\(\int x \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x)) \, dx\) [64]

Optimal result

Integrand size = 25, antiderivative size = 118 \[ \int x \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x)) \, dx=\frac {b x \sqrt {d-c^2 d x^2}}{3 c \sqrt {-1+c x} \sqrt {1+c x}}-\frac {b c x^3 \sqrt {d-c^2 d x^2}}{9 \sqrt {-1+c x} \sqrt {1+c x}}-\frac {\left (d-c^2 d x^2\right )^{3/2} (a+b \text {arccosh}(c x))}{3 c^2 d} \] Output:

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

Mathematica [A] (verified)

Time = 0.12 (sec) , antiderivative size = 98, normalized size of antiderivative = 0.83 \[ \int x \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x)) \, dx=\frac {\sqrt {d-c^2 d x^2} \left (b c x \sqrt {-1+c x} \sqrt {1+c x} \left (3-c^2 x^2\right )+3 a \left (-1+c^2 x^2\right )^2+3 b \left (-1+c^2 x^2\right )^2 \text {arccosh}(c x)\right )}{9 c^2 \left (-1+c^2 x^2\right )} \] Input:

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

Output:

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

Rubi [A] (verified)

Time = 0.31 (sec) , antiderivative size = 87, normalized size of antiderivative = 0.74, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.160, Rules used = {6329, 25, 39, 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.

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

\(\Big \downarrow \) 6329

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

\(\Big \downarrow \) 25

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

\(\Big \downarrow \) 39

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

\(\Big \downarrow \) 2009

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

Input:

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

Output:

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

Defintions of rubi rules used

rule 25

Int[-(Fx_), x_Symbol] :> Simp[Identity[-1]   Int[Fx, x], x]
 

rule 39

Int[((a_) + (b_.)*(x_))^(m_.)*((c_) + (d_.)*(x_))^(m_.), x_Symbol] :> Int[( 
a*c + b*d*x^2)^m, x] /; FreeQ[{a, b, c, d, m}, x] && EqQ[b*c + a*d, 0] && ( 
IntegerQ[m] || (GtQ[a, 0] && GtQ[c, 0]))
 

rule 2009

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

rule 6329

Int[((a_.) + ArcCosh[(c_.)*(x_)]*(b_.))^(n_.)*(x_)*((d_) + (e_.)*(x_)^2)^(p 
_.), x_Symbol] :> Simp[(d + e*x^2)^(p + 1)*((a + b*ArcCosh[c*x])^n/(2*e*(p 
+ 1))), x] - Simp[b*(n/(2*c*(p + 1)))*Simp[(d + e*x^2)^p/((1 + c*x)^p*(-1 + 
 c*x)^p)]   Int[(1 + c*x)^(p + 1/2)*(-1 + c*x)^(p + 1/2)*(a + b*ArcCosh[c*x 
])^(n - 1), x], x] /; FreeQ[{a, b, c, d, e, p}, x] && EqQ[c^2*d + e, 0] && 
GtQ[n, 0] && NeQ[p, -1]
 

Maple [A] (verified)

Time = 0.32 (sec) , antiderivative size = 155, normalized size of antiderivative = 1.31

method	result	size
orering	\(\frac {\left (5 c^{4} x^{4}-13 c^{2} x^{2}+6\right ) \sqrt {-c^{2} d \,x^{2}+d}\, \left (a +b \,\operatorname {arccosh}\left (c x \right )\right )}{9 c^{2} \left (c^{2} x^{2}-1\right )}-\frac {\left (c^{2} x^{2}-3\right ) \left (\sqrt {-c^{2} d \,x^{2}+d}\, \left (a +b \,\operatorname {arccosh}\left (c x \right )\right )-\frac {x^{2} \left (a +b \,\operatorname {arccosh}\left (c x \right )\right ) c^{2} d}{\sqrt {-c^{2} d \,x^{2}+d}}+\frac {b c x \sqrt {-c^{2} d \,x^{2}+d}}{\sqrt {c x -1}\, \sqrt {c x +1}}\right )}{9 c^{2}}\)	\(155\)
default	\(-\frac {a \left (-c^{2} d \,x^{2}+d \right )^{\frac {3}{2}}}{3 c^{2} d}+b \left (\frac {\sqrt {-d \left (c^{2} x^{2}-1\right )}\, \left (4 c^{4} x^{4}-5 c^{2} x^{2}+4 c^{3} x^{3} \sqrt {c x -1}\, \sqrt {c x +1}-3 \sqrt {c x -1}\, \sqrt {c x +1}\, c x +1\right ) \left (-1+3 \,\operatorname {arccosh}\left (c x \right )\right )}{72 \left (c x +1\right ) c^{2} \left (c x -1\right )}-\frac {\sqrt {-d \left (c^{2} x^{2}-1\right )}\, \left (\sqrt {c x -1}\, \sqrt {c x +1}\, c x +c^{2} x^{2}-1\right ) \left (-1+\operatorname {arccosh}\left (c x \right )\right )}{8 \left (c x +1\right ) c^{2} \left (c x -1\right )}-\frac {\sqrt {-d \left (c^{2} x^{2}-1\right )}\, \left (-\sqrt {c x -1}\, \sqrt {c x +1}\, c x +c^{2} x^{2}-1\right ) \left (1+\operatorname {arccosh}\left (c x \right )\right )}{8 \left (c x +1\right ) c^{2} \left (c x -1\right )}+\frac {\sqrt {-d \left (c^{2} x^{2}-1\right )}\, \left (-4 c^{3} x^{3} \sqrt {c x -1}\, \sqrt {c x +1}+4 c^{4} x^{4}+3 \sqrt {c x -1}\, \sqrt {c x +1}\, c x -5 c^{2} x^{2}+1\right ) \left (1+3 \,\operatorname {arccosh}\left (c x \right )\right )}{72 \left (c x +1\right ) c^{2} \left (c x -1\right )}\right )\)	\(356\)
parts	\(-\frac {a \left (-c^{2} d \,x^{2}+d \right )^{\frac {3}{2}}}{3 c^{2} d}+b \left (\frac {\sqrt {-d \left (c^{2} x^{2}-1\right )}\, \left (4 c^{4} x^{4}-5 c^{2} x^{2}+4 c^{3} x^{3} \sqrt {c x -1}\, \sqrt {c x +1}-3 \sqrt {c x -1}\, \sqrt {c x +1}\, c x +1\right ) \left (-1+3 \,\operatorname {arccosh}\left (c x \right )\right )}{72 \left (c x +1\right ) c^{2} \left (c x -1\right )}-\frac {\sqrt {-d \left (c^{2} x^{2}-1\right )}\, \left (\sqrt {c x -1}\, \sqrt {c x +1}\, c x +c^{2} x^{2}-1\right ) \left (-1+\operatorname {arccosh}\left (c x \right )\right )}{8 \left (c x +1\right ) c^{2} \left (c x -1\right )}-\frac {\sqrt {-d \left (c^{2} x^{2}-1\right )}\, \left (-\sqrt {c x -1}\, \sqrt {c x +1}\, c x +c^{2} x^{2}-1\right ) \left (1+\operatorname {arccosh}\left (c x \right )\right )}{8 \left (c x +1\right ) c^{2} \left (c x -1\right )}+\frac {\sqrt {-d \left (c^{2} x^{2}-1\right )}\, \left (-4 c^{3} x^{3} \sqrt {c x -1}\, \sqrt {c x +1}+4 c^{4} x^{4}+3 \sqrt {c x -1}\, \sqrt {c x +1}\, c x -5 c^{2} x^{2}+1\right ) \left (1+3 \,\operatorname {arccosh}\left (c x \right )\right )}{72 \left (c x +1\right ) c^{2} \left (c x -1\right )}\right )\)	\(356\)

Input:

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

Output:

1/9*(5*c^4*x^4-13*c^2*x^2+6)/c^2/(c^2*x^2-1)*(-c^2*d*x^2+d)^(1/2)*(a+b*arc 
cosh(c*x))-1/9*(c^2*x^2-3)/c^2*((-c^2*d*x^2+d)^(1/2)*(a+b*arccosh(c*x))-x^ 
2/(-c^2*d*x^2+d)^(1/2)*(a+b*arccosh(c*x))*c^2*d+b*c*x*(-c^2*d*x^2+d)^(1/2) 
/(c*x-1)^(1/2)/(c*x+1)^(1/2))
 

Fricas [A] (verification not implemented)

Time = 0.10 (sec) , antiderivative size = 142, normalized size of antiderivative = 1.20 \[ \int x \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x)) \, dx=\frac {3 \, {\left (b c^{4} x^{4} - 2 \, b c^{2} x^{2} + b\right )} \sqrt {-c^{2} d x^{2} + d} \log \left (c x + \sqrt {c^{2} x^{2} - 1}\right ) - {\left (b c^{3} x^{3} - 3 \, b c x\right )} \sqrt {-c^{2} d x^{2} + d} \sqrt {c^{2} x^{2} - 1} + 3 \, {\left (a c^{4} x^{4} - 2 \, a c^{2} x^{2} + a\right )} \sqrt {-c^{2} d x^{2} + d}}{9 \, {\left (c^{4} x^{2} - c^{2}\right )}} \] Input:

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

Output:

1/9*(3*(b*c^4*x^4 - 2*b*c^2*x^2 + b)*sqrt(-c^2*d*x^2 + d)*log(c*x + sqrt(c 
^2*x^2 - 1)) - (b*c^3*x^3 - 3*b*c*x)*sqrt(-c^2*d*x^2 + d)*sqrt(c^2*x^2 - 1 
) + 3*(a*c^4*x^4 - 2*a*c^2*x^2 + a)*sqrt(-c^2*d*x^2 + d))/(c^4*x^2 - c^2)
 

Sympy [F]

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

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

Output:

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

Maxima [A] (verification not implemented)

Time = 0.04 (sec) , antiderivative size = 81, normalized size of antiderivative = 0.69 \[ \int x \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x)) \, dx=-\frac {{\left (-c^{2} d x^{2} + d\right )}^{\frac {3}{2}} b \operatorname {arcosh}\left (c x\right )}{3 \, c^{2} d} - \frac {{\left (c^{2} \sqrt {-d} d x^{3} - 3 \, \sqrt {-d} d x\right )} b}{9 \, c d} - \frac {{\left (-c^{2} d x^{2} + d\right )}^{\frac {3}{2}} a}{3 \, c^{2} d} \] Input:

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

Output:

-1/3*(-c^2*d*x^2 + d)^(3/2)*b*arccosh(c*x)/(c^2*d) - 1/9*(c^2*sqrt(-d)*d*x 
^3 - 3*sqrt(-d)*d*x)*b/(c*d) - 1/3*(-c^2*d*x^2 + d)^(3/2)*a/(c^2*d)
 

Giac [F(-2)]

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

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

Output:

Exception raised: TypeError >> an error occurred running a Giac command:IN 
PUT:sage2:=int(sage0,sageVARx):;OUTPUT:sym2poly/r2sym(const gen & e,const 
index_m & i,const vecteur & l) Error: Bad Argument Value
 

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

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

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

Output:

(sqrt(d)*(sqrt( - c**2*x**2 + 1)*a*c**2*x**2 - sqrt( - c**2*x**2 + 1)*a + 
3*int(sqrt( - c**2*x**2 + 1)*acosh(c*x)*x,x)*b*c**2))/(3*c**2)