\(\int \frac {(a+b \text {arccosh}(c x))^2}{(\pi -c^2 \pi x^2)^{3/2}} \, dx\) [48]

Optimal result

Integrand size = 26, antiderivative size = 162 \[ \int \frac {(a+b \text {arccosh}(c x))^2}{\left (\pi -c^2 \pi x^2\right )^{3/2}} \, dx=\frac {\sqrt {-1+c x} (a+b \text {arccosh}(c x))^2}{c \pi ^{3/2} \sqrt {1-c x}}+\frac {x (a+b \text {arccosh}(c x))^2}{\pi \sqrt {\pi -c^2 \pi x^2}}-\frac {2 b \sqrt {-1+c x} (a+b \text {arccosh}(c x)) \log \left (1-e^{2 \text {arccosh}(c x)}\right )}{c \pi ^{3/2} \sqrt {1-c x}}-\frac {b^2 \sqrt {-1+c x} \operatorname {PolyLog}\left (2,e^{2 \text {arccosh}(c x)}\right )}{c \pi ^{3/2} \sqrt {1-c x}} \] Output:

(c*x-1)^(1/2)*(a+b*arccosh(c*x))^2/c/Pi^(3/2)/(-c*x+1)^(1/2)+x*(a+b*arccos 
h(c*x))^2/Pi/(-Pi*c^2*x^2+Pi)^(1/2)-2*b*(c*x-1)^(1/2)*(a+b*arccosh(c*x))*l 
n(1-(c*x+(c*x-1)^(1/2)*(c*x+1)^(1/2))^2)/c/Pi^(3/2)/(-c*x+1)^(1/2)-b^2*(c* 
x-1)^(1/2)*polylog(2,(c*x+(c*x-1)^(1/2)*(c*x+1)^(1/2))^2)/c/Pi^(3/2)/(-c*x 
+1)^(1/2)
 

Mathematica [A] (verified)

Time = 0.33 (sec) , antiderivative size = 127, normalized size of antiderivative = 0.78 \[ \int \frac {(a+b \text {arccosh}(c x))^2}{\left (\pi -c^2 \pi x^2\right )^{3/2}} \, dx=\frac {x (a+b \text {arccosh}(c x))^2+\frac {\sqrt {-1+c x} \sqrt {1+c x} \left ((a+b \text {arccosh}(c x)) \left (a+b \text {arccosh}(c x)-2 b \log \left (1-e^{\text {arccosh}(c x)}\right )-2 b \log \left (1+e^{\text {arccosh}(c x)}\right )\right )-2 b^2 \operatorname {PolyLog}\left (2,-e^{\text {arccosh}(c x)}\right )-2 b^2 \operatorname {PolyLog}\left (2,e^{\text {arccosh}(c x)}\right )\right )}{c}}{\pi ^{3/2} \sqrt {1-c^2 x^2}} \] Input:

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

Output:

(x*(a + b*ArcCosh[c*x])^2 + (Sqrt[-1 + c*x]*Sqrt[1 + c*x]*((a + b*ArcCosh[ 
c*x])*(a + b*ArcCosh[c*x] - 2*b*Log[1 - E^ArcCosh[c*x]] - 2*b*Log[1 + E^Ar 
cCosh[c*x]]) - 2*b^2*PolyLog[2, -E^ArcCosh[c*x]] - 2*b^2*PolyLog[2, E^ArcC 
osh[c*x]]))/c)/(Pi^(3/2)*Sqrt[1 - c^2*x^2])
 

Rubi [C] (verified)

Result contains complex when optimal does not.

Time = 0.67 (sec) , antiderivative size = 140, normalized size of antiderivative = 0.86, number of steps used = 10, number of rules used = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.346, Rules used = {6314, 6328, 3042, 26, 4199, 25, 2620, 2715, 2838}

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.

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

\(\Big \downarrow \) 6314

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

\(\Big \downarrow \) 6328

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 26

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

\(\Big \downarrow \) 4199

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

\(\Big \downarrow \) 25

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

\(\Big \downarrow \) 2620

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

\(\Big \downarrow \) 2715

\(\displaystyle \frac {x (a+b \text {arccosh}(c x))^2}{\pi \sqrt {\pi -\pi c^2 x^2}}+\frac {2 i b \sqrt {c x-1} \sqrt {c x+1} \left (-2 i \left (\frac {1}{4} b \int e^{-2 \text {arccosh}(c x)} \log \left (1-e^{2 \text {arccosh}(c x)}\right )de^{2 \text {arccosh}(c x)}-\frac {1}{2} \log \left (1-e^{2 \text {arccosh}(c x)}\right ) (a+b \text {arccosh}(c x))\right )-\frac {i (a+b \text {arccosh}(c x))^2}{2 b}\right )}{\pi c \sqrt {\pi -\pi c^2 x^2}}\)

\(\Big \downarrow \) 2838

\(\displaystyle \frac {x (a+b \text {arccosh}(c x))^2}{\pi \sqrt {\pi -\pi c^2 x^2}}+\frac {2 i b \sqrt {c x-1} \sqrt {c x+1} \left (-2 i \left (-\frac {1}{2} \log \left (1-e^{2 \text {arccosh}(c x)}\right ) (a+b \text {arccosh}(c x))-\frac {1}{4} b \operatorname {PolyLog}\left (2,e^{2 \text {arccosh}(c x)}\right )\right )-\frac {i (a+b \text {arccosh}(c x))^2}{2 b}\right )}{\pi c \sqrt {\pi -\pi c^2 x^2}}\)

Input:

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

Output:

(x*(a + b*ArcCosh[c*x])^2)/(Pi*Sqrt[Pi - c^2*Pi*x^2]) + ((2*I)*b*Sqrt[-1 + 
 c*x]*Sqrt[1 + c*x]*(((-1/2*I)*(a + b*ArcCosh[c*x])^2)/b - (2*I)*(-1/2*((a 
 + b*ArcCosh[c*x])*Log[1 - E^(2*ArcCosh[c*x])]) - (b*PolyLog[2, E^(2*ArcCo 
sh[c*x])])/4)))/(c*Pi*Sqrt[Pi - c^2*Pi*x^2])
 

Defintions of rubi rules used

rule 25

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

rule 26

Int[(Complex[0, a_])*(Fx_), x_Symbol] :> Simp[(Complex[Identity[0], a])   I 
nt[Fx, x], x] /; FreeQ[a, x] && EqQ[a^2, 1]
 

rule 2620

Int[(((F_)^((g_.)*((e_.) + (f_.)*(x_))))^(n_.)*((c_.) + (d_.)*(x_))^(m_.))/ 
((a_) + (b_.)*((F_)^((g_.)*((e_.) + (f_.)*(x_))))^(n_.)), x_Symbol] :> Simp 
[((c + d*x)^m/(b*f*g*n*Log[F]))*Log[1 + b*((F^(g*(e + f*x)))^n/a)], x] - Si 
mp[d*(m/(b*f*g*n*Log[F]))   Int[(c + d*x)^(m - 1)*Log[1 + b*((F^(g*(e + f*x 
)))^n/a)], x], x] /; FreeQ[{F, a, b, c, d, e, f, g, n}, x] && IGtQ[m, 0]
 

rule 2715

Int[Log[(a_) + (b_.)*((F_)^((e_.)*((c_.) + (d_.)*(x_))))^(n_.)], x_Symbol] 
:> Simp[1/(d*e*n*Log[F])   Subst[Int[Log[a + b*x]/x, x], x, (F^(e*(c + d*x) 
))^n], x] /; FreeQ[{F, a, b, c, d, e, n}, x] && GtQ[a, 0]
 

rule 2838

Int[Log[(c_.)*((d_) + (e_.)*(x_)^(n_.))]/(x_), x_Symbol] :> Simp[-PolyLog[2 
, (-c)*e*x^n]/n, x] /; FreeQ[{c, d, e, n}, x] && EqQ[c*d, 1]
 

rule 3042

Int[u_, x_Symbol] :> Int[DeactivateTrig[u, x], x] /; FunctionOfTrigOfLinear 
Q[u, x]
 

rule 4199

Int[((c_.) + (d_.)*(x_))^(m_.)*tan[(e_.) + Pi*(k_.) + (Complex[0, fz_])*(f_ 
.)*(x_)], x_Symbol] :> Simp[(-I)*((c + d*x)^(m + 1)/(d*(m + 1))), x] + Simp 
[2*I   Int[((c + d*x)^m*(E^(2*((-I)*e + f*fz*x))/(1 + E^(2*((-I)*e + f*fz*x 
))/E^(2*I*k*Pi))))/E^(2*I*k*Pi), x], x] /; FreeQ[{c, d, e, f, fz}, x] && In 
tegerQ[4*k] && IGtQ[m, 0]
 

rule 6314

Int[((a_.) + ArcCosh[(c_.)*(x_)]*(b_.))^(n_.)/((d_) + (e_.)*(x_)^2)^(3/2), 
x_Symbol] :> Simp[x*((a + b*ArcCosh[c*x])^n/(d*Sqrt[d + e*x^2])), x] + Simp 
[b*c*(n/d)*Simp[Sqrt[1 + c*x]*(Sqrt[-1 + c*x]/Sqrt[d + e*x^2])]   Int[x*((a 
 + b*ArcCosh[c*x])^(n - 1)/(1 - c^2*x^2)), x], x] /; FreeQ[{a, b, c, d, e}, 
 x] && EqQ[c^2*d + e, 0] && GtQ[n, 0]
 

rule 6328

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

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(559\) vs. \(2(168)=336\).

Time = 0.31 (sec) , antiderivative size = 560, normalized size of antiderivative = 3.46

method	result	size
default	\(\frac {a^{2} x}{\pi \sqrt {-\pi \,c^{2} x^{2}+\pi }}-\frac {b^{2} \sqrt {c x +1}\, \sqrt {c x -1}\, \sqrt {-c^{2} x^{2}+1}\, \operatorname {arccosh}\left (c x \right )^{2}}{\pi ^{\frac {3}{2}} c \left (c^{2} x^{2}-1\right )}-\frac {b^{2} \sqrt {-c^{2} x^{2}+1}\, \operatorname {arccosh}\left (c x \right )^{2} x}{\pi ^{\frac {3}{2}} \left (c^{2} x^{2}-1\right )}+\frac {2 b^{2} \sqrt {c x +1}\, \sqrt {c x -1}\, \sqrt {-c^{2} x^{2}+1}\, \operatorname {arccosh}\left (c x \right ) \ln \left (1-c x -\sqrt {c x -1}\, \sqrt {c x +1}\right )}{\pi ^{\frac {3}{2}} c \left (c^{2} x^{2}-1\right )}+\frac {2 b^{2} \sqrt {c x +1}\, \sqrt {c x -1}\, \sqrt {-c^{2} x^{2}+1}\, \operatorname {polylog}\left (2, c x +\sqrt {c x -1}\, \sqrt {c x +1}\right )}{\pi ^{\frac {3}{2}} c \left (c^{2} x^{2}-1\right )}+\frac {2 b^{2} \sqrt {c x +1}\, \sqrt {c x -1}\, \sqrt {-c^{2} x^{2}+1}\, \operatorname {arccosh}\left (c x \right ) \ln \left (1+c x +\sqrt {c x -1}\, \sqrt {c x +1}\right )}{\pi ^{\frac {3}{2}} c \left (c^{2} x^{2}-1\right )}+\frac {2 b^{2} \sqrt {c x +1}\, \sqrt {c x -1}\, \sqrt {-c^{2} x^{2}+1}\, \operatorname {polylog}\left (2, -c x -\sqrt {c x -1}\, \sqrt {c x +1}\right )}{\pi ^{\frac {3}{2}} c \left (c^{2} x^{2}-1\right )}-\frac {2 a b \sqrt {-c^{2} x^{2}+1}\, \sqrt {c x -1}\, \sqrt {c x +1}\, \operatorname {arccosh}\left (c x \right )}{\pi ^{\frac {3}{2}} c \left (c^{2} x^{2}-1\right )}-\frac {2 a b \sqrt {-c^{2} x^{2}+1}\, \operatorname {arccosh}\left (c x \right ) x}{\pi ^{\frac {3}{2}} \left (c^{2} x^{2}-1\right )}+\frac {2 a b \sqrt {-c^{2} x^{2}+1}\, \sqrt {c x -1}\, \sqrt {c x +1}\, \ln \left (\left (c x +\sqrt {c x -1}\, \sqrt {c x +1}\right )^{2}-1\right )}{\pi ^{\frac {3}{2}} c \left (c^{2} x^{2}-1\right )}\)	\(560\)
parts	\(\frac {a^{2} x}{\pi \sqrt {-\pi \,c^{2} x^{2}+\pi }}-\frac {b^{2} \sqrt {c x +1}\, \sqrt {c x -1}\, \sqrt {-c^{2} x^{2}+1}\, \operatorname {arccosh}\left (c x \right )^{2}}{\pi ^{\frac {3}{2}} c \left (c^{2} x^{2}-1\right )}-\frac {b^{2} \sqrt {-c^{2} x^{2}+1}\, \operatorname {arccosh}\left (c x \right )^{2} x}{\pi ^{\frac {3}{2}} \left (c^{2} x^{2}-1\right )}+\frac {2 b^{2} \sqrt {c x +1}\, \sqrt {c x -1}\, \sqrt {-c^{2} x^{2}+1}\, \operatorname {arccosh}\left (c x \right ) \ln \left (1-c x -\sqrt {c x -1}\, \sqrt {c x +1}\right )}{\pi ^{\frac {3}{2}} c \left (c^{2} x^{2}-1\right )}+\frac {2 b^{2} \sqrt {c x +1}\, \sqrt {c x -1}\, \sqrt {-c^{2} x^{2}+1}\, \operatorname {polylog}\left (2, c x +\sqrt {c x -1}\, \sqrt {c x +1}\right )}{\pi ^{\frac {3}{2}} c \left (c^{2} x^{2}-1\right )}+\frac {2 b^{2} \sqrt {c x +1}\, \sqrt {c x -1}\, \sqrt {-c^{2} x^{2}+1}\, \operatorname {arccosh}\left (c x \right ) \ln \left (1+c x +\sqrt {c x -1}\, \sqrt {c x +1}\right )}{\pi ^{\frac {3}{2}} c \left (c^{2} x^{2}-1\right )}+\frac {2 b^{2} \sqrt {c x +1}\, \sqrt {c x -1}\, \sqrt {-c^{2} x^{2}+1}\, \operatorname {polylog}\left (2, -c x -\sqrt {c x -1}\, \sqrt {c x +1}\right )}{\pi ^{\frac {3}{2}} c \left (c^{2} x^{2}-1\right )}-\frac {2 a b \sqrt {-c^{2} x^{2}+1}\, \sqrt {c x -1}\, \sqrt {c x +1}\, \operatorname {arccosh}\left (c x \right )}{\pi ^{\frac {3}{2}} c \left (c^{2} x^{2}-1\right )}-\frac {2 a b \sqrt {-c^{2} x^{2}+1}\, \operatorname {arccosh}\left (c x \right ) x}{\pi ^{\frac {3}{2}} \left (c^{2} x^{2}-1\right )}+\frac {2 a b \sqrt {-c^{2} x^{2}+1}\, \sqrt {c x -1}\, \sqrt {c x +1}\, \ln \left (\left (c x +\sqrt {c x -1}\, \sqrt {c x +1}\right )^{2}-1\right )}{\pi ^{\frac {3}{2}} c \left (c^{2} x^{2}-1\right )}\)	\(560\)

Input:

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

Output:

a^2/Pi*x/(-Pi*c^2*x^2+Pi)^(1/2)-b^2/Pi^(3/2)*(c*x+1)^(1/2)*(c*x-1)^(1/2)*( 
-c^2*x^2+1)^(1/2)/c/(c^2*x^2-1)*arccosh(c*x)^2-b^2/Pi^(3/2)*(-c^2*x^2+1)^( 
1/2)*arccosh(c*x)^2/(c^2*x^2-1)*x+2*b^2/Pi^(3/2)*(c*x+1)^(1/2)*(c*x-1)^(1/ 
2)*(-c^2*x^2+1)^(1/2)/c/(c^2*x^2-1)*arccosh(c*x)*ln(1-c*x-(c*x-1)^(1/2)*(c 
*x+1)^(1/2))+2*b^2/Pi^(3/2)*(c*x+1)^(1/2)*(c*x-1)^(1/2)*(-c^2*x^2+1)^(1/2) 
/c/(c^2*x^2-1)*polylog(2,c*x+(c*x-1)^(1/2)*(c*x+1)^(1/2))+2*b^2/Pi^(3/2)*( 
c*x+1)^(1/2)*(c*x-1)^(1/2)*(-c^2*x^2+1)^(1/2)/c/(c^2*x^2-1)*arccosh(c*x)*l 
n(1+c*x+(c*x-1)^(1/2)*(c*x+1)^(1/2))+2*b^2/Pi^(3/2)*(c*x+1)^(1/2)*(c*x-1)^ 
(1/2)*(-c^2*x^2+1)^(1/2)/c/(c^2*x^2-1)*polylog(2,-c*x-(c*x-1)^(1/2)*(c*x+1 
)^(1/2))-2*a*b/Pi^(3/2)*(-c^2*x^2+1)^(1/2)*(c*x-1)^(1/2)*(c*x+1)^(1/2)/c/( 
c^2*x^2-1)*arccosh(c*x)-2*a*b*(-c^2*x^2+1)^(1/2)/Pi^(3/2)*arccosh(c*x)/(c^ 
2*x^2-1)*x+2*a*b/Pi^(3/2)*(-c^2*x^2+1)^(1/2)*(c*x-1)^(1/2)*(c*x+1)^(1/2)/c 
/(c^2*x^2-1)*ln((c*x+(c*x-1)^(1/2)*(c*x+1)^(1/2))^2-1)
 

Fricas [F]

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

integrate((a+b*arccosh(c*x))^2/(-pi*c^2*x^2+pi)^(3/2),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)/(pi^2*c^4*x^4 - 2*pi^2*c^2*x^2 + pi^2), x)
 

Sympy [F]

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

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

Output:

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

Maxima [F]

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

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

Output:

-a*b*c*sqrt(-1/(pi*c^4))*log(x^2 - 1/c^2)/pi + b^2*integrate(log(c*x + sqr 
t(c*x + 1)*sqrt(c*x - 1))^2/(pi - pi*c^2*x^2)^(3/2), x) + 2*a*b*x*arccosh( 
c*x)/(pi*sqrt(pi - pi*c^2*x^2)) + a^2*x/(pi*sqrt(pi - pi*c^2*x^2))
 

Giac [F(-2)]

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

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

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

Output:

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

Reduce [F]

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

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

Output:

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