\(\int \frac {(a+b \text {arccosh}(c x))^n}{\sqrt {1-c^2 x^2}} \, dx\) [437]

Optimal result

Integrand size = 25, antiderivative size = 43 \[ \int \frac {(a+b \text {arccosh}(c x))^n}{\sqrt {1-c^2 x^2}} \, dx=\frac {\sqrt {-1+c x} (a+b \text {arccosh}(c x))^{1+n}}{b c (1+n) \sqrt {1-c x}} \]

[Out]

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

Rubi [A] (verified)

Time = 0.04 (sec) , antiderivative size = 43, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.040, Rules used = {5892} \[ \int \frac {(a+b \text {arccosh}(c x))^n}{\sqrt {1-c^2 x^2}} \, dx=\frac {\sqrt {c x-1} (a+b \text {arccosh}(c x))^{n+1}}{b c (n+1) \sqrt {1-c x}} \]

[In]

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

[Out]

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

Rule 5892

Int[((a_.) + ArcCosh[(c_.)*(x_)]*(b_.))^(n_.)/Sqrt[(d_) + (e_.)*(x_)^2], x_Symbol] :> Simp[(1/(b*c*(n + 1)))*S
imp[Sqrt[1 + c*x]*(Sqrt[-1 + c*x]/Sqrt[d + e*x^2])]*(a + b*ArcCosh[c*x])^(n + 1), x] /; FreeQ[{a, b, c, d, e,
n}, x] && EqQ[c^2*d + e, 0] && NeQ[n, -1]

Rubi steps \begin{align*} \text {integral}& = \frac {\sqrt {-1+c x} (a+b \text {arccosh}(c x))^{1+n}}{b c (1+n) \sqrt {1-c x}} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.03 (sec) , antiderivative size = 56, normalized size of antiderivative = 1.30 \[ \int \frac {(a+b \text {arccosh}(c x))^n}{\sqrt {1-c^2 x^2}} \, dx=\frac {\sqrt {-1+c x} \sqrt {1+c x} (a+b \text {arccosh}(c x))^{1+n}}{b c (1+n) \sqrt {1-c^2 x^2}} \]

[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])

Maple [A] (verified)

Time = 0.61 (sec) , antiderivative size = 53, normalized size of antiderivative = 1.23

method	result	size
default	\(\frac {\sqrt {c x -1}\, \sqrt {c x +1}\, \left (a +b \,\operatorname {arccosh}\left (c x \right )\right )^{1+n}}{b c \left (1+n \right ) \sqrt {-\left (c x -1\right ) \left (c x +1\right )}}\)	\(53\)

[In]

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

[Out]

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

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 213 vs. \(2 (39) = 78\).

Time = 0.28 (sec) , antiderivative size = 213, normalized size of antiderivative = 4.95 \[ \int \frac {(a+b \text {arccosh}(c x))^n}{\sqrt {1-c^2 x^2}} \, dx=\frac {{\left (\sqrt {c^{2} x^{2} - 1} \sqrt {-c^{2} x^{2} + 1} b \log \left (c x + \sqrt {c^{2} x^{2} - 1}\right ) + \sqrt {c^{2} x^{2} - 1} \sqrt {-c^{2} x^{2} + 1} a\right )} \cosh \left (n \log \left (b \log \left (c x + \sqrt {c^{2} x^{2} - 1}\right ) + a\right )\right ) + {\left (\sqrt {c^{2} x^{2} - 1} \sqrt {-c^{2} x^{2} + 1} b \log \left (c x + \sqrt {c^{2} x^{2} - 1}\right ) + \sqrt {c^{2} x^{2} - 1} \sqrt {-c^{2} x^{2} + 1} a\right )} \sinh \left (n \log \left (b \log \left (c x + \sqrt {c^{2} x^{2} - 1}\right ) + a\right )\right )}{b c n - {\left (b c^{3} n + b c^{3}\right )} x^{2} + b c} \]

[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)

Sympy [F]

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

[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)

Maxima [F]

\[ \int \frac {(a+b \text {arccosh}(c x))^n}{\sqrt {1-c^2 x^2}} \, dx=\int { \frac {{\left (b \operatorname {arcosh}\left (c x\right ) + a\right )}^{n}}{\sqrt {-c^{2} x^{2} + 1}} \,d x } \]

[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)

Giac [F]

\[ \int \frac {(a+b \text {arccosh}(c x))^n}{\sqrt {1-c^2 x^2}} \, dx=\int { \frac {{\left (b \operatorname {arcosh}\left (c x\right ) + a\right )}^{n}}{\sqrt {-c^{2} x^{2} + 1}} \,d x } \]

[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)

Mupad [F(-1)]

Timed out. \[ \int \frac {(a+b \text {arccosh}(c x))^n}{\sqrt {1-c^2 x^2}} \, dx=\int \frac {{\left (a+b\,\mathrm {acosh}\left (c\,x\right )\right )}^n}{\sqrt {1-c^2\,x^2}} \,d x \]

[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)