\(\int \frac {x^3 (a+b \text {arccosh}(c x))^n}{\sqrt {d-c^2 d x^2}} \, dx\) [339]

Optimal result

Integrand size = 29, antiderivative size = 379 \[ \int \frac {x^3 (a+b \text {arccosh}(c x))^n}{\sqrt {d-c^2 d x^2}} \, dx=\frac {3^{-1-n} e^{-\frac {3 a}{b}} \sqrt {-1+c x} \sqrt {1+c x} (a+b \text {arccosh}(c x))^n \left (-\frac {a+b \text {arccosh}(c x)}{b}\right )^{-n} \Gamma \left (1+n,-\frac {3 (a+b \text {arccosh}(c x))}{b}\right )}{8 c^4 \sqrt {d-c^2 d x^2}}+\frac {3 e^{-\frac {a}{b}} \sqrt {-1+c x} \sqrt {1+c x} (a+b \text {arccosh}(c x))^n \left (-\frac {a+b \text {arccosh}(c x)}{b}\right )^{-n} \Gamma \left (1+n,-\frac {a+b \text {arccosh}(c x)}{b}\right )}{8 c^4 \sqrt {d-c^2 d x^2}}-\frac {3 e^{a/b} \sqrt {-1+c x} \sqrt {1+c x} (a+b \text {arccosh}(c x))^n \left (\frac {a+b \text {arccosh}(c x)}{b}\right )^{-n} \Gamma \left (1+n,\frac {a+b \text {arccosh}(c x)}{b}\right )}{8 c^4 \sqrt {d-c^2 d x^2}}-\frac {3^{-1-n} e^{\frac {3 a}{b}} \sqrt {-1+c x} \sqrt {1+c x} (a+b \text {arccosh}(c x))^n \left (\frac {a+b \text {arccosh}(c x)}{b}\right )^{-n} \Gamma \left (1+n,\frac {3 (a+b \text {arccosh}(c x))}{b}\right )}{8 c^4 \sqrt {d-c^2 d x^2}} \] Output:

1/8*3^(-1-n)*(c*x-1)^(1/2)*(c*x+1)^(1/2)*(a+b*arccosh(c*x))^n*GAMMA(1+n,(- 
3*a-3*b*arccosh(c*x))/b)/c^4/exp(3*a/b)/(-c^2*d*x^2+d)^(1/2)/((-(a+b*arcco 
sh(c*x))/b)^n)+3/8*(c*x-1)^(1/2)*(c*x+1)^(1/2)*(a+b*arccosh(c*x))^n*GAMMA( 
1+n,-(a+b*arccosh(c*x))/b)/c^4/exp(a/b)/(-c^2*d*x^2+d)^(1/2)/((-(a+b*arcco 
sh(c*x))/b)^n)-3/8*exp(a/b)*(c*x-1)^(1/2)*(c*x+1)^(1/2)*(a+b*arccosh(c*x)) 
^n*GAMMA(1+n,(a+b*arccosh(c*x))/b)/c^4/(-c^2*d*x^2+d)^(1/2)/(((a+b*arccosh 
(c*x))/b)^n)-1/8*3^(-1-n)*exp(3*a/b)*(c*x-1)^(1/2)*(c*x+1)^(1/2)*(a+b*arcc 
osh(c*x))^n*GAMMA(1+n,3*(a+b*arccosh(c*x))/b)/c^4/(-c^2*d*x^2+d)^(1/2)/((( 
a+b*arccosh(c*x))/b)^n)
 

Mathematica [A] (warning: unable to verify)

Time = 0.84 (sec) , antiderivative size = 291, normalized size of antiderivative = 0.77 \[ \int \frac {x^3 (a+b \text {arccosh}(c x))^n}{\sqrt {d-c^2 d x^2}} \, dx=-\frac {3^{-1-n} e^{-\frac {3 a}{b}} \sqrt {\frac {-1+c x}{1+c x}} (1+c x) (a+b \text {arccosh}(c x))^n \left (-\frac {(a+b \text {arccosh}(c x))^2}{b^2}\right )^{-2 n} \left (3^{2+n} e^{\frac {4 a}{b}} \left (-\frac {a+b \text {arccosh}(c x)}{b}\right )^n \left (-\frac {(a+b \text {arccosh}(c x))^2}{b^2}\right )^n \Gamma \left (1+n,\frac {a}{b}+\text {arccosh}(c x)\right )-\left (\frac {a}{b}+\text {arccosh}(c x)\right )^n \left (\left (-\frac {(a+b \text {arccosh}(c x))^2}{b^2}\right )^n \Gamma \left (1+n,-\frac {3 (a+b \text {arccosh}(c x))}{b}\right )+3^{2+n} e^{\frac {2 a}{b}} \left (-\frac {(a+b \text {arccosh}(c x))^2}{b^2}\right )^n \Gamma \left (1+n,-\frac {a+b \text {arccosh}(c x)}{b}\right )-e^{\frac {6 a}{b}} \left (-\frac {a+b \text {arccosh}(c x)}{b}\right )^{2 n} \Gamma \left (1+n,\frac {3 (a+b \text {arccosh}(c x))}{b}\right )\right )\right )}{8 c^4 \sqrt {d-c^2 d x^2}} \] Input:

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

Output:

-1/8*(3^(-1 - n)*Sqrt[(-1 + c*x)/(1 + c*x)]*(1 + c*x)*(a + b*ArcCosh[c*x]) 
^n*(3^(2 + n)*E^((4*a)/b)*(-((a + b*ArcCosh[c*x])/b))^n*(-((a + b*ArcCosh[ 
c*x])^2/b^2))^n*Gamma[1 + n, a/b + ArcCosh[c*x]] - (a/b + ArcCosh[c*x])^n* 
((-((a + b*ArcCosh[c*x])^2/b^2))^n*Gamma[1 + n, (-3*(a + b*ArcCosh[c*x]))/ 
b] + 3^(2 + n)*E^((2*a)/b)*(-((a + b*ArcCosh[c*x])^2/b^2))^n*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])))/(c^4*E^((3*a)/b)*Sqrt[d - c^2 
*d*x^2]*(-((a + b*ArcCosh[c*x])^2/b^2))^(2*n))
 

Rubi [A] (verified)

Time = 0.70 (sec) , antiderivative size = 279, normalized size of antiderivative = 0.74, number of steps used = 5, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.138, Rules used = {6367, 3042, 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.

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

\(\Big \downarrow \) 6367

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 3793

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

\(\Big \downarrow \) 2009

\(\displaystyle \frac {\sqrt {c x-1} \sqrt {c x+1} \left (\frac {1}{8} b 3^{-n-1} e^{-\frac {3 a}{b}} (a+b \text {arccosh}(c x))^n \left (-\frac {a+b \text {arccosh}(c x)}{b}\right )^{-n} \Gamma \left (n+1,-\frac {3 (a+b \text {arccosh}(c x))}{b}\right )+\frac {3}{8} b e^{-\frac {a}{b}} (a+b \text {arccosh}(c x))^n \left (-\frac {a+b \text {arccosh}(c x)}{b}\right )^{-n} \Gamma \left (n+1,-\frac {a+b \text {arccosh}(c x)}{b}\right )-\frac {3}{8} b e^{a/b} (a+b \text {arccosh}(c x))^n \left (\frac {a+b \text {arccosh}(c x)}{b}\right )^{-n} \Gamma \left (n+1,\frac {a+b \text {arccosh}(c x)}{b}\right )-\frac {1}{8} b 3^{-n-1} e^{\frac {3 a}{b}} (a+b \text {arccosh}(c x))^n \left (\frac {a+b \text {arccosh}(c x)}{b}\right )^{-n} \Gamma \left (n+1,\frac {3 (a+b \text {arccosh}(c x))}{b}\right )\right )}{b c^4 \sqrt {d-c^2 d x^2}}\)

Input:

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

Output:

(Sqrt[-1 + c*x]*Sqrt[1 + c*x]*((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) + (3*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) - (3*b*E^(a/b)*(a + b*ArcC 
osh[c*x])^n*Gamma[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*ArcCosh[c*x]))/b])/(8*((a + b*ArcCosh[c*x])/b)^n)))/(b*c^4*Sqrt[d - 
 c^2*d*x^2])
 

Defintions of rubi rules used

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 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]

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

Input:

int(x^3*(a+b*arccosh(c*x))^n/(-c^2*d*x^2+d)^(1/2),x)
 

Output:

int(x^3*(a+b*arccosh(c*x))^n/(-c^2*d*x^2+d)^(1/2),x)
 

Fricas [F]

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

integrate(x^3*(a+b*arccosh(c*x))^n/(-c^2*d*x^2+d)^(1/2),x, algorithm="fric 
as")
 

Output:

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

Sympy [F]

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

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

Output:

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

Maxima [F]

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

integrate(x^3*(a+b*arccosh(c*x))^n/(-c^2*d*x^2+d)^(1/2),x, algorithm="maxi 
ma")
 

Output:

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

Giac [F(-2)]

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

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

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

Output:

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

Reduce [F]

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

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

Output:

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