\(\int \frac {\sqrt {1+c^2 x^2}}{(a+b \text {arcsinh}(c x))^2} \, dx\) [399]

Optimal result

Integrand size = 24, antiderivative size = 85 \[ \int \frac {\sqrt {1+c^2 x^2}}{(a+b \text {arcsinh}(c x))^2} \, dx=-\frac {1+c^2 x^2}{b c (a+b \text {arcsinh}(c x))}-\frac {\text {Chi}\left (\frac {2 (a+b \text {arcsinh}(c x))}{b}\right ) \sinh \left (\frac {2 a}{b}\right )}{b^2 c}+\frac {\cosh \left (\frac {2 a}{b}\right ) \text {Shi}\left (\frac {2 (a+b \text {arcsinh}(c x))}{b}\right )}{b^2 c} \] Output:

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

Mathematica [A] (verified)

Time = 0.15 (sec) , antiderivative size = 73, normalized size of antiderivative = 0.86 \[ \int \frac {\sqrt {1+c^2 x^2}}{(a+b \text {arcsinh}(c x))^2} \, dx=\frac {-\frac {b+b c^2 x^2}{a+b \text {arcsinh}(c x)}-\text {Chi}\left (2 \left (\frac {a}{b}+\text {arcsinh}(c x)\right )\right ) \sinh \left (\frac {2 a}{b}\right )+\cosh \left (\frac {2 a}{b}\right ) \text {Shi}\left (2 \left (\frac {a}{b}+\text {arcsinh}(c x)\right )\right )}{b^2 c} \] Input:

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

Output:

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

Rubi [C] (verified)

Result contains complex when optimal does not.

Time = 0.69 (sec) , antiderivative size = 89, normalized size of antiderivative = 1.05, number of steps used = 14, number of rules used = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.542, Rules used = {6205, 6195, 25, 5971, 27, 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.

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

\(\Big \downarrow \) 6205

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

\(\Big \downarrow \) 6195

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

\(\Big \downarrow \) 25

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

\(\Big \downarrow \) 5971

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 26

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

\(\Big \downarrow \) 3784

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

\(\Big \downarrow \) 26

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 26

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

\(\Big \downarrow \) 3779

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

\(\Big \downarrow \) 3782

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

Input:

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

Output:

-((1 + c^2*x^2)/(b*c*(a + b*ArcSinh[c*x]))) + (I*(I*CoshIntegral[(2*(a + b 
*ArcSinh[c*x]))/b]*Sinh[(2*a)/b] - I*Cosh[(2*a)/b]*SinhIntegral[(2*(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 27

Int[(a_)*(Fx_), x_Symbol] :> Simp[a   Int[Fx, x], x] /; FreeQ[a, x] &&  !Ma 
tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
 

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 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]
 

rule 6205

Int[((a_.) + ArcSinh[(c_.)*(x_)]*(b_.))^(n_)*((d_) + (e_.)*(x_)^2)^(p_.), x 
_Symbol] :> Simp[Simp[Sqrt[1 + c^2*x^2]*(d + e*x^2)^p]*((a + b*ArcSinh[c*x] 
)^(n + 1)/(b*c*(n + 1))), x] - Simp[c*((2*p + 1)/(b*(n + 1)))*Simp[(d + e*x 
^2)^p/(1 + c^2*x^2)^p]   Int[x*(1 + c^2*x^2)^(p - 1/2)*(a + b*ArcSinh[c*x]) 
^(n + 1), x], x] /; FreeQ[{a, b, c, d, e, p}, x] && EqQ[e, c^2*d] && LtQ[n, 
 -1]
 

Maple [A] (verified)

Time = 1.84 (sec) , antiderivative size = 138, normalized size of antiderivative = 1.62

method	result	size
default	\(-\frac {2 b \,c^{2} x^{2}+{\mathrm e}^{-\frac {2 a}{b}} \operatorname {expIntegral}_{1}\left (-2 \,\operatorname {arcsinh}\left (x c \right )-\frac {2 a}{b}\right ) b \,\operatorname {arcsinh}\left (x c \right )-{\mathrm e}^{\frac {2 a}{b}} \operatorname {expIntegral}_{1}\left (2 \,\operatorname {arcsinh}\left (x c \right )+\frac {2 a}{b}\right ) b \,\operatorname {arcsinh}\left (x c \right )+{\mathrm e}^{-\frac {2 a}{b}} \operatorname {expIntegral}_{1}\left (-2 \,\operatorname {arcsinh}\left (x c \right )-\frac {2 a}{b}\right ) a -{\mathrm e}^{\frac {2 a}{b}} \operatorname {expIntegral}_{1}\left (2 \,\operatorname {arcsinh}\left (x c \right )+\frac {2 a}{b}\right ) a +2 b}{2 c \,b^{2} \left (a +b \,\operatorname {arcsinh}\left (x c \right )\right )}\)	\(138\)

Input:

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

Output:

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

Fricas [F]

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

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

Output:

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

Sympy [F]

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

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

Output:

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

Maxima [F]

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

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

Output:

-((c^2*x^2 + 1)^2 + (c^3*x^3 + c*x)*sqrt(c^2*x^2 + 1))/(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(((2*c^2*x^2 - 1)*(c^ 
2*x^2 + 1)^(3/2) + 2*(2*c^3*x^3 + c*x)*(c^2*x^2 + 1) + (2*c^4*x^4 + 3*c^2* 
x^2 + 1)*sqrt(c^2*x^2 + 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]

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

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

Output:

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

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

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

Output:

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