3.146 \(\int \frac{1}{(a+b \cosh ^{-1}(c+d x))^3} \, dx\)

Optimal. Leaf size=132 \[ -\frac{\sinh \left (\frac{a}{b}\right ) \text{Chi}\left (\frac{a+b \cosh ^{-1}(c+d x)}{b}\right )}{2 b^3 d}+\frac{\cosh \left (\frac{a}{b}\right ) \text{Shi}\left (\frac{a+b \cosh ^{-1}(c+d x)}{b}\right )}{2 b^3 d}-\frac{c+d x}{2 b^2 d \left (a+b \cosh ^{-1}(c+d x)\right )}-\frac{\sqrt{c+d x-1} \sqrt{c+d x+1}}{2 b d \left (a+b \cosh ^{-1}(c+d x)\right )^2} \]

[Out]

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

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Rubi [A]  time = 0.259714, antiderivative size = 132, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 7, integrand size = 12, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.583, Rules used = {5864, 5656, 5775, 5658, 3303, 3298, 3301} \[ -\frac{\sinh \left (\frac{a}{b}\right ) \text{Chi}\left (\frac{a+b \cosh ^{-1}(c+d x)}{b}\right )}{2 b^3 d}+\frac{\cosh \left (\frac{a}{b}\right ) \text{Shi}\left (\frac{a+b \cosh ^{-1}(c+d x)}{b}\right )}{2 b^3 d}-\frac{c+d x}{2 b^2 d \left (a+b \cosh ^{-1}(c+d x)\right )}-\frac{\sqrt{c+d x-1} \sqrt{c+d x+1}}{2 b d \left (a+b \cosh ^{-1}(c+d x)\right )^2} \]

Antiderivative was successfully verified.

[In]

Int[(a + b*ArcCosh[c + d*x])^(-3),x]

[Out]

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

Rule 5864

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

Rule 5656

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

Rule 5775

Int[(((a_.) + ArcCosh[(c_.)*(x_)]*(b_.))^(n_)*((f_.)*(x_))^(m_.))/(Sqrt[(d1_) + (e1_.)*(x_)]*Sqrt[(d2_) + (e2_
.)*(x_)]), x_Symbol] :> Simp[((f*x)^m*(a + b*ArcCosh[c*x])^(n + 1))/(b*c*Sqrt[-(d1*d2)]*(n + 1)), x] - Dist[(f
*m)/(b*c*Sqrt[-(d1*d2)]*(n + 1)), Int[(f*x)^(m - 1)*(a + b*ArcCosh[c*x])^(n + 1), x], x] /; FreeQ[{a, b, c, d1
, e1, d2, e2, f, m}, x] && EqQ[e1 - c*d1, 0] && EqQ[e2 + c*d2, 0] && LtQ[n, -1] && GtQ[d1, 0] && LtQ[d2, 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 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 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]

Rubi steps

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

Mathematica [A]  time = 0.40856, size = 109, normalized size = 0.83 \[ -\frac{\sinh \left (\frac{a}{b}\right ) \text{Chi}\left (\frac{a}{b}+\cosh ^{-1}(c+d x)\right )-\cosh \left (\frac{a}{b}\right ) \text{Shi}\left (\frac{a}{b}+\cosh ^{-1}(c+d x)\right )+\frac{b \left (a c+a d x+b \sqrt{c+d x-1} \sqrt{c+d x+1}+b (c+d x) \cosh ^{-1}(c+d x)\right )}{\left (a+b \cosh ^{-1}(c+d x)\right )^2}}{2 b^3 d} \]

Antiderivative was successfully verified.

[In]

Integrate[(a + b*ArcCosh[c + d*x])^(-3),x]

[Out]

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

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Maple [A]  time = 0.059, size = 207, normalized size = 1.6 \begin{align*}{\frac{1}{d} \left ( -{\frac{b{\rm arccosh} \left (dx+c\right )+a-b}{4\,{b}^{2} \left ({b}^{2} \left ({\rm arccosh} \left (dx+c\right ) \right ) ^{2}+2\,ab{\rm arccosh} \left (dx+c\right )+{a}^{2} \right ) } \left ( -\sqrt{dx+c-1}\sqrt{dx+c+1}+dx+c \right ) }+{\frac{1}{4\,{b}^{3}}{{\rm e}^{{\frac{a}{b}}}}{\it Ei} \left ( 1,{\rm arccosh} \left (dx+c\right )+{\frac{a}{b}} \right ) }-{\frac{1}{4\,b \left ( a+b{\rm arccosh} \left (dx+c\right ) \right ) ^{2}} \left ( dx+c+\sqrt{dx+c-1}\sqrt{dx+c+1} \right ) }-{\frac{1}{4\,{b}^{2} \left ( a+b{\rm arccosh} \left (dx+c\right ) \right ) } \left ( dx+c+\sqrt{dx+c-1}\sqrt{dx+c+1} \right ) }-{\frac{1}{4\,{b}^{3}}{{\rm e}^{-{\frac{a}{b}}}}{\it Ei} \left ( 1,-{\rm arccosh} \left (dx+c\right )-{\frac{a}{b}} \right ) } \right ) } \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(a+b*arccosh(d*x+c))^3,x)

[Out]

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

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Maxima [F(-1)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Timed out

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Fricas [F]  time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (\frac{1}{b^{3} \operatorname{arcosh}\left (d x + c\right )^{3} + 3 \, a b^{2} \operatorname{arcosh}\left (d x + c\right )^{2} + 3 \, a^{2} b \operatorname{arcosh}\left (d x + c\right ) + a^{3}}, x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{\left (a + b \operatorname{acosh}{\left (c + d x \right )}\right )^{3}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Integral((a + b*acosh(c + d*x))**(-3), x)

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{{\left (b \operatorname{arcosh}\left (d x + c\right ) + a\right )}^{3}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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