\(\int \frac {(f+g x) (a+b \text {arccosh}(c x))^n}{\sqrt {1-c x} \sqrt {1+c x}} \, dx\) [64]

Optimal result

Integrand size = 35, antiderivative size = 200 \[ \int \frac {(f+g x) (a+b \text {arccosh}(c x))^n}{\sqrt {1-c x} \sqrt {1+c x}} \, dx=\frac {f \sqrt {-1+c x} (a+b \text {arccosh}(c x))^{1+n}}{b c (1+n) \sqrt {1-c x}}+\frac {e^{-\frac {a}{b}} g \sqrt {-1+c x} (a+b \text {arccosh}(c x))^n \left (-\frac {a+b \text {arccosh}(c x)}{b}\right )^{-n} \Gamma \left (1+n,-\frac {a+b \text {arccosh}(c x)}{b}\right )}{2 c^2 \sqrt {1-c x}}-\frac {e^{a/b} g \sqrt {-1+c x} (a+b \text {arccosh}(c x))^n \left (\frac {a+b \text {arccosh}(c x)}{b}\right )^{-n} \Gamma \left (1+n,\frac {a+b \text {arccosh}(c x)}{b}\right )}{2 c^2 \sqrt {1-c x}} \] Output:

f*(c*x-1)^(1/2)*(a+b*arccosh(c*x))^(1+n)/b/c/(1+n)/(-c*x+1)^(1/2)+1/2*g*(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)*g*(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] (warning: unable to verify)

Time = 0.21 (sec) , antiderivative size = 204, normalized size of antiderivative = 1.02 \[ \int \frac {(f+g x) (a+b \text {arccosh}(c x))^n}{\sqrt {1-c x} \sqrt {1+c x}} \, dx=\frac {e^{-\frac {a}{b}} \sqrt {\frac {-1+c x}{1+c x}} (1+c x) (a+b \text {arccosh}(c x))^n \left (-\frac {(a+b \text {arccosh}(c x))^2}{b^2}\right )^{-n} \left (2 c e^{a/b} f (a+b \text {arccosh}(c x)) \left (-\frac {(a+b \text {arccosh}(c x))^2}{b^2}\right )^n-b e^{\frac {2 a}{b}} g (1+n) \left (-\frac {a+b \text {arccosh}(c x)}{b}\right )^n \Gamma \left (1+n,\frac {a}{b}+\text {arccosh}(c x)\right )+b g (1+n) \left (\frac {a}{b}+\text {arccosh}(c x)\right )^n \Gamma \left (1+n,-\frac {a+b \text {arccosh}(c x)}{b}\right )\right )}{2 b c^2 (1+n) \sqrt {1-c^2 x^2}} \] Input:

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

Output:

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

Rubi [A] (verified)

Time = 1.28 (sec) , antiderivative size = 158, normalized size of antiderivative = 0.79, number of steps used = 6, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.143, Rules used = {6397, 6395, 3042, 3798, 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.

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

\(\Big \downarrow \) 6397

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

\(\Big \downarrow \) 6395

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 3798

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

\(\Big \downarrow \) 2009

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

Input:

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

Output:

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

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 3798

Int[((c_.) + (d_.)*(x_))^(m_.)*((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(n_.) 
, x_Symbol] :> Int[ExpandIntegrand[(c + d*x)^m, (a + b*Sin[e + f*x])^n, x], 
 x] /; FreeQ[{a, b, c, d, e, f, m}, x] && IGtQ[n, 0] && (EqQ[n, 1] || IGtQ[ 
m, 0] || NeQ[a^2 - b^2, 0])
 

rule 6395

Int[(((a_.) + ArcCosh[(c_.)*(x_)]*(b_.))^(n_.)*((f_) + (g_.)*(x_))^(m_.))/( 
Sqrt[(d1_) + (e1_.)*(x_)]*Sqrt[(d2_) + (e2_.)*(x_)]), x_Symbol] :> Simp[1/( 
c^(m + 1)*Sqrt[(-d1)*d2])   Subst[Int[(a + b*x)^n*(c*f + g*Cosh[x])^m, x], 
x, ArcCosh[c*x]], x] /; FreeQ[{a, b, c, d1, e1, d2, e2, f, g, n}, x] && EqQ 
[e1 - c*d1, 0] && EqQ[e2 + c*d2, 0] && IntegerQ[m] && GtQ[d1, 0] && LtQ[d2, 
 0] && (GtQ[m, 0] || IGtQ[n, 0])
 

rule 6397

Int[((a_.) + ArcCosh[(c_.)*(x_)]*(b_.))^(n_.)*((d1_) + (e1_.)*(x_))^(p_)*(( 
d2_) + (e2_.)*(x_))^(p_)*((f_) + (g_.)*(x_))^(m_.), x_Symbol] :> Simp[((-d1 
)*d2)^IntPart[p]*(d1 + e1*x)^FracPart[p]*((d2 + e2*x)^FracPart[p]/((-1 + c* 
x)^FracPart[p]*(1 + c*x)^FracPart[p]))   Int[(f + g*x)^m*(-1 + c*x)^p*(1 + 
c*x)^p*(a + b*ArcCosh[c*x])^n, x], x] /; FreeQ[{a, b, c, d1, e1, d2, e2, f, 
 g, n}, x] && EqQ[e1 - c*d1, 0] && EqQ[e2 + c*d2, 0] && IntegerQ[m] && Inte 
gerQ[p - 1/2] &&  !(GtQ[d1, 0] && LtQ[d2, 0])
 

Maple [F]

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

Input:

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

Output:

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

Fricas [F]

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

integrate((g*x+f)*(a+b*arccosh(c*x))^n/(-c*x+1)^(1/2)/(c*x+1)^(1/2),x, alg 
orithm="fricas")
 

Output:

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

Sympy [F]

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

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

Output:

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

Maxima [F]

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

integrate((g*x+f)*(a+b*arccosh(c*x))^n/(-c*x+1)^(1/2)/(c*x+1)^(1/2),x, alg 
orithm="maxima")
 

Output:

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

Giac [F]

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

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

Output:

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

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

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

Output:

int((acosh(c*x)*b + a)**n/(sqrt(c*x + 1)*sqrt( - c*x + 1)),x)*f + int(((ac 
osh(c*x)*b + a)**n*x)/(sqrt(c*x + 1)*sqrt( - c*x + 1)),x)*g