\(\int \frac {x (1-c^2 x^2)^{5/2}}{a+b \text {arccosh}(c x)} \, dx\) [286]

Optimal result

Integrand size = 26, antiderivative size = 397 \[ \int \frac {x \left (1-c^2 x^2\right )^{5/2}}{a+b \text {arccosh}(c x)} \, dx=-\frac {5 \sqrt {1-c x} \cosh \left (\frac {a}{b}\right ) \text {Chi}\left (\frac {a+b \text {arccosh}(c x)}{b}\right )}{64 b c^2 \sqrt {-1+c x}}+\frac {9 \sqrt {1-c x} \cosh \left (\frac {3 a}{b}\right ) \text {Chi}\left (\frac {3 (a+b \text {arccosh}(c x))}{b}\right )}{64 b c^2 \sqrt {-1+c x}}-\frac {5 \sqrt {1-c x} \cosh \left (\frac {5 a}{b}\right ) \text {Chi}\left (\frac {5 (a+b \text {arccosh}(c x))}{b}\right )}{64 b c^2 \sqrt {-1+c x}}+\frac {\sqrt {1-c x} \cosh \left (\frac {7 a}{b}\right ) \text {Chi}\left (\frac {7 (a+b \text {arccosh}(c x))}{b}\right )}{64 b c^2 \sqrt {-1+c x}}+\frac {5 \sqrt {1-c x} \sinh \left (\frac {a}{b}\right ) \text {Shi}\left (\frac {a+b \text {arccosh}(c x)}{b}\right )}{64 b c^2 \sqrt {-1+c x}}-\frac {9 \sqrt {1-c x} \sinh \left (\frac {3 a}{b}\right ) \text {Shi}\left (\frac {3 (a+b \text {arccosh}(c x))}{b}\right )}{64 b c^2 \sqrt {-1+c x}}+\frac {5 \sqrt {1-c x} \sinh \left (\frac {5 a}{b}\right ) \text {Shi}\left (\frac {5 (a+b \text {arccosh}(c x))}{b}\right )}{64 b c^2 \sqrt {-1+c x}}-\frac {\sqrt {1-c x} \sinh \left (\frac {7 a}{b}\right ) \text {Shi}\left (\frac {7 (a+b \text {arccosh}(c x))}{b}\right )}{64 b c^2 \sqrt {-1+c x}} \]

[Out]

-5/64*Chi((a+b*arccosh(c*x))/b)*cosh(a/b)*(-c*x+1)^(1/2)/b/c^2/(c*x-1)^(1/2)+9/64*Chi(3*(a+b*arccosh(c*x))/b)*
cosh(3*a/b)*(-c*x+1)^(1/2)/b/c^2/(c*x-1)^(1/2)-5/64*Chi(5*(a+b*arccosh(c*x))/b)*cosh(5*a/b)*(-c*x+1)^(1/2)/b/c
^2/(c*x-1)^(1/2)+1/64*Chi(7*(a+b*arccosh(c*x))/b)*cosh(7*a/b)*(-c*x+1)^(1/2)/b/c^2/(c*x-1)^(1/2)+5/64*Shi((a+b
*arccosh(c*x))/b)*sinh(a/b)*(-c*x+1)^(1/2)/b/c^2/(c*x-1)^(1/2)-9/64*Shi(3*(a+b*arccosh(c*x))/b)*sinh(3*a/b)*(-
c*x+1)^(1/2)/b/c^2/(c*x-1)^(1/2)+5/64*Shi(5*(a+b*arccosh(c*x))/b)*sinh(5*a/b)*(-c*x+1)^(1/2)/b/c^2/(c*x-1)^(1/
2)-1/64*Shi(7*(a+b*arccosh(c*x))/b)*sinh(7*a/b)*(-c*x+1)^(1/2)/b/c^2/(c*x-1)^(1/2)

Rubi [A] (verified)

Time = 0.31 (sec) , antiderivative size = 397, normalized size of antiderivative = 1.00, number of steps used = 15, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.192, Rules used = {5952, 5556, 3384, 3379, 3382} \[ \int \frac {x \left (1-c^2 x^2\right )^{5/2}}{a+b \text {arccosh}(c x)} \, dx=-\frac {5 \sqrt {1-c x} \cosh \left (\frac {a}{b}\right ) \text {Chi}\left (\frac {a+b \text {arccosh}(c x)}{b}\right )}{64 b c^2 \sqrt {c x-1}}+\frac {9 \sqrt {1-c x} \cosh \left (\frac {3 a}{b}\right ) \text {Chi}\left (\frac {3 (a+b \text {arccosh}(c x))}{b}\right )}{64 b c^2 \sqrt {c x-1}}-\frac {5 \sqrt {1-c x} \cosh \left (\frac {5 a}{b}\right ) \text {Chi}\left (\frac {5 (a+b \text {arccosh}(c x))}{b}\right )}{64 b c^2 \sqrt {c x-1}}+\frac {\sqrt {1-c x} \cosh \left (\frac {7 a}{b}\right ) \text {Chi}\left (\frac {7 (a+b \text {arccosh}(c x))}{b}\right )}{64 b c^2 \sqrt {c x-1}}+\frac {5 \sqrt {1-c x} \sinh \left (\frac {a}{b}\right ) \text {Shi}\left (\frac {a+b \text {arccosh}(c x)}{b}\right )}{64 b c^2 \sqrt {c x-1}}-\frac {9 \sqrt {1-c x} \sinh \left (\frac {3 a}{b}\right ) \text {Shi}\left (\frac {3 (a+b \text {arccosh}(c x))}{b}\right )}{64 b c^2 \sqrt {c x-1}}+\frac {5 \sqrt {1-c x} \sinh \left (\frac {5 a}{b}\right ) \text {Shi}\left (\frac {5 (a+b \text {arccosh}(c x))}{b}\right )}{64 b c^2 \sqrt {c x-1}}-\frac {\sqrt {1-c x} \sinh \left (\frac {7 a}{b}\right ) \text {Shi}\left (\frac {7 (a+b \text {arccosh}(c x))}{b}\right )}{64 b c^2 \sqrt {c x-1}} \]

[In]

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

[Out]

(-5*Sqrt[1 - c*x]*Cosh[a/b]*CoshIntegral[(a + b*ArcCosh[c*x])/b])/(64*b*c^2*Sqrt[-1 + c*x]) + (9*Sqrt[1 - c*x]
*Cosh[(3*a)/b]*CoshIntegral[(3*(a + b*ArcCosh[c*x]))/b])/(64*b*c^2*Sqrt[-1 + c*x]) - (5*Sqrt[1 - c*x]*Cosh[(5*
a)/b]*CoshIntegral[(5*(a + b*ArcCosh[c*x]))/b])/(64*b*c^2*Sqrt[-1 + c*x]) + (Sqrt[1 - c*x]*Cosh[(7*a)/b]*CoshI
ntegral[(7*(a + b*ArcCosh[c*x]))/b])/(64*b*c^2*Sqrt[-1 + c*x]) + (5*Sqrt[1 - c*x]*Sinh[a/b]*SinhIntegral[(a +
b*ArcCosh[c*x])/b])/(64*b*c^2*Sqrt[-1 + c*x]) - (9*Sqrt[1 - c*x]*Sinh[(3*a)/b]*SinhIntegral[(3*(a + b*ArcCosh[
c*x]))/b])/(64*b*c^2*Sqrt[-1 + c*x]) + (5*Sqrt[1 - c*x]*Sinh[(5*a)/b]*SinhIntegral[(5*(a + b*ArcCosh[c*x]))/b]
)/(64*b*c^2*Sqrt[-1 + c*x]) - (Sqrt[1 - c*x]*Sinh[(7*a)/b]*SinhIntegral[(7*(a + b*ArcCosh[c*x]))/b])/(64*b*c^2
*Sqrt[-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 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*} \text {integral}& = \frac {\sqrt {1-c x} \text {Subst}\left (\int \frac {\cosh \left (\frac {a}{b}-\frac {x}{b}\right ) \sinh ^6\left (\frac {a}{b}-\frac {x}{b}\right )}{x} \, dx,x,a+b \text {arccosh}(c x)\right )}{b c^2 \sqrt {-1+c x}} \\ & = \frac {\sqrt {1-c x} \text {Subst}\left (\int \left (\frac {\cosh \left (\frac {7 a}{b}-\frac {7 x}{b}\right )}{64 x}-\frac {5 \cosh \left (\frac {5 a}{b}-\frac {5 x}{b}\right )}{64 x}+\frac {9 \cosh \left (\frac {3 a}{b}-\frac {3 x}{b}\right )}{64 x}-\frac {5 \cosh \left (\frac {a}{b}-\frac {x}{b}\right )}{64 x}\right ) \, dx,x,a+b \text {arccosh}(c x)\right )}{b c^2 \sqrt {-1+c x}} \\ & = \frac {\sqrt {1-c x} \text {Subst}\left (\int \frac {\cosh \left (\frac {7 a}{b}-\frac {7 x}{b}\right )}{x} \, dx,x,a+b \text {arccosh}(c x)\right )}{64 b c^2 \sqrt {-1+c x}}-\frac {\left (5 \sqrt {1-c x}\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 )}{64 b c^2 \sqrt {-1+c x}}-\frac {\left (5 \sqrt {1-c x}\right ) \text {Subst}\left (\int \frac {\cosh \left (\frac {a}{b}-\frac {x}{b}\right )}{x} \, dx,x,a+b \text {arccosh}(c x)\right )}{64 b c^2 \sqrt {-1+c x}}+\frac {\left (9 \sqrt {1-c x}\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 )}{64 b c^2 \sqrt {-1+c x}} \\ & = -\frac {\left (5 \sqrt {1-c x} \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 )}{64 b c^2 \sqrt {-1+c x}}+\frac {\left (9 \sqrt {1-c x} \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 )}{64 b c^2 \sqrt {-1+c x}}-\frac {\left (5 \sqrt {1-c x} \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 )}{64 b c^2 \sqrt {-1+c x}}+\frac {\left (\sqrt {1-c x} \cosh \left (\frac {7 a}{b}\right )\right ) \text {Subst}\left (\int \frac {\cosh \left (\frac {7 x}{b}\right )}{x} \, dx,x,a+b \text {arccosh}(c x)\right )}{64 b c^2 \sqrt {-1+c x}}+\frac {\left (5 \sqrt {1-c x} \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 )}{64 b c^2 \sqrt {-1+c x}}-\frac {\left (9 \sqrt {1-c x} \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 )}{64 b c^2 \sqrt {-1+c x}}+\frac {\left (5 \sqrt {1-c x} \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 )}{64 b c^2 \sqrt {-1+c x}}-\frac {\left (\sqrt {1-c x} \sinh \left (\frac {7 a}{b}\right )\right ) \text {Subst}\left (\int \frac {\sinh \left (\frac {7 x}{b}\right )}{x} \, dx,x,a+b \text {arccosh}(c x)\right )}{64 b c^2 \sqrt {-1+c x}} \\ & = -\frac {5 \sqrt {1-c x} \cosh \left (\frac {a}{b}\right ) \text {Chi}\left (\frac {a+b \text {arccosh}(c x)}{b}\right )}{64 b c^2 \sqrt {-1+c x}}+\frac {9 \sqrt {1-c x} \cosh \left (\frac {3 a}{b}\right ) \text {Chi}\left (\frac {3 (a+b \text {arccosh}(c x))}{b}\right )}{64 b c^2 \sqrt {-1+c x}}-\frac {5 \sqrt {1-c x} \cosh \left (\frac {5 a}{b}\right ) \text {Chi}\left (\frac {5 (a+b \text {arccosh}(c x))}{b}\right )}{64 b c^2 \sqrt {-1+c x}}+\frac {\sqrt {1-c x} \cosh \left (\frac {7 a}{b}\right ) \text {Chi}\left (\frac {7 (a+b \text {arccosh}(c x))}{b}\right )}{64 b c^2 \sqrt {-1+c x}}+\frac {5 \sqrt {1-c x} \sinh \left (\frac {a}{b}\right ) \text {Shi}\left (\frac {a+b \text {arccosh}(c x)}{b}\right )}{64 b c^2 \sqrt {-1+c x}}-\frac {9 \sqrt {1-c x} \sinh \left (\frac {3 a}{b}\right ) \text {Shi}\left (\frac {3 (a+b \text {arccosh}(c x))}{b}\right )}{64 b c^2 \sqrt {-1+c x}}+\frac {5 \sqrt {1-c x} \sinh \left (\frac {5 a}{b}\right ) \text {Shi}\left (\frac {5 (a+b \text {arccosh}(c x))}{b}\right )}{64 b c^2 \sqrt {-1+c x}}-\frac {\sqrt {1-c x} \sinh \left (\frac {7 a}{b}\right ) \text {Shi}\left (\frac {7 (a+b \text {arccosh}(c x))}{b}\right )}{64 b c^2 \sqrt {-1+c x}} \\ \end{align*}

Mathematica [A] (warning: unable to verify)

Time = 0.98 (sec) , antiderivative size = 216, normalized size of antiderivative = 0.54 \[ \int \frac {x \left (1-c^2 x^2\right )^{5/2}}{a+b \text {arccosh}(c x)} \, dx=\frac {\sqrt {1-c^2 x^2} \left (-5 \cosh \left (\frac {a}{b}\right ) \text {Chi}\left (\frac {a}{b}+\text {arccosh}(c x)\right )+9 \cosh \left (\frac {3 a}{b}\right ) \text {Chi}\left (3 \left (\frac {a}{b}+\text {arccosh}(c x)\right )\right )-5 \cosh \left (\frac {5 a}{b}\right ) \text {Chi}\left (5 \left (\frac {a}{b}+\text {arccosh}(c x)\right )\right )+\cosh \left (\frac {7 a}{b}\right ) \text {Chi}\left (7 \left (\frac {a}{b}+\text {arccosh}(c x)\right )\right )+5 \sinh \left (\frac {a}{b}\right ) \text {Shi}\left (\frac {a}{b}+\text {arccosh}(c x)\right )-9 \sinh \left (\frac {3 a}{b}\right ) \text {Shi}\left (3 \left (\frac {a}{b}+\text {arccosh}(c x)\right )\right )+5 \sinh \left (\frac {5 a}{b}\right ) \text {Shi}\left (5 \left (\frac {a}{b}+\text {arccosh}(c x)\right )\right )-\sinh \left (\frac {7 a}{b}\right ) \text {Shi}\left (7 \left (\frac {a}{b}+\text {arccosh}(c x)\right )\right )\right )}{64 c^2 \sqrt {\frac {-1+c x}{1+c x}} (b+b c x)} \]

[In]

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

[Out]

(Sqrt[1 - c^2*x^2]*(-5*Cosh[a/b]*CoshIntegral[a/b + ArcCosh[c*x]] + 9*Cosh[(3*a)/b]*CoshIntegral[3*(a/b + ArcC
osh[c*x])] - 5*Cosh[(5*a)/b]*CoshIntegral[5*(a/b + ArcCosh[c*x])] + Cosh[(7*a)/b]*CoshIntegral[7*(a/b + ArcCos
h[c*x])] + 5*Sinh[a/b]*SinhIntegral[a/b + ArcCosh[c*x]] - 9*Sinh[(3*a)/b]*SinhIntegral[3*(a/b + ArcCosh[c*x])]
+ 5*Sinh[(5*a)/b]*SinhIntegral[5*(a/b + ArcCosh[c*x])] - Sinh[(7*a)/b]*SinhIntegral[7*(a/b + ArcCosh[c*x])]))
/(64*c^2*Sqrt[(-1 + c*x)/(1 + c*x)]*(b + b*c*x))

Maple [A] (verified)

Time = 0.51 (sec) , antiderivative size = 318, normalized size of antiderivative = 0.80

method	result	size
default	\(\frac {\sqrt {-c^{2} x^{2}+1}\, \left (-\sqrt {c x -1}\, \sqrt {c x +1}\, c x +c^{2} x^{2}-1\right ) \left (\operatorname {Ei}_{1}\left (7 \,\operatorname {arccosh}\left (c x \right )+\frac {7 a}{b}\right ) {\mathrm e}^{\frac {b \,\operatorname {arccosh}\left (c x \right )+7 a}{b}}+\operatorname {Ei}_{1}\left (-7 \,\operatorname {arccosh}\left (c x \right )-\frac {7 a}{b}\right ) {\mathrm e}^{-\frac {-b \,\operatorname {arccosh}\left (c x \right )+7 a}{b}}-5 \,\operatorname {Ei}_{1}\left (5 \,\operatorname {arccosh}\left (c x \right )+\frac {5 a}{b}\right ) {\mathrm e}^{\frac {b \,\operatorname {arccosh}\left (c x \right )+5 a}{b}}+9 \,\operatorname {Ei}_{1}\left (3 \,\operatorname {arccosh}\left (c x \right )+\frac {3 a}{b}\right ) {\mathrm e}^{\frac {b \,\operatorname {arccosh}\left (c x \right )+3 a}{b}}-5 \,\operatorname {Ei}_{1}\left (\operatorname {arccosh}\left (c x \right )+\frac {a}{b}\right ) {\mathrm e}^{\frac {a +b \,\operatorname {arccosh}\left (c x \right )}{b}}-5 \,\operatorname {Ei}_{1}\left (-\operatorname {arccosh}\left (c x \right )-\frac {a}{b}\right ) {\mathrm e}^{-\frac {-b \,\operatorname {arccosh}\left (c x \right )+a}{b}}+9 \,\operatorname {Ei}_{1}\left (-3 \,\operatorname {arccosh}\left (c x \right )-\frac {3 a}{b}\right ) {\mathrm e}^{-\frac {-b \,\operatorname {arccosh}\left (c x \right )+3 a}{b}}-5 \,\operatorname {Ei}_{1}\left (-5 \,\operatorname {arccosh}\left (c x \right )-\frac {5 a}{b}\right ) {\mathrm e}^{-\frac {-b \,\operatorname {arccosh}\left (c x \right )+5 a}{b}}\right )}{128 \left (c x +1\right ) c^{2} \left (c x -1\right ) b}\)	\(318\)

[In]

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

[Out]

1/128*(-c^2*x^2+1)^(1/2)*(-(c*x-1)^(1/2)*(c*x+1)^(1/2)*c*x+c^2*x^2-1)*(Ei(1,7*arccosh(c*x)+7*a/b)*exp((b*arcco
sh(c*x)+7*a)/b)+Ei(1,-7*arccosh(c*x)-7*a/b)*exp(-(-b*arccosh(c*x)+7*a)/b)-5*Ei(1,5*arccosh(c*x)+5*a/b)*exp((b*
arccosh(c*x)+5*a)/b)+9*Ei(1,3*arccosh(c*x)+3*a/b)*exp((b*arccosh(c*x)+3*a)/b)-5*Ei(1,arccosh(c*x)+a/b)*exp((a+
b*arccosh(c*x))/b)-5*Ei(1,-arccosh(c*x)-a/b)*exp(-(-b*arccosh(c*x)+a)/b)+9*Ei(1,-3*arccosh(c*x)-3*a/b)*exp(-(-
b*arccosh(c*x)+3*a)/b)-5*Ei(1,-5*arccosh(c*x)-5*a/b)*exp(-(-b*arccosh(c*x)+5*a)/b))/(c*x+1)/c^2/(c*x-1)/b

Fricas [F]

\[ \int \frac {x \left (1-c^2 x^2\right )^{5/2}}{a+b \text {arccosh}(c x)} \, dx=\int { \frac {{\left (-c^{2} x^{2} + 1\right )}^{\frac {5}{2}} x}{b \operatorname {arcosh}\left (c x\right ) + a} \,d x } \]

[In]

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

[Out]

integral((c^4*x^5 - 2*c^2*x^3 + x)*sqrt(-c^2*x^2 + 1)/(b*arccosh(c*x) + a), x)

Sympy [F(-1)]

Timed out. \[ \int \frac {x \left (1-c^2 x^2\right )^{5/2}}{a+b \text {arccosh}(c x)} \, dx=\text {Timed out} \]

[In]

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

[Out]

Timed out

Maxima [F]

\[ \int \frac {x \left (1-c^2 x^2\right )^{5/2}}{a+b \text {arccosh}(c x)} \, dx=\int { \frac {{\left (-c^{2} x^{2} + 1\right )}^{\frac {5}{2}} x}{b \operatorname {arcosh}\left (c x\right ) + a} \,d x } \]

[In]

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

[Out]

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

Giac [F]

\[ \int \frac {x \left (1-c^2 x^2\right )^{5/2}}{a+b \text {arccosh}(c x)} \, dx=\int { \frac {{\left (-c^{2} x^{2} + 1\right )}^{\frac {5}{2}} x}{b \operatorname {arcosh}\left (c x\right ) + a} \,d x } \]

[In]

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

[Out]

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

Mupad [F(-1)]

Timed out. \[ \int \frac {x \left (1-c^2 x^2\right )^{5/2}}{a+b \text {arccosh}(c x)} \, dx=\int \frac {x\,{\left (1-c^2\,x^2\right )}^{5/2}}{a+b\,\mathrm {acosh}\left (c\,x\right )} \,d x \]

[In]

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

[Out]

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