3.321 \(\int \frac {x^2 \sqrt {1-c^2 x^2}}{(a+b \cosh ^{-1}(c x))^2} \, dx\)

Optimal. Leaf size=154 \[ -\frac {\sqrt {1-c x} \sinh \left (\frac {4 a}{b}\right ) \text {Chi}\left (\frac {4 \left (a+b \cosh ^{-1}(c x)\right )}{b}\right )}{2 b^2 c^3 \sqrt {c x-1}}+\frac {\sqrt {1-c x} \cosh \left (\frac {4 a}{b}\right ) \text {Shi}\left (\frac {4 \left (a+b \cosh ^{-1}(c x)\right )}{b}\right )}{2 b^2 c^3 \sqrt {c x-1}}-\frac {x^2 \sqrt {c x-1} \sqrt {c x+1} \sqrt {1-c^2 x^2}}{b c \left (a+b \cosh ^{-1}(c x)\right )} \]

[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))

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Rubi [A]  time = 0.88, antiderivative size = 185, normalized size of antiderivative = 1.20, number of steps used = 17, number of rules used = 8, integrand size = 28, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.286, Rules used = {5798, 5778, 5670, 5448, 12, 3303, 3298, 3301} \[ -\frac {\sqrt {1-c^2 x^2} \sinh \left (\frac {4 a}{b}\right ) \text {Chi}\left (\frac {4 a}{b}+4 \cosh ^{-1}(c x)\right )}{2 b^2 c^3 \sqrt {c x-1} \sqrt {c x+1}}+\frac {\sqrt {1-c^2 x^2} \cosh \left (\frac {4 a}{b}\right ) \text {Shi}\left (\frac {4 a}{b}+4 \cosh ^{-1}(c x)\right )}{2 b^2 c^3 \sqrt {c x-1} \sqrt {c x+1}}+\frac {x^2 (1-c x) \sqrt {c x+1} \sqrt {1-c^2 x^2}}{b c \sqrt {c x-1} \left (a+b \cosh ^{-1}(c x)\right )} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(x^2*(1 - c*x)*Sqrt[1 + c*x]*Sqrt[1 - c^2*x^2])/(b*c*Sqrt[-1 + c*x]*(a + b*ArcCosh[c*x])) - (Sqrt[1 - c^2*x^2]
*CoshIntegral[(4*a)/b + 4*ArcCosh[c*x]]*Sinh[(4*a)/b])/(2*b^2*c^3*Sqrt[-1 + c*x]*Sqrt[1 + c*x]) + (Sqrt[1 - c^
2*x^2]*Cosh[(4*a)/b]*SinhIntegral[(4*a)/b + 4*ArcCosh[c*x]])/(2*b^2*c^3*Sqrt[-1 + c*x]*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 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 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 5670

Int[((a_.) + ArcCosh[(c_.)*(x_)]*(b_.))^(n_)*(x_)^(m_.), x_Symbol] :> Dist[1/c^(m + 1), Subst[Int[(a + b*x)^n*
Cosh[x]^m*Sinh[x], x], x, ArcCosh[c*x]], x] /; FreeQ[{a, b, c, n}, x] && IGtQ[m, 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 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^2 \sqrt {1-c^2 x^2}}{\left (a+b \cosh ^{-1}(c x)\right )^2} \, dx &=\frac {\sqrt {1-c^2 x^2} \int \frac {x^2 \sqrt {-1+c x} \sqrt {1+c x}}{\left (a+b \cosh ^{-1}(c x)\right )^2} \, dx}{\sqrt {-1+c x} \sqrt {1+c x}}\\ &=\frac {x^2 (1-c x) \sqrt {1+c x} \sqrt {1-c^2 x^2}}{b c \sqrt {-1+c x} \left (a+b \cosh ^{-1}(c x)\right )}-\frac {\left (2 \sqrt {1-c^2 x^2}\right ) \int \frac {x}{a+b \cosh ^{-1}(c x)} \, dx}{b c \sqrt {-1+c x} \sqrt {1+c x}}+\frac {\left (4 c \sqrt {1-c^2 x^2}\right ) \int \frac {x^3}{a+b \cosh ^{-1}(c x)} \, dx}{b \sqrt {-1+c x} \sqrt {1+c x}}\\ &=\frac {x^2 (1-c x) \sqrt {1+c x} \sqrt {1-c^2 x^2}}{b c \sqrt {-1+c x} \left (a+b \cosh ^{-1}(c x)\right )}-\frac {\left (2 \sqrt {1-c^2 x^2}\right ) \operatorname {Subst}\left (\int \frac {\cosh (x) \sinh (x)}{a+b x} \, dx,x,\cosh ^{-1}(c x)\right )}{b c^3 \sqrt {-1+c x} \sqrt {1+c x}}+\frac {\left (4 \sqrt {1-c^2 x^2}\right ) \operatorname {Subst}\left (\int \frac {\cosh ^3(x) \sinh (x)}{a+b x} \, dx,x,\cosh ^{-1}(c x)\right )}{b c^3 \sqrt {-1+c x} \sqrt {1+c x}}\\ &=\frac {x^2 (1-c x) \sqrt {1+c x} \sqrt {1-c^2 x^2}}{b c \sqrt {-1+c x} \left (a+b \cosh ^{-1}(c x)\right )}-\frac {\left (2 \sqrt {1-c^2 x^2}\right ) \operatorname {Subst}\left (\int \frac {\sinh (2 x)}{2 (a+b x)} \, dx,x,\cosh ^{-1}(c x)\right )}{b c^3 \sqrt {-1+c x} \sqrt {1+c x}}+\frac {\left (4 \sqrt {1-c^2 x^2}\right ) \operatorname {Subst}\left (\int \left (\frac {\sinh (2 x)}{4 (a+b x)}+\frac {\sinh (4 x)}{8 (a+b x)}\right ) \, dx,x,\cosh ^{-1}(c x)\right )}{b c^3 \sqrt {-1+c x} \sqrt {1+c x}}\\ &=\frac {x^2 (1-c x) \sqrt {1+c x} \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 (4 x)}{a+b x} \, dx,x,\cosh ^{-1}(c x)\right )}{2 b c^3 \sqrt {-1+c x} \sqrt {1+c x}}\\ &=\frac {x^2 (1-c x) \sqrt {1+c x} \sqrt {1-c^2 x^2}}{b c \sqrt {-1+c x} \left (a+b \cosh ^{-1}(c x)\right )}+\frac {\left (\sqrt {1-c^2 x^2} \cosh \left (\frac {4 a}{b}\right )\right ) \operatorname {Subst}\left (\int \frac {\sinh \left (\frac {4 a}{b}+4 x\right )}{a+b x} \, dx,x,\cosh ^{-1}(c x)\right )}{2 b c^3 \sqrt {-1+c x} \sqrt {1+c x}}-\frac {\left (\sqrt {1-c^2 x^2} \sinh \left (\frac {4 a}{b}\right )\right ) \operatorname {Subst}\left (\int \frac {\cosh \left (\frac {4 a}{b}+4 x\right )}{a+b x} \, dx,x,\cosh ^{-1}(c x)\right )}{2 b c^3 \sqrt {-1+c x} \sqrt {1+c x}}\\ &=\frac {x^2 (1-c x) \sqrt {1+c x} \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} \text {Chi}\left (\frac {4 a}{b}+4 \cosh ^{-1}(c x)\right ) \sinh \left (\frac {4 a}{b}\right )}{2 b^2 c^3 \sqrt {-1+c x} \sqrt {1+c x}}+\frac {\sqrt {1-c^2 x^2} \cosh \left (\frac {4 a}{b}\right ) \text {Shi}\left (\frac {4 a}{b}+4 \cosh ^{-1}(c x)\right )}{2 b^2 c^3 \sqrt {-1+c x} \sqrt {1+c x}}\\ \end {align*}

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Mathematica [A]  time = 0.49, size = 130, normalized size = 0.84 \[ \frac {\sqrt {1-c^2 x^2} \left (-\sinh \left (\frac {4 a}{b}\right ) \left (a+b \cosh ^{-1}(c x)\right ) \text {Chi}\left (4 \left (\frac {a}{b}+\cosh ^{-1}(c x)\right )\right )+\cosh \left (\frac {4 a}{b}\right ) \left (a+b \cosh ^{-1}(c x)\right ) \text {Shi}\left (4 \left (\frac {a}{b}+\cosh ^{-1}(c x)\right )\right )-2 b c^2 x^2 \left (c^2 x^2-1\right )\right )}{2 b^2 c^3 \sqrt {c x-1} \sqrt {c x+1} \left (a+b \cosh ^{-1}(c x)\right )} \]

Antiderivative was successfully verified.

[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]))

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fricas [F]  time = 0.56, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {\sqrt {-c^{2} x^{2} + 1} x^{2}}{b^{2} \operatorname {arcosh}\left (c x\right )^{2} + 2 \, a b \operatorname {arcosh}\left (c x\right ) + a^{2}}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^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)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^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)

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maple [B]  time = 0.57, size = 422, normalized size = 2.74 \[ \frac {\sqrt {-c^{2} x^{2}+1}\, \left (-8 \sqrt {c x +1}\, \sqrt {c x -1}\, x^{4} c^{4}+8 c^{5} x^{5}+8 \sqrt {c x +1}\, \sqrt {c x -1}\, x^{2} c^{2}-12 c^{3} x^{3}-\sqrt {c x -1}\, \sqrt {c x +1}+4 c x \right )}{16 \left (c x +1\right ) \left (c x -1\right ) c^{3} b \left (a +b \,\mathrm {arccosh}\left (c x \right )\right )}-\frac {\sqrt {-c^{2} x^{2}+1}\, \left (-\sqrt {c x +1}\, \sqrt {c x -1}\, x c +c^{2} x^{2}-1\right ) \Ei \left (1, 4 \,\mathrm {arccosh}\left (c x \right )+\frac {4 a}{b}\right ) {\mathrm e}^{\frac {b \,\mathrm {arccosh}\left (c x \right )+4 a}{b}}}{4 \left (c x +1\right ) \left (c x -1\right ) c^{3} b^{2}}-\frac {\sqrt {-c^{2} x^{2}+1}\, \left (8 \sqrt {c x -1}\, \sqrt {c x +1}\, x^{3} b \,c^{3}+8 x^{4} b \,c^{4}-4 \sqrt {c x +1}\, \sqrt {c x -1}\, x b c -8 x^{2} b \,c^{2}+4 \,\mathrm {arccosh}\left (c x \right ) \Ei \left (1, -4 \,\mathrm {arccosh}\left (c x \right )-\frac {4 a}{b}\right ) {\mathrm e}^{-\frac {4 a}{b}} b +4 \Ei \left (1, -4 \,\mathrm {arccosh}\left (c x \right )-\frac {4 a}{b}\right ) {\mathrm e}^{-\frac {4 a}{b}} a +b \right )}{16 \sqrt {c x -1}\, \sqrt {c x +1}\, c^{3} b^{2} \left (a +b \,\mathrm {arccosh}\left (c x \right )\right )}+\frac {\sqrt {-c^{2} x^{2}+1}}{8 \sqrt {c x -1}\, \sqrt {c x +1}\, c^{3} b \left (a +b \,\mathrm {arccosh}\left (c x \right )\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/16*(-c^2*x^2+1)^(1/2)*(-8*(c*x+1)^(1/2)*(c*x-1)^(1/2)*x^4*c^4+8*c^5*x^5+8*(c*x+1)^(1/2)*(c*x-1)^(1/2)*x^2*c^
2-12*c^3*x^3-(c*x-1)^(1/2)*(c*x+1)^(1/2)+4*c*x)/(c*x+1)/(c*x-1)/c^3/b/(a+b*arccosh(c*x))-1/4*(-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(1,4*arccosh(c*x)+4*a/b)*exp((b*arccosh(c*x)+4*a)/b)/(c*x+1)/
(c*x-1)/c^3/b^2-1/16*(-c^2*x^2+1)^(1/2)/(c*x-1)^(1/2)/(c*x+1)^(1/2)*(8*(c*x-1)^(1/2)*(c*x+1)^(1/2)*x^3*b*c^3+8
*x^4*b*c^4-4*(c*x+1)^(1/2)*(c*x-1)^(1/2)*x*b*c-8*x^2*b*c^2+4*arccosh(c*x)*Ei(1,-4*arccosh(c*x)-4*a/b)*exp(-4*a
/b)*b+4*Ei(1,-4*arccosh(c*x)-4*a/b)*exp(-4*a/b)*a+b)/c^3/b^2/(a+b*arccosh(c*x))+1/8*(-c^2*x^2+1)^(1/2)/(c*x-1)
^(1/2)/(c*x+1)^(1/2)/c^3/b/(a+b*arccosh(c*x))

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^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)

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mupad [F]  time = 0.00, size = -1, normalized size = -0.01 \[ \int \frac {x^2\,\sqrt {1-c^2\,x^2}}{{\left (a+b\,\mathrm {acosh}\left (c\,x\right )\right )}^2} \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((x^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)

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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {x^{2} \sqrt {- \left (c x - 1\right ) \left (c x + 1\right )}}{\left (a + b \operatorname {acosh}{\left (c x \right )}\right )^{2}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[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)

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