3.4.35 \(\int \frac {x^2 (1-c^2 x^2)^{5/2}}{(a+b \cosh ^{-1}(c x))^2} \, dx\) [335]

Optimal. Leaf size=454 \[ -\frac {x^2 \sqrt {-1+c x} \sqrt {1+c x} \left (1-c^2 x^2\right )^{5/2}}{b c \left (a+b \cosh ^{-1}(c x)\right )}-\frac {\sqrt {1-c x} \text {Chi}\left (\frac {2 \left (a+b \cosh ^{-1}(c x)\right )}{b}\right ) \sinh \left (\frac {2 a}{b}\right )}{16 b^2 c^3 \sqrt {-1+c x}}-\frac {\sqrt {1-c x} \text {Chi}\left (\frac {4 \left (a+b \cosh ^{-1}(c x)\right )}{b}\right ) \sinh \left (\frac {4 a}{b}\right )}{8 b^2 c^3 \sqrt {-1+c x}}+\frac {3 \sqrt {1-c x} \text {Chi}\left (\frac {6 \left (a+b \cosh ^{-1}(c x)\right )}{b}\right ) \sinh \left (\frac {6 a}{b}\right )}{16 b^2 c^3 \sqrt {-1+c x}}-\frac {\sqrt {1-c x} \text {Chi}\left (\frac {8 \left (a+b \cosh ^{-1}(c x)\right )}{b}\right ) \sinh \left (\frac {8 a}{b}\right )}{16 b^2 c^3 \sqrt {-1+c x}}+\frac {\sqrt {1-c x} \cosh \left (\frac {2 a}{b}\right ) \text {Shi}\left (\frac {2 \left (a+b \cosh ^{-1}(c x)\right )}{b}\right )}{16 b^2 c^3 \sqrt {-1+c x}}+\frac {\sqrt {1-c x} \cosh \left (\frac {4 a}{b}\right ) \text {Shi}\left (\frac {4 \left (a+b \cosh ^{-1}(c x)\right )}{b}\right )}{8 b^2 c^3 \sqrt {-1+c x}}-\frac {3 \sqrt {1-c x} \cosh \left (\frac {6 a}{b}\right ) \text {Shi}\left (\frac {6 \left (a+b \cosh ^{-1}(c x)\right )}{b}\right )}{16 b^2 c^3 \sqrt {-1+c x}}+\frac {\sqrt {1-c x} \cosh \left (\frac {8 a}{b}\right ) \text {Shi}\left (\frac {8 \left (a+b \cosh ^{-1}(c x)\right )}{b}\right )}{16 b^2 c^3 \sqrt {-1+c x}} \]

[Out]

1/16*cosh(2*a/b)*Shi(2*(a+b*arccosh(c*x))/b)*(-c*x+1)^(1/2)/b^2/c^3/(c*x-1)^(1/2)+1/8*cosh(4*a/b)*Shi(4*(a+b*a
rccosh(c*x))/b)*(-c*x+1)^(1/2)/b^2/c^3/(c*x-1)^(1/2)-3/16*cosh(6*a/b)*Shi(6*(a+b*arccosh(c*x))/b)*(-c*x+1)^(1/
2)/b^2/c^3/(c*x-1)^(1/2)+1/16*cosh(8*a/b)*Shi(8*(a+b*arccosh(c*x))/b)*(-c*x+1)^(1/2)/b^2/c^3/(c*x-1)^(1/2)-1/1
6*Chi(2*(a+b*arccosh(c*x))/b)*sinh(2*a/b)*(-c*x+1)^(1/2)/b^2/c^3/(c*x-1)^(1/2)-1/8*Chi(4*(a+b*arccosh(c*x))/b)
*sinh(4*a/b)*(-c*x+1)^(1/2)/b^2/c^3/(c*x-1)^(1/2)+3/16*Chi(6*(a+b*arccosh(c*x))/b)*sinh(6*a/b)*(-c*x+1)^(1/2)/
b^2/c^3/(c*x-1)^(1/2)-1/16*Chi(8*(a+b*arccosh(c*x))/b)*sinh(8*a/b)*(-c*x+1)^(1/2)/b^2/c^3/(c*x-1)^(1/2)-x^2*(-
c^2*x^2+1)^(5/2)*(c*x-1)^(1/2)*(c*x+1)^(1/2)/b/c/(a+b*arccosh(c*x))

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Rubi [A]
time = 0.83, antiderivative size = 454, normalized size of antiderivative = 1.00, number of steps used = 30, number of rules used = 7, integrand size = 28, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {5942, 5912, 5952, 5556, 3384, 3379, 3382} \begin {gather*} -\frac {\sqrt {1-c x} \sinh \left (\frac {2 a}{b}\right ) \text {Chi}\left (\frac {2 \left (a+b \cosh ^{-1}(c x)\right )}{b}\right )}{16 b^2 c^3 \sqrt {c x-1}}-\frac {\sqrt {1-c x} \sinh \left (\frac {4 a}{b}\right ) \text {Chi}\left (\frac {4 \left (a+b \cosh ^{-1}(c x)\right )}{b}\right )}{8 b^2 c^3 \sqrt {c x-1}}+\frac {3 \sqrt {1-c x} \sinh \left (\frac {6 a}{b}\right ) \text {Chi}\left (\frac {6 \left (a+b \cosh ^{-1}(c x)\right )}{b}\right )}{16 b^2 c^3 \sqrt {c x-1}}-\frac {\sqrt {1-c x} \sinh \left (\frac {8 a}{b}\right ) \text {Chi}\left (\frac {8 \left (a+b \cosh ^{-1}(c x)\right )}{b}\right )}{16 b^2 c^3 \sqrt {c x-1}}+\frac {\sqrt {1-c x} \cosh \left (\frac {2 a}{b}\right ) \text {Shi}\left (\frac {2 \left (a+b \cosh ^{-1}(c x)\right )}{b}\right )}{16 b^2 c^3 \sqrt {c x-1}}+\frac {\sqrt {1-c x} \cosh \left (\frac {4 a}{b}\right ) \text {Shi}\left (\frac {4 \left (a+b \cosh ^{-1}(c x)\right )}{b}\right )}{8 b^2 c^3 \sqrt {c x-1}}-\frac {3 \sqrt {1-c x} \cosh \left (\frac {6 a}{b}\right ) \text {Shi}\left (\frac {6 \left (a+b \cosh ^{-1}(c x)\right )}{b}\right )}{16 b^2 c^3 \sqrt {c x-1}}+\frac {\sqrt {1-c x} \cosh \left (\frac {8 a}{b}\right ) \text {Shi}\left (\frac {8 \left (a+b \cosh ^{-1}(c x)\right )}{b}\right )}{16 b^2 c^3 \sqrt {c x-1}}-\frac {x^2 \sqrt {c x-1} \sqrt {c x+1} \left (1-c^2 x^2\right )^{5/2}}{b c \left (a+b \cosh ^{-1}(c x)\right )} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(x^2*(1 - c^2*x^2)^(5/2))/(a + b*ArcCosh[c*x])^2,x]

[Out]

-((x^2*Sqrt[-1 + c*x]*Sqrt[1 + c*x]*(1 - c^2*x^2)^(5/2))/(b*c*(a + b*ArcCosh[c*x]))) - (Sqrt[1 - c*x]*CoshInte
gral[(2*(a + b*ArcCosh[c*x]))/b]*Sinh[(2*a)/b])/(16*b^2*c^3*Sqrt[-1 + c*x]) - (Sqrt[1 - c*x]*CoshIntegral[(4*(
a + b*ArcCosh[c*x]))/b]*Sinh[(4*a)/b])/(8*b^2*c^3*Sqrt[-1 + c*x]) + (3*Sqrt[1 - c*x]*CoshIntegral[(6*(a + b*Ar
cCosh[c*x]))/b]*Sinh[(6*a)/b])/(16*b^2*c^3*Sqrt[-1 + c*x]) - (Sqrt[1 - c*x]*CoshIntegral[(8*(a + b*ArcCosh[c*x
]))/b]*Sinh[(8*a)/b])/(16*b^2*c^3*Sqrt[-1 + c*x]) + (Sqrt[1 - c*x]*Cosh[(2*a)/b]*SinhIntegral[(2*(a + b*ArcCos
h[c*x]))/b])/(16*b^2*c^3*Sqrt[-1 + c*x]) + (Sqrt[1 - c*x]*Cosh[(4*a)/b]*SinhIntegral[(4*(a + b*ArcCosh[c*x]))/
b])/(8*b^2*c^3*Sqrt[-1 + c*x]) - (3*Sqrt[1 - c*x]*Cosh[(6*a)/b]*SinhIntegral[(6*(a + b*ArcCosh[c*x]))/b])/(16*
b^2*c^3*Sqrt[-1 + c*x]) + (Sqrt[1 - c*x]*Cosh[(8*a)/b]*SinhIntegral[(8*(a + b*ArcCosh[c*x]))/b])/(16*b^2*c^3*S
qrt[-1 + c*x])

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 5556

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 5912

Int[((a_.) + ArcCosh[(c_.)*(x_)]*(b_.))^(n_.)*((f_.)*(x_))^(m_.)*((d1_) + (e1_.)*(x_))^(p_.)*((d2_) + (e2_.)*(
x_))^(p_.), x_Symbol] :> Int[(f*x)^m*(d1*d2 + e1*e2*x^2)^p*(a + b*ArcCosh[c*x])^n, x] /; FreeQ[{a, b, c, d1, e
1, d2, e2, f, m, n}, x] && EqQ[d2*e1 + d1*e2, 0] && IntegerQ[p]

Rule 5942

Int[((a_.) + ArcCosh[(c_.)*(x_)]*(b_.))^(n_)*((f_.)*(x_))^(m_.)*((d_) + (e_.)*(x_)^2)^(p_.), x_Symbol] :> Simp
[(f*x)^m*Simp[Sqrt[1 + c*x]*Sqrt[-1 + c*x]*(d + e*x^2)^p]*((a + b*ArcCosh[c*x])^(n + 1)/(b*c*(n + 1))), x] + (
Dist[f*(m/(b*c*(n + 1)))*Simp[(d + e*x^2)^p/((1 + c*x)^p*(-1 + c*x)^p)], Int[(f*x)^(m - 1)*(1 + c*x)^(p - 1/2)
*(-1 + c*x)^(p - 1/2)*(a + b*ArcCosh[c*x])^(n + 1), x], x] - Dist[c*((m + 2*p + 1)/(b*f*(n + 1)))*Simp[(d + e*
x^2)^p/((1 + c*x)^p*(-1 + c*x)^p)], Int[(f*x)^(m + 1)*(1 + c*x)^(p - 1/2)*(-1 + c*x)^(p - 1/2)*(a + b*ArcCosh[
c*x])^(n + 1), x], x]) /; FreeQ[{a, b, c, d, e, f, m, p}, x] && EqQ[c^2*d + e, 0] && LtQ[n, -1] && IGtQ[2*p, 0
] && NeQ[m + 2*p + 1, 0] && IGtQ[m, -3]

Rule 5952

Int[((a_.) + ArcCosh[(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*x)^p*(-1 + c*x)^p)], Subst[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, d, e, n}, x] && EqQ[c^2*d + e, 0] && IGtQ[2*p + 2
, 0] && IGtQ[m, 0]

Rubi steps

\begin {align*} \int \frac {x^2 \left (1-c^2 x^2\right )^{5/2}}{\left (a+b \cosh ^{-1}(c x)\right )^2} \, dx &=\frac {\sqrt {1-c^2 x^2} \int \frac {x^2 (-1+c x)^{5/2} (1+c x)^{5/2}}{\left (a+b \cosh ^{-1}(c x)\right )^2} \, dx}{\sqrt {-1+c x} \sqrt {1+c x}}\\ &=\frac {x^2 (1-c x)^3 (1+c x)^{5/2} \sqrt {1-c^2 x^2}}{b c \sqrt {-1+c x} \left (a+b \cosh ^{-1}(c x)\right )}-\frac {\left (2 \sqrt {1-c^2 x^2}\right ) \int \frac {x \left (-1+c^2 x^2\right )^2}{a+b \cosh ^{-1}(c x)} \, dx}{b c \sqrt {-1+c x} \sqrt {1+c x}}+\frac {\left (8 c \sqrt {1-c^2 x^2}\right ) \int \frac {x^3 \left (-1+c^2 x^2\right )^2}{a+b \cosh ^{-1}(c x)} \, dx}{b \sqrt {-1+c x} \sqrt {1+c x}}\\ &=\frac {x^2 (1-c x)^3 (1+c x)^{5/2} \sqrt {1-c^2 x^2}}{b c \sqrt {-1+c x} \left (a+b \cosh ^{-1}(c x)\right )}-\frac {\left (2 \sqrt {1-c^2 x^2}\right ) \text {Subst}\left (\int \frac {\cosh (x) \sinh ^5(x)}{a+b x} \, dx,x,\cosh ^{-1}(c x)\right )}{b c^3 \sqrt {-1+c x} \sqrt {1+c x}}+\frac {\left (8 \sqrt {1-c^2 x^2}\right ) \text {Subst}\left (\int \frac {\cosh ^3(x) \sinh ^5(x)}{a+b x} \, dx,x,\cosh ^{-1}(c x)\right )}{b c^3 \sqrt {-1+c x} \sqrt {1+c x}}\\ &=\frac {x^2 (1-c x)^3 (1+c x)^{5/2} \sqrt {1-c^2 x^2}}{b c \sqrt {-1+c x} \left (a+b \cosh ^{-1}(c x)\right )}-\frac {\left (2 \sqrt {1-c^2 x^2}\right ) \text {Subst}\left (\int \left (\frac {5 \sinh (2 x)}{32 (a+b x)}-\frac {\sinh (4 x)}{8 (a+b x)}+\frac {\sinh (6 x)}{32 (a+b x)}\right ) \, dx,x,\cosh ^{-1}(c x)\right )}{b c^3 \sqrt {-1+c x} \sqrt {1+c x}}+\frac {\left (8 \sqrt {1-c^2 x^2}\right ) \text {Subst}\left (\int \left (\frac {3 \sinh (2 x)}{64 (a+b x)}-\frac {\sinh (4 x)}{64 (a+b x)}-\frac {\sinh (6 x)}{64 (a+b x)}+\frac {\sinh (8 x)}{128 (a+b x)}\right ) \, dx,x,\cosh ^{-1}(c x)\right )}{b c^3 \sqrt {-1+c x} \sqrt {1+c x}}\\ &=\frac {x^2 (1-c x)^3 (1+c x)^{5/2} \sqrt {1-c^2 x^2}}{b c \sqrt {-1+c x} \left (a+b \cosh ^{-1}(c x)\right )}-\frac {\sqrt {1-c^2 x^2} \text {Subst}\left (\int \frac {\sinh (6 x)}{a+b x} \, dx,x,\cosh ^{-1}(c x)\right )}{16 b c^3 \sqrt {-1+c x} \sqrt {1+c x}}+\frac {\sqrt {1-c^2 x^2} \text {Subst}\left (\int \frac {\sinh (8 x)}{a+b x} \, dx,x,\cosh ^{-1}(c x)\right )}{16 b c^3 \sqrt {-1+c x} \sqrt {1+c x}}-\frac {\sqrt {1-c^2 x^2} \text {Subst}\left (\int \frac {\sinh (4 x)}{a+b x} \, dx,x,\cosh ^{-1}(c x)\right )}{8 b c^3 \sqrt {-1+c x} \sqrt {1+c x}}-\frac {\sqrt {1-c^2 x^2} \text {Subst}\left (\int \frac {\sinh (6 x)}{a+b x} \, dx,x,\cosh ^{-1}(c x)\right )}{8 b c^3 \sqrt {-1+c x} \sqrt {1+c x}}+\frac {\sqrt {1-c^2 x^2} \text {Subst}\left (\int \frac {\sinh (4 x)}{a+b x} \, dx,x,\cosh ^{-1}(c x)\right )}{4 b c^3 \sqrt {-1+c x} \sqrt {1+c x}}-\frac {\left (5 \sqrt {1-c^2 x^2}\right ) \text {Subst}\left (\int \frac {\sinh (2 x)}{a+b x} \, dx,x,\cosh ^{-1}(c x)\right )}{16 b c^3 \sqrt {-1+c x} \sqrt {1+c x}}+\frac {\left (3 \sqrt {1-c^2 x^2}\right ) \text {Subst}\left (\int \frac {\sinh (2 x)}{a+b x} \, dx,x,\cosh ^{-1}(c x)\right )}{8 b c^3 \sqrt {-1+c x} \sqrt {1+c x}}\\ &=\frac {x^2 (1-c x)^3 (1+c x)^{5/2} \sqrt {1-c^2 x^2}}{b c \sqrt {-1+c x} \left (a+b \cosh ^{-1}(c x)\right )}-\frac {\left (5 \sqrt {1-c^2 x^2} \cosh \left (\frac {2 a}{b}\right )\right ) \text {Subst}\left (\int \frac {\sinh \left (\frac {2 a}{b}+2 x\right )}{a+b x} \, dx,x,\cosh ^{-1}(c x)\right )}{16 b c^3 \sqrt {-1+c x} \sqrt {1+c x}}+\frac {\left (3 \sqrt {1-c^2 x^2} \cosh \left (\frac {2 a}{b}\right )\right ) \text {Subst}\left (\int \frac {\sinh \left (\frac {2 a}{b}+2 x\right )}{a+b x} \, dx,x,\cosh ^{-1}(c x)\right )}{8 b c^3 \sqrt {-1+c x} \sqrt {1+c x}}-\frac {\left (\sqrt {1-c^2 x^2} \cosh \left (\frac {4 a}{b}\right )\right ) \text {Subst}\left (\int \frac {\sinh \left (\frac {4 a}{b}+4 x\right )}{a+b x} \, dx,x,\cosh ^{-1}(c x)\right )}{8 b c^3 \sqrt {-1+c x} \sqrt {1+c x}}+\frac {\left (\sqrt {1-c^2 x^2} \cosh \left (\frac {4 a}{b}\right )\right ) \text {Subst}\left (\int \frac {\sinh \left (\frac {4 a}{b}+4 x\right )}{a+b x} \, dx,x,\cosh ^{-1}(c x)\right )}{4 b c^3 \sqrt {-1+c x} \sqrt {1+c x}}-\frac {\left (\sqrt {1-c^2 x^2} \cosh \left (\frac {6 a}{b}\right )\right ) \text {Subst}\left (\int \frac {\sinh \left (\frac {6 a}{b}+6 x\right )}{a+b x} \, dx,x,\cosh ^{-1}(c x)\right )}{16 b c^3 \sqrt {-1+c x} \sqrt {1+c x}}-\frac {\left (\sqrt {1-c^2 x^2} \cosh \left (\frac {6 a}{b}\right )\right ) \text {Subst}\left (\int \frac {\sinh \left (\frac {6 a}{b}+6 x\right )}{a+b x} \, dx,x,\cosh ^{-1}(c x)\right )}{8 b c^3 \sqrt {-1+c x} \sqrt {1+c x}}+\frac {\left (\sqrt {1-c^2 x^2} \cosh \left (\frac {8 a}{b}\right )\right ) \text {Subst}\left (\int \frac {\sinh \left (\frac {8 a}{b}+8 x\right )}{a+b x} \, dx,x,\cosh ^{-1}(c x)\right )}{16 b c^3 \sqrt {-1+c x} \sqrt {1+c x}}+\frac {\left (5 \sqrt {1-c^2 x^2} \sinh \left (\frac {2 a}{b}\right )\right ) \text {Subst}\left (\int \frac {\cosh \left (\frac {2 a}{b}+2 x\right )}{a+b x} \, dx,x,\cosh ^{-1}(c x)\right )}{16 b c^3 \sqrt {-1+c x} \sqrt {1+c x}}-\frac {\left (3 \sqrt {1-c^2 x^2} \sinh \left (\frac {2 a}{b}\right )\right ) \text {Subst}\left (\int \frac {\cosh \left (\frac {2 a}{b}+2 x\right )}{a+b x} \, dx,x,\cosh ^{-1}(c x)\right )}{8 b c^3 \sqrt {-1+c x} \sqrt {1+c x}}+\frac {\left (\sqrt {1-c^2 x^2} \sinh \left (\frac {4 a}{b}\right )\right ) \text {Subst}\left (\int \frac {\cosh \left (\frac {4 a}{b}+4 x\right )}{a+b x} \, dx,x,\cosh ^{-1}(c x)\right )}{8 b c^3 \sqrt {-1+c x} \sqrt {1+c x}}-\frac {\left (\sqrt {1-c^2 x^2} \sinh \left (\frac {4 a}{b}\right )\right ) \text {Subst}\left (\int \frac {\cosh \left (\frac {4 a}{b}+4 x\right )}{a+b x} \, dx,x,\cosh ^{-1}(c x)\right )}{4 b c^3 \sqrt {-1+c x} \sqrt {1+c x}}+\frac {\left (\sqrt {1-c^2 x^2} \sinh \left (\frac {6 a}{b}\right )\right ) \text {Subst}\left (\int \frac {\cosh \left (\frac {6 a}{b}+6 x\right )}{a+b x} \, dx,x,\cosh ^{-1}(c x)\right )}{16 b c^3 \sqrt {-1+c x} \sqrt {1+c x}}+\frac {\left (\sqrt {1-c^2 x^2} \sinh \left (\frac {6 a}{b}\right )\right ) \text {Subst}\left (\int \frac {\cosh \left (\frac {6 a}{b}+6 x\right )}{a+b x} \, dx,x,\cosh ^{-1}(c x)\right )}{8 b c^3 \sqrt {-1+c x} \sqrt {1+c x}}-\frac {\left (\sqrt {1-c^2 x^2} \sinh \left (\frac {8 a}{b}\right )\right ) \text {Subst}\left (\int \frac {\cosh \left (\frac {8 a}{b}+8 x\right )}{a+b x} \, dx,x,\cosh ^{-1}(c x)\right )}{16 b c^3 \sqrt {-1+c x} \sqrt {1+c x}}\\ &=\frac {x^2 (1-c x)^3 (1+c x)^{5/2} \sqrt {1-c^2 x^2}}{b c \sqrt {-1+c x} \left (a+b \cosh ^{-1}(c x)\right )}-\frac {\sqrt {1-c^2 x^2} \text {Chi}\left (\frac {2 a}{b}+2 \cosh ^{-1}(c x)\right ) \sinh \left (\frac {2 a}{b}\right )}{16 b^2 c^3 \sqrt {-1+c x} \sqrt {1+c x}}-\frac {\sqrt {1-c^2 x^2} \text {Chi}\left (\frac {4 a}{b}+4 \cosh ^{-1}(c x)\right ) \sinh \left (\frac {4 a}{b}\right )}{8 b^2 c^3 \sqrt {-1+c x} \sqrt {1+c x}}+\frac {3 \sqrt {1-c^2 x^2} \text {Chi}\left (\frac {6 a}{b}+6 \cosh ^{-1}(c x)\right ) \sinh \left (\frac {6 a}{b}\right )}{16 b^2 c^3 \sqrt {-1+c x} \sqrt {1+c x}}-\frac {\sqrt {1-c^2 x^2} \text {Chi}\left (\frac {8 a}{b}+8 \cosh ^{-1}(c x)\right ) \sinh \left (\frac {8 a}{b}\right )}{16 b^2 c^3 \sqrt {-1+c x} \sqrt {1+c x}}+\frac {\sqrt {1-c^2 x^2} \cosh \left (\frac {2 a}{b}\right ) \text {Shi}\left (\frac {2 a}{b}+2 \cosh ^{-1}(c x)\right )}{16 b^2 c^3 \sqrt {-1+c x} \sqrt {1+c x}}+\frac {\sqrt {1-c^2 x^2} \cosh \left (\frac {4 a}{b}\right ) \text {Shi}\left (\frac {4 a}{b}+4 \cosh ^{-1}(c x)\right )}{8 b^2 c^3 \sqrt {-1+c x} \sqrt {1+c x}}-\frac {3 \sqrt {1-c^2 x^2} \cosh \left (\frac {6 a}{b}\right ) \text {Shi}\left (\frac {6 a}{b}+6 \cosh ^{-1}(c x)\right )}{16 b^2 c^3 \sqrt {-1+c x} \sqrt {1+c x}}+\frac {\sqrt {1-c^2 x^2} \cosh \left (\frac {8 a}{b}\right ) \text {Shi}\left (\frac {8 a}{b}+8 \cosh ^{-1}(c x)\right )}{16 b^2 c^3 \sqrt {-1+c x} \sqrt {1+c x}}\\ \end {align*}

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Mathematica [A]
time = 1.20, size = 446, normalized size = 0.98 \begin {gather*} \frac {\sqrt {-1+c x} \sqrt {1+c x} \left (-16 b c^2 x^2+48 b c^4 x^4-48 b c^6 x^6+16 b c^8 x^8+\left (a+b \cosh ^{-1}(c x)\right ) \text {Chi}\left (2 \left (\frac {a}{b}+\cosh ^{-1}(c x)\right )\right ) \sinh \left (\frac {2 a}{b}\right )+2 \left (a+b \cosh ^{-1}(c x)\right ) \text {Chi}\left (4 \left (\frac {a}{b}+\cosh ^{-1}(c x)\right )\right ) \sinh \left (\frac {4 a}{b}\right )-3 a \text {Chi}\left (6 \left (\frac {a}{b}+\cosh ^{-1}(c x)\right )\right ) \sinh \left (\frac {6 a}{b}\right )-3 b \cosh ^{-1}(c x) \text {Chi}\left (6 \left (\frac {a}{b}+\cosh ^{-1}(c x)\right )\right ) \sinh \left (\frac {6 a}{b}\right )+a \text {Chi}\left (8 \left (\frac {a}{b}+\cosh ^{-1}(c x)\right )\right ) \sinh \left (\frac {8 a}{b}\right )+b \cosh ^{-1}(c x) \text {Chi}\left (8 \left (\frac {a}{b}+\cosh ^{-1}(c x)\right )\right ) \sinh \left (\frac {8 a}{b}\right )-a \cosh \left (\frac {2 a}{b}\right ) \text {Shi}\left (2 \left (\frac {a}{b}+\cosh ^{-1}(c x)\right )\right )-b \cosh ^{-1}(c x) \cosh \left (\frac {2 a}{b}\right ) \text {Shi}\left (2 \left (\frac {a}{b}+\cosh ^{-1}(c x)\right )\right )-2 a \cosh \left (\frac {4 a}{b}\right ) \text {Shi}\left (4 \left (\frac {a}{b}+\cosh ^{-1}(c x)\right )\right )-2 b \cosh ^{-1}(c x) \cosh \left (\frac {4 a}{b}\right ) \text {Shi}\left (4 \left (\frac {a}{b}+\cosh ^{-1}(c x)\right )\right )+3 a \cosh \left (\frac {6 a}{b}\right ) \text {Shi}\left (6 \left (\frac {a}{b}+\cosh ^{-1}(c x)\right )\right )+3 b \cosh ^{-1}(c x) \cosh \left (\frac {6 a}{b}\right ) \text {Shi}\left (6 \left (\frac {a}{b}+\cosh ^{-1}(c x)\right )\right )-a \cosh \left (\frac {8 a}{b}\right ) \text {Shi}\left (8 \left (\frac {a}{b}+\cosh ^{-1}(c x)\right )\right )-b \cosh ^{-1}(c x) \cosh \left (\frac {8 a}{b}\right ) \text {Shi}\left (8 \left (\frac {a}{b}+\cosh ^{-1}(c x)\right )\right )\right )}{16 b^2 c^3 \sqrt {1-c^2 x^2} \left (a+b \cosh ^{-1}(c x)\right )} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(x^2*(1 - c^2*x^2)^(5/2))/(a + b*ArcCosh[c*x])^2,x]

[Out]

(Sqrt[-1 + c*x]*Sqrt[1 + c*x]*(-16*b*c^2*x^2 + 48*b*c^4*x^4 - 48*b*c^6*x^6 + 16*b*c^8*x^8 + (a + b*ArcCosh[c*x
])*CoshIntegral[2*(a/b + ArcCosh[c*x])]*Sinh[(2*a)/b] + 2*(a + b*ArcCosh[c*x])*CoshIntegral[4*(a/b + ArcCosh[c
*x])]*Sinh[(4*a)/b] - 3*a*CoshIntegral[6*(a/b + ArcCosh[c*x])]*Sinh[(6*a)/b] - 3*b*ArcCosh[c*x]*CoshIntegral[6
*(a/b + ArcCosh[c*x])]*Sinh[(6*a)/b] + a*CoshIntegral[8*(a/b + ArcCosh[c*x])]*Sinh[(8*a)/b] + b*ArcCosh[c*x]*C
oshIntegral[8*(a/b + ArcCosh[c*x])]*Sinh[(8*a)/b] - a*Cosh[(2*a)/b]*SinhIntegral[2*(a/b + ArcCosh[c*x])] - b*A
rcCosh[c*x]*Cosh[(2*a)/b]*SinhIntegral[2*(a/b + ArcCosh[c*x])] - 2*a*Cosh[(4*a)/b]*SinhIntegral[4*(a/b + ArcCo
sh[c*x])] - 2*b*ArcCosh[c*x]*Cosh[(4*a)/b]*SinhIntegral[4*(a/b + ArcCosh[c*x])] + 3*a*Cosh[(6*a)/b]*SinhIntegr
al[6*(a/b + ArcCosh[c*x])] + 3*b*ArcCosh[c*x]*Cosh[(6*a)/b]*SinhIntegral[6*(a/b + ArcCosh[c*x])] - a*Cosh[(8*a
)/b]*SinhIntegral[8*(a/b + ArcCosh[c*x])] - b*ArcCosh[c*x]*Cosh[(8*a)/b]*SinhIntegral[8*(a/b + ArcCosh[c*x])])
)/(16*b^2*c^3*Sqrt[1 - c^2*x^2]*(a + b*ArcCosh[c*x]))

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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(1675\) vs. \(2(400)=800\).
time = 5.31, size = 1676, normalized size = 3.69

method	result	size
default	\(\text {Expression too large to display}\)	\(1676\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^2*(-c^2*x^2+1)^(5/2)/(a+b*arccosh(c*x))^2,x,method=_RETURNVERBOSE)

[Out]

1/256*(-c^2*x^2+1)^(1/2)*(-128*(c*x+1)^(1/2)*(c*x-1)^(1/2)*x^8*c^8+128*c^9*x^9+256*(c*x+1)^(1/2)*(c*x-1)^(1/2)
*x^6*c^6-320*c^7*x^7-160*(c*x+1)^(1/2)*(c*x-1)^(1/2)*x^4*c^4+272*c^5*x^5+32*(c*x+1)^(1/2)*(c*x-1)^(1/2)*x^2*c^
2-88*c^3*x^3-(c*x-1)^(1/2)*(c*x+1)^(1/2)+8*c*x)/(c*x+1)/(c*x-1)/c^3/(a+b*arccosh(c*x))/b-1/32*(-c^2*x^2+1)^(1/
2)*(-(c*x+1)^(1/2)*(c*x-1)^(1/2)*x*c+c^2*x^2-1)*Ei(1,8*arccosh(c*x)+8*a/b)*exp((b*arccosh(c*x)+8*a)/b)/(c*x+1)
/(c*x-1)/c^3/b^2-1/256/(c*x+1)^(1/2)/(c*x-1)^(1/2)*(-c^2*x^2+1)^(1/2)*(128*(c*x-1)^(1/2)*(c*x+1)^(1/2)*b*c^7*x
^7+128*b*c^8*x^8-192*(c*x-1)^(1/2)*(c*x+1)^(1/2)*b*c^5*x^5-256*b*c^6*x^6+80*(c*x+1)^(1/2)*(c*x-1)^(1/2)*b*c^3*
x^3+160*b*c^4*x^4-8*(c*x+1)^(1/2)*(c*x-1)^(1/2)*b*c*x-32*b*c^2*x^2+8*arccosh(c*x)*exp(-8*a/b)*Ei(1,-8*arccosh(
c*x)-8*a/b)*b+8*exp(-8*a/b)*Ei(1,-8*arccosh(c*x)-8*a/b)*a+b)/c^3/b^2/(a+b*arccosh(c*x))+5/128*(-c^2*x^2+1)^(1/
2)/(c*x-1)^(1/2)/(c*x+1)^(1/2)/c^3/(a+b*arccosh(c*x))/b-1/64*(-c^2*x^2+1)^(1/2)*(-32*(c*x+1)^(1/2)*(c*x-1)^(1/
2)*x^6*c^6+32*c^7*x^7+48*(c*x+1)^(1/2)*(c*x-1)^(1/2)*x^4*c^4-64*c^5*x^5-18*(c*x+1)^(1/2)*(c*x-1)^(1/2)*x^2*c^2
+38*c^3*x^3+(c*x-1)^(1/2)*(c*x+1)^(1/2)-6*c*x)/(c*x+1)/(c*x-1)/c^3/(a+b*arccosh(c*x))/b+3/32*(-c^2*x^2+1)^(1/2
)*(-(c*x+1)^(1/2)*(c*x-1)^(1/2)*x*c+c^2*x^2-1)*Ei(1,6*arccosh(c*x)+6*a/b)*exp((b*arccosh(c*x)+6*a)/b)/(c*x+1)/
(c*x-1)/c^3/b^2+1/64*(-c^2*x^2+1)^(1/2)*(-8*(c*x+1)^(1/2)*(c*x-1)^(1/2)*x^4*c^4+8*c^5*x^5+8*(c*x+1)^(1/2)*(c*x
-1)^(1/2)*x^2*c^2-12*c^3*x^3-(c*x-1)^(1/2)*(c*x+1)^(1/2)+4*c*x)/(c*x+1)/(c*x-1)/c^3/(a+b*arccosh(c*x))/b-1/16*
(-c^2*x^2+1)^(1/2)*(-(c*x+1)^(1/2)*(c*x-1)^(1/2)*x*c+c^2*x^2-1)*Ei(1,4*arccosh(c*x)+4*a/b)*exp((b*arccosh(c*x)
+4*a)/b)/(c*x+1)/(c*x-1)/c^3/b^2+1/64*(-c^2*x^2+1)^(1/2)*(-2*(c*x+1)^(1/2)*(c*x-1)^(1/2)*x^2*c^2+2*c^3*x^3+(c*
x-1)^(1/2)*(c*x+1)^(1/2)-2*c*x)/(c*x+1)/(c*x-1)/c^3/(a+b*arccosh(c*x))/b-1/32*(-c^2*x^2+1)^(1/2)*(-(c*x+1)^(1/
2)*(c*x-1)^(1/2)*x*c+c^2*x^2-1)*Ei(1,2*arccosh(c*x)+2*a/b)*exp((b*arccosh(c*x)+2*a)/b)/(c*x+1)/(c*x-1)/c^3/b^2
-1/64/(c*x+1)^(1/2)/(c*x-1)^(1/2)*(-c^2*x^2+1)^(1/2)*(2*(c*x+1)^(1/2)*(c*x-1)^(1/2)*b*c*x+2*b*c^2*x^2+2*arccos
h(c*x)*Ei(1,-2*arccosh(c*x)-2*a/b)*exp(-2*a/b)*b+2*Ei(1,-2*arccosh(c*x)-2*a/b)*exp(-2*a/b)*a-b)/c^3/b^2/(a+b*a
rccosh(c*x))-1/64*(-c^2*x^2+1)^(1/2)/(c*x-1)^(1/2)/(c*x+1)^(1/2)*(8*(c*x+1)^(1/2)*(c*x-1)^(1/2)*b*c^3*x^3+8*b*
c^4*x^4-4*(c*x+1)^(1/2)*(c*x-1)^(1/2)*b*c*x-8*b*c^2*x^2+4*arccosh(c*x)*exp(-4*a/b)*Ei(1,-4*arccosh(c*x)-4*a/b)
*b+4*exp(-4*a/b)*Ei(1,-4*arccosh(c*x)-4*a/b)*a+b)/c^3/b^2/(a+b*arccosh(c*x))+1/64/(c*x+1)^(1/2)/(c*x-1)^(1/2)*
(-c^2*x^2+1)^(1/2)*(32*(c*x-1)^(1/2)*(c*x+1)^(1/2)*b*c^5*x^5+32*b*c^6*x^6-32*(c*x+1)^(1/2)*(c*x-1)^(1/2)*b*c^3
*x^3-48*b*c^4*x^4+6*(c*x+1)^(1/2)*(c*x-1)^(1/2)*b*c*x+18*b*c^2*x^2+6*arccosh(c*x)*Ei(1,-6*arccosh(c*x)-6*a/b)*
exp(-6*a/b)*b+6*Ei(1,-6*arccosh(c*x)-6*a/b)*exp(-6*a/b)*a-b)/c^3/b^2/(a+b*arccosh(c*x))

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Maxima [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^2*(-c^2*x^2+1)^(5/2)/(a+b*arccosh(c*x))^2,x, algorithm="maxima")

[Out]

-((c^6*x^8 - 3*c^4*x^6 + 3*c^2*x^4 - x^2)*(c*x + 1)*sqrt(c*x - 1) + (c^7*x^9 - 3*c^5*x^7 + 3*c^3*x^5 - c*x^3)*
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 + sq
rt(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(((8*c^7*x^8 -
17*c^5*x^6 + 10*c^3*x^4 - c*x^2)*(c*x + 1)^(3/2)*(c*x - 1) + 2*(8*c^8*x^9 - 22*c^6*x^7 + 21*c^4*x^5 - 8*c^2*x
^3 + x)*(c*x + 1)*sqrt(c*x - 1) + (8*c^9*x^10 - 27*c^7*x^8 + 33*c^5*x^6 - 17*c^3*x^4 + 3*c*x^2)*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)

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Fricas [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^2*(-c^2*x^2+1)^(5/2)/(a+b*arccosh(c*x))^2,x, algorithm="fricas")

[Out]

integral((c^4*x^6 - 2*c^2*x^4 + x^2)*sqrt(-c^2*x^2 + 1)/(b^2*arccosh(c*x)^2 + 2*a*b*arccosh(c*x) + a^2), x)

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Sympy [F(-1)] Timed out
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x**2*(-c**2*x**2+1)**(5/2)/(a+b*acosh(c*x))**2,x)

[Out]

Timed out

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Giac [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^2*(-c^2*x^2+1)^(5/2)/(a+b*arccosh(c*x))^2,x, algorithm="giac")

[Out]

integrate((-c^2*x^2 + 1)^(5/2)*x^2/(b*arccosh(c*x) + a)^2, x)

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Mupad [F]
time = 0.00, size = -1, normalized size = -0.00 \begin {gather*} \int \frac {x^2\,{\left (1-c^2\,x^2\right )}^{5/2}}{{\left (a+b\,\mathrm {acosh}\left (c\,x\right )\right )}^2} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((x^2*(1 - c^2*x^2)^(5/2))/(a + b*acosh(c*x))^2,x)

[Out]

int((x^2*(1 - c^2*x^2)^(5/2))/(a + b*acosh(c*x))^2, x)

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