\(\int \frac {\sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))^2}{x^2} \, dx\) [175]

Optimal result

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

[Out]

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

Rubi [A] (verified)

Time = 0.29 (sec) , antiderivative size = 234, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.241, Rules used = {5924, 5882, 3799, 2221, 2317, 2438, 5893} \[ \int \frac {\sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))^2}{x^2} \, dx=\frac {c \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))^3}{3 b \sqrt {c x-1} \sqrt {c x+1}}+\frac {c \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))^2}{\sqrt {c x-1} \sqrt {c x+1}}-\frac {\sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))^2}{x}+\frac {2 b c \sqrt {d-c^2 d x^2} \log \left (e^{-2 \text {arccosh}(c x)}+1\right ) (a+b \text {arccosh}(c x))}{\sqrt {c x-1} \sqrt {c x+1}}-\frac {b^2 c \sqrt {d-c^2 d x^2} \operatorname {PolyLog}\left (2,-e^{-2 \text {arccosh}(c x)}\right )}{\sqrt {c x-1} \sqrt {c x+1}} \]

[In]

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

[Out]

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

Rule 2221

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]
- Dist[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 2317

Int[Log[(a_) + (b_.)*((F_)^((e_.)*((c_.) + (d_.)*(x_))))^(n_.)], x_Symbol] :> Dist[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 2438

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 3799

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] + Dist[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 5882

Int[((a_.) + ArcCosh[(c_.)*(x_)]*(b_.))^(n_.)/(x_), x_Symbol] :> Dist[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 5893

Int[((a_.) + ArcCosh[(c_.)*(x_)]*(b_.))^(n_.)/(Sqrt[(d1_) + (e1_.)*(x_)]*Sqrt[(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*Arc
Cosh[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 5924

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*ArcCosh[c*x])^n/(f*(m + 1))), x] + (-Dist[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] - Dist[(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]

Rubi steps \begin{align*} \text {integral}& = -\frac {\sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))^2}{x}+\frac {\left (2 b c \sqrt {d-c^2 d x^2}\right ) \int \frac {a+b \text {arccosh}(c x)}{x} \, dx}{\sqrt {-1+c x} \sqrt {1+c x}}+\frac {\left (c^2 \sqrt {d-c^2 d x^2}\right ) \int \frac {(a+b \text {arccosh}(c x))^2}{\sqrt {-1+c x} \sqrt {1+c x}} \, dx}{\sqrt {-1+c x} \sqrt {1+c x}} \\ & = -\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))^3}{3 b \sqrt {-1+c x} \sqrt {1+c x}}-\frac {\left (2 c \sqrt {d-c^2 d x^2}\right ) \text {Subst}\left (\int x \tanh \left (\frac {a}{b}-\frac {x}{b}\right ) \, dx,x,a+b \text {arccosh}(c x)\right )}{\sqrt {-1+c x} \sqrt {1+c x}} \\ & = -\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 {\left (4 c \sqrt {d-c^2 d x^2}\right ) \text {Subst}\left (\int \frac {e^{2 \left (\frac {a}{b}-\frac {x}{b}\right )} x}{1+e^{2 \left (\frac {a}{b}-\frac {x}{b}\right )}} \, dx,x,a+b \text {arccosh}(c x)\right )}{\sqrt {-1+c x} \sqrt {1+c x}} \\ & = -\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 {\left (2 b c \sqrt {d-c^2 d x^2}\right ) \text {Subst}\left (\int \log \left (1+e^{2 \left (\frac {a}{b}-\frac {x}{b}\right )}\right ) \, dx,x,a+b \text {arccosh}(c x)\right )}{\sqrt {-1+c x} \sqrt {1+c x}} \\ & = -\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 {\left (b^2 c \sqrt {d-c^2 d x^2}\right ) \text {Subst}\left (\int \frac {\log (1+x)}{x} \, dx,x,e^{2 \left (\frac {a}{b}-\frac {a+b \text {arccosh}(c x)}{b}\right )}\right )}{\sqrt {-1+c x} \sqrt {1+c x}} \\ & = -\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 \left (\frac {a}{b}-\frac {a+b \text {arccosh}(c x)}{b}\right )}\right )}{\sqrt {-1+c x} \sqrt {1+c x}} \\ \end{align*}

Mathematica [A] (warning: unable to verify)

Time = 1.69 (sec) , antiderivative size = 270, normalized size of antiderivative = 1.15 \[ \int \frac {\sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))^2}{x^2} \, dx=-\frac {a^2 \sqrt {d-c^2 d x^2}}{x}+a^2 c \sqrt {d} \arctan \left (\frac {c x \sqrt {d-c^2 d x^2}}{\sqrt {d} \left (-1+c^2 x^2\right )}\right )+a b c \sqrt {d-c^2 d x^2} \left (-\frac {2 \text {arccosh}(c x)}{c x}+\frac {\text {arccosh}(c x)^2+2 \log (c x)}{\sqrt {\frac {-1+c x}{1+c x}} (1+c x)}\right )+\frac {1}{3} b^2 c \sqrt {d-c^2 d x^2} \left (\text {arccosh}(c x) \left (-\frac {3 \text {arccosh}(c x)}{c x}+\frac {\text {arccosh}(c x) (3+\text {arccosh}(c x))+6 \log \left (1+e^{-2 \text {arccosh}(c x)}\right )}{\sqrt {\frac {-1+c x}{1+c x}} (1+c x)}\right )+\frac {3 \sqrt {\frac {-1+c x}{1+c x}} \operatorname {PolyLog}\left (2,-e^{-2 \text {arccosh}(c x)}\right )}{1-c x}\right ) \]

[In]

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

[Out]

-((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])])/(Sqrt[(-1 + c*x)/(1 + c*x)]*(1 + c*x))) + (3*Sqrt[(-1 + c*x)/(1 + c*x)]*
PolyLog[2, -E^(-2*ArcCosh[c*x])])/(1 - c*x)))/3

Maple [B] (verified)

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

Time = 0.86 (sec) , antiderivative size = 582, normalized size of antiderivative = 2.49

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

[In]

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

[Out]

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

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

[In]

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

[Out]

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]

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

[In]

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

[Out]

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

Maxima [F]

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

[In]

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

[Out]

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

[In]

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

[Out]

Exception raised: TypeError >> an error occurred running a Giac command:INPUT:sage2:=int(sage0,sageVARx):;OUTP
UT:sym2poly/r2sym(const gen & e,const index_m & i,const vecteur & l) Error: Bad Argument Value

Mupad [F(-1)]

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

[In]

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

[Out]

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