\(\int (\pi -c^2 \pi x^2)^{3/2} (a+b \text {arccosh}(c x)) \, dx\) [39]

Optimal result

Integrand size = 24, antiderivative size = 179 \[ \int \left (\pi -c^2 \pi x^2\right )^{3/2} (a+b \text {arccosh}(c x)) \, dx=-\frac {3 b c \pi ^{3/2} x^2 \sqrt {1-c x}}{16 \sqrt {-1+c x}}+\frac {b \pi ^{3/2} \sqrt {1-c x} \left (1-c^2 x^2\right )^2}{16 c \sqrt {-1+c x}}+\frac {3}{8} \pi x \sqrt {\pi -c^2 \pi x^2} (a+b \text {arccosh}(c x))+\frac {1}{4} x \left (\pi -c^2 \pi x^2\right )^{3/2} (a+b \text {arccosh}(c x))-\frac {3 \pi ^{3/2} \sqrt {1-c x} (a+b \text {arccosh}(c x))^2}{16 b c \sqrt {-1+c x}} \] Output:

-3/16*b*c*Pi^(3/2)*x^2*(-c*x+1)^(1/2)/(c*x-1)^(1/2)+1/16*b*Pi^(3/2)*(-c*x+ 
1)^(1/2)*(-c^2*x^2+1)^2/c/(c*x-1)^(1/2)+3/8*Pi*x*(-Pi*c^2*x^2+Pi)^(1/2)*(a 
+b*arccosh(c*x))+1/4*x*(-Pi*c^2*x^2+Pi)^(3/2)*(a+b*arccosh(c*x))-3/16*Pi^( 
3/2)*(-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 = 1.26 (sec) , antiderivative size = 192, normalized size of antiderivative = 1.07 \[ \int \left (\pi -c^2 \pi x^2\right )^{3/2} (a+b \text {arccosh}(c x)) \, dx=\frac {1}{128} \pi ^{3/2} \left (-16 a x \sqrt {1-c^2 x^2} \left (-5+2 c^2 x^2\right )+\frac {48 a \arcsin (c x)}{c}-\frac {16 b \sqrt {1-c^2 x^2} (\cosh (2 \text {arccosh}(c x))+2 \text {arccosh}(c x) (\text {arccosh}(c x)-\sinh (2 \text {arccosh}(c x))))}{c \sqrt {\frac {-1+c x}{1+c x}} (1+c x)}+\frac {b \sqrt {1-c^2 x^2} \left (8 \text {arccosh}(c x)^2+\cosh (4 \text {arccosh}(c x))-4 \text {arccosh}(c x) \sinh (4 \text {arccosh}(c x))\right )}{c \sqrt {\frac {-1+c x}{1+c x}} (1+c x)}\right ) \] Input:

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

Output:

(Pi^(3/2)*(-16*a*x*Sqrt[1 - c^2*x^2]*(-5 + 2*c^2*x^2) + (48*a*ArcSin[c*x]) 
/c - (16*b*Sqrt[1 - c^2*x^2]*(Cosh[2*ArcCosh[c*x]] + 2*ArcCosh[c*x]*(ArcCo 
sh[c*x] - Sinh[2*ArcCosh[c*x]])))/(c*Sqrt[(-1 + c*x)/(1 + c*x)]*(1 + c*x)) 
 + (b*Sqrt[1 - c^2*x^2]*(8*ArcCosh[c*x]^2 + Cosh[4*ArcCosh[c*x]] - 4*ArcCo 
sh[c*x]*Sinh[4*ArcCosh[c*x]]))/(c*Sqrt[(-1 + c*x)/(1 + c*x)]*(1 + c*x))))/ 
128
 

Rubi [A] (verified)

Time = 0.66 (sec) , antiderivative size = 216, normalized size of antiderivative = 1.21, number of steps used = 8, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.333, Rules used = {6312, 25, 82, 244, 2009, 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.

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

\(\Big \downarrow \) 6312

\(\displaystyle \frac {3}{4} \pi \int \sqrt {\pi -c^2 \pi x^2} (a+b \text {arccosh}(c x))dx+\frac {\pi b c \sqrt {\pi -\pi c^2 x^2} \int -x (1-c x) (c x+1)dx}{4 \sqrt {c x-1} \sqrt {c x+1}}+\frac {1}{4} x \left (\pi -\pi c^2 x^2\right )^{3/2} (a+b \text {arccosh}(c x))\)

\(\Big \downarrow \) 25

\(\displaystyle \frac {3}{4} \pi \int \sqrt {\pi -c^2 \pi x^2} (a+b \text {arccosh}(c x))dx-\frac {\pi b c \sqrt {\pi -\pi c^2 x^2} \int x (1-c x) (c x+1)dx}{4 \sqrt {c x-1} \sqrt {c x+1}}+\frac {1}{4} x \left (\pi -\pi c^2 x^2\right )^{3/2} (a+b \text {arccosh}(c x))\)

\(\Big \downarrow \) 82

\(\displaystyle \frac {3}{4} \pi \int \sqrt {\pi -c^2 \pi x^2} (a+b \text {arccosh}(c x))dx-\frac {\pi b c \sqrt {\pi -\pi c^2 x^2} \int x \left (1-c^2 x^2\right )dx}{4 \sqrt {c x-1} \sqrt {c x+1}}+\frac {1}{4} x \left (\pi -\pi c^2 x^2\right )^{3/2} (a+b \text {arccosh}(c x))\)

\(\Big \downarrow \) 244

\(\displaystyle \frac {3}{4} \pi \int \sqrt {\pi -c^2 \pi x^2} (a+b \text {arccosh}(c x))dx-\frac {\pi b c \sqrt {\pi -\pi c^2 x^2} \int \left (x-c^2 x^3\right )dx}{4 \sqrt {c x-1} \sqrt {c x+1}}+\frac {1}{4} x \left (\pi -\pi c^2 x^2\right )^{3/2} (a+b \text {arccosh}(c x))\)

\(\Big \downarrow \) 2009

\(\displaystyle \frac {3}{4} \pi \int \sqrt {\pi -c^2 \pi x^2} (a+b \text {arccosh}(c x))dx+\frac {1}{4} x \left (\pi -\pi c^2 x^2\right )^{3/2} (a+b \text {arccosh}(c x))-\frac {\pi b c \left (\frac {x^2}{2}-\frac {c^2 x^4}{4}\right ) \sqrt {\pi -\pi c^2 x^2}}{4 \sqrt {c x-1} \sqrt {c x+1}}\)

\(\Big \downarrow \) 6310

\(\displaystyle \frac {3}{4} \pi \left (-\frac {\sqrt {\pi -\pi c^2 x^2} \int \frac {a+b \text {arccosh}(c x)}{\sqrt {c x-1} \sqrt {c x+1}}dx}{2 \sqrt {c x-1} \sqrt {c x+1}}-\frac {b c \sqrt {\pi -\pi c^2 x^2} \int xdx}{2 \sqrt {c x-1} \sqrt {c x+1}}+\frac {1}{2} x \sqrt {\pi -\pi c^2 x^2} (a+b \text {arccosh}(c x))\right )+\frac {1}{4} x \left (\pi -\pi c^2 x^2\right )^{3/2} (a+b \text {arccosh}(c x))-\frac {\pi b c \left (\frac {x^2}{2}-\frac {c^2 x^4}{4}\right ) \sqrt {\pi -\pi c^2 x^2}}{4 \sqrt {c x-1} \sqrt {c x+1}}\)

\(\Big \downarrow \) 15

\(\displaystyle \frac {3}{4} \pi \left (-\frac {\sqrt {\pi -\pi c^2 x^2} \int \frac {a+b \text {arccosh}(c x)}{\sqrt {c x-1} \sqrt {c x+1}}dx}{2 \sqrt {c x-1} \sqrt {c x+1}}+\frac {1}{2} x \sqrt {\pi -\pi c^2 x^2} (a+b \text {arccosh}(c x))-\frac {b c x^2 \sqrt {\pi -\pi c^2 x^2}}{4 \sqrt {c x-1} \sqrt {c x+1}}\right )+\frac {1}{4} x \left (\pi -\pi c^2 x^2\right )^{3/2} (a+b \text {arccosh}(c x))-\frac {\pi b c \left (\frac {x^2}{2}-\frac {c^2 x^4}{4}\right ) \sqrt {\pi -\pi c^2 x^2}}{4 \sqrt {c x-1} \sqrt {c x+1}}\)

\(\Big \downarrow \) 6308

\(\displaystyle \frac {1}{4} x \left (\pi -\pi c^2 x^2\right )^{3/2} (a+b \text {arccosh}(c x))+\frac {3}{4} \pi \left (\frac {1}{2} x \sqrt {\pi -\pi c^2 x^2} (a+b \text {arccosh}(c x))-\frac {\sqrt {\pi -\pi c^2 x^2} (a+b \text {arccosh}(c x))^2}{4 b c \sqrt {c x-1} \sqrt {c x+1}}-\frac {b c x^2 \sqrt {\pi -\pi c^2 x^2}}{4 \sqrt {c x-1} \sqrt {c x+1}}\right )-\frac {\pi b c \left (\frac {x^2}{2}-\frac {c^2 x^4}{4}\right ) \sqrt {\pi -\pi c^2 x^2}}{4 \sqrt {c x-1} \sqrt {c x+1}}\)

Input:

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

Output:

-1/4*(b*c*Pi*Sqrt[Pi - c^2*Pi*x^2]*(x^2/2 - (c^2*x^4)/4))/(Sqrt[-1 + c*x]* 
Sqrt[1 + c*x]) + (x*(Pi - c^2*Pi*x^2)^(3/2)*(a + b*ArcCosh[c*x]))/4 + (3*P 
i*(-1/4*(b*c*x^2*Sqrt[Pi - c^2*Pi*x^2])/(Sqrt[-1 + c*x]*Sqrt[1 + c*x]) + ( 
x*Sqrt[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])))/4
 

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 25

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

rule 82

Int[((a_) + (b_.)*(x_))^(m_.)*((c_) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_) 
)^(p_.), x_] :> Int[(a*c + b*d*x^2)^m*(e + f*x)^p, x] /; FreeQ[{a, b, c, d, 
 e, f, m, n, p}, x] && EqQ[b*c + a*d, 0] && EqQ[n, m] && IntegerQ[m]
 

rule 244

Int[((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^2)^(p_.), x_Symbol] :> Int[Expand 
Integrand[(c*x)^m*(a + b*x^2)^p, x], x] /; FreeQ[{a, b, c, m}, x] && IGtQ[p 
, 0]
 

rule 2009

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

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]
 

rule 6312

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

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(545\) vs. \(2(147)=294\).

Time = 0.24 (sec) , antiderivative size = 546, normalized size of antiderivative = 3.05

method	result	size
default	\(\frac {a x \left (-\pi \,c^{2} x^{2}+\pi \right )^{\frac {3}{2}}}{4}+\frac {3 a \pi x \sqrt {-\pi \,c^{2} x^{2}+\pi }}{8}+\frac {3 a \,\pi ^{2} \arctan \left (\frac {\sqrt {\pi \,c^{2}}\, x}{\sqrt {-\pi \,c^{2} x^{2}+\pi }}\right )}{8 \sqrt {\pi \,c^{2}}}+b \left (-\frac {3 \pi ^{\frac {3}{2}} \sqrt {-c^{2} x^{2}+1}\, \operatorname {arccosh}\left (c x \right )^{2}}{16 \sqrt {c x -1}\, \sqrt {c x +1}\, c}-\frac {\pi ^{\frac {3}{2}} \sqrt {-c^{2} x^{2}+1}\, \left (8 c^{5} x^{5}-12 c^{3} x^{3}+8 c^{4} x^{4} \sqrt {c x -1}\, \sqrt {c x +1}+4 c x -8 \sqrt {c x -1}\, \sqrt {c x +1}\, c^{2} x^{2}+\sqrt {c x -1}\, \sqrt {c x +1}\right ) \left (-1+4 \,\operatorname {arccosh}\left (c x \right )\right )}{256 \left (c x -1\right ) \left (c x +1\right ) c}+\frac {\pi ^{\frac {3}{2}} \sqrt {-c^{2} x^{2}+1}\, \left (2 c^{3} x^{3}-2 c x +2 \sqrt {c x -1}\, \sqrt {c x +1}\, c^{2} x^{2}-\sqrt {c x -1}\, \sqrt {c x +1}\right ) \left (-1+2 \,\operatorname {arccosh}\left (c x \right )\right )}{16 \left (c x -1\right ) \left (c x +1\right ) c}+\frac {\pi ^{\frac {3}{2}} \sqrt {-c^{2} x^{2}+1}\, \left (-2 \sqrt {c x -1}\, \sqrt {c x +1}\, c^{2} x^{2}+2 c^{3} x^{3}+\sqrt {c x -1}\, \sqrt {c x +1}-2 c x \right ) \left (1+2 \,\operatorname {arccosh}\left (c x \right )\right )}{16 \left (c x -1\right ) \left (c x +1\right ) c}-\frac {\pi ^{\frac {3}{2}} \sqrt {-c^{2} x^{2}+1}\, \left (-8 c^{4} x^{4} \sqrt {c x -1}\, \sqrt {c x +1}+8 c^{5} x^{5}+8 \sqrt {c x -1}\, \sqrt {c x +1}\, c^{2} x^{2}-12 c^{3} x^{3}-\sqrt {c x -1}\, \sqrt {c x +1}+4 c x \right ) \left (1+4 \,\operatorname {arccosh}\left (c x \right )\right )}{256 \left (c x -1\right ) \left (c x +1\right ) c}\right )\)	\(546\)
parts	\(\frac {a x \left (-\pi \,c^{2} x^{2}+\pi \right )^{\frac {3}{2}}}{4}+\frac {3 a \pi x \sqrt {-\pi \,c^{2} x^{2}+\pi }}{8}+\frac {3 a \,\pi ^{2} \arctan \left (\frac {\sqrt {\pi \,c^{2}}\, x}{\sqrt {-\pi \,c^{2} x^{2}+\pi }}\right )}{8 \sqrt {\pi \,c^{2}}}+b \left (-\frac {3 \pi ^{\frac {3}{2}} \sqrt {-c^{2} x^{2}+1}\, \operatorname {arccosh}\left (c x \right )^{2}}{16 \sqrt {c x -1}\, \sqrt {c x +1}\, c}-\frac {\pi ^{\frac {3}{2}} \sqrt {-c^{2} x^{2}+1}\, \left (8 c^{5} x^{5}-12 c^{3} x^{3}+8 c^{4} x^{4} \sqrt {c x -1}\, \sqrt {c x +1}+4 c x -8 \sqrt {c x -1}\, \sqrt {c x +1}\, c^{2} x^{2}+\sqrt {c x -1}\, \sqrt {c x +1}\right ) \left (-1+4 \,\operatorname {arccosh}\left (c x \right )\right )}{256 \left (c x -1\right ) \left (c x +1\right ) c}+\frac {\pi ^{\frac {3}{2}} \sqrt {-c^{2} x^{2}+1}\, \left (2 c^{3} x^{3}-2 c x +2 \sqrt {c x -1}\, \sqrt {c x +1}\, c^{2} x^{2}-\sqrt {c x -1}\, \sqrt {c x +1}\right ) \left (-1+2 \,\operatorname {arccosh}\left (c x \right )\right )}{16 \left (c x -1\right ) \left (c x +1\right ) c}+\frac {\pi ^{\frac {3}{2}} \sqrt {-c^{2} x^{2}+1}\, \left (-2 \sqrt {c x -1}\, \sqrt {c x +1}\, c^{2} x^{2}+2 c^{3} x^{3}+\sqrt {c x -1}\, \sqrt {c x +1}-2 c x \right ) \left (1+2 \,\operatorname {arccosh}\left (c x \right )\right )}{16 \left (c x -1\right ) \left (c x +1\right ) c}-\frac {\pi ^{\frac {3}{2}} \sqrt {-c^{2} x^{2}+1}\, \left (-8 c^{4} x^{4} \sqrt {c x -1}\, \sqrt {c x +1}+8 c^{5} x^{5}+8 \sqrt {c x -1}\, \sqrt {c x +1}\, c^{2} x^{2}-12 c^{3} x^{3}-\sqrt {c x -1}\, \sqrt {c x +1}+4 c x \right ) \left (1+4 \,\operatorname {arccosh}\left (c x \right )\right )}{256 \left (c x -1\right ) \left (c x +1\right ) c}\right )\)	\(546\)

Input:

int((-Pi*c^2*x^2+Pi)^(3/2)*(a+b*arccosh(c*x)),x,method=_RETURNVERBOSE)
 

Output:

1/4*a*x*(-Pi*c^2*x^2+Pi)^(3/2)+3/8*a*Pi*x*(-Pi*c^2*x^2+Pi)^(1/2)+3/8*a*Pi^ 
2/(Pi*c^2)^(1/2)*arctan((Pi*c^2)^(1/2)*x/(-Pi*c^2*x^2+Pi)^(1/2))+b*(-3/16* 
Pi^(3/2)*(-c^2*x^2+1)^(1/2)/(c*x-1)^(1/2)/(c*x+1)^(1/2)/c*arccosh(c*x)^2-1 
/256*Pi^(3/2)*(-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))*(-1+4*arccosh(c*x))/(c*x-1)/(c*x+1)/c+1/16*Pi^(3/2)*(-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^(3/ 
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-1/25 
6*Pi^(3/2)*(-c^2*x^2+1)^(1/2)*(-8*c^4*x^4*(c*x-1)^(1/2)*(c*x+1)^(1/2)+8*c^ 
5*x^5+8*(c*x-1)^(1/2)*(c*x+1)^(1/2)*c^2*x^2-12*c^3*x^3-(c*x-1)^(1/2)*(c*x+ 
1)^(1/2)+4*c*x)*(1+4*arccosh(c*x))/(c*x-1)/(c*x+1)/c)
 

Fricas [F]

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

integrate((-pi*c^2*x^2+pi)^(3/2)*(a+b*arccosh(c*x)),x, algorithm="fricas")
 

Output:

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

Sympy [F(-1)]

Timed out. \[ \int \left (\pi -c^2 \pi x^2\right )^{3/2} (a+b \text {arccosh}(c x)) \, dx=\text {Timed out} \] Input:

integrate((-pi*c**2*x**2+pi)**(3/2)*(a+b*acosh(c*x)),x)
 

Output:

Timed out
 

Maxima [F]

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

integrate((-pi*c^2*x^2+pi)^(3/2)*(a+b*arccosh(c*x)),x, algorithm="maxima")
 

Output:

1/8*(3*pi*sqrt(pi - pi*c^2*x^2)*x + 2*(pi - pi*c^2*x^2)^(3/2)*x + 3*pi^(3/ 
2)*arcsin(c*x)/c)*a + b*integrate((pi - pi*c^2*x^2)^(3/2)*log(c*x + sqrt(c 
*x + 1)*sqrt(c*x - 1)), x)
 

Giac [F(-2)]

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

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

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

Output:

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

Reduce [F]

\[ \int \left (\pi -c^2 \pi x^2\right )^{3/2} (a+b \text {arccosh}(c x)) \, dx=\frac {\sqrt {\pi }\, \pi \left (3 \mathit {asin} \left (c x \right ) a -2 \sqrt {-c^{2} x^{2}+1}\, a \,c^{3} x^{3}+5 \sqrt {-c^{2} x^{2}+1}\, a c x -8 \left (\int \sqrt {-c^{2} x^{2}+1}\, \mathit {acosh} \left (c x \right ) x^{2}d x \right ) b \,c^{3}+8 \left (\int \sqrt {-c^{2} x^{2}+1}\, \mathit {acosh} \left (c x \right )d x \right ) b c \right )}{8 c} \] Input:

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

Output:

(sqrt(pi)*pi*(3*asin(c*x)*a - 2*sqrt( - c**2*x**2 + 1)*a*c**3*x**3 + 5*sqr 
t( - c**2*x**2 + 1)*a*c*x - 8*int(sqrt( - c**2*x**2 + 1)*acosh(c*x)*x**2,x 
)*b*c**3 + 8*int(sqrt( - c**2*x**2 + 1)*acosh(c*x),x)*b*c))/(8*c)