∫ (a+b cosh−1(cx))n _____________________

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

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

________________________________________________________________________________________

Rubi [A]
time = 0.04, antiderivative size = 57, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, integrand size = 26, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.038, Rules used = {5892} \begin {gather*} \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}} \end {gather*}

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

(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])

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,

\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 = 0.03, size = 57, normalized size = 1.00 \begin {gather*} \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 {gather*}

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

(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])

________________________________________________________________________________________


method	result	size

default	\(\frac {\left (a +b \,\mathrm {arccosh}\left (c x \right )\right )^{1+n} \sqrt {c x -1}\, \sqrt {c x +1}}{b \left (1+n \right ) c \sqrt {-d \left (c x -1\right ) \left (c x +1\right )}}\)	\(54\)

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

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

________________________________________________________________________________________

Maxima [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}

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

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

________________________________________________________________________________________

Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 221 vs. \(2 (51) = 102\).
time = 0.37, size = 221, normalized size = 3.88 \begin {gather*} \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 {gather*}

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

((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)

________________________________________________________________________________________

Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \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 {gather*}

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

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

________________________________________________________________________________________

Giac [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}

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

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

________________________________________________________________________________________

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

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

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

________________________________________________________________________________________

3.5.43 \(\int \frac {(a+b \cosh ^{-1}(c x))^n}{\sqrt {d-c^2 d x^2}} \, dx\) [443]