\(\int \frac {x^2 \sqrt {1-c^2 x^2}}{(a+b \text {arccosh}(c x))^2} \, dx\) [321]

Optimal result

Integrand size = 28, antiderivative size = 154 \[ \int \frac {x^2 \sqrt {1-c^2 x^2}}{(a+b \text {arccosh}(c x))^2} \, dx=-\frac {x^2 \sqrt {-1+c x} \sqrt {1+c x} \sqrt {1-c^2 x^2}}{b c (a+b \text {arccosh}(c x))}-\frac {\sqrt {1-c x} \text {Chi}\left (\frac {4 (a+b \text {arccosh}(c x))}{b}\right ) \sinh \left (\frac {4 a}{b}\right )}{2 b^2 c^3 \sqrt {-1+c x}}+\frac {\sqrt {1-c x} \cosh \left (\frac {4 a}{b}\right ) \text {Shi}\left (\frac {4 (a+b \text {arccosh}(c x))}{b}\right )}{2 b^2 c^3 \sqrt {-1+c x}} \]

[Out]

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

Rubi [A] (verified)

Time = 0.36 (sec) , antiderivative size = 154, normalized size of antiderivative = 1.00, number of steps used = 16, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {5942, 5887, 5556, 12, 3384, 3379, 3382} \[ \int \frac {x^2 \sqrt {1-c^2 x^2}}{(a+b \text {arccosh}(c x))^2} \, dx=-\frac {\sqrt {1-c x} \sinh \left (\frac {4 a}{b}\right ) \text {Chi}\left (\frac {4 (a+b \text {arccosh}(c x))}{b}\right )}{2 b^2 c^3 \sqrt {c x-1}}+\frac {\sqrt {1-c x} \cosh \left (\frac {4 a}{b}\right ) \text {Shi}\left (\frac {4 (a+b \text {arccosh}(c x))}{b}\right )}{2 b^2 c^3 \sqrt {c x-1}}-\frac {x^2 \sqrt {c x-1} \sqrt {c x+1} \sqrt {1-c^2 x^2}}{b c (a+b \text {arccosh}(c x))} \]

[In]

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

[Out]

-((x^2*Sqrt[-1 + c*x]*Sqrt[1 + c*x]*Sqrt[1 - c^2*x^2])/(b*c*(a + b*ArcCosh[c*x]))) - (Sqrt[1 - c*x]*CoshIntegr
al[(4*(a + b*ArcCosh[c*x]))/b]*Sinh[(4*a)/b])/(2*b^2*c^3*Sqrt[-1 + c*x]) + (Sqrt[1 - c*x]*Cosh[(4*a)/b]*SinhIn
tegral[(4*(a + b*ArcCosh[c*x]))/b])/(2*b^2*c^3*Sqrt[-1 + c*x])

Rule 12

Int[(a_)*(u_), x_Symbol] :> Dist[a, Int[u, x], x] /; FreeQ[a, x] &&  !MatchQ[u, (b_)*(v_) /; FreeQ[b, 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 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]

Rubi steps \begin{align*} \text {integral}& = -\frac {x^2 \sqrt {-1+c x} \sqrt {1+c x} \sqrt {1-c^2 x^2}}{b c (a+b \text {arccosh}(c x))}-\frac {\left (2 \sqrt {1-c x}\right ) \int \frac {x}{a+b \text {arccosh}(c x)} \, dx}{b c \sqrt {-1+c x}}+\frac {\left (4 c \sqrt {1-c x}\right ) \int \frac {x^3}{a+b \text {arccosh}(c x)} \, dx}{b \sqrt {-1+c x}} \\ & = -\frac {x^2 \sqrt {-1+c x} \sqrt {1+c x} \sqrt {1-c^2 x^2}}{b c (a+b \text {arccosh}(c x))}+\frac {\left (2 \sqrt {1-c x}\right ) \text {Subst}\left (\int \frac {\cosh \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^3 \sqrt {-1+c x}}-\frac {\left (4 \sqrt {1-c x}\right ) \text {Subst}\left (\int \frac {\cosh ^3\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^3 \sqrt {-1+c x}} \\ & = -\frac {x^2 \sqrt {-1+c x} \sqrt {1+c x} \sqrt {1-c^2 x^2}}{b c (a+b \text {arccosh}(c x))}+\frac {\left (2 \sqrt {1-c x}\right ) \text {Subst}\left (\int \frac {\sinh \left (\frac {2 a}{b}-\frac {2 x}{b}\right )}{2 x} \, dx,x,a+b \text {arccosh}(c x)\right )}{b^2 c^3 \sqrt {-1+c x}}-\frac {\left (4 \sqrt {1-c x}\right ) \text {Subst}\left (\int \left (\frac {\sinh \left (\frac {4 a}{b}-\frac {4 x}{b}\right )}{8 x}+\frac {\sinh \left (\frac {2 a}{b}-\frac {2 x}{b}\right )}{4 x}\right ) \, dx,x,a+b \text {arccosh}(c x)\right )}{b^2 c^3 \sqrt {-1+c x}} \\ & = -\frac {x^2 \sqrt {-1+c x} \sqrt {1+c x} \sqrt {1-c^2 x^2}}{b c (a+b \text {arccosh}(c x))}-\frac {\sqrt {1-c x} \text {Subst}\left (\int \frac {\sinh \left (\frac {4 a}{b}-\frac {4 x}{b}\right )}{x} \, dx,x,a+b \text {arccosh}(c x)\right )}{2 b^2 c^3 \sqrt {-1+c x}} \\ & = -\frac {x^2 \sqrt {-1+c x} \sqrt {1+c x} \sqrt {1-c^2 x^2}}{b c (a+b \text {arccosh}(c x))}+\frac {\left (\sqrt {1-c x} \cosh \left (\frac {4 a}{b}\right )\right ) \text {Subst}\left (\int \frac {\sinh \left (\frac {4 x}{b}\right )}{x} \, dx,x,a+b \text {arccosh}(c x)\right )}{2 b^2 c^3 \sqrt {-1+c x}}-\frac {\left (\sqrt {1-c x} \sinh \left (\frac {4 a}{b}\right )\right ) \text {Subst}\left (\int \frac {\cosh \left (\frac {4 x}{b}\right )}{x} \, dx,x,a+b \text {arccosh}(c x)\right )}{2 b^2 c^3 \sqrt {-1+c x}} \\ & = -\frac {x^2 \sqrt {-1+c x} \sqrt {1+c x} \sqrt {1-c^2 x^2}}{b c (a+b \text {arccosh}(c x))}-\frac {\sqrt {1-c x} \text {Chi}\left (\frac {4 (a+b \text {arccosh}(c x))}{b}\right ) \sinh \left (\frac {4 a}{b}\right )}{2 b^2 c^3 \sqrt {-1+c x}}+\frac {\sqrt {1-c x} \cosh \left (\frac {4 a}{b}\right ) \text {Shi}\left (\frac {4 (a+b \text {arccosh}(c x))}{b}\right )}{2 b^2 c^3 \sqrt {-1+c x}} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.44 (sec) , antiderivative size = 130, normalized size of antiderivative = 0.84 \[ \int \frac {x^2 \sqrt {1-c^2 x^2}}{(a+b \text {arccosh}(c x))^2} \, dx=\frac {\sqrt {1-c^2 x^2} \left (-2 b c^2 x^2 \left (-1+c^2 x^2\right )-(a+b \text {arccosh}(c x)) \text {Chi}\left (4 \left (\frac {a}{b}+\text {arccosh}(c x)\right )\right ) \sinh \left (\frac {4 a}{b}\right )+(a+b \text {arccosh}(c x)) \cosh \left (\frac {4 a}{b}\right ) \text {Shi}\left (4 \left (\frac {a}{b}+\text {arccosh}(c x)\right )\right )\right )}{2 b^2 c^3 \sqrt {-1+c x} \sqrt {1+c x} (a+b \text {arccosh}(c x))} \]

[In]

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

[Out]

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

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(278\) vs. \(2(136)=272\).

Time = 0.58 (sec) , antiderivative size = 279, normalized size of antiderivative = 1.81

method	result	size
default	\(\frac {\sqrt {-c^{2} x^{2}+1}\, \left (-\sqrt {c x -1}\, \sqrt {c x +1}\, c x +c^{2} x^{2}-1\right ) \left (4 x^{4} c^{4} b \sqrt {c x -1}\, \sqrt {c x +1}+4 b \,c^{5} x^{5}-4 \sqrt {c x -1}\, \sqrt {c x +1}\, b \,c^{2} x^{2}-4 b \,c^{3} x^{3}+\operatorname {arccosh}\left (c x \right ) b \,\operatorname {Ei}_{1}\left (-4 \,\operatorname {arccosh}\left (c x \right )-\frac {4 a}{b}\right ) {\mathrm e}^{-\frac {-b \,\operatorname {arccosh}\left (c x \right )+4 a}{b}}-\operatorname {Ei}_{1}\left (4 \,\operatorname {arccosh}\left (c x \right )+\frac {4 a}{b}\right ) {\mathrm e}^{\frac {b \,\operatorname {arccosh}\left (c x \right )+4 a}{b}} b \,\operatorname {arccosh}\left (c x \right )+a \,\operatorname {Ei}_{1}\left (-4 \,\operatorname {arccosh}\left (c x \right )-\frac {4 a}{b}\right ) {\mathrm e}^{-\frac {-b \,\operatorname {arccosh}\left (c x \right )+4 a}{b}}-\operatorname {Ei}_{1}\left (4 \,\operatorname {arccosh}\left (c x \right )+\frac {4 a}{b}\right ) {\mathrm e}^{\frac {b \,\operatorname {arccosh}\left (c x \right )+4 a}{b}} a \right )}{4 \left (c x +1\right ) c^{3} \left (c x -1\right ) b^{2} \left (a +b \,\operatorname {arccosh}\left (c x \right )\right )}\)	\(279\)

[In]

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

[Out]

1/4*(-c^2*x^2+1)^(1/2)*(-(c*x-1)^(1/2)*(c*x+1)^(1/2)*c*x+c^2*x^2-1)*(4*x^4*c^4*b*(c*x-1)^(1/2)*(c*x+1)^(1/2)+4
*b*c^5*x^5-4*(c*x-1)^(1/2)*(c*x+1)^(1/2)*b*c^2*x^2-4*b*c^3*x^3+arccosh(c*x)*b*Ei(1,-4*arccosh(c*x)-4*a/b)*exp(
-(-b*arccosh(c*x)+4*a)/b)-Ei(1,4*arccosh(c*x)+4*a/b)*exp((b*arccosh(c*x)+4*a)/b)*b*arccosh(c*x)+a*Ei(1,-4*arcc
osh(c*x)-4*a/b)*exp(-(-b*arccosh(c*x)+4*a)/b)-Ei(1,4*arccosh(c*x)+4*a/b)*exp((b*arccosh(c*x)+4*a)/b)*a)/(c*x+1
)/c^3/(c*x-1)/b^2/(a+b*arccosh(c*x))

Fricas [F]

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

[In]

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

[Out]

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

Sympy [F]

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

[In]

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

[Out]

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

Maxima [F]

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

[In]

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

[Out]

-((c^2*x^4 - x^2)*(c*x + 1)*sqrt(c*x - 1) + (c^3*x^5 - c*x^3)*sqrt(c*x + 1))*sqrt(-c*x + 1)/(a*b*c^3*x^2 + sqr
t(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(((4*c^3*x^4 - c*x^2)*(c*x + 1)^(3/2)*(c*x - 1) + 2*(4*c^4*x^5
- 4*c^2*x^3 + x)*(c*x + 1)*sqrt(c*x - 1) + (4*c^5*x^6 - 7*c^3*x^4 + 3*c*x^2)*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)

Giac [F]

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

[In]

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

[Out]

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

Mupad [F(-1)]

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

[In]

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

[Out]

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