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

Optimal. Leaf size=248 \[ \frac {\sqrt {1-c x} \sinh \left (\frac {a}{b}\right ) \text {Chi}\left (\frac {a+b \cosh ^{-1}(c x)}{b}\right )}{4 b^2 c^2 \sqrt {c x-1}}-\frac {3 \sqrt {1-c x} \sinh \left (\frac {3 a}{b}\right ) \text {Chi}\left (\frac {3 \left (a+b \cosh ^{-1}(c x)\right )}{b}\right )}{4 b^2 c^2 \sqrt {c x-1}}-\frac {\sqrt {1-c x} \cosh \left (\frac {a}{b}\right ) \text {Shi}\left (\frac {a+b \cosh ^{-1}(c x)}{b}\right )}{4 b^2 c^2 \sqrt {c x-1}}+\frac {3 \sqrt {1-c x} \cosh \left (\frac {3 a}{b}\right ) \text {Shi}\left (\frac {3 \left (a+b \cosh ^{-1}(c x)\right )}{b}\right )}{4 b^2 c^2 \sqrt {c x-1}}-\frac {x \sqrt {c x-1} \sqrt {c x+1} \sqrt {1-c^2 x^2}}{b c \left (a+b \cosh ^{-1}(c x)\right )} \]

[Out]

-1/4*cosh(a/b)*Shi((a+b*arccosh(c*x))/b)*(-c*x+1)^(1/2)/b^2/c^2/(c*x-1)^(1/2)+3/4*cosh(3*a/b)*Shi(3*(a+b*arcco
sh(c*x))/b)*(-c*x+1)^(1/2)/b^2/c^2/(c*x-1)^(1/2)+1/4*Chi((a+b*arccosh(c*x))/b)*sinh(a/b)*(-c*x+1)^(1/2)/b^2/c^
2/(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^2/(c*x-1)^(1/2)-x*(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 = 0.69, antiderivative size = 418, normalized size of antiderivative = 1.69, number of steps used = 15, number of rules used = 8, integrand size = 26, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.308, Rules used = {5798, 5778, 5658, 3303, 3298, 3301, 5670, 5448} \[ -\frac {3 \sqrt {1-c^2 x^2} \sinh \left (\frac {a}{b}\right ) \text {Chi}\left (\frac {a}{b}+\cosh ^{-1}(c x)\right )}{4 b^2 c^2 \sqrt {c x-1} \sqrt {c x+1}}+\frac {\sqrt {1-c^2 x^2} \sinh \left (\frac {a}{b}\right ) \text {Chi}\left (\frac {a+b \cosh ^{-1}(c x)}{b}\right )}{b^2 c^2 \sqrt {c x-1} \sqrt {c x+1}}-\frac {3 \sqrt {1-c^2 x^2} \sinh \left (\frac {3 a}{b}\right ) \text {Chi}\left (\frac {3 a}{b}+3 \cosh ^{-1}(c x)\right )}{4 b^2 c^2 \sqrt {c x-1} \sqrt {c x+1}}+\frac {3 \sqrt {1-c^2 x^2} \cosh \left (\frac {a}{b}\right ) \text {Shi}\left (\frac {a}{b}+\cosh ^{-1}(c x)\right )}{4 b^2 c^2 \sqrt {c x-1} \sqrt {c x+1}}+\frac {3 \sqrt {1-c^2 x^2} \cosh \left (\frac {3 a}{b}\right ) \text {Shi}\left (\frac {3 a}{b}+3 \cosh ^{-1}(c x)\right )}{4 b^2 c^2 \sqrt {c x-1} \sqrt {c x+1}}-\frac {\sqrt {1-c^2 x^2} \cosh \left (\frac {a}{b}\right ) \text {Shi}\left (\frac {a+b \cosh ^{-1}(c x)}{b}\right )}{b^2 c^2 \sqrt {c x-1} \sqrt {c x+1}}+\frac {x \sqrt {c x+1} \sqrt {1-c^2 x^2} (1-c x)}{b c \sqrt {c x-1} \left (a+b \cosh ^{-1}(c x)\right )} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(x*(1 - c*x)*Sqrt[1 + c*x]*Sqrt[1 - c^2*x^2])/(b*c*Sqrt[-1 + c*x]*(a + b*ArcCosh[c*x])) - (3*Sqrt[1 - c^2*x^2]
*CoshIntegral[a/b + ArcCosh[c*x]]*Sinh[a/b])/(4*b^2*c^2*Sqrt[-1 + c*x]*Sqrt[1 + c*x]) + (Sqrt[1 - c^2*x^2]*Cos
hIntegral[(a + b*ArcCosh[c*x])/b]*Sinh[a/b])/(b^2*c^2*Sqrt[-1 + c*x]*Sqrt[1 + c*x]) - (3*Sqrt[1 - c^2*x^2]*Cos
hIntegral[(3*a)/b + 3*ArcCosh[c*x]]*Sinh[(3*a)/b])/(4*b^2*c^2*Sqrt[-1 + c*x]*Sqrt[1 + c*x]) + (3*Sqrt[1 - c^2*
x^2]*Cosh[a/b]*SinhIntegral[a/b + ArcCosh[c*x]])/(4*b^2*c^2*Sqrt[-1 + c*x]*Sqrt[1 + c*x]) + (3*Sqrt[1 - c^2*x^
2]*Cosh[(3*a)/b]*SinhIntegral[(3*a)/b + 3*ArcCosh[c*x]])/(4*b^2*c^2*Sqrt[-1 + c*x]*Sqrt[1 + c*x]) - (Sqrt[1 -
c^2*x^2]*Cosh[a/b]*SinhIntegral[(a + b*ArcCosh[c*x])/b])/(b^2*c^2*Sqrt[-1 + c*x]*Sqrt[1 + c*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 5658

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

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

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Mathematica [A]  time = 0.43, size = 217, normalized size = 0.88 \[ \frac {\sqrt {1-c^2 x^2} \left (\sinh \left (\frac {a}{b}\right ) \left (a+b \cosh ^{-1}(c x)\right ) \text {Chi}\left (\frac {a}{b}+\cosh ^{-1}(c x)\right )-3 \sinh \left (\frac {3 a}{b}\right ) \left (a+b \cosh ^{-1}(c x)\right ) \text {Chi}\left (3 \left (\frac {a}{b}+\cosh ^{-1}(c x)\right )\right )-a \cosh \left (\frac {a}{b}\right ) \text {Shi}\left (\frac {a}{b}+\cosh ^{-1}(c x)\right )-b \cosh \left (\frac {a}{b}\right ) \cosh ^{-1}(c x) \text {Shi}\left (\frac {a}{b}+\cosh ^{-1}(c x)\right )+3 a \cosh \left (\frac {3 a}{b}\right ) \text {Shi}\left (3 \left (\frac {a}{b}+\cosh ^{-1}(c x)\right )\right )+3 b \cosh \left (\frac {3 a}{b}\right ) \cosh ^{-1}(c x) \text {Shi}\left (3 \left (\frac {a}{b}+\cosh ^{-1}(c x)\right )\right )-4 b c^3 x^3+4 b c x\right )}{4 b^2 c^2 \sqrt {c x-1} \sqrt {c x+1} \left (a+b \cosh ^{-1}(c x)\right )} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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fricas [F]  time = 0.66, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {\sqrt {-c^{2} x^{2} + 1} x}{b^{2} \operatorname {arcosh}\left (c x\right )^{2} + 2 \, a b \operatorname {arcosh}\left (c x\right ) + a^{2}}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x*(-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/(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 \[ \int \frac {\sqrt {-c^{2} x^{2} + 1} x}{{\left (b \operatorname {arcosh}\left (c x\right ) + a\right )}^{2}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

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maple [B]  time = 0.32, size = 622, normalized size = 2.51 \[ \frac {\sqrt {-c^{2} x^{2}+1}\, \left (-4 \sqrt {c x +1}\, \sqrt {c x -1}\, x^{3} c^{3}+4 c^{4} x^{4}+3 \sqrt {c x +1}\, \sqrt {c x -1}\, x c -5 c^{2} x^{2}+1\right )}{8 \left (c x +1\right ) c^{2} \left (c x -1\right ) \left (a +b \,\mathrm {arccosh}\left (c x \right )\right ) b}-\frac {3 \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, 3 \,\mathrm {arccosh}\left (c x \right )+\frac {3 a}{b}\right ) {\mathrm e}^{\frac {b \,\mathrm {arccosh}\left (c x \right )+3 a}{b}}}{8 \left (c x +1\right ) c^{2} \left (c x -1\right ) b^{2}}-\frac {\sqrt {-c^{2} x^{2}+1}\, \left (4 \sqrt {c x +1}\, \sqrt {c x -1}\, x^{2} b \,c^{2}+4 x^{3} b \,c^{3}+3 \,\mathrm {arccosh}\left (c x \right ) \Ei \left (1, -3 \,\mathrm {arccosh}\left (c x \right )-\frac {3 a}{b}\right ) {\mathrm e}^{-\frac {3 a}{b}} b +3 \Ei \left (1, -3 \,\mathrm {arccosh}\left (c x \right )-\frac {3 a}{b}\right ) {\mathrm e}^{-\frac {3 a}{b}} a -\sqrt {c x +1}\, \sqrt {c x -1}\, b -3 x b c \right )}{8 \sqrt {c x -1}\, \sqrt {c x +1}\, c^{2} b^{2} \left (a +b \,\mathrm {arccosh}\left (c x \right )\right )}+\frac {\sqrt {-c^{2} x^{2}+1}\, \left (\mathrm {arccosh}\left (c x \right ) {\mathrm e}^{-\frac {a}{b}} \Ei \left (1, -\mathrm {arccosh}\left (c x \right )-\frac {a}{b}\right ) b +\sqrt {c x +1}\, \sqrt {c x -1}\, b +{\mathrm e}^{-\frac {a}{b}} \Ei \left (1, -\mathrm {arccosh}\left (c x \right )-\frac {a}{b}\right ) a +x b c \right )}{8 \sqrt {c x -1}\, \sqrt {c x +1}\, c^{2} b^{2} \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 )}{8 \left (c x +1\right ) c^{2} \left (c x -1\right ) \left (a +b \,\mathrm {arccosh}\left (c x \right )\right ) b}+\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, \mathrm {arccosh}\left (c x \right )+\frac {a}{b}\right ) {\mathrm e}^{\frac {a +b \,\mathrm {arccosh}\left (c x \right )}{b}}}{8 \left (c x +1\right ) c^{2} \left (c x -1\right ) b^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/8*(-c^2*x^2+1)^(1/2)*(-4*(c*x+1)^(1/2)*(c*x-1)^(1/2)*x^3*c^3+4*c^4*x^4+3*(c*x+1)^(1/2)*(c*x-1)^(1/2)*x*c-5*c
^2*x^2+1)/(c*x+1)/c^2/(c*x-1)/(a+b*arccosh(c*x))/b-3/8*(-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(1,3*arccosh(c*x)+3*a/b)*exp((b*arccosh(c*x)+3*a)/b)/(c*x+1)/c^2/(c*x-1)/b^2-1/8*(-c^2*x^2+1)^(1/2)
/(c*x-1)^(1/2)/(c*x+1)^(1/2)*(4*(c*x+1)^(1/2)*(c*x-1)^(1/2)*x^2*b*c^2+4*x^3*b*c^3+3*arccosh(c*x)*Ei(1,-3*arcco
sh(c*x)-3*a/b)*exp(-3*a/b)*b+3*Ei(1,-3*arccosh(c*x)-3*a/b)*exp(-3*a/b)*a-(c*x+1)^(1/2)*(c*x-1)^(1/2)*b-3*x*b*c
)/c^2/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)*(arccosh(c*x)*exp(-a/b)*Ei(1,-
arccosh(c*x)-a/b)*b+(c*x+1)^(1/2)*(c*x-1)^(1/2)*b+exp(-a/b)*Ei(1,-arccosh(c*x)-a/b)*a+x*b*c)/c^2/b^2/(a+b*arcc
osh(c*x))-1/8*(-c^2*x^2+1)^(1/2)*(-(c*x+1)^(1/2)*(c*x-1)^(1/2)*x*c+c^2*x^2-1)/(c*x+1)/c^2/(c*x-1)/(a+b*arccosh
(c*x))/b+1/8*(-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,arccosh(c*x)+a/b)*exp((a+b*a
rccosh(c*x))/b)/(c*x+1)/c^2/(c*x-1)/b^2

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-((c^2*x^3 - x)*(c*x + 1)*sqrt(c*x - 1) + (c^3*x^4 - c*x^2)*sqrt(c*x + 1))*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 + (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((3*(c*x + 1)^(3/2)*(c*x - 1)*c^3*x^3 + (6*c^4*x^4 - 5*c^2*x^2 +
1)*(c*x + 1)*sqrt(c*x - 1) + (3*c^5*x^5 - 5*c^3*x^3 + 2*c*x)*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 = -0.00 \[ \int \frac {x\,\sqrt {1-c^2\,x^2}}{{\left (a+b\,\mathrm {acosh}\left (c\,x\right )\right )}^2} \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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