\(\int \frac {x^3}{\sqrt {1-c^2 x^2} (a+b \text {arccosh}(c x))^2} \, dx\) [344]

Optimal result

Integrand size = 28, antiderivative size = 237 \[ \int \frac {x^3}{\sqrt {1-c^2 x^2} (a+b \text {arccosh}(c x))^2} \, dx=-\frac {x^3 \sqrt {-1+c x}}{b c \sqrt {1-c x} (a+b \text {arccosh}(c x))}-\frac {3 \sqrt {-1+c x} \text {Chi}\left (\frac {a+b \text {arccosh}(c x)}{b}\right ) \sinh \left (\frac {a}{b}\right )}{4 b^2 c^4 \sqrt {1-c x}}-\frac {3 \sqrt {-1+c x} \text {Chi}\left (\frac {3 (a+b \text {arccosh}(c x))}{b}\right ) \sinh \left (\frac {3 a}{b}\right )}{4 b^2 c^4 \sqrt {1-c x}}+\frac {3 \sqrt {-1+c x} \cosh \left (\frac {a}{b}\right ) \text {Shi}\left (\frac {a+b \text {arccosh}(c x)}{b}\right )}{4 b^2 c^4 \sqrt {1-c x}}+\frac {3 \sqrt {-1+c x} \cosh \left (\frac {3 a}{b}\right ) \text {Shi}\left (\frac {3 (a+b \text {arccosh}(c x))}{b}\right )}{4 b^2 c^4 \sqrt {1-c x}} \]

[Out]

-x^3*(c*x-1)^(1/2)/b/c/(a+b*arccosh(c*x))/(-c*x+1)^(1/2)+3/4*cosh(a/b)*Shi((a+b*arccosh(c*x))/b)*(c*x-1)^(1/2)
/b^2/c^4/(-c*x+1)^(1/2)+3/4*cosh(3*a/b)*Shi(3*(a+b*arccosh(c*x))/b)*(c*x-1)^(1/2)/b^2/c^4/(-c*x+1)^(1/2)-3/4*C
hi((a+b*arccosh(c*x))/b)*sinh(a/b)*(c*x-1)^(1/2)/b^2/c^4/(-c*x+1)^(1/2)-3/4*Chi(3*(a+b*arccosh(c*x))/b)*sinh(3
*a/b)*(c*x-1)^(1/2)/b^2/c^4/(-c*x+1)^(1/2)

Rubi [A] (verified)

Time = 0.24 (sec) , antiderivative size = 237, normalized size of antiderivative = 1.00, number of steps used = 10, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.214, Rules used = {5950, 5887, 5556, 3384, 3379, 3382} \[ \int \frac {x^3}{\sqrt {1-c^2 x^2} (a+b \text {arccosh}(c x))^2} \, dx=-\frac {3 \sqrt {c x-1} \sinh \left (\frac {a}{b}\right ) \text {Chi}\left (\frac {a+b \text {arccosh}(c x)}{b}\right )}{4 b^2 c^4 \sqrt {1-c x}}-\frac {3 \sqrt {c x-1} \sinh \left (\frac {3 a}{b}\right ) \text {Chi}\left (\frac {3 (a+b \text {arccosh}(c x))}{b}\right )}{4 b^2 c^4 \sqrt {1-c x}}+\frac {3 \sqrt {c x-1} \cosh \left (\frac {a}{b}\right ) \text {Shi}\left (\frac {a+b \text {arccosh}(c x)}{b}\right )}{4 b^2 c^4 \sqrt {1-c x}}+\frac {3 \sqrt {c x-1} \cosh \left (\frac {3 a}{b}\right ) \text {Shi}\left (\frac {3 (a+b \text {arccosh}(c x))}{b}\right )}{4 b^2 c^4 \sqrt {1-c x}}-\frac {x^3 \sqrt {c x-1}}{b c \sqrt {1-c x} (a+b \text {arccosh}(c x))} \]

[In]

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

[Out]

-((x^3*Sqrt[-1 + c*x])/(b*c*Sqrt[1 - c*x]*(a + b*ArcCosh[c*x]))) - (3*Sqrt[-1 + c*x]*CoshIntegral[(a + b*ArcCo
sh[c*x])/b]*Sinh[a/b])/(4*b^2*c^4*Sqrt[1 - c*x]) - (3*Sqrt[-1 + c*x]*CoshIntegral[(3*(a + b*ArcCosh[c*x]))/b]*
Sinh[(3*a)/b])/(4*b^2*c^4*Sqrt[1 - c*x]) + (3*Sqrt[-1 + c*x]*Cosh[a/b]*SinhIntegral[(a + b*ArcCosh[c*x])/b])/(
4*b^2*c^4*Sqrt[1 - c*x]) + (3*Sqrt[-1 + c*x]*Cosh[(3*a)/b]*SinhIntegral[(3*(a + b*ArcCosh[c*x]))/b])/(4*b^2*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 5887

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

Rule 5950

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

Rubi steps \begin{align*} \text {integral}& = -\frac {x^3 \sqrt {-1+c x}}{b c \sqrt {1-c x} (a+b \text {arccosh}(c x))}+\frac {\left (3 \sqrt {-1+c x}\right ) \int \frac {x^2}{a+b \text {arccosh}(c x)} \, dx}{b c \sqrt {1-c x}} \\ & = -\frac {x^3 \sqrt {-1+c x}}{b c \sqrt {1-c x} (a+b \text {arccosh}(c x))}-\frac {\left (3 \sqrt {-1+c x}\right ) \text {Subst}\left (\int \frac {\cosh ^2\left (\frac {a}{b}-\frac {x}{b}\right ) \sinh \left (\frac {a}{b}-\frac {x}{b}\right )}{x} \, dx,x,a+b \text {arccosh}(c x)\right )}{b^2 c^4 \sqrt {1-c x}} \\ & = -\frac {x^3 \sqrt {-1+c x}}{b c \sqrt {1-c x} (a+b \text {arccosh}(c x))}-\frac {\left (3 \sqrt {-1+c x}\right ) \text {Subst}\left (\int \left (\frac {\sinh \left (\frac {3 a}{b}-\frac {3 x}{b}\right )}{4 x}+\frac {\sinh \left (\frac {a}{b}-\frac {x}{b}\right )}{4 x}\right ) \, dx,x,a+b \text {arccosh}(c x)\right )}{b^2 c^4 \sqrt {1-c x}} \\ & = -\frac {x^3 \sqrt {-1+c x}}{b c \sqrt {1-c x} (a+b \text {arccosh}(c x))}-\frac {\left (3 \sqrt {-1+c x}\right ) \text {Subst}\left (\int \frac {\sinh \left (\frac {3 a}{b}-\frac {3 x}{b}\right )}{x} \, dx,x,a+b \text {arccosh}(c x)\right )}{4 b^2 c^4 \sqrt {1-c x}}-\frac {\left (3 \sqrt {-1+c x}\right ) \text {Subst}\left (\int \frac {\sinh \left (\frac {a}{b}-\frac {x}{b}\right )}{x} \, dx,x,a+b \text {arccosh}(c x)\right )}{4 b^2 c^4 \sqrt {1-c x}} \\ & = -\frac {x^3 \sqrt {-1+c x}}{b c \sqrt {1-c x} (a+b \text {arccosh}(c x))}+\frac {\left (3 \sqrt {-1+c x} \cosh \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 )}{4 b^2 c^4 \sqrt {1-c x}}+\frac {\left (3 \sqrt {-1+c x} \cosh \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 )}{4 b^2 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 {\cosh \left (\frac {x}{b}\right )}{x} \, dx,x,a+b \text {arccosh}(c x)\right )}{4 b^2 c^4 \sqrt {1-c x}}-\frac {\left (3 \sqrt {-1+c x} \sinh \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 )}{4 b^2 c^4 \sqrt {1-c x}} \\ & = -\frac {x^3 \sqrt {-1+c x}}{b c \sqrt {1-c x} (a+b \text {arccosh}(c x))}-\frac {3 \sqrt {-1+c x} \text {Chi}\left (\frac {a+b \text {arccosh}(c x)}{b}\right ) \sinh \left (\frac {a}{b}\right )}{4 b^2 c^4 \sqrt {1-c x}}-\frac {3 \sqrt {-1+c x} \text {Chi}\left (\frac {3 (a+b \text {arccosh}(c x))}{b}\right ) \sinh \left (\frac {3 a}{b}\right )}{4 b^2 c^4 \sqrt {1-c x}}+\frac {3 \sqrt {-1+c x} \cosh \left (\frac {a}{b}\right ) \text {Shi}\left (\frac {a+b \text {arccosh}(c x)}{b}\right )}{4 b^2 c^4 \sqrt {1-c x}}+\frac {3 \sqrt {-1+c x} \cosh \left (\frac {3 a}{b}\right ) \text {Shi}\left (\frac {3 (a+b \text {arccosh}(c x))}{b}\right )}{4 b^2 c^4 \sqrt {1-c x}} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.33 (sec) , antiderivative size = 144, normalized size of antiderivative = 0.61 \[ \int \frac {x^3}{\sqrt {1-c^2 x^2} (a+b \text {arccosh}(c x))^2} \, dx=\frac {\sqrt {1-c^2 x^2} \left (\frac {4 b c^3 x^3}{a+b \text {arccosh}(c x)}+3 \text {Chi}\left (\frac {a}{b}+\text {arccosh}(c x)\right ) \sinh \left (\frac {a}{b}\right )+3 \text {Chi}\left (3 \left (\frac {a}{b}+\text {arccosh}(c x)\right )\right ) \sinh \left (\frac {3 a}{b}\right )-3 \cosh \left (\frac {a}{b}\right ) \text {Shi}\left (\frac {a}{b}+\text {arccosh}(c x)\right )-3 \cosh \left (\frac {3 a}{b}\right ) \text {Shi}\left (3 \left (\frac {a}{b}+\text {arccosh}(c x)\right )\right )\right )}{4 b^2 c^4 \sqrt {-1+c x} \sqrt {1+c x}} \]

[In]

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

[Out]

(Sqrt[1 - c^2*x^2]*((4*b*c^3*x^3)/(a + b*ArcCosh[c*x]) + 3*CoshIntegral[a/b + ArcCosh[c*x]]*Sinh[a/b] + 3*Cosh
Integral[3*(a/b + ArcCosh[c*x])]*Sinh[(3*a)/b] - 3*Cosh[a/b]*SinhIntegral[a/b + ArcCosh[c*x]] - 3*Cosh[(3*a)/b
]*SinhIntegral[3*(a/b + ArcCosh[c*x])]))/(4*b^2*c^4*Sqrt[-1 + c*x]*Sqrt[1 + c*x])

Maple [A] (verified)

Time = 1.17 (sec) , antiderivative size = 375, normalized size of antiderivative = 1.58

method	result	size
default	\(\frac {\sqrt {-c^{2} x^{2}+1}\, \left (\sqrt {c x -1}\, \sqrt {c x +1}\, c x +c^{2} x^{2}-1\right ) \left (-8 \sqrt {c x -1}\, \sqrt {c x +1}\, b \,c^{3} x^{3}+8 b \,c^{4} x^{4}+3 \,\operatorname {arccosh}\left (c x \right ) b \,\operatorname {Ei}_{1}\left (-3 \,\operatorname {arccosh}\left (c x \right )-\frac {3 a}{b}\right ) {\mathrm e}^{-\frac {b \,\operatorname {arccosh}\left (c x \right )+3 a}{b}}+3 \,\operatorname {arccosh}\left (c x \right ) b \,\operatorname {Ei}_{1}\left (-\operatorname {arccosh}\left (c x \right )-\frac {a}{b}\right ) {\mathrm e}^{-\frac {a +b \,\operatorname {arccosh}\left (c x \right )}{b}}-3 \,\operatorname {Ei}_{1}\left (3 \,\operatorname {arccosh}\left (c x \right )+\frac {3 a}{b}\right ) {\mathrm e}^{\frac {-b \,\operatorname {arccosh}\left (c x \right )+3 a}{b}} b \,\operatorname {arccosh}\left (c x \right )-3 \,\operatorname {Ei}_{1}\left (\operatorname {arccosh}\left (c x \right )+\frac {a}{b}\right ) {\mathrm e}^{\frac {-b \,\operatorname {arccosh}\left (c x \right )+a}{b}} b \,\operatorname {arccosh}\left (c x \right )+3 a \,\operatorname {Ei}_{1}\left (-3 \,\operatorname {arccosh}\left (c x \right )-\frac {3 a}{b}\right ) {\mathrm e}^{-\frac {b \,\operatorname {arccosh}\left (c x \right )+3 a}{b}}+3 a \,\operatorname {Ei}_{1}\left (-\operatorname {arccosh}\left (c x \right )-\frac {a}{b}\right ) {\mathrm e}^{-\frac {a +b \,\operatorname {arccosh}\left (c x \right )}{b}}-3 \,\operatorname {Ei}_{1}\left (3 \,\operatorname {arccosh}\left (c x \right )+\frac {3 a}{b}\right ) {\mathrm e}^{\frac {-b \,\operatorname {arccosh}\left (c x \right )+3 a}{b}} a -3 \,\operatorname {Ei}_{1}\left (\operatorname {arccosh}\left (c x \right )+\frac {a}{b}\right ) {\mathrm e}^{\frac {-b \,\operatorname {arccosh}\left (c x \right )+a}{b}} a \right )}{8 c^{4} \left (c^{2} x^{2}-1\right ) b^{2} \left (a +b \,\operatorname {arccosh}\left (c x \right )\right )}\)	\(375\)

[In]

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

[Out]

1/8*(-c^2*x^2+1)^(1/2)*((c*x-1)^(1/2)*(c*x+1)^(1/2)*c*x+c^2*x^2-1)*(-8*(c*x-1)^(1/2)*(c*x+1)^(1/2)*b*c^3*x^3+8
*b*c^4*x^4+3*arccosh(c*x)*b*Ei(1,-3*arccosh(c*x)-3*a/b)*exp(-(b*arccosh(c*x)+3*a)/b)+3*arccosh(c*x)*b*Ei(1,-ar
ccosh(c*x)-a/b)*exp(-(a+b*arccosh(c*x))/b)-3*Ei(1,3*arccosh(c*x)+3*a/b)*exp((-b*arccosh(c*x)+3*a)/b)*b*arccosh
(c*x)-3*Ei(1,arccosh(c*x)+a/b)*exp((-b*arccosh(c*x)+a)/b)*b*arccosh(c*x)+3*a*Ei(1,-3*arccosh(c*x)-3*a/b)*exp(-
(b*arccosh(c*x)+3*a)/b)+3*a*Ei(1,-arccosh(c*x)-a/b)*exp(-(a+b*arccosh(c*x))/b)-3*Ei(1,3*arccosh(c*x)+3*a/b)*ex
p((-b*arccosh(c*x)+3*a)/b)*a-3*Ei(1,arccosh(c*x)+a/b)*exp((-b*arccosh(c*x)+a)/b)*a)/c^4/(c^2*x^2-1)/b^2/(a+b*a
rccosh(c*x))

Fricas [F]

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

[In]

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

[Out]

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

Sympy [F]

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

[In]

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

[Out]

Integral(x**3/(sqrt(-(c*x - 1)*(c*x + 1))*(a + b*acosh(c*x))**2), x)

Maxima [F]

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

[In]

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

[Out]

-(c^3*x^6 - c*x^4 + (c^2*x^5 - x^3)*sqrt(c*x + 1)*sqrt(c*x - 1))/(((c*x + 1)*sqrt(c*x - 1)*b^2*c^2*x + (b^2*c^
3*x^2 - b^2*c)*sqrt(c*x + 1))*sqrt(-c*x + 1)*log(c*x + sqrt(c*x + 1)*sqrt(c*x - 1)) + ((c*x + 1)*sqrt(c*x - 1)
*a*b*c^2*x + (a*b*c^3*x^2 - a*b*c)*sqrt(c*x + 1))*sqrt(-c*x + 1)) + integrate((3*c^5*x^7 - 7*c^3*x^5 + 4*c*x^3
+ (3*c^3*x^5 - 2*c*x^3)*(c*x + 1)*(c*x - 1) + 3*(2*c^4*x^6 - 3*c^2*x^4 + x^2)*sqrt(c*x + 1)*sqrt(c*x - 1))/((
(c*x + 1)^(3/2)*(c*x - 1)*b^2*c^3*x^2 + 2*(b^2*c^4*x^3 - b^2*c^2*x)*(c*x + 1)*sqrt(c*x - 1) + (b^2*c^5*x^4 - 2
*b^2*c^3*x^2 + b^2*c)*sqrt(c*x + 1))*sqrt(-c*x + 1)*log(c*x + sqrt(c*x + 1)*sqrt(c*x - 1)) + ((c*x + 1)^(3/2)*
(c*x - 1)*a*b*c^3*x^2 + 2*(a*b*c^4*x^3 - a*b*c^2*x)*(c*x + 1)*sqrt(c*x - 1) + (a*b*c^5*x^4 - 2*a*b*c^3*x^2 + a
*b*c)*sqrt(c*x + 1))*sqrt(-c*x + 1)), x)

Giac [F(-2)]

Exception generated. \[ \int \frac {x^3}{\sqrt {1-c^2 x^2} (a+b \text {arccosh}(c x))^2} \, dx=\text {Exception raised: TypeError} \]

[In]

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

[Out]

Exception raised: TypeError >> an error occurred running a Giac command:INPUT:sage2:=int(sage0,sageVARx):;OUTP
UT:sym2poly/r2sym(const gen & e,const index_m & i,const vecteur & l) Error: Bad Argument Value

Mupad [F(-1)]

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

[In]

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

[Out]

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