3.185 \(\int \frac{(d-c^2 d x^2)^{3/2} (a+b \cosh ^{-1}(c x))^2}{x^4} \, dx\)

Optimal. Leaf size=426 \[ \frac{4 b^2 c^3 d \sqrt{d-c^2 d x^2} \text{PolyLog}\left (2,-e^{-2 \cosh ^{-1}(c x)}\right )}{3 \sqrt{c x-1} \sqrt{c x+1}}-\frac{c^3 d \sqrt{d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )^3}{3 b \sqrt{c x-1} \sqrt{c x+1}}-\frac{4 c^3 d \sqrt{d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )^2}{3 \sqrt{c x-1} \sqrt{c x+1}}+\frac{c^2 d \sqrt{d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )^2}{x}-\frac{b c d \left (1-c^2 x^2\right ) \sqrt{d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )}{3 x^2 \sqrt{c x-1} \sqrt{c x+1}}-\frac{\left (d-c^2 d x^2\right )^{3/2} \left (a+b \cosh ^{-1}(c x)\right )^2}{3 x^3}-\frac{8 b c^3 d \sqrt{d-c^2 d x^2} \log \left (e^{-2 \cosh ^{-1}(c x)}+1\right ) \left (a+b \cosh ^{-1}(c x)\right )}{3 \sqrt{c x-1} \sqrt{c x+1}}+\frac{b^2 c^2 d \sqrt{d-c^2 d x^2}}{3 x}-\frac{b^2 c^3 d \sqrt{d-c^2 d x^2} \cosh ^{-1}(c x)}{3 \sqrt{c x-1} \sqrt{c x+1}} \]

[Out]

(b^2*c^2*d*Sqrt[d - c^2*d*x^2])/(3*x) - (b^2*c^3*d*Sqrt[d - c^2*d*x^2]*ArcCosh[c*x])/(3*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]))/(3*x^2*Sqrt[-1 + c*x]*Sqrt[1 + c*x])
+ (c^2*d*Sqrt[d - c^2*d*x^2]*(a + b*ArcCosh[c*x])^2)/x - (4*c^3*d*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*x^3) - (c^3*d*Sqrt[d - c^
2*d*x^2]*(a + b*ArcCosh[c*x])^3)/(3*b*Sqrt[-1 + c*x]*Sqrt[1 + c*x]) - (8*b*c^3*d*Sqrt[d - c^2*d*x^2]*(a + b*Ar
cCosh[c*x])*Log[1 + E^(-2*ArcCosh[c*x])])/(3*Sqrt[-1 + c*x]*Sqrt[1 + c*x]) + (4*b^2*c^3*d*Sqrt[d - c^2*d*x^2]*
PolyLog[2, -E^(-2*ArcCosh[c*x])])/(3*Sqrt[-1 + c*x]*Sqrt[1 + c*x])

________________________________________________________________________________________

Rubi [A]  time = 1.16697, antiderivative size = 438, normalized size of antiderivative = 1.03, number of steps used = 18, number of rules used = 13, integrand size = 29, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.448, Rules used = {5798, 5740, 5738, 5660, 3718, 2190, 2279, 2391, 5676, 5729, 97, 12, 52} \[ -\frac{4 b^2 c^3 d \sqrt{d-c^2 d x^2} \text{PolyLog}\left (2,-e^{2 \cosh ^{-1}(c x)}\right )}{3 \sqrt{c x-1} \sqrt{c x+1}}-\frac{c^3 d \sqrt{d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )^3}{3 b \sqrt{c x-1} \sqrt{c x+1}}+\frac{4 c^3 d \sqrt{d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )^2}{3 \sqrt{c x-1} \sqrt{c x+1}}+\frac{c^2 d \sqrt{d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )^2}{x}-\frac{b c d \left (1-c^2 x^2\right ) \sqrt{d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )}{3 x^2 \sqrt{c x-1} \sqrt{c x+1}}-\frac{d (1-c x) (c x+1) \sqrt{d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )^2}{3 x^3}-\frac{8 b c^3 d \sqrt{d-c^2 d x^2} \log \left (e^{2 \cosh ^{-1}(c x)}+1\right ) \left (a+b \cosh ^{-1}(c x)\right )}{3 \sqrt{c x-1} \sqrt{c x+1}}+\frac{b^2 c^2 d \sqrt{d-c^2 d x^2}}{3 x}-\frac{b^2 c^3 d \sqrt{d-c^2 d x^2} \cosh ^{-1}(c x)}{3 \sqrt{c x-1} \sqrt{c x+1}} \]

Warning: Unable to verify antiderivative.

[In]

Int[((d - c^2*d*x^2)^(3/2)*(a + b*ArcCosh[c*x])^2)/x^4,x]

[Out]

(b^2*c^2*d*Sqrt[d - c^2*d*x^2])/(3*x) - (b^2*c^3*d*Sqrt[d - c^2*d*x^2]*ArcCosh[c*x])/(3*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]))/(3*x^2*Sqrt[-1 + c*x]*Sqrt[1 + c*x])
+ (c^2*d*Sqrt[d - c^2*d*x^2]*(a + b*ArcCosh[c*x])^2)/x + (4*c^3*d*Sqrt[d - c^2*d*x^2]*(a + b*ArcCosh[c*x])^2)/
(3*Sqrt[-1 + c*x]*Sqrt[1 + c*x]) - (d*(1 - c*x)*(1 + c*x)*Sqrt[d - c^2*d*x^2]*(a + b*ArcCosh[c*x])^2)/(3*x^3)
- (c^3*d*Sqrt[d - c^2*d*x^2]*(a + b*ArcCosh[c*x])^3)/(3*b*Sqrt[-1 + c*x]*Sqrt[1 + c*x]) - (8*b*c^3*d*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]) - (4*b^2*c^3*d*S
qrt[d - c^2*d*x^2]*PolyLog[2, -E^(2*ArcCosh[c*x])])/(3*Sqrt[-1 + c*x]*Sqrt[1 + c*x])

Rule 5798

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

Rule 5740

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

Rule 5738

Int[((a_.) + ArcCosh[(c_.)*(x_)]*(b_.))^(n_.)*((f_.)*(x_))^(m_)*Sqrt[(d1_) + (e1_.)*(x_)]*Sqrt[(d2_) + (e2_.)*
(x_)], x_Symbol] :> Simp[((f*x)^(m + 1)*Sqrt[d1 + e1*x]*Sqrt[d2 + e2*x]*(a + b*ArcCosh[c*x])^n)/(f*(m + 1)), x
] + (-Dist[(b*c*n*Sqrt[d1 + e1*x]*Sqrt[d2 + e2*x])/(f*(m + 1)*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*Sqrt[d1 + e1*x]*Sqrt[d2 + e2*x])/(f^2*(m + 1)*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, d1, e1, d2, e2, f}, x] && EqQ[e1 - c*d1, 0] && EqQ[e2 + c*d2, 0] && GtQ[n, 0] && LtQ[m, -1]

Rule 5660

Int[((a_.) + ArcCosh[(c_.)*(x_)]*(b_.))^(n_.)/(x_), x_Symbol] :> Subst[Int[(a + b*x)^n/Coth[x], x], x, ArcCosh
[c*x]] /; FreeQ[{a, b, c}, x] && IGtQ[n, 0]

Rule 3718

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 2190

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*Log[1 + (b*(F^(g*(e + f*x)))^n)/a])/(b*f*g*n*Log[F]), 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 2279

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 2391

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 5676

Int[((a_.) + ArcCosh[(c_.)*(x_)]*(b_.))^(n_.)/(Sqrt[(d1_) + (e1_.)*(x_)]*Sqrt[(d2_) + (e2_.)*(x_)]), x_Symbol]
:> Simp[(a + b*ArcCosh[c*x])^(n + 1)/(b*c*Sqrt[-(d1*d2)]*(n + 1)), x] /; FreeQ[{a, b, c, d1, e1, d2, e2, n},
x] && EqQ[e1, c*d1] && EqQ[e2, -(c*d2)] && GtQ[d1, 0] && LtQ[d2, 0] && NeQ[n, -1]

Rule 5729

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]

Rule 97

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 12

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

Rule 52

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]

Rubi steps

\begin{align*} \int \frac{\left (d-c^2 d x^2\right )^{3/2} \left (a+b \cosh ^{-1}(c x)\right )^2}{x^4} \, dx &=-\frac{\left (d \sqrt{d-c^2 d x^2}\right ) \int \frac{(-1+c x)^{3/2} (1+c x)^{3/2} \left (a+b \cosh ^{-1}(c x)\right )^2}{x^4} \, dx}{\sqrt{-1+c x} \sqrt{1+c x}}\\ &=-\frac{d (1-c x) (1+c x) \sqrt{d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )^2}{3 x^3}-\frac{\left (2 b c d \sqrt{d-c^2 d x^2}\right ) \int \frac{\left (-1+c^2 x^2\right ) \left (a+b \cosh ^{-1}(c x)\right )}{x^3} \, dx}{3 \sqrt{-1+c x} \sqrt{1+c x}}-\frac{\left (c^2 d \sqrt{d-c^2 d x^2}\right ) \int \frac{\sqrt{-1+c x} \sqrt{1+c x} \left (a+b \cosh ^{-1}(c x)\right )^2}{x^2} \, dx}{\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} \left (a+b \cosh ^{-1}(c x)\right )}{3 x^2 \sqrt{-1+c x} \sqrt{1+c x}}+\frac{c^2 d \sqrt{d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )^2}{x}-\frac{d (1-c x) (1+c x) \sqrt{d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )^2}{3 x^3}-\frac{\left (b^2 c^2 d \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 d \sqrt{d-c^2 d x^2}\right ) \int \frac{a+b \cosh ^{-1}(c x)}{x} \, dx}{3 \sqrt{-1+c x} \sqrt{1+c x}}-\frac{\left (2 b c^3 d \sqrt{d-c^2 d x^2}\right ) \int \frac{a+b \cosh ^{-1}(c x)}{x} \, dx}{\sqrt{-1+c x} \sqrt{1+c x}}-\frac{\left (c^4 d \sqrt{d-c^2 d x^2}\right ) \int \frac{\left (a+b \cosh ^{-1}(c x)\right )^2}{\sqrt{-1+c x} \sqrt{1+c x}} \, dx}{\sqrt{-1+c x} \sqrt{1+c x}}\\ &=\frac{b^2 c^2 d \sqrt{d-c^2 d x^2}}{3 x}-\frac{b c d \left (1-c^2 x^2\right ) \sqrt{d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )}{3 x^2 \sqrt{-1+c x} \sqrt{1+c x}}+\frac{c^2 d \sqrt{d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )^2}{x}-\frac{d (1-c x) (1+c x) \sqrt{d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )^2}{3 x^3}-\frac{c^3 d \sqrt{d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )^3}{3 b \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{c^2}{\sqrt{-1+c x} \sqrt{1+c x}} \, dx}{3 \sqrt{-1+c x} \sqrt{1+c x}}-\frac{\left (2 b c^3 d \sqrt{d-c^2 d x^2}\right ) \operatorname{Subst}\left (\int (a+b x) \tanh (x) \, dx,x,\cosh ^{-1}(c x)\right )}{3 \sqrt{-1+c x} \sqrt{1+c x}}-\frac{\left (2 b c^3 d \sqrt{d-c^2 d x^2}\right ) \operatorname{Subst}\left (\int (a+b x) \tanh (x) \, dx,x,\cosh ^{-1}(c x)\right )}{\sqrt{-1+c x} \sqrt{1+c x}}\\ &=\frac{b^2 c^2 d \sqrt{d-c^2 d x^2}}{3 x}-\frac{b c d \left (1-c^2 x^2\right ) \sqrt{d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )}{3 x^2 \sqrt{-1+c x} \sqrt{1+c x}}+\frac{c^2 d \sqrt{d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )^2}{x}+\frac{4 c^3 d \sqrt{d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )^2}{3 \sqrt{-1+c x} \sqrt{1+c x}}-\frac{d (1-c x) (1+c x) \sqrt{d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )^2}{3 x^3}-\frac{c^3 d \sqrt{d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )^3}{3 b \sqrt{-1+c x} \sqrt{1+c x}}-\frac{\left (4 b c^3 d \sqrt{d-c^2 d x^2}\right ) \operatorname{Subst}\left (\int \frac{e^{2 x} (a+b x)}{1+e^{2 x}} \, dx,x,\cosh ^{-1}(c x)\right )}{3 \sqrt{-1+c x} \sqrt{1+c x}}-\frac{\left (4 b c^3 d \sqrt{d-c^2 d x^2}\right ) \operatorname{Subst}\left (\int \frac{e^{2 x} (a+b x)}{1+e^{2 x}} \, dx,x,\cosh ^{-1}(c x)\right )}{\sqrt{-1+c x} \sqrt{1+c x}}-\frac{\left (b^2 c^4 d \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 d \sqrt{d-c^2 d x^2}}{3 x}-\frac{b^2 c^3 d \sqrt{d-c^2 d x^2} \cosh ^{-1}(c x)}{3 \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} \left (a+b \cosh ^{-1}(c x)\right )}{3 x^2 \sqrt{-1+c x} \sqrt{1+c x}}+\frac{c^2 d \sqrt{d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )^2}{x}+\frac{4 c^3 d \sqrt{d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )^2}{3 \sqrt{-1+c x} \sqrt{1+c x}}-\frac{d (1-c x) (1+c x) \sqrt{d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )^2}{3 x^3}-\frac{c^3 d \sqrt{d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )^3}{3 b \sqrt{-1+c x} \sqrt{1+c x}}-\frac{8 b c^3 d \sqrt{d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right ) \log \left (1+e^{2 \cosh ^{-1}(c x)}\right )}{3 \sqrt{-1+c x} \sqrt{1+c x}}+\frac{\left (2 b^2 c^3 d \sqrt{d-c^2 d x^2}\right ) \operatorname{Subst}\left (\int \log \left (1+e^{2 x}\right ) \, dx,x,\cosh ^{-1}(c x)\right )}{3 \sqrt{-1+c x} \sqrt{1+c x}}+\frac{\left (2 b^2 c^3 d \sqrt{d-c^2 d x^2}\right ) \operatorname{Subst}\left (\int \log \left (1+e^{2 x}\right ) \, dx,x,\cosh ^{-1}(c x)\right )}{\sqrt{-1+c x} \sqrt{1+c x}}\\ &=\frac{b^2 c^2 d \sqrt{d-c^2 d x^2}}{3 x}-\frac{b^2 c^3 d \sqrt{d-c^2 d x^2} \cosh ^{-1}(c x)}{3 \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} \left (a+b \cosh ^{-1}(c x)\right )}{3 x^2 \sqrt{-1+c x} \sqrt{1+c x}}+\frac{c^2 d \sqrt{d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )^2}{x}+\frac{4 c^3 d \sqrt{d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )^2}{3 \sqrt{-1+c x} \sqrt{1+c x}}-\frac{d (1-c x) (1+c x) \sqrt{d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )^2}{3 x^3}-\frac{c^3 d \sqrt{d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )^3}{3 b \sqrt{-1+c x} \sqrt{1+c x}}-\frac{8 b c^3 d \sqrt{d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right ) \log \left (1+e^{2 \cosh ^{-1}(c x)}\right )}{3 \sqrt{-1+c x} \sqrt{1+c x}}+\frac{\left (b^2 c^3 d \sqrt{d-c^2 d x^2}\right ) \operatorname{Subst}\left (\int \frac{\log (1+x)}{x} \, dx,x,e^{2 \cosh ^{-1}(c x)}\right )}{3 \sqrt{-1+c x} \sqrt{1+c x}}+\frac{\left (b^2 c^3 d \sqrt{d-c^2 d x^2}\right ) \operatorname{Subst}\left (\int \frac{\log (1+x)}{x} \, dx,x,e^{2 \cosh ^{-1}(c x)}\right )}{\sqrt{-1+c x} \sqrt{1+c x}}\\ &=\frac{b^2 c^2 d \sqrt{d-c^2 d x^2}}{3 x}-\frac{b^2 c^3 d \sqrt{d-c^2 d x^2} \cosh ^{-1}(c x)}{3 \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} \left (a+b \cosh ^{-1}(c x)\right )}{3 x^2 \sqrt{-1+c x} \sqrt{1+c x}}+\frac{c^2 d \sqrt{d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )^2}{x}+\frac{4 c^3 d \sqrt{d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )^2}{3 \sqrt{-1+c x} \sqrt{1+c x}}-\frac{d (1-c x) (1+c x) \sqrt{d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )^2}{3 x^3}-\frac{c^3 d \sqrt{d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )^3}{3 b \sqrt{-1+c x} \sqrt{1+c x}}-\frac{8 b c^3 d \sqrt{d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right ) \log \left (1+e^{2 \cosh ^{-1}(c x)}\right )}{3 \sqrt{-1+c x} \sqrt{1+c x}}-\frac{4 b^2 c^3 d \sqrt{d-c^2 d x^2} \text{Li}_2\left (-e^{2 \cosh ^{-1}(c x)}\right )}{3 \sqrt{-1+c x} \sqrt{1+c x}}\\ \end{align*}

Mathematica [A]  time = 2.05603, size = 583, normalized size = 1.37 \[ \frac{-4 b^2 c^3 d^2 x^3 (c x-1) \text{PolyLog}\left (2,-e^{-2 \cosh ^{-1}(c x)}\right )-4 a^2 c^4 d^2 x^4 \sqrt{\frac{c x-1}{c x+1}}+5 a^2 c^2 d^2 x^2 \sqrt{\frac{c x-1}{c x+1}}-3 a^2 c^3 d^{3/2} x^3 \sqrt{\frac{c x-1}{c x+1}} \sqrt{d-c^2 d x^2} \tan ^{-1}\left (\frac{c x \sqrt{d-c^2 d x^2}}{\sqrt{d} \left (c^2 x^2-1\right )}\right )-a^2 d^2 \sqrt{\frac{c x-1}{c x+1}}+a b c^2 d^2 x^2+8 a b c^4 d^2 x^4 \log (c x)-8 a b c^3 d^2 x^3 \log (c x)-b d^2 (c x-1) \cosh ^{-1}(c x)^2 \left (b \left (4 c^3 x^3 \left (\sqrt{\frac{c x-1}{c x+1}}-1\right )+4 c^2 x^2 \sqrt{\frac{c x-1}{c x+1}}-c x \sqrt{\frac{c x-1}{c x+1}}-\sqrt{\frac{c x-1}{c x+1}}\right )-3 a c^3 x^3\right )+b d^2 (c x-1) \cosh ^{-1}(c x) \left (2 a \sqrt{\frac{c x-1}{c x+1}} \left (-4 c^3 x^3-4 c^2 x^2+c x+1\right )+8 b c^3 x^3 \log \left (e^{-2 \cosh ^{-1}(c x)}+1\right )+b c x\right )-a b c d^2 x-b^2 c^4 d^2 x^4 \sqrt{\frac{c x-1}{c x+1}}+b^2 c^2 d^2 x^2 \sqrt{\frac{c x-1}{c x+1}}+b^2 c^3 d^2 x^3 (c x-1) \cosh ^{-1}(c x)^3}{3 x^3 \sqrt{\frac{c x-1}{c x+1}} \sqrt{d-c^2 d x^2}} \]

Warning: Unable to verify antiderivative.

[In]

Integrate[((d - c^2*d*x^2)^(3/2)*(a + b*ArcCosh[c*x])^2)/x^4,x]

[Out]

(-(a*b*c*d^2*x) + a*b*c^2*d^2*x^2 - a^2*d^2*Sqrt[(-1 + c*x)/(1 + c*x)] + 5*a^2*c^2*d^2*x^2*Sqrt[(-1 + c*x)/(1
+ c*x)] + b^2*c^2*d^2*x^2*Sqrt[(-1 + c*x)/(1 + c*x)] - 4*a^2*c^4*d^2*x^4*Sqrt[(-1 + c*x)/(1 + c*x)] - b^2*c^4*
d^2*x^4*Sqrt[(-1 + c*x)/(1 + c*x)] - b*d^2*(-1 + c*x)*(-3*a*c^3*x^3 + b*(-Sqrt[(-1 + c*x)/(1 + c*x)] - c*x*Sqr
t[(-1 + c*x)/(1 + c*x)] + 4*c^2*x^2*Sqrt[(-1 + c*x)/(1 + c*x)] + 4*c^3*x^3*(-1 + Sqrt[(-1 + c*x)/(1 + c*x)])))
*ArcCosh[c*x]^2 + b^2*c^3*d^2*x^3*(-1 + c*x)*ArcCosh[c*x]^3 - 3*a^2*c^3*d^(3/2)*x^3*Sqrt[(-1 + c*x)/(1 + c*x)]
*Sqrt[d - c^2*d*x^2]*ArcTan[(c*x*Sqrt[d - c^2*d*x^2])/(Sqrt[d]*(-1 + c^2*x^2))] + b*d^2*(-1 + c*x)*ArcCosh[c*x
]*(b*c*x + 2*a*Sqrt[(-1 + c*x)/(1 + c*x)]*(1 + c*x - 4*c^2*x^2 - 4*c^3*x^3) + 8*b*c^3*x^3*Log[1 + E^(-2*ArcCos
h[c*x])]) - 8*a*b*c^3*d^2*x^3*Log[c*x] + 8*a*b*c^4*d^2*x^4*Log[c*x] - 4*b^2*c^3*d^2*x^3*(-1 + c*x)*PolyLog[2,
-E^(-2*ArcCosh[c*x])])/(3*x^3*Sqrt[(-1 + c*x)/(1 + c*x)]*Sqrt[d - c^2*d*x^2])

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Maple [B]  time = 0.342, size = 2879, normalized size = 6.8 \begin{align*} \text{output too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((-c^2*d*x^2+d)^(3/2)*(a+b*arccosh(c*x))^2/x^4,x)

[Out]

2/3*a^2*c^2/d/x*(-c^2*d*x^2+d)^(5/2)+16/3*a*b*(-d*(c^2*x^2-1))^(1/2)/(c*x-1)^(1/2)/(c*x+1)^(1/2)*arccosh(c*x)*
c^3*d-8/3*a*b*(-d*(c^2*x^2-1))^(1/2)/(c*x-1)^(1/2)/(c*x+1)^(1/2)*ln((c*x+(c*x-1)^(1/2)*(c*x+1)^(1/2))^2+1)*c^3
*d+20/3*b^2*(-d*(c^2*x^2-1))^(1/2)*d/(24*c^4*x^4-9*c^2*x^2+1)*x^5/(c*x+1)/(c*x-1)*c^8-29/3*b^2*(-d*(c^2*x^2-1)
)^(1/2)*d/(24*c^4*x^4-9*c^2*x^2+1)*x^3/(c*x+1)/(c*x-1)*c^6+10/3*b^2*(-d*(c^2*x^2-1))^(1/2)*d/(24*c^4*x^4-9*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)*d/(24*c^4*x^4-9*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)*d/(24*c^4*x^4-9*c^2*x^2+1)/x^3/(c*x+1)/(c*x-1)*arccosh(c*x)^2-8/3*b^2*(-d*(c^2*
x^2-1))^(1/2)/(c*x-1)^(1/2)/(c*x+1)^(1/2)*arccosh(c*x)*ln((c*x+(c*x-1)^(1/2)*(c*x+1)^(1/2))^2+1)*c^3*d+3*b^2*(
-d*(c^2*x^2-1))^(1/2)*d/(24*c^4*x^4-9*c^2*x^2+1)/(c*x+1)^(1/2)/(c*x-1)^(1/2)*arccosh(c*x)*c^3-8*b^2*(-d*(c^2*x
^2-1))^(1/2)*d/(24*c^4*x^4-9*c^2*x^2+1)*x^4/(c*x+1)^(1/2)/(c*x-1)^(1/2)*c^7+3*b^2*(-d*(c^2*x^2-1))^(1/2)*d/(24
*c^4*x^4-9*c^2*x^2+1)*x^2/(c*x+1)^(1/2)/(c*x-1)^(1/2)*c^5-4/3*b^2*(-d*(c^2*x^2-1))^(1/2)*d/(24*c^4*x^4-9*c^2*x
^2+1)/(c*x+1)^(1/2)/(c*x-1)^(1/2)*arccosh(c*x)^2*c^3+3*a*b*(-d*(c^2*x^2-1))^(1/2)*d/(24*c^4*x^4-9*c^2*x^2+1)/(
c*x+1)^(1/2)/(c*x-1)^(1/2)*c^3-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^3*d-1/3
*a^2/d/x^3*(-c^2*d*x^2+d)^(5/2)+2/3*a^2*c^4*x*(-c^2*d*x^2+d)^(3/2)+64*a*b*(-d*(c^2*x^2-1))^(1/2)*d/(24*c^4*x^4
-9*c^2*x^2+1)*x^5/(c*x+1)/(c*x-1)*arccosh(c*x)*c^8-104*a*b*(-d*(c^2*x^2-1))^(1/2)*d/(24*c^4*x^4-9*c^2*x^2+1)*x
^3/(c*x+1)/(c*x-1)*arccosh(c*x)*c^6+146/3*a*b*(-d*(c^2*x^2-1))^(1/2)*d/(24*c^4*x^4-9*c^2*x^2+1)*x/(c*x+1)/(c*x
-1)*arccosh(c*x)*c^4+8/3*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*d-1/3*b^2*(
-d*(c^2*x^2-1))^(1/2)*d/(24*c^4*x^4-9*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/(24*c^4*x^4-9*c^2*x^2+1)*x*arccosh(c*x)*c^4-16/3*a*b*(-d*(c^2*x^2-1))^(1/2)*d/(24*c^4*x^4-9*c^2*x^2+1)*x^3
*c^6+4/3*a*b*(-d*(c^2*x^2-1))^(1/2)*d/(24*c^4*x^4-9*c^2*x^2+1)*x*c^4-16/3*b^2*(-d*(c^2*x^2-1))^(1/2)*d/(24*c^4
*x^4-9*c^2*x^2+1)*x^3*arccosh(c*x)*c^6-4/3*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*d-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^3*d-64*a*b*(-d*(c^2*x^2-1))^(1/2)*d/(24*c^4*x^4-9*c^2*x^2+1)*x^4/(c*x+1)^(1/2)/(c*x-1)^(1/2)*arccosh(c
*x)*c^7+24*a*b*(-d*(c^2*x^2-1))^(1/2)*d/(24*c^4*x^4-9*c^2*x^2+1)*x^2/(c*x+1)^(1/2)/(c*x-1)^(1/2)*arccosh(c*x)*
c^5-28/3*a*b*(-d*(c^2*x^2-1))^(1/2)*d/(24*c^4*x^4-9*c^2*x^2+1)/x/(c*x+1)/(c*x-1)*arccosh(c*x)*c^2-52*b^2*(-d*(
c^2*x^2-1))^(1/2)*d/(24*c^4*x^4-9*c^2*x^2+1)*x^3/(c*x+1)/(c*x-1)*arccosh(c*x)^2*c^6-20/3*b^2*(-d*(c^2*x^2-1))^
(1/2)*d/(24*c^4*x^4-9*c^2*x^2+1)*x^3/(c*x+1)/(c*x-1)*arccosh(c*x)*c^6+73/3*b^2*(-d*(c^2*x^2-1))^(1/2)*d/(24*c^
4*x^4-9*c^2*x^2+1)*x/(c*x+1)/(c*x-1)*arccosh(c*x)^2*c^4+4/3*b^2*(-d*(c^2*x^2-1))^(1/2)*d/(24*c^4*x^4-9*c^2*x^2
+1)*x/(c*x+1)/(c*x-1)*arccosh(c*x)*c^4-14/3*b^2*(-d*(c^2*x^2-1))^(1/2)*d/(24*c^4*x^4-9*c^2*x^2+1)/x/(c*x+1)/(c
*x-1)*arccosh(c*x)^2*c^2+16/3*a*b*(-d*(c^2*x^2-1))^(1/2)*d/(24*c^4*x^4-9*c^2*x^2+1)*x^5/(c*x+1)/(c*x-1)*c^8-20
/3*a*b*(-d*(c^2*x^2-1))^(1/2)*d/(24*c^4*x^4-9*c^2*x^2+1)*x^3/(c*x+1)/(c*x-1)*c^6+4/3*a*b*(-d*(c^2*x^2-1))^(1/2
)*d/(24*c^4*x^4-9*c^2*x^2+1)*x/(c*x+1)/(c*x-1)*c^4+2/3*a*b*(-d*(c^2*x^2-1))^(1/2)*d/(24*c^4*x^4-9*c^2*x^2+1)/x
^3/(c*x+1)/(c*x-1)*arccosh(c*x)+12*b^2*(-d*(c^2*x^2-1))^(1/2)*d/(24*c^4*x^4-9*c^2*x^2+1)*x^2/(c*x+1)^(1/2)/(c*
x-1)^(1/2)*arccosh(c*x)^2*c^5-8*b^2*(-d*(c^2*x^2-1))^(1/2)*d/(24*c^4*x^4-9*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)*d/(24*c^4*x^4-9*c^2*x^2+1)/x^2/(c*x+1)^(1/2)/(c*x-1)^(
1/2)*arccosh(c*x)*c-32*b^2*(-d*(c^2*x^2-1))^(1/2)*d/(24*c^4*x^4-9*c^2*x^2+1)*x^4/(c*x+1)^(1/2)/(c*x-1)^(1/2)*a
rccosh(c*x)^2*c^7-8*a*b*(-d*(c^2*x^2-1))^(1/2)*d/(24*c^4*x^4-9*c^2*x^2+1)*x^2/(c*x+1)^(1/2)/(c*x-1)^(1/2)*c^5-
8/3*a*b*(-d*(c^2*x^2-1))^(1/2)*d/(24*c^4*x^4-9*c^2*x^2+1)/(c*x+1)^(1/2)/(c*x-1)^(1/2)*arccosh(c*x)*c^3-1/3*a*b
*(-d*(c^2*x^2-1))^(1/2)*d/(24*c^4*x^4-9*c^2*x^2+1)/x^2/(c*x+1)^(1/2)/(c*x-1)^(1/2)*c+a^2*c^4*d*x*(-c^2*d*x^2+d
)^(1/2)+a^2*c^4*d^2/(c^2*d)^(1/2)*arctan((c^2*d)^(1/2)*x/(-c^2*d*x^2+d)^(1/2))+4/3*b^2*(-d*(c^2*x^2-1))^(1/2)*
d/(24*c^4*x^4-9*c^2*x^2+1)*x^3*c^6+32*b^2*(-d*(c^2*x^2-1))^(1/2)*d/(24*c^4*x^4-9*c^2*x^2+1)*x^5/(c*x+1)/(c*x-1
)*arccosh(c*x)^2*c^8+16/3*b^2*(-d*(c^2*x^2-1))^(1/2)*d/(24*c^4*x^4-9*c^2*x^2+1)*x^5/(c*x+1)/(c*x-1)*arccosh(c*
x)*c^8

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Maxima [F(-2)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: ValueError} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-c^2*d*x^2+d)^(3/2)*(a+b*arccosh(c*x))^2/x^4,x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [F]  time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (-\frac{{\left (a^{2} c^{2} d x^{2} - a^{2} d +{\left (b^{2} c^{2} d x^{2} - b^{2} d\right )} \operatorname{arcosh}\left (c x\right )^{2} + 2 \,{\left (a b c^{2} d x^{2} - a b d\right )} \operatorname{arcosh}\left (c x\right )\right )} \sqrt{-c^{2} d x^{2} + d}}{x^{4}}, x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \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^{4}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-c**2*d*x**2+d)**(3/2)*(a+b*acosh(c*x))**2/x**4,x)

[Out]

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

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \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^{4}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-c^2*d*x^2+d)^(3/2)*(a+b*arccosh(c*x))^2/x^4,x, algorithm="giac")

[Out]

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