\(\int \frac {(d+e x^2)^2}{(a+b \text {arccosh}(c x))^2} \, dx\) [543]

Optimal result

Integrand size = 20, antiderivative size = 510 \[ \int \frac {\left (d+e x^2\right )^2}{(a+b \text {arccosh}(c x))^2} \, dx=-\frac {d^2 \sqrt {-1+c x} \sqrt {1+c x}}{b c (a+b \text {arccosh}(c x))}-\frac {2 d e x^2 \sqrt {-1+c x} \sqrt {1+c x}}{b c (a+b \text {arccosh}(c x))}-\frac {e^2 x^4 \sqrt {-1+c x} \sqrt {1+c x}}{b c (a+b \text {arccosh}(c x))}+\frac {d^2 \cosh \left (\frac {a}{b}\right ) \text {Chi}\left (\frac {a+b \text {arccosh}(c x)}{b}\right )}{b^2 c}+\frac {d e \cosh \left (\frac {a}{b}\right ) \text {Chi}\left (\frac {a+b \text {arccosh}(c x)}{b}\right )}{2 b^2 c^3}+\frac {e^2 \cosh \left (\frac {a}{b}\right ) \text {Chi}\left (\frac {a+b \text {arccosh}(c x)}{b}\right )}{8 b^2 c^5}+\frac {3 d e \cosh \left (\frac {3 a}{b}\right ) \text {Chi}\left (\frac {3 (a+b \text {arccosh}(c x))}{b}\right )}{2 b^2 c^3}+\frac {9 e^2 \cosh \left (\frac {3 a}{b}\right ) \text {Chi}\left (\frac {3 (a+b \text {arccosh}(c x))}{b}\right )}{16 b^2 c^5}+\frac {5 e^2 \cosh \left (\frac {5 a}{b}\right ) \text {Chi}\left (\frac {5 (a+b \text {arccosh}(c x))}{b}\right )}{16 b^2 c^5}-\frac {d^2 \sinh \left (\frac {a}{b}\right ) \text {Shi}\left (\frac {a+b \text {arccosh}(c x)}{b}\right )}{b^2 c}-\frac {d e \sinh \left (\frac {a}{b}\right ) \text {Shi}\left (\frac {a+b \text {arccosh}(c x)}{b}\right )}{2 b^2 c^3}-\frac {e^2 \sinh \left (\frac {a}{b}\right ) \text {Shi}\left (\frac {a+b \text {arccosh}(c x)}{b}\right )}{8 b^2 c^5}-\frac {3 d e \sinh \left (\frac {3 a}{b}\right ) \text {Shi}\left (\frac {3 (a+b \text {arccosh}(c x))}{b}\right )}{2 b^2 c^3}-\frac {9 e^2 \sinh \left (\frac {3 a}{b}\right ) \text {Shi}\left (\frac {3 (a+b \text {arccosh}(c x))}{b}\right )}{16 b^2 c^5}-\frac {5 e^2 \sinh \left (\frac {5 a}{b}\right ) \text {Shi}\left (\frac {5 (a+b \text {arccosh}(c x))}{b}\right )}{16 b^2 c^5} \]

[Out]

d^2*Chi((a+b*arccosh(c*x))/b)*cosh(a/b)/b^2/c+1/2*d*e*Chi((a+b*arccosh(c*x))/b)*cosh(a/b)/b^2/c^3+1/8*e^2*Chi(
(a+b*arccosh(c*x))/b)*cosh(a/b)/b^2/c^5+3/2*d*e*Chi(3*(a+b*arccosh(c*x))/b)*cosh(3*a/b)/b^2/c^3+9/16*e^2*Chi(3
*(a+b*arccosh(c*x))/b)*cosh(3*a/b)/b^2/c^5+5/16*e^2*Chi(5*(a+b*arccosh(c*x))/b)*cosh(5*a/b)/b^2/c^5-d^2*Shi((a
+b*arccosh(c*x))/b)*sinh(a/b)/b^2/c-1/2*d*e*Shi((a+b*arccosh(c*x))/b)*sinh(a/b)/b^2/c^3-1/8*e^2*Shi((a+b*arcco
sh(c*x))/b)*sinh(a/b)/b^2/c^5-3/2*d*e*Shi(3*(a+b*arccosh(c*x))/b)*sinh(3*a/b)/b^2/c^3-9/16*e^2*Shi(3*(a+b*arcc
osh(c*x))/b)*sinh(3*a/b)/b^2/c^5-5/16*e^2*Shi(5*(a+b*arccosh(c*x))/b)*sinh(5*a/b)/b^2/c^5-d^2*(c*x-1)^(1/2)*(c
*x+1)^(1/2)/b/c/(a+b*arccosh(c*x))-2*d*e*x^2*(c*x-1)^(1/2)*(c*x+1)^(1/2)/b/c/(a+b*arccosh(c*x))-e^2*x^4*(c*x-1
)^(1/2)*(c*x+1)^(1/2)/b/c/(a+b*arccosh(c*x))

Rubi [A] (verified)

Time = 0.62 (sec) , antiderivative size = 510, 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 = {5909, 5880, 5953, 3384, 3379, 3382, 5885} \[ \int \frac {\left (d+e x^2\right )^2}{(a+b \text {arccosh}(c x))^2} \, dx=\frac {e^2 \cosh \left (\frac {a}{b}\right ) \text {Chi}\left (\frac {a+b \text {arccosh}(c x)}{b}\right )}{8 b^2 c^5}+\frac {9 e^2 \cosh \left (\frac {3 a}{b}\right ) \text {Chi}\left (\frac {3 (a+b \text {arccosh}(c x))}{b}\right )}{16 b^2 c^5}+\frac {5 e^2 \cosh \left (\frac {5 a}{b}\right ) \text {Chi}\left (\frac {5 (a+b \text {arccosh}(c x))}{b}\right )}{16 b^2 c^5}-\frac {e^2 \sinh \left (\frac {a}{b}\right ) \text {Shi}\left (\frac {a+b \text {arccosh}(c x)}{b}\right )}{8 b^2 c^5}-\frac {9 e^2 \sinh \left (\frac {3 a}{b}\right ) \text {Shi}\left (\frac {3 (a+b \text {arccosh}(c x))}{b}\right )}{16 b^2 c^5}-\frac {5 e^2 \sinh \left (\frac {5 a}{b}\right ) \text {Shi}\left (\frac {5 (a+b \text {arccosh}(c x))}{b}\right )}{16 b^2 c^5}+\frac {d e \cosh \left (\frac {a}{b}\right ) \text {Chi}\left (\frac {a+b \text {arccosh}(c x)}{b}\right )}{2 b^2 c^3}+\frac {3 d e \cosh \left (\frac {3 a}{b}\right ) \text {Chi}\left (\frac {3 (a+b \text {arccosh}(c x))}{b}\right )}{2 b^2 c^3}-\frac {d e \sinh \left (\frac {a}{b}\right ) \text {Shi}\left (\frac {a+b \text {arccosh}(c x)}{b}\right )}{2 b^2 c^3}-\frac {3 d e \sinh \left (\frac {3 a}{b}\right ) \text {Shi}\left (\frac {3 (a+b \text {arccosh}(c x))}{b}\right )}{2 b^2 c^3}+\frac {d^2 \cosh \left (\frac {a}{b}\right ) \text {Chi}\left (\frac {a+b \text {arccosh}(c x)}{b}\right )}{b^2 c}-\frac {d^2 \sinh \left (\frac {a}{b}\right ) \text {Shi}\left (\frac {a+b \text {arccosh}(c x)}{b}\right )}{b^2 c}-\frac {d^2 \sqrt {c x-1} \sqrt {c x+1}}{b c (a+b \text {arccosh}(c x))}-\frac {2 d e x^2 \sqrt {c x-1} \sqrt {c x+1}}{b c (a+b \text {arccosh}(c x))}-\frac {e^2 x^4 \sqrt {c x-1} \sqrt {c x+1}}{b c (a+b \text {arccosh}(c x))} \]

[In]

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

[Out]

-((d^2*Sqrt[-1 + c*x]*Sqrt[1 + c*x])/(b*c*(a + b*ArcCosh[c*x]))) - (2*d*e*x^2*Sqrt[-1 + c*x]*Sqrt[1 + c*x])/(b
*c*(a + b*ArcCosh[c*x])) - (e^2*x^4*Sqrt[-1 + c*x]*Sqrt[1 + c*x])/(b*c*(a + b*ArcCosh[c*x])) + (d^2*Cosh[a/b]*
CoshIntegral[(a + b*ArcCosh[c*x])/b])/(b^2*c) + (d*e*Cosh[a/b]*CoshIntegral[(a + b*ArcCosh[c*x])/b])/(2*b^2*c^
3) + (e^2*Cosh[a/b]*CoshIntegral[(a + b*ArcCosh[c*x])/b])/(8*b^2*c^5) + (3*d*e*Cosh[(3*a)/b]*CoshIntegral[(3*(
a + b*ArcCosh[c*x]))/b])/(2*b^2*c^3) + (9*e^2*Cosh[(3*a)/b]*CoshIntegral[(3*(a + b*ArcCosh[c*x]))/b])/(16*b^2*
c^5) + (5*e^2*Cosh[(5*a)/b]*CoshIntegral[(5*(a + b*ArcCosh[c*x]))/b])/(16*b^2*c^5) - (d^2*Sinh[a/b]*SinhIntegr
al[(a + b*ArcCosh[c*x])/b])/(b^2*c) - (d*e*Sinh[a/b]*SinhIntegral[(a + b*ArcCosh[c*x])/b])/(2*b^2*c^3) - (e^2*
Sinh[a/b]*SinhIntegral[(a + b*ArcCosh[c*x])/b])/(8*b^2*c^5) - (3*d*e*Sinh[(3*a)/b]*SinhIntegral[(3*(a + b*ArcC
osh[c*x]))/b])/(2*b^2*c^3) - (9*e^2*Sinh[(3*a)/b]*SinhIntegral[(3*(a + b*ArcCosh[c*x]))/b])/(16*b^2*c^5) - (5*
e^2*Sinh[(5*a)/b]*SinhIntegral[(5*(a + b*ArcCosh[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 5880

Int[((a_.) + ArcCosh[(c_.)*(x_)]*(b_.))^(n_), x_Symbol] :> Simp[Sqrt[1 + c*x]*Sqrt[-1 + c*x]*((a + b*ArcCosh[c
*x])^(n + 1)/(b*c*(n + 1))), x] - Dist[c/(b*(n + 1)), Int[x*((a + b*ArcCosh[c*x])^(n + 1)/(Sqrt[1 + c*x]*Sqrt[
-1 + c*x])), x], x] /; FreeQ[{a, b, c}, x] && LtQ[n, -1]

Rule 5885

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

Rule 5909

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

Rule 5953

Int[((a_.) + ArcCosh[(c_.)*(x_)]*(b_.))^(n_.)*(x_)^(m_.)*((d1_) + (e1_.)*(x_))^(p_.)*((d2_) + (e2_.)*(x_))^(p_
.), x_Symbol] :> Dist[(1/(b*c^(m + 1)))*Simp[(d1 + e1*x)^p/(1 + c*x)^p]*Simp[(d2 + e2*x)^p/(-1 + c*x)^p], Subs
t[Int[x^n*Cosh[-a/b + x/b]^m*Sinh[-a/b + x/b]^(2*p + 1), x], x, a + b*ArcCosh[c*x]], x] /; FreeQ[{a, b, c, d1,
e1, d2, e2, n}, x] && EqQ[e1, c*d1] && EqQ[e2, (-c)*d2] && IGtQ[p + 3/2, 0] && IGtQ[m, 0]

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

Mathematica [A] (warning: unable to verify)

Time = 2.58 (sec) , antiderivative size = 663, normalized size of antiderivative = 1.30 \[ \int \frac {\left (d+e x^2\right )^2}{(a+b \text {arccosh}(c x))^2} \, dx=-\frac {16 b c^4 d^2 \sqrt {\frac {-1+c x}{1+c x}}+16 b c^5 d^2 x \sqrt {\frac {-1+c x}{1+c x}}+32 b c^4 d e x^2 \sqrt {\frac {-1+c x}{1+c x}}+32 b c^5 d e x^3 \sqrt {\frac {-1+c x}{1+c x}}+16 b c^4 e^2 x^4 \sqrt {\frac {-1+c x}{1+c x}}+16 b c^5 e^2 x^5 \sqrt {\frac {-1+c x}{1+c x}}-2 \left (8 c^4 d^2+4 c^2 d e+e^2\right ) (a+b \text {arccosh}(c x)) \cosh \left (\frac {a}{b}\right ) \text {Chi}\left (\frac {a}{b}+\text {arccosh}(c x)\right )-3 e \left (8 c^2 d+3 e\right ) (a+b \text {arccosh}(c x)) \cosh \left (\frac {3 a}{b}\right ) \text {Chi}\left (3 \left (\frac {a}{b}+\text {arccosh}(c x)\right )\right )-5 a e^2 \cosh \left (\frac {5 a}{b}\right ) \text {Chi}\left (5 \left (\frac {a}{b}+\text {arccosh}(c x)\right )\right )-5 b e^2 \text {arccosh}(c x) \cosh \left (\frac {5 a}{b}\right ) \text {Chi}\left (5 \left (\frac {a}{b}+\text {arccosh}(c x)\right )\right )+16 a c^4 d^2 \sinh \left (\frac {a}{b}\right ) \text {Shi}\left (\frac {a}{b}+\text {arccosh}(c x)\right )+8 a c^2 d e \sinh \left (\frac {a}{b}\right ) \text {Shi}\left (\frac {a}{b}+\text {arccosh}(c x)\right )+2 a e^2 \sinh \left (\frac {a}{b}\right ) \text {Shi}\left (\frac {a}{b}+\text {arccosh}(c x)\right )+16 b c^4 d^2 \text {arccosh}(c x) \sinh \left (\frac {a}{b}\right ) \text {Shi}\left (\frac {a}{b}+\text {arccosh}(c x)\right )+8 b c^2 d e \text {arccosh}(c x) \sinh \left (\frac {a}{b}\right ) \text {Shi}\left (\frac {a}{b}+\text {arccosh}(c x)\right )+2 b e^2 \text {arccosh}(c x) \sinh \left (\frac {a}{b}\right ) \text {Shi}\left (\frac {a}{b}+\text {arccosh}(c x)\right )+24 a c^2 d e \sinh \left (\frac {3 a}{b}\right ) \text {Shi}\left (3 \left (\frac {a}{b}+\text {arccosh}(c x)\right )\right )+9 a e^2 \sinh \left (\frac {3 a}{b}\right ) \text {Shi}\left (3 \left (\frac {a}{b}+\text {arccosh}(c x)\right )\right )+24 b c^2 d e \text {arccosh}(c x) \sinh \left (\frac {3 a}{b}\right ) \text {Shi}\left (3 \left (\frac {a}{b}+\text {arccosh}(c x)\right )\right )+9 b e^2 \text {arccosh}(c x) \sinh \left (\frac {3 a}{b}\right ) \text {Shi}\left (3 \left (\frac {a}{b}+\text {arccosh}(c x)\right )\right )+5 a e^2 \sinh \left (\frac {5 a}{b}\right ) \text {Shi}\left (5 \left (\frac {a}{b}+\text {arccosh}(c x)\right )\right )+5 b e^2 \text {arccosh}(c x) \sinh \left (\frac {5 a}{b}\right ) \text {Shi}\left (5 \left (\frac {a}{b}+\text {arccosh}(c x)\right )\right )}{16 b^2 c^5 (a+b \text {arccosh}(c x))} \]

[In]

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

[Out]

-1/16*(16*b*c^4*d^2*Sqrt[(-1 + c*x)/(1 + c*x)] + 16*b*c^5*d^2*x*Sqrt[(-1 + c*x)/(1 + c*x)] + 32*b*c^4*d*e*x^2*
Sqrt[(-1 + c*x)/(1 + c*x)] + 32*b*c^5*d*e*x^3*Sqrt[(-1 + c*x)/(1 + c*x)] + 16*b*c^4*e^2*x^4*Sqrt[(-1 + c*x)/(1
+ c*x)] + 16*b*c^5*e^2*x^5*Sqrt[(-1 + c*x)/(1 + c*x)] - 2*(8*c^4*d^2 + 4*c^2*d*e + e^2)*(a + b*ArcCosh[c*x])*
Cosh[a/b]*CoshIntegral[a/b + ArcCosh[c*x]] - 3*e*(8*c^2*d + 3*e)*(a + b*ArcCosh[c*x])*Cosh[(3*a)/b]*CoshIntegr
al[3*(a/b + ArcCosh[c*x])] - 5*a*e^2*Cosh[(5*a)/b]*CoshIntegral[5*(a/b + ArcCosh[c*x])] - 5*b*e^2*ArcCosh[c*x]
*Cosh[(5*a)/b]*CoshIntegral[5*(a/b + ArcCosh[c*x])] + 16*a*c^4*d^2*Sinh[a/b]*SinhIntegral[a/b + ArcCosh[c*x]]
+ 8*a*c^2*d*e*Sinh[a/b]*SinhIntegral[a/b + ArcCosh[c*x]] + 2*a*e^2*Sinh[a/b]*SinhIntegral[a/b + ArcCosh[c*x]]
+ 16*b*c^4*d^2*ArcCosh[c*x]*Sinh[a/b]*SinhIntegral[a/b + ArcCosh[c*x]] + 8*b*c^2*d*e*ArcCosh[c*x]*Sinh[a/b]*Si
nhIntegral[a/b + ArcCosh[c*x]] + 2*b*e^2*ArcCosh[c*x]*Sinh[a/b]*SinhIntegral[a/b + ArcCosh[c*x]] + 24*a*c^2*d*
e*Sinh[(3*a)/b]*SinhIntegral[3*(a/b + ArcCosh[c*x])] + 9*a*e^2*Sinh[(3*a)/b]*SinhIntegral[3*(a/b + ArcCosh[c*x
])] + 24*b*c^2*d*e*ArcCosh[c*x]*Sinh[(3*a)/b]*SinhIntegral[3*(a/b + ArcCosh[c*x])] + 9*b*e^2*ArcCosh[c*x]*Sinh
[(3*a)/b]*SinhIntegral[3*(a/b + ArcCosh[c*x])] + 5*a*e^2*Sinh[(5*a)/b]*SinhIntegral[5*(a/b + ArcCosh[c*x])] +
5*b*e^2*ArcCosh[c*x]*Sinh[(5*a)/b]*SinhIntegral[5*(a/b + ArcCosh[c*x])])/(b^2*c^5*(a + b*ArcCosh[c*x]))

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(1101\) vs. \(2(478)=956\).

Time = 1.98 (sec) , antiderivative size = 1102, normalized size of antiderivative = 2.16

method	result	size
derivativedivides	\(\text {Expression too large to display}\)	\(1102\)
default	\(\text {Expression too large to display}\)	\(1102\)

[In]

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

[Out]

1/c*(1/32*(-16*(c*x+1)^(1/2)*(c*x-1)^(1/2)*c^4*x^4+12*(c*x-1)^(1/2)*(c*x+1)^(1/2)*c^2*x^2-(c*x-1)^(1/2)*(c*x+1
)^(1/2)+16*c^5*x^5-20*c^3*x^3+5*c*x)*e^2/c^4/b/(a+b*arccosh(c*x))-5/32*e^2/c^4/b^2*exp(5*a/b)*Ei(1,5*arccosh(c
*x)+5*a/b)-1/32/b*e^2/c^4*(16*c^5*x^5-20*c^3*x^3+16*(c*x+1)^(1/2)*(c*x-1)^(1/2)*c^4*x^4+5*c*x-12*(c*x-1)^(1/2)
*(c*x+1)^(1/2)*c^2*x^2+(c*x-1)^(1/2)*(c*x+1)^(1/2))/(a+b*arccosh(c*x))-5/32/b^2*e^2/c^4*exp(-5*a/b)*Ei(1,-5*ar
ccosh(c*x)-5*a/b)+1/2*(-(c*x-1)^(1/2)*(c*x+1)^(1/2)+c*x)*d^2/b/(a+b*arccosh(c*x))-1/2*d^2/b^2*exp(a/b)*Ei(1,ar
ccosh(c*x)+a/b)+1/4*(-(c*x-1)^(1/2)*(c*x+1)^(1/2)+c*x)*d*e/c^2/b/(a+b*arccosh(c*x))-1/4/c^2*d*e/b^2*exp(a/b)*E
i(1,arccosh(c*x)+a/b)+1/16*(-(c*x-1)^(1/2)*(c*x+1)^(1/2)+c*x)*e^2/c^4/b/(a+b*arccosh(c*x))-1/16/c^4*e^2/b^2*ex
p(a/b)*Ei(1,arccosh(c*x)+a/b)-1/2/b*d^2*(c*x+(c*x-1)^(1/2)*(c*x+1)^(1/2))/(a+b*arccosh(c*x))-1/2/b^2*d^2*exp(-
a/b)*Ei(1,-arccosh(c*x)-a/b)-1/4/c^2/b*d*e*(c*x+(c*x-1)^(1/2)*(c*x+1)^(1/2))/(a+b*arccosh(c*x))-1/4/c^2/b^2*d*
e*exp(-a/b)*Ei(1,-arccosh(c*x)-a/b)-1/16/c^4/b*e^2*(c*x+(c*x-1)^(1/2)*(c*x+1)^(1/2))/(a+b*arccosh(c*x))-1/16/c
^4/b^2*e^2*exp(-a/b)*Ei(1,-arccosh(c*x)-a/b)+1/4*(-4*(c*x-1)^(1/2)*(c*x+1)^(1/2)*c^2*x^2+(c*x-1)^(1/2)*(c*x+1)
^(1/2)+4*c^3*x^3-3*c*x)*d*e/c^2/b/(a+b*arccosh(c*x))+3/32*(-4*(c*x-1)^(1/2)*(c*x+1)^(1/2)*c^2*x^2+(c*x-1)^(1/2
)*(c*x+1)^(1/2)+4*c^3*x^3-3*c*x)*e^2/c^4/b/(a+b*arccosh(c*x))-3/4*e/c^2/b^2*exp(3*a/b)*Ei(1,3*arccosh(c*x)+3*a
/b)*d-9/32*e^2/c^4/b^2*exp(3*a/b)*Ei(1,3*arccosh(c*x)+3*a/b)-1/4/c^2*e/b*(4*c^3*x^3-3*c*x+4*(c*x-1)^(1/2)*(c*x
+1)^(1/2)*c^2*x^2-(c*x-1)^(1/2)*(c*x+1)^(1/2))/(a+b*arccosh(c*x))*d-3/32/c^4*e^2/b*(4*c^3*x^3-3*c*x+4*(c*x-1)^
(1/2)*(c*x+1)^(1/2)*c^2*x^2-(c*x-1)^(1/2)*(c*x+1)^(1/2))/(a+b*arccosh(c*x))-3/4/c^2*e/b^2*exp(-3*a/b)*Ei(1,-3*
arccosh(c*x)-3*a/b)*d-9/32/c^4*e^2/b^2*exp(-3*a/b)*Ei(1,-3*arccosh(c*x)-3*a/b))

Fricas [F]

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

[In]

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

[Out]

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

Sympy [F]

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

[In]

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

[Out]

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

Maxima [F]

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

[In]

integrate((e*x^2+d)^2/(a+b*arccosh(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*x + 1)*sqrt(c*x - 1))/(a*b*c^3*x^2 + sqrt(c*x + 1)*sqrt(c*x - 1)*a*
b*c^2*x - a*b*c + (b^2*c^3*x^2 + sqrt(c*x + 1)*sqrt(c*x - 1)*b^2*c^2*x - b^2*c)*log(c*x + sqrt(c*x + 1)*sqrt(c
*x - 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 +
(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*x + 1)*(c*x - 1) + c*d^2 - 2*
(c^3*d^2 - 3*c*d*e)*x^2 + (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*x + 1)*sqrt(c*x - 1))/(a*b*c^5*x^4 + (c*x + 1)*(c*x - 1)*a*b*c^3*x^2 - 2*a*b*c
^3*x^2 + a*b*c + 2*(a*b*c^4*x^3 - a*b*c^2*x)*sqrt(c*x + 1)*sqrt(c*x - 1) + (b^2*c^5*x^4 + (c*x + 1)*(c*x - 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*x + 1)*sqrt(c*x - 1))*log(c*x + sqrt(
c*x + 1)*sqrt(c*x - 1))), x)

Giac [F]

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

[In]

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

[Out]

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

Mupad [F(-1)]

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

[In]

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

[Out]

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