\(\int \frac {\sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))^2}{x^4} \, dx\) [177]

Optimal result

Integrand size = 29, antiderivative size = 336 \[ \int \frac {\sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))^2}{x^4} \, dx=\frac {b^2 c^2 \sqrt {d-c^2 d x^2}}{3 x}-\frac {b^2 c^3 \sqrt {d-c^2 d x^2} \text {arccosh}(c x)}{3 \sqrt {-1+c x} \sqrt {1+c x}}-\frac {b c \left (1-c^2 x^2\right ) \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))}{3 x^2 \sqrt {-1+c x} \sqrt {1+c x}}-\frac {c^3 \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))^2}{3 \sqrt {-1+c x} \sqrt {1+c x}}-\frac {\left (d-c^2 d x^2\right )^{3/2} (a+b \text {arccosh}(c x))^2}{3 d x^3}-\frac {2 b c^3 \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x)) \log \left (1+e^{-2 \text {arccosh}(c x)}\right )}{3 \sqrt {-1+c x} \sqrt {1+c x}}+\frac {b^2 c^3 \sqrt {d-c^2 d x^2} \operatorname {PolyLog}\left (2,-e^{-2 \text {arccosh}(c x)}\right )}{3 \sqrt {-1+c x} \sqrt {1+c x}} \]

[Out]

-1/3*(-c^2*d*x^2+d)^(3/2)*(a+b*arccosh(c*x))^2/d/x^3+1/3*b^2*c^2*(-c^2*d*x^2+d)^(1/2)/x-1/3*b^2*c^3*arccosh(c*
x)*(-c^2*d*x^2+d)^(1/2)/(c*x-1)^(1/2)/(c*x+1)^(1/2)-1/3*b*c*(-c^2*x^2+1)*(a+b*arccosh(c*x))*(-c^2*d*x^2+d)^(1/
2)/x^2/(c*x-1)^(1/2)/(c*x+1)^(1/2)-1/3*c^3*(a+b*arccosh(c*x))^2*(-c^2*d*x^2+d)^(1/2)/(c*x-1)^(1/2)/(c*x+1)^(1/
2)-2/3*b*c^3*(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)+1/3*b^2*c^3*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 = 0.30 (sec) , antiderivative size = 336, normalized size of antiderivative = 1.00, number of steps used = 11, number of rules used = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.379, Rules used = {5917, 5912, 5920, 99, 12, 54, 5882, 3799, 2221, 2317, 2438} \[ \int \frac {\sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))^2}{x^4} \, dx=-\frac {b c \left (1-c^2 x^2\right ) \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))}{3 x^2 \sqrt {c x-1} \sqrt {c x+1}}-\frac {\left (d-c^2 d x^2\right )^{3/2} (a+b \text {arccosh}(c x))^2}{3 d x^3}-\frac {c^3 \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))^2}{3 \sqrt {c x-1} \sqrt {c x+1}}-\frac {2 b c^3 \sqrt {d-c^2 d x^2} \log \left (e^{-2 \text {arccosh}(c x)}+1\right ) (a+b \text {arccosh}(c x))}{3 \sqrt {c x-1} \sqrt {c x+1}}+\frac {b^2 c^3 \sqrt {d-c^2 d x^2} \operatorname {PolyLog}\left (2,-e^{-2 \text {arccosh}(c x)}\right )}{3 \sqrt {c x-1} \sqrt {c x+1}}-\frac {b^2 c^3 \text {arccosh}(c x) \sqrt {d-c^2 d x^2}}{3 \sqrt {c x-1} \sqrt {c x+1}}+\frac {b^2 c^2 \sqrt {d-c^2 d x^2}}{3 x} \]

[In]

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

[Out]

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

Rule 12

Int[(a_)*(u_), x_Symbol] :> Dist[a, Int[u, x], x] /; FreeQ[a, x] &&  !MatchQ[u, (b_)*(v_) /; FreeQ[b, x]]

Rule 54

Int[1/(Sqrt[(a_) + (b_.)*(x_)]*Sqrt[(c_) + (d_.)*(x_)]), x_Symbol] :> Simp[ArcCosh[b*(x/a)]/b, x] /; FreeQ[{a,
b, c, d}, x] && EqQ[a + c, 0] && EqQ[b - d, 0] && GtQ[a, 0]

Rule 99

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

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 5912

Int[((a_.) + ArcCosh[(c_.)*(x_)]*(b_.))^(n_.)*((f_.)*(x_))^(m_.)*((d1_) + (e1_.)*(x_))^(p_.)*((d2_) + (e2_.)*(
x_))^(p_.), x_Symbol] :> Int[(f*x)^m*(d1*d2 + e1*e2*x^2)^p*(a + b*ArcCosh[c*x])^n, x] /; FreeQ[{a, b, c, d1, e
1, d2, e2, f, m, n}, x] && EqQ[d2*e1 + d1*e2, 0] && IntegerQ[p]

Rule 5917

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] + Dist[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*A
rcCosh[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 5920

Int[((a_.) + ArcCosh[(c_.)*(x_)]*(b_.))*((f_.)*(x_))^(m_)*((d_) + (e_.)*(x_)^2)^(p_.), x_Symbol] :> Simp[(f*x)
^(m + 1)*(d + e*x^2)^p*((a + b*ArcCosh[c*x])/(f*(m + 1))), x] + (-Dist[b*c*((-d)^p/(f*(m + 1))), Int[(f*x)^(m
+ 1)*(1 + c*x)^(p - 1/2)*(-1 + c*x)^(p - 1/2), x], x] - Dist[2*e*(p/(f^2*(m + 1))), Int[(f*x)^(m + 2)*(d + e*x
^2)^(p - 1)*(a + b*ArcCosh[c*x]), x], x]) /; FreeQ[{a, b, c, d, e, f}, x] && EqQ[c^2*d + e, 0] && IGtQ[p, 0] &
& ILtQ[(m + 1)/2, 0]

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

Mathematica [A] (warning: unable to verify)

Time = 0.90 (sec) , antiderivative size = 304, normalized size of antiderivative = 0.90 \[ \int \frac {\sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))^2}{x^4} \, dx=-\frac {d (1+c x) \left (a^2-a^2 c x-a^2 c^2 x^2-b^2 c^2 x^2+a^2 c^3 x^3+b^2 c^3 x^3-a b c x \sqrt {\frac {-1+c x}{1+c x}}-b^2 \left (-1+c x+c^2 x^2+c^3 x^3 \left (-1+\sqrt {\frac {-1+c x}{1+c x}}\right )\right ) \text {arccosh}(c x)^2-b \text {arccosh}(c x) \left (b c x \sqrt {\frac {-1+c x}{1+c x}}-2 a (-1+c x)^2 (1+c x)+2 b c^3 x^3 \sqrt {\frac {-1+c x}{1+c x}} \log \left (1+e^{-2 \text {arccosh}(c x)}\right )\right )-2 a b c^3 x^3 \sqrt {\frac {-1+c x}{1+c x}} \log (c x)+b^2 c^3 x^3 \sqrt {\frac {-1+c x}{1+c x}} \operatorname {PolyLog}\left (2,-e^{-2 \text {arccosh}(c x)}\right )\right )}{3 x^3 \sqrt {d-c^2 d x^2}} \]

[In]

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

[Out]

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

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(1750\) vs. \(2(314)=628\).

Time = 1.22 (sec) , antiderivative size = 1751, normalized size of antiderivative = 5.21

method	result	size
default	\(\text {Expression too large to display}\)	\(1751\)
parts	\(\text {Expression too large to display}\)	\(1751\)

[In]

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

[Out]

1/3*b^2*(-d*(c^2*x^2-1))^(1/2)/(3*c^4*x^4-3*c^2*x^2+1)*x*arccosh(c*x)*c^4-1/3*b^2*(-d*(c^2*x^2-1))^(1/2)/(3*c^
4*x^4-3*c^2*x^2+1)/(c*x+1)^(1/2)/(c*x-1)^(1/2)*c^3-1/3*b^2*(-d*(c^2*x^2-1))^(1/2)/(3*c^4*x^4-3*c^2*x^2+1)*x^3*
arccosh(c*x)*c^6+1/3*b^2*(-d*(c^2*x^2-1))^(1/2)/(3*c^4*x^4-3*c^2*x^2+1)*x^3*c^6-5/3*b^2*(-d*(c^2*x^2-1))^(1/2)
/(3*c^4*x^4-3*c^2*x^2+1)*x^3/(c*x+1)/(c*x-1)*c^6+4/3*b^2*(-d*(c^2*x^2-1))^(1/2)/(3*c^4*x^4-3*c^2*x^2+1)*x/(c*x
+1)/(c*x-1)*c^4-1/3*b^2*(-d*(c^2*x^2-1))^(1/2)/(3*c^4*x^4-3*c^2*x^2+1)/x/(c*x+1)/(c*x-1)*c^2+1/3*b^2*(-d*(c^2*
x^2-1))^(1/2)/(3*c^4*x^4-3*c^2*x^2+1)/x^3/(c*x+1)/(c*x-1)*arccosh(c*x)^2-2/3*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^3+2/3*b^2*(-d*(c^2*x^2-1))^(1/2
)/(3*c^4*x^4-3*c^2*x^2+1)*x^5/(c*x+1)/(c*x-1)*c^8-b^2*(-d*(c^2*x^2-1))^(1/2)/(3*c^4*x^4-3*c^2*x^2+1)*x^4/(c*x+
1)^(1/2)/(c*x-1)^(1/2)*c^7+b^2*(-d*(c^2*x^2-1))^(1/2)/(3*c^4*x^4-3*c^2*x^2+1)/(c*x+1)^(1/2)/(c*x-1)^(1/2)*arcc
osh(c*x)*c^3+b^2*(-d*(c^2*x^2-1))^(1/2)/(3*c^4*x^4-3*c^2*x^2+1)*x^2/(c*x+1)^(1/2)/(c*x-1)^(1/2)*c^5-1/3*b^2*(-
d*(c^2*x^2-1))^(1/2)/(3*c^4*x^4-3*c^2*x^2+1)/(c*x+1)^(1/2)/(c*x-1)^(1/2)*arccosh(c*x)^2*c^3+1/3*a*b*(-d*(c^2*x
^2-1))^(1/2)*(2*(c*x+1)^(1/2)*arccosh(c*x)*(c*x-1)^(1/2)*c^2*x^2+2*c^3*x^3*arccosh(c*x)-2*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)/(c*x-1)^(1/2)/(c*x+1)^(1/2)/x^3-
1/3*a^2/d/x^3*(-c^2*d*x^2+d)^(3/2)-3*b^2*(-d*(c^2*x^2-1))^(1/2)/(3*c^4*x^4-3*c^2*x^2+1)*x^3/(c*x+1)/(c*x-1)*ar
ccosh(c*x)^2*c^6-2/3*b^2*(-d*(c^2*x^2-1))^(1/2)/(3*c^4*x^4-3*c^2*x^2+1)*x^3/(c*x+1)/(c*x-1)*arccosh(c*x)*c^6-b
^2*(-d*(c^2*x^2-1))^(1/2)/(3*c^4*x^4-3*c^2*x^2+1)*x^4/(c*x+1)^(1/2)/(c*x-1)^(1/2)*arccosh(c*x)^2*c^7+b^2*(-d*(
c^2*x^2-1))^(1/2)/(3*c^4*x^4-3*c^2*x^2+1)*x^2/(c*x+1)^(1/2)/(c*x-1)^(1/2)*arccosh(c*x)^2*c^5-b^2*(-d*(c^2*x^2-
1))^(1/2)/(3*c^4*x^4-3*c^2*x^2+1)*x^2/(c*x+1)^(1/2)/(c*x-1)^(1/2)*arccosh(c*x)*c^5-1/3*b^2*(-d*(c^2*x^2-1))^(1
/2)/(3*c^4*x^4-3*c^2*x^2+1)/x^2/(c*x+1)^(1/2)/(c*x-1)^(1/2)*arccosh(c*x)*c+10/3*b^2*(-d*(c^2*x^2-1))^(1/2)/(3*
c^4*x^4-3*c^2*x^2+1)*x/(c*x+1)/(c*x-1)*arccosh(c*x)^2*c^4+1/3*b^2*(-d*(c^2*x^2-1))^(1/2)/(3*c^4*x^4-3*c^2*x^2+
1)*x/(c*x+1)/(c*x-1)*arccosh(c*x)*c^4-5/3*b^2*(-d*(c^2*x^2-1))^(1/2)/(3*c^4*x^4-3*c^2*x^2+1)/x/(c*x+1)/(c*x-1)
*arccosh(c*x)^2*c^2+b^2*(-d*(c^2*x^2-1))^(1/2)/(3*c^4*x^4-3*c^2*x^2+1)*x^5/(c*x+1)/(c*x-1)*arccosh(c*x)^2*c^8+
1/3*b^2*(-d*(c^2*x^2-1))^(1/2)/(3*c^4*x^4-3*c^2*x^2+1)*x^5/(c*x+1)/(c*x-1)*arccosh(c*x)*c^8+2/3*b^2*(-d*(c^2*x
^2-1))^(1/2)/(c*x-1)^(1/2)/(c*x+1)^(1/2)*arccosh(c*x)^2*c^3-1/3*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^3

Fricas [F]

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

[In]

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

Sympy [F]

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

[In]

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

[Out]

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

Maxima [F]

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

[In]

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

[Out]

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

Giac [F(-2)]

Exception generated. \[ \int \frac {\sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))^2}{x^4} \, dx=\text {Exception raised: TypeError} \]

[In]

integrate((a+b*arccosh(c*x))^2*(-c^2*d*x^2+d)^(1/2)/x^4,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. \[ \int \frac {\sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))^2}{x^4} \, dx=\int \frac {{\left (a+b\,\mathrm {acosh}\left (c\,x\right )\right )}^2\,\sqrt {d-c^2\,d\,x^2}}{x^4} \,d x \]

[In]

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

[Out]

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