\(\int \frac {(d+e x^2)^2}{(a+b \text {arcsinh}(c x))^2} \, dx\) [161]

Optimal result

Integrand size = 20, antiderivative size = 495 \[ \int \frac {\left (d+e x^2\right )^2}{(a+b \text {arcsinh}(c x))^2} \, dx=-\frac {d^2 \sqrt {1+c^2 x^2}}{b c (a+b \text {arcsinh}(c x))}-\frac {2 d e x^2 \sqrt {1+c^2 x^2}}{b c (a+b \text {arcsinh}(c x))}-\frac {e^2 x^4 \sqrt {1+c^2 x^2}}{b c (a+b \text {arcsinh}(c x))}-\frac {d^2 \text {Chi}\left (\frac {a+b \text {arcsinh}(c x)}{b}\right ) \sinh \left (\frac {a}{b}\right )}{b^2 c}+\frac {d e \text {Chi}\left (\frac {a+b \text {arcsinh}(c x)}{b}\right ) \sinh \left (\frac {a}{b}\right )}{2 b^2 c^3}-\frac {e^2 \text {Chi}\left (\frac {a+b \text {arcsinh}(c x)}{b}\right ) \sinh \left (\frac {a}{b}\right )}{8 b^2 c^5}-\frac {3 d e \text {Chi}\left (\frac {3 (a+b \text {arcsinh}(c x))}{b}\right ) \sinh \left (\frac {3 a}{b}\right )}{2 b^2 c^3}+\frac {9 e^2 \text {Chi}\left (\frac {3 (a+b \text {arcsinh}(c x))}{b}\right ) \sinh \left (\frac {3 a}{b}\right )}{16 b^2 c^5}-\frac {5 e^2 \text {Chi}\left (\frac {5 (a+b \text {arcsinh}(c x))}{b}\right ) \sinh \left (\frac {5 a}{b}\right )}{16 b^2 c^5}+\frac {d^2 \cosh \left (\frac {a}{b}\right ) \text {Shi}\left (\frac {a+b \text {arcsinh}(c x)}{b}\right )}{b^2 c}-\frac {d e \cosh \left (\frac {a}{b}\right ) \text {Shi}\left (\frac {a+b \text {arcsinh}(c x)}{b}\right )}{2 b^2 c^3}+\frac {e^2 \cosh \left (\frac {a}{b}\right ) \text {Shi}\left (\frac {a+b \text {arcsinh}(c x)}{b}\right )}{8 b^2 c^5}+\frac {3 d e \cosh \left (\frac {3 a}{b}\right ) \text {Shi}\left (\frac {3 (a+b \text {arcsinh}(c x))}{b}\right )}{2 b^2 c^3}-\frac {9 e^2 \cosh \left (\frac {3 a}{b}\right ) \text {Shi}\left (\frac {3 (a+b \text {arcsinh}(c x))}{b}\right )}{16 b^2 c^5}+\frac {5 e^2 \cosh \left (\frac {5 a}{b}\right ) \text {Shi}\left (\frac {5 (a+b \text {arcsinh}(c x))}{b}\right )}{16 b^2 c^5} \] Output:

-d^2*(c^2*x^2+1)^(1/2)/b/c/(a+b*arcsinh(c*x))-2*d*e*x^2*(c^2*x^2+1)^(1/2)/ 
b/c/(a+b*arcsinh(c*x))-e^2*x^4*(c^2*x^2+1)^(1/2)/b/c/(a+b*arcsinh(c*x))-d^ 
2*Chi((a+b*arcsinh(c*x))/b)*sinh(a/b)/b^2/c+1/2*d*e*Chi((a+b*arcsinh(c*x)) 
/b)*sinh(a/b)/b^2/c^3-1/8*e^2*Chi((a+b*arcsinh(c*x))/b)*sinh(a/b)/b^2/c^5- 
3/2*d*e*Chi(3*(a+b*arcsinh(c*x))/b)*sinh(3*a/b)/b^2/c^3+9/16*e^2*Chi(3*(a+ 
b*arcsinh(c*x))/b)*sinh(3*a/b)/b^2/c^5-5/16*e^2*Chi(5*(a+b*arcsinh(c*x))/b 
)*sinh(5*a/b)/b^2/c^5+d^2*cosh(a/b)*Shi((a+b*arcsinh(c*x))/b)/b^2/c-1/2*d* 
e*cosh(a/b)*Shi((a+b*arcsinh(c*x))/b)/b^2/c^3+1/8*e^2*cosh(a/b)*Shi((a+b*a 
rcsinh(c*x))/b)/b^2/c^5+3/2*d*e*cosh(3*a/b)*Shi(3*(a+b*arcsinh(c*x))/b)/b^ 
2/c^3-9/16*e^2*cosh(3*a/b)*Shi(3*(a+b*arcsinh(c*x))/b)/b^2/c^5+5/16*e^2*co 
sh(5*a/b)*Shi(5*(a+b*arcsinh(c*x))/b)/b^2/c^5
 

Mathematica [A] (verified)

Time = 1.82 (sec) , antiderivative size = 356, normalized size of antiderivative = 0.72 \[ \int \frac {\left (d+e x^2\right )^2}{(a+b \text {arcsinh}(c x))^2} \, dx=-\frac {\frac {16 b c^4 d^2 \sqrt {1+c^2 x^2}}{a+b \text {arcsinh}(c x)}+\frac {32 b c^4 d e x^2 \sqrt {1+c^2 x^2}}{a+b \text {arcsinh}(c x)}+\frac {16 b c^4 e^2 x^4 \sqrt {1+c^2 x^2}}{a+b \text {arcsinh}(c x)}+2 \left (8 c^4 d^2-4 c^2 d e+e^2\right ) \text {Chi}\left (\frac {a}{b}+\text {arcsinh}(c x)\right ) \sinh \left (\frac {a}{b}\right )+3 \left (8 c^2 d-3 e\right ) e \text {Chi}\left (3 \left (\frac {a}{b}+\text {arcsinh}(c x)\right )\right ) \sinh \left (\frac {3 a}{b}\right )+5 e^2 \text {Chi}\left (5 \left (\frac {a}{b}+\text {arcsinh}(c x)\right )\right ) \sinh \left (\frac {5 a}{b}\right )-16 c^4 d^2 \cosh \left (\frac {a}{b}\right ) \text {Shi}\left (\frac {a}{b}+\text {arcsinh}(c x)\right )+8 c^2 d e \cosh \left (\frac {a}{b}\right ) \text {Shi}\left (\frac {a}{b}+\text {arcsinh}(c x)\right )-2 e^2 \cosh \left (\frac {a}{b}\right ) \text {Shi}\left (\frac {a}{b}+\text {arcsinh}(c x)\right )-24 c^2 d e \cosh \left (\frac {3 a}{b}\right ) \text {Shi}\left (3 \left (\frac {a}{b}+\text {arcsinh}(c x)\right )\right )+9 e^2 \cosh \left (\frac {3 a}{b}\right ) \text {Shi}\left (3 \left (\frac {a}{b}+\text {arcsinh}(c x)\right )\right )-5 e^2 \cosh \left (\frac {5 a}{b}\right ) \text {Shi}\left (5 \left (\frac {a}{b}+\text {arcsinh}(c x)\right )\right )}{16 b^2 c^5} \] Input:

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

Output:

-1/16*((16*b*c^4*d^2*Sqrt[1 + c^2*x^2])/(a + b*ArcSinh[c*x]) + (32*b*c^4*d 
*e*x^2*Sqrt[1 + c^2*x^2])/(a + b*ArcSinh[c*x]) + (16*b*c^4*e^2*x^4*Sqrt[1 
+ c^2*x^2])/(a + b*ArcSinh[c*x]) + 2*(8*c^4*d^2 - 4*c^2*d*e + e^2)*CoshInt 
egral[a/b + ArcSinh[c*x]]*Sinh[a/b] + 3*(8*c^2*d - 3*e)*e*CoshIntegral[3*( 
a/b + ArcSinh[c*x])]*Sinh[(3*a)/b] + 5*e^2*CoshIntegral[5*(a/b + ArcSinh[c 
*x])]*Sinh[(5*a)/b] - 16*c^4*d^2*Cosh[a/b]*SinhIntegral[a/b + ArcSinh[c*x] 
] + 8*c^2*d*e*Cosh[a/b]*SinhIntegral[a/b + ArcSinh[c*x]] - 2*e^2*Cosh[a/b] 
*SinhIntegral[a/b + ArcSinh[c*x]] - 24*c^2*d*e*Cosh[(3*a)/b]*SinhIntegral[ 
3*(a/b + ArcSinh[c*x])] + 9*e^2*Cosh[(3*a)/b]*SinhIntegral[3*(a/b + ArcSin 
h[c*x])] - 5*e^2*Cosh[(5*a)/b]*SinhIntegral[5*(a/b + ArcSinh[c*x])])/(b^2* 
c^5)
 

Rubi [A] (verified)

Time = 1.09 (sec) , antiderivative size = 495, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.100, Rules used = {6208, 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.

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

\(\Big \downarrow \) 6208

\(\displaystyle \int \left (\frac {d^2}{(a+b \text {arcsinh}(c x))^2}+\frac {2 d e x^2}{(a+b \text {arcsinh}(c x))^2}+\frac {e^2 x^4}{(a+b \text {arcsinh}(c x))^2}\right )dx\)

\(\Big \downarrow \) 2009

\(\displaystyle -\frac {e^2 \sinh \left (\frac {a}{b}\right ) \text {Chi}\left (\frac {a+b \text {arcsinh}(c x)}{b}\right )}{8 b^2 c^5}+\frac {9 e^2 \sinh \left (\frac {3 a}{b}\right ) \text {Chi}\left (\frac {3 (a+b \text {arcsinh}(c x))}{b}\right )}{16 b^2 c^5}-\frac {5 e^2 \sinh \left (\frac {5 a}{b}\right ) \text {Chi}\left (\frac {5 (a+b \text {arcsinh}(c x))}{b}\right )}{16 b^2 c^5}+\frac {e^2 \cosh \left (\frac {a}{b}\right ) \text {Shi}\left (\frac {a+b \text {arcsinh}(c x)}{b}\right )}{8 b^2 c^5}-\frac {9 e^2 \cosh \left (\frac {3 a}{b}\right ) \text {Shi}\left (\frac {3 (a+b \text {arcsinh}(c x))}{b}\right )}{16 b^2 c^5}+\frac {5 e^2 \cosh \left (\frac {5 a}{b}\right ) \text {Shi}\left (\frac {5 (a+b \text {arcsinh}(c x))}{b}\right )}{16 b^2 c^5}+\frac {d e \sinh \left (\frac {a}{b}\right ) \text {Chi}\left (\frac {a+b \text {arcsinh}(c x)}{b}\right )}{2 b^2 c^3}-\frac {3 d e \sinh \left (\frac {3 a}{b}\right ) \text {Chi}\left (\frac {3 (a+b \text {arcsinh}(c x))}{b}\right )}{2 b^2 c^3}-\frac {d e \cosh \left (\frac {a}{b}\right ) \text {Shi}\left (\frac {a+b \text {arcsinh}(c x)}{b}\right )}{2 b^2 c^3}+\frac {3 d e \cosh \left (\frac {3 a}{b}\right ) \text {Shi}\left (\frac {3 (a+b \text {arcsinh}(c x))}{b}\right )}{2 b^2 c^3}-\frac {d^2 \sinh \left (\frac {a}{b}\right ) \text {Chi}\left (\frac {a+b \text {arcsinh}(c x)}{b}\right )}{b^2 c}+\frac {d^2 \cosh \left (\frac {a}{b}\right ) \text {Shi}\left (\frac {a+b \text {arcsinh}(c x)}{b}\right )}{b^2 c}-\frac {d^2 \sqrt {c^2 x^2+1}}{b c (a+b \text {arcsinh}(c x))}-\frac {2 d e x^2 \sqrt {c^2 x^2+1}}{b c (a+b \text {arcsinh}(c x))}-\frac {e^2 x^4 \sqrt {c^2 x^2+1}}{b c (a+b \text {arcsinh}(c x))}\)

Input:

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

Output:

-((d^2*Sqrt[1 + c^2*x^2])/(b*c*(a + b*ArcSinh[c*x]))) - (2*d*e*x^2*Sqrt[1 
+ c^2*x^2])/(b*c*(a + b*ArcSinh[c*x])) - (e^2*x^4*Sqrt[1 + c^2*x^2])/(b*c* 
(a + b*ArcSinh[c*x])) - (d^2*CoshIntegral[(a + b*ArcSinh[c*x])/b]*Sinh[a/b 
])/(b^2*c) + (d*e*CoshIntegral[(a + b*ArcSinh[c*x])/b]*Sinh[a/b])/(2*b^2*c 
^3) - (e^2*CoshIntegral[(a + b*ArcSinh[c*x])/b]*Sinh[a/b])/(8*b^2*c^5) - ( 
3*d*e*CoshIntegral[(3*(a + b*ArcSinh[c*x]))/b]*Sinh[(3*a)/b])/(2*b^2*c^3) 
+ (9*e^2*CoshIntegral[(3*(a + b*ArcSinh[c*x]))/b]*Sinh[(3*a)/b])/(16*b^2*c 
^5) - (5*e^2*CoshIntegral[(5*(a + b*ArcSinh[c*x]))/b]*Sinh[(5*a)/b])/(16*b 
^2*c^5) + (d^2*Cosh[a/b]*SinhIntegral[(a + b*ArcSinh[c*x])/b])/(b^2*c) - ( 
d*e*Cosh[a/b]*SinhIntegral[(a + b*ArcSinh[c*x])/b])/(2*b^2*c^3) + (e^2*Cos 
h[a/b]*SinhIntegral[(a + b*ArcSinh[c*x])/b])/(8*b^2*c^5) + (3*d*e*Cosh[(3* 
a)/b]*SinhIntegral[(3*(a + b*ArcSinh[c*x]))/b])/(2*b^2*c^3) - (9*e^2*Cosh[ 
(3*a)/b]*SinhIntegral[(3*(a + b*ArcSinh[c*x]))/b])/(16*b^2*c^5) + (5*e^2*C 
osh[(5*a)/b]*SinhIntegral[(5*(a + b*ArcSinh[c*x]))/b])/(16*b^2*c^5)
 

Defintions of rubi rules used

rule 2009

Int[u_, x_Symbol] :> Simp[IntSum[u, x], x] /; SumQ[u]
 

rule 6208

Int[((a_.) + ArcSinh[(c_.)*(x_)]*(b_.))^(n_.)*((d_) + (e_.)*(x_)^2)^(p_.), 
x_Symbol] :> Int[ExpandIntegrand[(a + b*ArcSinh[c*x])^n, (d + e*x^2)^p, x], 
 x] /; FreeQ[{a, b, c, d, e, n}, x] && NeQ[e, c^2*d] && IntegerQ[p] && (p > 
 0 || IGtQ[n, 0])
 

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(1035\) vs. \(2(469)=938\).

Time = 3.77 (sec) , antiderivative size = 1036, normalized size of antiderivative = 2.09

method	result	size
derivativedivides	\(\text {Expression too large to display}\)	\(1036\)
default	\(\text {Expression too large to display}\)	\(1036\)

Input:

int((e*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)*e^2/c^4/b/(a+b*arcsinh(x*c))+5/ 
32*e^2/c^4/b^2*exp(5*a/b)*Ei(1,5*arcsinh(x*c)+5*a/b)-1/32/b*e^2/c^4*(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*arcsinh(x*c))-5/32/b^2*e^2/c^4*exp(-5*a/b)* 
Ei(1,-5*arcsinh(x*c)-5*a/b)+1/2*(x*c-(c^2*x^2+1)^(1/2))*d^2/b/(a+b*arcsinh 
(x*c))+1/2*d^2/b^2*exp(a/b)*Ei(1,arcsinh(x*c)+a/b)-1/4*(x*c-(c^2*x^2+1)^(1 
/2))*d*e/c^2/b/(a+b*arcsinh(x*c))-1/4/c^2*d*e/b^2*exp(a/b)*Ei(1,arcsinh(x* 
c)+a/b)+1/16*(x*c-(c^2*x^2+1)^(1/2))*e^2/c^4/b/(a+b*arcsinh(x*c))+1/16/c^4 
*e^2/b^2*exp(a/b)*Ei(1,arcsinh(x*c)+a/b)-1/2/b*d^2*(x*c+(c^2*x^2+1)^(1/2)) 
/(a+b*arcsinh(x*c))-1/2/b^2*d^2*exp(-a/b)*Ei(1,-arcsinh(x*c)-a/b)+1/4/c^2/ 
b*d*e*(x*c+(c^2*x^2+1)^(1/2))/(a+b*arcsinh(x*c))+1/4/c^2/b^2*d*e*exp(-a/b) 
*Ei(1,-arcsinh(x*c)-a/b)-1/16/c^4/b*e^2*(x*c+(c^2*x^2+1)^(1/2))/(a+b*arcsi 
nh(x*c))-1/16/c^4/b^2*e^2*exp(-a/b)*Ei(1,-arcsinh(x*c)-a/b)+1/4*(-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*e/c^2/b/(a+b*arcs 
inh(x*c))-3/32*(-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)*e^2/c^4/b/(a+b*arcsinh(x*c))+3/4*e/c^2/b^2*exp(3*a/b)*Ei(1,3*arcsinh 
(x*c)+3*a/b)*d-9/32*e^2/c^4/b^2*exp(3*a/b)*Ei(1,3*arcsinh(x*c)+3*a/b)-1/4/ 
c^2*e/b*(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))*d+3/32/c^4*e^2/b*(4*x^3*c^3+3*x*c+4*x^2*c^2*(c^2*x^2+1...
 

Fricas [F]

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

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

Output:

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

Sympy [F]

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

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

Output:

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

Maxima [F]

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

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

Output:

-(c^3*e^2*x^7 + (2*c^3*d*e + c*e^2)*x^5 + c*d^2*x + (c^3*d^2 + 2*c*d*e)*x^ 
3 + (c^2*e^2*x^6 + (2*c^2*d*e + e^2)*x^4 + (c^2*d^2 + 2*d*e)*x^2 + d^2)*sq 
rt(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^5*e^2*x^8 + 2*(3*c^5*d*e + 5*c^3*e^2)*x^6 + (c^5*d^2 
+ 12*c^3*d*e + 5*c*e^2)*x^4 + c*d^2 + 2*(c^3*d^2 + 3*c*d*e)*x^2 + (5*c^3*e 
^2*x^6 + 3*(2*c^3*d*e + c*e^2)*x^4 - c*d^2 + (c^3*d^2 + 2*c*d*e)*x^2)*(c^2 
*x^2 + 1) + (10*c^4*e^2*x^7 + (12*c^4*d*e + 13*c^2*e^2)*x^5 + 2*(c^4*d^2 + 
 7*c^2*d*e + 2*e^2)*x^3 + (c^2*d^2 + 4*d*e)*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]

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

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

Output:

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

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

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

Output:

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