\(\int \frac {x (a+b \text {arccosh}(c x))^n}{\sqrt {1-c^2 x^2}} \, dx\) [335]

Optimal result

Integrand size = 26, antiderivative size = 154 \[ \int \frac {x (a+b \text {arccosh}(c x))^n}{\sqrt {1-c^2 x^2}} \, dx=\frac {e^{-\frac {a}{b}} \sqrt {-1+c x} (a+b \text {arccosh}(c x))^n \left (-\frac {a+b \text {arccosh}(c x)}{b}\right )^{-n} \Gamma \left (1+n,-\frac {a+b \text {arccosh}(c x)}{b}\right )}{2 c^2 \sqrt {1-c x}}-\frac {e^{a/b} \sqrt {-1+c x} (a+b \text {arccosh}(c x))^n \left (\frac {a+b \text {arccosh}(c x)}{b}\right )^{-n} \Gamma \left (1+n,\frac {a+b \text {arccosh}(c x)}{b}\right )}{2 c^2 \sqrt {1-c x}} \] Output:

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

Mathematica [A] (verified)

Time = 0.20 (sec) , antiderivative size = 154, normalized size of antiderivative = 1.00 \[ \int \frac {x (a+b \text {arccosh}(c x))^n}{\sqrt {1-c^2 x^2}} \, dx=-\frac {e^{-\frac {a}{b}} \sqrt {-((-1+c x) (1+c x))} (a+b \text {arccosh}(c x))^n \left (-\frac {(a+b \text {arccosh}(c x))^2}{b^2}\right )^{-n} \left (-e^{\frac {2 a}{b}} \left (-\frac {a+b \text {arccosh}(c x)}{b}\right )^n \Gamma \left (1+n,\frac {a}{b}+\text {arccosh}(c x)\right )+\left (\frac {a}{b}+\text {arccosh}(c x)\right )^n \Gamma \left (1+n,-\frac {a+b \text {arccosh}(c x)}{b}\right )\right )}{2 c^2 \sqrt {\frac {-1+c x}{1+c x}} (1+c x)} \] Input:

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

Output:

-1/2*(Sqrt[-((-1 + c*x)*(1 + c*x))]*(a + b*ArcCosh[c*x])^n*(-(E^((2*a)/b)* 
(-((a + b*ArcCosh[c*x])/b))^n*Gamma[1 + n, a/b + ArcCosh[c*x]]) + (a/b + A 
rcCosh[c*x])^n*Gamma[1 + n, -((a + b*ArcCosh[c*x])/b)]))/(c^2*E^(a/b)*Sqrt 
[(-1 + c*x)/(1 + c*x)]*(1 + c*x)*(-((a + b*ArcCosh[c*x])^2/b^2))^n)
 

Rubi [A] (verified)

Time = 0.46 (sec) , antiderivative size = 138, normalized size of antiderivative = 0.90, number of steps used = 6, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.192, Rules used = {6367, 3042, 3788, 26, 2612}

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.

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

\(\Big \downarrow \) 6367

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 3788

\(\displaystyle \frac {\sqrt {c x-1} \left (\frac {1}{2} i \int -i e^{-\text {arccosh}(c x)} (a+b \text {arccosh}(c x))^nd(a+b \text {arccosh}(c x))-\frac {1}{2} i \int i e^{\text {arccosh}(c x)} (a+b \text {arccosh}(c x))^nd(a+b \text {arccosh}(c x))\right )}{b c^2 \sqrt {1-c x}}\)

\(\Big \downarrow \) 26

\(\displaystyle \frac {\sqrt {c x-1} \left (\frac {1}{2} \int e^{-\text {arccosh}(c x)} (a+b \text {arccosh}(c x))^nd(a+b \text {arccosh}(c x))+\frac {1}{2} \int e^{\text {arccosh}(c x)} (a+b \text {arccosh}(c x))^nd(a+b \text {arccosh}(c x))\right )}{b c^2 \sqrt {1-c x}}\)

\(\Big \downarrow \) 2612

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

Input:

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

Output:

(Sqrt[-1 + c*x]*((b*(a + b*ArcCosh[c*x])^n*Gamma[1 + n, -((a + b*ArcCosh[c 
*x])/b)])/(2*E^(a/b)*(-((a + b*ArcCosh[c*x])/b))^n) - (b*E^(a/b)*(a + b*Ar 
cCosh[c*x])^n*Gamma[1 + n, (a + b*ArcCosh[c*x])/b])/(2*((a + b*ArcCosh[c*x 
])/b)^n)))/(b*c^2*Sqrt[1 - c*x])
 

Defintions of rubi rules used

rule 26

Int[(Complex[0, a_])*(Fx_), x_Symbol] :> Simp[(Complex[Identity[0], a])   I 
nt[Fx, x], x] /; FreeQ[a, x] && EqQ[a^2, 1]
 

rule 2612

Int[(F_)^((g_.)*((e_.) + (f_.)*(x_)))*((c_.) + (d_.)*(x_))^(m_), x_Symbol] 
:> Simp[(-F^(g*(e - c*(f/d))))*((c + d*x)^FracPart[m]/(d*((-f)*g*(Log[F]/d) 
)^(IntPart[m] + 1)*((-f)*g*Log[F]*((c + d*x)/d))^FracPart[m]))*Gamma[m + 1, 
 ((-f)*g*(Log[F]/d))*(c + d*x)], x] /; FreeQ[{F, c, d, e, f, g, m}, x] && 
!IntegerQ[m]
 

rule 3042

Int[u_, x_Symbol] :> Int[DeactivateTrig[u, x], x] /; FunctionOfTrigOfLinear 
Q[u, x]
 

rule 3788

Int[((c_.) + (d_.)*(x_))^(m_.)*sin[(e_.) + Pi*(k_.) + (f_.)*(x_)], x_Symbol 
] :> Simp[I/2   Int[(c + d*x)^m/(E^(I*k*Pi)*E^(I*(e + f*x))), x], x] - Simp 
[I/2   Int[(c + d*x)^m*E^(I*k*Pi)*E^(I*(e + f*x)), x], x] /; FreeQ[{c, d, e 
, f, m}, x] && IntegerQ[2*k]
 

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]

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

Input:

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

Output:

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

Fricas [F]

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

integrate(x*(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/(c^2*x^2 - 1), x)
 

Sympy [F]

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

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

Output:

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

Maxima [F]

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

integrate(x*(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/sqrt(-c^2*x^2 + 1), x)
 

Giac [F]

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

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

Output:

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

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

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

Output:

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