\(\int \frac {(d-c^2 d x^2)^{3/2} (a+b \text {arccosh}(c x))^2}{x^2} \, dx\) [183]

Optimal result

Integrand size = 29, antiderivative size = 453 \[ \int \frac {\left (d-c^2 d x^2\right )^{3/2} (a+b \text {arccosh}(c x))^2}{x^2} \, dx=-\frac {1}{4} b^2 c^2 d x \sqrt {d-c^2 d x^2}-\frac {5 b^2 c d \sqrt {d-c^2 d x^2} \text {arccosh}(c x)}{4 \sqrt {-1+c x} \sqrt {1+c x}}+\frac {3 b c^3 d x^2 \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))}{2 \sqrt {-1+c x} \sqrt {1+c x}}+\frac {b c d \left (1-c^2 x^2\right ) \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))}{\sqrt {-1+c x} \sqrt {1+c x}}-\frac {3}{2} c^2 d x \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))^2+\frac {c d \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))^2}{\sqrt {-1+c x} \sqrt {1+c x}}-\frac {\left (d-c^2 d x^2\right )^{3/2} (a+b \text {arccosh}(c x))^2}{x}+\frac {c d \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))^3}{2 b \sqrt {-1+c x} \sqrt {1+c x}}+\frac {2 b c d \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x)) \log \left (1+e^{-2 \text {arccosh}(c x)}\right )}{\sqrt {-1+c x} \sqrt {1+c x}}-\frac {b^2 c d \sqrt {d-c^2 d x^2} \operatorname {PolyLog}\left (2,-e^{-2 \text {arccosh}(c x)}\right )}{\sqrt {-1+c x} \sqrt {1+c x}} \]

[Out]

-(-c^2*d*x^2+d)^(3/2)*(a+b*arccosh(c*x))^2/x-1/4*b^2*c^2*d*x*(-c^2*d*x^2+d)^(1/2)-3/2*c^2*d*x*(a+b*arccosh(c*x
))^2*(-c^2*d*x^2+d)^(1/2)-5/4*b^2*c*d*arccosh(c*x)*(-c^2*d*x^2+d)^(1/2)/(c*x-1)^(1/2)/(c*x+1)^(1/2)+3/2*b*c^3*
d*x^2*(a+b*arccosh(c*x))*(-c^2*d*x^2+d)^(1/2)/(c*x-1)^(1/2)/(c*x+1)^(1/2)+b*c*d*(-c^2*x^2+1)*(a+b*arccosh(c*x)
)*(-c^2*d*x^2+d)^(1/2)/(c*x-1)^(1/2)/(c*x+1)^(1/2)+c*d*(a+b*arccosh(c*x))^2*(-c^2*d*x^2+d)^(1/2)/(c*x-1)^(1/2)
/(c*x+1)^(1/2)+1/2*c*d*(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*d*(a+b*ar
ccosh(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
*d*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 = 0.45 (sec) , antiderivative size = 453, normalized size of antiderivative = 1.00, number of steps used = 15, number of rules used = 14, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.483, Rules used = {5928, 5895, 5893, 5883, 92, 54, 5912, 5919, 5882, 3799, 2221, 2317, 2438, 38} \[ \int \frac {\left (d-c^2 d x^2\right )^{3/2} (a+b \text {arccosh}(c x))^2}{x^2} \, dx=-\frac {3}{2} c^2 d x \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))^2+\frac {c d \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))^3}{2 b \sqrt {c x-1} \sqrt {c x+1}}+\frac {c d \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))^2}{\sqrt {c x-1} \sqrt {c x+1}}+\frac {b c d \left (1-c^2 x^2\right ) \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))}{\sqrt {c x-1} \sqrt {c x+1}}-\frac {\left (d-c^2 d x^2\right )^{3/2} (a+b \text {arccosh}(c x))^2}{x}+\frac {2 b c d \sqrt {d-c^2 d x^2} \log \left (e^{-2 \text {arccosh}(c x)}+1\right ) (a+b \text {arccosh}(c x))}{\sqrt {c x-1} \sqrt {c x+1}}+\frac {3 b c^3 d x^2 \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))}{2 \sqrt {c x-1} \sqrt {c x+1}}-\frac {b^2 c d \sqrt {d-c^2 d x^2} \operatorname {PolyLog}\left (2,-e^{-2 \text {arccosh}(c x)}\right )}{\sqrt {c x-1} \sqrt {c x+1}}-\frac {5 b^2 c d \text {arccosh}(c x) \sqrt {d-c^2 d x^2}}{4 \sqrt {c x-1} \sqrt {c x+1}}-\frac {1}{4} b^2 c^2 d x \sqrt {d-c^2 d x^2} \]

[In]

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

[Out]

-1/4*(b^2*c^2*d*x*Sqrt[d - c^2*d*x^2]) - (5*b^2*c*d*Sqrt[d - c^2*d*x^2]*ArcCosh[c*x])/(4*Sqrt[-1 + c*x]*Sqrt[1
+ c*x]) + (3*b*c^3*d*x^2*Sqrt[d - c^2*d*x^2]*(a + b*ArcCosh[c*x]))/(2*Sqrt[-1 + c*x]*Sqrt[1 + c*x]) + (b*c*d*
(1 - c^2*x^2)*Sqrt[d - c^2*d*x^2]*(a + b*ArcCosh[c*x]))/(Sqrt[-1 + c*x]*Sqrt[1 + c*x]) - (3*c^2*d*x*Sqrt[d - c
^2*d*x^2]*(a + b*ArcCosh[c*x])^2)/2 + (c*d*Sqrt[d - c^2*d*x^2]*(a + b*ArcCosh[c*x])^2)/(Sqrt[-1 + c*x]*Sqrt[1
+ c*x]) - ((d - c^2*d*x^2)^(3/2)*(a + b*ArcCosh[c*x])^2)/x + (c*d*Sqrt[d - c^2*d*x^2]*(a + b*ArcCosh[c*x])^3)/
(2*b*Sqrt[-1 + c*x]*Sqrt[1 + c*x]) + (2*b*c*d*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*d*Sqrt[d - c^2*d*x^2]*PolyLog[2, -E^(-2*ArcCosh[c*x])])/(Sqrt[-
1 + c*x]*Sqrt[1 + c*x])

Rule 38

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

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 92

Int[((a_.) + (b_.)*(x_))^2*((c_.) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p_.), x_Symbol] :> Simp[b*(a + b*x
)*(c + d*x)^(n + 1)*((e + f*x)^(p + 1)/(d*f*(n + p + 3))), x] + Dist[1/(d*f*(n + p + 3)), Int[(c + d*x)^n*(e +
f*x)^p*Simp[a^2*d*f*(n + p + 3) - b*(b*c*e + a*(d*e*(n + 1) + c*f*(p + 1))) + b*(a*d*f*(n + p + 4) - b*(d*e*(
n + 2) + c*f*(p + 2)))*x, x], x], x] /; FreeQ[{a, b, c, d, e, f, n, p}, x] && NeQ[n + p + 3, 0]

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 5883

Int[((a_.) + ArcCosh[(c_.)*(x_)]*(b_.))^(n_.)*((d_.)*(x_))^(m_.), x_Symbol] :> Simp[(d*x)^(m + 1)*((a + b*ArcC
osh[c*x])^n/(d*(m + 1))), x] - Dist[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 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 5895

Int[((a_.) + ArcCosh[(c_.)*(x_)]*(b_.))^(n_.)*Sqrt[(d_) + (e_.)*(x_)^2], x_Symbol] :> Simp[x*Sqrt[d + e*x^2]*(
(a + b*ArcCosh[c*x])^n/2), x] + (-Dist[(1/2)*Simp[Sqrt[d + e*x^2]/(Sqrt[1 + c*x]*Sqrt[-1 + c*x])], Int[(a + b*
ArcCosh[c*x])^n/(Sqrt[1 + c*x]*Sqrt[-1 + c*x]), x], x] - Dist[b*c*(n/2)*Simp[Sqrt[d + e*x^2]/(Sqrt[1 + c*x]*Sq
rt[-1 + c*x])], Int[x*(a + b*ArcCosh[c*x])^(n - 1), x], x]) /; FreeQ[{a, b, c, d, e}, x] && EqQ[c^2*d + e, 0]
&& GtQ[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 5919

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

Rule 5928

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*((a + b*ArcCosh[c*x])^n/(f*(m + 1))), 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])^n, x], 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*ArcCosh[c*x])^(n -
1), x], x]) /; FreeQ[{a, b, c, d, e, f}, x] && EqQ[c^2*d + e, 0] && GtQ[n, 0] && GtQ[p, 0] && LtQ[m, -1]

Rubi steps \begin{align*} \text {integral}& = -\frac {\left (d-c^2 d x^2\right )^{3/2} (a+b \text {arccosh}(c x))^2}{x}-\left (3 c^2 d\right ) \int \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))^2 \, dx-\frac {\left (2 b c d \sqrt {d-c^2 d x^2}\right ) \int \frac {(-1+c x) (1+c x) (a+b \text {arccosh}(c x))}{x} \, dx}{\sqrt {-1+c x} \sqrt {1+c x}} \\ & = -\frac {3}{2} c^2 d x \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))^2-\frac {\left (d-c^2 d x^2\right )^{3/2} (a+b \text {arccosh}(c x))^2}{x}-\frac {\left (2 b c d \sqrt {d-c^2 d x^2}\right ) \int \frac {\left (-1+c^2 x^2\right ) (a+b \text {arccosh}(c x))}{x} \, dx}{\sqrt {-1+c x} \sqrt {1+c x}}+\frac {\left (3 c^2 d \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}{2 \sqrt {-1+c x} \sqrt {1+c x}}+\frac {\left (3 b c^3 d \sqrt {d-c^2 d x^2}\right ) \int x (a+b \text {arccosh}(c x)) \, dx}{\sqrt {-1+c x} \sqrt {1+c x}} \\ & = \frac {3 b c^3 d x^2 \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))}{2 \sqrt {-1+c x} \sqrt {1+c x}}+\frac {b c d \left (1-c^2 x^2\right ) \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))}{\sqrt {-1+c x} \sqrt {1+c x}}-\frac {3}{2} c^2 d x \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))^2-\frac {\left (d-c^2 d x^2\right )^{3/2} (a+b \text {arccosh}(c x))^2}{x}+\frac {c d \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))^3}{2 b \sqrt {-1+c x} \sqrt {1+c x}}+\frac {\left (2 b c d \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 (b^2 c^2 d \sqrt {d-c^2 d x^2}\right ) \int \sqrt {-1+c x} \sqrt {1+c x} \, dx}{\sqrt {-1+c x} \sqrt {1+c x}}-\frac {\left (3 b^2 c^4 d \sqrt {d-c^2 d x^2}\right ) \int \frac {x^2}{\sqrt {-1+c x} \sqrt {1+c x}} \, dx}{2 \sqrt {-1+c x} \sqrt {1+c x}} \\ & = -\frac {1}{4} b^2 c^2 d x \sqrt {d-c^2 d x^2}+\frac {3 b c^3 d x^2 \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))}{2 \sqrt {-1+c x} \sqrt {1+c x}}+\frac {b c d \left (1-c^2 x^2\right ) \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))}{\sqrt {-1+c x} \sqrt {1+c x}}-\frac {3}{2} c^2 d x \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))^2-\frac {\left (d-c^2 d x^2\right )^{3/2} (a+b \text {arccosh}(c x))^2}{x}+\frac {c d \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))^3}{2 b \sqrt {-1+c x} \sqrt {1+c x}}-\frac {\left (2 c d \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 {\left (b^2 c^2 d \sqrt {d-c^2 d x^2}\right ) \int \frac {1}{\sqrt {-1+c x} \sqrt {1+c x}} \, dx}{2 \sqrt {-1+c x} \sqrt {1+c x}}-\frac {\left (3 b^2 c^2 d \sqrt {d-c^2 d x^2}\right ) \int \frac {1}{\sqrt {-1+c x} \sqrt {1+c x}} \, dx}{4 \sqrt {-1+c x} \sqrt {1+c x}} \\ & = -\frac {1}{4} b^2 c^2 d x \sqrt {d-c^2 d x^2}-\frac {5 b^2 c d \sqrt {d-c^2 d x^2} \text {arccosh}(c x)}{4 \sqrt {-1+c x} \sqrt {1+c x}}+\frac {3 b c^3 d x^2 \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))}{2 \sqrt {-1+c x} \sqrt {1+c x}}+\frac {b c d \left (1-c^2 x^2\right ) \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))}{\sqrt {-1+c x} \sqrt {1+c x}}-\frac {3}{2} c^2 d x \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))^2+\frac {c d \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c 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}{x}+\frac {c d \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))^3}{2 b \sqrt {-1+c x} \sqrt {1+c x}}-\frac {\left (4 c d \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 {1}{4} b^2 c^2 d x \sqrt {d-c^2 d x^2}-\frac {5 b^2 c d \sqrt {d-c^2 d x^2} \text {arccosh}(c x)}{4 \sqrt {-1+c x} \sqrt {1+c x}}+\frac {3 b c^3 d x^2 \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))}{2 \sqrt {-1+c x} \sqrt {1+c x}}+\frac {b c d \left (1-c^2 x^2\right ) \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))}{\sqrt {-1+c x} \sqrt {1+c x}}-\frac {3}{2} c^2 d x \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))^2+\frac {c d \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c 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}{x}+\frac {c d \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))^3}{2 b \sqrt {-1+c x} \sqrt {1+c x}}+\frac {2 b c d \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 d \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 {1}{4} b^2 c^2 d x \sqrt {d-c^2 d x^2}-\frac {5 b^2 c d \sqrt {d-c^2 d x^2} \text {arccosh}(c x)}{4 \sqrt {-1+c x} \sqrt {1+c x}}+\frac {3 b c^3 d x^2 \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))}{2 \sqrt {-1+c x} \sqrt {1+c x}}+\frac {b c d \left (1-c^2 x^2\right ) \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))}{\sqrt {-1+c x} \sqrt {1+c x}}-\frac {3}{2} c^2 d x \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))^2+\frac {c d \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c 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}{x}+\frac {c d \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))^3}{2 b \sqrt {-1+c x} \sqrt {1+c x}}+\frac {2 b c d \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 d \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 {1}{4} b^2 c^2 d x \sqrt {d-c^2 d x^2}-\frac {5 b^2 c d \sqrt {d-c^2 d x^2} \text {arccosh}(c x)}{4 \sqrt {-1+c x} \sqrt {1+c x}}+\frac {3 b c^3 d x^2 \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))}{2 \sqrt {-1+c x} \sqrt {1+c x}}+\frac {b c d \left (1-c^2 x^2\right ) \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))}{\sqrt {-1+c x} \sqrt {1+c x}}-\frac {3}{2} c^2 d x \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))^2+\frac {c d \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c 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}{x}+\frac {c d \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))^3}{2 b \sqrt {-1+c x} \sqrt {1+c x}}+\frac {2 b c d \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 d \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 = 4.19 (sec) , antiderivative size = 433, normalized size of antiderivative = 0.96 \[ \int \frac {\left (d-c^2 d x^2\right )^{3/2} (a+b \text {arccosh}(c x))^2}{x^2} \, dx=\frac {-12 a^2 d \sqrt {\frac {-1+c x}{1+c x}} (1+c x) \left (2+c^2 x^2\right ) \sqrt {d-c^2 d x^2}+36 a^2 c d^{3/2} x \sqrt {\frac {-1+c x}{1+c x}} (1+c x) \arctan \left (\frac {c x \sqrt {d-c^2 d x^2}}{\sqrt {d} \left (-1+c^2 x^2\right )}\right )-24 a b d \sqrt {d-c^2 d x^2} \left (2 \sqrt {\frac {-1+c x}{1+c x}} (1+c x) \text {arccosh}(c x)-c x \left (\text {arccosh}(c x)^2+2 \log (c x)\right )\right )-8 b^2 d \sqrt {d-c^2 d x^2} \left (\text {arccosh}(c x) \left (3 \sqrt {\frac {-1+c x}{1+c x}} (1+c x) \text {arccosh}(c x)-c x \left (\text {arccosh}(c x) (3+\text {arccosh}(c x))+6 \log \left (1+e^{-2 \text {arccosh}(c x)}\right )\right )\right )+3 c x \operatorname {PolyLog}\left (2,-e^{-2 \text {arccosh}(c x)}\right )\right )+6 a b c d x \sqrt {d-c^2 d x^2} (\cosh (2 \text {arccosh}(c x))+2 \text {arccosh}(c x) (\text {arccosh}(c x)-\sinh (2 \text {arccosh}(c x))))+b^2 c d x \sqrt {d-c^2 d x^2} \left (4 \text {arccosh}(c x)^3+6 \text {arccosh}(c x) \cosh (2 \text {arccosh}(c x))-3 \left (1+2 \text {arccosh}(c x)^2\right ) \sinh (2 \text {arccosh}(c x))\right )}{24 x \sqrt {\frac {-1+c x}{1+c x}} (1+c x)} \]

[In]

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

[Out]

(-12*a^2*d*Sqrt[(-1 + c*x)/(1 + c*x)]*(1 + c*x)*(2 + c^2*x^2)*Sqrt[d - c^2*d*x^2] + 36*a^2*c*d^(3/2)*x*Sqrt[(-
1 + c*x)/(1 + c*x)]*(1 + c*x)*ArcTan[(c*x*Sqrt[d - c^2*d*x^2])/(Sqrt[d]*(-1 + c^2*x^2))] - 24*a*b*d*Sqrt[d - c
^2*d*x^2]*(2*Sqrt[(-1 + c*x)/(1 + c*x)]*(1 + c*x)*ArcCosh[c*x] - c*x*(ArcCosh[c*x]^2 + 2*Log[c*x])) - 8*b^2*d*
Sqrt[d - c^2*d*x^2]*(ArcCosh[c*x]*(3*Sqrt[(-1 + c*x)/(1 + c*x)]*(1 + c*x)*ArcCosh[c*x] - c*x*(ArcCosh[c*x]*(3
+ ArcCosh[c*x]) + 6*Log[1 + E^(-2*ArcCosh[c*x])])) + 3*c*x*PolyLog[2, -E^(-2*ArcCosh[c*x])]) + 6*a*b*c*d*x*Sqr
t[d - c^2*d*x^2]*(Cosh[2*ArcCosh[c*x]] + 2*ArcCosh[c*x]*(ArcCosh[c*x] - Sinh[2*ArcCosh[c*x]])) + b^2*c*d*x*Sqr
t[d - c^2*d*x^2]*(4*ArcCosh[c*x]^3 + 6*ArcCosh[c*x]*Cosh[2*ArcCosh[c*x]] - 3*(1 + 2*ArcCosh[c*x]^2)*Sinh[2*Arc
Cosh[c*x]]))/(24*x*Sqrt[(-1 + c*x)/(1 + c*x)]*(1 + c*x))

Maple [A] (verified)

Time = 1.08 (sec) , antiderivative size = 464, normalized size of antiderivative = 1.02

method	result	size
default	\(-\frac {a^{2} \left (-c^{2} d \,x^{2}+d \right )^{\frac {5}{2}}}{d x}-a^{2} c^{2} x \left (-c^{2} d \,x^{2}+d \right )^{\frac {3}{2}}-\frac {3 a^{2} c^{2} d x \sqrt {-c^{2} d \,x^{2}+d}}{2}-\frac {3 a^{2} c^{2} d^{2} \arctan \left (\frac {\sqrt {c^{2} d}\, x}{\sqrt {-c^{2} d \,x^{2}+d}}\right )}{2 \sqrt {c^{2} d}}+\frac {b^{2} \sqrt {-d \left (c^{2} x^{2}-1\right )}\, \left (-2 \operatorname {arccosh}\left (c x \right )^{2} \sqrt {c x +1}\, \sqrt {c x -1}\, x^{2} c^{2}+2 c^{3} x^{3} \operatorname {arccosh}\left (c x \right )-\sqrt {c x -1}\, \sqrt {c x +1}\, c^{2} x^{2}+2 \operatorname {arccosh}\left (c x \right )^{3} x c -4 \sqrt {c x -1}\, \sqrt {c x +1}\, \operatorname {arccosh}\left (c x \right )^{2}-4 \operatorname {arccosh}\left (c x \right )^{2} x c +8 \,\operatorname {arccosh}\left (c x \right ) \ln \left (1+\left (c x +\sqrt {c x -1}\, \sqrt {c x +1}\right )^{2}\right ) x c -c x \,\operatorname {arccosh}\left (c x \right )+4 \operatorname {polylog}\left (2, -\left (c x +\sqrt {c x -1}\, \sqrt {c x +1}\right )^{2}\right ) x c \right ) d}{4 \sqrt {c x -1}\, \sqrt {c x +1}\, x}+\frac {a b \sqrt {-d \left (c^{2} x^{2}-1\right )}\, \left (-4 \sqrt {c x +1}\, \operatorname {arccosh}\left (c x \right ) \sqrt {c x -1}\, c^{2} x^{2}+2 c^{3} x^{3}+6 \operatorname {arccosh}\left (c x \right )^{2} x c -8 \,\operatorname {arccosh}\left (c x \right ) \sqrt {c x -1}\, \sqrt {c x +1}-8 c x \,\operatorname {arccosh}\left (c x \right )+8 \ln \left (1+\left (c x +\sqrt {c x -1}\, \sqrt {c x +1}\right )^{2}\right ) x c -c x \right ) d}{4 \sqrt {c x -1}\, \sqrt {c x +1}\, x}\)	\(464\)
parts	\(-\frac {a^{2} \left (-c^{2} d \,x^{2}+d \right )^{\frac {5}{2}}}{d x}-a^{2} c^{2} x \left (-c^{2} d \,x^{2}+d \right )^{\frac {3}{2}}-\frac {3 a^{2} c^{2} d x \sqrt {-c^{2} d \,x^{2}+d}}{2}-\frac {3 a^{2} c^{2} d^{2} \arctan \left (\frac {\sqrt {c^{2} d}\, x}{\sqrt {-c^{2} d \,x^{2}+d}}\right )}{2 \sqrt {c^{2} d}}+\frac {b^{2} \sqrt {-d \left (c^{2} x^{2}-1\right )}\, \left (-2 \operatorname {arccosh}\left (c x \right )^{2} \sqrt {c x +1}\, \sqrt {c x -1}\, x^{2} c^{2}+2 c^{3} x^{3} \operatorname {arccosh}\left (c x \right )-\sqrt {c x -1}\, \sqrt {c x +1}\, c^{2} x^{2}+2 \operatorname {arccosh}\left (c x \right )^{3} x c -4 \sqrt {c x -1}\, \sqrt {c x +1}\, \operatorname {arccosh}\left (c x \right )^{2}-4 \operatorname {arccosh}\left (c x \right )^{2} x c +8 \,\operatorname {arccosh}\left (c x \right ) \ln \left (1+\left (c x +\sqrt {c x -1}\, \sqrt {c x +1}\right )^{2}\right ) x c -c x \,\operatorname {arccosh}\left (c x \right )+4 \operatorname {polylog}\left (2, -\left (c x +\sqrt {c x -1}\, \sqrt {c x +1}\right )^{2}\right ) x c \right ) d}{4 \sqrt {c x -1}\, \sqrt {c x +1}\, x}+\frac {a b \sqrt {-d \left (c^{2} x^{2}-1\right )}\, \left (-4 \sqrt {c x +1}\, \operatorname {arccosh}\left (c x \right ) \sqrt {c x -1}\, c^{2} x^{2}+2 c^{3} x^{3}+6 \operatorname {arccosh}\left (c x \right )^{2} x c -8 \,\operatorname {arccosh}\left (c x \right ) \sqrt {c x -1}\, \sqrt {c x +1}-8 c x \,\operatorname {arccosh}\left (c x \right )+8 \ln \left (1+\left (c x +\sqrt {c x -1}\, \sqrt {c x +1}\right )^{2}\right ) x c -c x \right ) d}{4 \sqrt {c x -1}\, \sqrt {c x +1}\, x}\)	\(464\)

[In]

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

[Out]

-a^2/d/x*(-c^2*d*x^2+d)^(5/2)-a^2*c^2*x*(-c^2*d*x^2+d)^(3/2)-3/2*a^2*c^2*d*x*(-c^2*d*x^2+d)^(1/2)-3/2*a^2*c^2*
d^2/(c^2*d)^(1/2)*arctan((c^2*d)^(1/2)*x/(-c^2*d*x^2+d)^(1/2))+1/4*b^2*(-d*(c^2*x^2-1))^(1/2)/(c*x-1)^(1/2)/(c
*x+1)^(1/2)/x*(-2*arccosh(c*x)^2*(c*x+1)^(1/2)*(c*x-1)^(1/2)*x^2*c^2+2*c^3*x^3*arccosh(c*x)-(c*x-1)^(1/2)*(c*x
+1)^(1/2)*c^2*x^2+2*arccosh(c*x)^3*x*c-4*(c*x-1)^(1/2)*(c*x+1)^(1/2)*arccosh(c*x)^2-4*arccosh(c*x)^2*x*c+8*arc
cosh(c*x)*ln(1+(c*x+(c*x-1)^(1/2)*(c*x+1)^(1/2))^2)*x*c-c*x*arccosh(c*x)+4*polylog(2,-(c*x+(c*x-1)^(1/2)*(c*x+
1)^(1/2))^2)*x*c)*d+1/4*a*b*(-d*(c^2*x^2-1))^(1/2)/(c*x-1)^(1/2)/(c*x+1)^(1/2)/x*(-4*(c*x+1)^(1/2)*arccosh(c*x
)*(c*x-1)^(1/2)*c^2*x^2+2*c^3*x^3+6*arccosh(c*x)^2*x*c-8*arccosh(c*x)*(c*x-1)^(1/2)*(c*x+1)^(1/2)-8*c*x*arccos
h(c*x)+8*ln(1+(c*x+(c*x-1)^(1/2)*(c*x+1)^(1/2))^2)*x*c-c*x)*d

Fricas [F]

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

[In]

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

[Out]

integral(-(a^2*c^2*d*x^2 - a^2*d + (b^2*c^2*d*x^2 - b^2*d)*arccosh(c*x)^2 + 2*(a*b*c^2*d*x^2 - a*b*d)*arccosh(
c*x))*sqrt(-c^2*d*x^2 + d)/x^2, x)

Sympy [F]

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

[In]

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

[Out]

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

Maxima [F]

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

[In]

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

[Out]

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

Giac [F(-2)]

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

[In]

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

[In]

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

[Out]

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