3.5.34 \(\int \frac {x^3 (a+b \text {arccosh}(c x))^n}{\sqrt {1-c^2 x^2}} \, dx\) [434]

3.5.34.1 Optimal result

Integrand size = 28, antiderivative size = 323 \[ \int \frac {x^3 (a+b \text {arccosh}(c x))^n}{\sqrt {1-c^2 x^2}} \, dx=\frac {3^{-1-n} e^{-\frac {3 a}{b}} \sqrt {-1+c x} (a+b \text {arccosh}(c x))^n \left (-\frac {a+b \text {arccosh}(c x)}{b}\right )^{-n} \Gamma \left (1+n,-\frac {3 (a+b \text {arccosh}(c x))}{b}\right )}{8 c^4 \sqrt {1-c x}}+\frac {3 e^{-\frac {a}{b}} \sqrt {-1+c x} (a+b \text {arccosh}(c x))^n \left (-\frac {a+b \text {arccosh}(c x)}{b}\right )^{-n} \Gamma \left (1+n,-\frac {a+b \text {arccosh}(c x)}{b}\right )}{8 c^4 \sqrt {1-c x}}-\frac {3 e^{a/b} \sqrt {-1+c x} (a+b \text {arccosh}(c x))^n \left (\frac {a+b \text {arccosh}(c x)}{b}\right )^{-n} \Gamma \left (1+n,\frac {a+b \text {arccosh}(c x)}{b}\right )}{8 c^4 \sqrt {1-c x}}-\frac {3^{-1-n} e^{\frac {3 a}{b}} \sqrt {-1+c x} (a+b \text {arccosh}(c x))^n \left (\frac {a+b \text {arccosh}(c x)}{b}\right )^{-n} \Gamma \left (1+n,\frac {3 (a+b \text {arccosh}(c x))}{b}\right )}{8 c^4 \sqrt {1-c x}} \]

output

1/8*3^(-1-n)*(a+b*arccosh(c*x))^n*GAMMA(1+n,-3*(a+b*arccosh(c*x))/b)*(c*x- 
1)^(1/2)/c^4/exp(3*a/b)/(((-a-b*arccosh(c*x))/b)^n)/(-c*x+1)^(1/2)+3/8*(a+ 
b*arccosh(c*x))^n*GAMMA(1+n,(-a-b*arccosh(c*x))/b)*(c*x-1)^(1/2)/c^4/exp(a 
/b)/(((-a-b*arccosh(c*x))/b)^n)/(-c*x+1)^(1/2)-3/8*exp(a/b)*(a+b*arccosh(c 
*x))^n*GAMMA(1+n,(a+b*arccosh(c*x))/b)*(c*x-1)^(1/2)/c^4/(((a+b*arccosh(c* 
x))/b)^n)/(-c*x+1)^(1/2)-1/8*3^(-1-n)*exp(3*a/b)*(a+b*arccosh(c*x))^n*GAMM 
A(1+n,3*(a+b*arccosh(c*x))/b)*(c*x-1)^(1/2)/c^4/(((a+b*arccosh(c*x))/b)^n) 
/(-c*x+1)^(1/2)
 

3.5.34.2 Mathematica [A] (verified)

Time = 0.97 (sec) , antiderivative size = 292, normalized size of antiderivative = 0.90 \[ \int \frac {x^3 (a+b \text {arccosh}(c x))^n}{\sqrt {1-c^2 x^2}} \, dx=\frac {3^{-1-n} e^{-\frac {3 a}{b}} \sqrt {1-c^2 x^2} (a+b \text {arccosh}(c x))^n \left (-\frac {(a+b \text {arccosh}(c x))^2}{b^2}\right )^{-2 n} \left (3^{2+n} e^{\frac {4 a}{b}} \left (-\frac {a+b \text {arccosh}(c x)}{b}\right )^n \left (-\frac {(a+b \text {arccosh}(c x))^2}{b^2}\right )^n \Gamma \left (1+n,\frac {a}{b}+\text {arccosh}(c x)\right )-\left (\frac {a}{b}+\text {arccosh}(c x)\right )^n \left (\left (-\frac {(a+b \text {arccosh}(c x))^2}{b^2}\right )^n \Gamma \left (1+n,-\frac {3 (a+b \text {arccosh}(c x))}{b}\right )+3^{2+n} e^{\frac {2 a}{b}} \left (-\frac {(a+b \text {arccosh}(c x))^2}{b^2}\right )^n \Gamma \left (1+n,-\frac {a+b \text {arccosh}(c x)}{b}\right )-e^{\frac {6 a}{b}} \left (-\frac {a+b \text {arccosh}(c x)}{b}\right )^{2 n} \Gamma \left (1+n,\frac {3 (a+b \text {arccosh}(c x))}{b}\right )\right )\right )}{8 c^4 \sqrt {\frac {-1+c x}{1+c x}} (1+c x)} \]

input

Integrate[(x^3*(a + b*ArcCosh[c*x])^n)/Sqrt[1 - c^2*x^2],x]
 

output

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

3.5.34.3 Rubi [A] (verified)

Time = 0.64 (sec) , antiderivative size = 265, normalized size of antiderivative = 0.82, number of steps used = 5, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.143, 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.

\(\displaystyle \int \frac {x^3 (a+b \text {arccosh}(c x))^n}{\sqrt {1-c^2 x^2}} \, dx\)

\(\Big \downarrow \) 6367

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 3793

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

\(\Big \downarrow \) 2009

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

input

Int[(x^3*(a + b*ArcCosh[c*x])^n)/Sqrt[1 - c^2*x^2],x]
 

output

(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*ArcCosh[c*x])^n*Ga 
mma[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[1 - c*x])
 

3.5.34.3.1 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]
 

3.5.34.4 Maple [F]

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

input

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

output

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

3.5.34.5 Fricas [F]

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

input

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

output

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

3.5.34.6 Sympy [F]

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

input

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

output

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

3.5.34.7 Maxima [F]

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

input

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

output

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

3.5.34.8 Giac [F(-2)]

Exception generated. \[ \int \frac {x^3 (a+b \text {arccosh}(c x))^n}{\sqrt {1-c^2 x^2}} \, dx=\text {Exception raised: TypeError} \]

input

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

3.5.34.9 Mupad [F(-1)]

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

input

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

output

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