\(\int \frac {(d+c^2 d x^2)^2}{(a+b \text {arcsinh}(c x))^2} \, dx\) [33]

Optimal result

Integrand size = 23, antiderivative size = 235 \[ \int \frac {\left (d+c^2 d x^2\right )^2}{(a+b \text {arcsinh}(c x))^2} \, dx=-\frac {d^2 \left (1+c^2 x^2\right )^{5/2}}{b c (a+b \text {arcsinh}(c x))}-\frac {5 d^2 \text {Chi}\left (\frac {a+b \text {arcsinh}(c x)}{b}\right ) \sinh \left (\frac {a}{b}\right )}{8 b^2 c}-\frac {15 d^2 \text {Chi}\left (\frac {3 (a+b \text {arcsinh}(c x))}{b}\right ) \sinh \left (\frac {3 a}{b}\right )}{16 b^2 c}-\frac {5 d^2 \text {Chi}\left (\frac {5 (a+b \text {arcsinh}(c x))}{b}\right ) \sinh \left (\frac {5 a}{b}\right )}{16 b^2 c}+\frac {5 d^2 \cosh \left (\frac {a}{b}\right ) \text {Shi}\left (\frac {a+b \text {arcsinh}(c x)}{b}\right )}{8 b^2 c}+\frac {15 d^2 \cosh \left (\frac {3 a}{b}\right ) \text {Shi}\left (\frac {3 (a+b \text {arcsinh}(c x))}{b}\right )}{16 b^2 c}+\frac {5 d^2 \cosh \left (\frac {5 a}{b}\right ) \text {Shi}\left (\frac {5 (a+b \text {arcsinh}(c x))}{b}\right )}{16 b^2 c} \] Output:

-d^2*(c^2*x^2+1)^(5/2)/b/c/(a+b*arcsinh(c*x))-5/8*d^2*Chi((a+b*arcsinh(c*x 
))/b)*sinh(a/b)/b^2/c-15/16*d^2*Chi(3*(a+b*arcsinh(c*x))/b)*sinh(3*a/b)/b^ 
2/c-5/16*d^2*Chi(5*(a+b*arcsinh(c*x))/b)*sinh(5*a/b)/b^2/c+5/8*d^2*cosh(a/ 
b)*Shi((a+b*arcsinh(c*x))/b)/b^2/c+15/16*d^2*cosh(3*a/b)*Shi(3*(a+b*arcsin 
h(c*x))/b)/b^2/c+5/16*d^2*cosh(5*a/b)*Shi(5*(a+b*arcsinh(c*x))/b)/b^2/c
 

Mathematica [A] (verified)

Time = 0.76 (sec) , antiderivative size = 230, normalized size of antiderivative = 0.98 \[ \int \frac {\left (d+c^2 d x^2\right )^2}{(a+b \text {arcsinh}(c x))^2} \, dx=\frac {d^2 \left (-\frac {16 b \sqrt {1+c^2 x^2}}{a+b \text {arcsinh}(c x)}-\frac {32 b c^2 x^2 \sqrt {1+c^2 x^2}}{a+b \text {arcsinh}(c x)}-\frac {16 b c^4 x^4 \sqrt {1+c^2 x^2}}{a+b \text {arcsinh}(c x)}-10 \text {Chi}\left (\frac {a}{b}+\text {arcsinh}(c x)\right ) \sinh \left (\frac {a}{b}\right )-15 \text {Chi}\left (3 \left (\frac {a}{b}+\text {arcsinh}(c x)\right )\right ) \sinh \left (\frac {3 a}{b}\right )-5 \text {Chi}\left (5 \left (\frac {a}{b}+\text {arcsinh}(c x)\right )\right ) \sinh \left (\frac {5 a}{b}\right )+10 \cosh \left (\frac {a}{b}\right ) \text {Shi}\left (\frac {a}{b}+\text {arcsinh}(c x)\right )+15 \cosh \left (\frac {3 a}{b}\right ) \text {Shi}\left (3 \left (\frac {a}{b}+\text {arcsinh}(c x)\right )\right )+5 \cosh \left (\frac {5 a}{b}\right ) \text {Shi}\left (5 \left (\frac {a}{b}+\text {arcsinh}(c x)\right )\right )\right )}{16 b^2 c} \] Input:

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

Output:

(d^2*((-16*b*Sqrt[1 + c^2*x^2])/(a + b*ArcSinh[c*x]) - (32*b*c^2*x^2*Sqrt[ 
1 + c^2*x^2])/(a + b*ArcSinh[c*x]) - (16*b*c^4*x^4*Sqrt[1 + c^2*x^2])/(a + 
 b*ArcSinh[c*x]) - 10*CoshIntegral[a/b + ArcSinh[c*x]]*Sinh[a/b] - 15*Cosh 
Integral[3*(a/b + ArcSinh[c*x])]*Sinh[(3*a)/b] - 5*CoshIntegral[5*(a/b + A 
rcSinh[c*x])]*Sinh[(5*a)/b] + 10*Cosh[a/b]*SinhIntegral[a/b + ArcSinh[c*x] 
] + 15*Cosh[(3*a)/b]*SinhIntegral[3*(a/b + ArcSinh[c*x])] + 5*Cosh[(5*a)/b 
]*SinhIntegral[5*(a/b + ArcSinh[c*x])]))/(16*b^2*c)
 

Rubi [A] (verified)

Time = 1.00 (sec) , antiderivative size = 193, normalized size of antiderivative = 0.82, number of steps used = 6, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.217, Rules used = {6205, 6234, 25, 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.

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

\(\Big \downarrow \) 6205

\(\displaystyle \frac {5 c d^2 \int \frac {x \left (c^2 x^2+1\right )^{3/2}}{a+b \text {arcsinh}(c x)}dx}{b}-\frac {d^2 \left (c^2 x^2+1\right )^{5/2}}{b c (a+b \text {arcsinh}(c x))}\)

\(\Big \downarrow \) 6234

\(\displaystyle \frac {5 d^2 \int -\frac {\cosh ^4\left (\frac {a}{b}-\frac {a+b \text {arcsinh}(c x)}{b}\right ) \sinh \left (\frac {a}{b}-\frac {a+b \text {arcsinh}(c x)}{b}\right )}{a+b \text {arcsinh}(c x)}d(a+b \text {arcsinh}(c x))}{b^2 c}-\frac {d^2 \left (c^2 x^2+1\right )^{5/2}}{b c (a+b \text {arcsinh}(c x))}\)

\(\Big \downarrow \) 25

\(\displaystyle -\frac {5 d^2 \int \frac {\cosh ^4\left (\frac {a}{b}-\frac {a+b \text {arcsinh}(c x)}{b}\right ) \sinh \left (\frac {a}{b}-\frac {a+b \text {arcsinh}(c x)}{b}\right )}{a+b \text {arcsinh}(c x)}d(a+b \text {arcsinh}(c x))}{b^2 c}-\frac {d^2 \left (c^2 x^2+1\right )^{5/2}}{b c (a+b \text {arcsinh}(c x))}\)

\(\Big \downarrow \) 5971

\(\displaystyle -\frac {5 d^2 \int \left (\frac {\sinh \left (\frac {5 a}{b}-\frac {5 (a+b \text {arcsinh}(c x))}{b}\right )}{16 (a+b \text {arcsinh}(c x))}+\frac {3 \sinh \left (\frac {3 a}{b}-\frac {3 (a+b \text {arcsinh}(c x))}{b}\right )}{16 (a+b \text {arcsinh}(c x))}+\frac {\sinh \left (\frac {a}{b}-\frac {a+b \text {arcsinh}(c x)}{b}\right )}{8 (a+b \text {arcsinh}(c x))}\right )d(a+b \text {arcsinh}(c x))}{b^2 c}-\frac {d^2 \left (c^2 x^2+1\right )^{5/2}}{b c (a+b \text {arcsinh}(c x))}\)

\(\Big \downarrow \) 2009

\(\displaystyle \frac {5 d^2 \left (-\frac {1}{8} \sinh \left (\frac {a}{b}\right ) \text {Chi}\left (\frac {a+b \text {arcsinh}(c x)}{b}\right )-\frac {3}{16} \sinh \left (\frac {3 a}{b}\right ) \text {Chi}\left (\frac {3 (a+b \text {arcsinh}(c x))}{b}\right )-\frac {1}{16} \sinh \left (\frac {5 a}{b}\right ) \text {Chi}\left (\frac {5 (a+b \text {arcsinh}(c x))}{b}\right )+\frac {1}{8} \cosh \left (\frac {a}{b}\right ) \text {Shi}\left (\frac {a+b \text {arcsinh}(c x)}{b}\right )+\frac {3}{16} \cosh \left (\frac {3 a}{b}\right ) \text {Shi}\left (\frac {3 (a+b \text {arcsinh}(c x))}{b}\right )+\frac {1}{16} \cosh \left (\frac {5 a}{b}\right ) \text {Shi}\left (\frac {5 (a+b \text {arcsinh}(c x))}{b}\right )\right )}{b^2 c}-\frac {d^2 \left (c^2 x^2+1\right )^{5/2}}{b c (a+b \text {arcsinh}(c x))}\)

Input:

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

Output:

-((d^2*(1 + c^2*x^2)^(5/2))/(b*c*(a + b*ArcSinh[c*x]))) + (5*d^2*(-1/8*(Co 
shIntegral[(a + b*ArcSinh[c*x])/b]*Sinh[a/b]) - (3*CoshIntegral[(3*(a + b* 
ArcSinh[c*x]))/b]*Sinh[(3*a)/b])/16 - (CoshIntegral[(5*(a + b*ArcSinh[c*x] 
))/b]*Sinh[(5*a)/b])/16 + (Cosh[a/b]*SinhIntegral[(a + b*ArcSinh[c*x])/b]) 
/8 + (3*Cosh[(3*a)/b]*SinhIntegral[(3*(a + b*ArcSinh[c*x]))/b])/16 + (Cosh 
[(5*a)/b]*SinhIntegral[(5*(a + b*ArcSinh[c*x]))/b])/16))/(b^2*c)
 

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 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 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]
 

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 [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(551\) vs. \(2(221)=442\).

Time = 3.07 (sec) , antiderivative size = 552, normalized size of antiderivative = 2.35

method	result	size
derivativedivides	\(\frac {\frac {\left (-16 x^{4} c^{4} \sqrt {c^{2} x^{2}+1}+16 x^{5} c^{5}-12 x^{2} c^{2} \sqrt {c^{2} x^{2}+1}+20 x^{3} c^{3}-\sqrt {c^{2} x^{2}+1}+5 x c \right ) d^{2}}{32 b \left (a +b \,\operatorname {arcsinh}\left (x c \right )\right )}+\frac {5 d^{2} {\mathrm e}^{\frac {5 a}{b}} \operatorname {expIntegral}_{1}\left (5 \,\operatorname {arcsinh}\left (x c \right )+\frac {5 a}{b}\right )}{32 b^{2}}+\frac {5 \left (-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 \right ) d^{2}}{32 b \left (a +b \,\operatorname {arcsinh}\left (x c \right )\right )}+\frac {15 d^{2} {\mathrm e}^{\frac {3 a}{b}} \operatorname {expIntegral}_{1}\left (3 \,\operatorname {arcsinh}\left (x c \right )+\frac {3 a}{b}\right )}{32 b^{2}}+\frac {5 \left (x c -\sqrt {c^{2} x^{2}+1}\right ) d^{2}}{16 b \left (a +b \,\operatorname {arcsinh}\left (x c \right )\right )}+\frac {5 d^{2} {\mathrm e}^{\frac {a}{b}} \operatorname {expIntegral}_{1}\left (\operatorname {arcsinh}\left (x c \right )+\frac {a}{b}\right )}{16 b^{2}}-\frac {5 d^{2} \left (x c +\sqrt {c^{2} x^{2}+1}\right )}{16 b \left (a +b \,\operatorname {arcsinh}\left (x c \right )\right )}-\frac {5 d^{2} {\mathrm e}^{-\frac {a}{b}} \operatorname {expIntegral}_{1}\left (-\operatorname {arcsinh}\left (x c \right )-\frac {a}{b}\right )}{16 b^{2}}-\frac {5 d^{2} \left (4 x^{3} c^{3}+3 x c +4 x^{2} c^{2} \sqrt {c^{2} x^{2}+1}+\sqrt {c^{2} x^{2}+1}\right )}{32 b \left (a +b \,\operatorname {arcsinh}\left (x c \right )\right )}-\frac {15 d^{2} {\mathrm e}^{-\frac {3 a}{b}} \operatorname {expIntegral}_{1}\left (-3 \,\operatorname {arcsinh}\left (x c \right )-\frac {3 a}{b}\right )}{32 b^{2}}-\frac {d^{2} \left (16 x^{5} c^{5}+20 x^{3} c^{3}+16 x^{4} c^{4} \sqrt {c^{2} x^{2}+1}+5 x c +12 x^{2} c^{2} \sqrt {c^{2} x^{2}+1}+\sqrt {c^{2} x^{2}+1}\right )}{32 b \left (a +b \,\operatorname {arcsinh}\left (x c \right )\right )}-\frac {5 d^{2} {\mathrm e}^{-\frac {5 a}{b}} \operatorname {expIntegral}_{1}\left (-5 \,\operatorname {arcsinh}\left (x c \right )-\frac {5 a}{b}\right )}{32 b^{2}}}{c}\)	\(552\)
default	\(\frac {\frac {\left (-16 x^{4} c^{4} \sqrt {c^{2} x^{2}+1}+16 x^{5} c^{5}-12 x^{2} c^{2} \sqrt {c^{2} x^{2}+1}+20 x^{3} c^{3}-\sqrt {c^{2} x^{2}+1}+5 x c \right ) d^{2}}{32 b \left (a +b \,\operatorname {arcsinh}\left (x c \right )\right )}+\frac {5 d^{2} {\mathrm e}^{\frac {5 a}{b}} \operatorname {expIntegral}_{1}\left (5 \,\operatorname {arcsinh}\left (x c \right )+\frac {5 a}{b}\right )}{32 b^{2}}+\frac {5 \left (-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 \right ) d^{2}}{32 b \left (a +b \,\operatorname {arcsinh}\left (x c \right )\right )}+\frac {15 d^{2} {\mathrm e}^{\frac {3 a}{b}} \operatorname {expIntegral}_{1}\left (3 \,\operatorname {arcsinh}\left (x c \right )+\frac {3 a}{b}\right )}{32 b^{2}}+\frac {5 \left (x c -\sqrt {c^{2} x^{2}+1}\right ) d^{2}}{16 b \left (a +b \,\operatorname {arcsinh}\left (x c \right )\right )}+\frac {5 d^{2} {\mathrm e}^{\frac {a}{b}} \operatorname {expIntegral}_{1}\left (\operatorname {arcsinh}\left (x c \right )+\frac {a}{b}\right )}{16 b^{2}}-\frac {5 d^{2} \left (x c +\sqrt {c^{2} x^{2}+1}\right )}{16 b \left (a +b \,\operatorname {arcsinh}\left (x c \right )\right )}-\frac {5 d^{2} {\mathrm e}^{-\frac {a}{b}} \operatorname {expIntegral}_{1}\left (-\operatorname {arcsinh}\left (x c \right )-\frac {a}{b}\right )}{16 b^{2}}-\frac {5 d^{2} \left (4 x^{3} c^{3}+3 x c +4 x^{2} c^{2} \sqrt {c^{2} x^{2}+1}+\sqrt {c^{2} x^{2}+1}\right )}{32 b \left (a +b \,\operatorname {arcsinh}\left (x c \right )\right )}-\frac {15 d^{2} {\mathrm e}^{-\frac {3 a}{b}} \operatorname {expIntegral}_{1}\left (-3 \,\operatorname {arcsinh}\left (x c \right )-\frac {3 a}{b}\right )}{32 b^{2}}-\frac {d^{2} \left (16 x^{5} c^{5}+20 x^{3} c^{3}+16 x^{4} c^{4} \sqrt {c^{2} x^{2}+1}+5 x c +12 x^{2} c^{2} \sqrt {c^{2} x^{2}+1}+\sqrt {c^{2} x^{2}+1}\right )}{32 b \left (a +b \,\operatorname {arcsinh}\left (x c \right )\right )}-\frac {5 d^{2} {\mathrm e}^{-\frac {5 a}{b}} \operatorname {expIntegral}_{1}\left (-5 \,\operatorname {arcsinh}\left (x c \right )-\frac {5 a}{b}\right )}{32 b^{2}}}{c}\)	\(552\)

Input:

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

Output:

1/c*(1/32*(-16*x^4*c^4*(c^2*x^2+1)^(1/2)+16*x^5*c^5-12*x^2*c^2*(c^2*x^2+1) 
^(1/2)+20*x^3*c^3-(c^2*x^2+1)^(1/2)+5*x*c)*d^2/b/(a+b*arcsinh(x*c))+5/32*d 
^2/b^2*exp(5*a/b)*Ei(1,5*arcsinh(x*c)+5*a/b)+5/32*(-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)*d^2/b/(a+b*arcsinh(x*c))+15/32*d^ 
2/b^2*exp(3*a/b)*Ei(1,3*arcsinh(x*c)+3*a/b)+5/16*(x*c-(c^2*x^2+1)^(1/2))*d 
^2/b/(a+b*arcsinh(x*c))+5/16*d^2/b^2*exp(a/b)*Ei(1,arcsinh(x*c)+a/b)-5/16/ 
b*d^2*(x*c+(c^2*x^2+1)^(1/2))/(a+b*arcsinh(x*c))-5/16/b^2*d^2*exp(-a/b)*Ei 
(1,-arcsinh(x*c)-a/b)-5/32/b*d^2*(4*x^3*c^3+3*x*c+4*x^2*c^2*(c^2*x^2+1)^(1 
/2)+(c^2*x^2+1)^(1/2))/(a+b*arcsinh(x*c))-15/32/b^2*d^2*exp(-3*a/b)*Ei(1,- 
3*arcsinh(x*c)-3*a/b)-1/32/b*d^2*(16*x^5*c^5+20*x^3*c^3+16*x^4*c^4*(c^2*x^ 
2+1)^(1/2)+5*x*c+12*x^2*c^2*(c^2*x^2+1)^(1/2)+(c^2*x^2+1)^(1/2))/(a+b*arcs 
inh(x*c))-5/32/b^2*d^2*exp(-5*a/b)*Ei(1,-5*arcsinh(x*c)-5*a/b))
 

Fricas [F]

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

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

Output:

integral((c^4*d^2*x^4 + 2*c^2*d^2*x^2 + d^2)/(b^2*arcsinh(c*x)^2 + 2*a*b*a 
rcsinh(c*x) + a^2), x)
                                                                                    
                                                                                    
 

Sympy [F]

\[ \int \frac {\left (d+c^2 d x^2\right )^2}{(a+b \text {arcsinh}(c x))^2} \, dx=d^{2} \left (\int \frac {2 c^{2} x^{2}}{a^{2} + 2 a b \operatorname {asinh}{\left (c x \right )} + b^{2} \operatorname {asinh}^{2}{\left (c x \right )}}\, dx + \int \frac {c^{4} x^{4}}{a^{2} + 2 a b \operatorname {asinh}{\left (c x \right )} + b^{2} \operatorname {asinh}^{2}{\left (c x \right )}}\, dx + \int \frac {1}{a^{2} + 2 a b \operatorname {asinh}{\left (c x \right )} + b^{2} \operatorname {asinh}^{2}{\left (c x \right )}}\, dx\right ) \] Input:

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

Output:

d**2*(Integral(2*c**2*x**2/(a**2 + 2*a*b*asinh(c*x) + b**2*asinh(c*x)**2), 
 x) + Integral(c**4*x**4/(a**2 + 2*a*b*asinh(c*x) + b**2*asinh(c*x)**2), x 
) + Integral(1/(a**2 + 2*a*b*asinh(c*x) + b**2*asinh(c*x)**2), x))
 

Maxima [F]

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

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

Output:

-(c^7*d^2*x^7 + 3*c^5*d^2*x^5 + 3*c^3*d^2*x^3 + c*d^2*x + (c^6*d^2*x^6 + 3 
*c^4*d^2*x^4 + 3*c^2*d^2*x^2 + d^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((5*c^8*d^2*x^8 + 16* 
c^6*d^2*x^6 + 18*c^4*d^2*x^4 + 8*c^2*d^2*x^2 + (5*c^6*d^2*x^6 + 9*c^4*d^2* 
x^4 + 3*c^2*d^2*x^2 - d^2)*(c^2*x^2 + 1) + d^2 + 5*(2*c^7*d^2*x^7 + 5*c^5* 
d^2*x^5 + 4*c^3*d^2*x^3 + c*d^2*x)*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]

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

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

Output:

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

\[ \int \frac {\left (d+c^2 d x^2\right )^2}{(a+b \text {arcsinh}(c x))^2} \, dx=d^{2} \left (\left (\int \frac {x^{4}}{\mathit {asinh} \left (c x \right )^{2} b^{2}+2 \mathit {asinh} \left (c x \right ) a b +a^{2}}d x \right ) c^{4}+2 \left (\int \frac {x^{2}}{\mathit {asinh} \left (c x \right )^{2} b^{2}+2 \mathit {asinh} \left (c x \right ) a b +a^{2}}d x \right ) c^{2}+\int \frac {1}{\mathit {asinh} \left (c x \right )^{2} b^{2}+2 \mathit {asinh} \left (c x \right ) a b +a^{2}}d x \right ) \] Input:

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

Output:

d**2*(int(x**4/(asinh(c*x)**2*b**2 + 2*asinh(c*x)*a*b + a**2),x)*c**4 + 2* 
int(x**2/(asinh(c*x)**2*b**2 + 2*asinh(c*x)*a*b + a**2),x)*c**2 + int(1/(a 
sinh(c*x)**2*b**2 + 2*asinh(c*x)*a*b + a**2),x))