex[0, fz_])*(f_.)*(x_)]/((c_.) + (d_.)*(x_)), x_Symbo 
l] :> Simp[CoshIntegral[c*f*(fz/d) + f*fz*x]/d, x] /; FreeQ[{c, d, e, f, fz 
}, x] && EqQ[d*(e - Pi/2) - c*f*fz*I, 0]
 

rule 3784

Int[sin[(e_.) + (f_.)*(x_)]/((c_.) + (d_.)*(x_)), x_Symbol] :> Simp[Cos[(d* 
e - c*f)/d]   Int[Sin[c*(f/d) + f*x]/(c + d*x), x], x] + Simp[Sin[(d*e - c* 
f)/d]   Int[Cos[c*(f/d) + f*x]/(c + d*x), x], x] /; FreeQ[{c, d, e, f}, x] 
&& NeQ[d*e - c*f, 0]
 

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 [A] (verified)

Time = 1.12 (sec) , antiderivative size = 118, normalized size of antiderivative = 1.39

method	result	size
derivativedivides	\(\frac {\frac {x c -\sqrt {c^{2} x^{2}+1}}{2 b \left (a +b \,\operatorname {arcsinh}\left (x c \right )\right )}+\frac {{\mathrm e}^{\frac {a}{b}} \operatorname {expIntegral}_{1}\left (\operatorname {arcsinh}\left (x c \right )+\frac {a}{b}\right )}{2 b^{2}}-\frac {x c +\sqrt {c^{2} x^{2}+1}}{2 b \left (a +b \,\operatorname {arcsinh}\left (x c \right )\right )}-\frac {{\mathrm e}^{-\frac {a}{b}} \operatorname {expIntegral}_{1}\left (-\operatorname {arcsinh}\left (x c \right )-\frac {a}{b}\right )}{2 b^{2}}}{c}\)	\(118\)
default	\(\frac {\frac {x c -\sqrt {c^{2} x^{2}+1}}{2 b \left (a +b \,\operatorname {arcsinh}\left (x c \right )\right )}+\frac {{\mathrm e}^{\frac {a}{b}} \operatorname {expIntegral}_{1}\left (\operatorname {arcsinh}\left (x c \right )+\frac {a}{b}\right )}{2 b^{2}}-\frac {x c +\sqrt {c^{2} x^{2}+1}}{2 b \left (a +b \,\operatorname {arcsinh}\left (x c \right )\right )}-\frac {{\mathrm e}^{-\frac {a}{b}} \operatorname {expIntegral}_{1}\left (-\operatorname {arcsinh}\left (x c \right )-\frac {a}{b}\right )}{2 b^{2}}}{c}\)	\(118\)

Input:

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

Output:

1/c*(1/2*(x*c-(c^2*x^2+1)^(1/2))/b/(a+b*arcsinh(x*c))+1/2/b^2*exp(a/b)*Ei( 
1,arcsinh(x*c)+a/b)-1/2/b*(x*c+(c^2*x^2+1)^(1/2))/(a+b*arcsinh(x*c))-1/2/b 
^2*exp(-a/b)*Ei(1,-arcsinh(x*c)-a/b))
 

Fricas [F]

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

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

Output:

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

Sympy [F]

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

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

Output:

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

Maxima [F]

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

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

Output:

-(c^3*x^3 + c*x + (c^2*x^2 + 1)^(3/2))/(a*b*c^3*x^2 + sqrt(c^2*x^2 + 1)*a* 
b*c^2*x + a*b*c + (b^2*c^3*x^2 + sqrt(c^2*x^2 + 1)*b^2*c^2*x + b^2*c)*log( 
c*x + sqrt(c^2*x^2 + 1))) + integrate((c^4*x^4 + 2*c^2*x^2 + (c^2*x^2 + 1) 
*(c^2*x^2 - 1) + (2*c^3*x^3 + c*x)*sqrt(c^2*x^2 + 1) + 1)/(a*b*c^4*x^4 + ( 
c^2*x^2 + 1)*a*b*c^2*x^2 + 2*a*b*c^2*x^2 + a*b + (b^2*c^4*x^4 + (c^2*x^2 + 
 1)*b^2*c^2*x^2 + 2*b^2*c^2*x^2 + b^2 + 2*(b^2*c^3*x^3 + b^2*c*x)*sqrt(c^2 
*x^2 + 1))*log(c*x + sqrt(c^2*x^2 + 1)) + 2*(a*b*c^3*x^3 + a*b*c*x)*sqrt(c 
^2*x^2 + 1)), x)
 

Giac [F]

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

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

Output:

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

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

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

Output:

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