\(\int \sqrt {\pi -c^2 \pi x^2} (a+b \text {arccosh}(c x)) \, dx\) [40]

Optimal result

Integrand size = 24, antiderivative size = 106 \[ \int \sqrt {\pi -c^2 \pi x^2} (a+b \text {arccosh}(c x)) \, dx=-\frac {b c \sqrt {\pi } x^2 \sqrt {1-c x}}{4 \sqrt {-1+c x}}+\frac {1}{2} x \sqrt {\pi -c^2 \pi x^2} (a+b \text {arccosh}(c x))-\frac {\sqrt {\pi } \sqrt {1-c x} (a+b \text {arccosh}(c x))^2}{4 b c \sqrt {-1+c x}} \] Output:

-1/4*b*c*Pi^(1/2)*x^2*(-c*x+1)^(1/2)/(c*x-1)^(1/2)+1/2*x*(-Pi*c^2*x^2+Pi)^ 
(1/2)*(a+b*arccosh(c*x))-1/4*Pi^(1/2)*(-c*x+1)^(1/2)*(a+b*arccosh(c*x))^2/ 
b/c/(c*x-1)^(1/2)
 

Mathematica [A] (warning: unable to verify)

Time = 0.35 (sec) , antiderivative size = 110, normalized size of antiderivative = 1.04 \[ \int \sqrt {\pi -c^2 \pi x^2} (a+b \text {arccosh}(c x)) \, dx=\frac {1}{8} \sqrt {\pi } \left (4 a x \sqrt {1-c^2 x^2}+\frac {4 a \arcsin (c x)}{c}-\frac {b \sqrt {1-c^2 x^2} (\cosh (2 \text {arccosh}(c x))+2 \text {arccosh}(c x) (\text {arccosh}(c x)-\sinh (2 \text {arccosh}(c x))))}{c \sqrt {\frac {-1+c x}{1+c x}} (1+c x)}\right ) \] Input:

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

Output:

(Sqrt[Pi]*(4*a*x*Sqrt[1 - c^2*x^2] + (4*a*ArcSin[c*x])/c - (b*Sqrt[1 - c^2 
*x^2]*(Cosh[2*ArcCosh[c*x]] + 2*ArcCosh[c*x]*(ArcCosh[c*x] - Sinh[2*ArcCos 
h[c*x]])))/(c*Sqrt[(-1 + c*x)/(1 + c*x)]*(1 + c*x))))/8
 

Rubi [A] (verified)

Time = 0.43 (sec) , antiderivative size = 124, normalized size of antiderivative = 1.17, 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.

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

\(\Big \downarrow \) 6310

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

\(\Big \downarrow \) 15

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

\(\Big \downarrow \) 6308

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

Input:

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

Output:

-1/4*(b*c*x^2*Sqrt[Pi - c^2*Pi*x^2])/(Sqrt[-1 + c*x]*Sqrt[1 + c*x]) + (x*S 
qrt[Pi - c^2*Pi*x^2]*(a + b*ArcCosh[c*x]))/2 - (Sqrt[Pi - c^2*Pi*x^2]*(a + 
 b*ArcCosh[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. \(280\) vs. \(2(86)=172\).

Time = 0.21 (sec) , antiderivative size = 281, normalized size of antiderivative = 2.65

method	result	size
default	\(\frac {a x \sqrt {-\pi \,c^{2} x^{2}+\pi }}{2}+\frac {a \pi \arctan \left (\frac {\sqrt {\pi \,c^{2}}\, x}{\sqrt {-\pi \,c^{2} x^{2}+\pi }}\right )}{2 \sqrt {\pi \,c^{2}}}+b \left (-\frac {\sqrt {\pi }\, \sqrt {-c^{2} x^{2}+1}\, \operatorname {arccosh}\left (c x \right )^{2}}{4 \sqrt {c x -1}\, \sqrt {c x +1}\, c}+\frac {\sqrt {\pi }\, \sqrt {-c^{2} x^{2}+1}\, \left (2 c^{3} x^{3}-2 c x +2 \sqrt {c x -1}\, \sqrt {c x +1}\, c^{2} x^{2}-\sqrt {c x -1}\, \sqrt {c x +1}\right ) \left (-1+2 \,\operatorname {arccosh}\left (c x \right )\right )}{16 \left (c x -1\right ) \left (c x +1\right ) c}+\frac {\sqrt {\pi }\, \sqrt {-c^{2} x^{2}+1}\, \left (-2 \sqrt {c x -1}\, \sqrt {c x +1}\, c^{2} x^{2}+2 c^{3} x^{3}+\sqrt {c x -1}\, \sqrt {c x +1}-2 c x \right ) \left (1+2 \,\operatorname {arccosh}\left (c x \right )\right )}{16 \left (c x -1\right ) \left (c x +1\right ) c}\right )\)	\(281\)
parts	\(\frac {a x \sqrt {-\pi \,c^{2} x^{2}+\pi }}{2}+\frac {a \pi \arctan \left (\frac {\sqrt {\pi \,c^{2}}\, x}{\sqrt {-\pi \,c^{2} x^{2}+\pi }}\right )}{2 \sqrt {\pi \,c^{2}}}+b \left (-\frac {\sqrt {\pi }\, \sqrt {-c^{2} x^{2}+1}\, \operatorname {arccosh}\left (c x \right )^{2}}{4 \sqrt {c x -1}\, \sqrt {c x +1}\, c}+\frac {\sqrt {\pi }\, \sqrt {-c^{2} x^{2}+1}\, \left (2 c^{3} x^{3}-2 c x +2 \sqrt {c x -1}\, \sqrt {c x +1}\, c^{2} x^{2}-\sqrt {c x -1}\, \sqrt {c x +1}\right ) \left (-1+2 \,\operatorname {arccosh}\left (c x \right )\right )}{16 \left (c x -1\right ) \left (c x +1\right ) c}+\frac {\sqrt {\pi }\, \sqrt {-c^{2} x^{2}+1}\, \left (-2 \sqrt {c x -1}\, \sqrt {c x +1}\, c^{2} x^{2}+2 c^{3} x^{3}+\sqrt {c x -1}\, \sqrt {c x +1}-2 c x \right ) \left (1+2 \,\operatorname {arccosh}\left (c x \right )\right )}{16 \left (c x -1\right ) \left (c x +1\right ) c}\right )\)	\(281\)

Input:

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

Output:

1/2*a*x*(-Pi*c^2*x^2+Pi)^(1/2)+1/2*a*Pi/(Pi*c^2)^(1/2)*arctan((Pi*c^2)^(1/ 
2)*x/(-Pi*c^2*x^2+Pi)^(1/2))+b*(-1/4*Pi^(1/2)*(-c^2*x^2+1)^(1/2)/(c*x-1)^( 
1/2)/(c*x+1)^(1/2)/c*arccosh(c*x)^2+1/16*Pi^(1/2)*(-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*arccosh(c*x))/(c*x-1)/(c*x+1)/c+1/16*Pi^(1/2)*(-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*arccosh(c*x))/(c*x-1)/(c*x+1)/c)
 

Fricas [F]

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

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

Output:

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

Sympy [F]

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

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

Output:

sqrt(pi)*(Integral(a*sqrt(-c**2*x**2 + 1), x) + Integral(b*sqrt(-c**2*x**2 
 + 1)*acosh(c*x), x))
 

Maxima [F]

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

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

Output:

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

Giac [F(-2)]

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

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

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

Output:

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

Reduce [F]

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

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

Output:

(sqrt(pi)*(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)