\(\int (d-c^2 d x^2)^{5/2} (a+b \text {arccosh}(c x))^n \, dx\) [330]

Optimal result

Integrand size = 26, antiderivative size = 674 \[ \int \left (d-c^2 d x^2\right )^{5/2} (a+b \text {arccosh}(c x))^n \, dx=-\frac {5 d^2 \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))^{1+n}}{16 b c (1+n) \sqrt {-1+c x} \sqrt {1+c x}}+\frac {2^{-7-n} 3^{-1-n} d^2 e^{-\frac {6 a}{b}} \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))^n \left (-\frac {a+b \text {arccosh}(c x)}{b}\right )^{-n} \Gamma \left (1+n,-\frac {6 (a+b \text {arccosh}(c x))}{b}\right )}{c \sqrt {-1+c x} \sqrt {1+c x}}-\frac {3\ 2^{-7-2 n} d^2 e^{-\frac {4 a}{b}} \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))^n \left (-\frac {a+b \text {arccosh}(c x)}{b}\right )^{-n} \Gamma \left (1+n,-\frac {4 (a+b \text {arccosh}(c x))}{b}\right )}{c \sqrt {-1+c x} \sqrt {1+c x}}+\frac {15\ 2^{-7-n} d^2 e^{-\frac {2 a}{b}} \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))^n \left (-\frac {a+b \text {arccosh}(c x)}{b}\right )^{-n} \Gamma \left (1+n,-\frac {2 (a+b \text {arccosh}(c x))}{b}\right )}{c \sqrt {-1+c x} \sqrt {1+c x}}-\frac {15\ 2^{-7-n} d^2 e^{\frac {2 a}{b}} \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))^n \left (\frac {a+b \text {arccosh}(c x)}{b}\right )^{-n} \Gamma \left (1+n,\frac {2 (a+b \text {arccosh}(c x))}{b}\right )}{c \sqrt {-1+c x} \sqrt {1+c x}}+\frac {3\ 2^{-7-2 n} d^2 e^{\frac {4 a}{b}} \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))^n \left (\frac {a+b \text {arccosh}(c x)}{b}\right )^{-n} \Gamma \left (1+n,\frac {4 (a+b \text {arccosh}(c x))}{b}\right )}{c \sqrt {-1+c x} \sqrt {1+c x}}-\frac {2^{-7-n} 3^{-1-n} d^2 e^{\frac {6 a}{b}} \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))^n \left (\frac {a+b \text {arccosh}(c x)}{b}\right )^{-n} \Gamma \left (1+n,\frac {6 (a+b \text {arccosh}(c x))}{b}\right )}{c \sqrt {-1+c x} \sqrt {1+c x}} \] Output:

-5/16*d^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^(-7-n)*3^(-1-n)*d^2*(-c^2*d*x^2+d)^(1/2)*(a+b*arccos 
h(c*x))^n*GAMMA(1+n,(-6*a-6*b*arccosh(c*x))/b)/c/exp(6*a/b)/(c*x-1)^(1/2)/ 
(c*x+1)^(1/2)/((-(a+b*arccosh(c*x))/b)^n)-3*2^(-7-2*n)*d^2*(-c^2*d*x^2+d)^ 
(1/2)*(a+b*arccosh(c*x))^n*GAMMA(1+n,(-4*a-4*b*arccosh(c*x))/b)/c/exp(4*a/ 
b)/(c*x-1)^(1/2)/(c*x+1)^(1/2)/((-(a+b*arccosh(c*x))/b)^n)+15*2^(-7-n)*d^2 
*(-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) 
-15*2^(-7-n)*d^2*exp(2*a/b)*(-c^2*d*x^2+d)^(1/2)*(a+b*arccosh(c*x))^n*GAMM 
A(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)+3*2^(-7-2*n)*d^2*exp(4*a/b)*(-c^2*d*x^2+d)^(1/2)*(a+b*arccosh 
(c*x))^n*GAMMA(1+n,4*(a+b*arccosh(c*x))/b)/c/(c*x-1)^(1/2)/(c*x+1)^(1/2)/( 
((a+b*arccosh(c*x))/b)^n)-2^(-7-n)*3^(-1-n)*d^2*exp(6*a/b)*(-c^2*d*x^2+d)^ 
(1/2)*(a+b*arccosh(c*x))^n*GAMMA(1+n,6*(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 = 3.48 (sec) , antiderivative size = 538, normalized size of antiderivative = 0.80 \[ \int \left (d-c^2 d x^2\right )^{5/2} (a+b \text {arccosh}(c x))^n \, dx=\frac {2^{-7-2 n} 3^{-1-n} d^3 e^{-\frac {6 a}{b}} \sqrt {\frac {-1+c x}{1+c x}} (1+c x) (a+b \text {arccosh}(c x))^n \left (-\frac {(a+b \text {arccosh}(c x))^2}{b^2}\right )^{-2 n} \left (-2^n b (1+n) \left (\frac {a}{b}+\text {arccosh}(c x)\right )^{2 n} \left (-\frac {a+b \text {arccosh}(c x)}{b}\right )^n \Gamma \left (1+n,-\frac {6 (a+b \text {arccosh}(c x))}{b}\right )+3^{2+n} b e^{\frac {2 a}{b}} (1+n) \left (\frac {a}{b}+\text {arccosh}(c x)\right )^{2 n} \left (-\frac {a+b \text {arccosh}(c x)}{b}\right )^n \Gamma \left (1+n,-\frac {4 (a+b \text {arccosh}(c x))}{b}\right )-5\ 2^n 3^{2+n} b e^{\frac {4 a}{b}} (1+n) \left (\frac {a}{b}+\text {arccosh}(c x)\right )^n \left (-\frac {(a+b \text {arccosh}(c x))^2}{b^2}\right )^n \Gamma \left (1+n,-\frac {2 (a+b \text {arccosh}(c x))}{b}\right )+5\ 2^n 3^{2+n} b e^{\frac {8 a}{b}} (1+n) \left (-\frac {a+b \text {arccosh}(c x)}{b}\right )^n \left (-\frac {(a+b \text {arccosh}(c x))^2}{b^2}\right )^n \Gamma \left (1+n,\frac {2 (a+b \text {arccosh}(c x))}{b}\right )-3^{2+n} b e^{\frac {10 a}{b}} (1+n) \left (\frac {a}{b}+\text {arccosh}(c x)\right )^n \left (-\frac {a+b \text {arccosh}(c x)}{b}\right )^{2 n} \Gamma \left (1+n,\frac {4 (a+b \text {arccosh}(c x))}{b}\right )+2^n e^{\frac {6 a}{b}} \left (5\ 2^{3+n} 3^{1+n} (a+b \text {arccosh}(c x)) \left (-\frac {(a+b \text {arccosh}(c x))^2}{b^2}\right )^{2 n}+b e^{\frac {6 a}{b}} (1+n) \left (\frac {a}{b}+\text {arccosh}(c x)\right )^n \left (-\frac {a+b \text {arccosh}(c x)}{b}\right )^{2 n} \Gamma \left (1+n,\frac {6 (a+b \text {arccosh}(c x))}{b}\right )\right )\right )}{b c (1+n) \sqrt {d-c^2 d x^2}} \] Input:

Integrate[(d - c^2*d*x^2)^(5/2)*(a + b*ArcCosh[c*x])^n,x]
 

Output:

(2^(-7 - 2*n)*3^(-1 - n)*d^3*Sqrt[(-1 + c*x)/(1 + c*x)]*(1 + c*x)*(a + b*A 
rcCosh[c*x])^n*(-(2^n*b*(1 + n)*(a/b + ArcCosh[c*x])^(2*n)*(-((a + b*ArcCo 
sh[c*x])/b))^n*Gamma[1 + n, (-6*(a + b*ArcCosh[c*x]))/b]) + 3^(2 + n)*b*E^ 
((2*a)/b)*(1 + n)*(a/b + ArcCosh[c*x])^(2*n)*(-((a + b*ArcCosh[c*x])/b))^n 
*Gamma[1 + n, (-4*(a + b*ArcCosh[c*x]))/b] - 5*2^n*3^(2 + n)*b*E^((4*a)/b) 
*(1 + n)*(a/b + ArcCosh[c*x])^n*(-((a + b*ArcCosh[c*x])^2/b^2))^n*Gamma[1 
+ n, (-2*(a + b*ArcCosh[c*x]))/b] + 5*2^n*3^(2 + n)*b*E^((8*a)/b)*(1 + n)* 
(-((a + b*ArcCosh[c*x])/b))^n*(-((a + b*ArcCosh[c*x])^2/b^2))^n*Gamma[1 + 
n, (2*(a + b*ArcCosh[c*x]))/b] - 3^(2 + n)*b*E^((10*a)/b)*(1 + n)*(a/b + A 
rcCosh[c*x])^n*(-((a + b*ArcCosh[c*x])/b))^(2*n)*Gamma[1 + n, (4*(a + b*Ar 
cCosh[c*x]))/b] + 2^n*E^((6*a)/b)*(5*2^(3 + n)*3^(1 + n)*(a + b*ArcCosh[c* 
x])*(-((a + b*ArcCosh[c*x])^2/b^2))^(2*n) + b*E^((6*a)/b)*(1 + n)*(a/b + A 
rcCosh[c*x])^n*(-((a + b*ArcCosh[c*x])/b))^(2*n)*Gamma[1 + n, (6*(a + b*Ar 
cCosh[c*x]))/b])))/(b*c*E^((6*a)/b)*(1 + n)*Sqrt[d - c^2*d*x^2]*(-((a + b* 
ArcCosh[c*x])^2/b^2))^(2*n))
 

Rubi [A] (verified)

Time = 0.80 (sec) , antiderivative size = 447, normalized size of antiderivative = 0.66, 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.

\(\displaystyle \int \left (d-c^2 d x^2\right )^{5/2} (a+b \text {arccosh}(c x))^n \, dx\)

\(\Big \downarrow \) 6321

\(\displaystyle \frac {d^2 \sqrt {d-c^2 d x^2} \int (a+b \text {arccosh}(c x))^n \sinh ^6\left (\frac {a}{b}-\frac {a+b \text {arccosh}(c x)}{b}\right )d(a+b \text {arccosh}(c x))}{b c \sqrt {c x-1} \sqrt {c x+1}}\)

\(\Big \downarrow \) 3042

\(\displaystyle \frac {d^2 \sqrt {d-c^2 d x^2} \int -(a+b \text {arccosh}(c x))^n \sin \left (\frac {i a}{b}-\frac {i (a+b \text {arccosh}(c x))}{b}\right )^6d(a+b \text {arccosh}(c x))}{b c \sqrt {c x-1} \sqrt {c x+1}}\)

\(\Big \downarrow \) 25

\(\displaystyle -\frac {d^2 \sqrt {d-c^2 d x^2} \int (a+b \text {arccosh}(c x))^n \sin \left (\frac {i a}{b}-\frac {i (a+b \text {arccosh}(c x))}{b}\right )^6d(a+b \text {arccosh}(c x))}{b c \sqrt {c x-1} \sqrt {c x+1}}\)

\(\Big \downarrow \) 3793

\(\displaystyle -\frac {d^2 \sqrt {d-c^2 d x^2} \int \left (-\frac {1}{32} \cosh \left (\frac {6 a}{b}-\frac {6 (a+b \text {arccosh}(c x))}{b}\right ) (a+b \text {arccosh}(c x))^n+\frac {3}{16} \cosh \left (\frac {4 a}{b}-\frac {4 (a+b \text {arccosh}(c x))}{b}\right ) (a+b \text {arccosh}(c x))^n-\frac {15}{32} \cosh \left (\frac {2 a}{b}-\frac {2 (a+b \text {arccosh}(c x))}{b}\right ) (a+b \text {arccosh}(c x))^n+\frac {5}{16} (a+b \text {arccosh}(c x))^n\right )d(a+b \text {arccosh}(c x))}{b c \sqrt {c x-1} \sqrt {c x+1}}\)

\(\Big \downarrow \) 2009

\(\displaystyle \frac {d^2 \sqrt {d-c^2 d x^2} \left (-\frac {5 (a+b \text {arccosh}(c x))^{n+1}}{16 (n+1)}+b 2^{-n-7} 3^{-n-1} e^{-\frac {6 a}{b}} (a+b \text {arccosh}(c x))^n \left (-\frac {a+b \text {arccosh}(c x)}{b}\right )^{-n} \Gamma \left (n+1,-\frac {6 (a+b \text {arccosh}(c x))}{b}\right )-3 b 2^{-2 n-7} e^{-\frac {4 a}{b}} (a+b \text {arccosh}(c x))^n \left (-\frac {a+b \text {arccosh}(c x)}{b}\right )^{-n} \Gamma \left (n+1,-\frac {4 (a+b \text {arccosh}(c x))}{b}\right )+15 b 2^{-n-7} e^{-\frac {2 a}{b}} (a+b \text {arccosh}(c x))^n \left (-\frac {a+b \text {arccosh}(c x)}{b}\right )^{-n} \Gamma \left (n+1,-\frac {2 (a+b \text {arccosh}(c x))}{b}\right )-15 b 2^{-n-7} e^{\frac {2 a}{b}} (a+b \text {arccosh}(c x))^n \left (\frac {a+b \text {arccosh}(c x)}{b}\right )^{-n} \Gamma \left (n+1,\frac {2 (a+b \text {arccosh}(c x))}{b}\right )+3 b 2^{-2 n-7} e^{\frac {4 a}{b}} (a+b \text {arccosh}(c x))^n \left (\frac {a+b \text {arccosh}(c x)}{b}\right )^{-n} \Gamma \left (n+1,\frac {4 (a+b \text {arccosh}(c x))}{b}\right )-b 2^{-n-7} 3^{-n-1} e^{\frac {6 a}{b}} (a+b \text {arccosh}(c x))^n \left (\frac {a+b \text {arccosh}(c x)}{b}\right )^{-n} \Gamma \left (n+1,\frac {6 (a+b \text {arccosh}(c x))}{b}\right )\right )}{b c \sqrt {c x-1} \sqrt {c x+1}}\)

Input:

Int[(d - c^2*d*x^2)^(5/2)*(a + b*ArcCosh[c*x])^n,x]
 

Output:

(d^2*Sqrt[d - c^2*d*x^2]*((-5*(a + b*ArcCosh[c*x])^(1 + n))/(16*(1 + n)) + 
 (2^(-7 - n)*3^(-1 - n)*b*(a + b*ArcCosh[c*x])^n*Gamma[1 + n, (-6*(a + b*A 
rcCosh[c*x]))/b])/(E^((6*a)/b)*(-((a + b*ArcCosh[c*x])/b))^n) - (3*2^(-7 - 
 2*n)*b*(a + b*ArcCosh[c*x])^n*Gamma[1 + n, (-4*(a + b*ArcCosh[c*x]))/b])/ 
(E^((4*a)/b)*(-((a + b*ArcCosh[c*x])/b))^n) + (15*2^(-7 - n)*b*(a + b*ArcC 
osh[c*x])^n*Gamma[1 + n, (-2*(a + b*ArcCosh[c*x]))/b])/(E^((2*a)/b)*(-((a 
+ b*ArcCosh[c*x])/b))^n) - (15*2^(-7 - n)*b*E^((2*a)/b)*(a + b*ArcCosh[c*x 
])^n*Gamma[1 + n, (2*(a + b*ArcCosh[c*x]))/b])/((a + b*ArcCosh[c*x])/b)^n 
+ (3*2^(-7 - 2*n)*b*E^((4*a)/b)*(a + b*ArcCosh[c*x])^n*Gamma[1 + n, (4*(a 
+ b*ArcCosh[c*x]))/b])/((a + b*ArcCosh[c*x])/b)^n - (2^(-7 - n)*3^(-1 - n) 
*b*E^((6*a)/b)*(a + b*ArcCosh[c*x])^n*Gamma[1 + n, (6*(a + b*ArcCosh[c*x]) 
)/b])/((a + b*ArcCosh[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]

\[\int \left (-c^{2} d \,x^{2}+d \right )^{\frac {5}{2}} \left (a +b \,\operatorname {arccosh}\left (c x \right )\right )^{n}d x\]

Input:

int((-c^2*d*x^2+d)^(5/2)*(a+b*arccosh(c*x))^n,x)
 

Output:

int((-c^2*d*x^2+d)^(5/2)*(a+b*arccosh(c*x))^n,x)
 

Fricas [F]

\[ \int \left (d-c^2 d x^2\right )^{5/2} (a+b \text {arccosh}(c x))^n \, dx=\int { {\left (-c^{2} d x^{2} + d\right )}^{\frac {5}{2}} {\left (b \operatorname {arcosh}\left (c x\right ) + a\right )}^{n} \,d x } \] Input:

integrate((-c^2*d*x^2+d)^(5/2)*(a+b*arccosh(c*x))^n,x, algorithm="fricas")
 

Output:

integral((c^4*d^2*x^4 - 2*c^2*d^2*x^2 + d^2)*sqrt(-c^2*d*x^2 + d)*(b*arcco 
sh(c*x) + a)^n, x)
 

Sympy [F(-1)]

Timed out. \[ \int \left (d-c^2 d x^2\right )^{5/2} (a+b \text {arccosh}(c x))^n \, dx=\text {Timed out} \] Input:

integrate((-c**2*d*x**2+d)**(5/2)*(a+b*acosh(c*x))**n,x)
 

Output:

Timed out
 

Maxima [F]

\[ \int \left (d-c^2 d x^2\right )^{5/2} (a+b \text {arccosh}(c x))^n \, dx=\int { {\left (-c^{2} d x^{2} + d\right )}^{\frac {5}{2}} {\left (b \operatorname {arcosh}\left (c x\right ) + a\right )}^{n} \,d x } \] Input:

integrate((-c^2*d*x^2+d)^(5/2)*(a+b*arccosh(c*x))^n,x, algorithm="maxima")
 

Output:

integrate((-c^2*d*x^2 + d)^(5/2)*(b*arccosh(c*x) + a)^n, x)
 

Giac [F(-2)]

Exception generated. \[ \int \left (d-c^2 d x^2\right )^{5/2} (a+b \text {arccosh}(c x))^n \, dx=\text {Exception raised: TypeError} \] Input:

integrate((-c^2*d*x^2+d)^(5/2)*(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. \[ \int \left (d-c^2 d x^2\right )^{5/2} (a+b \text {arccosh}(c x))^n \, dx=\int {\left (a+b\,\mathrm {acosh}\left (c\,x\right )\right )}^n\,{\left (d-c^2\,d\,x^2\right )}^{5/2} \,d x \] Input:

int((a + b*acosh(c*x))^n*(d - c^2*d*x^2)^(5/2),x)
 

Output:

int((a + b*acosh(c*x))^n*(d - c^2*d*x^2)^(5/2), x)
 

Reduce [F]

\[ \int \left (d-c^2 d x^2\right )^{5/2} (a+b \text {arccosh}(c x))^n \, dx=\sqrt {d}\, d^{2} \left (\left (\int \left (\mathit {acosh} \left (c x \right ) b +a \right )^{n} \sqrt {-c^{2} x^{2}+1}\, x^{4}d x \right ) c^{4}-2 \left (\int \left (\mathit {acosh} \left (c x \right ) b +a \right )^{n} \sqrt {-c^{2} x^{2}+1}\, x^{2}d x \right ) c^{2}+\int \left (\mathit {acosh} \left (c x \right ) b +a \right )^{n} \sqrt {-c^{2} x^{2}+1}d x \right ) \] Input:

int((-c^2*d*x^2+d)^(5/2)*(a+b*acosh(c*x))^n,x)
 

Output:

sqrt(d)*d**2*(int((acosh(c*x)*b + a)**n*sqrt( - c**2*x**2 + 1)*x**4,x)*c** 
4 - 2*int((acosh(c*x)*b + a)**n*sqrt( - c**2*x**2 + 1)*x**2,x)*c**2 + int( 
(acosh(c*x)*b + a)**n*sqrt( - c**2*x**2 + 1),x))