\(\int (f+g x) (d-c^2 d x^2)^{3/2} (a+b \text {arccosh}(c x)) \, dx\) [48]

Optimal result

Integrand size = 29, antiderivative size = 395 \[ \int (f+g x) \left (d-c^2 d x^2\right )^{3/2} (a+b \text {arccosh}(c x)) \, dx=\frac {b d g x \sqrt {d-c^2 d x^2}}{5 c \sqrt {-1+c x} \sqrt {1+c x}}-\frac {3 b c d f x^2 \sqrt {d-c^2 d x^2}}{16 \sqrt {-1+c x} \sqrt {1+c x}}-\frac {2 b c d g x^3 \sqrt {d-c^2 d x^2}}{15 \sqrt {-1+c x} \sqrt {1+c x}}+\frac {b c^3 d g x^5 \sqrt {d-c^2 d x^2}}{25 \sqrt {-1+c x} \sqrt {1+c x}}+\frac {b d f (-1+c x)^{3/2} (1+c x)^{3/2} \sqrt {d-c^2 d x^2}}{16 c}+\frac {3}{8} d f x \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))+\frac {1}{4} d f x (1-c x) (1+c x) \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))-\frac {d g (1-c x)^2 (1+c x)^2 \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))}{5 c^2}-\frac {3 d f \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))^2}{16 b c \sqrt {-1+c x} \sqrt {1+c x}} \] Output:

1/5*b*d*g*x*(-c^2*d*x^2+d)^(1/2)/c/(c*x-1)^(1/2)/(c*x+1)^(1/2)-3/16*b*c*d* 
f*x^2*(-c^2*d*x^2+d)^(1/2)/(c*x-1)^(1/2)/(c*x+1)^(1/2)-2/15*b*c*d*g*x^3*(- 
c^2*d*x^2+d)^(1/2)/(c*x-1)^(1/2)/(c*x+1)^(1/2)+1/25*b*c^3*d*g*x^5*(-c^2*d* 
x^2+d)^(1/2)/(c*x-1)^(1/2)/(c*x+1)^(1/2)+1/16*b*d*f*(c*x-1)^(3/2)*(c*x+1)^ 
(3/2)*(-c^2*d*x^2+d)^(1/2)/c+3/8*d*f*x*(-c^2*d*x^2+d)^(1/2)*(a+b*arccosh(c 
*x))+1/4*d*f*x*(-c*x+1)*(c*x+1)*(-c^2*d*x^2+d)^(1/2)*(a+b*arccosh(c*x))-1/ 
5*d*g*(-c*x+1)^2*(c*x+1)^2*(-c^2*d*x^2+d)^(1/2)*(a+b*arccosh(c*x))/c^2-3/1 
6*d*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 = 1.27 (sec) , antiderivative size = 432, normalized size of antiderivative = 1.09 \[ \int (f+g x) \left (d-c^2 d x^2\right )^{3/2} (a+b \text {arccosh}(c x)) \, dx=\frac {-720 a d \sqrt {\frac {-1+c x}{1+c x}} (1+c x) \sqrt {d-c^2 d x^2} \left (8 g \left (-1+c^2 x^2\right )^2+5 c^2 f x \left (-5+2 c^2 x^2\right )\right )-10800 a c d^{3/2} f \sqrt {\frac {-1+c x}{1+c x}} (1+c x) \arctan \left (\frac {c x \sqrt {d-c^2 d x^2}}{\sqrt {d} \left (-1+c^2 x^2\right )}\right )+800 b d g \sqrt {d-c^2 d x^2} \left (9 c x+12 \left (\frac {-1+c x}{1+c x}\right )^{3/2} (1+c x)^3 \text {arccosh}(c x)-\cosh (3 \text {arccosh}(c x))\right )-3600 b c d f \sqrt {d-c^2 d x^2} (\cosh (2 \text {arccosh}(c x))+2 \text {arccosh}(c x) (\text {arccosh}(c x)-\sinh (2 \text {arccosh}(c x))))+225 b c d f \sqrt {d-c^2 d x^2} \left (8 \text {arccosh}(c x)^2+\cosh (4 \text {arccosh}(c x))-4 \text {arccosh}(c x) \sinh (4 \text {arccosh}(c x))\right )-8 b d g \sqrt {d-c^2 d x^2} \left (450 c x-450 \sqrt {\frac {-1+c x}{1+c x}} (1+c x) \text {arccosh}(c x)-25 \cosh (3 \text {arccosh}(c x))-9 \cosh (5 \text {arccosh}(c x))+75 \text {arccosh}(c x) \sinh (3 \text {arccosh}(c x))+45 \text {arccosh}(c x) \sinh (5 \text {arccosh}(c x))\right )}{28800 c^2 \sqrt {\frac {-1+c x}{1+c x}} (1+c x)} \] Input:

Integrate[(f + g*x)*(d - c^2*d*x^2)^(3/2)*(a + b*ArcCosh[c*x]),x]
 

Output:

(-720*a*d*Sqrt[(-1 + c*x)/(1 + c*x)]*(1 + c*x)*Sqrt[d - c^2*d*x^2]*(8*g*(- 
1 + c^2*x^2)^2 + 5*c^2*f*x*(-5 + 2*c^2*x^2)) - 10800*a*c*d^(3/2)*f*Sqrt[(- 
1 + c*x)/(1 + c*x)]*(1 + c*x)*ArcTan[(c*x*Sqrt[d - c^2*d*x^2])/(Sqrt[d]*(- 
1 + c^2*x^2))] + 800*b*d*g*Sqrt[d - c^2*d*x^2]*(9*c*x + 12*((-1 + c*x)/(1 
+ c*x))^(3/2)*(1 + c*x)^3*ArcCosh[c*x] - Cosh[3*ArcCosh[c*x]]) - 3600*b*c* 
d*f*Sqrt[d - c^2*d*x^2]*(Cosh[2*ArcCosh[c*x]] + 2*ArcCosh[c*x]*(ArcCosh[c* 
x] - Sinh[2*ArcCosh[c*x]])) + 225*b*c*d*f*Sqrt[d - c^2*d*x^2]*(8*ArcCosh[c 
*x]^2 + Cosh[4*ArcCosh[c*x]] - 4*ArcCosh[c*x]*Sinh[4*ArcCosh[c*x]]) - 8*b* 
d*g*Sqrt[d - c^2*d*x^2]*(450*c*x - 450*Sqrt[(-1 + c*x)/(1 + c*x)]*(1 + c*x 
)*ArcCosh[c*x] - 25*Cosh[3*ArcCosh[c*x]] - 9*Cosh[5*ArcCosh[c*x]] + 75*Arc 
Cosh[c*x]*Sinh[3*ArcCosh[c*x]] + 45*ArcCosh[c*x]*Sinh[5*ArcCosh[c*x]]))/(2 
8800*c^2*Sqrt[(-1 + c*x)/(1 + c*x)]*(1 + c*x))
 

Rubi [A] (verified)

Time = 0.99 (sec) , antiderivative size = 210, normalized size of antiderivative = 0.53, 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.

\(\displaystyle \int \left (d-c^2 d x^2\right )^{3/2} (f+g x) (a+b \text {arccosh}(c x)) \, dx\)

\(\Big \downarrow \) 6387

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

\(\Big \downarrow \) 6390

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

\(\Big \downarrow \) 2009

\(\displaystyle -\frac {d \sqrt {d-c^2 d x^2} \left (\frac {g (c x-1)^{5/2} (c x+1)^{5/2} (a+b \text {arccosh}(c x))}{5 c^2}+\frac {1}{4} f x (c x-1)^{3/2} (c x+1)^{3/2} (a+b \text {arccosh}(c x))-\frac {3}{8} f x \sqrt {c x-1} \sqrt {c x+1} (a+b \text {arccosh}(c x))+\frac {3 f (a+b \text {arccosh}(c x))^2}{16 b c}-\frac {1}{16} b c^3 f x^4-\frac {1}{25} b c^3 g x^5+\frac {5}{16} b c f x^2+\frac {2}{15} b c g x^3-\frac {b g x}{5 c}\right )}{\sqrt {c x-1} \sqrt {c x+1}}\)

Input:

Int[(f + g*x)*(d - c^2*d*x^2)^(3/2)*(a + b*ArcCosh[c*x]),x]
 

Output:

-((d*Sqrt[d - c^2*d*x^2]*(-1/5*(b*g*x)/c + (5*b*c*f*x^2)/16 + (2*b*c*g*x^3 
)/15 - (b*c^3*f*x^4)/16 - (b*c^3*g*x^5)/25 - (3*f*x*Sqrt[-1 + c*x]*Sqrt[1 
+ c*x]*(a + b*ArcCosh[c*x]))/8 + (f*x*(-1 + c*x)^(3/2)*(1 + c*x)^(3/2)*(a 
+ b*ArcCosh[c*x]))/4 + (g*(-1 + c*x)^(5/2)*(1 + c*x)^(5/2)*(a + b*ArcCosh[ 
c*x]))/(5*c^2) + (3*f*(a + b*ArcCosh[c*x])^2)/(16*b*c)))/(Sqrt[-1 + c*x]*S 
qrt[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. \(1175\) vs. \(2(335)=670\).

Time = 0.78 (sec) , antiderivative size = 1176, normalized size of antiderivative = 2.98

method	result	size
default	\(\text {Expression too large to display}\)	\(1176\)
parts	\(\text {Expression too large to display}\)	\(1176\)

Input:

int((g*x+f)*(-c^2*d*x^2+d)^(3/2)*(a+b*arccosh(c*x)),x,method=_RETURNVERBOS 
E)
 

Output:

1/4*a*f*x*(-c^2*d*x^2+d)^(3/2)+3/8*a*f*d*x*(-c^2*d*x^2+d)^(1/2)+3/8*a*f*d^ 
2/(c^2*d)^(1/2)*arctan((c^2*d)^(1/2)*x/(-c^2*d*x^2+d)^(1/2))-1/5*a*g*(-c^2 
*d*x^2+d)^(5/2)/c^2/d+b*(-3/16*(-d*(c^2*x^2-1))^(1/2)/(c*x-1)^(1/2)/(c*x+1 
)^(1/2)/c*arccosh(c*x)^2*f*d-1/800*(-d*(c^2*x^2-1))^(1/2)*(16*c^6*x^6-28*c 
^4*x^4+16*c^5*x^5*(c*x-1)^(1/2)*(c*x+1)^(1/2)+13*c^2*x^2-20*c^3*x^3*(c*x-1 
)^(1/2)*(c*x+1)^(1/2)+5*(c*x-1)^(1/2)*(c*x+1)^(1/2)*c*x-1)*g*(-1+5*arccosh 
(c*x))*d/(c*x+1)/c^2/(c*x-1)-1/256*(-d*(c^2*x^2-1))^(1/2)*(8*c^5*x^5-12*c^ 
3*x^3+8*c^4*x^4*(c*x-1)^(1/2)*(c*x+1)^(1/2)+4*c*x-8*(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+4*arccosh(c*x))*d/(c*x-1) 
/(c*x+1)/c+1/96*(-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*arcco 
sh(c*x))*d/(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))*d/(c*x-1)/(c*x+1)/c-1/16*(-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))*d/(c*x+1)/c^2/(c*x- 
1)-1/16*(-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))*d/(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))*d/(c*x-1)/(c*x+1)/c+1/96*(-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...
 

Fricas [F]

\[ \int (f+g x) \left (d-c^2 d x^2\right )^{3/2} (a+b \text {arccosh}(c x)) \, dx=\int { {\left (-c^{2} d x^{2} + d\right )}^{\frac {3}{2}} {\left (g x + f\right )} {\left (b \operatorname {arcosh}\left (c x\right ) + a\right )} \,d x } \] Input:

integrate((g*x+f)*(-c^2*d*x^2+d)^(3/2)*(a+b*arccosh(c*x)),x, algorithm="fr 
icas")
 

Output:

integral(-(a*c^2*d*g*x^3 + a*c^2*d*f*x^2 - a*d*g*x - a*d*f + (b*c^2*d*g*x^ 
3 + b*c^2*d*f*x^2 - b*d*g*x - b*d*f)*arccosh(c*x))*sqrt(-c^2*d*x^2 + d), x 
)
 

Sympy [F]

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

integrate((g*x+f)*(-c**2*d*x**2+d)**(3/2)*(a+b*acosh(c*x)),x)
 

Output:

Integral((-d*(c*x - 1)*(c*x + 1))**(3/2)*(a + b*acosh(c*x))*(f + g*x), x)
 

Maxima [F]

\[ \int (f+g x) \left (d-c^2 d x^2\right )^{3/2} (a+b \text {arccosh}(c x)) \, dx=\int { {\left (-c^{2} d x^{2} + d\right )}^{\frac {3}{2}} {\left (g x + f\right )} {\left (b \operatorname {arcosh}\left (c x\right ) + a\right )} \,d x } \] Input:

integrate((g*x+f)*(-c^2*d*x^2+d)^(3/2)*(a+b*arccosh(c*x)),x, algorithm="ma 
xima")
 

Output:

1/8*(2*(-c^2*d*x^2 + d)^(3/2)*x + 3*sqrt(-c^2*d*x^2 + d)*d*x + 3*d^(3/2)*a 
rcsin(c*x)/c)*a*f - 1/5*(-c^2*d*x^2 + d)^(5/2)*a*g/(c^2*d) + integrate((-c 
^2*d*x^2 + d)^(3/2)*b*g*x*log(c*x + sqrt(c*x + 1)*sqrt(c*x - 1)) + (-c^2*d 
*x^2 + d)^(3/2)*b*f*log(c*x + sqrt(c*x + 1)*sqrt(c*x - 1)), x)
 

Giac [F(-2)]

Exception generated. \[ \int (f+g x) \left (d-c^2 d x^2\right )^{3/2} (a+b \text {arccosh}(c x)) \, dx=\text {Exception raised: RuntimeError} \] Input:

integrate((g*x+f)*(-c^2*d*x^2+d)^(3/2)*(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. \[ \int (f+g x) \left (d-c^2 d x^2\right )^{3/2} (a+b \text {arccosh}(c x)) \, dx=\int \left (f+g\,x\right )\,\left (a+b\,\mathrm {acosh}\left (c\,x\right )\right )\,{\left (d-c^2\,d\,x^2\right )}^{3/2} \,d x \] Input:

int((f + g*x)*(a + b*acosh(c*x))*(d - c^2*d*x^2)^(3/2),x)
 

Output:

int((f + g*x)*(a + b*acosh(c*x))*(d - c^2*d*x^2)^(3/2), x)
 

Reduce [F]

\[ \int (f+g x) \left (d-c^2 d x^2\right )^{3/2} (a+b \text {arccosh}(c x)) \, dx=\frac {\sqrt {d}\, d \left (15 \mathit {asin} \left (c x \right ) a c f -10 \sqrt {-c^{2} x^{2}+1}\, a \,c^{4} f \,x^{3}-8 \sqrt {-c^{2} x^{2}+1}\, a \,c^{4} g \,x^{4}+25 \sqrt {-c^{2} x^{2}+1}\, a \,c^{2} f x +16 \sqrt {-c^{2} x^{2}+1}\, a \,c^{2} g \,x^{2}-8 \sqrt {-c^{2} x^{2}+1}\, a g -40 \left (\int \sqrt {-c^{2} x^{2}+1}\, \mathit {acosh} \left (c x \right ) x^{3}d x \right ) b \,c^{4} g -40 \left (\int \sqrt {-c^{2} x^{2}+1}\, \mathit {acosh} \left (c x \right ) x^{2}d x \right ) b \,c^{4} f +40 \left (\int \sqrt {-c^{2} x^{2}+1}\, \mathit {acosh} \left (c x \right ) x d x \right ) b \,c^{2} g +40 \left (\int \sqrt {-c^{2} x^{2}+1}\, \mathit {acosh} \left (c x \right )d x \right ) b \,c^{2} f +8 a g \right )}{40 c^{2}} \] Input:

int((g*x+f)*(-c^2*d*x^2+d)^(3/2)*(a+b*acosh(c*x)),x)
 

Output:

(sqrt(d)*d*(15*asin(c*x)*a*c*f - 10*sqrt( - c**2*x**2 + 1)*a*c**4*f*x**3 - 
 8*sqrt( - c**2*x**2 + 1)*a*c**4*g*x**4 + 25*sqrt( - c**2*x**2 + 1)*a*c**2 
*f*x + 16*sqrt( - c**2*x**2 + 1)*a*c**2*g*x**2 - 8*sqrt( - c**2*x**2 + 1)* 
a*g - 40*int(sqrt( - c**2*x**2 + 1)*acosh(c*x)*x**3,x)*b*c**4*g - 40*int(s 
qrt( - c**2*x**2 + 1)*acosh(c*x)*x**2,x)*b*c**4*f + 40*int(sqrt( - c**2*x* 
*2 + 1)*acosh(c*x)*x,x)*b*c**2*g + 40*int(sqrt( - c**2*x**2 + 1)*acosh(c*x 
),x)*b*c**2*f + 8*a*g))/(40*c**2)