\(\int \frac {(a+b \text {arccosh}(c x))^2}{x^2 \sqrt {d-c^2 d x^2}} \, dx\) [190]

Optimal result

Integrand size = 29, antiderivative size = 195 \[ \int \frac {(a+b \text {arccosh}(c x))^2}{x^2 \sqrt {d-c^2 d x^2}} \, dx=-\frac {\sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))^2}{d x}-\frac {c \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))^2}{d \sqrt {-1+c x} \sqrt {1+c x}}+\frac {2 b c \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x)) \log \left (1+e^{2 \text {arccosh}(c x)}\right )}{d \sqrt {-1+c x} \sqrt {1+c x}}+\frac {b^2 c \sqrt {d-c^2 d x^2} \operatorname {PolyLog}\left (2,-e^{2 \text {arccosh}(c x)}\right )}{d \sqrt {-1+c x} \sqrt {1+c x}} \] Output:

-(-c^2*d*x^2+d)^(1/2)*(a+b*arccosh(c*x))^2/d/x-c*(-c^2*d*x^2+d)^(1/2)*(a+b 
*arccosh(c*x))^2/d/(c*x-1)^(1/2)/(c*x+1)^(1/2)+2*b*c*(-c^2*d*x^2+d)^(1/2)* 
(a+b*arccosh(c*x))*ln(1+(c*x+(c*x-1)^(1/2)*(c*x+1)^(1/2))^2)/d/(c*x-1)^(1/ 
2)/(c*x+1)^(1/2)+b^2*c*(-c^2*d*x^2+d)^(1/2)*polylog(2,-(c*x+(c*x-1)^(1/2)* 
(c*x+1)^(1/2))^2)/d/(c*x-1)^(1/2)/(c*x+1)^(1/2)
 

Mathematica [A] (verified)

Time = 0.89 (sec) , antiderivative size = 237, normalized size of antiderivative = 1.22 \[ \int \frac {(a+b \text {arccosh}(c x))^2}{x^2 \sqrt {d-c^2 d x^2}} \, dx=-\frac {a^2 \sqrt {-d \left (-1+c^2 x^2\right )}}{d x}-2 a b c \left (\frac {\sqrt {d-c^2 d x^2} \text {arccosh}(c x)}{c d x}+\frac {\sqrt {-1+c x} \sqrt {1+c x} \left (\log \left (-1+\sqrt {1+c x}\right )+\log \left (1+\sqrt {1+c x}\right )\right )}{\sqrt {d-c^2 d x^2}}\right )+\frac {b^2 c \sqrt {\frac {-1+c x}{1+c x}} (1+c x) \left (\text {arccosh}(c x) \left (-\text {arccosh}(c x)+\frac {\sqrt {\frac {-1+c x}{1+c x}} (1+c x) \text {arccosh}(c x)}{c x}-2 \log \left (1+e^{-2 \text {arccosh}(c x)}\right )\right )+\operatorname {PolyLog}\left (2,-e^{-2 \text {arccosh}(c x)}\right )\right )}{\sqrt {-d (-1+c x) (1+c x)}} \] Input:

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

Output:

-((a^2*Sqrt[-(d*(-1 + c^2*x^2))])/(d*x)) - 2*a*b*c*((Sqrt[d - c^2*d*x^2]*A 
rcCosh[c*x])/(c*d*x) + (Sqrt[-1 + c*x]*Sqrt[1 + c*x]*(Log[-1 + Sqrt[1 + c* 
x]] + Log[1 + Sqrt[1 + c*x]]))/Sqrt[d - c^2*d*x^2]) + (b^2*c*Sqrt[(-1 + c* 
x)/(1 + c*x)]*(1 + c*x)*(ArcCosh[c*x]*(-ArcCosh[c*x] + (Sqrt[(-1 + c*x)/(1 
 + c*x)]*(1 + c*x)*ArcCosh[c*x])/(c*x) - 2*Log[1 + E^(-2*ArcCosh[c*x])]) + 
 PolyLog[2, -E^(-2*ArcCosh[c*x])]))/Sqrt[-(d*(-1 + c*x)*(1 + c*x))]
 

Rubi [C] (warning: unable to verify)

Result contains complex when optimal does not.

Time = 0.75 (sec) , antiderivative size = 138, normalized size of antiderivative = 0.71, number of steps used = 10, number of rules used = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.310, Rules used = {6332, 6297, 25, 3042, 26, 4201, 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.

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

\(\Big \downarrow \) 6332

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

\(\Big \downarrow \) 6297

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

\(\Big \downarrow \) 25

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 26

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

\(\Big \downarrow \) 4201

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

\(\Big \downarrow \) 2620

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

\(\Big \downarrow \) 2715

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

\(\Big \downarrow \) 2838

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

Input:

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

Output:

-((Sqrt[d - c^2*d*x^2]*(a + b*ArcCosh[c*x])^2)/(d*x)) - ((2*I)*c*Sqrt[-1 + 
 c*x]*Sqrt[1 + c*x]*((-1/2*I)*(a + b*ArcCosh[c*x])^2 + (2*I)*(-1/2*(b*(a + 
 b*ArcCosh[c*x])*Log[1 + E^(-2*ArcCosh[c*x])]) + (b^2*PolyLog[2, -a - b*Ar 
cCosh[c*x]])/4)))/Sqrt[d - c^2*d*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 4201

Int[((c_.) + (d_.)*(x_))^(m_.)*tan[(e_.) + (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)))), x], x] 
 /; FreeQ[{c, d, e, f, fz}, x] && IGtQ[m, 0]
 

rule 6297

Int[((a_.) + ArcCosh[(c_.)*(x_)]*(b_.))^(n_.)/(x_), x_Symbol] :> Simp[1/b 
 Subst[Int[x^n*Tanh[-a/b + x/b], x], x, a + b*ArcCosh[c*x]], x] /; FreeQ[{a 
, b, c}, x] && IGtQ[n, 0]
 

rule 6332

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

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(475\) vs. \(2(201)=402\).

Time = 0.58 (sec) , antiderivative size = 476, normalized size of antiderivative = 2.44

method	result	size
default	\(-\frac {a^{2} \sqrt {-c^{2} d \,x^{2}+d}}{d x}+b^{2} \left (-\frac {\sqrt {-d \left (c^{2} x^{2}-1\right )}\, \left (-\sqrt {c x -1}\, \sqrt {c x +1}\, c x +c^{2} x^{2}-1\right ) \operatorname {arccosh}\left (c x \right )^{2}}{x \left (c^{2} x^{2}-1\right ) d}-\frac {2 \sqrt {-d \left (c^{2} x^{2}-1\right )}\, \sqrt {c x -1}\, \sqrt {c x +1}\, \operatorname {arccosh}\left (c x \right )^{2} c}{d \left (c^{2} x^{2}-1\right )}+\frac {2 \sqrt {-d \left (c^{2} x^{2}-1\right )}\, \sqrt {c x -1}\, \sqrt {c x +1}\, \operatorname {arccosh}\left (c x \right ) \ln \left (1+\left (c x +\sqrt {c x -1}\, \sqrt {c x +1}\right )^{2}\right ) c}{d \left (c^{2} x^{2}-1\right )}+\frac {\sqrt {-d \left (c^{2} x^{2}-1\right )}\, \sqrt {c x -1}\, \sqrt {c x +1}\, \operatorname {polylog}\left (2, -\left (c x +\sqrt {c x -1}\, \sqrt {c x +1}\right )^{2}\right ) c}{d \left (c^{2} x^{2}-1\right )}\right )+2 a b \left (-\frac {2 \sqrt {-d \left (c^{2} x^{2}-1\right )}\, \sqrt {c x -1}\, \sqrt {c x +1}\, \operatorname {arccosh}\left (c x \right ) c}{d \left (c^{2} x^{2}-1\right )}-\frac {\sqrt {-d \left (c^{2} x^{2}-1\right )}\, \left (-\sqrt {c x -1}\, \sqrt {c x +1}\, c x +c^{2} x^{2}-1\right ) \operatorname {arccosh}\left (c x \right )}{x \left (c^{2} x^{2}-1\right ) d}+\frac {\sqrt {-d \left (c^{2} x^{2}-1\right )}\, \sqrt {c x -1}\, \sqrt {c x +1}\, \ln \left (1+\left (c x +\sqrt {c x -1}\, \sqrt {c x +1}\right )^{2}\right ) c}{d \left (c^{2} x^{2}-1\right )}\right )\)	\(476\)
parts	\(-\frac {a^{2} \sqrt {-c^{2} d \,x^{2}+d}}{d x}+b^{2} \left (-\frac {\sqrt {-d \left (c^{2} x^{2}-1\right )}\, \left (-\sqrt {c x -1}\, \sqrt {c x +1}\, c x +c^{2} x^{2}-1\right ) \operatorname {arccosh}\left (c x \right )^{2}}{x \left (c^{2} x^{2}-1\right ) d}-\frac {2 \sqrt {-d \left (c^{2} x^{2}-1\right )}\, \sqrt {c x -1}\, \sqrt {c x +1}\, \operatorname {arccosh}\left (c x \right )^{2} c}{d \left (c^{2} x^{2}-1\right )}+\frac {2 \sqrt {-d \left (c^{2} x^{2}-1\right )}\, \sqrt {c x -1}\, \sqrt {c x +1}\, \operatorname {arccosh}\left (c x \right ) \ln \left (1+\left (c x +\sqrt {c x -1}\, \sqrt {c x +1}\right )^{2}\right ) c}{d \left (c^{2} x^{2}-1\right )}+\frac {\sqrt {-d \left (c^{2} x^{2}-1\right )}\, \sqrt {c x -1}\, \sqrt {c x +1}\, \operatorname {polylog}\left (2, -\left (c x +\sqrt {c x -1}\, \sqrt {c x +1}\right )^{2}\right ) c}{d \left (c^{2} x^{2}-1\right )}\right )+2 a b \left (-\frac {2 \sqrt {-d \left (c^{2} x^{2}-1\right )}\, \sqrt {c x -1}\, \sqrt {c x +1}\, \operatorname {arccosh}\left (c x \right ) c}{d \left (c^{2} x^{2}-1\right )}-\frac {\sqrt {-d \left (c^{2} x^{2}-1\right )}\, \left (-\sqrt {c x -1}\, \sqrt {c x +1}\, c x +c^{2} x^{2}-1\right ) \operatorname {arccosh}\left (c x \right )}{x \left (c^{2} x^{2}-1\right ) d}+\frac {\sqrt {-d \left (c^{2} x^{2}-1\right )}\, \sqrt {c x -1}\, \sqrt {c x +1}\, \ln \left (1+\left (c x +\sqrt {c x -1}\, \sqrt {c x +1}\right )^{2}\right ) c}{d \left (c^{2} x^{2}-1\right )}\right )\)	\(476\)

Input:

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

Output:

-a^2/d/x*(-c^2*d*x^2+d)^(1/2)+b^2*(-(-d*(c^2*x^2-1))^(1/2)*(-(c*x-1)^(1/2) 
*(c*x+1)^(1/2)*c*x+c^2*x^2-1)*arccosh(c*x)^2/x/(c^2*x^2-1)/d-2*(-d*(c^2*x^ 
2-1))^(1/2)*(c*x-1)^(1/2)*(c*x+1)^(1/2)/d/(c^2*x^2-1)*arccosh(c*x)^2*c+2*( 
-d*(c^2*x^2-1))^(1/2)*(c*x-1)^(1/2)*(c*x+1)^(1/2)/d/(c^2*x^2-1)*arccosh(c* 
x)*ln(1+(c*x+(c*x-1)^(1/2)*(c*x+1)^(1/2))^2)*c+(-d*(c^2*x^2-1))^(1/2)*(c*x 
-1)^(1/2)*(c*x+1)^(1/2)/d/(c^2*x^2-1)*polylog(2,-(c*x+(c*x-1)^(1/2)*(c*x+1 
)^(1/2))^2)*c)+2*a*b*(-2*(-d*(c^2*x^2-1))^(1/2)*(c*x-1)^(1/2)*(c*x+1)^(1/2 
)/d/(c^2*x^2-1)*arccosh(c*x)*c-(-d*(c^2*x^2-1))^(1/2)*(-(c*x-1)^(1/2)*(c*x 
+1)^(1/2)*c*x+c^2*x^2-1)*arccosh(c*x)/x/(c^2*x^2-1)/d+(-d*(c^2*x^2-1))^(1/ 
2)*(c*x-1)^(1/2)*(c*x+1)^(1/2)/d/(c^2*x^2-1)*ln(1+(c*x+(c*x-1)^(1/2)*(c*x+ 
1)^(1/2))^2)*c)
 

Fricas [F]

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

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

Output:

integral(-sqrt(-c^2*d*x^2 + d)*(b^2*arccosh(c*x)^2 + 2*a*b*arccosh(c*x) + 
a^2)/(c^2*d*x^4 - d*x^2), x)
 

Sympy [F]

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

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

Output:

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

Maxima [F]

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

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

Output:

-(c^2*d*sqrt(-1/(c^4*d))*log(x^2 - 1/c^2) + I*(-1)^(-2*c^2*d*x^2 + 2*d)*sq 
rt(d)*log(-2*c^2*d + 2*d/x^2))*a*b*c/d + b^2*integrate(log(c*x + sqrt(c*x 
+ 1)*sqrt(c*x - 1))^2/(sqrt(-c^2*d*x^2 + d)*x^2), x) - 2*sqrt(-c^2*d*x^2 + 
 d)*a*b*arccosh(c*x)/(d*x) - sqrt(-c^2*d*x^2 + d)*a^2/(d*x)
 

Giac [F]

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

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

Output:

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

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

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

Output:

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