\(\int x \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))^n \, dx\) [319]

Optimal result

Integrand size = 27, antiderivative size = 379 \[ \int x \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))^n \, dx=\frac {3^{-1-n} e^{-\frac {3 a}{b}} \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))^n \left (-\frac {a+b \text {arccosh}(c x)}{b}\right )^{-n} \Gamma \left (1+n,-\frac {3 (a+b \text {arccosh}(c x))}{b}\right )}{8 c^2 \sqrt {-1+c x} \sqrt {1+c x}}-\frac {e^{-\frac {a}{b}} \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))^n \left (-\frac {a+b \text {arccosh}(c x)}{b}\right )^{-n} \Gamma \left (1+n,-\frac {a+b \text {arccosh}(c x)}{b}\right )}{8 c^2 \sqrt {-1+c x} \sqrt {1+c x}}+\frac {e^{a/b} \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))^n \left (\frac {a+b \text {arccosh}(c x)}{b}\right )^{-n} \Gamma \left (1+n,\frac {a+b \text {arccosh}(c x)}{b}\right )}{8 c^2 \sqrt {-1+c x} \sqrt {1+c x}}-\frac {3^{-1-n} e^{\frac {3 a}{b}} \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))^n \left (\frac {a+b \text {arccosh}(c x)}{b}\right )^{-n} \Gamma \left (1+n,\frac {3 (a+b \text {arccosh}(c x))}{b}\right )}{8 c^2 \sqrt {-1+c x} \sqrt {1+c x}} \] Output:

1/8*3^(-1-n)*(-c^2*d*x^2+d)^(1/2)*(a+b*arccosh(c*x))^n*GAMMA(1+n,(-3*a-3*b 
*arccosh(c*x))/b)/c^2/exp(3*a/b)/(c*x-1)^(1/2)/(c*x+1)^(1/2)/((-(a+b*arcco 
sh(c*x))/b)^n)-1/8*(-c^2*d*x^2+d)^(1/2)*(a+b*arccosh(c*x))^n*GAMMA(1+n,-(a 
+b*arccosh(c*x))/b)/c^2/exp(a/b)/(c*x-1)^(1/2)/(c*x+1)^(1/2)/((-(a+b*arcco 
sh(c*x))/b)^n)+1/8*exp(a/b)*(-c^2*d*x^2+d)^(1/2)*(a+b*arccosh(c*x))^n*GAMM 
A(1+n,(a+b*arccosh(c*x))/b)/c^2/(c*x-1)^(1/2)/(c*x+1)^(1/2)/(((a+b*arccosh 
(c*x))/b)^n)-1/8*3^(-1-n)*exp(3*a/b)*(-c^2*d*x^2+d)^(1/2)*(a+b*arccosh(c*x 
))^n*GAMMA(1+n,3*(a+b*arccosh(c*x))/b)/c^2/(c*x-1)^(1/2)/(c*x+1)^(1/2)/((( 
a+b*arccosh(c*x))/b)^n)
 

Mathematica [A] (warning: unable to verify)

Time = 0.92 (sec) , antiderivative size = 241, normalized size of antiderivative = 0.64 \[ \int x \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))^n \, dx=-\frac {d e^{-\frac {3 a}{b}} \sqrt {\frac {-1+c x}{1+c x}} (1+c x) (a+b \text {arccosh}(c x))^n \left (3 e^{\frac {4 a}{b}} \left (\frac {a}{b}+\text {arccosh}(c x)\right )^{-n} \Gamma \left (1+n,\frac {a}{b}+\text {arccosh}(c x)\right )+\left (-\frac {a+b \text {arccosh}(c x)}{b}\right )^{-n} \left (3^{-n} \Gamma \left (1+n,-\frac {3 (a+b \text {arccosh}(c x))}{b}\right )-3 e^{\frac {2 a}{b}} \Gamma \left (1+n,-\frac {a+b \text {arccosh}(c x)}{b}\right )-3^{-n} e^{\frac {6 a}{b}} \left (-\frac {a+b \text {arccosh}(c x)}{b}\right )^{2 n} \left (-\frac {(a+b \text {arccosh}(c x))^2}{b^2}\right )^{-n} \Gamma \left (1+n,\frac {3 (a+b \text {arccosh}(c x))}{b}\right )\right )\right )}{24 c^2 \sqrt {-d (-1+c x) (1+c x)}} \] Input:

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

Output:

-1/24*(d*Sqrt[(-1 + c*x)/(1 + c*x)]*(1 + c*x)*(a + b*ArcCosh[c*x])^n*((3*E 
^((4*a)/b)*Gamma[1 + n, a/b + ArcCosh[c*x]])/(a/b + ArcCosh[c*x])^n + (Gam 
ma[1 + n, (-3*(a + b*ArcCosh[c*x]))/b]/3^n - 3*E^((2*a)/b)*Gamma[1 + n, -( 
(a + b*ArcCosh[c*x])/b)] - (E^((6*a)/b)*(-((a + b*ArcCosh[c*x])/b))^(2*n)* 
Gamma[1 + n, (3*(a + b*ArcCosh[c*x]))/b])/(3^n*(-((a + b*ArcCosh[c*x])^2/b 
^2))^n))/(-((a + b*ArcCosh[c*x])/b))^n))/(c^2*E^((3*a)/b)*Sqrt[-(d*(-1 + c 
*x)*(1 + c*x))])
 

Rubi [A] (verified)

Time = 1.09 (sec) , antiderivative size = 279, normalized size of antiderivative = 0.74, number of steps used = 4, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.111, Rules used = {6367, 5971, 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.

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

\(\Big \downarrow \) 6367

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

\(\Big \downarrow \) 5971

\(\displaystyle \frac {\sqrt {d-c^2 d x^2} \int \left (\frac {1}{4} (a+b \text {arccosh}(c x))^n \cosh \left (\frac {3 a}{b}-\frac {3 (a+b \text {arccosh}(c x))}{b}\right )-\frac {1}{4} (a+b \text {arccosh}(c x))^n \cosh \left (\frac {a}{b}-\frac {a+b \text {arccosh}(c x)}{b}\right )\right )d(a+b \text {arccosh}(c x))}{b c^2 \sqrt {c x-1} \sqrt {c x+1}}\)

\(\Big \downarrow \) 2009

\(\displaystyle \frac {\sqrt {d-c^2 d x^2} \left (\frac {1}{8} b 3^{-n-1} e^{-\frac {3 a}{b}} (a+b \text {arccosh}(c x))^n \left (-\frac {a+b \text {arccosh}(c x)}{b}\right )^{-n} \Gamma \left (n+1,-\frac {3 (a+b \text {arccosh}(c x))}{b}\right )-\frac {1}{8} b e^{-\frac {a}{b}} (a+b \text {arccosh}(c x))^n \left (-\frac {a+b \text {arccosh}(c x)}{b}\right )^{-n} \Gamma \left (n+1,-\frac {a+b \text {arccosh}(c x)}{b}\right )+\frac {1}{8} b e^{a/b} (a+b \text {arccosh}(c x))^n \left (\frac {a+b \text {arccosh}(c x)}{b}\right )^{-n} \Gamma \left (n+1,\frac {a+b \text {arccosh}(c x)}{b}\right )-\frac {1}{8} b 3^{-n-1} e^{\frac {3 a}{b}} (a+b \text {arccosh}(c x))^n \left (\frac {a+b \text {arccosh}(c x)}{b}\right )^{-n} \Gamma \left (n+1,\frac {3 (a+b \text {arccosh}(c x))}{b}\right )\right )}{b c^2 \sqrt {c x-1} \sqrt {c x+1}}\)

Input:

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

Output:

(Sqrt[d - c^2*d*x^2]*((3^(-1 - n)*b*(a + b*ArcCosh[c*x])^n*Gamma[1 + n, (- 
3*(a + b*ArcCosh[c*x]))/b])/(8*E^((3*a)/b)*(-((a + b*ArcCosh[c*x])/b))^n) 
- (b*(a + b*ArcCosh[c*x])^n*Gamma[1 + n, -((a + b*ArcCosh[c*x])/b)])/(8*E^ 
(a/b)*(-((a + b*ArcCosh[c*x])/b))^n) + (b*E^(a/b)*(a + b*ArcCosh[c*x])^n*G 
amma[1 + n, (a + b*ArcCosh[c*x])/b])/(8*((a + b*ArcCosh[c*x])/b)^n) - (3^( 
-1 - n)*b*E^((3*a)/b)*(a + b*ArcCosh[c*x])^n*Gamma[1 + n, (3*(a + b*ArcCos 
h[c*x]))/b])/(8*((a + b*ArcCosh[c*x])/b)^n)))/(b*c^2*Sqrt[-1 + c*x]*Sqrt[1 
 + c*x])
 

Defintions of rubi rules used

rule 2009

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

rule 5971

Int[Cosh[(a_.) + (b_.)*(x_)]^(p_.)*((c_.) + (d_.)*(x_))^(m_.)*Sinh[(a_.) + 
(b_.)*(x_)]^(n_.), x_Symbol] :> Int[ExpandTrigReduce[(c + d*x)^m, Sinh[a + 
b*x]^n*Cosh[a + b*x]^p, x], x] /; FreeQ[{a, b, c, d, m}, x] && IGtQ[n, 0] & 
& IGtQ[p, 0]
 

rule 6367

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

Maple [F]

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

Input:

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

Output:

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

Fricas [F]

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

integrate(x*(-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, x)
 

Sympy [F]

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

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

Output:

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

Maxima [F]

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

integrate(x*(-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, x)
 

Giac [F(-2)]

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

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

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

Output:

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

Reduce [F]

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

int(x*(-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,x)