\(\int \frac {x \sqrt {1+c^2 x^2}}{(a+b \text {arcsinh}(c x))^2} \, dx\) [398]

Optimal result

Integrand size = 25, antiderivative size = 149 \[ \int \frac {x \sqrt {1+c^2 x^2}}{(a+b \text {arcsinh}(c x))^2} \, dx=-\frac {x \left (1+c^2 x^2\right )}{b c (a+b \text {arcsinh}(c x))}+\frac {\cosh \left (\frac {a}{b}\right ) \text {Chi}\left (\frac {a+b \text {arcsinh}(c x)}{b}\right )}{4 b^2 c^2}+\frac {3 \cosh \left (\frac {3 a}{b}\right ) \text {Chi}\left (\frac {3 (a+b \text {arcsinh}(c x))}{b}\right )}{4 b^2 c^2}-\frac {\sinh \left (\frac {a}{b}\right ) \text {Shi}\left (\frac {a+b \text {arcsinh}(c x)}{b}\right )}{4 b^2 c^2}-\frac {3 \sinh \left (\frac {3 a}{b}\right ) \text {Shi}\left (\frac {3 (a+b \text {arcsinh}(c x))}{b}\right )}{4 b^2 c^2} \] Output:

-x*(c^2*x^2+1)/b/c/(a+b*arcsinh(c*x))+1/4*cosh(a/b)*Chi((a+b*arcsinh(c*x)) 
/b)/b^2/c^2+3/4*cosh(3*a/b)*Chi(3*(a+b*arcsinh(c*x))/b)/b^2/c^2-1/4*sinh(a 
/b)*Shi((a+b*arcsinh(c*x))/b)/b^2/c^2-3/4*sinh(3*a/b)*Shi(3*(a+b*arcsinh(c 
*x))/b)/b^2/c^2
 

Mathematica [A] (verified)

Time = 0.30 (sec) , antiderivative size = 126, normalized size of antiderivative = 0.85 \[ \int \frac {x \sqrt {1+c^2 x^2}}{(a+b \text {arcsinh}(c x))^2} \, dx=-\frac {\frac {4 b c x}{a+b \text {arcsinh}(c x)}+\frac {4 b c^3 x^3}{a+b \text {arcsinh}(c x)}-\cosh \left (\frac {a}{b}\right ) \text {Chi}\left (\frac {a}{b}+\text {arcsinh}(c x)\right )-3 \cosh \left (\frac {3 a}{b}\right ) \text {Chi}\left (3 \left (\frac {a}{b}+\text {arcsinh}(c x)\right )\right )+\sinh \left (\frac {a}{b}\right ) \text {Shi}\left (\frac {a}{b}+\text {arcsinh}(c x)\right )+3 \sinh \left (\frac {3 a}{b}\right ) \text {Shi}\left (3 \left (\frac {a}{b}+\text {arcsinh}(c x)\right )\right )}{4 b^2 c^2} \] Input:

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

Output:

-1/4*((4*b*c*x)/(a + b*ArcSinh[c*x]) + (4*b*c^3*x^3)/(a + b*ArcSinh[c*x]) 
- Cosh[a/b]*CoshIntegral[a/b + ArcSinh[c*x]] - 3*Cosh[(3*a)/b]*CoshIntegra 
l[3*(a/b + ArcSinh[c*x])] + Sinh[a/b]*SinhIntegral[a/b + ArcSinh[c*x]] + 3 
*Sinh[(3*a)/b]*SinhIntegral[3*(a/b + ArcSinh[c*x])])/(b^2*c^2)
 

Rubi [A] (verified)

Time = 1.22 (sec) , antiderivative size = 183, normalized size of antiderivative = 1.23, number of steps used = 13, number of rules used = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.480, Rules used = {6229, 6189, 3042, 3784, 26, 3042, 26, 3779, 3782, 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.

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

\(\Big \downarrow \) 6229

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

\(\Big \downarrow \) 6189

\(\displaystyle \frac {\int \frac {\cosh \left (\frac {a}{b}-\frac {a+b \text {arcsinh}(c x)}{b}\right )}{a+b \text {arcsinh}(c x)}d(a+b \text {arcsinh}(c x))}{b^2 c^2}+\frac {3 c \int \frac {x^2}{a+b \text {arcsinh}(c x)}dx}{b}-\frac {x \left (c^2 x^2+1\right )}{b c (a+b \text {arcsinh}(c x))}\)

\(\Big \downarrow \) 3042

\(\displaystyle \frac {\int \frac {\sin \left (\frac {i a}{b}-\frac {i (a+b \text {arcsinh}(c x))}{b}+\frac {\pi }{2}\right )}{a+b \text {arcsinh}(c x)}d(a+b \text {arcsinh}(c x))}{b^2 c^2}+\frac {3 c \int \frac {x^2}{a+b \text {arcsinh}(c x)}dx}{b}-\frac {x \left (c^2 x^2+1\right )}{b c (a+b \text {arcsinh}(c x))}\)

\(\Big \downarrow \) 3784

\(\displaystyle \frac {\cosh \left (\frac {a}{b}\right ) \int \frac {\cosh \left (\frac {a+b \text {arcsinh}(c x)}{b}\right )}{a+b \text {arcsinh}(c x)}d(a+b \text {arcsinh}(c x))-i \sinh \left (\frac {a}{b}\right ) \int -\frac {i \sinh \left (\frac {a+b \text {arcsinh}(c x)}{b}\right )}{a+b \text {arcsinh}(c x)}d(a+b \text {arcsinh}(c x))}{b^2 c^2}+\frac {3 c \int \frac {x^2}{a+b \text {arcsinh}(c x)}dx}{b}-\frac {x \left (c^2 x^2+1\right )}{b c (a+b \text {arcsinh}(c x))}\)

\(\Big \downarrow \) 26

\(\displaystyle \frac {\cosh \left (\frac {a}{b}\right ) \int \frac {\cosh \left (\frac {a+b \text {arcsinh}(c x)}{b}\right )}{a+b \text {arcsinh}(c x)}d(a+b \text {arcsinh}(c x))-\sinh \left (\frac {a}{b}\right ) \int \frac {\sinh \left (\frac {a+b \text {arcsinh}(c x)}{b}\right )}{a+b \text {arcsinh}(c x)}d(a+b \text {arcsinh}(c x))}{b^2 c^2}+\frac {3 c \int \frac {x^2}{a+b \text {arcsinh}(c x)}dx}{b}-\frac {x \left (c^2 x^2+1\right )}{b c (a+b \text {arcsinh}(c x))}\)

\(\Big \downarrow \) 3042

\(\displaystyle \frac {\cosh \left (\frac {a}{b}\right ) \int \frac {\sin \left (\frac {i (a+b \text {arcsinh}(c x))}{b}+\frac {\pi }{2}\right )}{a+b \text {arcsinh}(c x)}d(a+b \text {arcsinh}(c x))-\sinh \left (\frac {a}{b}\right ) \int -\frac {i \sin \left (\frac {i (a+b \text {arcsinh}(c x))}{b}\right )}{a+b \text {arcsinh}(c x)}d(a+b \text {arcsinh}(c x))}{b^2 c^2}+\frac {3 c \int \frac {x^2}{a+b \text {arcsinh}(c x)}dx}{b}-\frac {x \left (c^2 x^2+1\right )}{b c (a+b \text {arcsinh}(c x))}\)

\(\Big \downarrow \) 26

\(\displaystyle \frac {i \sinh \left (\frac {a}{b}\right ) \int \frac {\sin \left (\frac {i (a+b \text {arcsinh}(c x))}{b}\right )}{a+b \text {arcsinh}(c x)}d(a+b \text {arcsinh}(c x))+\cosh \left (\frac {a}{b}\right ) \int \frac {\sin \left (\frac {i (a+b \text {arcsinh}(c x))}{b}+\frac {\pi }{2}\right )}{a+b \text {arcsinh}(c x)}d(a+b \text {arcsinh}(c x))}{b^2 c^2}+\frac {3 c \int \frac {x^2}{a+b \text {arcsinh}(c x)}dx}{b}-\frac {x \left (c^2 x^2+1\right )}{b c (a+b \text {arcsinh}(c x))}\)

\(\Big \downarrow \) 3779

\(\displaystyle \frac {-\sinh \left (\frac {a}{b}\right ) \text {Shi}\left (\frac {a+b \text {arcsinh}(c x)}{b}\right )+\cosh \left (\frac {a}{b}\right ) \int \frac {\sin \left (\frac {i (a+b \text {arcsinh}(c x))}{b}+\frac {\pi }{2}\right )}{a+b \text {arcsinh}(c x)}d(a+b \text {arcsinh}(c x))}{b^2 c^2}+\frac {3 c \int \frac {x^2}{a+b \text {arcsinh}(c x)}dx}{b}-\frac {x \left (c^2 x^2+1\right )}{b c (a+b \text {arcsinh}(c x))}\)

\(\Big \downarrow \) 3782

\(\displaystyle \frac {3 c \int \frac {x^2}{a+b \text {arcsinh}(c x)}dx}{b}+\frac {\cosh \left (\frac {a}{b}\right ) \text {Chi}\left (\frac {a+b \text {arcsinh}(c x)}{b}\right )-\sinh \left (\frac {a}{b}\right ) \text {Shi}\left (\frac {a+b \text {arcsinh}(c x)}{b}\right )}{b^2 c^2}-\frac {x \left (c^2 x^2+1\right )}{b c (a+b \text {arcsinh}(c x))}\)

\(\Big \downarrow \) 6195

\(\displaystyle \frac {3 \int \frac {\cosh \left (\frac {a}{b}-\frac {a+b \text {arcsinh}(c x)}{b}\right ) \sinh ^2\left (\frac {a}{b}-\frac {a+b \text {arcsinh}(c x)}{b}\right )}{a+b \text {arcsinh}(c x)}d(a+b \text {arcsinh}(c x))}{b^2 c^2}+\frac {\cosh \left (\frac {a}{b}\right ) \text {Chi}\left (\frac {a+b \text {arcsinh}(c x)}{b}\right )-\sinh \left (\frac {a}{b}\right ) \text {Shi}\left (\frac {a+b \text {arcsinh}(c x)}{b}\right )}{b^2 c^2}-\frac {x \left (c^2 x^2+1\right )}{b c (a+b \text {arcsinh}(c x))}\)

\(\Big \downarrow \) 5971

\(\displaystyle \frac {3 \int \left (\frac {\cosh \left (\frac {3 a}{b}-\frac {3 (a+b \text {arcsinh}(c x))}{b}\right )}{4 (a+b \text {arcsinh}(c x))}-\frac {\cosh \left (\frac {a}{b}-\frac {a+b \text {arcsinh}(c x)}{b}\right )}{4 (a+b \text {arcsinh}(c x))}\right )d(a+b \text {arcsinh}(c x))}{b^2 c^2}+\frac {\cosh \left (\frac {a}{b}\right ) \text {Chi}\left (\frac {a+b \text {arcsinh}(c x)}{b}\right )-\sinh \left (\frac {a}{b}\right ) \text {Shi}\left (\frac {a+b \text {arcsinh}(c x)}{b}\right )}{b^2 c^2}-\frac {x \left (c^2 x^2+1\right )}{b c (a+b \text {arcsinh}(c x))}\)

\(\Big \downarrow \) 2009

\(\displaystyle \frac {\cosh \left (\frac {a}{b}\right ) \text {Chi}\left (\frac {a+b \text {arcsinh}(c x)}{b}\right )-\sinh \left (\frac {a}{b}\right ) \text {Shi}\left (\frac {a+b \text {arcsinh}(c x)}{b}\right )}{b^2 c^2}+\frac {3 \left (-\frac {1}{4} \cosh \left (\frac {a}{b}\right ) \text {Chi}\left (\frac {a+b \text {arcsinh}(c x)}{b}\right )+\frac {1}{4} \cosh \left (\frac {3 a}{b}\right ) \text {Chi}\left (\frac {3 (a+b \text {arcsinh}(c x))}{b}\right )+\frac {1}{4} \sinh \left (\frac {a}{b}\right ) \text {Shi}\left (\frac {a+b \text {arcsinh}(c x)}{b}\right )-\frac {1}{4} \sinh \left (\frac {3 a}{b}\right ) \text {Shi}\left (\frac {3 (a+b \text {arcsinh}(c x))}{b}\right )\right )}{b^2 c^2}-\frac {x \left (c^2 x^2+1\right )}{b c (a+b \text {arcsinh}(c x))}\)

Input:

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

Output:

-((x*(1 + c^2*x^2))/(b*c*(a + b*ArcSinh[c*x]))) + (Cosh[a/b]*CoshIntegral[ 
(a + b*ArcSinh[c*x])/b] - Sinh[a/b]*SinhIntegral[(a + b*ArcSinh[c*x])/b])/ 
(b^2*c^2) + (3*(-1/4*(Cosh[a/b]*CoshIntegral[(a + b*ArcSinh[c*x])/b]) + (C 
osh[(3*a)/b]*CoshIntegral[(3*(a + b*ArcSinh[c*x]))/b])/4 + (Sinh[a/b]*Sinh 
Integral[(a + b*ArcSinh[c*x])/b])/4 - (Sinh[(3*a)/b]*SinhIntegral[(3*(a + 
b*ArcSinh[c*x]))/b])/4))/(b^2*c^2)
 

Defintions of rubi rules used

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 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 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 6189

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

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 6229

Int[((a_.) + ArcSinh[(c_.)*(x_)]*(b_.))^(n_)*((f_.)*(x_))^(m_.)*((d_) + (e_ 
.)*(x_)^2)^(p_.), x_Symbol] :> Simp[(f*x)^m*Sqrt[1 + c^2*x^2]*(d + e*x^2)^p 
*((a + b*ArcSinh[c*x])^(n + 1)/(b*c*(n + 1))), x] + (-Simp[f*(m/(b*c*(n + 1 
)))*Simp[(d + e*x^2)^p/(1 + c^2*x^2)^p]   Int[(f*x)^(m - 1)*(1 + c^2*x^2)^( 
p - 1/2)*(a + b*ArcSinh[c*x])^(n + 1), x], x] - Simp[c*((m + 2*p + 1)/(b*f* 
(n + 1)))*Simp[(d + e*x^2)^p/(1 + c^2*x^2)^p]   Int[(f*x)^(m + 1)*(1 + c^2* 
x^2)^(p - 1/2)*(a + b*ArcSinh[c*x])^(n + 1), x], x]) /; FreeQ[{a, b, c, d, 
e, f}, x] && EqQ[e, c^2*d] && LtQ[n, -1] && IGtQ[2*p, 0] && NeQ[m + 2*p + 1 
, 0] && IGtQ[m, -3]
 

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(363\) vs. \(2(141)=282\).

Time = 1.88 (sec) , antiderivative size = 364, normalized size of antiderivative = 2.44

method	result	size
default	\(-\frac {-4 x^{2} c^{2} \sqrt {c^{2} x^{2}+1}+4 x^{3} c^{3}-\sqrt {c^{2} x^{2}+1}+3 x c}{8 c^{2} b \left (a +b \,\operatorname {arcsinh}\left (x c \right )\right )}-\frac {3 \,{\mathrm e}^{\frac {3 a}{b}} \operatorname {expIntegral}_{1}\left (3 \,\operatorname {arcsinh}\left (x c \right )+\frac {3 a}{b}\right )}{8 c^{2} b^{2}}-\frac {x c -\sqrt {c^{2} x^{2}+1}}{8 c^{2} b \left (a +b \,\operatorname {arcsinh}\left (x c \right )\right )}-\frac {{\mathrm e}^{\frac {a}{b}} \operatorname {expIntegral}_{1}\left (\operatorname {arcsinh}\left (x c \right )+\frac {a}{b}\right )}{8 c^{2} b^{2}}-\frac {\operatorname {arcsinh}\left (x c \right ) {\mathrm e}^{-\frac {a}{b}} \operatorname {expIntegral}_{1}\left (-\operatorname {arcsinh}\left (x c \right )-\frac {a}{b}\right ) b +{\mathrm e}^{-\frac {a}{b}} \operatorname {expIntegral}_{1}\left (-\operatorname {arcsinh}\left (x c \right )-\frac {a}{b}\right ) a +b x c +\sqrt {c^{2} x^{2}+1}\, b}{8 c^{2} b^{2} \left (a +b \,\operatorname {arcsinh}\left (x c \right )\right )}-\frac {4 b \,c^{3} x^{3}+4 \sqrt {c^{2} x^{2}+1}\, b \,c^{2} x^{2}+3 \,\operatorname {arcsinh}\left (x c \right ) {\mathrm e}^{-\frac {3 a}{b}} \operatorname {expIntegral}_{1}\left (-3 \,\operatorname {arcsinh}\left (x c \right )-\frac {3 a}{b}\right ) b +3 \,{\mathrm e}^{-\frac {3 a}{b}} \operatorname {expIntegral}_{1}\left (-3 \,\operatorname {arcsinh}\left (x c \right )-\frac {3 a}{b}\right ) a +3 b x c +\sqrt {c^{2} x^{2}+1}\, b}{8 c^{2} b^{2} \left (a +b \,\operatorname {arcsinh}\left (x c \right )\right )}\)	\(364\)

Input:

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

Output:

-1/8*(-4*x^2*c^2*(c^2*x^2+1)^(1/2)+4*x^3*c^3-(c^2*x^2+1)^(1/2)+3*x*c)/c^2/ 
b/(a+b*arcsinh(x*c))-3/8/c^2/b^2*exp(3*a/b)*Ei(1,3*arcsinh(x*c)+3*a/b)-1/8 
*(x*c-(c^2*x^2+1)^(1/2))/c^2/b/(a+b*arcsinh(x*c))-1/8/c^2/b^2*exp(a/b)*Ei( 
1,arcsinh(x*c)+a/b)-1/8/c^2/b^2*(arcsinh(x*c)*exp(-a/b)*Ei(1,-arcsinh(x*c) 
-a/b)*b+exp(-a/b)*Ei(1,-arcsinh(x*c)-a/b)*a+b*x*c+(c^2*x^2+1)^(1/2)*b)/(a+ 
b*arcsinh(x*c))-1/8/c^2/b^2*(4*b*c^3*x^3+4*(c^2*x^2+1)^(1/2)*b*c^2*x^2+3*a 
rcsinh(x*c)*exp(-3*a/b)*Ei(1,-3*arcsinh(x*c)-3*a/b)*b+3*exp(-3*a/b)*Ei(1,- 
3*arcsinh(x*c)-3*a/b)*a+3*b*x*c+(c^2*x^2+1)^(1/2)*b)/(a+b*arcsinh(x*c))
 

Fricas [F]

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

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

Output:

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

Sympy [F]

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

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

Output:

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

Maxima [F]

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

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

Output:

-((c^2*x^3 + x)*(c^2*x^2 + 1) + (c^3*x^4 + c*x^2)*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((3*(c^ 
2*x^2 + 1)^(3/2)*c^3*x^3 + (6*c^4*x^4 + 5*c^2*x^2 + 1)*(c^2*x^2 + 1) + (3* 
c^5*x^5 + 5*c^3*x^3 + 2*c*x)*sqrt(c^2*x^2 + 1))/(a*b*c^5*x^4 + (c^2*x^2 + 
1)*a*b*c^3*x^2 + 2*a*b*c^3*x^2 + a*b*c + (b^2*c^5*x^4 + (c^2*x^2 + 1)*b^2* 
c^3*x^2 + 2*b^2*c^3*x^2 + b^2*c + 2*(b^2*c^4*x^3 + b^2*c^2*x)*sqrt(c^2*x^2 
 + 1))*log(c*x + sqrt(c^2*x^2 + 1)) + 2*(a*b*c^4*x^3 + a*b*c^2*x)*sqrt(c^2 
*x^2 + 1)), x)
 

Giac [F]

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

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

Output:

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

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

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

Output:

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