3.421 \(\int \sqrt{d-c^2 d x^2} (a+b \cosh ^{-1}(c x))^n \, dx\)

Optimal. Leaf size=253 \[ \frac{2^{-n-3} e^{-\frac{2 a}{b}} \sqrt{d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )^n \left (-\frac{a+b \cosh ^{-1}(c x)}{b}\right )^{-n} \text{Gamma}\left (n+1,-\frac{2 \left (a+b \cosh ^{-1}(c x)\right )}{b}\right )}{c \sqrt{c x-1} \sqrt{c x+1}}-\frac{2^{-n-3} e^{\frac{2 a}{b}} \sqrt{d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )^n \left (\frac{a+b \cosh ^{-1}(c x)}{b}\right )^{-n} \text{Gamma}\left (n+1,\frac{2 \left (a+b \cosh ^{-1}(c x)\right )}{b}\right )}{c \sqrt{c x-1} \sqrt{c x+1}}-\frac{\sqrt{d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )^{n+1}}{2 b c (n+1) \sqrt{c x-1} \sqrt{c x+1}} \]

[Out]

-(Sqrt[d - c^2*d*x^2]*(a + b*ArcCosh[c*x])^(1 + n))/(2*b*c*(1 + n)*Sqrt[-1 + c*x]*Sqrt[1 + c*x]) + (2^(-3 - n)
*Sqrt[d - c^2*d*x^2]*(a + b*ArcCosh[c*x])^n*Gamma[1 + n, (-2*(a + b*ArcCosh[c*x]))/b])/(c*E^((2*a)/b)*Sqrt[-1
+ c*x]*Sqrt[1 + c*x]*(-((a + b*ArcCosh[c*x])/b))^n) - (2^(-3 - n)*E^((2*a)/b)*Sqrt[d - c^2*d*x^2]*(a + b*ArcCo
sh[c*x])^n*Gamma[1 + n, (2*(a + b*ArcCosh[c*x]))/b])/(c*Sqrt[-1 + c*x]*Sqrt[1 + c*x]*((a + b*ArcCosh[c*x])/b)^
n)

________________________________________________________________________________________

Rubi [A]  time = 0.375673, antiderivative size = 253, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 5, integrand size = 26, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.192, Rules used = {5713, 5701, 3312, 3307, 2181} \[ \frac{2^{-n-3} e^{-\frac{2 a}{b}} \sqrt{d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )^n \left (-\frac{a+b \cosh ^{-1}(c x)}{b}\right )^{-n} \text{Gamma}\left (n+1,-\frac{2 \left (a+b \cosh ^{-1}(c x)\right )}{b}\right )}{c \sqrt{c x-1} \sqrt{c x+1}}-\frac{2^{-n-3} e^{\frac{2 a}{b}} \sqrt{d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )^n \left (\frac{a+b \cosh ^{-1}(c x)}{b}\right )^{-n} \text{Gamma}\left (n+1,\frac{2 \left (a+b \cosh ^{-1}(c x)\right )}{b}\right )}{c \sqrt{c x-1} \sqrt{c x+1}}-\frac{\sqrt{d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )^{n+1}}{2 b c (n+1) \sqrt{c x-1} \sqrt{c x+1}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-(Sqrt[d - c^2*d*x^2]*(a + b*ArcCosh[c*x])^(1 + n))/(2*b*c*(1 + n)*Sqrt[-1 + c*x]*Sqrt[1 + c*x]) + (2^(-3 - n)
*Sqrt[d - c^2*d*x^2]*(a + b*ArcCosh[c*x])^n*Gamma[1 + n, (-2*(a + b*ArcCosh[c*x]))/b])/(c*E^((2*a)/b)*Sqrt[-1
+ c*x]*Sqrt[1 + c*x]*(-((a + b*ArcCosh[c*x])/b))^n) - (2^(-3 - n)*E^((2*a)/b)*Sqrt[d - c^2*d*x^2]*(a + b*ArcCo
sh[c*x])^n*Gamma[1 + n, (2*(a + b*ArcCosh[c*x]))/b])/(c*Sqrt[-1 + c*x]*Sqrt[1 + c*x]*((a + b*ArcCosh[c*x])/b)^
n)

Rule 5713

Int[((a_.) + ArcCosh[(c_.)*(x_)]*(b_.))^(n_.)*((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[(1 + c*x)^p*(-1 + c*x)^p*(a + b*Ar
cCosh[c*x])^n, x], x] /; FreeQ[{a, b, c, d, e, n, p}, x] && EqQ[c^2*d + e, 0] &&  !IntegerQ[p]

Rule 5701

Int[((a_.) + ArcCosh[(c_.)*(x_)]*(b_.))^(n_.)*((d1_) + (e1_.)*(x_))^(p_.)*((d2_) + (e2_.)*(x_))^(p_.), x_Symbo
l] :> Dist[(-(d1*d2))^p/c, Subst[Int[(a + b*x)^n*Sinh[x]^(2*p + 1), x], x, ArcCosh[c*x]], x] /; FreeQ[{a, b, c
, d1, e1, d2, e2, n}, x] && EqQ[e1, c*d1] && EqQ[e2, -(c*d2)] && IGtQ[p + 1/2, 0] && (GtQ[d1, 0] && LtQ[d2, 0]
)

Rule 3312

Int[((c_.) + (d_.)*(x_))^(m_)*sin[(e_.) + (f_.)*(x_)]^(n_), x_Symbol] :> Int[ExpandTrigReduce[(c + d*x)^m, Sin
[e + f*x]^n, x], x] /; FreeQ[{c, d, e, f, m}, x] && IGtQ[n, 1] && ( !RationalQ[m] || (GeQ[m, -1] && LtQ[m, 1])
)

Rule 3307

Int[((c_.) + (d_.)*(x_))^(m_.)*sin[(e_.) + Pi*(k_.) + (f_.)*(x_)], x_Symbol] :> Dist[I/2, Int[(c + d*x)^m/(E^(
I*k*Pi)*E^(I*(e + f*x))), x], x] - Dist[I/2, Int[(c + d*x)^m*E^(I*k*Pi)*E^(I*(e + f*x)), x], x] /; FreeQ[{c, d
, e, f, m}, x] && IntegerQ[2*k]

Rule 2181

Int[(F_)^((g_.)*((e_.) + (f_.)*(x_)))*((c_.) + (d_.)*(x_))^(m_), x_Symbol] :> -Simp[(F^(g*(e - (c*f)/d))*(c +
d*x)^FracPart[m]*Gamma[m + 1, (-((f*g*Log[F])/d))*(c + d*x)])/(d*(-((f*g*Log[F])/d))^(IntPart[m] + 1)*(-((f*g*
Log[F]*(c + d*x))/d))^FracPart[m]), x] /; FreeQ[{F, c, d, e, f, g, m}, x] &&  !IntegerQ[m]

Rubi steps

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

Mathematica [A]  time = 0.687661, size = 214, normalized size = 0.85 \[ \frac{d 2^{-n-3} e^{-\frac{2 a}{b}} \sqrt{\frac{c x-1}{c x+1}} (c x+1) \left (a+b \cosh ^{-1}(c x)\right )^n \left (-\frac{\left (a+b \cosh ^{-1}(c x)\right )^2}{b^2}\right )^{-n} \left (-b (n+1) \left (\frac{a}{b}+\cosh ^{-1}(c x)\right )^n \text{Gamma}\left (n+1,-\frac{2 \left (a+b \cosh ^{-1}(c x)\right )}{b}\right )+b (n+1) e^{\frac{4 a}{b}} \left (-\frac{a+b \cosh ^{-1}(c x)}{b}\right )^n \text{Gamma}\left (n+1,\frac{2 \left (a+b \cosh ^{-1}(c x)\right )}{b}\right )+2^{n+2} e^{\frac{2 a}{b}} \left (a+b \cosh ^{-1}(c x)\right ) \left (-\frac{\left (a+b \cosh ^{-1}(c x)\right )^2}{b^2}\right )^n\right )}{b c (n+1) \sqrt{d-c^2 d x^2}} \]

Warning: Unable to verify antiderivative.

[In]

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

[Out]

(2^(-3 - n)*d*Sqrt[(-1 + c*x)/(1 + c*x)]*(1 + c*x)*(a + b*ArcCosh[c*x])^n*(2^(2 + n)*E^((2*a)/b)*(a + b*ArcCos
h[c*x])*(-((a + b*ArcCosh[c*x])^2/b^2))^n - b*(1 + n)*(a/b + ArcCosh[c*x])^n*Gamma[1 + n, (-2*(a + b*ArcCosh[c
*x]))/b] + b*E^((4*a)/b)*(1 + n)*(-((a + b*ArcCosh[c*x])/b))^n*Gamma[1 + n, (2*(a + b*ArcCosh[c*x]))/b]))/(b*c
*E^((2*a)/b)*(1 + n)*Sqrt[d - c^2*d*x^2]*(-((a + b*ArcCosh[c*x])^2/b^2))^n)

________________________________________________________________________________________

Maple [F]  time = 0.286, size = 0, normalized size = 0. \begin{align*} \int \left ( a+b{\rm arccosh} \left (cx\right ) \right ) ^{n}\sqrt{-{c}^{2}d{x}^{2}+d}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \sqrt{-c^{2} d x^{2} + d}{\left (b \operatorname{arcosh}\left (c x\right ) + a\right )}^{n}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate(sqrt(-c^2*d*x^2 + d)*(b*arccosh(c*x) + a)^n, x)

________________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integral(sqrt(-c^2*d*x^2 + d)*(b*arccosh(c*x) + a)^n, x)

________________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Integral(sqrt(-d*(c*x - 1)*(c*x + 1))*(a + b*acosh(c*x))**n, x)

________________________________________________________________________________________

Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \mathit{sage}_{0} x \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

sage0*x