\(\int x (d-c^2 d x^2)^{3/2} (a+b \text {arccosh}(c x))^n \, dx\) [324]

Optimal result

Integrand size = 27, antiderivative size = 578 \[ \int x \left (d-c^2 d x^2\right )^{3/2} (a+b \text {arccosh}(c x))^n \, dx=-\frac {5^{-1-n} d e^{-\frac {5 a}{b}} \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))^n \left (-\frac {a+b \text {arccosh}(c x)}{b}\right )^{-n} \Gamma \left (1+n,-\frac {5 (a+b \text {arccosh}(c x))}{b}\right )}{32 c^2 \sqrt {-1+c x} \sqrt {1+c x}}+\frac {3^{-n} d e^{-\frac {3 a}{b}} \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))^n \left (-\frac {a+b \text {arccosh}(c x)}{b}\right )^{-n} \Gamma \left (1+n,-\frac {3 (a+b \text {arccosh}(c x))}{b}\right )}{32 c^2 \sqrt {-1+c x} \sqrt {1+c x}}-\frac {d e^{-\frac {a}{b}} \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))^n \left (-\frac {a+b \text {arccosh}(c x)}{b}\right )^{-n} \Gamma \left (1+n,-\frac {a+b \text {arccosh}(c x)}{b}\right )}{16 c^2 \sqrt {-1+c x} \sqrt {1+c x}}+\frac {d e^{a/b} \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))^n \left (\frac {a+b \text {arccosh}(c x)}{b}\right )^{-n} \Gamma \left (1+n,\frac {a+b \text {arccosh}(c x)}{b}\right )}{16 c^2 \sqrt {-1+c x} \sqrt {1+c x}}-\frac {3^{-n} d e^{\frac {3 a}{b}} \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))^n \left (\frac {a+b \text {arccosh}(c x)}{b}\right )^{-n} \Gamma \left (1+n,\frac {3 (a+b \text {arccosh}(c x))}{b}\right )}{32 c^2 \sqrt {-1+c x} \sqrt {1+c x}}+\frac {5^{-1-n} d e^{\frac {5 a}{b}} \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))^n \left (\frac {a+b \text {arccosh}(c x)}{b}\right )^{-n} \Gamma \left (1+n,\frac {5 (a+b \text {arccosh}(c x))}{b}\right )}{32 c^2 \sqrt {-1+c x} \sqrt {1+c x}} \] Output:

-1/32*5^(-1-n)*d*(-c^2*d*x^2+d)^(1/2)*(a+b*arccosh(c*x))^n*GAMMA(1+n,(-5*a 
-5*b*arccosh(c*x))/b)/c^2/exp(5*a/b)/(c*x-1)^(1/2)/(c*x+1)^(1/2)/((-(a+b*a 
rccosh(c*x))/b)^n)+1/32*d*(-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)/(3^n)/c^2/exp(3*a/b)/(c*x-1)^(1/2)/(c*x+1)^ 
(1/2)/((-(a+b*arccosh(c*x))/b)^n)-1/16*d*(-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*arccosh(c*x))/b)^n)+1/16*d*exp(a/b)*(-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/(c*x-1)^(1/2)/(c* 
x+1)^(1/2)/(((a+b*arccosh(c*x))/b)^n)-1/32*d*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)/(3^n)/c^2/(c*x-1 
)^(1/2)/(c*x+1)^(1/2)/(((a+b*arccosh(c*x))/b)^n)+1/32*5^(-1-n)*d*exp(5*a/b 
)*(-c^2*d*x^2+d)^(1/2)*(a+b*arccosh(c*x))^n*GAMMA(1+n,5*(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 = 1.44 (sec) , antiderivative size = 500, normalized size of antiderivative = 0.87 \[ \int x \left (d-c^2 d x^2\right )^{3/2} (a+b \text {arccosh}(c x))^n \, dx=-\frac {15^{-1-n} d^2 e^{-\frac {5 a}{b}} \sqrt {\frac {-1+c x}{1+c x}} (1+c x) (a+b \text {arccosh}(c x))^n \left (-\frac {(a+b \text {arccosh}(c x))^2}{b^2}\right )^{-3 n} \left (2\ 15^{1+n} e^{\frac {6 a}{b}} \left (-\frac {a+b \text {arccosh}(c x)}{b}\right )^n \left (-\frac {(a+b \text {arccosh}(c x))^2}{b^2}\right )^{2 n} \Gamma \left (1+n,\frac {a}{b}+\text {arccosh}(c x)\right )+\left (\frac {a}{b}+\text {arccosh}(c x)\right )^n \left (-3^{1+n} \left (-\frac {(a+b \text {arccosh}(c x))^2}{b^2}\right )^{2 n} \Gamma \left (1+n,-\frac {5 (a+b \text {arccosh}(c x))}{b}\right )+3\ 5^{1+n} e^{\frac {2 a}{b}} \left (-\frac {(a+b \text {arccosh}(c x))^2}{b^2}\right )^{2 n} \Gamma \left (1+n,-\frac {3 (a+b \text {arccosh}(c x))}{b}\right )-2\ 15^{1+n} e^{\frac {4 a}{b}} \left (-\frac {(a+b \text {arccosh}(c x))^2}{b^2}\right )^{2 n} \Gamma \left (1+n,-\frac {a+b \text {arccosh}(c x)}{b}\right )+5^{1+n} e^{\frac {8 a}{b}} \left (\frac {a}{b}+\text {arccosh}(c x)\right )^n \left (-\frac {a+b \text {arccosh}(c x)}{b}\right )^{3 n} \Gamma \left (1+n,\frac {3 (a+b \text {arccosh}(c x))}{b}\right )-4\ 5^{1+n} e^{\frac {8 a}{b}} \left (-\frac {a+b \text {arccosh}(c x)}{b}\right )^{2 n} \left (-\frac {(a+b \text {arccosh}(c x))^2}{b^2}\right )^n \Gamma \left (1+n,\frac {3 (a+b \text {arccosh}(c x))}{b}\right )+3^{1+n} e^{\frac {10 a}{b}} \left (\frac {a}{b}+\text {arccosh}(c x)\right )^n \left (-\frac {a+b \text {arccosh}(c x)}{b}\right )^{3 n} \Gamma \left (1+n,\frac {5 (a+b \text {arccosh}(c x))}{b}\right )\right )\right )}{32 c^2 \sqrt {d-c^2 d x^2}} \] Input:

Integrate[x*(d - c^2*d*x^2)^(3/2)*(a + b*ArcCosh[c*x])^n,x]
 

Output:

-1/32*(15^(-1 - n)*d^2*Sqrt[(-1 + c*x)/(1 + c*x)]*(1 + c*x)*(a + b*ArcCosh 
[c*x])^n*(2*15^(1 + n)*E^((6*a)/b)*(-((a + b*ArcCosh[c*x])/b))^n*(-((a + b 
*ArcCosh[c*x])^2/b^2))^(2*n)*Gamma[1 + n, a/b + ArcCosh[c*x]] + (a/b + Arc 
Cosh[c*x])^n*(-(3^(1 + n)*(-((a + b*ArcCosh[c*x])^2/b^2))^(2*n)*Gamma[1 + 
n, (-5*(a + b*ArcCosh[c*x]))/b]) + 3*5^(1 + n)*E^((2*a)/b)*(-((a + b*ArcCo 
sh[c*x])^2/b^2))^(2*n)*Gamma[1 + n, (-3*(a + b*ArcCosh[c*x]))/b] - 2*15^(1 
 + n)*E^((4*a)/b)*(-((a + b*ArcCosh[c*x])^2/b^2))^(2*n)*Gamma[1 + n, -((a 
+ b*ArcCosh[c*x])/b)] + 5^(1 + n)*E^((8*a)/b)*(a/b + ArcCosh[c*x])^n*(-((a 
 + b*ArcCosh[c*x])/b))^(3*n)*Gamma[1 + n, (3*(a + b*ArcCosh[c*x]))/b] - 4* 
5^(1 + n)*E^((8*a)/b)*(-((a + b*ArcCosh[c*x])/b))^(2*n)*(-((a + b*ArcCosh[ 
c*x])^2/b^2))^n*Gamma[1 + n, (3*(a + b*ArcCosh[c*x]))/b] + 3^(1 + n)*E^((1 
0*a)/b)*(a/b + ArcCosh[c*x])^n*(-((a + b*ArcCosh[c*x])/b))^(3*n)*Gamma[1 + 
 n, (5*(a + b*ArcCosh[c*x]))/b])))/(c^2*E^((5*a)/b)*Sqrt[d - c^2*d*x^2]*(- 
((a + b*ArcCosh[c*x])^2/b^2))^(3*n))
 

Rubi [A] (verified)

Time = 0.80 (sec) , antiderivative size = 404, normalized size of antiderivative = 0.70, 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.

\(\displaystyle \int x \left (d-c^2 d x^2\right )^{3/2} (a+b \text {arccosh}(c x))^n \, dx\)

\(\Big \downarrow \) 6367

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

\(\Big \downarrow \) 5971

\(\displaystyle -\frac {d \sqrt {d-c^2 d x^2} \int \left (\frac {1}{16} \cosh \left (\frac {5 a}{b}-\frac {5 (a+b \text {arccosh}(c x))}{b}\right ) (a+b \text {arccosh}(c x))^n-\frac {3}{16} \cosh \left (\frac {3 a}{b}-\frac {3 (a+b \text {arccosh}(c x))}{b}\right ) (a+b \text {arccosh}(c x))^n+\frac {1}{8} \cosh \left (\frac {a}{b}-\frac {a+b \text {arccosh}(c x)}{b}\right ) (a+b \text {arccosh}(c x))^n\right )d(a+b \text {arccosh}(c x))}{b c^2 \sqrt {c x-1} \sqrt {c x+1}}\)

\(\Big \downarrow \) 2009

\(\displaystyle -\frac {d \sqrt {d-c^2 d x^2} \left (\frac {1}{32} b 5^{-n-1} e^{-\frac {5 a}{b}} (a+b \text {arccosh}(c x))^n \left (-\frac {a+b \text {arccosh}(c x)}{b}\right )^{-n} \Gamma \left (n+1,-\frac {5 (a+b \text {arccosh}(c x))}{b}\right )-\frac {1}{32} b 3^{-n} e^{-\frac {3 a}{b}} (a+b \text {arccosh}(c x))^n \left (-\frac {a+b \text {arccosh}(c x)}{b}\right )^{-n} \Gamma \left (n+1,-\frac {3 (a+b \text {arccosh}(c x))}{b}\right )+\frac {1}{16} b e^{-\frac {a}{b}} (a+b \text {arccosh}(c x))^n \left (-\frac {a+b \text {arccosh}(c x)}{b}\right )^{-n} \Gamma \left (n+1,-\frac {a+b \text {arccosh}(c x)}{b}\right )-\frac {1}{16} b e^{a/b} (a+b \text {arccosh}(c x))^n \left (\frac {a+b \text {arccosh}(c x)}{b}\right )^{-n} \Gamma \left (n+1,\frac {a+b \text {arccosh}(c x)}{b}\right )+\frac {1}{32} b 3^{-n} e^{\frac {3 a}{b}} (a+b \text {arccosh}(c x))^n \left (\frac {a+b \text {arccosh}(c x)}{b}\right )^{-n} \Gamma \left (n+1,\frac {3 (a+b \text {arccosh}(c x))}{b}\right )-\frac {1}{32} b 5^{-n-1} e^{\frac {5 a}{b}} (a+b \text {arccosh}(c x))^n \left (\frac {a+b \text {arccosh}(c x)}{b}\right )^{-n} \Gamma \left (n+1,\frac {5 (a+b \text {arccosh}(c x))}{b}\right )\right )}{b c^2 \sqrt {c x-1} \sqrt {c x+1}}\)

Input:

Int[x*(d - c^2*d*x^2)^(3/2)*(a + b*ArcCosh[c*x])^n,x]
 

Output:

-((d*Sqrt[d - c^2*d*x^2]*((5^(-1 - n)*b*(a + b*ArcCosh[c*x])^n*Gamma[1 + n 
, (-5*(a + b*ArcCosh[c*x]))/b])/(32*E^((5*a)/b)*(-((a + b*ArcCosh[c*x])/b) 
)^n) - (b*(a + b*ArcCosh[c*x])^n*Gamma[1 + n, (-3*(a + b*ArcCosh[c*x]))/b] 
)/(32*3^n*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)])/(16*E^(a/b)*(-((a + b*ArcC 
osh[c*x])/b))^n) - (b*E^(a/b)*(a + b*ArcCosh[c*x])^n*Gamma[1 + n, (a + b*A 
rcCosh[c*x])/b])/(16*((a + b*ArcCosh[c*x])/b)^n) + (b*E^((3*a)/b)*(a + b*A 
rcCosh[c*x])^n*Gamma[1 + n, (3*(a + b*ArcCosh[c*x]))/b])/(32*3^n*((a + b*A 
rcCosh[c*x])/b)^n) - (5^(-1 - n)*b*E^((5*a)/b)*(a + b*ArcCosh[c*x])^n*Gamm 
a[1 + n, (5*(a + b*ArcCosh[c*x]))/b])/(32*((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]

\[\int x \left (-c^{2} d \,x^{2}+d \right )^{\frac {3}{2}} \left (a +b \,\operatorname {arccosh}\left (c x \right )\right )^{n}d x\]

Input:

int(x*(-c^2*d*x^2+d)^(3/2)*(a+b*arccosh(c*x))^n,x)
 

Output:

int(x*(-c^2*d*x^2+d)^(3/2)*(a+b*arccosh(c*x))^n,x)
 

Fricas [F]

\[ \int x \left (d-c^2 d x^2\right )^{3/2} (a+b \text {arccosh}(c x))^n \, dx=\int { {\left (-c^{2} d x^{2} + d\right )}^{\frac {3}{2}} {\left (b \operatorname {arcosh}\left (c x\right ) + a\right )}^{n} x \,d x } \] Input:

integrate(x*(-c^2*d*x^2+d)^(3/2)*(a+b*arccosh(c*x))^n,x, algorithm="fricas 
")
 

Output:

integral(-(c^2*d*x^3 - d*x)*sqrt(-c^2*d*x^2 + d)*(b*arccosh(c*x) + a)^n, x 
)
 

Sympy [F(-1)]

Timed out. \[ \int x \left (d-c^2 d x^2\right )^{3/2} (a+b \text {arccosh}(c x))^n \, dx=\text {Timed out} \] Input:

integrate(x*(-c**2*d*x**2+d)**(3/2)*(a+b*acosh(c*x))**n,x)
 

Output:

Timed out
 

Maxima [F]

\[ \int x \left (d-c^2 d x^2\right )^{3/2} (a+b \text {arccosh}(c x))^n \, dx=\int { {\left (-c^{2} d x^{2} + d\right )}^{\frac {3}{2}} {\left (b \operatorname {arcosh}\left (c x\right ) + a\right )}^{n} x \,d x } \] Input:

integrate(x*(-c^2*d*x^2+d)^(3/2)*(a+b*arccosh(c*x))^n,x, algorithm="maxima 
")
 

Output:

integrate((-c^2*d*x^2 + d)^(3/2)*(b*arccosh(c*x) + a)^n*x, x)
 

Giac [F(-2)]

Exception generated. \[ \int x \left (d-c^2 d x^2\right )^{3/2} (a+b \text {arccosh}(c x))^n \, dx=\text {Exception raised: TypeError} \] Input:

integrate(x*(-c^2*d*x^2+d)^(3/2)*(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. \[ \int x \left (d-c^2 d x^2\right )^{3/2} (a+b \text {arccosh}(c x))^n \, dx=\int x\,{\left (a+b\,\mathrm {acosh}\left (c\,x\right )\right )}^n\,{\left (d-c^2\,d\,x^2\right )}^{3/2} \,d x \] Input:

int(x*(a + b*acosh(c*x))^n*(d - c^2*d*x^2)^(3/2),x)
 

Output:

int(x*(a + b*acosh(c*x))^n*(d - c^2*d*x^2)^(3/2), x)
 

Reduce [F]

\[ \int x \left (d-c^2 d x^2\right )^{3/2} (a+b \text {arccosh}(c x))^n \, dx=\sqrt {d}\, d \left (-\left (\int \left (\mathit {acosh} \left (c x \right ) b +a \right )^{n} \sqrt {-c^{2} x^{2}+1}\, x^{3}d x \right ) c^{2}+\int \left (\mathit {acosh} \left (c x \right ) b +a \right )^{n} \sqrt {-c^{2} x^{2}+1}\, x d x \right ) \] Input:

int(x*(-c^2*d*x^2+d)^(3/2)*(a+b*acosh(c*x))^n,x)
 

Output:

sqrt(d)*d*( - int((acosh(c*x)*b + a)**n*sqrt( - c**2*x**2 + 1)*x**3,x)*c** 
2 + int((acosh(c*x)*b + a)**n*sqrt( - c**2*x**2 + 1)*x,x))