3.2.76 \(\int \frac {\sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))^2}{x^3} \, dx\) [176]

3.2.76.1 Optimal result

Integrand size = 29, antiderivative size = 427 \[ \int \frac {\sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))^2}{x^3} \, dx=-\frac {b c \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))}{x \sqrt {-1+c x} \sqrt {1+c x}}-\frac {\sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))^2}{2 x^2}+\frac {c^2 \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))^2 \arctan \left (e^{\text {arccosh}(c x)}\right )}{\sqrt {-1+c x} \sqrt {1+c x}}+\frac {b^2 c^2 \sqrt {d-c^2 d x^2} \arctan \left (\sqrt {-1+c x} \sqrt {1+c x}\right )}{\sqrt {-1+c x} \sqrt {1+c x}}-\frac {i b c^2 \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x)) \operatorname {PolyLog}\left (2,-i e^{\text {arccosh}(c x)}\right )}{\sqrt {-1+c x} \sqrt {1+c x}}+\frac {i b c^2 \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x)) \operatorname {PolyLog}\left (2,i e^{\text {arccosh}(c x)}\right )}{\sqrt {-1+c x} \sqrt {1+c x}}+\frac {i b^2 c^2 \sqrt {d-c^2 d x^2} \operatorname {PolyLog}\left (3,-i e^{\text {arccosh}(c x)}\right )}{\sqrt {-1+c x} \sqrt {1+c x}}-\frac {i b^2 c^2 \sqrt {d-c^2 d x^2} \operatorname {PolyLog}\left (3,i e^{\text {arccosh}(c x)}\right )}{\sqrt {-1+c x} \sqrt {1+c x}} \]

output

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

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

Both result and optimal contain complex but leaf count is larger than twice the leaf count of optimal. \(5035\) vs. \(2(427)=854\).

Time = 66.46 (sec) , antiderivative size = 5035, normalized size of antiderivative = 11.79 \[ \int \frac {\sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))^2}{x^3} \, dx=\text {Result too large to show} \]

input

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

output

Result too large to show
 

3.2.76.3 Rubi [A] (verified)

Time = 1.86 (sec) , antiderivative size = 250, 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 = {6339, 6298, 103, 218, 6362, 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.

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

\(\Big \downarrow \) 6339

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

\(\Big \downarrow \) 6298

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

\(\Big \downarrow \) 103

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

\(\Big \downarrow \) 218

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

\(\Big \downarrow \) 6362

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 4668

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

\(\Big \downarrow \) 3011

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

\(\Big \downarrow \) 2720

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

\(\Big \downarrow \) 7143

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

input

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

output

-1/2*(Sqrt[d - c^2*d*x^2]*(a + b*ArcCosh[c*x])^2)/x^2 + (b*c*Sqrt[d - c^2* 
d*x^2]*(-((a + b*ArcCosh[c*x])/x) + b*c*ArcTan[Sqrt[-1 + c*x]*Sqrt[1 + c*x 
]]))/(Sqrt[-1 + c*x]*Sqrt[1 + c*x]) + (c^2*Sqrt[d - c^2*d*x^2]*(2*(a + b*A 
rcCosh[c*x])^2*ArcTan[E^ArcCosh[c*x]] + (2*I)*b*(-((a + b*ArcCosh[c*x])*Po 
lyLog[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*PolyLog[3, I 
*E^ArcCosh[c*x]])))/(2*Sqrt[-1 + c*x]*Sqrt[1 + c*x])
 

3.2.76.3.1 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 6339

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

rule 6362

Int[(((a_.) + ArcCosh[(c_.)*(x_)]*(b_.))^(n_.)*(x_)^(m_))/(Sqrt[(d1_) + (e1 
_.)*(x_)]*Sqrt[(d2_) + (e2_.)*(x_)]), x_Symbol] :> Simp[(1/c^(m + 1))*Simp[ 
Sqrt[1 + c*x]/Sqrt[d1 + e1*x]]*Simp[Sqrt[-1 + c*x]/Sqrt[d2 + e2*x]]   Subst 
[Int[(a + b*x)^n*Cosh[x]^m, x], x, ArcCosh[c*x]], x] /; FreeQ[{a, b, c, d1, 
 e1, d2, e2}, x] && EqQ[e1, c*d1] && EqQ[e2, (-c)*d2] && IGtQ[n, 0] && Inte 
gerQ[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]
 

3.2.76.4 Maple [F]

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

input

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

output

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

3.2.76.5 Fricas [F]

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

input

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

3.2.76.6 Sympy [F]

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

input

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

output

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

3.2.76.7 Maxima [F]

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

input

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

output

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

3.2.76.8 Giac [F(-2)]

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

input

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

3.2.76.9 Mupad [F(-1)]

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

input

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

output

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