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

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

[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 = 1.33, antiderivative size = 555, normalized size of antiderivative = 1.24, number of steps used = 29, number of rules used = 9, integrand size = 26, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.346, Rules used = {5798, 5778, 5700, 3312, 3303, 3298, 3301, 5780, 5448} \[ \frac {5 \sqrt {1-c^2 x^2} \sinh \left (\frac {a}{b}\right ) \text {Chi}\left (\frac {a}{b}+\cosh ^{-1}(c x)\right )}{64 b^2 c^2 \sqrt {c x-1} \sqrt {c x+1}}-\frac {27 \sqrt {1-c^2 x^2} \sinh \left (\frac {3 a}{b}\right ) \text {Chi}\left (\frac {3 a}{b}+3 \cosh ^{-1}(c x)\right )}{64 b^2 c^2 \sqrt {c x-1} \sqrt {c x+1}}+\frac {25 \sqrt {1-c^2 x^2} \sinh \left (\frac {5 a}{b}\right ) \text {Chi}\left (\frac {5 a}{b}+5 \cosh ^{-1}(c x)\right )}{64 b^2 c^2 \sqrt {c x-1} \sqrt {c x+1}}-\frac {7 \sqrt {1-c^2 x^2} \sinh \left (\frac {7 a}{b}\right ) \text {Chi}\left (\frac {7 a}{b}+7 \cosh ^{-1}(c x)\right )}{64 b^2 c^2 \sqrt {c x-1} \sqrt {c x+1}}-\frac {5 \sqrt {1-c^2 x^2} \cosh \left (\frac {a}{b}\right ) \text {Shi}\left (\frac {a}{b}+\cosh ^{-1}(c x)\right )}{64 b^2 c^2 \sqrt {c x-1} \sqrt {c x+1}}+\frac {27 \sqrt {1-c^2 x^2} \cosh \left (\frac {3 a}{b}\right ) \text {Shi}\left (\frac {3 a}{b}+3 \cosh ^{-1}(c x)\right )}{64 b^2 c^2 \sqrt {c x-1} \sqrt {c x+1}}-\frac {25 \sqrt {1-c^2 x^2} \cosh \left (\frac {5 a}{b}\right ) \text {Shi}\left (\frac {5 a}{b}+5 \cosh ^{-1}(c x)\right )}{64 b^2 c^2 \sqrt {c x-1} \sqrt {c x+1}}+\frac {7 \sqrt {1-c^2 x^2} \cosh \left (\frac {7 a}{b}\right ) \text {Shi}\left (\frac {7 a}{b}+7 \cosh ^{-1}(c x)\right )}{64 b^2 c^2 \sqrt {c x-1} \sqrt {c x+1}}+\frac {x (c x+1)^{5/2} \sqrt {1-c^2 x^2} (1-c x)^3}{b c \sqrt {c x-1} \left (a+b \cosh ^{-1}(c x)\right )} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(x*(1 - c*x)^3*(1 + c*x)^(5/2)*Sqrt[1 - c^2*x^2])/(b*c*Sqrt[-1 + c*x]*(a + b*ArcCosh[c*x])) + (5*Sqrt[1 - c^2*
x^2]*CoshIntegral[a/b + ArcCosh[c*x]]*Sinh[a/b])/(64*b^2*c^2*Sqrt[-1 + c*x]*Sqrt[1 + c*x]) - (27*Sqrt[1 - c^2*
x^2]*CoshIntegral[(3*a)/b + 3*ArcCosh[c*x]]*Sinh[(3*a)/b])/(64*b^2*c^2*Sqrt[-1 + c*x]*Sqrt[1 + c*x]) + (25*Sqr
t[1 - c^2*x^2]*CoshIntegral[(5*a)/b + 5*ArcCosh[c*x]]*Sinh[(5*a)/b])/(64*b^2*c^2*Sqrt[-1 + c*x]*Sqrt[1 + c*x])
- (7*Sqrt[1 - c^2*x^2]*CoshIntegral[(7*a)/b + 7*ArcCosh[c*x]]*Sinh[(7*a)/b])/(64*b^2*c^2*Sqrt[-1 + c*x]*Sqrt[
1 + c*x]) - (5*Sqrt[1 - c^2*x^2]*Cosh[a/b]*SinhIntegral[a/b + ArcCosh[c*x]])/(64*b^2*c^2*Sqrt[-1 + c*x]*Sqrt[1
+ c*x]) + (27*Sqrt[1 - c^2*x^2]*Cosh[(3*a)/b]*SinhIntegral[(3*a)/b + 3*ArcCosh[c*x]])/(64*b^2*c^2*Sqrt[-1 + c
*x]*Sqrt[1 + c*x]) - (25*Sqrt[1 - c^2*x^2]*Cosh[(5*a)/b]*SinhIntegral[(5*a)/b + 5*ArcCosh[c*x]])/(64*b^2*c^2*S
qrt[-1 + c*x]*Sqrt[1 + c*x]) + (7*Sqrt[1 - c^2*x^2]*Cosh[(7*a)/b]*SinhIntegral[(7*a)/b + 7*ArcCosh[c*x]])/(64*
b^2*c^2*Sqrt[-1 + c*x]*Sqrt[1 + c*x])

Rule 3298

Int[sin[(e_.) + (Complex[0, fz_])*(f_.)*(x_)]/((c_.) + (d_.)*(x_)), x_Symbol] :> Simp[(I*SinhIntegral[(c*f*fz)
/d + f*fz*x])/d, x] /; FreeQ[{c, d, e, f, fz}, x] && EqQ[d*e - c*f*fz*I, 0]

Rule 3301

Int[sin[(e_.) + (Complex[0, fz_])*(f_.)*(x_)]/((c_.) + (d_.)*(x_)), x_Symbol] :> Simp[CoshIntegral[(c*f*fz)/d
+ f*fz*x]/d, x] /; FreeQ[{c, d, e, f, fz}, x] && EqQ[d*(e - Pi/2) - c*f*fz*I, 0]

Rule 3303

Int[sin[(e_.) + (f_.)*(x_)]/((c_.) + (d_.)*(x_)), x_Symbol] :> Dist[Cos[(d*e - c*f)/d], Int[Sin[(c*f)/d + f*x]
/(c + d*x), x], x] + Dist[Sin[(d*e - c*f)/d], Int[Cos[(c*f)/d + f*x]/(c + d*x), x], x] /; FreeQ[{c, d, e, f},
x] && NeQ[d*e - c*f, 0]

Rule 3312

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 5448

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 5700

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

Rule 5778

Int[((a_.) + ArcCosh[(c_.)*(x_)]*(b_.))^(n_)*((f_.)*(x_))^(m_.)*((d1_) + (e1_.)*(x_))^(p_.)*((d2_) + (e2_.)*(x
_))^(p_.), x_Symbol] :> Simp[((f*x)^m*Sqrt[1 + c*x]*Sqrt[-1 + c*x]*(d1 + e1*x)^p*(d2 + e2*x)^p*(a + b*ArcCosh[
c*x])^(n + 1))/(b*c*(n + 1)), x] + (Dist[(f*m*(-(d1*d2))^IntPart[p]*(d1 + e1*x)^FracPart[p]*(d2 + e2*x)^FracPa
rt[p])/(b*c*(n + 1)*(1 + c*x)^FracPart[p]*(-1 + c*x)^FracPart[p]), Int[(f*x)^(m - 1)*(-1 + c^2*x^2)^(p - 1/2)*
(a + b*ArcCosh[c*x])^(n + 1), x], x] - Dist[(c*(m + 2*p + 1)*(-(d1*d2))^IntPart[p]*(d1 + e1*x)^FracPart[p]*(d2
+ e2*x)^FracPart[p])/(b*f*(n + 1)*(1 + c*x)^FracPart[p]*(-1 + c*x)^FracPart[p]), Int[(f*x)^(m + 1)*(-1 + c^2*
x^2)^(p - 1/2)*(a + b*ArcCosh[c*x])^(n + 1), x], x]) /; FreeQ[{a, b, c, d1, e1, d2, e2, f}, x] && EqQ[e1 - c*d
1, 0] && EqQ[e2 + c*d2, 0] && LtQ[n, -1] && IGtQ[m, -3] && IGtQ[p + 1/2, 0]

Rule 5780

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

Rule 5798

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

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} \operatorname {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 ) \operatorname {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 ) \operatorname {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 ) \operatorname {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} \operatorname {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 ) \operatorname {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 ) \operatorname {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 ) \operatorname {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 ) \operatorname {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 ) \operatorname {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 ) \operatorname {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 ) \operatorname {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 ) \operatorname {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 ) \operatorname {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 ) \operatorname {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 ) \operatorname {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 ) \operatorname {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 ) \operatorname {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 ) \operatorname {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 ) \operatorname {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 ) \operatorname {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 ) \operatorname {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 ) \operatorname {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 ) \operatorname {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 ) \operatorname {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 = 1.43, size = 436, normalized size = 0.97 \[ \frac {\sqrt {c x-1} \sqrt {c x+1} \left (-5 \sinh \left (\frac {a}{b}\right ) \left (a+b \cosh ^{-1}(c x)\right ) \text {Chi}\left (\frac {a}{b}+\cosh ^{-1}(c x)\right )+27 \sinh \left (\frac {3 a}{b}\right ) \left (a+b \cosh ^{-1}(c x)\right ) \text {Chi}\left (3 \left (\frac {a}{b}+\cosh ^{-1}(c x)\right )\right )-25 a \sinh \left (\frac {5 a}{b}\right ) \text {Chi}\left (5 \left (\frac {a}{b}+\cosh ^{-1}(c x)\right )\right )-25 b \sinh \left (\frac {5 a}{b}\right ) \cosh ^{-1}(c x) \text {Chi}\left (5 \left (\frac {a}{b}+\cosh ^{-1}(c x)\right )\right )+7 a \sinh \left (\frac {7 a}{b}\right ) \text {Chi}\left (7 \left (\frac {a}{b}+\cosh ^{-1}(c x)\right )\right )+7 b \sinh \left (\frac {7 a}{b}\right ) \cosh ^{-1}(c x) \text {Chi}\left (7 \left (\frac {a}{b}+\cosh ^{-1}(c x)\right )\right )+5 a \cosh \left (\frac {a}{b}\right ) \text {Shi}\left (\frac {a}{b}+\cosh ^{-1}(c x)\right )+5 b \cosh \left (\frac {a}{b}\right ) \cosh ^{-1}(c x) \text {Shi}\left (\frac {a}{b}+\cosh ^{-1}(c x)\right )-27 a \cosh \left (\frac {3 a}{b}\right ) \text {Shi}\left (3 \left (\frac {a}{b}+\cosh ^{-1}(c x)\right )\right )-27 b \cosh \left (\frac {3 a}{b}\right ) \cosh ^{-1}(c x) \text {Shi}\left (3 \left (\frac {a}{b}+\cosh ^{-1}(c x)\right )\right )+25 a \cosh \left (\frac {5 a}{b}\right ) \text {Shi}\left (5 \left (\frac {a}{b}+\cosh ^{-1}(c x)\right )\right )+25 b \cosh \left (\frac {5 a}{b}\right ) \cosh ^{-1}(c x) \text {Shi}\left (5 \left (\frac {a}{b}+\cosh ^{-1}(c x)\right )\right )-7 a \cosh \left (\frac {7 a}{b}\right ) \text {Shi}\left (7 \left (\frac {a}{b}+\cosh ^{-1}(c x)\right )\right )-7 b \cosh \left (\frac {7 a}{b}\right ) \cosh ^{-1}(c x) \text {Shi}\left (7 \left (\frac {a}{b}+\cosh ^{-1}(c x)\right )\right )+64 b c^7 x^7-192 b c^5 x^5+192 b c^3 x^3-64 b c x\right )}{64 b^2 c^2 \sqrt {1-c^2 x^2} \left (a+b \cosh ^{-1}(c x)\right )} \]

Warning: Unable to verify antiderivative.

[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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fricas [F]  time = 0.48, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {{\left (c^{4} x^{5} - 2 \, c^{2} x^{3} + x\right )} \sqrt {-c^{2} x^{2} + 1}}{b^{2} \operatorname {arcosh}\left (c x\right )^{2} + 2 \, a b \operatorname {arcosh}\left (c x\right ) + a^{2}}, x\right ) \]

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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giac [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {{\left (-c^{2} x^{2} + 1\right )}^{\frac {5}{2}} x}{{\left (b \operatorname {arcosh}\left (c x\right ) + a\right )}^{2}}\,{d x} \]

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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maple [B]  time = 0.56, size = 1499, normalized size = 3.35 \[ \text {result too large to display} \]

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)

[Out]

1/128*(-c^2*x^2+1)^(1/2)*(-64*(c*x+1)^(1/2)*(c*x-1)^(1/2)*x^7*c^7+64*c^8*x^8+112*(c*x+1)^(1/2)*(c*x-1)^(1/2)*x
^5*c^5-144*c^6*x^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^2*x^2+1)^(1/
2)/(c*x-1)^(1/2)/(c*x+1)^(1/2)*(64*(c*x+1)^(1/2)*(c*x-1)^(1/2)*x^6*b*c^6+64*x^7*b*c^7-80*(c*x+1)^(1/2)*(c*x-1)
^(1/2)*x^4*b*c^4-112*x^5*b*c^5+24*(c*x+1)^(1/2)*(c*x-1)^(1/2)*x^2*b*c^2+56*x^3*b*c^3+7*arccosh(c*x)*Ei(1,-7*ar
ccosh(c*x)-7*a/b)*exp(-7*a/b)*b+7*Ei(1,-7*arccosh(c*x)-7*a/b)*exp(-7*a/b)*a-(c*x+1)^(1/2)*(c*x-1)^(1/2)*b-7*x*
b*c)/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*c^6*x^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)*(arcco
sh(c*x)*exp(-a/b)*Ei(1,-arccosh(c*x)-a/b)*b+(c*x+1)^(1/2)*(c*x-1)^(1/2)*b+exp(-a/b)*Ei(1,-arccosh(c*x)-a/b)*a+
x*b*c)/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)*x^2*b*c^2+4*x^3*b*c^3+3*arccosh(c*x)*Ei(1,-3*arccosh(c*x)-3*a/b)*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*x*b*c)/c^2/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)*(16*(c*x+1)^(1/2)*(c*x-1)^(1/2)*x^4*b*c^4+16*x^5*b*c^5-12*(c*x+1)^(1/2)*(c*x-1)^
(1/2)*x^2*b*c^2-20*x^3*b*c^3+5*arccosh(c*x)*exp(-5*a/b)*Ei(1,-5*arccosh(c*x)-5*a/b)*b+(c*x+1)^(1/2)*(c*x-1)^(1
/2)*b+5*exp(-5*a/b)*Ei(1,-5*arccosh(c*x)-5*a/b)*a+5*x*b*c)/c^2/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)*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

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maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \[ -\frac {{\left ({\left (c^{6} x^{7} - 3 \, c^{4} x^{5} + 3 \, c^{2} x^{3} - x\right )} {\left (c x + 1\right )} \sqrt {c x - 1} + {\left (c^{7} x^{8} - 3 \, c^{5} x^{6} + 3 \, c^{3} x^{4} - c x^{2}\right )} \sqrt {c x + 1}\right )} \sqrt {-c x + 1}}{a b c^{3} x^{2} + \sqrt {c x + 1} \sqrt {c x - 1} a b c^{2} x - a b c + {\left (b^{2} c^{3} x^{2} + \sqrt {c x + 1} \sqrt {c x - 1} b^{2} c^{2} x - b^{2} c\right )} \log \left (c x + \sqrt {c x + 1} \sqrt {c x - 1}\right )} + \int \frac {{\left (7 \, {\left (c^{7} x^{7} - 2 \, c^{5} x^{5} + c^{3} x^{3}\right )} {\left (c x + 1\right )}^{\frac {3}{2}} {\left (c x - 1\right )} + {\left (14 \, c^{8} x^{8} - 37 \, c^{6} x^{6} + 33 \, c^{4} x^{4} - 11 \, c^{2} x^{2} + 1\right )} {\left (c x + 1\right )} \sqrt {c x - 1} + {\left (7 \, c^{9} x^{9} - 23 \, c^{7} x^{7} + 27 \, c^{5} x^{5} - 13 \, c^{3} x^{3} + 2 \, c x\right )} \sqrt {c x + 1}\right )} \sqrt {-c x + 1}}{a b c^{5} x^{4} + {\left (c x + 1\right )} {\left (c x - 1\right )} a b c^{3} x^{2} - 2 \, a b c^{3} x^{2} + a b c + 2 \, {\left (a b c^{4} x^{3} - a b c^{2} x\right )} \sqrt {c x + 1} \sqrt {c x - 1} + {\left (b^{2} c^{5} x^{4} + {\left (c x + 1\right )} {\left (c x - 1\right )} b^{2} c^{3} x^{2} - 2 \, b^{2} c^{3} x^{2} + b^{2} c + 2 \, {\left (b^{2} c^{4} x^{3} - b^{2} c^{2} x\right )} \sqrt {c x + 1} \sqrt {c x - 1}\right )} \log \left (c x + \sqrt {c x + 1} \sqrt {c x - 1}\right )}\,{d x} \]

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

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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sympy [F(-1)]  time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]

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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