\(\int (f+g x) \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x)) \, dx\) [43]

Optimal result

Integrand size = 29, antiderivative size = 255 \[ \int (f+g x) \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x)) \, dx=\frac {b g x \sqrt {d-c^2 d x^2}}{3 c \sqrt {-1+c x} \sqrt {1+c x}}-\frac {b c f x^2 \sqrt {d-c^2 d x^2}}{4 \sqrt {-1+c x} \sqrt {1+c x}}-\frac {b c g x^3 \sqrt {d-c^2 d x^2}}{9 \sqrt {-1+c x} \sqrt {1+c x}}+\frac {1}{2} f x \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))-\frac {g (1-c x) (1+c x) \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))}{3 c^2}-\frac {f \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))^2}{4 b c \sqrt {-1+c x} \sqrt {1+c x}} \] Output:

1/3*b*g*x*(-c^2*d*x^2+d)^(1/2)/c/(c*x-1)^(1/2)/(c*x+1)^(1/2)-1/4*b*c*f*x^2 
*(-c^2*d*x^2+d)^(1/2)/(c*x-1)^(1/2)/(c*x+1)^(1/2)-1/9*b*c*g*x^3*(-c^2*d*x^ 
2+d)^(1/2)/(c*x-1)^(1/2)/(c*x+1)^(1/2)+1/2*f*x*(-c^2*d*x^2+d)^(1/2)*(a+b*a 
rccosh(c*x))-1/3*g*(-c*x+1)*(c*x+1)*(-c^2*d*x^2+d)^(1/2)*(a+b*arccosh(c*x) 
)/c^2-1/4*f*(-c^2*d*x^2+d)^(1/2)*(a+b*arccosh(c*x))^2/b/c/(c*x-1)^(1/2)/(c 
*x+1)^(1/2)
 

Mathematica [A] (warning: unable to verify)

Time = 0.78 (sec) , antiderivative size = 251, normalized size of antiderivative = 0.98 \[ \int (f+g x) \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x)) \, dx=\frac {12 a \sqrt {d-c^2 d x^2} \left (3 c^2 f x+2 g \left (-1+c^2 x^2\right )\right )-36 a c \sqrt {d} f \arctan \left (\frac {c x \sqrt {d-c^2 d x^2}}{\sqrt {d} \left (-1+c^2 x^2\right )}\right )+\frac {2 b g \sqrt {d-c^2 d x^2} \left (9 c x+12 \left (\frac {-1+c x}{1+c x}\right )^{3/2} (1+c x)^3 \text {arccosh}(c x)-\cosh (3 \text {arccosh}(c x))\right )}{\sqrt {\frac {-1+c x}{1+c x}} (1+c x)}-\frac {9 b c f \sqrt {d-c^2 d x^2} \left (2 \text {arccosh}(c x)^2+\cosh (2 \text {arccosh}(c x))-2 \text {arccosh}(c x) \sinh (2 \text {arccosh}(c x))\right )}{\sqrt {\frac {-1+c x}{1+c x}} (1+c x)}}{72 c^2} \] Input:

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

Output:

(12*a*Sqrt[d - c^2*d*x^2]*(3*c^2*f*x + 2*g*(-1 + c^2*x^2)) - 36*a*c*Sqrt[d 
]*f*ArcTan[(c*x*Sqrt[d - c^2*d*x^2])/(Sqrt[d]*(-1 + c^2*x^2))] + (2*b*g*Sq 
rt[d - c^2*d*x^2]*(9*c*x + 12*((-1 + c*x)/(1 + c*x))^(3/2)*(1 + c*x)^3*Arc 
Cosh[c*x] - Cosh[3*ArcCosh[c*x]]))/(Sqrt[(-1 + c*x)/(1 + c*x)]*(1 + c*x)) 
- (9*b*c*f*Sqrt[d - c^2*d*x^2]*(2*ArcCosh[c*x]^2 + Cosh[2*ArcCosh[c*x]] - 
2*ArcCosh[c*x]*Sinh[2*ArcCosh[c*x]]))/(Sqrt[(-1 + c*x)/(1 + c*x)]*(1 + c*x 
)))/(72*c^2)
 

Rubi [A] (verified)

Time = 0.82 (sec) , antiderivative size = 152, normalized size of antiderivative = 0.60, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.103, Rules used = {6387, 6390, 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.

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

\(\Big \downarrow \) 6387

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

\(\Big \downarrow \) 6390

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

\(\Big \downarrow \) 2009

\(\displaystyle \frac {\sqrt {d-c^2 d x^2} \left (\frac {g (c x-1)^{3/2} (c x+1)^{3/2} (a+b \text {arccosh}(c x))}{3 c^2}+\frac {1}{2} f x \sqrt {c x-1} \sqrt {c x+1} (a+b \text {arccosh}(c x))-\frac {f (a+b \text {arccosh}(c x))^2}{4 b c}-\frac {1}{4} b c f x^2-\frac {1}{9} b c g x^3+\frac {b g x}{3 c}\right )}{\sqrt {c x-1} \sqrt {c x+1}}\)

Input:

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

Output:

(Sqrt[d - c^2*d*x^2]*((b*g*x)/(3*c) - (b*c*f*x^2)/4 - (b*c*g*x^3)/9 + (f*x 
*Sqrt[-1 + c*x]*Sqrt[1 + c*x]*(a + b*ArcCosh[c*x]))/2 + (g*(-1 + c*x)^(3/2 
)*(1 + c*x)^(3/2)*(a + b*ArcCosh[c*x]))/(3*c^2) - (f*(a + b*ArcCosh[c*x])^ 
2)/(4*b*c)))/(Sqrt[-1 + c*x]*Sqrt[1 + c*x])
 

Defintions of rubi rules used

rule 2009

Int[u_, x_Symbol] :> Simp[IntSum[u, x], x] /; SumQ[u]
 

rule 6387

Int[((a_.) + ArcCosh[(c_.)*(x_)]*(b_.))^(n_.)*((f_) + (g_.)*(x_))^(m_.)*((d 
_) + (e_.)*(x_)^2)^(p_), x_Symbol] :> Simp[(-d)^IntPart[p]*((d + e*x^2)^Fra 
cPart[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, 
d, e, f, g, n}, x] && EqQ[c^2*d + e, 0] && IntegerQ[p - 1/2] && IntegerQ[m]
 

rule 6390

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

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(638\) vs. \(2(215)=430\).

Time = 0.62 (sec) , antiderivative size = 639, normalized size of antiderivative = 2.51

method	result	size
default	\(\frac {a f x \sqrt {-c^{2} d \,x^{2}+d}}{2}+\frac {a f d \arctan \left (\frac {\sqrt {c^{2} d}\, x}{\sqrt {-c^{2} d \,x^{2}+d}}\right )}{2 \sqrt {c^{2} d}}-\frac {a g \left (-c^{2} d \,x^{2}+d \right )^{\frac {3}{2}}}{3 c^{2} d}+b \left (-\frac {\sqrt {-d \left (c^{2} x^{2}-1\right )}\, \operatorname {arccosh}\left (c x \right )^{2} f}{4 \sqrt {c x -1}\, \sqrt {c x +1}\, c}+\frac {\sqrt {-d \left (c^{2} x^{2}-1\right )}\, \left (4 c^{4} x^{4}-5 c^{2} x^{2}+4 c^{3} x^{3} \sqrt {c x -1}\, \sqrt {c x +1}-3 \sqrt {c x -1}\, \sqrt {c x +1}\, c x +1\right ) g \left (-1+3 \,\operatorname {arccosh}\left (c x \right )\right )}{72 \left (c x +1\right ) c^{2} \left (c x -1\right )}+\frac {\sqrt {-d \left (c^{2} x^{2}-1\right )}\, \left (2 c^{3} x^{3}-2 c x +2 \sqrt {c x -1}\, \sqrt {c x +1}\, c^{2} x^{2}-\sqrt {c x -1}\, \sqrt {c x +1}\right ) f \left (-1+2 \,\operatorname {arccosh}\left (c x \right )\right )}{16 \left (c x -1\right ) \left (c x +1\right ) c}-\frac {\sqrt {-d \left (c^{2} x^{2}-1\right )}\, \left (\sqrt {c x -1}\, \sqrt {c x +1}\, c x +c^{2} x^{2}-1\right ) g \left (-1+\operatorname {arccosh}\left (c x \right )\right )}{8 \left (c x +1\right ) c^{2} \left (c x -1\right )}-\frac {\sqrt {-d \left (c^{2} x^{2}-1\right )}\, \left (-\sqrt {c x -1}\, \sqrt {c x +1}\, c x +c^{2} x^{2}-1\right ) g \left (1+\operatorname {arccosh}\left (c x \right )\right )}{8 \left (c x +1\right ) c^{2} \left (c x -1\right )}+\frac {\sqrt {-d \left (c^{2} x^{2}-1\right )}\, \left (-2 \sqrt {c x -1}\, \sqrt {c x +1}\, c^{2} x^{2}+2 c^{3} x^{3}+\sqrt {c x -1}\, \sqrt {c x +1}-2 c x \right ) f \left (1+2 \,\operatorname {arccosh}\left (c x \right )\right )}{16 \left (c x -1\right ) \left (c x +1\right ) c}+\frac {\sqrt {-d \left (c^{2} x^{2}-1\right )}\, \left (-4 c^{3} x^{3} \sqrt {c x -1}\, \sqrt {c x +1}+4 c^{4} x^{4}+3 \sqrt {c x -1}\, \sqrt {c x +1}\, c x -5 c^{2} x^{2}+1\right ) g \left (1+3 \,\operatorname {arccosh}\left (c x \right )\right )}{72 \left (c x +1\right ) c^{2} \left (c x -1\right )}\right )\)	\(639\)
parts	\(\frac {a f x \sqrt {-c^{2} d \,x^{2}+d}}{2}+\frac {a f d \arctan \left (\frac {\sqrt {c^{2} d}\, x}{\sqrt {-c^{2} d \,x^{2}+d}}\right )}{2 \sqrt {c^{2} d}}-\frac {a g \left (-c^{2} d \,x^{2}+d \right )^{\frac {3}{2}}}{3 c^{2} d}+b \left (-\frac {\sqrt {-d \left (c^{2} x^{2}-1\right )}\, \operatorname {arccosh}\left (c x \right )^{2} f}{4 \sqrt {c x -1}\, \sqrt {c x +1}\, c}+\frac {\sqrt {-d \left (c^{2} x^{2}-1\right )}\, \left (4 c^{4} x^{4}-5 c^{2} x^{2}+4 c^{3} x^{3} \sqrt {c x -1}\, \sqrt {c x +1}-3 \sqrt {c x -1}\, \sqrt {c x +1}\, c x +1\right ) g \left (-1+3 \,\operatorname {arccosh}\left (c x \right )\right )}{72 \left (c x +1\right ) c^{2} \left (c x -1\right )}+\frac {\sqrt {-d \left (c^{2} x^{2}-1\right )}\, \left (2 c^{3} x^{3}-2 c x +2 \sqrt {c x -1}\, \sqrt {c x +1}\, c^{2} x^{2}-\sqrt {c x -1}\, \sqrt {c x +1}\right ) f \left (-1+2 \,\operatorname {arccosh}\left (c x \right )\right )}{16 \left (c x -1\right ) \left (c x +1\right ) c}-\frac {\sqrt {-d \left (c^{2} x^{2}-1\right )}\, \left (\sqrt {c x -1}\, \sqrt {c x +1}\, c x +c^{2} x^{2}-1\right ) g \left (-1+\operatorname {arccosh}\left (c x \right )\right )}{8 \left (c x +1\right ) c^{2} \left (c x -1\right )}-\frac {\sqrt {-d \left (c^{2} x^{2}-1\right )}\, \left (-\sqrt {c x -1}\, \sqrt {c x +1}\, c x +c^{2} x^{2}-1\right ) g \left (1+\operatorname {arccosh}\left (c x \right )\right )}{8 \left (c x +1\right ) c^{2} \left (c x -1\right )}+\frac {\sqrt {-d \left (c^{2} x^{2}-1\right )}\, \left (-2 \sqrt {c x -1}\, \sqrt {c x +1}\, c^{2} x^{2}+2 c^{3} x^{3}+\sqrt {c x -1}\, \sqrt {c x +1}-2 c x \right ) f \left (1+2 \,\operatorname {arccosh}\left (c x \right )\right )}{16 \left (c x -1\right ) \left (c x +1\right ) c}+\frac {\sqrt {-d \left (c^{2} x^{2}-1\right )}\, \left (-4 c^{3} x^{3} \sqrt {c x -1}\, \sqrt {c x +1}+4 c^{4} x^{4}+3 \sqrt {c x -1}\, \sqrt {c x +1}\, c x -5 c^{2} x^{2}+1\right ) g \left (1+3 \,\operatorname {arccosh}\left (c x \right )\right )}{72 \left (c x +1\right ) c^{2} \left (c x -1\right )}\right )\)	\(639\)

Input:

int((g*x+f)*(-c^2*d*x^2+d)^(1/2)*(a+b*arccosh(c*x)),x,method=_RETURNVERBOS 
E)
 

Output:

1/2*a*f*x*(-c^2*d*x^2+d)^(1/2)+1/2*a*f*d/(c^2*d)^(1/2)*arctan((c^2*d)^(1/2 
)*x/(-c^2*d*x^2+d)^(1/2))-1/3*a*g*(-c^2*d*x^2+d)^(3/2)/c^2/d+b*(-1/4*(-d*( 
c^2*x^2-1))^(1/2)/(c*x-1)^(1/2)/(c*x+1)^(1/2)/c*arccosh(c*x)^2*f+1/72*(-d* 
(c^2*x^2-1))^(1/2)*(4*c^4*x^4-5*c^2*x^2+4*c^3*x^3*(c*x-1)^(1/2)*(c*x+1)^(1 
/2)-3*(c*x-1)^(1/2)*(c*x+1)^(1/2)*c*x+1)*g*(-1+3*arccosh(c*x))/(c*x+1)/c^2 
/(c*x-1)+1/16*(-d*(c^2*x^2-1))^(1/2)*(2*c^3*x^3-2*c*x+2*(c*x-1)^(1/2)*(c*x 
+1)^(1/2)*c^2*x^2-(c*x-1)^(1/2)*(c*x+1)^(1/2))*f*(-1+2*arccosh(c*x))/(c*x- 
1)/(c*x+1)/c-1/8*(-d*(c^2*x^2-1))^(1/2)*((c*x-1)^(1/2)*(c*x+1)^(1/2)*c*x+c 
^2*x^2-1)*g*(-1+arccosh(c*x))/(c*x+1)/c^2/(c*x-1)-1/8*(-d*(c^2*x^2-1))^(1/ 
2)*(-(c*x-1)^(1/2)*(c*x+1)^(1/2)*c*x+c^2*x^2-1)*g*(1+arccosh(c*x))/(c*x+1) 
/c^2/(c*x-1)+1/16*(-d*(c^2*x^2-1))^(1/2)*(-2*(c*x-1)^(1/2)*(c*x+1)^(1/2)*c 
^2*x^2+2*c^3*x^3+(c*x-1)^(1/2)*(c*x+1)^(1/2)-2*c*x)*f*(1+2*arccosh(c*x))/( 
c*x-1)/(c*x+1)/c+1/72*(-d*(c^2*x^2-1))^(1/2)*(-4*c^3*x^3*(c*x-1)^(1/2)*(c* 
x+1)^(1/2)+4*c^4*x^4+3*(c*x-1)^(1/2)*(c*x+1)^(1/2)*c*x-5*c^2*x^2+1)*g*(1+3 
*arccosh(c*x))/(c*x+1)/c^2/(c*x-1))
 

Fricas [F]

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

integrate((g*x+f)*(-c^2*d*x^2+d)^(1/2)*(a+b*arccosh(c*x)),x, algorithm="fr 
icas")
 

Output:

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

Sympy [F]

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

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

Output:

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

Maxima [F]

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

integrate((g*x+f)*(-c^2*d*x^2+d)^(1/2)*(a+b*arccosh(c*x)),x, algorithm="ma 
xima")
 

Output:

1/2*(sqrt(-c^2*d*x^2 + d)*x + sqrt(d)*arcsin(c*x)/c)*a*f - 1/3*(-c^2*d*x^2 
 + d)^(3/2)*a*g/(c^2*d) + integrate(sqrt(-c^2*d*x^2 + d)*b*g*x*log(c*x + s 
qrt(c*x + 1)*sqrt(c*x - 1)) + sqrt(-c^2*d*x^2 + d)*b*f*log(c*x + sqrt(c*x 
+ 1)*sqrt(c*x - 1)), x)
 

Giac [F(-2)]

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

integrate((g*x+f)*(-c^2*d*x^2+d)^(1/2)*(a+b*arccosh(c*x)),x, algorithm="gi 
ac")
 

Output:

Exception raised: RuntimeError >> an error occurred running a Giac command 
:INPUT:sage2OUTPUT:sym2poly/r2sym(const gen & e,const index_m & i,const ve 
cteur & l) Error: Bad Argument Value
 

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

\[ \int (f+g x) \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x)) \, dx=\frac {\sqrt {d}\, \left (3 \mathit {asin} \left (c x \right ) a c f +3 \sqrt {-c^{2} x^{2}+1}\, a \,c^{2} f x +2 \sqrt {-c^{2} x^{2}+1}\, a \,c^{2} g \,x^{2}-2 \sqrt {-c^{2} x^{2}+1}\, a g +6 \left (\int \sqrt {-c^{2} x^{2}+1}\, \mathit {acosh} \left (c x \right ) x d x \right ) b \,c^{2} g +6 \left (\int \sqrt {-c^{2} x^{2}+1}\, \mathit {acosh} \left (c x \right )d x \right ) b \,c^{2} f +2 a g \right )}{6 c^{2}} \] Input:

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

Output:

(sqrt(d)*(3*asin(c*x)*a*c*f + 3*sqrt( - c**2*x**2 + 1)*a*c**2*f*x + 2*sqrt 
( - c**2*x**2 + 1)*a*c**2*g*x**2 - 2*sqrt( - c**2*x**2 + 1)*a*g + 6*int(sq 
rt( - c**2*x**2 + 1)*acosh(c*x)*x,x)*b*c**2*g + 6*int(sqrt( - c**2*x**2 + 
1)*acosh(c*x),x)*b*c**2*f + 2*a*g))/(6*c**2)