\(\int \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x)) \, dx\) [57]

Optimal result

Integrand size = 24, antiderivative size = 124 \[ \int \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x)) \, dx=-\frac {b c x^2 \sqrt {d-c^2 d x^2}}{4 \sqrt {-1+c x} \sqrt {1+c x}}+\frac {1}{2} x \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))-\frac {\sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))^2}{4 b c \sqrt {-1+c x} \sqrt {1+c x}} \] Output:

-1/4*b*c*x^2*(-c^2*d*x^2+d)^(1/2)/(c*x-1)^(1/2)/(c*x+1)^(1/2)+1/2*x*(-c^2* 
d*x^2+d)^(1/2)*(a+b*arccosh(c*x))-1/4*(-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 = 0.39 (sec) , antiderivative size = 144, normalized size of antiderivative = 1.16 \[ \int \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x)) \, dx=\frac {1}{8} \left (4 a x \sqrt {d-c^2 d x^2}-\frac {4 a \sqrt {d} \arctan \left (\frac {c x \sqrt {d-c^2 d x^2}}{\sqrt {d} \left (-1+c^2 x^2\right )}\right )}{c}-\frac {b \sqrt {d-c^2 d x^2} \left (2 \text {arccosh}(c x)^2+\cosh (2 \text {arccosh}(c x))-2 \text {arccosh}(c x) \sinh (2 \text {arccosh}(c x))\right )}{c \sqrt {\frac {-1+c x}{1+c x}} (1+c x)}\right ) \] Input:

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

Output:

(4*a*x*Sqrt[d - c^2*d*x^2] - (4*a*Sqrt[d]*ArcTan[(c*x*Sqrt[d - c^2*d*x^2]) 
/(Sqrt[d]*(-1 + c^2*x^2))])/c - (b*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]]))/(c*Sqrt[(-1 
+ c*x)/(1 + c*x)]*(1 + c*x)))/8
 

Rubi [A] (verified)

Time = 0.44 (sec) , antiderivative size = 124, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.125, Rules used = {6310, 15, 6308}

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.

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

\(\Big \downarrow \) 6310

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

\(\Big \downarrow \) 15

\(\displaystyle -\frac {\sqrt {d-c^2 d x^2} \int \frac {a+b \text {arccosh}(c x)}{\sqrt {c x-1} \sqrt {c x+1}}dx}{2 \sqrt {c x-1} \sqrt {c x+1}}+\frac {1}{2} x \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))-\frac {b c x^2 \sqrt {d-c^2 d x^2}}{4 \sqrt {c x-1} \sqrt {c x+1}}\)

\(\Big \downarrow \) 6308

\(\displaystyle \frac {1}{2} x \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))-\frac {\sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))^2}{4 b c \sqrt {c x-1} \sqrt {c x+1}}-\frac {b c x^2 \sqrt {d-c^2 d x^2}}{4 \sqrt {c x-1} \sqrt {c x+1}}\)

Input:

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

Output:

-1/4*(b*c*x^2*Sqrt[d - c^2*d*x^2])/(Sqrt[-1 + c*x]*Sqrt[1 + c*x]) + (x*Sqr 
t[d - c^2*d*x^2]*(a + b*ArcCosh[c*x]))/2 - (Sqrt[d - c^2*d*x^2]*(a + b*Arc 
Cosh[c*x])^2)/(4*b*c*Sqrt[-1 + c*x]*Sqrt[1 + c*x])
 

Defintions of rubi rules used

rule 15

Int[(a_.)*(x_)^(m_.), x_Symbol] :> Simp[a*(x^(m + 1)/(m + 1)), x] /; FreeQ[ 
{a, m}, x] && NeQ[m, -1]
 

rule 6308

Int[((a_.) + ArcCosh[(c_.)*(x_)]*(b_.))^(n_.)/(Sqrt[(d1_) + (e1_.)*(x_)]*Sq 
rt[(d2_) + (e2_.)*(x_)]), x_Symbol] :> Simp[(1/(b*c*(n + 1)))*Simp[Sqrt[1 + 
 c*x]/Sqrt[d1 + e1*x]]*Simp[Sqrt[-1 + c*x]/Sqrt[d2 + e2*x]]*(a + b*ArcCosh[ 
c*x])^(n + 1), x] /; FreeQ[{a, b, c, d1, e1, d2, e2, n}, x] && EqQ[e1, c*d1 
] && EqQ[e2, (-c)*d2] && NeQ[n, -1]
 

rule 6310

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

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(277\) vs. \(2(104)=208\).

Time = 0.00 (sec) , antiderivative size = 278, normalized size of antiderivative = 2.24

method	result	size
default	\(\frac {a x \sqrt {-c^{2} d \,x^{2}+d}}{2}+\frac {a d \arctan \left (\frac {\sqrt {c^{2} d}\, x}{\sqrt {-c^{2} d \,x^{2}+d}}\right )}{2 \sqrt {c^{2} d}}+b \left (-\frac {\sqrt {-d \left (c^{2} x^{2}-1\right )}\, \operatorname {arccosh}\left (c x \right )^{2}}{4 \sqrt {c x -1}\, \sqrt {c x +1}\, c}+\frac {\sqrt {-d \left (c^{2} x^{2}-1\right )}\, \left (2 c^{3} x^{3}-2 c x +2 \sqrt {c x -1}\, \sqrt {c x +1}\, c^{2} x^{2}-\sqrt {c x -1}\, \sqrt {c x +1}\right ) \left (-1+2 \,\operatorname {arccosh}\left (c x \right )\right )}{16 \left (c x -1\right ) \left (c x +1\right ) c}+\frac {\sqrt {-d \left (c^{2} x^{2}-1\right )}\, \left (-2 \sqrt {c x -1}\, \sqrt {c x +1}\, c^{2} x^{2}+2 c^{3} x^{3}+\sqrt {c x -1}\, \sqrt {c x +1}-2 c x \right ) \left (1+2 \,\operatorname {arccosh}\left (c x \right )\right )}{16 \left (c x -1\right ) \left (c x +1\right ) c}\right )\)	\(278\)
parts	\(\frac {a x \sqrt {-c^{2} d \,x^{2}+d}}{2}+\frac {a d \arctan \left (\frac {\sqrt {c^{2} d}\, x}{\sqrt {-c^{2} d \,x^{2}+d}}\right )}{2 \sqrt {c^{2} d}}+b \left (-\frac {\sqrt {-d \left (c^{2} x^{2}-1\right )}\, \operatorname {arccosh}\left (c x \right )^{2}}{4 \sqrt {c x -1}\, \sqrt {c x +1}\, c}+\frac {\sqrt {-d \left (c^{2} x^{2}-1\right )}\, \left (2 c^{3} x^{3}-2 c x +2 \sqrt {c x -1}\, \sqrt {c x +1}\, c^{2} x^{2}-\sqrt {c x -1}\, \sqrt {c x +1}\right ) \left (-1+2 \,\operatorname {arccosh}\left (c x \right )\right )}{16 \left (c x -1\right ) \left (c x +1\right ) c}+\frac {\sqrt {-d \left (c^{2} x^{2}-1\right )}\, \left (-2 \sqrt {c x -1}\, \sqrt {c x +1}\, c^{2} x^{2}+2 c^{3} x^{3}+\sqrt {c x -1}\, \sqrt {c x +1}-2 c x \right ) \left (1+2 \,\operatorname {arccosh}\left (c x \right )\right )}{16 \left (c x -1\right ) \left (c x +1\right ) c}\right )\)	\(278\)

Input:

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

Output:

1/2*a*x*(-c^2*d*x^2+d)^(1/2)+1/2*a*d/(c^2*d)^(1/2)*arctan((c^2*d)^(1/2)*x/ 
(-c^2*d*x^2+d)^(1/2))+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+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))*(-1+2*arccos 
h(c*x))/(c*x-1)/(c*x+1)/c+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)*(1+2*arcc 
osh(c*x))/(c*x-1)/(c*x+1)/c)
 

Fricas [F]

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

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

Output:

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

Sympy [F]

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

integrate((-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)), x)
 

Maxima [F]

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

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

Output:

1/2*(sqrt(-c^2*d*x^2 + d)*x + sqrt(d)*arcsin(c*x)/c)*a + b*integrate(sqrt( 
-c^2*d*x^2 + d)*log(c*x + sqrt(c*x + 1)*sqrt(c*x - 1)), x)
 

Giac [F(-2)]

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

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

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

Output:

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

Reduce [F]

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

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

Output:

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