\(\int \sqrt {\pi -c^2 \pi x^2} (a+b \text {arccosh}(c x))^2 \, dx\) [46]

Optimal result

Integrand size = 26, antiderivative size = 186 \[ \int \sqrt {\pi -c^2 \pi x^2} (a+b \text {arccosh}(c x))^2 \, dx=\frac {1}{4} b^2 \sqrt {\pi } x \sqrt {1-c x} \sqrt {1+c x}+\frac {b^2 \sqrt {\pi } \sqrt {1-c x} \text {arccosh}(c x)}{4 c \sqrt {-1+c x}}-\frac {b c \sqrt {\pi } x^2 \sqrt {1-c x} (a+b \text {arccosh}(c x))}{2 \sqrt {-1+c x}}+\frac {1}{2} x \sqrt {\pi -c^2 \pi x^2} (a+b \text {arccosh}(c x))^2-\frac {\sqrt {\pi } \sqrt {1-c x} (a+b \text {arccosh}(c x))^3}{6 b c \sqrt {-1+c x}} \] Output:

1/4*b^2*Pi^(1/2)*x*(-c*x+1)^(1/2)*(c*x+1)^(1/2)+1/4*b^2*Pi^(1/2)*(-c*x+1)^ 
(1/2)*arccosh(c*x)/c/(c*x-1)^(1/2)-1/2*b*c*Pi^(1/2)*x^2*(-c*x+1)^(1/2)*(a+ 
b*arccosh(c*x))/(c*x-1)^(1/2)+1/2*x*(-Pi*c^2*x^2+Pi)^(1/2)*(a+b*arccosh(c* 
x))^2-1/6*Pi^(1/2)*(-c*x+1)^(1/2)*(a+b*arccosh(c*x))^3/b/c/(c*x-1)^(1/2)
 

Mathematica [A] (warning: unable to verify)

Time = 1.01 (sec) , antiderivative size = 202, normalized size of antiderivative = 1.09 \[ \int \sqrt {\pi -c^2 \pi x^2} (a+b \text {arccosh}(c x))^2 \, dx=\frac {1}{24} \sqrt {\pi } \left (12 a^2 x \sqrt {1-c^2 x^2}+\frac {12 a^2 \arcsin (c x)}{c}-\frac {6 a b \sqrt {1-c^2 x^2} (\cosh (2 \text {arccosh}(c x))+2 \text {arccosh}(c x) (\text {arccosh}(c x)-\sinh (2 \text {arccosh}(c x))))}{c \sqrt {\frac {-1+c x}{1+c x}} (1+c x)}-\frac {b^2 \sqrt {1-c^2 x^2} \left (4 \text {arccosh}(c x)^3+6 \text {arccosh}(c x) \cosh (2 \text {arccosh}(c x))-3 \left (1+2 \text {arccosh}(c x)^2\right ) \sinh (2 \text {arccosh}(c x))\right )}{c \sqrt {\frac {-1+c x}{1+c x}} (1+c x)}\right ) \] Input:

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

Output:

(Sqrt[Pi]*(12*a^2*x*Sqrt[1 - c^2*x^2] + (12*a^2*ArcSin[c*x])/c - (6*a*b*Sq 
rt[1 - c^2*x^2]*(Cosh[2*ArcCosh[c*x]] + 2*ArcCosh[c*x]*(ArcCosh[c*x] - Sin 
h[2*ArcCosh[c*x]])))/(c*Sqrt[(-1 + c*x)/(1 + c*x)]*(1 + c*x)) - (b^2*Sqrt[ 
1 - c^2*x^2]*(4*ArcCosh[c*x]^3 + 6*ArcCosh[c*x]*Cosh[2*ArcCosh[c*x]] - 3*( 
1 + 2*ArcCosh[c*x]^2)*Sinh[2*ArcCosh[c*x]]))/(c*Sqrt[(-1 + c*x)/(1 + c*x)] 
*(1 + c*x))))/24
 

Rubi [A] (verified)

Time = 0.80 (sec) , antiderivative size = 181, normalized size of antiderivative = 0.97, number of steps used = 5, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.192, Rules used = {6310, 6298, 101, 43, 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.

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

\(\Big \downarrow \) 6310

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

\(\Big \downarrow \) 6298

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

\(\Big \downarrow \) 101

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

\(\Big \downarrow \) 43

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

\(\Big \downarrow \) 6308

\(\displaystyle -\frac {\sqrt {\pi -\pi c^2 x^2} (a+b \text {arccosh}(c x))^3}{6 b c \sqrt {c x-1} \sqrt {c x+1}}+\frac {1}{2} x \sqrt {\pi -\pi c^2 x^2} (a+b \text {arccosh}(c x))^2-\frac {b c \sqrt {\pi -\pi c^2 x^2} \left (\frac {1}{2} x^2 (a+b \text {arccosh}(c x))-\frac {1}{2} b c \left (\frac {\text {arccosh}(c x)}{2 c^3}+\frac {x \sqrt {c x-1} \sqrt {c x+1}}{2 c^2}\right )\right )}{\sqrt {c x-1} \sqrt {c x+1}}\)

Input:

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

Output:

(x*Sqrt[Pi - c^2*Pi*x^2]*(a + b*ArcCosh[c*x])^2)/2 - (Sqrt[Pi - c^2*Pi*x^2 
]*(a + b*ArcCosh[c*x])^3)/(6*b*c*Sqrt[-1 + c*x]*Sqrt[1 + c*x]) - (b*c*Sqrt 
[Pi - c^2*Pi*x^2]*((x^2*(a + b*ArcCosh[c*x]))/2 - (b*c*((x*Sqrt[-1 + c*x]* 
Sqrt[1 + c*x])/(2*c^2) + ArcCosh[c*x]/(2*c^3)))/2))/(Sqrt[-1 + c*x]*Sqrt[1 
 + c*x])
 

Defintions of rubi rules used

rule 43

Int[1/(Sqrt[(a_) + (b_.)*(x_)]*Sqrt[(c_) + (d_.)*(x_)]), x_Symbol] :> Simp[ 
ArcCosh[b*(x/a)]/(b*Sqrt[d/b]), x] /; FreeQ[{a, b, c, d}, x] && EqQ[b*c + a 
*d, 0] && GtQ[a, 0] && GtQ[d/b, 0]
 

rule 101

Int[((a_.) + (b_.)*(x_))^2*((c_.) + (d_.)*(x_))^(n_)*((e_.) + (f_.)*(x_))^( 
p_), x_] :> Simp[b*(a + b*x)*(c + d*x)^(n + 1)*((e + f*x)^(p + 1)/(d*f*(n + 
 p + 3))), x] + Simp[1/(d*f*(n + p + 3))   Int[(c + d*x)^n*(e + f*x)^p*Simp 
[a^2*d*f*(n + p + 3) - b*(b*c*e + a*(d*e*(n + 1) + c*f*(p + 1))) + b*(a*d*f 
*(n + p + 4) - b*(d*e*(n + 2) + c*f*(p + 2)))*x, x], x], x] /; FreeQ[{a, b, 
 c, d, e, f, n, p}, x] && NeQ[n + p + 3, 0]
 

rule 6298

Int[((a_.) + ArcCosh[(c_.)*(x_)]*(b_.))^(n_.)*((d_.)*(x_))^(m_.), x_Symbol] 
 :> Simp[(d*x)^(m + 1)*((a + b*ArcCosh[c*x])^n/(d*(m + 1))), x] - Simp[b*c* 
(n/(d*(m + 1)))   Int[(d*x)^(m + 1)*((a + b*ArcCosh[c*x])^(n - 1)/(Sqrt[1 + 
 c*x]*Sqrt[-1 + c*x])), x], x] /; FreeQ[{a, b, c, d, m}, x] && IGtQ[n, 0] & 
& 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. \(532\) vs. \(2(150)=300\).

Time = 0.25 (sec) , antiderivative size = 533, normalized size of antiderivative = 2.87

method	result	size
default	\(\frac {a^{2} x \sqrt {-\pi \,c^{2} x^{2}+\pi }}{2}+\frac {a^{2} \pi \arctan \left (\frac {\sqrt {\pi \,c^{2}}\, x}{\sqrt {-\pi \,c^{2} x^{2}+\pi }}\right )}{2 \sqrt {\pi \,c^{2}}}+b^{2} \left (-\frac {\sqrt {\pi }\, \sqrt {-c^{2} x^{2}+1}\, \operatorname {arccosh}\left (c x \right )^{3}}{6 \sqrt {c x -1}\, \sqrt {c x +1}\, c}+\frac {\sqrt {\pi }\, \sqrt {-c^{2} x^{2}+1}\, \left (2 c^{3} x^{3}-2 c x +2 \sqrt {c x -1}\, \sqrt {c x +1}\, c^{2} x^{2}-\sqrt {c x -1}\, \sqrt {c x +1}\right ) \left (2 \operatorname {arccosh}\left (c x \right )^{2}-2 \,\operatorname {arccosh}\left (c x \right )+1\right )}{16 \left (c x -1\right ) \left (c x +1\right ) c}+\frac {\sqrt {\pi }\, \sqrt {-c^{2} x^{2}+1}\, \left (-2 \sqrt {c x -1}\, \sqrt {c x +1}\, c^{2} x^{2}+2 c^{3} x^{3}+\sqrt {c x -1}\, \sqrt {c x +1}-2 c x \right ) \left (2 \operatorname {arccosh}\left (c x \right )^{2}+2 \,\operatorname {arccosh}\left (c x \right )+1\right )}{16 \left (c x -1\right ) \left (c x +1\right ) c}\right )+2 a b \left (-\frac {\sqrt {\pi }\, \sqrt {-c^{2} x^{2}+1}\, \operatorname {arccosh}\left (c x \right )^{2}}{4 \sqrt {c x -1}\, \sqrt {c x +1}\, c}+\frac {\sqrt {\pi }\, \sqrt {-c^{2} x^{2}+1}\, \left (2 c^{3} x^{3}-2 c x +2 \sqrt {c x -1}\, \sqrt {c x +1}\, c^{2} x^{2}-\sqrt {c x -1}\, \sqrt {c x +1}\right ) \left (-1+2 \,\operatorname {arccosh}\left (c x \right )\right )}{16 \left (c x -1\right ) \left (c x +1\right ) c}+\frac {\sqrt {\pi }\, \sqrt {-c^{2} x^{2}+1}\, \left (-2 \sqrt {c x -1}\, \sqrt {c x +1}\, c^{2} x^{2}+2 c^{3} x^{3}+\sqrt {c x -1}\, \sqrt {c x +1}-2 c x \right ) \left (1+2 \,\operatorname {arccosh}\left (c x \right )\right )}{16 \left (c x -1\right ) \left (c x +1\right ) c}\right )\)	\(533\)
parts	\(\frac {a^{2} x \sqrt {-\pi \,c^{2} x^{2}+\pi }}{2}+\frac {a^{2} \pi \arctan \left (\frac {\sqrt {\pi \,c^{2}}\, x}{\sqrt {-\pi \,c^{2} x^{2}+\pi }}\right )}{2 \sqrt {\pi \,c^{2}}}+b^{2} \left (-\frac {\sqrt {\pi }\, \sqrt {-c^{2} x^{2}+1}\, \operatorname {arccosh}\left (c x \right )^{3}}{6 \sqrt {c x -1}\, \sqrt {c x +1}\, c}+\frac {\sqrt {\pi }\, \sqrt {-c^{2} x^{2}+1}\, \left (2 c^{3} x^{3}-2 c x +2 \sqrt {c x -1}\, \sqrt {c x +1}\, c^{2} x^{2}-\sqrt {c x -1}\, \sqrt {c x +1}\right ) \left (2 \operatorname {arccosh}\left (c x \right )^{2}-2 \,\operatorname {arccosh}\left (c x \right )+1\right )}{16 \left (c x -1\right ) \left (c x +1\right ) c}+\frac {\sqrt {\pi }\, \sqrt {-c^{2} x^{2}+1}\, \left (-2 \sqrt {c x -1}\, \sqrt {c x +1}\, c^{2} x^{2}+2 c^{3} x^{3}+\sqrt {c x -1}\, \sqrt {c x +1}-2 c x \right ) \left (2 \operatorname {arccosh}\left (c x \right )^{2}+2 \,\operatorname {arccosh}\left (c x \right )+1\right )}{16 \left (c x -1\right ) \left (c x +1\right ) c}\right )+2 a b \left (-\frac {\sqrt {\pi }\, \sqrt {-c^{2} x^{2}+1}\, \operatorname {arccosh}\left (c x \right )^{2}}{4 \sqrt {c x -1}\, \sqrt {c x +1}\, c}+\frac {\sqrt {\pi }\, \sqrt {-c^{2} x^{2}+1}\, \left (2 c^{3} x^{3}-2 c x +2 \sqrt {c x -1}\, \sqrt {c x +1}\, c^{2} x^{2}-\sqrt {c x -1}\, \sqrt {c x +1}\right ) \left (-1+2 \,\operatorname {arccosh}\left (c x \right )\right )}{16 \left (c x -1\right ) \left (c x +1\right ) c}+\frac {\sqrt {\pi }\, \sqrt {-c^{2} x^{2}+1}\, \left (-2 \sqrt {c x -1}\, \sqrt {c x +1}\, c^{2} x^{2}+2 c^{3} x^{3}+\sqrt {c x -1}\, \sqrt {c x +1}-2 c x \right ) \left (1+2 \,\operatorname {arccosh}\left (c x \right )\right )}{16 \left (c x -1\right ) \left (c x +1\right ) c}\right )\)	\(533\)

Input:

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

Output:

1/2*a^2*x*(-Pi*c^2*x^2+Pi)^(1/2)+1/2*a^2*Pi/(Pi*c^2)^(1/2)*arctan((Pi*c^2) 
^(1/2)*x/(-Pi*c^2*x^2+Pi)^(1/2))+b^2*(-1/6*Pi^(1/2)*(-c^2*x^2+1)^(1/2)/(c* 
x-1)^(1/2)/(c*x+1)^(1/2)/c*arccosh(c*x)^3+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))*(2*arccosh(c*x)^2-2*arccosh(c*x)+1)/(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)*(2*arccosh(c*x)^2+2*arccosh(c*x)+1)/(c* 
x-1)/(c*x+1)/c)+2*a*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]

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

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

Output:

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

Sympy [F]

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

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

Output:

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

Maxima [F]

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

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

Output:

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

Giac [F(-2)]

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

integrate((-pi*c^2*x^2+pi)^(1/2)*(a+b*arccosh(c*x))^2,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. \[ \int \sqrt {\pi -c^2 \pi x^2} (a+b \text {arccosh}(c x))^2 \, dx=\int {\left (a+b\,\mathrm {acosh}\left (c\,x\right )\right )}^2\,\sqrt {\Pi -\Pi \,c^2\,x^2} \,d x \] Input:

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

Output:

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

Reduce [F]

\[ \int \sqrt {\pi -c^2 \pi x^2} (a+b \text {arccosh}(c x))^2 \, dx=\frac {\sqrt {\pi }\, \left (\mathit {asin} \left (c x \right ) a^{2}+\sqrt {-c^{2} x^{2}+1}\, a^{2} c x +4 \left (\int \sqrt {-c^{2} x^{2}+1}\, \mathit {acosh} \left (c x \right )d x \right ) a b c +2 \left (\int \sqrt {-c^{2} x^{2}+1}\, \mathit {acosh} \left (c x \right )^{2}d x \right ) b^{2} c \right )}{2 c} \] Input:

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

Output:

(sqrt(pi)*(asin(c*x)*a**2 + sqrt( - c**2*x**2 + 1)*a**2*c*x + 4*int(sqrt( 
- c**2*x**2 + 1)*acosh(c*x),x)*a*b*c + 2*int(sqrt( - c**2*x**2 + 1)*acosh( 
c*x)**2,x)*b**2*c))/(2*c)