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

Optimal. Leaf size=57 \[ \frac{\sqrt{c x-1} \sqrt{c x+1} \left (a+b \cosh ^{-1}(c x)\right )^{n+1}}{b c (n+1) \sqrt{d-c^2 d x^2}} \]

[Out]

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

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Rubi [A]  time = 0.193784, antiderivative size = 57, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 26, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.077, Rules used = {5713, 5676} \[ \frac{\sqrt{c x-1} \sqrt{c x+1} \left (a+b \cosh ^{-1}(c x)\right )^{n+1}}{b c (n+1) \sqrt{d-c^2 d x^2}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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 5676

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

Rubi steps

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

Mathematica [A]  time = 0.0448806, size = 57, normalized size = 1. \[ \frac{\sqrt{c x-1} \sqrt{c x+1} \left (a+b \cosh ^{-1}(c x)\right )^{n+1}}{b c (n+1) \sqrt{d-c^2 d x^2}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [A]  time = 0.035, size = 54, normalized size = 1. \begin{align*}{\frac{ \left ( a+b{\rm arccosh} \left (cx\right ) \right ) ^{1+n}}{cb \left ( 1+n \right ) }\sqrt{cx-1}\sqrt{cx+1}{\frac{1}{\sqrt{- \left ( cx-1 \right ) \left ( cx+1 \right ) d}}}} \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]

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

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

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Fricas [B]  time = 2.68469, size = 509, normalized size = 8.93 \begin{align*} \frac{{\left (\sqrt{-c^{2} d x^{2} + d} \sqrt{c^{2} x^{2} - 1} b \log \left (c x + \sqrt{c^{2} x^{2} - 1}\right ) + \sqrt{-c^{2} d x^{2} + d} \sqrt{c^{2} x^{2} - 1} a\right )} \cosh \left (n \log \left (b \log \left (c x + \sqrt{c^{2} x^{2} - 1}\right ) + a\right )\right ) +{\left (\sqrt{-c^{2} d x^{2} + d} \sqrt{c^{2} x^{2} - 1} b \log \left (c x + \sqrt{c^{2} x^{2} - 1}\right ) + \sqrt{-c^{2} d x^{2} + d} \sqrt{c^{2} x^{2} - 1} a\right )} \sinh \left (n \log \left (b \log \left (c x + \sqrt{c^{2} x^{2} - 1}\right ) + a\right )\right )}{b c d n + b c d -{\left (b c^{3} d n + b c^{3} d\right )} x^{2}} \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]

((sqrt(-c^2*d*x^2 + d)*sqrt(c^2*x^2 - 1)*b*log(c*x + sqrt(c^2*x^2 - 1)) + sqrt(-c^2*d*x^2 + d)*sqrt(c^2*x^2 -
1)*a)*cosh(n*log(b*log(c*x + sqrt(c^2*x^2 - 1)) + a)) + (sqrt(-c^2*d*x^2 + d)*sqrt(c^2*x^2 - 1)*b*log(c*x + sq
rt(c^2*x^2 - 1)) + sqrt(-c^2*d*x^2 + d)*sqrt(c^2*x^2 - 1)*a)*sinh(n*log(b*log(c*x + sqrt(c^2*x^2 - 1)) + a)))/
(b*c*d*n + b*c*d - (b*c^3*d*n + b*c^3*d)*x^2)

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

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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