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

Optimal. Leaf size=43 \[ \frac {\sqrt {c x-1} \left (a+b \cosh ^{-1}(c x)\right )^{n+1}}{b c (n+1) \sqrt {1-c x}} \]

[Out]

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

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Rubi [A]  time = 0.21, antiderivative size = 56, normalized size of antiderivative = 1.30, number of steps used = 2, number of rules used = 2, integrand size = 25, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.080, Rules used = {5713, 5676} \[ \frac {\sqrt {c x-1} \sqrt {c x+1} \left (a+b \cosh ^{-1}(c x)\right )^{n+1}}{b c (n+1) \sqrt {1-c^2 x^2}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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]

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]

Rubi steps

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

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Mathematica [A]  time = 0.04, size = 56, normalized size = 1.30 \[ \frac {\sqrt {c x-1} \sqrt {c x+1} \left (a+b \cosh ^{-1}(c x)\right )^{n+1}}{b c (n+1) \sqrt {1-c^2 x^2}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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fricas [B]  time = 0.55, size = 213, normalized size = 4.95 \[ \frac {{\left (\sqrt {c^{2} x^{2} - 1} \sqrt {-c^{2} x^{2} + 1} b \log \left (c x + \sqrt {c^{2} x^{2} - 1}\right ) + \sqrt {c^{2} x^{2} - 1} \sqrt {-c^{2} x^{2} + 1} a\right )} \cosh \left (n \log \left (b \log \left (c x + \sqrt {c^{2} x^{2} - 1}\right ) + a\right )\right ) + {\left (\sqrt {c^{2} x^{2} - 1} \sqrt {-c^{2} x^{2} + 1} b \log \left (c x + \sqrt {c^{2} x^{2} - 1}\right ) + \sqrt {c^{2} x^{2} - 1} \sqrt {-c^{2} x^{2} + 1} a\right )} \sinh \left (n \log \left (b \log \left (c x + \sqrt {c^{2} x^{2} - 1}\right ) + a\right )\right )}{b c n - {\left (b c^{3} n + b c^{3}\right )} x^{2} + b c} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maple [A]  time = 0.06, size = 53, normalized size = 1.23 \[ \frac {\left (a +b \,\mathrm {arccosh}\left (c x \right )\right )^{1+n} \sqrt {c x -1}\, \sqrt {c x +1}}{b \left (1+n \right ) c \sqrt {-\left (c x -1\right ) \left (c x +1\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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mupad [F]  time = 0.00, size = -1, normalized size = -0.02 \[ \int \frac {{\left (a+b\,\mathrm {acosh}\left (c\,x\right )\right )}^n}{\sqrt {1-c^2\,x^2}} \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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