\(\int \frac {(d+e x^2)^2}{(a+b \text {arcsinh}(c x))^2} \, dx\) [624]

Optimal result

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

[Out]

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*arcsinh(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*cosh(5*a/b)*Shi(5*(a+b*arcsinh(c*x))/b)/b^2/c^5-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*arcsi
nh(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*arcs
inh(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*(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))

Rubi [A] (verified)

Time = 0.56 (sec) , antiderivative size = 495, normalized size of antiderivative = 1.00, number of steps used = 26, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.350, Rules used = {5793, 5773, 5819, 3384, 3379, 3382, 5778} \[ \int \frac {\left (d+e x^2\right )^2}{(a+b \text {arcsinh}(c x))^2} \, dx=-\frac {e^2 \sinh \left (\frac {a}{b}\right ) \text {Chi}\left (\frac {a+b \text {arcsinh}(c x)}{b}\right )}{8 b^2 c^5}+\frac {9 e^2 \sinh \left (\frac {3 a}{b}\right ) \text {Chi}\left (\frac {3 (a+b \text {arcsinh}(c x))}{b}\right )}{16 b^2 c^5}-\frac {5 e^2 \sinh \left (\frac {5 a}{b}\right ) \text {Chi}\left (\frac {5 (a+b \text {arcsinh}(c x))}{b}\right )}{16 b^2 c^5}+\frac {e^2 \cosh \left (\frac {a}{b}\right ) \text {Shi}\left (\frac {a+b \text {arcsinh}(c x)}{b}\right )}{8 b^2 c^5}-\frac {9 e^2 \cosh \left (\frac {3 a}{b}\right ) \text {Shi}\left (\frac {3 (a+b \text {arcsinh}(c x))}{b}\right )}{16 b^2 c^5}+\frac {5 e^2 \cosh \left (\frac {5 a}{b}\right ) \text {Shi}\left (\frac {5 (a+b \text {arcsinh}(c x))}{b}\right )}{16 b^2 c^5}+\frac {d e \sinh \left (\frac {a}{b}\right ) \text {Chi}\left (\frac {a+b \text {arcsinh}(c x)}{b}\right )}{2 b^2 c^3}-\frac {3 d e \sinh \left (\frac {3 a}{b}\right ) \text {Chi}\left (\frac {3 (a+b \text {arcsinh}(c x))}{b}\right )}{2 b^2 c^3}-\frac {d e \cosh \left (\frac {a}{b}\right ) \text {Shi}\left (\frac {a+b \text {arcsinh}(c x)}{b}\right )}{2 b^2 c^3}+\frac {3 d e \cosh \left (\frac {3 a}{b}\right ) \text {Shi}\left (\frac {3 (a+b \text {arcsinh}(c x))}{b}\right )}{2 b^2 c^3}-\frac {d^2 \sinh \left (\frac {a}{b}\right ) \text {Chi}\left (\frac {a+b \text {arcsinh}(c x)}{b}\right )}{b^2 c}+\frac {d^2 \cosh \left (\frac {a}{b}\right ) \text {Shi}\left (\frac {a+b \text {arcsinh}(c x)}{b}\right )}{b^2 c}-\frac {d^2 \sqrt {c^2 x^2+1}}{b c (a+b \text {arcsinh}(c x))}-\frac {2 d e x^2 \sqrt {c^2 x^2+1}}{b c (a+b \text {arcsinh}(c x))}-\frac {e^2 x^4 \sqrt {c^2 x^2+1}}{b c (a+b \text {arcsinh}(c x))} \]

[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])/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)

Rule 3379

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 3382

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 3384

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 5773

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 5778

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^2*c^(m + 1)*(n + 1)), Subst[Int[ExpandTrigReduce[x^(n + 1), Si
nh[-a/b + x/b]^(m - 1)*(m + (m + 1)*Sinh[-a/b + x/b]^2), x], x], x, a + b*ArcSinh[c*x]], x] /; FreeQ[{a, b, c}
, x] && IGtQ[m, 0] && GeQ[n, -2] && LtQ[n, -1]

Rule 5793

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 5819

Int[((a_.) + ArcSinh[(c_.)*(x_)]*(b_.))^(n_.)*(x_)^(m_.)*((d_) + (e_.)*(x_)^2)^(p_.), x_Symbol] :> Dist[(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]

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

Mathematica [A] (verified)

Time = 2.18 (sec) , antiderivative size = 356, normalized size of antiderivative = 0.72 \[ \int \frac {\left (d+e x^2\right )^2}{(a+b \text {arcsinh}(c x))^2} \, dx=-\frac {\frac {16 b c^4 d^2 \sqrt {1+c^2 x^2}}{a+b \text {arcsinh}(c x)}+\frac {32 b c^4 d e x^2 \sqrt {1+c^2 x^2}}{a+b \text {arcsinh}(c x)}+\frac {16 b c^4 e^2 x^4 \sqrt {1+c^2 x^2}}{a+b \text {arcsinh}(c x)}+2 \left (8 c^4 d^2-4 c^2 d e+e^2\right ) \text {Chi}\left (\frac {a}{b}+\text {arcsinh}(c x)\right ) \sinh \left (\frac {a}{b}\right )+3 \left (8 c^2 d-3 e\right ) e \text {Chi}\left (3 \left (\frac {a}{b}+\text {arcsinh}(c x)\right )\right ) \sinh \left (\frac {3 a}{b}\right )+5 e^2 \text {Chi}\left (5 \left (\frac {a}{b}+\text {arcsinh}(c x)\right )\right ) \sinh \left (\frac {5 a}{b}\right )-16 c^4 d^2 \cosh \left (\frac {a}{b}\right ) \text {Shi}\left (\frac {a}{b}+\text {arcsinh}(c x)\right )+8 c^2 d e \cosh \left (\frac {a}{b}\right ) \text {Shi}\left (\frac {a}{b}+\text {arcsinh}(c x)\right )-2 e^2 \cosh \left (\frac {a}{b}\right ) \text {Shi}\left (\frac {a}{b}+\text {arcsinh}(c x)\right )-24 c^2 d e \cosh \left (\frac {3 a}{b}\right ) \text {Shi}\left (3 \left (\frac {a}{b}+\text {arcsinh}(c x)\right )\right )+9 e^2 \cosh \left (\frac {3 a}{b}\right ) \text {Shi}\left (3 \left (\frac {a}{b}+\text {arcsinh}(c x)\right )\right )-5 e^2 \cosh \left (\frac {5 a}{b}\right ) \text {Shi}\left (5 \left (\frac {a}{b}+\text {arcsinh}(c x)\right )\right )}{16 b^2 c^5} \]

[In]

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

[Out]

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

Maple [B] (verified)

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

Time = 0.97 (sec) , antiderivative size = 1036, normalized size of antiderivative = 2.09

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

[In]

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

[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*e^2/c^4/b^2*exp(5*a/b)*Ei(1,5*arcsinh(c*x)+5*a/b)-1/32/b*e^2/c^4*(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/b^2*e^2/c^4*exp(-5*a/b)*Ei(1,-5*arcsinh(c*x)-5*a/b)+1/2*(-(c^2*x^2+1)^(1/2)+c*x)*d^2/b/(a+b*arc
sinh(c*x))+1/2*d^2/b^2*exp(a/b)*Ei(1,arcsinh(c*x)+a/b)-1/4*(-(c^2*x^2+1)^(1/2)+c*x)*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^2*x^2+1)^(1/2)+c*x)*e^2/c^4/b/(a+b*arcsinh(c*x))+
1/16/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
*exp(-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*arcsi
nh(c*x))+3/4*e/c^2/b^2*exp(3*a/b)*Ei(1,3*arcsinh(c*x)+3*a/b)*d-9/32*e^2/c^4/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))

Fricas [F]

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

[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)

Sympy [F]

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

[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)

Maxima [F]

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

[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)

Giac [F]

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

[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)

Mupad [F(-1)]

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

[In]

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

[Out]

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