3.4.36 \(\int \frac {x (1-c^2 x^2)^{5/2}}{(a+b \cosh ^{-1}(c x))^2} \, dx\) [336]

Optimal. Leaf size=448 \[ -\frac {x \sqrt {-1+c x} \sqrt {1+c x} \left (1-c^2 x^2\right )^{5/2}}{b c \left (a+b \cosh ^{-1}(c x)\right )}+\frac {5 \sqrt {1-c x} \text {Chi}\left (\frac {a+b \cosh ^{-1}(c x)}{b}\right ) \sinh \left (\frac {a}{b}\right )}{64 b^2 c^2 \sqrt {-1+c x}}-\frac {27 \sqrt {1-c x} \text {Chi}\left (\frac {3 \left (a+b \cosh ^{-1}(c x)\right )}{b}\right ) \sinh \left (\frac {3 a}{b}\right )}{64 b^2 c^2 \sqrt {-1+c x}}+\frac {25 \sqrt {1-c x} \text {Chi}\left (\frac {5 \left (a+b \cosh ^{-1}(c x)\right )}{b}\right ) \sinh \left (\frac {5 a}{b}\right )}{64 b^2 c^2 \sqrt {-1+c x}}-\frac {7 \sqrt {1-c x} \text {Chi}\left (\frac {7 \left (a+b \cosh ^{-1}(c x)\right )}{b}\right ) \sinh \left (\frac {7 a}{b}\right )}{64 b^2 c^2 \sqrt {-1+c x}}-\frac {5 \sqrt {1-c x} \cosh \left (\frac {a}{b}\right ) \text {Shi}\left (\frac {a+b \cosh ^{-1}(c x)}{b}\right )}{64 b^2 c^2 \sqrt {-1+c x}}+\frac {27 \sqrt {1-c x} \cosh \left (\frac {3 a}{b}\right ) \text {Shi}\left (\frac {3 \left (a+b \cosh ^{-1}(c x)\right )}{b}\right )}{64 b^2 c^2 \sqrt {-1+c x}}-\frac {25 \sqrt {1-c x} \cosh \left (\frac {5 a}{b}\right ) \text {Shi}\left (\frac {5 \left (a+b \cosh ^{-1}(c x)\right )}{b}\right )}{64 b^2 c^2 \sqrt {-1+c x}}+\frac {7 \sqrt {1-c x} \cosh \left (\frac {7 a}{b}\right ) \text {Shi}\left (\frac {7 \left (a+b \cosh ^{-1}(c x)\right )}{b}\right )}{64 b^2 c^2 \sqrt {-1+c x}} \]

[Out]

-5/64*cosh(a/b)*Shi((a+b*arccosh(c*x))/b)*(-c*x+1)^(1/2)/b^2/c^2/(c*x-1)^(1/2)+27/64*cosh(3*a/b)*Shi(3*(a+b*ar
ccosh(c*x))/b)*(-c*x+1)^(1/2)/b^2/c^2/(c*x-1)^(1/2)-25/64*cosh(5*a/b)*Shi(5*(a+b*arccosh(c*x))/b)*(-c*x+1)^(1/
2)/b^2/c^2/(c*x-1)^(1/2)+7/64*cosh(7*a/b)*Shi(7*(a+b*arccosh(c*x))/b)*(-c*x+1)^(1/2)/b^2/c^2/(c*x-1)^(1/2)+5/6
4*Chi((a+b*arccosh(c*x))/b)*sinh(a/b)*(-c*x+1)^(1/2)/b^2/c^2/(c*x-1)^(1/2)-27/64*Chi(3*(a+b*arccosh(c*x))/b)*s
inh(3*a/b)*(-c*x+1)^(1/2)/b^2/c^2/(c*x-1)^(1/2)+25/64*Chi(5*(a+b*arccosh(c*x))/b)*sinh(5*a/b)*(-c*x+1)^(1/2)/b
^2/c^2/(c*x-1)^(1/2)-7/64*Chi(7*(a+b*arccosh(c*x))/b)*sinh(7*a/b)*(-c*x+1)^(1/2)/b^2/c^2/(c*x-1)^(1/2)-x*(-c^2
*x^2+1)^(5/2)*(c*x-1)^(1/2)*(c*x+1)^(1/2)/b/c/(a+b*arccosh(c*x))

________________________________________________________________________________________

Rubi [A]
time = 0.79, antiderivative size = 448, normalized size of antiderivative = 1.00, number of steps used = 30, number of rules used = 10, integrand size = 26, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.385, Rules used = {5942, 5889, 5906, 3393, 3384, 3379, 3382, 5912, 5952, 5556} \begin {gather*} \frac {5 \sqrt {1-c x} \sinh \left (\frac {a}{b}\right ) \text {Chi}\left (\frac {a+b \cosh ^{-1}(c x)}{b}\right )}{64 b^2 c^2 \sqrt {c x-1}}-\frac {27 \sqrt {1-c x} \sinh \left (\frac {3 a}{b}\right ) \text {Chi}\left (\frac {3 \left (a+b \cosh ^{-1}(c x)\right )}{b}\right )}{64 b^2 c^2 \sqrt {c x-1}}+\frac {25 \sqrt {1-c x} \sinh \left (\frac {5 a}{b}\right ) \text {Chi}\left (\frac {5 \left (a+b \cosh ^{-1}(c x)\right )}{b}\right )}{64 b^2 c^2 \sqrt {c x-1}}-\frac {7 \sqrt {1-c x} \sinh \left (\frac {7 a}{b}\right ) \text {Chi}\left (\frac {7 \left (a+b \cosh ^{-1}(c x)\right )}{b}\right )}{64 b^2 c^2 \sqrt {c x-1}}-\frac {5 \sqrt {1-c x} \cosh \left (\frac {a}{b}\right ) \text {Shi}\left (\frac {a+b \cosh ^{-1}(c x)}{b}\right )}{64 b^2 c^2 \sqrt {c x-1}}+\frac {27 \sqrt {1-c x} \cosh \left (\frac {3 a}{b}\right ) \text {Shi}\left (\frac {3 \left (a+b \cosh ^{-1}(c x)\right )}{b}\right )}{64 b^2 c^2 \sqrt {c x-1}}-\frac {25 \sqrt {1-c x} \cosh \left (\frac {5 a}{b}\right ) \text {Shi}\left (\frac {5 \left (a+b \cosh ^{-1}(c x)\right )}{b}\right )}{64 b^2 c^2 \sqrt {c x-1}}+\frac {7 \sqrt {1-c x} \cosh \left (\frac {7 a}{b}\right ) \text {Shi}\left (\frac {7 \left (a+b \cosh ^{-1}(c x)\right )}{b}\right )}{64 b^2 c^2 \sqrt {c x-1}}-\frac {x \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*(1 - c^2*x^2)^(5/2))/(a + b*ArcCosh[c*x])^2,x]

[Out]

-((x*Sqrt[-1 + c*x]*Sqrt[1 + c*x]*(1 - c^2*x^2)^(5/2))/(b*c*(a + b*ArcCosh[c*x]))) + (5*Sqrt[1 - c*x]*CoshInte
gral[(a + b*ArcCosh[c*x])/b]*Sinh[a/b])/(64*b^2*c^2*Sqrt[-1 + c*x]) - (27*Sqrt[1 - c*x]*CoshIntegral[(3*(a + b
*ArcCosh[c*x]))/b]*Sinh[(3*a)/b])/(64*b^2*c^2*Sqrt[-1 + c*x]) + (25*Sqrt[1 - c*x]*CoshIntegral[(5*(a + b*ArcCo
sh[c*x]))/b]*Sinh[(5*a)/b])/(64*b^2*c^2*Sqrt[-1 + c*x]) - (7*Sqrt[1 - c*x]*CoshIntegral[(7*(a + b*ArcCosh[c*x]
))/b]*Sinh[(7*a)/b])/(64*b^2*c^2*Sqrt[-1 + c*x]) - (5*Sqrt[1 - c*x]*Cosh[a/b]*SinhIntegral[(a + b*ArcCosh[c*x]
)/b])/(64*b^2*c^2*Sqrt[-1 + c*x]) + (27*Sqrt[1 - c*x]*Cosh[(3*a)/b]*SinhIntegral[(3*(a + b*ArcCosh[c*x]))/b])/
(64*b^2*c^2*Sqrt[-1 + c*x]) - (25*Sqrt[1 - c*x]*Cosh[(5*a)/b]*SinhIntegral[(5*(a + b*ArcCosh[c*x]))/b])/(64*b^
2*c^2*Sqrt[-1 + c*x]) + (7*Sqrt[1 - c*x]*Cosh[(7*a)/b]*SinhIntegral[(7*(a + b*ArcCosh[c*x]))/b])/(64*b^2*c^2*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 3393

Int[((c_.) + (d_.)*(x_))^(m_)*sin[(e_.) + (f_.)*(x_)]^(n_), x_Symbol] :> Int[ExpandTrigReduce[(c + d*x)^m, Sin
[e + f*x]^n, x], x] /; FreeQ[{c, d, e, f, m}, x] && IGtQ[n, 1] && ( !RationalQ[m] || (GeQ[m, -1] && LtQ[m, 1])
)

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 5889

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

Rule 5906

Int[((a_.) + ArcCosh[(c_.)*(x_)]*(b_.))^(n_.)*((d_) + (e_.)*(x_)^2)^(p_.), x_Symbol] :> Dist[(1/(b*c))*Simp[(d
+ e*x^2)^p/((1 + c*x)^p*(-1 + c*x)^p)], Subst[Int[x^n*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, 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 \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 (-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 (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} \int \frac {\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 (7 c \sqrt {1-c^2 x^2}\right ) \int \frac {x^2 \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 (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 ^5(x)}{a+b x} \, dx,x,\cosh ^{-1}(c x)\right )}{b c^2 \sqrt {-1+c x} \sqrt {1+c x}}+\frac {\left (7 \sqrt {1-c^2 x^2}\right ) \text {Subst}\left (\int \frac {\cosh ^2(x) \sinh ^5(x)}{a+b x} \, dx,x,\cosh ^{-1}(c x)\right )}{b c^2 \sqrt {-1+c x} \sqrt {1+c x}}\\ &=\frac {x (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 (i \sqrt {1-c^2 x^2}\right ) \text {Subst}\left (\int \left (\frac {5 i \sinh (x)}{8 (a+b x)}-\frac {5 i \sinh (3 x)}{16 (a+b x)}+\frac {i \sinh (5 x)}{16 (a+b x)}\right ) \, dx,x,\cosh ^{-1}(c x)\right )}{b c^2 \sqrt {-1+c x} \sqrt {1+c x}}+\frac {\left (7 \sqrt {1-c^2 x^2}\right ) \text {Subst}\left (\int \left (\frac {5 \sinh (x)}{64 (a+b x)}+\frac {\sinh (3 x)}{64 (a+b x)}-\frac {3 \sinh (5 x)}{64 (a+b x)}+\frac {\sinh (7 x)}{64 (a+b x)}\right ) \, dx,x,\cosh ^{-1}(c x)\right )}{b c^2 \sqrt {-1+c x} \sqrt {1+c x}}\\ &=\frac {x (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 (5 x)}{a+b x} \, dx,x,\cosh ^{-1}(c x)\right )}{16 b c^2 \sqrt {-1+c x} \sqrt {1+c x}}+\frac {\left (7 \sqrt {1-c^2 x^2}\right ) \text {Subst}\left (\int \frac {\sinh (3 x)}{a+b x} \, dx,x,\cosh ^{-1}(c x)\right )}{64 b c^2 \sqrt {-1+c x} \sqrt {1+c x}}+\frac {\left (7 \sqrt {1-c^2 x^2}\right ) \text {Subst}\left (\int \frac {\sinh (7 x)}{a+b x} \, dx,x,\cosh ^{-1}(c x)\right )}{64 b c^2 \sqrt {-1+c x} \sqrt {1+c x}}+\frac {\left (5 \sqrt {1-c^2 x^2}\right ) \text {Subst}\left (\int \frac {\sinh (3 x)}{a+b x} \, dx,x,\cosh ^{-1}(c x)\right )}{16 b c^2 \sqrt {-1+c x} \sqrt {1+c x}}-\frac {\left (21 \sqrt {1-c^2 x^2}\right ) \text {Subst}\left (\int \frac {\sinh (5 x)}{a+b x} \, dx,x,\cosh ^{-1}(c x)\right )}{64 b c^2 \sqrt {-1+c x} \sqrt {1+c x}}+\frac {\left (35 \sqrt {1-c^2 x^2}\right ) \text {Subst}\left (\int \frac {\sinh (x)}{a+b x} \, dx,x,\cosh ^{-1}(c x)\right )}{64 b c^2 \sqrt {-1+c x} \sqrt {1+c x}}-\frac {\left (5 \sqrt {1-c^2 x^2}\right ) \text {Subst}\left (\int \frac {\sinh (x)}{a+b x} \, dx,x,\cosh ^{-1}(c x)\right )}{8 b c^2 \sqrt {-1+c x} \sqrt {1+c x}}\\ &=\frac {x (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 (35 \sqrt {1-c^2 x^2} \cosh \left (\frac {a}{b}\right )\right ) \text {Subst}\left (\int \frac {\sinh \left (\frac {a}{b}+x\right )}{a+b x} \, dx,x,\cosh ^{-1}(c x)\right )}{64 b c^2 \sqrt {-1+c x} \sqrt {1+c x}}-\frac {\left (5 \sqrt {1-c^2 x^2} \cosh \left (\frac {a}{b}\right )\right ) \text {Subst}\left (\int \frac {\sinh \left (\frac {a}{b}+x\right )}{a+b x} \, dx,x,\cosh ^{-1}(c x)\right )}{8 b c^2 \sqrt {-1+c x} \sqrt {1+c x}}+\frac {\left (7 \sqrt {1-c^2 x^2} \cosh \left (\frac {3 a}{b}\right )\right ) \text {Subst}\left (\int \frac {\sinh \left (\frac {3 a}{b}+3 x\right )}{a+b x} \, dx,x,\cosh ^{-1}(c x)\right )}{64 b c^2 \sqrt {-1+c x} \sqrt {1+c x}}+\frac {\left (5 \sqrt {1-c^2 x^2} \cosh \left (\frac {3 a}{b}\right )\right ) \text {Subst}\left (\int \frac {\sinh \left (\frac {3 a}{b}+3 x\right )}{a+b x} \, dx,x,\cosh ^{-1}(c x)\right )}{16 b c^2 \sqrt {-1+c x} \sqrt {1+c x}}-\frac {\left (\sqrt {1-c^2 x^2} \cosh \left (\frac {5 a}{b}\right )\right ) \text {Subst}\left (\int \frac {\sinh \left (\frac {5 a}{b}+5 x\right )}{a+b x} \, dx,x,\cosh ^{-1}(c x)\right )}{16 b c^2 \sqrt {-1+c x} \sqrt {1+c x}}-\frac {\left (21 \sqrt {1-c^2 x^2} \cosh \left (\frac {5 a}{b}\right )\right ) \text {Subst}\left (\int \frac {\sinh \left (\frac {5 a}{b}+5 x\right )}{a+b x} \, dx,x,\cosh ^{-1}(c x)\right )}{64 b c^2 \sqrt {-1+c x} \sqrt {1+c x}}+\frac {\left (7 \sqrt {1-c^2 x^2} \cosh \left (\frac {7 a}{b}\right )\right ) \text {Subst}\left (\int \frac {\sinh \left (\frac {7 a}{b}+7 x\right )}{a+b x} \, dx,x,\cosh ^{-1}(c x)\right )}{64 b c^2 \sqrt {-1+c x} \sqrt {1+c x}}-\frac {\left (35 \sqrt {1-c^2 x^2} \sinh \left (\frac {a}{b}\right )\right ) \text {Subst}\left (\int \frac {\cosh \left (\frac {a}{b}+x\right )}{a+b x} \, dx,x,\cosh ^{-1}(c x)\right )}{64 b c^2 \sqrt {-1+c x} \sqrt {1+c x}}+\frac {\left (5 \sqrt {1-c^2 x^2} \sinh \left (\frac {a}{b}\right )\right ) \text {Subst}\left (\int \frac {\cosh \left (\frac {a}{b}+x\right )}{a+b x} \, dx,x,\cosh ^{-1}(c x)\right )}{8 b c^2 \sqrt {-1+c x} \sqrt {1+c x}}-\frac {\left (7 \sqrt {1-c^2 x^2} \sinh \left (\frac {3 a}{b}\right )\right ) \text {Subst}\left (\int \frac {\cosh \left (\frac {3 a}{b}+3 x\right )}{a+b x} \, dx,x,\cosh ^{-1}(c x)\right )}{64 b c^2 \sqrt {-1+c x} \sqrt {1+c x}}-\frac {\left (5 \sqrt {1-c^2 x^2} \sinh \left (\frac {3 a}{b}\right )\right ) \text {Subst}\left (\int \frac {\cosh \left (\frac {3 a}{b}+3 x\right )}{a+b x} \, dx,x,\cosh ^{-1}(c x)\right )}{16 b c^2 \sqrt {-1+c x} \sqrt {1+c x}}+\frac {\left (\sqrt {1-c^2 x^2} \sinh \left (\frac {5 a}{b}\right )\right ) \text {Subst}\left (\int \frac {\cosh \left (\frac {5 a}{b}+5 x\right )}{a+b x} \, dx,x,\cosh ^{-1}(c x)\right )}{16 b c^2 \sqrt {-1+c x} \sqrt {1+c x}}+\frac {\left (21 \sqrt {1-c^2 x^2} \sinh \left (\frac {5 a}{b}\right )\right ) \text {Subst}\left (\int \frac {\cosh \left (\frac {5 a}{b}+5 x\right )}{a+b x} \, dx,x,\cosh ^{-1}(c x)\right )}{64 b c^2 \sqrt {-1+c x} \sqrt {1+c x}}-\frac {\left (7 \sqrt {1-c^2 x^2} \sinh \left (\frac {7 a}{b}\right )\right ) \text {Subst}\left (\int \frac {\cosh \left (\frac {7 a}{b}+7 x\right )}{a+b x} \, dx,x,\cosh ^{-1}(c x)\right )}{64 b c^2 \sqrt {-1+c x} \sqrt {1+c x}}\\ &=\frac {x (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 {5 \sqrt {1-c^2 x^2} \text {Chi}\left (\frac {a}{b}+\cosh ^{-1}(c x)\right ) \sinh \left (\frac {a}{b}\right )}{64 b^2 c^2 \sqrt {-1+c x} \sqrt {1+c x}}-\frac {27 \sqrt {1-c^2 x^2} \text {Chi}\left (\frac {3 a}{b}+3 \cosh ^{-1}(c x)\right ) \sinh \left (\frac {3 a}{b}\right )}{64 b^2 c^2 \sqrt {-1+c x} \sqrt {1+c x}}+\frac {25 \sqrt {1-c^2 x^2} \text {Chi}\left (\frac {5 a}{b}+5 \cosh ^{-1}(c x)\right ) \sinh \left (\frac {5 a}{b}\right )}{64 b^2 c^2 \sqrt {-1+c x} \sqrt {1+c x}}-\frac {7 \sqrt {1-c^2 x^2} \text {Chi}\left (\frac {7 a}{b}+7 \cosh ^{-1}(c x)\right ) \sinh \left (\frac {7 a}{b}\right )}{64 b^2 c^2 \sqrt {-1+c x} \sqrt {1+c x}}-\frac {5 \sqrt {1-c^2 x^2} \cosh \left (\frac {a}{b}\right ) \text {Shi}\left (\frac {a}{b}+\cosh ^{-1}(c x)\right )}{64 b^2 c^2 \sqrt {-1+c x} \sqrt {1+c x}}+\frac {27 \sqrt {1-c^2 x^2} \cosh \left (\frac {3 a}{b}\right ) \text {Shi}\left (\frac {3 a}{b}+3 \cosh ^{-1}(c x)\right )}{64 b^2 c^2 \sqrt {-1+c x} \sqrt {1+c x}}-\frac {25 \sqrt {1-c^2 x^2} \cosh \left (\frac {5 a}{b}\right ) \text {Shi}\left (\frac {5 a}{b}+5 \cosh ^{-1}(c x)\right )}{64 b^2 c^2 \sqrt {-1+c x} \sqrt {1+c x}}+\frac {7 \sqrt {1-c^2 x^2} \cosh \left (\frac {7 a}{b}\right ) \text {Shi}\left (\frac {7 a}{b}+7 \cosh ^{-1}(c x)\right )}{64 b^2 c^2 \sqrt {-1+c x} \sqrt {1+c x}}\\ \end {align*}

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Mathematica [A]
time = 1.01, size = 436, normalized size = 0.97 \begin {gather*} \frac {\sqrt {-1+c x} \sqrt {1+c x} \left (-64 b c x+192 b c^3 x^3-192 b c^5 x^5+64 b c^7 x^7-5 \left (a+b \cosh ^{-1}(c x)\right ) \text {Chi}\left (\frac {a}{b}+\cosh ^{-1}(c x)\right ) \sinh \left (\frac {a}{b}\right )+27 \left (a+b \cosh ^{-1}(c x)\right ) \text {Chi}\left (3 \left (\frac {a}{b}+\cosh ^{-1}(c x)\right )\right ) \sinh \left (\frac {3 a}{b}\right )-25 a \text {Chi}\left (5 \left (\frac {a}{b}+\cosh ^{-1}(c x)\right )\right ) \sinh \left (\frac {5 a}{b}\right )-25 b \cosh ^{-1}(c x) \text {Chi}\left (5 \left (\frac {a}{b}+\cosh ^{-1}(c x)\right )\right ) \sinh \left (\frac {5 a}{b}\right )+7 a \text {Chi}\left (7 \left (\frac {a}{b}+\cosh ^{-1}(c x)\right )\right ) \sinh \left (\frac {7 a}{b}\right )+7 b \cosh ^{-1}(c x) \text {Chi}\left (7 \left (\frac {a}{b}+\cosh ^{-1}(c x)\right )\right ) \sinh \left (\frac {7 a}{b}\right )+5 a \cosh \left (\frac {a}{b}\right ) \text {Shi}\left (\frac {a}{b}+\cosh ^{-1}(c x)\right )+5 b \cosh ^{-1}(c x) \cosh \left (\frac {a}{b}\right ) \text {Shi}\left (\frac {a}{b}+\cosh ^{-1}(c x)\right )-27 a \cosh \left (\frac {3 a}{b}\right ) \text {Shi}\left (3 \left (\frac {a}{b}+\cosh ^{-1}(c x)\right )\right )-27 b \cosh ^{-1}(c x) \cosh \left (\frac {3 a}{b}\right ) \text {Shi}\left (3 \left (\frac {a}{b}+\cosh ^{-1}(c x)\right )\right )+25 a \cosh \left (\frac {5 a}{b}\right ) \text {Shi}\left (5 \left (\frac {a}{b}+\cosh ^{-1}(c x)\right )\right )+25 b \cosh ^{-1}(c x) \cosh \left (\frac {5 a}{b}\right ) \text {Shi}\left (5 \left (\frac {a}{b}+\cosh ^{-1}(c x)\right )\right )-7 a \cosh \left (\frac {7 a}{b}\right ) \text {Shi}\left (7 \left (\frac {a}{b}+\cosh ^{-1}(c x)\right )\right )-7 b \cosh ^{-1}(c x) \cosh \left (\frac {7 a}{b}\right ) \text {Shi}\left (7 \left (\frac {a}{b}+\cosh ^{-1}(c x)\right )\right )\right )}{64 b^2 c^2 \sqrt {1-c^2 x^2} \left (a+b \cosh ^{-1}(c x)\right )} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(Sqrt[-1 + c*x]*Sqrt[1 + c*x]*(-64*b*c*x + 192*b*c^3*x^3 - 192*b*c^5*x^5 + 64*b*c^7*x^7 - 5*(a + b*ArcCosh[c*x
])*CoshIntegral[a/b + ArcCosh[c*x]]*Sinh[a/b] + 27*(a + b*ArcCosh[c*x])*CoshIntegral[3*(a/b + ArcCosh[c*x])]*S
inh[(3*a)/b] - 25*a*CoshIntegral[5*(a/b + ArcCosh[c*x])]*Sinh[(5*a)/b] - 25*b*ArcCosh[c*x]*CoshIntegral[5*(a/b
+ ArcCosh[c*x])]*Sinh[(5*a)/b] + 7*a*CoshIntegral[7*(a/b + ArcCosh[c*x])]*Sinh[(7*a)/b] + 7*b*ArcCosh[c*x]*Co
shIntegral[7*(a/b + ArcCosh[c*x])]*Sinh[(7*a)/b] + 5*a*Cosh[a/b]*SinhIntegral[a/b + ArcCosh[c*x]] + 5*b*ArcCos
h[c*x]*Cosh[a/b]*SinhIntegral[a/b + ArcCosh[c*x]] - 27*a*Cosh[(3*a)/b]*SinhIntegral[3*(a/b + ArcCosh[c*x])] -
27*b*ArcCosh[c*x]*Cosh[(3*a)/b]*SinhIntegral[3*(a/b + ArcCosh[c*x])] + 25*a*Cosh[(5*a)/b]*SinhIntegral[5*(a/b
+ ArcCosh[c*x])] + 25*b*ArcCosh[c*x]*Cosh[(5*a)/b]*SinhIntegral[5*(a/b + ArcCosh[c*x])] - 7*a*Cosh[(7*a)/b]*Si
nhIntegral[7*(a/b + ArcCosh[c*x])] - 7*b*ArcCosh[c*x]*Cosh[(7*a)/b]*SinhIntegral[7*(a/b + ArcCosh[c*x])]))/(64
*b^2*c^2*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. \(1498\) vs. \(2(394)=788\).
time = 4.50, size = 1499, normalized size = 3.35

method	result	size
default	\(\text {Expression too large to display}\)	\(1499\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/128*(-c^2*x^2+1)^(1/2)*(-64*(c*x+1)^(1/2)*(c*x-1)^(1/2)*c^7*x^7+64*c^8*x^8+112*(c*x+1)^(1/2)*(c*x-1)^(1/2)*x
^5*c^5-144*x^6*c^6-56*(c*x+1)^(1/2)*(c*x-1)^(1/2)*x^3*c^3+104*c^4*x^4+7*(c*x+1)^(1/2)*(c*x-1)^(1/2)*x*c-25*c^2
*x^2+1)/(c*x+1)/c^2/(c*x-1)/(a+b*arccosh(c*x))/b-7/128*(-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(1,7*arccosh(c*x)+7*a/b)*exp((b*arccosh(c*x)+7*a)/b)/(c*x+1)/c^2/(c*x-1)/b^2-1/128/(c*x+1)^(1/2)/(c
*x-1)^(1/2)*(-c^2*x^2+1)^(1/2)*(64*(c*x-1)^(1/2)*(c*x+1)^(1/2)*b*c^6*x^6+64*b*c^7*x^7-80*(c*x+1)^(1/2)*(c*x-1)
^(1/2)*b*c^4*x^4-112*b*c^5*x^5+24*(c*x+1)^(1/2)*(c*x-1)^(1/2)*b*c^2*x^2+56*b*c^3*x^3+7*exp(-7*a/b)*arccosh(c*x
)*Ei(1,-7*arccosh(c*x)-7*a/b)*b+7*exp(-7*a/b)*Ei(1,-7*arccosh(c*x)-7*a/b)*a-(c*x+1)^(1/2)*(c*x-1)^(1/2)*b-7*b*
c*x)/c^2/b^2/(a+b*arccosh(c*x))-5/128*(-c^2*x^2+1)^(1/2)*(-16*(c*x+1)^(1/2)*(c*x-1)^(1/2)*x^5*c^5+16*x^6*c^6+2
0*(c*x+1)^(1/2)*(c*x-1)^(1/2)*x^3*c^3-28*c^4*x^4-5*(c*x+1)^(1/2)*(c*x-1)^(1/2)*x*c+13*c^2*x^2-1)/(c*x+1)/c^2/(
c*x-1)/(a+b*arccosh(c*x))/b+25/128*(-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(1,5*arcc
osh(c*x)+5*a/b)*exp((b*arccosh(c*x)+5*a)/b)/(c*x+1)/c^2/(c*x-1)/b^2+9/128*(-c^2*x^2+1)^(1/2)*(-4*(c*x+1)^(1/2)
*(c*x-1)^(1/2)*x^3*c^3+4*c^4*x^4+3*(c*x+1)^(1/2)*(c*x-1)^(1/2)*x*c-5*c^2*x^2+1)/(c*x+1)/c^2/(c*x-1)/(a+b*arcco
sh(c*x))/b-27/128*(-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(1,3*arccosh(c*x)+3*a/b)*e
xp((b*arccosh(c*x)+3*a)/b)/(c*x+1)/c^2/(c*x-1)/b^2-5/128*(-c^2*x^2+1)^(1/2)*(-(c*x+1)^(1/2)*(c*x-1)^(1/2)*x*c+
c^2*x^2-1)/(c*x+1)/c^2/(c*x-1)/(a+b*arccosh(c*x))/b+5/128*(-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,arccosh(c*x)+a/b)*exp((a+b*arccosh(c*x))/b)/(c*x+1)/c^2/(c*x-1)/b^2+5/128*(-c^2*x^2+1)^(1/2)/
(c*x-1)^(1/2)/(c*x+1)^(1/2)*(arccosh(c*x)*exp(-a/b)*Ei(1,-arccosh(c*x)-a/b)*b+(c*x+1)^(1/2)*(c*x-1)^(1/2)*b+ex
p(-a/b)*Ei(1,-arccosh(c*x)-a/b)*a+b*c*x)/c^2/b^2/(a+b*arccosh(c*x))-9/128*(-c^2*x^2+1)^(1/2)/(c*x-1)^(1/2)/(c*
x+1)^(1/2)*(4*(c*x+1)^(1/2)*(c*x-1)^(1/2)*b*c^2*x^2+4*b*c^3*x^3+3*Ei(1,-3*arccosh(c*x)-3*a/b)*arccosh(c*x)*exp
(-3*a/b)*b+3*Ei(1,-3*arccosh(c*x)-3*a/b)*exp(-3*a/b)*a-(c*x+1)^(1/2)*(c*x-1)^(1/2)*b-3*b*c*x)/c^2/b^2/(a+b*arc
cosh(c*x))+5/128/(c*x+1)^(1/2)/(c*x-1)^(1/2)*(-c^2*x^2+1)^(1/2)*(16*(c*x+1)^(1/2)*(c*x-1)^(1/2)*b*c^4*x^4+16*b
*c^5*x^5-12*(c*x+1)^(1/2)*(c*x-1)^(1/2)*b*c^2*x^2-20*b*c^3*x^3+5*Ei(1,-5*arccosh(c*x)-5*a/b)*arccosh(c*x)*exp(
-5*a/b)*b+5*Ei(1,-5*arccosh(c*x)-5*a/b)*exp(-5*a/b)*a+(c*x+1)^(1/2)*(c*x-1)^(1/2)*b+5*b*c*x)/c^2/b^2/(a+b*arcc
osh(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*(-c^2*x^2+1)^(5/2)/(a+b*arccosh(c*x))^2,x, algorithm="maxima")

[Out]

-((c^6*x^7 - 3*c^4*x^5 + 3*c^2*x^3 - x)*(c*x + 1)*sqrt(c*x - 1) + (c^7*x^8 - 3*c^5*x^6 + 3*c^3*x^4 - c*x^2)*sq
rt(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((7*(c^7*x^7 - 2
*c^5*x^5 + c^3*x^3)*(c*x + 1)^(3/2)*(c*x - 1) + (14*c^8*x^8 - 37*c^6*x^6 + 33*c^4*x^4 - 11*c^2*x^2 + 1)*(c*x +
1)*sqrt(c*x - 1) + (7*c^9*x^9 - 23*c^7*x^7 + 27*c^5*x^5 - 13*c^3*x^3 + 2*c*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)

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

[Out]

integral((c^4*x^5 - 2*c^2*x^3 + x)*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*(-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*(-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/(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\,{\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*(1 - c^2*x^2)^(5/2))/(a + b*acosh(c*x))^2,x)

[Out]

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

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