\(\int \frac {\sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))^2}{x^2} \, dx\) [164]

Optimal result

Integrand size = 29, antiderivative size = 234 \[ \int \frac {\sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))^2}{x^2} \, dx=-\frac {\sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))^2}{x}-\frac {c \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))^2}{\sqrt {-1+c x} \sqrt {1+c x}}+\frac {c \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))^3}{3 b \sqrt {-1+c x} \sqrt {1+c x}}+\frac {2 b c \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x)) \log \left (1+e^{2 \text {arccosh}(c x)}\right )}{\sqrt {-1+c x} \sqrt {1+c x}}+\frac {b^2 c \sqrt {d-c^2 d x^2} \operatorname {PolyLog}\left (2,-e^{2 \text {arccosh}(c x)}\right )}{\sqrt {-1+c x} \sqrt {1+c x}} \] Output:

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

Mathematica [A] (warning: unable to verify)

Time = 1.48 (sec) , antiderivative size = 270, normalized size of antiderivative = 1.15 \[ \int \frac {\sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))^2}{x^2} \, dx=-\frac {a^2 \sqrt {d-c^2 d x^2}}{x}+a^2 c \sqrt {d} \arctan \left (\frac {c x \sqrt {d-c^2 d x^2}}{\sqrt {d} \left (-1+c^2 x^2\right )}\right )+a b c \sqrt {d-c^2 d x^2} \left (-\frac {2 \text {arccosh}(c x)}{c x}+\frac {\text {arccosh}(c x)^2+2 \log (c x)}{\sqrt {\frac {-1+c x}{1+c x}} (1+c x)}\right )+\frac {1}{3} b^2 c \sqrt {d-c^2 d x^2} \left (\text {arccosh}(c x) \left (-\frac {3 \text {arccosh}(c x)}{c x}+\frac {\text {arccosh}(c x) (3+\text {arccosh}(c x))+6 \log \left (1+e^{-2 \text {arccosh}(c x)}\right )}{\sqrt {\frac {-1+c x}{1+c x}} (1+c x)}\right )+\frac {3 \sqrt {\frac {-1+c x}{1+c x}} \operatorname {PolyLog}\left (2,-e^{-2 \text {arccosh}(c x)}\right )}{1-c x}\right ) \] Input:

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

Output:

-((a^2*Sqrt[d - c^2*d*x^2])/x) + a^2*c*Sqrt[d]*ArcTan[(c*x*Sqrt[d - c^2*d* 
x^2])/(Sqrt[d]*(-1 + c^2*x^2))] + a*b*c*Sqrt[d - c^2*d*x^2]*((-2*ArcCosh[c 
*x])/(c*x) + (ArcCosh[c*x]^2 + 2*Log[c*x])/(Sqrt[(-1 + c*x)/(1 + c*x)]*(1 
+ c*x))) + (b^2*c*Sqrt[d - c^2*d*x^2]*(ArcCosh[c*x]*((-3*ArcCosh[c*x])/(c* 
x) + (ArcCosh[c*x]*(3 + ArcCosh[c*x]) + 6*Log[1 + E^(-2*ArcCosh[c*x])])/(S 
qrt[(-1 + c*x)/(1 + c*x)]*(1 + c*x))) + (3*Sqrt[(-1 + c*x)/(1 + c*x)]*Poly 
Log[2, -E^(-2*ArcCosh[c*x])])/(1 - c*x)))/3
 

Rubi [C] (warning: unable to verify)

Result contains complex when optimal does not.

Time = 1.82 (sec) , antiderivative size = 186, normalized size of antiderivative = 0.79, number of steps used = 11, number of rules used = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.345, Rules used = {6339, 6297, 25, 3042, 26, 4201, 2620, 2715, 2838, 6308}

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.

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

\(\Big \downarrow \) 6339

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

\(\Big \downarrow \) 6297

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

\(\Big \downarrow \) 25

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 26

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

\(\Big \downarrow \) 4201

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

\(\Big \downarrow \) 2620

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

\(\Big \downarrow \) 2715

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

\(\Big \downarrow \) 2838

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

\(\Big \downarrow \) 6308

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

Input:

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

Output:

-((Sqrt[d - c^2*d*x^2]*(a + b*ArcCosh[c*x])^2)/x) + (c*Sqrt[d - c^2*d*x^2] 
*(a + b*ArcCosh[c*x])^3)/(3*b*Sqrt[-1 + c*x]*Sqrt[1 + c*x]) + ((2*I)*c*Sqr 
t[d - c^2*d*x^2]*((-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*ArcCo 
sh[c*x]])/4)))/(Sqrt[-1 + c*x]*Sqrt[1 + c*x])
 

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 6308

Int[((a_.) + ArcCosh[(c_.)*(x_)]*(b_.))^(n_.)/(Sqrt[(d1_) + (e1_.)*(x_)]*Sq 
rt[(d2_) + (e2_.)*(x_)]), x_Symbol] :> Simp[(1/(b*c*(n + 1)))*Simp[Sqrt[1 + 
 c*x]/Sqrt[d1 + e1*x]]*Simp[Sqrt[-1 + c*x]/Sqrt[d2 + e2*x]]*(a + b*ArcCosh[ 
c*x])^(n + 1), x] /; FreeQ[{a, b, c, d1, e1, d2, e2, n}, x] && EqQ[e1, c*d1 
] && EqQ[e2, (-c)*d2] && NeQ[n, -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]
 

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(581\) vs. \(2(232)=464\).

Time = 0.47 (sec) , antiderivative size = 582, normalized size of antiderivative = 2.49

method	result	size
default	\(-\frac {a^{2} \left (-c^{2} d \,x^{2}+d \right )^{\frac {3}{2}}}{d x}-a^{2} c^{2} x \sqrt {-c^{2} d \,x^{2}+d}-\frac {a^{2} c^{2} d \arctan \left (\frac {\sqrt {c^{2} d}\, x}{\sqrt {-c^{2} d \,x^{2}+d}}\right )}{\sqrt {c^{2} d}}+\frac {b^{2} \sqrt {-d \left (c^{2} x^{2}-1\right )}\, \operatorname {arccosh}\left (c x \right )^{3} c}{3 \sqrt {c x -1}\, \sqrt {c x +1}}-\frac {b^{2} \sqrt {-d \left (c^{2} x^{2}-1\right )}\, \operatorname {arccosh}\left (c x \right )^{2} c}{\sqrt {c x -1}\, \sqrt {c x +1}}-\frac {b^{2} \sqrt {-d \left (c^{2} x^{2}-1\right )}\, \operatorname {arccosh}\left (c x \right )^{2} x \,c^{2}}{\left (c x -1\right ) \left (c x +1\right )}+\frac {b^{2} \sqrt {-d \left (c^{2} x^{2}-1\right )}\, \operatorname {arccosh}\left (c x \right )^{2}}{\left (c x -1\right ) \left (c x +1\right ) x}+\frac {2 b^{2} \sqrt {-d \left (c^{2} x^{2}-1\right )}\, \operatorname {arccosh}\left (c x \right ) \ln \left (1+\left (c x +\sqrt {c x -1}\, \sqrt {c x +1}\right )^{2}\right ) c}{\sqrt {c x -1}\, \sqrt {c x +1}}+\frac {b^{2} \sqrt {-d \left (c^{2} x^{2}-1\right )}\, \operatorname {polylog}\left (2, -\left (c x +\sqrt {c x -1}\, \sqrt {c x +1}\right )^{2}\right ) c}{\sqrt {c x -1}\, \sqrt {c x +1}}+\frac {a b \sqrt {-d \left (c^{2} x^{2}-1\right )}\, \operatorname {arccosh}\left (c x \right )^{2} c}{\sqrt {c x -1}\, \sqrt {c x +1}}-\frac {2 a b \sqrt {-d \left (c^{2} x^{2}-1\right )}\, \operatorname {arccosh}\left (c x \right ) c}{\sqrt {c x -1}\, \sqrt {c x +1}}-\frac {2 a b \sqrt {-d \left (c^{2} x^{2}-1\right )}\, \operatorname {arccosh}\left (c x \right ) x \,c^{2}}{\left (c x -1\right ) \left (c x +1\right )}+\frac {2 a b \sqrt {-d \left (c^{2} x^{2}-1\right )}\, \operatorname {arccosh}\left (c x \right )}{\left (c x -1\right ) \left (c x +1\right ) x}+\frac {2 a b \sqrt {-d \left (c^{2} x^{2}-1\right )}\, \ln \left (1+\left (c x +\sqrt {c x -1}\, \sqrt {c x +1}\right )^{2}\right ) c}{\sqrt {c x -1}\, \sqrt {c x +1}}\)	\(582\)
parts	\(-\frac {a^{2} \left (-c^{2} d \,x^{2}+d \right )^{\frac {3}{2}}}{d x}-a^{2} c^{2} x \sqrt {-c^{2} d \,x^{2}+d}-\frac {a^{2} c^{2} d \arctan \left (\frac {\sqrt {c^{2} d}\, x}{\sqrt {-c^{2} d \,x^{2}+d}}\right )}{\sqrt {c^{2} d}}+\frac {b^{2} \sqrt {-d \left (c^{2} x^{2}-1\right )}\, \operatorname {arccosh}\left (c x \right )^{3} c}{3 \sqrt {c x -1}\, \sqrt {c x +1}}-\frac {b^{2} \sqrt {-d \left (c^{2} x^{2}-1\right )}\, \operatorname {arccosh}\left (c x \right )^{2} c}{\sqrt {c x -1}\, \sqrt {c x +1}}-\frac {b^{2} \sqrt {-d \left (c^{2} x^{2}-1\right )}\, \operatorname {arccosh}\left (c x \right )^{2} x \,c^{2}}{\left (c x -1\right ) \left (c x +1\right )}+\frac {b^{2} \sqrt {-d \left (c^{2} x^{2}-1\right )}\, \operatorname {arccosh}\left (c x \right )^{2}}{\left (c x -1\right ) \left (c x +1\right ) x}+\frac {2 b^{2} \sqrt {-d \left (c^{2} x^{2}-1\right )}\, \operatorname {arccosh}\left (c x \right ) \ln \left (1+\left (c x +\sqrt {c x -1}\, \sqrt {c x +1}\right )^{2}\right ) c}{\sqrt {c x -1}\, \sqrt {c x +1}}+\frac {b^{2} \sqrt {-d \left (c^{2} x^{2}-1\right )}\, \operatorname {polylog}\left (2, -\left (c x +\sqrt {c x -1}\, \sqrt {c x +1}\right )^{2}\right ) c}{\sqrt {c x -1}\, \sqrt {c x +1}}+\frac {a b \sqrt {-d \left (c^{2} x^{2}-1\right )}\, \operatorname {arccosh}\left (c x \right )^{2} c}{\sqrt {c x -1}\, \sqrt {c x +1}}-\frac {2 a b \sqrt {-d \left (c^{2} x^{2}-1\right )}\, \operatorname {arccosh}\left (c x \right ) c}{\sqrt {c x -1}\, \sqrt {c x +1}}-\frac {2 a b \sqrt {-d \left (c^{2} x^{2}-1\right )}\, \operatorname {arccosh}\left (c x \right ) x \,c^{2}}{\left (c x -1\right ) \left (c x +1\right )}+\frac {2 a b \sqrt {-d \left (c^{2} x^{2}-1\right )}\, \operatorname {arccosh}\left (c x \right )}{\left (c x -1\right ) \left (c x +1\right ) x}+\frac {2 a b \sqrt {-d \left (c^{2} x^{2}-1\right )}\, \ln \left (1+\left (c x +\sqrt {c x -1}\, \sqrt {c x +1}\right )^{2}\right ) c}{\sqrt {c x -1}\, \sqrt {c x +1}}\)	\(582\)

Input:

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

Output:

-a^2/d/x*(-c^2*d*x^2+d)^(3/2)-a^2*c^2*x*(-c^2*d*x^2+d)^(1/2)-a^2*c^2*d/(c^ 
2*d)^(1/2)*arctan((c^2*d)^(1/2)*x/(-c^2*d*x^2+d)^(1/2))+1/3*b^2*(-d*(c^2*x 
^2-1))^(1/2)/(c*x-1)^(1/2)/(c*x+1)^(1/2)*arccosh(c*x)^3*c-b^2*(-d*(c^2*x^2 
-1))^(1/2)*arccosh(c*x)^2/(c*x-1)^(1/2)/(c*x+1)^(1/2)*c-b^2*(-d*(c^2*x^2-1 
))^(1/2)*arccosh(c*x)^2/(c*x-1)/(c*x+1)*x*c^2+b^2*(-d*(c^2*x^2-1))^(1/2)*a 
rccosh(c*x)^2/(c*x-1)/(c*x+1)/x+2*b^2*(-d*(c^2*x^2-1))^(1/2)/(c*x-1)^(1/2) 
/(c*x+1)^(1/2)*arccosh(c*x)*ln(1+(c*x+(c*x-1)^(1/2)*(c*x+1)^(1/2))^2)*c+b^ 
2*(-d*(c^2*x^2-1))^(1/2)/(c*x-1)^(1/2)/(c*x+1)^(1/2)*polylog(2,-(c*x+(c*x- 
1)^(1/2)*(c*x+1)^(1/2))^2)*c+a*b*(-d*(c^2*x^2-1))^(1/2)/(c*x-1)^(1/2)/(c*x 
+1)^(1/2)*arccosh(c*x)^2*c-2*a*b*(-d*(c^2*x^2-1))^(1/2)/(c*x-1)^(1/2)/(c*x 
+1)^(1/2)*arccosh(c*x)*c-2*a*b*(-d*(c^2*x^2-1))^(1/2)*arccosh(c*x)/(c*x-1) 
/(c*x+1)*x*c^2+2*a*b*(-d*(c^2*x^2-1))^(1/2)*arccosh(c*x)/(c*x-1)/(c*x+1)/x 
+2*a*b*(-d*(c^2*x^2-1))^(1/2)/(c*x-1)^(1/2)/(c*x+1)^(1/2)*ln(1+(c*x+(c*x-1 
)^(1/2)*(c*x+1)^(1/2))^2)*c
 

Fricas [F]

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

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

Sympy [F]

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

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

Output:

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

Maxima [F]

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

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

Output:

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

Giac [F(-2)]

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

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

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

Output:

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

Reduce [F]

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

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

Output:

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