3.5.42 \(\int \frac {x (a+b \text {arccosh}(c x))^n}{\sqrt {d-c^2 d x^2}} \, dx\) [442]

3.5.42.1 Optimal result

Integrand size = 27, antiderivative size = 182 \[ \int \frac {x (a+b \text {arccosh}(c x))^n}{\sqrt {d-c^2 d x^2}} \, dx=\frac {e^{-\frac {a}{b}} \sqrt {-1+c x} \sqrt {1+c x} (a+b \text {arccosh}(c x))^n \left (-\frac {a+b \text {arccosh}(c x)}{b}\right )^{-n} \Gamma \left (1+n,-\frac {a+b \text {arccosh}(c x)}{b}\right )}{2 c^2 \sqrt {d-c^2 d x^2}}-\frac {e^{a/b} \sqrt {-1+c x} \sqrt {1+c x} (a+b \text {arccosh}(c x))^n \left (\frac {a+b \text {arccosh}(c x)}{b}\right )^{-n} \Gamma \left (1+n,\frac {a+b \text {arccosh}(c x)}{b}\right )}{2 c^2 \sqrt {d-c^2 d x^2}} \]

output

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

3.5.42.2 Mathematica [A] (verified)

Time = 0.24 (sec) , antiderivative size = 153, normalized size of antiderivative = 0.84 \[ \int \frac {x (a+b \text {arccosh}(c x))^n}{\sqrt {d-c^2 d x^2}} \, dx=\frac {e^{-\frac {a}{b}} \sqrt {\frac {-1+c x}{1+c x}} (1+c x) (a+b \text {arccosh}(c x))^n \left (-\frac {(a+b \text {arccosh}(c x))^2}{b^2}\right )^{-n} \left (-e^{\frac {2 a}{b}} \left (-\frac {a+b \text {arccosh}(c x)}{b}\right )^n \Gamma \left (1+n,\frac {a}{b}+\text {arccosh}(c x)\right )+\left (\frac {a}{b}+\text {arccosh}(c x)\right )^n \Gamma \left (1+n,-\frac {a+b \text {arccosh}(c x)}{b}\right )\right )}{2 c^2 \sqrt {-d (-1+c x) (1+c x)}} \]

input

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

output

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

3.5.42.3 Rubi [A] (verified)

Time = 0.48 (sec) , antiderivative size = 152, normalized size of antiderivative = 0.84, number of steps used = 6, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.185, 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.

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

\(\Big \downarrow \) 6367

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 3788

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

\(\Big \downarrow \) 26

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

\(\Big \downarrow \) 2612

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

input

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

output

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

3.5.42.3.1 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]
 

3.5.42.4 Maple [F]

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

input

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

output

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

3.5.42.5 Fricas [F]

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

input

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

output

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

3.5.42.6 Sympy [F]

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

input

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

output

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

3.5.42.7 Maxima [F]

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

input

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

output

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

3.5.42.8 Giac [F]

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

input

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

output

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

3.5.42.9 Mupad [F(-1)]

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

input

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

output

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