3.3.2 \(\int \frac {(a+b \text {arccosh}(c x))^2}{x^3 \sqrt {d-c^2 d x^2}} \, dx\) [202]

3.3.2.1 Optimal result

Integrand size = 29, antiderivative size = 430 \[ \int \frac {(a+b \text {arccosh}(c x))^2}{x^3 \sqrt {d-c^2 d x^2}} \, dx=\frac {b c \sqrt {-1+c x} \sqrt {1+c x} (a+b \text {arccosh}(c x))}{x \sqrt {d-c^2 d x^2}}-\frac {\sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))^2}{2 d x^2}+\frac {c^2 \sqrt {-1+c x} \sqrt {1+c x} (a+b \text {arccosh}(c x))^2 \arctan \left (e^{\text {arccosh}(c x)}\right )}{\sqrt {d-c^2 d x^2}}-\frac {b^2 c^2 \sqrt {-1+c x} \sqrt {1+c x} \arctan \left (\sqrt {-1+c x} \sqrt {1+c x}\right )}{\sqrt {d-c^2 d x^2}}-\frac {i b c^2 \sqrt {-1+c x} \sqrt {1+c x} (a+b \text {arccosh}(c x)) \operatorname {PolyLog}\left (2,-i e^{\text {arccosh}(c x)}\right )}{\sqrt {d-c^2 d x^2}}+\frac {i b c^2 \sqrt {-1+c x} \sqrt {1+c x} (a+b \text {arccosh}(c x)) \operatorname {PolyLog}\left (2,i e^{\text {arccosh}(c x)}\right )}{\sqrt {d-c^2 d x^2}}+\frac {i b^2 c^2 \sqrt {-1+c x} \sqrt {1+c x} \operatorname {PolyLog}\left (3,-i e^{\text {arccosh}(c x)}\right )}{\sqrt {d-c^2 d x^2}}-\frac {i b^2 c^2 \sqrt {-1+c x} \sqrt {1+c x} \operatorname {PolyLog}\left (3,i e^{\text {arccosh}(c x)}\right )}{\sqrt {d-c^2 d x^2}} \]

output

b*c*(a+b*arccosh(c*x))*(c*x-1)^(1/2)*(c*x+1)^(1/2)/x/(-c^2*d*x^2+d)^(1/2)+ 
c^2*(a+b*arccosh(c*x))^2*arctan(c*x+(c*x-1)^(1/2)*(c*x+1)^(1/2))*(c*x-1)^( 
1/2)*(c*x+1)^(1/2)/(-c^2*d*x^2+d)^(1/2)-b^2*c^2*arctan((c*x-1)^(1/2)*(c*x+ 
1)^(1/2))*(c*x-1)^(1/2)*(c*x+1)^(1/2)/(-c^2*d*x^2+d)^(1/2)-I*b*c^2*(a+b*ar 
ccosh(c*x))*polylog(2,-I*(c*x+(c*x-1)^(1/2)*(c*x+1)^(1/2)))*(c*x-1)^(1/2)* 
(c*x+1)^(1/2)/(-c^2*d*x^2+d)^(1/2)+I*b*c^2*(a+b*arccosh(c*x))*polylog(2,I* 
(c*x+(c*x-1)^(1/2)*(c*x+1)^(1/2)))*(c*x-1)^(1/2)*(c*x+1)^(1/2)/(-c^2*d*x^2 
+d)^(1/2)+I*b^2*c^2*polylog(3,-I*(c*x+(c*x-1)^(1/2)*(c*x+1)^(1/2)))*(c*x-1 
)^(1/2)*(c*x+1)^(1/2)/(-c^2*d*x^2+d)^(1/2)-I*b^2*c^2*polylog(3,I*(c*x+(c*x 
-1)^(1/2)*(c*x+1)^(1/2)))*(c*x-1)^(1/2)*(c*x+1)^(1/2)/(-c^2*d*x^2+d)^(1/2) 
-1/2*(a+b*arccosh(c*x))^2*(-c^2*d*x^2+d)^(1/2)/d/x^2
 

3.3.2.2 Mathematica [B] (warning: unable to verify)

Both result and optimal contain complex but leaf count is larger than twice the leaf count of optimal. \(5036\) vs. \(2(430)=860\).

Time = 63.66 (sec) , antiderivative size = 5036, normalized size of antiderivative = 11.71 \[ \int \frac {(a+b \text {arccosh}(c x))^2}{x^3 \sqrt {d-c^2 d x^2}} \, dx=\text {Result too large to show} \]

input

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

output

Result too large to show
 

3.3.2.3 Rubi [A] (verified)

Time = 1.41 (sec) , antiderivative size = 254, normalized size of antiderivative = 0.59, number of steps used = 11, number of rules used = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.345, Rules used = {6347, 6298, 103, 218, 6361, 3042, 4668, 3011, 2720, 7143}

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.

\(\displaystyle \int \frac {(a+b \text {arccosh}(c x))^2}{x^3 \sqrt {d-c^2 d x^2}} \, dx\)

\(\Big \downarrow \) 6347

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

\(\Big \downarrow \) 6298

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

\(\Big \downarrow \) 103

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

\(\Big \downarrow \) 218

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

\(\Big \downarrow \) 6361

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 4668

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

\(\Big \downarrow \) 3011

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

\(\Big \downarrow \) 2720

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

\(\Big \downarrow \) 7143

\(\displaystyle \frac {c^2 \sqrt {c x-1} \sqrt {c x+1} \left (2 \arctan \left (e^{\text {arccosh}(c x)}\right ) (a+b \text {arccosh}(c x))^2+2 i b \left (b \operatorname {PolyLog}\left (3,-i e^{\text {arccosh}(c x)}\right )-\operatorname {PolyLog}\left (2,-i e^{\text {arccosh}(c x)}\right ) (a+b \text {arccosh}(c x))\right )-2 i b \left (b \operatorname {PolyLog}\left (3,i e^{\text {arccosh}(c x)}\right )-\operatorname {PolyLog}\left (2,i e^{\text {arccosh}(c x)}\right ) (a+b \text {arccosh}(c x))\right )\right )}{2 \sqrt {d-c^2 d x^2}}-\frac {b c \sqrt {c x-1} \sqrt {c x+1} \left (b c \arctan \left (\sqrt {c x-1} \sqrt {c x+1}\right )-\frac {a+b \text {arccosh}(c x)}{x}\right )}{\sqrt {d-c^2 d x^2}}-\frac {\sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))^2}{2 d x^2}\)

input

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

output

-1/2*(Sqrt[d - c^2*d*x^2]*(a + b*ArcCosh[c*x])^2)/(d*x^2) - (b*c*Sqrt[-1 + 
 c*x]*Sqrt[1 + c*x]*(-((a + b*ArcCosh[c*x])/x) + b*c*ArcTan[Sqrt[-1 + c*x] 
*Sqrt[1 + c*x]]))/Sqrt[d - c^2*d*x^2] + (c^2*Sqrt[-1 + c*x]*Sqrt[1 + c*x]* 
(2*(a + b*ArcCosh[c*x])^2*ArcTan[E^ArcCosh[c*x]] + (2*I)*b*(-((a + b*ArcCo 
sh[c*x])*PolyLog[2, (-I)*E^ArcCosh[c*x]]) + b*PolyLog[3, (-I)*E^ArcCosh[c* 
x]]) - (2*I)*b*(-((a + b*ArcCosh[c*x])*PolyLog[2, I*E^ArcCosh[c*x]]) + b*P 
olyLog[3, I*E^ArcCosh[c*x]])))/(2*Sqrt[d - c^2*d*x^2])
 

3.3.2.3.1 Defintions of rubi rules used

rule 103

Int[1/(Sqrt[(a_.) + (b_.)*(x_)]*Sqrt[(c_.) + (d_.)*(x_)]*((e_.) + (f_.)*(x_ 
))), x_] :> Simp[b*f   Subst[Int[1/(d*(b*e - a*f)^2 + b*f^2*x^2), x], x, Sq 
rt[a + b*x]*Sqrt[c + d*x]], x] /; FreeQ[{a, b, c, d, e, f}, x] && EqQ[2*b*d 
*e - f*(b*c + a*d), 0]
 

rule 218

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(Rt[a/b, 2]/a)*ArcTan[x/R 
t[a/b, 2]], x] /; FreeQ[{a, b}, x] && PosQ[a/b]
 

rule 2720

Int[u_, x_Symbol] :> With[{v = FunctionOfExponential[u, x]}, Simp[v/D[v, x] 
   Subst[Int[FunctionOfExponentialFunction[u, x]/x, x], x, v], x]] /; Funct 
ionOfExponentialQ[u, x] &&  !MatchQ[u, (w_)*((a_.)*(v_)^(n_))^(m_) /; FreeQ 
[{a, m, n}, x] && IntegerQ[m*n]] &&  !MatchQ[u, E^((c_.)*((a_.) + (b_.)*x)) 
*(F_)[v_] /; FreeQ[{a, b, c}, x] && InverseFunctionQ[F[x]]]
 

rule 3011

Int[Log[1 + (e_.)*((F_)^((c_.)*((a_.) + (b_.)*(x_))))^(n_.)]*((f_.) + (g_.) 
*(x_))^(m_.), x_Symbol] :> Simp[(-(f + g*x)^m)*(PolyLog[2, (-e)*(F^(c*(a + 
b*x)))^n]/(b*c*n*Log[F])), x] + Simp[g*(m/(b*c*n*Log[F]))   Int[(f + g*x)^( 
m - 1)*PolyLog[2, (-e)*(F^(c*(a + b*x)))^n], x], x] /; FreeQ[{F, a, b, c, e 
, f, g, n}, x] && GtQ[m, 0]
 

rule 3042

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

rule 4668

Int[csc[(e_.) + Pi*(k_.) + (Complex[0, fz_])*(f_.)*(x_)]*((c_.) + (d_.)*(x_ 
))^(m_.), x_Symbol] :> Simp[-2*(c + d*x)^m*(ArcTanh[E^((-I)*e + f*fz*x)/E^( 
I*k*Pi)]/(f*fz*I)), x] + (-Simp[d*(m/(f*fz*I))   Int[(c + d*x)^(m - 1)*Log[ 
1 - E^((-I)*e + f*fz*x)/E^(I*k*Pi)], x], x] + Simp[d*(m/(f*fz*I))   Int[(c 
+ d*x)^(m - 1)*Log[1 + E^((-I)*e + f*fz*x)/E^(I*k*Pi)], x], x]) /; FreeQ[{c 
, d, e, f, fz}, x] && IntegerQ[2*k] && IGtQ[m, 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 6347

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[c^2*((m + 2*p + 3)/(f^2*(m + 1 
)))   Int[(f*x)^(m + 2)*(d + e*x^2)^p*(a + b*ArcCosh[c*x])^n, x], 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, p}, x] && EqQ[c^2*d + e, 0] && 
 GtQ[n, 0] && ILtQ[m, -1]
 

rule 6361

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

rule 7143

Int[PolyLog[n_, (c_.)*((a_.) + (b_.)*(x_))^(p_.)]/((d_.) + (e_.)*(x_)), x_S 
ymbol] :> Simp[PolyLog[n + 1, c*(a + b*x)^p]/(e*p), x] /; FreeQ[{a, b, c, d 
, e, n, p}, x] && EqQ[b*d, a*e]
 

3.3.2.4 Maple [F]

\[\int \frac {\left (a +b \,\operatorname {arccosh}\left (c x \right )\right )^{2}}{x^{3} \sqrt {-c^{2} d \,x^{2}+d}}d x\]

input

int((a+b*arccosh(c*x))^2/x^3/(-c^2*d*x^2+d)^(1/2),x)
 

output

int((a+b*arccosh(c*x))^2/x^3/(-c^2*d*x^2+d)^(1/2),x)
 

3.3.2.5 Fricas [F]

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

input

integrate((a+b*arccosh(c*x))^2/x^3/(-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^5 - d*x^3), x)
 

3.3.2.6 Sympy [F]

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

input

integrate((a+b*acosh(c*x))**2/x**3/(-c**2*d*x**2+d)**(1/2),x)
 

output

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

3.3.2.7 Maxima [F]

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

input

integrate((a+b*arccosh(c*x))^2/x^3/(-c^2*d*x^2+d)^(1/2),x, algorithm="maxi 
ma")
 

output

-1/2*(c^2*log(2*sqrt(-c^2*d*x^2 + d)*sqrt(d)/abs(x) + 2*d/abs(x))/sqrt(d) 
+ sqrt(-c^2*d*x^2 + d)/(d*x^2))*a^2 + integrate(b^2*log(c*x + sqrt(c*x + 1 
)*sqrt(c*x - 1))^2/(sqrt(-c^2*d*x^2 + d)*x^3) + 2*a*b*log(c*x + sqrt(c*x + 
 1)*sqrt(c*x - 1))/(sqrt(-c^2*d*x^2 + d)*x^3), x)
 

3.3.2.8 Giac [F]

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

input

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

3.3.2.9 Mupad [F(-1)]

Timed out. \[ \int \frac {(a+b \text {arccosh}(c x))^2}{x^3 \sqrt {d-c^2 d x^2}} \, dx=\int \frac {{\left (a+b\,\mathrm {acosh}\left (c\,x\right )\right )}^2}{x^3\,\sqrt {d-c^2\,d\,x^2}} \,d x \]

input

int((a + b*acosh(c*x))^2/(x^3*(d - c^2*d*x^2)^(1/2)),x)
 

output

int((a + b*acosh(c*x))^2/(x^3*(d - c^2*d*x^2)^(1/2)), x)