\(\int \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))^n \, dx\) [320]

Optimal result

Integrand size = 26, antiderivative size = 253 \[ \int \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))^n \, dx=-\frac {\sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))^{1+n}}{2 b c (1+n) \sqrt {-1+c x} \sqrt {1+c x}}+\frac {2^{-3-n} e^{-\frac {2 a}{b}} \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))^n \left (-\frac {a+b \text {arccosh}(c x)}{b}\right )^{-n} \Gamma \left (1+n,-\frac {2 (a+b \text {arccosh}(c x))}{b}\right )}{c \sqrt {-1+c x} \sqrt {1+c x}}-\frac {2^{-3-n} e^{\frac {2 a}{b}} \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))^n \left (\frac {a+b \text {arccosh}(c x)}{b}\right )^{-n} \Gamma \left (1+n,\frac {2 (a+b \text {arccosh}(c x))}{b}\right )}{c \sqrt {-1+c x} \sqrt {1+c x}} \] Output:

-1/2*(-c^2*d*x^2+d)^(1/2)*(a+b*arccosh(c*x))^(1+n)/b/c/(1+n)/(c*x-1)^(1/2) 
/(c*x+1)^(1/2)+2^(-3-n)*(-c^2*d*x^2+d)^(1/2)*(a+b*arccosh(c*x))^n*GAMMA(1+ 
n,(-2*a-2*b*arccosh(c*x))/b)/c/exp(2*a/b)/(c*x-1)^(1/2)/(c*x+1)^(1/2)/((-( 
a+b*arccosh(c*x))/b)^n)-2^(-3-n)*exp(2*a/b)*(-c^2*d*x^2+d)^(1/2)*(a+b*arcc 
osh(c*x))^n*GAMMA(1+n,2*(a+b*arccosh(c*x))/b)/c/(c*x-1)^(1/2)/(c*x+1)^(1/2 
)/(((a+b*arccosh(c*x))/b)^n)
 

Mathematica [A] (warning: unable to verify)

Time = 0.55 (sec) , antiderivative size = 214, normalized size of antiderivative = 0.85 \[ \int \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))^n \, dx=\frac {2^{-3-n} d e^{-\frac {2 a}{b}} \sqrt {\frac {-1+c x}{1+c x}} (1+c x) (a+b \text {arccosh}(c x))^n \left (-\frac {(a+b \text {arccosh}(c x))^2}{b^2}\right )^{-n} \left (2^{2+n} e^{\frac {2 a}{b}} (a+b \text {arccosh}(c x)) \left (-\frac {(a+b \text {arccosh}(c x))^2}{b^2}\right )^n-b (1+n) \left (\frac {a}{b}+\text {arccosh}(c x)\right )^n \Gamma \left (1+n,-\frac {2 (a+b \text {arccosh}(c x))}{b}\right )+b e^{\frac {4 a}{b}} (1+n) \left (-\frac {a+b \text {arccosh}(c x)}{b}\right )^n \Gamma \left (1+n,\frac {2 (a+b \text {arccosh}(c x))}{b}\right )\right )}{b c (1+n) \sqrt {d-c^2 d x^2}} \] Input:

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

Output:

(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*ArcCosh[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)
 

Rubi [A] (verified)

Time = 0.82 (sec) , antiderivative size = 184, normalized size of antiderivative = 0.73, number of steps used = 6, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.192, Rules used = {6321, 3042, 25, 3793, 2009}

Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.

\(\displaystyle \int \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))^n \, dx\)

\(\Big \downarrow \) 6321

\(\displaystyle \frac {\sqrt {d-c^2 d x^2} \int (a+b \text {arccosh}(c x))^n \sinh ^2\left (\frac {a}{b}-\frac {a+b \text {arccosh}(c x)}{b}\right )d(a+b \text {arccosh}(c x))}{b c \sqrt {c x-1} \sqrt {c x+1}}\)

\(\Big \downarrow \) 3042

\(\displaystyle \frac {\sqrt {d-c^2 d x^2} \int -(a+b \text {arccosh}(c x))^n \sin \left (\frac {i a}{b}-\frac {i (a+b \text {arccosh}(c x))}{b}\right )^2d(a+b \text {arccosh}(c x))}{b c \sqrt {c x-1} \sqrt {c x+1}}\)

\(\Big \downarrow \) 25

\(\displaystyle -\frac {\sqrt {d-c^2 d x^2} \int (a+b \text {arccosh}(c x))^n \sin \left (\frac {i a}{b}-\frac {i (a+b \text {arccosh}(c x))}{b}\right )^2d(a+b \text {arccosh}(c x))}{b c \sqrt {c x-1} \sqrt {c x+1}}\)

\(\Big \downarrow \) 3793

\(\displaystyle -\frac {\sqrt {d-c^2 d x^2} \int \left (\frac {1}{2} (a+b \text {arccosh}(c x))^n-\frac {1}{2} (a+b \text {arccosh}(c x))^n \cosh \left (\frac {2 a}{b}-\frac {2 (a+b \text {arccosh}(c x))}{b}\right )\right )d(a+b \text {arccosh}(c x))}{b c \sqrt {c x-1} \sqrt {c x+1}}\)

\(\Big \downarrow \) 2009

\(\displaystyle \frac {\sqrt {d-c^2 d x^2} \left (-\frac {(a+b \text {arccosh}(c x))^{n+1}}{2 (n+1)}+b 2^{-n-3} e^{-\frac {2 a}{b}} (a+b \text {arccosh}(c x))^n \left (-\frac {a+b \text {arccosh}(c x)}{b}\right )^{-n} \Gamma \left (n+1,-\frac {2 (a+b \text {arccosh}(c x))}{b}\right )-b 2^{-n-3} e^{\frac {2 a}{b}} (a+b \text {arccosh}(c x))^n \left (\frac {a+b \text {arccosh}(c x)}{b}\right )^{-n} \Gamma \left (n+1,\frac {2 (a+b \text {arccosh}(c x))}{b}\right )\right )}{b c \sqrt {c x-1} \sqrt {c x+1}}\)

Input:

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

Output:

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

Defintions of rubi rules used

rule 25

Int[-(Fx_), x_Symbol] :> Simp[Identity[-1]   Int[Fx, x], x]
 

rule 2009

Int[u_, x_Symbol] :> Simp[IntSum[u, x], x] /; SumQ[u]
 

rule 3042

Int[u_, x_Symbol] :> Int[DeactivateTrig[u, x], x] /; FunctionOfTrigOfLinear 
Q[u, x]
 

rule 3793

Int[((c_.) + (d_.)*(x_))^(m_)*sin[(e_.) + (f_.)*(x_)]^(n_), x_Symbol] :> In 
t[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 6321

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

Maple [F]

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

Input:

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

Output:

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

Fricas [F]

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

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

Output:

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

Sympy [F]

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

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

Output:

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

Maxima [F]

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

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

Output:

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

Giac [F(-2)]

Exception generated. \[ \int \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))^n \, dx=\text {Exception raised: TypeError} \] Input:

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

Output:

Exception raised: TypeError >> an error occurred running a Giac command:IN 
PUT:sage2:=int(sage0,sageVARx):;OUTPUT:sym2poly/r2sym(const gen & e,const 
index_m & i,const vecteur & l) Error: Bad Argument Value
 

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

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

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

Output:

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