\(\int \frac {(a+b \text {arccosh}(c x))^2}{x^4 \sqrt {d-c^2 d x^2}} \, dx\) [192]

Optimal result

Integrand size = 29, antiderivative size = 332 \[ \int \frac {(a+b \text {arccosh}(c x))^2}{x^4 \sqrt {d-c^2 d x^2}} \, dx=\frac {b^2 c^2 \sqrt {d-c^2 d x^2}}{3 d x}-\frac {b c \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))}{3 d x^2 \sqrt {-1+c x} \sqrt {1+c x}}-\frac {\sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))^2}{3 d x^3}-\frac {2 c^2 \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))^2}{3 d x}-\frac {2 c^3 \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))^2}{3 d \sqrt {-1+c x} \sqrt {1+c x}}+\frac {4 b c^3 \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x)) \log \left (1+e^{2 \text {arccosh}(c x)}\right )}{3 d \sqrt {-1+c x} \sqrt {1+c x}}+\frac {2 b^2 c^3 \sqrt {d-c^2 d x^2} \operatorname {PolyLog}\left (2,-e^{2 \text {arccosh}(c x)}\right )}{3 d \sqrt {-1+c x} \sqrt {1+c x}} \] Output:

1/3*b^2*c^2*(-c^2*d*x^2+d)^(1/2)/d/x-1/3*b*c*(-c^2*d*x^2+d)^(1/2)*(a+b*arc 
cosh(c*x))/d/x^2/(c*x-1)^(1/2)/(c*x+1)^(1/2)-1/3*(-c^2*d*x^2+d)^(1/2)*(a+b 
*arccosh(c*x))^2/d/x^3-2/3*c^2*(-c^2*d*x^2+d)^(1/2)*(a+b*arccosh(c*x))^2/d 
/x-2/3*c^3*(-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)+4/3*b*c^3*(-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)+2/3*b^2*c^3*(-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] (warning: unable to verify)

Time = 0.93 (sec) , antiderivative size = 370, normalized size of antiderivative = 1.11 \[ \int \frac {(a+b \text {arccosh}(c x))^2}{x^4 \sqrt {d-c^2 d x^2}} \, dx=\frac {-a^2-a^2 c^2 x^2+b^2 c^2 x^2+2 a^2 c^4 x^4-b^2 c^4 x^4+a b c x \sqrt {-1+c x} \sqrt {1+c x}-b^2 (1+c x) \left (1-c x+2 c^2 x^2+2 c^3 x^3 \left (-1+\sqrt {\frac {-1+c x}{1+c x}}\right )\right ) \text {arccosh}(c x)^2+b (1+c x) \text {arccosh}(c x) \left (b c x \sqrt {\frac {-1+c x}{1+c x}}+2 a \left (-1+c x-2 c^2 x^2+2 c^3 x^3\right )-4 b c^3 x^3 \sqrt {\frac {-1+c x}{1+c x}} \log \left (1+e^{-2 \text {arccosh}(c x)}\right )\right )-4 a b c^3 x^3 \sqrt {-1+c x} \sqrt {1+c x} \log \left (-1+\sqrt {1+c x}\right )-4 a b c^3 x^3 \sqrt {-1+c x} \sqrt {1+c x} \log \left (1+\sqrt {1+c x}\right )+2 b^2 c^3 x^3 \sqrt {\frac {-1+c x}{1+c x}} (1+c x) \operatorname {PolyLog}\left (2,-e^{-2 \text {arccosh}(c x)}\right )}{3 x^3 \sqrt {d-c^2 d x^2}} \] Input:

Integrate[(a + b*ArcCosh[c*x])^2/(x^4*Sqrt[d - c^2*d*x^2]),x]
 

Output:

(-a^2 - a^2*c^2*x^2 + b^2*c^2*x^2 + 2*a^2*c^4*x^4 - b^2*c^4*x^4 + a*b*c*x* 
Sqrt[-1 + c*x]*Sqrt[1 + c*x] - b^2*(1 + c*x)*(1 - c*x + 2*c^2*x^2 + 2*c^3* 
x^3*(-1 + Sqrt[(-1 + c*x)/(1 + c*x)]))*ArcCosh[c*x]^2 + b*(1 + c*x)*ArcCos 
h[c*x]*(b*c*x*Sqrt[(-1 + c*x)/(1 + c*x)] + 2*a*(-1 + c*x - 2*c^2*x^2 + 2*c 
^3*x^3) - 4*b*c^3*x^3*Sqrt[(-1 + c*x)/(1 + c*x)]*Log[1 + E^(-2*ArcCosh[c*x 
])]) - 4*a*b*c^3*x^3*Sqrt[-1 + c*x]*Sqrt[1 + c*x]*Log[-1 + Sqrt[1 + c*x]] 
- 4*a*b*c^3*x^3*Sqrt[-1 + c*x]*Sqrt[1 + c*x]*Log[1 + Sqrt[1 + c*x]] + 2*b^ 
2*c^3*x^3*Sqrt[(-1 + c*x)/(1 + c*x)]*(1 + c*x)*PolyLog[2, -E^(-2*ArcCosh[c 
*x])])/(3*x^3*Sqrt[d - c^2*d*x^2])
 

Rubi [C] (warning: unable to verify)

Result contains complex when optimal does not.

Time = 2.20 (sec) , antiderivative size = 263, normalized size of antiderivative = 0.79, number of steps used = 13, number of rules used = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.414, Rules used = {6347, 6298, 106, 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.

\(\displaystyle \int \frac {(a+b \text {arccosh}(c x))^2}{x^4 \sqrt {d-c^2 d x^2}} \, dx\)

\(\Big \downarrow \) 6347

\(\displaystyle \frac {2}{3} c^2 \int \frac {(a+b \text {arccosh}(c x))^2}{x^2 \sqrt {d-c^2 d x^2}}dx-\frac {2 b c \sqrt {c x-1} \sqrt {c x+1} \int \frac {a+b \text {arccosh}(c x)}{x^3}dx}{3 \sqrt {d-c^2 d x^2}}-\frac {\sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))^2}{3 d x^3}\)

\(\Big \downarrow \) 6298

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

\(\Big \downarrow \) 106

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

\(\Big \downarrow \) 6332

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

\(\Big \downarrow \) 6297

\(\displaystyle \frac {2}{3} c^2 \left (-\frac {2 c \sqrt {c x-1} \sqrt {c x+1} \int -\left ((a+b \text {arccosh}(c x)) \tanh \left (\frac {a}{b}-\frac {a+b \text {arccosh}(c x)}{b}\right )\right )d(a+b \text {arccosh}(c x))}{\sqrt {d-c^2 d x^2}}-\frac {\sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))^2}{d x}\right )-\frac {2 b c \sqrt {c x-1} \sqrt {c x+1} \left (\frac {b c \sqrt {c x-1} \sqrt {c x+1}}{2 x}-\frac {a+b \text {arccosh}(c x)}{2 x^2}\right )}{3 \sqrt {d-c^2 d x^2}}-\frac {\sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))^2}{3 d x^3}\)

\(\Big \downarrow \) 25

\(\displaystyle \frac {2}{3} c^2 \left (\frac {2 c \sqrt {c x-1} \sqrt {c x+1} \int (a+b \text {arccosh}(c x)) \tanh \left (\frac {a}{b}-\frac {a+b \text {arccosh}(c x)}{b}\right )d(a+b \text {arccosh}(c x))}{\sqrt {d-c^2 d x^2}}-\frac {\sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))^2}{d x}\right )-\frac {2 b c \sqrt {c x-1} \sqrt {c x+1} \left (\frac {b c \sqrt {c x-1} \sqrt {c x+1}}{2 x}-\frac {a+b \text {arccosh}(c x)}{2 x^2}\right )}{3 \sqrt {d-c^2 d x^2}}-\frac {\sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))^2}{3 d x^3}\)

\(\Big \downarrow \) 3042

\(\displaystyle \frac {2}{3} c^2 \left (-\frac {\sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))^2}{d x}+\frac {2 c \sqrt {c x-1} \sqrt {c x+1} \int -i (a+b \text {arccosh}(c x)) \tan \left (\frac {i a}{b}-\frac {i (a+b \text {arccosh}(c x))}{b}\right )d(a+b \text {arccosh}(c x))}{\sqrt {d-c^2 d x^2}}\right )-\frac {2 b c \sqrt {c x-1} \sqrt {c x+1} \left (\frac {b c \sqrt {c x-1} \sqrt {c x+1}}{2 x}-\frac {a+b \text {arccosh}(c x)}{2 x^2}\right )}{3 \sqrt {d-c^2 d x^2}}-\frac {\sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))^2}{3 d x^3}\)

\(\Big \downarrow \) 26

\(\displaystyle \frac {2}{3} c^2 \left (-\frac {\sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))^2}{d x}-\frac {2 i c \sqrt {c x-1} \sqrt {c x+1} \int (a+b \text {arccosh}(c x)) \tan \left (\frac {i a}{b}-\frac {i (a+b \text {arccosh}(c x))}{b}\right )d(a+b \text {arccosh}(c x))}{\sqrt {d-c^2 d x^2}}\right )-\frac {2 b c \sqrt {c x-1} \sqrt {c x+1} \left (\frac {b c \sqrt {c x-1} \sqrt {c x+1}}{2 x}-\frac {a+b \text {arccosh}(c x)}{2 x^2}\right )}{3 \sqrt {d-c^2 d x^2}}-\frac {\sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))^2}{3 d x^3}\)

\(\Big \downarrow \) 4201

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

\(\Big \downarrow \) 2620

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

\(\Big \downarrow \) 2715

\(\displaystyle \frac {2}{3} c^2 \left (-\frac {\sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))^2}{d x}-\frac {2 i c \sqrt {c x-1} \sqrt {c x+1} \left (2 i \left (-\frac {1}{4} b^2 \int e^{2 \text {arccosh}(c x)} \log \left (1+e^{-2 \text {arccosh}(c x)}\right )de^{-2 \text {arccosh}(c x)}-\frac {1}{2} b \log \left (e^{-2 \text {arccosh}(c x)}+1\right ) (a+b \text {arccosh}(c x))\right )-\frac {1}{2} i (a+b \text {arccosh}(c x))^2\right )}{\sqrt {d-c^2 d x^2}}\right )-\frac {2 b c \sqrt {c x-1} \sqrt {c x+1} \left (\frac {b c \sqrt {c x-1} \sqrt {c x+1}}{2 x}-\frac {a+b \text {arccosh}(c x)}{2 x^2}\right )}{3 \sqrt {d-c^2 d x^2}}-\frac {\sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))^2}{3 d x^3}\)

\(\Big \downarrow \) 2838

\(\displaystyle \frac {2}{3} c^2 \left (-\frac {\sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))^2}{d x}-\frac {2 i c \sqrt {c x-1} \sqrt {c x+1} \left (2 i \left (\frac {1}{4} b^2 \operatorname {PolyLog}(2,-a-b \text {arccosh}(c x))-\frac {1}{2} b \log \left (e^{-2 \text {arccosh}(c x)}+1\right ) (a+b \text {arccosh}(c x))\right )-\frac {1}{2} i (a+b \text {arccosh}(c x))^2\right )}{\sqrt {d-c^2 d x^2}}\right )-\frac {2 b c \sqrt {c x-1} \sqrt {c x+1} \left (\frac {b c \sqrt {c x-1} \sqrt {c x+1}}{2 x}-\frac {a+b \text {arccosh}(c x)}{2 x^2}\right )}{3 \sqrt {d-c^2 d x^2}}-\frac {\sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))^2}{3 d x^3}\)

Input:

Int[(a + b*ArcCosh[c*x])^2/(x^4*Sqrt[d - c^2*d*x^2]),x]
 

Output:

-1/3*(Sqrt[d - c^2*d*x^2]*(a + b*ArcCosh[c*x])^2)/(d*x^3) - (2*b*c*Sqrt[-1 
 + c*x]*Sqrt[1 + c*x]*((b*c*Sqrt[-1 + c*x]*Sqrt[1 + c*x])/(2*x) - (a + b*A 
rcCosh[c*x])/(2*x^2)))/(3*Sqrt[d - c^2*d*x^2]) + (2*c^2*(-((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*ArcCosh[c*x]])/4))) 
/Sqrt[d - c^2*d*x^2]))/3
 

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 106

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

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]
 

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(1519\) vs. \(2(314)=628\).

Time = 0.67 (sec) , antiderivative size = 1520, normalized size of antiderivative = 4.58

method	result	size
default	\(\text {Expression too large to display}\)	\(1520\)
parts	\(\text {Expression too large to display}\)	\(1520\)

Input:

int((a+b*arccosh(c*x))^2/x^4/(-c^2*d*x^2+d)^(1/2),x,method=_RETURNVERBOSE)
 

Output:

2/3*b^2*(-d*(c^2*x^2-1))^(1/2)/d/(3*c^4*x^4-2*c^2*x^2-1)*x^5*c^8+1/3*b^2*( 
-d*(c^2*x^2-1))^(1/2)/d/(3*c^4*x^4-2*c^2*x^2-1)*x^3*c^6-2/3*b^2*(-d*(c^2*x 
^2-1))^(1/2)/d/(3*c^4*x^4-2*c^2*x^2-1)*x*c^4-1/3*b^2*(-d*(c^2*x^2-1))^(1/2 
)/d/(3*c^4*x^4-2*c^2*x^2-1)/x*c^2-1/3*a*b*(-d*(c^2*x^2-1))^(1/2)*(c*x-1)^( 
1/2)*(c*x+1)^(1/2)*(4*arccosh(c*x)*(c*x+1)^(1/2)*(c*x-1)^(1/2)*c^2*x^2+4*c 
^3*x^3*arccosh(c*x)-4*ln(1+(c*x+(c*x-1)^(1/2)*(c*x+1)^(1/2))^2)*x^3*c^3+2* 
arccosh(c*x)*(c*x-1)^(1/2)*(c*x+1)^(1/2)+c*x)/d/x^3/(c^2*x^2-1)+a^2*(-1/3/ 
d/x^3*(-c^2*d*x^2+d)^(1/2)-2/3*c^2/d/x*(-c^2*d*x^2+d)^(1/2))+4/3*b^2*(-d*( 
c^2*x^2-1))^(1/2)/d/(3*c^4*x^4-2*c^2*x^2-1)*x^5*arccosh(c*x)*c^8-2*b^2*(-d 
*(c^2*x^2-1))^(1/2)/d/(3*c^4*x^4-2*c^2*x^2-1)*x^3*arccosh(c*x)^2*c^6-2/3*b 
^2*(-d*(c^2*x^2-1))^(1/2)/d/(3*c^4*x^4-2*c^2*x^2-1)*x^3*arccosh(c*x)*c^6+1 
/3*b^2*(-d*(c^2*x^2-1))^(1/2)/d/(3*c^4*x^4-2*c^2*x^2-1)*x*arccosh(c*x)^2*c 
^4-2/3*b^2*(-d*(c^2*x^2-1))^(1/2)/d/(3*c^4*x^4-2*c^2*x^2-1)*x*arccosh(c*x) 
*c^4-1/3*b^2*(-d*(c^2*x^2-1))^(1/2)/d/(3*c^4*x^4-2*c^2*x^2-1)*(c*x+1)^(1/2 
)*(c*x-1)^(1/2)*c^3+4/3*b^2*(-d*(c^2*x^2-1))^(1/2)/d/(3*c^4*x^4-2*c^2*x^2- 
1)/x*arccosh(c*x)^2*c^2+1/3*b^2*(-d*(c^2*x^2-1))^(1/2)/d/(3*c^4*x^4-2*c^2* 
x^2-1)/x^3*arccosh(c*x)^2-2/3*b^2*(-d*(c^2*x^2-1))^(1/2)/d/(3*c^4*x^4-2*c^ 
2*x^2-1)*x^3*(c*x+1)*(c*x-1)*c^6-b^2*(-d*(c^2*x^2-1))^(1/2)/d/(3*c^4*x^4-2 
*c^2*x^2-1)*x^2*(c*x+1)^(1/2)*(c*x-1)^(1/2)*c^5+2/3*b^2*(-d*(c^2*x^2-1))^( 
1/2)/d/(3*c^4*x^4-2*c^2*x^2-1)*arccosh(c*x)^2*(c*x+1)^(1/2)*(c*x-1)^(1/...
 

Fricas [F]

\[ \int \frac {(a+b \text {arccosh}(c x))^2}{x^4 \sqrt {d-c^2 d x^2}} \, dx=\int { \frac {{\left (b \operatorname {arcosh}\left (c x\right ) + a\right )}^{2}}{\sqrt {-c^{2} d x^{2} + d} x^{4}} \,d x } \] Input:

integrate((a+b*arccosh(c*x))^2/x^4/(-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^6 - d*x^4), x)
 

Sympy [F]

\[ \int \frac {(a+b \text {arccosh}(c x))^2}{x^4 \sqrt {d-c^2 d x^2}} \, dx=\int \frac {\left (a + b \operatorname {acosh}{\left (c x \right )}\right )^{2}}{x^{4} \sqrt {- d \left (c x - 1\right ) \left (c x + 1\right )}}\, dx \] Input:

integrate((a+b*acosh(c*x))**2/x**4/(-c**2*d*x**2+d)**(1/2),x)
 

Output:

Integral((a + b*acosh(c*x))**2/(x**4*sqrt(-d*(c*x - 1)*(c*x + 1))), x)
 

Maxima [F]

\[ \int \frac {(a+b \text {arccosh}(c x))^2}{x^4 \sqrt {d-c^2 d x^2}} \, dx=\int { \frac {{\left (b \operatorname {arcosh}\left (c x\right ) + a\right )}^{2}}{\sqrt {-c^{2} d x^{2} + d} x^{4}} \,d x } \] Input:

integrate((a+b*arccosh(c*x))^2/x^4/(-c^2*d*x^2+d)^(1/2),x, algorithm="maxi 
ma")
 

Output:

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

Giac [F]

\[ \int \frac {(a+b \text {arccosh}(c x))^2}{x^4 \sqrt {d-c^2 d x^2}} \, dx=\int { \frac {{\left (b \operatorname {arcosh}\left (c x\right ) + a\right )}^{2}}{\sqrt {-c^{2} d x^{2} + d} x^{4}} \,d x } \] Input:

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

Mupad [F(-1)]

Timed out. \[ \int \frac {(a+b \text {arccosh}(c x))^2}{x^4 \sqrt {d-c^2 d x^2}} \, dx=\int \frac {{\left (a+b\,\mathrm {acosh}\left (c\,x\right )\right )}^2}{x^4\,\sqrt {d-c^2\,d\,x^2}} \,d x \] Input:

int((a + b*acosh(c*x))^2/(x^4*(d - c^2*d*x^2)^(1/2)),x)
 

Output:

int((a + b*acosh(c*x))^2/(x^4*(d - c^2*d*x^2)^(1/2)), x)
 

Reduce [F]

\[ \int \frac {(a+b \text {arccosh}(c x))^2}{x^4 \sqrt {d-c^2 d x^2}} \, dx=\frac {-2 \sqrt {-c^{2} x^{2}+1}\, a^{2} c^{2} x^{2}-\sqrt {-c^{2} x^{2}+1}\, a^{2}+6 \left (\int \frac {\mathit {acosh} \left (c x \right )}{\sqrt {-c^{2} x^{2}+1}\, x^{4}}d x \right ) a b \,x^{3}+3 \left (\int \frac {\mathit {acosh} \left (c x \right )^{2}}{\sqrt {-c^{2} x^{2}+1}\, x^{4}}d x \right ) b^{2} x^{3}}{3 \sqrt {d}\, x^{3}} \] Input:

int((a+b*acosh(c*x))^2/x^4/(-c^2*d*x^2+d)^(1/2),x)
 

Output:

( - 2*sqrt( - c**2*x**2 + 1)*a**2*c**2*x**2 - sqrt( - c**2*x**2 + 1)*a**2 
+ 6*int(acosh(c*x)/(sqrt( - c**2*x**2 + 1)*x**4),x)*a*b*x**3 + 3*int(acosh 
(c*x)**2/(sqrt( - c**2*x**2 + 1)*x**4),x)*b**2*x**3)/(3*sqrt(d)*x**3)