3.624 \(\int \frac{(d+e x^2)^2}{(a+b \sinh ^{-1}(c x))^2} \, dx\)

Optimal. Leaf size=495 \[ \frac{d e \sinh \left (\frac{a}{b}\right ) \text{Chi}\left (\frac{a+b \sinh ^{-1}(c x)}{b}\right )}{2 b^2 c^3}-\frac{3 d e \sinh \left (\frac{3 a}{b}\right ) \text{Chi}\left (\frac{3 \left (a+b \sinh ^{-1}(c x)\right )}{b}\right )}{2 b^2 c^3}-\frac{e^2 \sinh \left (\frac{a}{b}\right ) \text{Chi}\left (\frac{a+b \sinh ^{-1}(c x)}{b}\right )}{8 b^2 c^5}+\frac{9 e^2 \sinh \left (\frac{3 a}{b}\right ) \text{Chi}\left (\frac{3 \left (a+b \sinh ^{-1}(c x)\right )}{b}\right )}{16 b^2 c^5}-\frac{5 e^2 \sinh \left (\frac{5 a}{b}\right ) \text{Chi}\left (\frac{5 \left (a+b \sinh ^{-1}(c x)\right )}{b}\right )}{16 b^2 c^5}-\frac{d e \cosh \left (\frac{a}{b}\right ) \text{Shi}\left (\frac{a+b \sinh ^{-1}(c x)}{b}\right )}{2 b^2 c^3}+\frac{3 d e \cosh \left (\frac{3 a}{b}\right ) \text{Shi}\left (\frac{3 \left (a+b \sinh ^{-1}(c x)\right )}{b}\right )}{2 b^2 c^3}+\frac{e^2 \cosh \left (\frac{a}{b}\right ) \text{Shi}\left (\frac{a+b \sinh ^{-1}(c x)}{b}\right )}{8 b^2 c^5}-\frac{9 e^2 \cosh \left (\frac{3 a}{b}\right ) \text{Shi}\left (\frac{3 \left (a+b \sinh ^{-1}(c x)\right )}{b}\right )}{16 b^2 c^5}+\frac{5 e^2 \cosh \left (\frac{5 a}{b}\right ) \text{Shi}\left (\frac{5 \left (a+b \sinh ^{-1}(c x)\right )}{b}\right )}{16 b^2 c^5}-\frac{d^2 \sinh \left (\frac{a}{b}\right ) \text{Chi}\left (\frac{a+b \sinh ^{-1}(c x)}{b}\right )}{b^2 c}+\frac{d^2 \cosh \left (\frac{a}{b}\right ) \text{Shi}\left (\frac{a+b \sinh ^{-1}(c x)}{b}\right )}{b^2 c}-\frac{d^2 \sqrt{c^2 x^2+1}}{b c \left (a+b \sinh ^{-1}(c x)\right )}-\frac{2 d e x^2 \sqrt{c^2 x^2+1}}{b c \left (a+b \sinh ^{-1}(c x)\right )}-\frac{e^2 x^4 \sqrt{c^2 x^2+1}}{b c \left (a+b \sinh ^{-1}(c x)\right )} \]

[Out]

-((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*Arc
Sinh[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*Cosh[a/b]*SinhIntegral[(a + b*Arc
Sinh[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*Cosh[(5*a)/b]*SinhIntegral[(5
*(a + b*ArcSinh[c*x]))/b])/(16*b^2*c^5)

________________________________________________________________________________________

Rubi [A]  time = 0.874567, antiderivative size = 483, normalized size of antiderivative = 0.98, number of steps used = 26, number of rules used = 7, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.35, Rules used = {5706, 5655, 5779, 3303, 3298, 3301, 5665} \[ \frac{d e \sinh \left (\frac{a}{b}\right ) \text{Chi}\left (\frac{a}{b}+\sinh ^{-1}(c x)\right )}{2 b^2 c^3}-\frac{3 d e \sinh \left (\frac{3 a}{b}\right ) \text{Chi}\left (\frac{3 a}{b}+3 \sinh ^{-1}(c x)\right )}{2 b^2 c^3}-\frac{e^2 \sinh \left (\frac{a}{b}\right ) \text{Chi}\left (\frac{a}{b}+\sinh ^{-1}(c x)\right )}{8 b^2 c^5}+\frac{9 e^2 \sinh \left (\frac{3 a}{b}\right ) \text{Chi}\left (\frac{3 a}{b}+3 \sinh ^{-1}(c x)\right )}{16 b^2 c^5}-\frac{5 e^2 \sinh \left (\frac{5 a}{b}\right ) \text{Chi}\left (\frac{5 a}{b}+5 \sinh ^{-1}(c x)\right )}{16 b^2 c^5}-\frac{d e \cosh \left (\frac{a}{b}\right ) \text{Shi}\left (\frac{a}{b}+\sinh ^{-1}(c x)\right )}{2 b^2 c^3}+\frac{3 d e \cosh \left (\frac{3 a}{b}\right ) \text{Shi}\left (\frac{3 a}{b}+3 \sinh ^{-1}(c x)\right )}{2 b^2 c^3}+\frac{e^2 \cosh \left (\frac{a}{b}\right ) \text{Shi}\left (\frac{a}{b}+\sinh ^{-1}(c x)\right )}{8 b^2 c^5}-\frac{9 e^2 \cosh \left (\frac{3 a}{b}\right ) \text{Shi}\left (\frac{3 a}{b}+3 \sinh ^{-1}(c x)\right )}{16 b^2 c^5}+\frac{5 e^2 \cosh \left (\frac{5 a}{b}\right ) \text{Shi}\left (\frac{5 a}{b}+5 \sinh ^{-1}(c x)\right )}{16 b^2 c^5}-\frac{d^2 \sinh \left (\frac{a}{b}\right ) \text{Chi}\left (\frac{a}{b}+\sinh ^{-1}(c x)\right )}{b^2 c}+\frac{d^2 \cosh \left (\frac{a}{b}\right ) \text{Shi}\left (\frac{a}{b}+\sinh ^{-1}(c x)\right )}{b^2 c}-\frac{d^2 \sqrt{c^2 x^2+1}}{b c \left (a+b \sinh ^{-1}(c x)\right )}-\frac{2 d e x^2 \sqrt{c^2 x^2+1}}{b c \left (a+b \sinh ^{-1}(c x)\right )}-\frac{e^2 x^4 \sqrt{c^2 x^2+1}}{b c \left (a+b \sinh ^{-1}(c x)\right )} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-((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]]*Sinh[a/b])
/(b^2*c) + (d*e*CoshIntegral[a/b + ArcSinh[c*x]]*Sinh[a/b])/(2*b^2*c^3) - (e^2*CoshIntegral[a/b + ArcSinh[c*x]
]*Sinh[a/b])/(8*b^2*c^5) - (3*d*e*CoshIntegral[(3*a)/b + 3*ArcSinh[c*x]]*Sinh[(3*a)/b])/(2*b^2*c^3) + (9*e^2*C
oshIntegral[(3*a)/b + 3*ArcSinh[c*x]]*Sinh[(3*a)/b])/(16*b^2*c^5) - (5*e^2*CoshIntegral[(5*a)/b + 5*ArcSinh[c*
x]]*Sinh[(5*a)/b])/(16*b^2*c^5) + (d^2*Cosh[a/b]*SinhIntegral[a/b + ArcSinh[c*x]])/(b^2*c) - (d*e*Cosh[a/b]*Si
nhIntegral[a/b + ArcSinh[c*x]])/(2*b^2*c^3) + (e^2*Cosh[a/b]*SinhIntegral[a/b + ArcSinh[c*x]])/(8*b^2*c^5) + (
3*d*e*Cosh[(3*a)/b]*SinhIntegral[(3*a)/b + 3*ArcSinh[c*x]])/(2*b^2*c^3) - (9*e^2*Cosh[(3*a)/b]*SinhIntegral[(3
*a)/b + 3*ArcSinh[c*x]])/(16*b^2*c^5) + (5*e^2*Cosh[(5*a)/b]*SinhIntegral[(5*a)/b + 5*ArcSinh[c*x]])/(16*b^2*c
^5)

Rule 5706

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])

Rule 5655

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] - Dist[c/(b*(n + 1)), Int[(x*(a + b*ArcSinh[c*x])^(n + 1))/Sqrt[1 + c^2*x^2], x], x] /; F
reeQ[{a, b, c}, x] && LtQ[n, -1]

Rule 5779

Int[((a_.) + ArcSinh[(c_.)*(x_)]*(b_.))^(n_.)*(x_)^(m_.)*((d_) + (e_.)*(x_)^2)^(p_.), x_Symbol] :> Dist[d^p/c^
(m + 1), Subst[Int[(a + b*x)^n*Sinh[x]^m*Cosh[x]^(2*p + 1), x], x, ArcSinh[c*x]], x] /; FreeQ[{a, b, c, d, e,
n}, x] && EqQ[e, c^2*d] && IntegerQ[2*p] && GtQ[p, -1] && IGtQ[m, 0] && (IntegerQ[p] || GtQ[d, 0])

Rule 3303

Int[sin[(e_.) + (f_.)*(x_)]/((c_.) + (d_.)*(x_)), x_Symbol] :> Dist[Cos[(d*e - c*f)/d], Int[Sin[(c*f)/d + f*x]
/(c + d*x), x], x] + Dist[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 3298

Int[sin[(e_.) + (Complex[0, fz_])*(f_.)*(x_)]/((c_.) + (d_.)*(x_)), x_Symbol] :> 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 3301

Int[sin[(e_.) + (Complex[0, fz_])*(f_.)*(x_)]/((c_.) + (d_.)*(x_)), x_Symbol] :> 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 5665

Int[((a_.) + ArcSinh[(c_.)*(x_)]*(b_.))^(n_)*(x_)^(m_.), x_Symbol] :> Simp[(x^m*Sqrt[1 + c^2*x^2]*(a + b*ArcSi
nh[c*x])^(n + 1))/(b*c*(n + 1)), x] - Dist[1/(b*c^(m + 1)*(n + 1)), Subst[Int[ExpandTrigReduce[(a + b*x)^(n +
1), Sinh[x]^(m - 1)*(m + (m + 1)*Sinh[x]^2), x], x], x, ArcSinh[c*x]], x] /; FreeQ[{a, b, c}, x] && IGtQ[m, 0]
&& GeQ[n, -2] && LtQ[n, -1]

Rubi steps

\begin{align*} \int \frac{\left (d+e x^2\right )^2}{\left (a+b \sinh ^{-1}(c x)\right )^2} \, dx &=\int \left (\frac{d^2}{\left (a+b \sinh ^{-1}(c x)\right )^2}+\frac{2 d e x^2}{\left (a+b \sinh ^{-1}(c x)\right )^2}+\frac{e^2 x^4}{\left (a+b \sinh ^{-1}(c x)\right )^2}\right ) \, dx\\ &=d^2 \int \frac{1}{\left (a+b \sinh ^{-1}(c x)\right )^2} \, dx+(2 d e) \int \frac{x^2}{\left (a+b \sinh ^{-1}(c x)\right )^2} \, dx+e^2 \int \frac{x^4}{\left (a+b \sinh ^{-1}(c x)\right )^2} \, dx\\ &=-\frac{d^2 \sqrt{1+c^2 x^2}}{b c \left (a+b \sinh ^{-1}(c x)\right )}-\frac{2 d e x^2 \sqrt{1+c^2 x^2}}{b c \left (a+b \sinh ^{-1}(c x)\right )}-\frac{e^2 x^4 \sqrt{1+c^2 x^2}}{b c \left (a+b \sinh ^{-1}(c x)\right )}+\frac{\left (c d^2\right ) \int \frac{x}{\sqrt{1+c^2 x^2} \left (a+b \sinh ^{-1}(c x)\right )} \, dx}{b}+\frac{(2 d e) \operatorname{Subst}\left (\int \left (-\frac{\sinh (x)}{4 (a+b x)}+\frac{3 \sinh (3 x)}{4 (a+b x)}\right ) \, dx,x,\sinh ^{-1}(c x)\right )}{b c^3}+\frac{e^2 \operatorname{Subst}\left (\int \left (\frac{\sinh (x)}{8 (a+b x)}-\frac{9 \sinh (3 x)}{16 (a+b x)}+\frac{5 \sinh (5 x)}{16 (a+b x)}\right ) \, dx,x,\sinh ^{-1}(c x)\right )}{b c^5}\\ &=-\frac{d^2 \sqrt{1+c^2 x^2}}{b c \left (a+b \sinh ^{-1}(c x)\right )}-\frac{2 d e x^2 \sqrt{1+c^2 x^2}}{b c \left (a+b \sinh ^{-1}(c x)\right )}-\frac{e^2 x^4 \sqrt{1+c^2 x^2}}{b c \left (a+b \sinh ^{-1}(c x)\right )}+\frac{d^2 \operatorname{Subst}\left (\int \frac{\sinh (x)}{a+b x} \, dx,x,\sinh ^{-1}(c x)\right )}{b c}-\frac{(d e) \operatorname{Subst}\left (\int \frac{\sinh (x)}{a+b x} \, dx,x,\sinh ^{-1}(c x)\right )}{2 b c^3}+\frac{(3 d e) \operatorname{Subst}\left (\int \frac{\sinh (3 x)}{a+b x} \, dx,x,\sinh ^{-1}(c x)\right )}{2 b c^3}+\frac{e^2 \operatorname{Subst}\left (\int \frac{\sinh (x)}{a+b x} \, dx,x,\sinh ^{-1}(c x)\right )}{8 b c^5}+\frac{\left (5 e^2\right ) \operatorname{Subst}\left (\int \frac{\sinh (5 x)}{a+b x} \, dx,x,\sinh ^{-1}(c x)\right )}{16 b c^5}-\frac{\left (9 e^2\right ) \operatorname{Subst}\left (\int \frac{\sinh (3 x)}{a+b x} \, dx,x,\sinh ^{-1}(c x)\right )}{16 b c^5}\\ &=-\frac{d^2 \sqrt{1+c^2 x^2}}{b c \left (a+b \sinh ^{-1}(c x)\right )}-\frac{2 d e x^2 \sqrt{1+c^2 x^2}}{b c \left (a+b \sinh ^{-1}(c x)\right )}-\frac{e^2 x^4 \sqrt{1+c^2 x^2}}{b c \left (a+b \sinh ^{-1}(c x)\right )}+\frac{\left (d^2 \cosh \left (\frac{a}{b}\right )\right ) \operatorname{Subst}\left (\int \frac{\sinh \left (\frac{a}{b}+x\right )}{a+b x} \, dx,x,\sinh ^{-1}(c x)\right )}{b c}-\frac{\left (d e \cosh \left (\frac{a}{b}\right )\right ) \operatorname{Subst}\left (\int \frac{\sinh \left (\frac{a}{b}+x\right )}{a+b x} \, dx,x,\sinh ^{-1}(c x)\right )}{2 b c^3}+\frac{\left (e^2 \cosh \left (\frac{a}{b}\right )\right ) \operatorname{Subst}\left (\int \frac{\sinh \left (\frac{a}{b}+x\right )}{a+b x} \, dx,x,\sinh ^{-1}(c x)\right )}{8 b c^5}+\frac{\left (3 d e \cosh \left (\frac{3 a}{b}\right )\right ) \operatorname{Subst}\left (\int \frac{\sinh \left (\frac{3 a}{b}+3 x\right )}{a+b x} \, dx,x,\sinh ^{-1}(c x)\right )}{2 b c^3}-\frac{\left (9 e^2 \cosh \left (\frac{3 a}{b}\right )\right ) \operatorname{Subst}\left (\int \frac{\sinh \left (\frac{3 a}{b}+3 x\right )}{a+b x} \, dx,x,\sinh ^{-1}(c x)\right )}{16 b c^5}+\frac{\left (5 e^2 \cosh \left (\frac{5 a}{b}\right )\right ) \operatorname{Subst}\left (\int \frac{\sinh \left (\frac{5 a}{b}+5 x\right )}{a+b x} \, dx,x,\sinh ^{-1}(c x)\right )}{16 b c^5}-\frac{\left (d^2 \sinh \left (\frac{a}{b}\right )\right ) \operatorname{Subst}\left (\int \frac{\cosh \left (\frac{a}{b}+x\right )}{a+b x} \, dx,x,\sinh ^{-1}(c x)\right )}{b c}+\frac{\left (d e \sinh \left (\frac{a}{b}\right )\right ) \operatorname{Subst}\left (\int \frac{\cosh \left (\frac{a}{b}+x\right )}{a+b x} \, dx,x,\sinh ^{-1}(c x)\right )}{2 b c^3}-\frac{\left (e^2 \sinh \left (\frac{a}{b}\right )\right ) \operatorname{Subst}\left (\int \frac{\cosh \left (\frac{a}{b}+x\right )}{a+b x} \, dx,x,\sinh ^{-1}(c x)\right )}{8 b c^5}-\frac{\left (3 d e \sinh \left (\frac{3 a}{b}\right )\right ) \operatorname{Subst}\left (\int \frac{\cosh \left (\frac{3 a}{b}+3 x\right )}{a+b x} \, dx,x,\sinh ^{-1}(c x)\right )}{2 b c^3}+\frac{\left (9 e^2 \sinh \left (\frac{3 a}{b}\right )\right ) \operatorname{Subst}\left (\int \frac{\cosh \left (\frac{3 a}{b}+3 x\right )}{a+b x} \, dx,x,\sinh ^{-1}(c x)\right )}{16 b c^5}-\frac{\left (5 e^2 \sinh \left (\frac{5 a}{b}\right )\right ) \operatorname{Subst}\left (\int \frac{\cosh \left (\frac{5 a}{b}+5 x\right )}{a+b x} \, dx,x,\sinh ^{-1}(c x)\right )}{16 b c^5}\\ &=-\frac{d^2 \sqrt{1+c^2 x^2}}{b c \left (a+b \sinh ^{-1}(c x)\right )}-\frac{2 d e x^2 \sqrt{1+c^2 x^2}}{b c \left (a+b \sinh ^{-1}(c x)\right )}-\frac{e^2 x^4 \sqrt{1+c^2 x^2}}{b c \left (a+b \sinh ^{-1}(c x)\right )}-\frac{d^2 \text{Chi}\left (\frac{a}{b}+\sinh ^{-1}(c x)\right ) \sinh \left (\frac{a}{b}\right )}{b^2 c}+\frac{d e \text{Chi}\left (\frac{a}{b}+\sinh ^{-1}(c x)\right ) \sinh \left (\frac{a}{b}\right )}{2 b^2 c^3}-\frac{e^2 \text{Chi}\left (\frac{a}{b}+\sinh ^{-1}(c x)\right ) \sinh \left (\frac{a}{b}\right )}{8 b^2 c^5}-\frac{3 d e \text{Chi}\left (\frac{3 a}{b}+3 \sinh ^{-1}(c x)\right ) \sinh \left (\frac{3 a}{b}\right )}{2 b^2 c^3}+\frac{9 e^2 \text{Chi}\left (\frac{3 a}{b}+3 \sinh ^{-1}(c x)\right ) \sinh \left (\frac{3 a}{b}\right )}{16 b^2 c^5}-\frac{5 e^2 \text{Chi}\left (\frac{5 a}{b}+5 \sinh ^{-1}(c x)\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}+\sinh ^{-1}(c x)\right )}{b^2 c}-\frac{d e \cosh \left (\frac{a}{b}\right ) \text{Shi}\left (\frac{a}{b}+\sinh ^{-1}(c x)\right )}{2 b^2 c^3}+\frac{e^2 \cosh \left (\frac{a}{b}\right ) \text{Shi}\left (\frac{a}{b}+\sinh ^{-1}(c x)\right )}{8 b^2 c^5}+\frac{3 d e \cosh \left (\frac{3 a}{b}\right ) \text{Shi}\left (\frac{3 a}{b}+3 \sinh ^{-1}(c x)\right )}{2 b^2 c^3}-\frac{9 e^2 \cosh \left (\frac{3 a}{b}\right ) \text{Shi}\left (\frac{3 a}{b}+3 \sinh ^{-1}(c x)\right )}{16 b^2 c^5}+\frac{5 e^2 \cosh \left (\frac{5 a}{b}\right ) \text{Shi}\left (\frac{5 a}{b}+5 \sinh ^{-1}(c x)\right )}{16 b^2 c^5}\\ \end{align*}

Mathematica [A]  time = 2.02929, size = 356, normalized size = 0.72 \[ -\frac{2 \sinh \left (\frac{a}{b}\right ) \left (8 c^4 d^2-4 c^2 d e+e^2\right ) \text{Chi}\left (\frac{a}{b}+\sinh ^{-1}(c x)\right )+3 e \sinh \left (\frac{3 a}{b}\right ) \left (8 c^2 d-3 e\right ) \text{Chi}\left (3 \left (\frac{a}{b}+\sinh ^{-1}(c x)\right )\right )-16 c^4 d^2 \cosh \left (\frac{a}{b}\right ) \text{Shi}\left (\frac{a}{b}+\sinh ^{-1}(c x)\right )+\frac{16 b c^4 d^2 \sqrt{c^2 x^2+1}}{a+b \sinh ^{-1}(c x)}+8 c^2 d e \cosh \left (\frac{a}{b}\right ) \text{Shi}\left (\frac{a}{b}+\sinh ^{-1}(c x)\right )-24 c^2 d e \cosh \left (\frac{3 a}{b}\right ) \text{Shi}\left (3 \left (\frac{a}{b}+\sinh ^{-1}(c x)\right )\right )+\frac{32 b c^4 d e x^2 \sqrt{c^2 x^2+1}}{a+b \sinh ^{-1}(c x)}+\frac{16 b c^4 e^2 x^4 \sqrt{c^2 x^2+1}}{a+b \sinh ^{-1}(c x)}+5 e^2 \sinh \left (\frac{5 a}{b}\right ) \text{Chi}\left (5 \left (\frac{a}{b}+\sinh ^{-1}(c x)\right )\right )-2 e^2 \cosh \left (\frac{a}{b}\right ) \text{Shi}\left (\frac{a}{b}+\sinh ^{-1}(c x)\right )+9 e^2 \cosh \left (\frac{3 a}{b}\right ) \text{Shi}\left (3 \left (\frac{a}{b}+\sinh ^{-1}(c x)\right )\right )-5 e^2 \cosh \left (\frac{5 a}{b}\right ) \text{Shi}\left (5 \left (\frac{a}{b}+\sinh ^{-1}(c x)\right )\right )}{16 b^2 c^5} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-((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)*CoshIntegr
al[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]] - 2
4*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 + ArcSi
nh[c*x])] - 5*e^2*Cosh[(5*a)/b]*SinhIntegral[5*(a/b + ArcSinh[c*x])])/(16*b^2*c^5)

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Maple [B]  time = 0.221, size = 1036, normalized size = 2.1 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/c*(1/32*(16*c^5*x^5-16*c^4*x^4*(c^2*x^2+1)^(1/2)+20*c^3*x^3-12*c^2*x^2*(c^2*x^2+1)^(1/2)+5*c*x-(c^2*x^2+1)^(
1/2))*e^2/c^4/b/(a+b*arcsinh(c*x))+5/32/c^4*e^2/b^2*exp(5*a/b)*Ei(1,5*arcsinh(c*x)+5*a/b)-1/32/c^4*e^2/b*(16*c
^5*x^5+20*c^3*x^3+16*c^4*x^4*(c^2*x^2+1)^(1/2)+5*c*x+12*c^2*x^2*(c^2*x^2+1)^(1/2)+(c^2*x^2+1)^(1/2))/(a+b*arcs
inh(c*x))-5/32/c^4*e^2/b^2*exp(-5*a/b)*Ei(1,-5*arcsinh(c*x)-5*a/b)+1/2*(c*x-(c^2*x^2+1)^(1/2))*d^2/b/(a+b*arcs
inh(c*x))+1/2*d^2/b^2*exp(a/b)*Ei(1,arcsinh(c*x)+a/b)-1/4*(c*x-(c^2*x^2+1)^(1/2))*d*e/c^2/b/(a+b*arcsinh(c*x))
-1/4/c^2*d*e/b^2*exp(a/b)*Ei(1,arcsinh(c*x)+a/b)+1/16*(c*x-(c^2*x^2+1)^(1/2))*e^2/c^4/b/(a+b*arcsinh(c*x))+1/1
6/c^4*e^2/b^2*exp(a/b)*Ei(1,arcsinh(c*x)+a/b)-1/2/b*d^2*(c*x+(c^2*x^2+1)^(1/2))/(a+b*arcsinh(c*x))-1/2/b^2*d^2
*exp(-a/b)*Ei(1,-arcsinh(c*x)-a/b)+1/4/c^2/b*d*e*(c*x+(c^2*x^2+1)^(1/2))/(a+b*arcsinh(c*x))+1/4/c^2/b^2*d*e*ex
p(-a/b)*Ei(1,-arcsinh(c*x)-a/b)-1/16/c^4/b*e^2*(c*x+(c^2*x^2+1)^(1/2))/(a+b*arcsinh(c*x))-1/16/c^4/b^2*e^2*exp
(-a/b)*Ei(1,-arcsinh(c*x)-a/b)+1/4*(4*c^3*x^3-4*c^2*x^2*(c^2*x^2+1)^(1/2)+3*c*x-(c^2*x^2+1)^(1/2))*d*e/c^2/b/(
a+b*arcsinh(c*x))-3/32*(4*c^3*x^3-4*c^2*x^2*(c^2*x^2+1)^(1/2)+3*c*x-(c^2*x^2+1)^(1/2))*e^2/c^4/b/(a+b*arcsinh(
c*x))+3/4/c^2*e/b^2*exp(3*a/b)*Ei(1,3*arcsinh(c*x)+3*a/b)*d-9/32/c^4*e^2/b^2*exp(3*a/b)*Ei(1,3*arcsinh(c*x)+3*
a/b)-1/4/c^2*e/b*(4*c^3*x^3+3*c*x+4*c^2*x^2*(c^2*x^2+1)^(1/2)+(c^2*x^2+1)^(1/2))/(a+b*arcsinh(c*x))*d+3/32/c^4
*e^2/b*(4*c^3*x^3+3*c*x+4*c^2*x^2*(c^2*x^2+1)^(1/2)+(c^2*x^2+1)^(1/2))/(a+b*arcsinh(c*x))-3/4/c^2*e/b^2*exp(-3
*a/b)*Ei(1,-3*arcsinh(c*x)-3*a/b)*d+9/32/c^4*e^2/b^2*exp(-3*a/b)*Ei(1,-3*arcsinh(c*x)-3*a/b))

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} -\frac{c^{3} e^{2} x^{7} +{\left (2 \, c^{3} d e + c e^{2}\right )} x^{5} + c d^{2} x +{\left (c^{3} d^{2} + 2 \, c d e\right )} x^{3} +{\left (c^{2} e^{2} x^{6} +{\left (2 \, c^{2} d e + e^{2}\right )} x^{4} +{\left (c^{2} d^{2} + 2 \, d e\right )} x^{2} + d^{2}\right )} \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 +{\left (b^{2} c^{3} x^{2} + \sqrt{c^{2} x^{2} + 1} b^{2} c^{2} x + b^{2} c\right )} \log \left (c x + \sqrt{c^{2} x^{2} + 1}\right )} + \int \frac{5 \, c^{5} e^{2} x^{8} + 2 \,{\left (3 \, c^{5} d e + 5 \, c^{3} e^{2}\right )} x^{6} +{\left (c^{5} d^{2} + 12 \, c^{3} d e + 5 \, c e^{2}\right )} x^{4} + c d^{2} + 2 \,{\left (c^{3} d^{2} + 3 \, c d e\right )} x^{2} +{\left (5 \, c^{3} e^{2} x^{6} + 3 \,{\left (2 \, c^{3} d e + c e^{2}\right )} x^{4} - c d^{2} +{\left (c^{3} d^{2} + 2 \, c d e\right )} x^{2}\right )}{\left (c^{2} x^{2} + 1\right )} +{\left (10 \, c^{4} e^{2} x^{7} +{\left (12 \, c^{4} d e + 13 \, c^{2} e^{2}\right )} x^{5} + 2 \,{\left (c^{4} d^{2} + 7 \, c^{2} d e + 2 \, e^{2}\right )} x^{3} +{\left (c^{2} d^{2} + 4 \, d e\right )} x\right )} \sqrt{c^{2} x^{2} + 1}}{a b c^{5} x^{4} +{\left (c^{2} x^{2} + 1\right )} a b c^{3} x^{2} + 2 \, a b c^{3} x^{2} + a b c +{\left (b^{2} c^{5} x^{4} +{\left (c^{2} x^{2} + 1\right )} b^{2} c^{3} x^{2} + 2 \, b^{2} c^{3} x^{2} + b^{2} c + 2 \,{\left (b^{2} c^{4} x^{3} + b^{2} c^{2} x\right )} \sqrt{c^{2} x^{2} + 1}\right )} \log \left (c x + \sqrt{c^{2} x^{2} + 1}\right ) + 2 \,{\left (a b c^{4} x^{3} + a b c^{2} x\right )} \sqrt{c^{2} x^{2} + 1}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-(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)*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^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 + (1
2*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)

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Fricas [F]  time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (\frac{e^{2} x^{4} + 2 \, d e x^{2} + d^{2}}{b^{2} \operatorname{arsinh}\left (c x\right )^{2} + 2 \, a b \operatorname{arsinh}\left (c x\right ) + a^{2}}, x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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)

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\left (d + e x^{2}\right )^{2}}{\left (a + b \operatorname{asinh}{\left (c x \right )}\right )^{2}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{{\left (e x^{2} + d\right )}^{2}}{{\left (b \operatorname{arsinh}\left (c x\right ) + a\right )}^{2}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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