\(\int \frac {x^3 (1-c^2 x^2)^{5/2}}{a+b \text {arccosh}(c x)} \, dx\) [284]

Optimal result

Integrand size = 28, antiderivative size = 397 \[ \int \frac {x^3 \left (1-c^2 x^2\right )^{5/2}}{a+b \text {arccosh}(c x)} \, dx=-\frac {3 \sqrt {1-c x} \cosh \left (\frac {a}{b}\right ) \text {Chi}\left (\frac {a+b \text {arccosh}(c x)}{b}\right )}{128 b c^4 \sqrt {-1+c x}}+\frac {\sqrt {1-c x} \cosh \left (\frac {3 a}{b}\right ) \text {Chi}\left (\frac {3 (a+b \text {arccosh}(c x))}{b}\right )}{32 b c^4 \sqrt {-1+c x}}-\frac {3 \sqrt {1-c x} \cosh \left (\frac {7 a}{b}\right ) \text {Chi}\left (\frac {7 (a+b \text {arccosh}(c x))}{b}\right )}{256 b c^4 \sqrt {-1+c x}}+\frac {\sqrt {1-c x} \cosh \left (\frac {9 a}{b}\right ) \text {Chi}\left (\frac {9 (a+b \text {arccosh}(c x))}{b}\right )}{256 b c^4 \sqrt {-1+c x}}+\frac {3 \sqrt {1-c x} \sinh \left (\frac {a}{b}\right ) \text {Shi}\left (\frac {a+b \text {arccosh}(c x)}{b}\right )}{128 b c^4 \sqrt {-1+c x}}-\frac {\sqrt {1-c x} \sinh \left (\frac {3 a}{b}\right ) \text {Shi}\left (\frac {3 (a+b \text {arccosh}(c x))}{b}\right )}{32 b c^4 \sqrt {-1+c x}}+\frac {3 \sqrt {1-c x} \sinh \left (\frac {7 a}{b}\right ) \text {Shi}\left (\frac {7 (a+b \text {arccosh}(c x))}{b}\right )}{256 b c^4 \sqrt {-1+c x}}-\frac {\sqrt {1-c x} \sinh \left (\frac {9 a}{b}\right ) \text {Shi}\left (\frac {9 (a+b \text {arccosh}(c x))}{b}\right )}{256 b c^4 \sqrt {-1+c x}} \]

[Out]

-3/128*Chi((a+b*arccosh(c*x))/b)*cosh(a/b)*(-c*x+1)^(1/2)/b/c^4/(c*x-1)^(1/2)+1/32*Chi(3*(a+b*arccosh(c*x))/b)
*cosh(3*a/b)*(-c*x+1)^(1/2)/b/c^4/(c*x-1)^(1/2)-3/256*Chi(7*(a+b*arccosh(c*x))/b)*cosh(7*a/b)*(-c*x+1)^(1/2)/b
/c^4/(c*x-1)^(1/2)+1/256*Chi(9*(a+b*arccosh(c*x))/b)*cosh(9*a/b)*(-c*x+1)^(1/2)/b/c^4/(c*x-1)^(1/2)+3/128*Shi(
(a+b*arccosh(c*x))/b)*sinh(a/b)*(-c*x+1)^(1/2)/b/c^4/(c*x-1)^(1/2)-1/32*Shi(3*(a+b*arccosh(c*x))/b)*sinh(3*a/b
)*(-c*x+1)^(1/2)/b/c^4/(c*x-1)^(1/2)+3/256*Shi(7*(a+b*arccosh(c*x))/b)*sinh(7*a/b)*(-c*x+1)^(1/2)/b/c^4/(c*x-1
)^(1/2)-1/256*Shi(9*(a+b*arccosh(c*x))/b)*sinh(9*a/b)*(-c*x+1)^(1/2)/b/c^4/(c*x-1)^(1/2)

Rubi [A] (verified)

Time = 0.37 (sec) , antiderivative size = 397, normalized size of antiderivative = 1.00, number of steps used = 15, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.179, Rules used = {5952, 5556, 3384, 3379, 3382} \[ \int \frac {x^3 \left (1-c^2 x^2\right )^{5/2}}{a+b \text {arccosh}(c x)} \, dx=-\frac {3 \sqrt {1-c x} \cosh \left (\frac {a}{b}\right ) \text {Chi}\left (\frac {a+b \text {arccosh}(c x)}{b}\right )}{128 b c^4 \sqrt {c x-1}}+\frac {\sqrt {1-c x} \cosh \left (\frac {3 a}{b}\right ) \text {Chi}\left (\frac {3 (a+b \text {arccosh}(c x))}{b}\right )}{32 b c^4 \sqrt {c x-1}}-\frac {3 \sqrt {1-c x} \cosh \left (\frac {7 a}{b}\right ) \text {Chi}\left (\frac {7 (a+b \text {arccosh}(c x))}{b}\right )}{256 b c^4 \sqrt {c x-1}}+\frac {\sqrt {1-c x} \cosh \left (\frac {9 a}{b}\right ) \text {Chi}\left (\frac {9 (a+b \text {arccosh}(c x))}{b}\right )}{256 b c^4 \sqrt {c x-1}}+\frac {3 \sqrt {1-c x} \sinh \left (\frac {a}{b}\right ) \text {Shi}\left (\frac {a+b \text {arccosh}(c x)}{b}\right )}{128 b c^4 \sqrt {c x-1}}-\frac {\sqrt {1-c x} \sinh \left (\frac {3 a}{b}\right ) \text {Shi}\left (\frac {3 (a+b \text {arccosh}(c x))}{b}\right )}{32 b c^4 \sqrt {c x-1}}+\frac {3 \sqrt {1-c x} \sinh \left (\frac {7 a}{b}\right ) \text {Shi}\left (\frac {7 (a+b \text {arccosh}(c x))}{b}\right )}{256 b c^4 \sqrt {c x-1}}-\frac {\sqrt {1-c x} \sinh \left (\frac {9 a}{b}\right ) \text {Shi}\left (\frac {9 (a+b \text {arccosh}(c x))}{b}\right )}{256 b c^4 \sqrt {c x-1}} \]

[In]

Int[(x^3*(1 - c^2*x^2)^(5/2))/(a + b*ArcCosh[c*x]),x]

[Out]

(-3*Sqrt[1 - c*x]*Cosh[a/b]*CoshIntegral[(a + b*ArcCosh[c*x])/b])/(128*b*c^4*Sqrt[-1 + c*x]) + (Sqrt[1 - c*x]*
Cosh[(3*a)/b]*CoshIntegral[(3*(a + b*ArcCosh[c*x]))/b])/(32*b*c^4*Sqrt[-1 + c*x]) - (3*Sqrt[1 - c*x]*Cosh[(7*a
)/b]*CoshIntegral[(7*(a + b*ArcCosh[c*x]))/b])/(256*b*c^4*Sqrt[-1 + c*x]) + (Sqrt[1 - c*x]*Cosh[(9*a)/b]*CoshI
ntegral[(9*(a + b*ArcCosh[c*x]))/b])/(256*b*c^4*Sqrt[-1 + c*x]) + (3*Sqrt[1 - c*x]*Sinh[a/b]*SinhIntegral[(a +
b*ArcCosh[c*x])/b])/(128*b*c^4*Sqrt[-1 + c*x]) - (Sqrt[1 - c*x]*Sinh[(3*a)/b]*SinhIntegral[(3*(a + b*ArcCosh[
c*x]))/b])/(32*b*c^4*Sqrt[-1 + c*x]) + (3*Sqrt[1 - c*x]*Sinh[(7*a)/b]*SinhIntegral[(7*(a + b*ArcCosh[c*x]))/b]
)/(256*b*c^4*Sqrt[-1 + c*x]) - (Sqrt[1 - c*x]*Sinh[(9*a)/b]*SinhIntegral[(9*(a + b*ArcCosh[c*x]))/b])/(256*b*c
^4*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 ^3\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^4 \sqrt {-1+c x}} \\ & = \frac {\sqrt {1-c x} \text {Subst}\left (\int \left (\frac {\cosh \left (\frac {9 a}{b}-\frac {9 x}{b}\right )}{256 x}-\frac {3 \cosh \left (\frac {7 a}{b}-\frac {7 x}{b}\right )}{256 x}+\frac {\cosh \left (\frac {3 a}{b}-\frac {3 x}{b}\right )}{32 x}-\frac {3 \cosh \left (\frac {a}{b}-\frac {x}{b}\right )}{128 x}\right ) \, dx,x,a+b \text {arccosh}(c x)\right )}{b c^4 \sqrt {-1+c x}} \\ & = \frac {\sqrt {1-c x} \text {Subst}\left (\int \frac {\cosh \left (\frac {9 a}{b}-\frac {9 x}{b}\right )}{x} \, dx,x,a+b \text {arccosh}(c x)\right )}{256 b c^4 \sqrt {-1+c x}}-\frac {\left (3 \sqrt {1-c x}\right ) \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 )}{256 b c^4 \sqrt {-1+c x}}-\frac {\left (3 \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 )}{128 b c^4 \sqrt {-1+c x}}+\frac {\sqrt {1-c x} \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 )}{32 b c^4 \sqrt {-1+c x}} \\ & = -\frac {\left (3 \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 )}{128 b c^4 \sqrt {-1+c x}}+\frac {\left (\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 )}{32 b c^4 \sqrt {-1+c x}}-\frac {\left (3 \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 )}{256 b c^4 \sqrt {-1+c x}}+\frac {\left (\sqrt {1-c x} \cosh \left (\frac {9 a}{b}\right )\right ) \text {Subst}\left (\int \frac {\cosh \left (\frac {9 x}{b}\right )}{x} \, dx,x,a+b \text {arccosh}(c x)\right )}{256 b c^4 \sqrt {-1+c x}}+\frac {\left (3 \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 )}{128 b c^4 \sqrt {-1+c x}}-\frac {\left (\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 )}{32 b c^4 \sqrt {-1+c x}}+\frac {\left (3 \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 )}{256 b c^4 \sqrt {-1+c x}}-\frac {\left (\sqrt {1-c x} \sinh \left (\frac {9 a}{b}\right )\right ) \text {Subst}\left (\int \frac {\sinh \left (\frac {9 x}{b}\right )}{x} \, dx,x,a+b \text {arccosh}(c x)\right )}{256 b c^4 \sqrt {-1+c x}} \\ & = -\frac {3 \sqrt {1-c x} \cosh \left (\frac {a}{b}\right ) \text {Chi}\left (\frac {a+b \text {arccosh}(c x)}{b}\right )}{128 b c^4 \sqrt {-1+c x}}+\frac {\sqrt {1-c x} \cosh \left (\frac {3 a}{b}\right ) \text {Chi}\left (\frac {3 (a+b \text {arccosh}(c x))}{b}\right )}{32 b c^4 \sqrt {-1+c x}}-\frac {3 \sqrt {1-c x} \cosh \left (\frac {7 a}{b}\right ) \text {Chi}\left (\frac {7 (a+b \text {arccosh}(c x))}{b}\right )}{256 b c^4 \sqrt {-1+c x}}+\frac {\sqrt {1-c x} \cosh \left (\frac {9 a}{b}\right ) \text {Chi}\left (\frac {9 (a+b \text {arccosh}(c x))}{b}\right )}{256 b c^4 \sqrt {-1+c x}}+\frac {3 \sqrt {1-c x} \sinh \left (\frac {a}{b}\right ) \text {Shi}\left (\frac {a+b \text {arccosh}(c x)}{b}\right )}{128 b c^4 \sqrt {-1+c x}}-\frac {\sqrt {1-c x} \sinh \left (\frac {3 a}{b}\right ) \text {Shi}\left (\frac {3 (a+b \text {arccosh}(c x))}{b}\right )}{32 b c^4 \sqrt {-1+c x}}+\frac {3 \sqrt {1-c x} \sinh \left (\frac {7 a}{b}\right ) \text {Shi}\left (\frac {7 (a+b \text {arccosh}(c x))}{b}\right )}{256 b c^4 \sqrt {-1+c x}}-\frac {\sqrt {1-c x} \sinh \left (\frac {9 a}{b}\right ) \text {Shi}\left (\frac {9 (a+b \text {arccosh}(c x))}{b}\right )}{256 b c^4 \sqrt {-1+c x}} \\ \end{align*}

Mathematica [A] (warning: unable to verify)

Time = 1.05 (sec) , antiderivative size = 216, normalized size of antiderivative = 0.54 \[ \int \frac {x^3 \left (1-c^2 x^2\right )^{5/2}}{a+b \text {arccosh}(c x)} \, dx=\frac {\sqrt {1-c^2 x^2} \left (-6 \cosh \left (\frac {a}{b}\right ) \text {Chi}\left (\frac {a}{b}+\text {arccosh}(c x)\right )+8 \cosh \left (\frac {3 a}{b}\right ) \text {Chi}\left (3 \left (\frac {a}{b}+\text {arccosh}(c x)\right )\right )-3 \cosh \left (\frac {7 a}{b}\right ) \text {Chi}\left (7 \left (\frac {a}{b}+\text {arccosh}(c x)\right )\right )+\cosh \left (\frac {9 a}{b}\right ) \text {Chi}\left (9 \left (\frac {a}{b}+\text {arccosh}(c x)\right )\right )+6 \sinh \left (\frac {a}{b}\right ) \text {Shi}\left (\frac {a}{b}+\text {arccosh}(c x)\right )-8 \sinh \left (\frac {3 a}{b}\right ) \text {Shi}\left (3 \left (\frac {a}{b}+\text {arccosh}(c x)\right )\right )+3 \sinh \left (\frac {7 a}{b}\right ) \text {Shi}\left (7 \left (\frac {a}{b}+\text {arccosh}(c x)\right )\right )-\sinh \left (\frac {9 a}{b}\right ) \text {Shi}\left (9 \left (\frac {a}{b}+\text {arccosh}(c x)\right )\right )\right )}{256 c^4 \sqrt {\frac {-1+c x}{1+c x}} (b+b c x)} \]

[In]

Integrate[(x^3*(1 - c^2*x^2)^(5/2))/(a + b*ArcCosh[c*x]),x]

[Out]

(Sqrt[1 - c^2*x^2]*(-6*Cosh[a/b]*CoshIntegral[a/b + ArcCosh[c*x]] + 8*Cosh[(3*a)/b]*CoshIntegral[3*(a/b + ArcC
osh[c*x])] - 3*Cosh[(7*a)/b]*CoshIntegral[7*(a/b + ArcCosh[c*x])] + Cosh[(9*a)/b]*CoshIntegral[9*(a/b + ArcCos
h[c*x])] + 6*Sinh[a/b]*SinhIntegral[a/b + ArcCosh[c*x]] - 8*Sinh[(3*a)/b]*SinhIntegral[3*(a/b + ArcCosh[c*x])]
+ 3*Sinh[(7*a)/b]*SinhIntegral[7*(a/b + ArcCosh[c*x])] - Sinh[(9*a)/b]*SinhIntegral[9*(a/b + ArcCosh[c*x])]))
/(256*c^4*Sqrt[(-1 + c*x)/(1 + c*x)]*(b + b*c*x))

Maple [A] (verified)

Time = 0.60 (sec) , antiderivative size = 320, normalized size of antiderivative = 0.81

method	result	size
default	\(-\frac {\sqrt {-c^{2} x^{2}+1}\, \left (-\sqrt {c x -1}\, \sqrt {c x +1}\, c x +c^{2} x^{2}-1\right ) \left (3 \,\operatorname {Ei}_{1}\left (7 \,\operatorname {arccosh}\left (c x \right )+\frac {7 a}{b}\right ) {\mathrm e}^{\frac {b \,\operatorname {arccosh}\left (c x \right )+7 a}{b}}-\operatorname {Ei}_{1}\left (9 \,\operatorname {arccosh}\left (c x \right )+\frac {9 a}{b}\right ) {\mathrm e}^{\frac {b \,\operatorname {arccosh}\left (c x \right )+9 a}{b}}-\operatorname {Ei}_{1}\left (-9 \,\operatorname {arccosh}\left (c x \right )-\frac {9 a}{b}\right ) {\mathrm e}^{-\frac {-b \,\operatorname {arccosh}\left (c x \right )+9 a}{b}}-8 \,\operatorname {Ei}_{1}\left (3 \,\operatorname {arccosh}\left (c x \right )+\frac {3 a}{b}\right ) {\mathrm e}^{\frac {b \,\operatorname {arccosh}\left (c x \right )+3 a}{b}}+6 \,\operatorname {Ei}_{1}\left (\operatorname {arccosh}\left (c x \right )+\frac {a}{b}\right ) {\mathrm e}^{\frac {a +b \,\operatorname {arccosh}\left (c x \right )}{b}}+6 \,\operatorname {Ei}_{1}\left (-\operatorname {arccosh}\left (c x \right )-\frac {a}{b}\right ) {\mathrm e}^{-\frac {-b \,\operatorname {arccosh}\left (c x \right )+a}{b}}-8 \,\operatorname {Ei}_{1}\left (-3 \,\operatorname {arccosh}\left (c x \right )-\frac {3 a}{b}\right ) {\mathrm e}^{-\frac {-b \,\operatorname {arccosh}\left (c x \right )+3 a}{b}}+3 \,\operatorname {Ei}_{1}\left (-7 \,\operatorname {arccosh}\left (c x \right )-\frac {7 a}{b}\right ) {\mathrm e}^{-\frac {-b \,\operatorname {arccosh}\left (c x \right )+7 a}{b}}\right )}{512 \left (c x +1\right ) c^{4} \left (c x -1\right ) b}\)	\(320\)

[In]

int(x^3*(-c^2*x^2+1)^(5/2)/(a+b*arccosh(c*x)),x,method=_RETURNVERBOSE)

[Out]

-1/512*(-c^2*x^2+1)^(1/2)*(-(c*x-1)^(1/2)*(c*x+1)^(1/2)*c*x+c^2*x^2-1)*(3*Ei(1,7*arccosh(c*x)+7*a/b)*exp((b*ar
ccosh(c*x)+7*a)/b)-Ei(1,9*arccosh(c*x)+9*a/b)*exp((b*arccosh(c*x)+9*a)/b)-Ei(1,-9*arccosh(c*x)-9*a/b)*exp(-(-b
*arccosh(c*x)+9*a)/b)-8*Ei(1,3*arccosh(c*x)+3*a/b)*exp((b*arccosh(c*x)+3*a)/b)+6*Ei(1,arccosh(c*x)+a/b)*exp((a
+b*arccosh(c*x))/b)+6*Ei(1,-arccosh(c*x)-a/b)*exp(-(-b*arccosh(c*x)+a)/b)-8*Ei(1,-3*arccosh(c*x)-3*a/b)*exp(-(
-b*arccosh(c*x)+3*a)/b)+3*Ei(1,-7*arccosh(c*x)-7*a/b)*exp(-(-b*arccosh(c*x)+7*a)/b))/(c*x+1)/c^4/(c*x-1)/b

Fricas [F]

\[ \int \frac {x^3 \left (1-c^2 x^2\right )^{5/2}}{a+b \text {arccosh}(c x)} \, dx=\int { \frac {{\left (-c^{2} x^{2} + 1\right )}^{\frac {5}{2}} x^{3}}{b \operatorname {arcosh}\left (c x\right ) + a} \,d x } \]

[In]

integrate(x^3*(-c^2*x^2+1)^(5/2)/(a+b*arccosh(c*x)),x, algorithm="fricas")

[Out]

integral((c^4*x^7 - 2*c^2*x^5 + x^3)*sqrt(-c^2*x^2 + 1)/(b*arccosh(c*x) + a), x)

Sympy [F(-1)]

Timed out. \[ \int \frac {x^3 \left (1-c^2 x^2\right )^{5/2}}{a+b \text {arccosh}(c x)} \, dx=\text {Timed out} \]

[In]

integrate(x**3*(-c**2*x**2+1)**(5/2)/(a+b*acosh(c*x)),x)

[Out]

Timed out

Maxima [F]

\[ \int \frac {x^3 \left (1-c^2 x^2\right )^{5/2}}{a+b \text {arccosh}(c x)} \, dx=\int { \frac {{\left (-c^{2} x^{2} + 1\right )}^{\frac {5}{2}} x^{3}}{b \operatorname {arcosh}\left (c x\right ) + a} \,d x } \]

[In]

integrate(x^3*(-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^3/(b*arccosh(c*x) + a), x)

Giac [F]

\[ \int \frac {x^3 \left (1-c^2 x^2\right )^{5/2}}{a+b \text {arccosh}(c x)} \, dx=\int { \frac {{\left (-c^{2} x^{2} + 1\right )}^{\frac {5}{2}} x^{3}}{b \operatorname {arcosh}\left (c x\right ) + a} \,d x } \]

[In]

integrate(x^3*(-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^3/(b*arccosh(c*x) + a), x)

Mupad [F(-1)]

Timed out. \[ \int \frac {x^3 \left (1-c^2 x^2\right )^{5/2}}{a+b \text {arccosh}(c x)} \, dx=\int \frac {x^3\,{\left (1-c^2\,x^2\right )}^{5/2}}{a+b\,\mathrm {acosh}\left (c\,x\right )} \,d x \]

[In]

int((x^3*(1 - c^2*x^2)^(5/2))/(a + b*acosh(c*x)),x)

[Out]

int((x^3*(1 - c^2*x^2)^(5/2))/(a + b*acosh(c*x)), x)