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

Optimal. Leaf size=427 \[ -\frac{i b c^2 \sqrt{d-c^2 d x^2} \text{PolyLog}\left (2,-i e^{\cosh ^{-1}(c x)}\right ) \left (a+b \cosh ^{-1}(c x)\right )}{\sqrt{c x-1} \sqrt{c x+1}}+\frac{i b c^2 \sqrt{d-c^2 d x^2} \text{PolyLog}\left (2,i e^{\cosh ^{-1}(c x)}\right ) \left (a+b \cosh ^{-1}(c x)\right )}{\sqrt{c x-1} \sqrt{c x+1}}+\frac{i b^2 c^2 \sqrt{d-c^2 d x^2} \text{PolyLog}\left (3,-i e^{\cosh ^{-1}(c x)}\right )}{\sqrt{c x-1} \sqrt{c x+1}}-\frac{i b^2 c^2 \sqrt{d-c^2 d x^2} \text{PolyLog}\left (3,i e^{\cosh ^{-1}(c x)}\right )}{\sqrt{c x-1} \sqrt{c x+1}}-\frac{b c \sqrt{d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )}{x \sqrt{c x-1} \sqrt{c x+1}}-\frac{\sqrt{d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )^2}{2 x^2}+\frac{c^2 \sqrt{d-c^2 d x^2} \tan ^{-1}\left (e^{\cosh ^{-1}(c x)}\right ) \left (a+b \cosh ^{-1}(c x)\right )^2}{\sqrt{c x-1} \sqrt{c x+1}}+\frac{b^2 c^2 \sqrt{d-c^2 d x^2} \tan ^{-1}\left (\sqrt{c x-1} \sqrt{c x+1}\right )}{\sqrt{c x-1} \sqrt{c x+1}} \]

[Out]

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

________________________________________________________________________________________

Rubi [A]  time = 0.876632, antiderivative size = 427, normalized size of antiderivative = 1., number of steps used = 13, number of rules used = 10, integrand size = 29, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.345, Rules used = {5798, 5738, 5662, 92, 205, 5761, 4180, 2531, 2282, 6589} \[ -\frac{i b c^2 \sqrt{d-c^2 d x^2} \text{PolyLog}\left (2,-i e^{\cosh ^{-1}(c x)}\right ) \left (a+b \cosh ^{-1}(c x)\right )}{\sqrt{c x-1} \sqrt{c x+1}}+\frac{i b c^2 \sqrt{d-c^2 d x^2} \text{PolyLog}\left (2,i e^{\cosh ^{-1}(c x)}\right ) \left (a+b \cosh ^{-1}(c x)\right )}{\sqrt{c x-1} \sqrt{c x+1}}+\frac{i b^2 c^2 \sqrt{d-c^2 d x^2} \text{PolyLog}\left (3,-i e^{\cosh ^{-1}(c x)}\right )}{\sqrt{c x-1} \sqrt{c x+1}}-\frac{i b^2 c^2 \sqrt{d-c^2 d x^2} \text{PolyLog}\left (3,i e^{\cosh ^{-1}(c x)}\right )}{\sqrt{c x-1} \sqrt{c x+1}}-\frac{b c \sqrt{d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )}{x \sqrt{c x-1} \sqrt{c x+1}}-\frac{\sqrt{d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )^2}{2 x^2}+\frac{c^2 \sqrt{d-c^2 d x^2} \tan ^{-1}\left (e^{\cosh ^{-1}(c x)}\right ) \left (a+b \cosh ^{-1}(c x)\right )^2}{\sqrt{c x-1} \sqrt{c x+1}}+\frac{b^2 c^2 \sqrt{d-c^2 d x^2} \tan ^{-1}\left (\sqrt{c x-1} \sqrt{c x+1}\right )}{\sqrt{c x-1} \sqrt{c x+1}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-((b*c*Sqrt[d - c^2*d*x^2]*(a + b*ArcCosh[c*x]))/(x*Sqrt[-1 + c*x]*Sqrt[1 + c*x])) - (Sqrt[d - c^2*d*x^2]*(a +
b*ArcCosh[c*x])^2)/(2*x^2) + (c^2*Sqrt[d - c^2*d*x^2]*(a + b*ArcCosh[c*x])^2*ArcTan[E^ArcCosh[c*x]])/(Sqrt[-1
+ c*x]*Sqrt[1 + c*x]) + (b^2*c^2*Sqrt[d - c^2*d*x^2]*ArcTan[Sqrt[-1 + c*x]*Sqrt[1 + c*x]])/(Sqrt[-1 + c*x]*Sq
rt[1 + c*x]) - (I*b*c^2*Sqrt[d - c^2*d*x^2]*(a + b*ArcCosh[c*x])*PolyLog[2, (-I)*E^ArcCosh[c*x]])/(Sqrt[-1 + c
*x]*Sqrt[1 + c*x]) + (I*b*c^2*Sqrt[d - c^2*d*x^2]*(a + b*ArcCosh[c*x])*PolyLog[2, I*E^ArcCosh[c*x]])/(Sqrt[-1
+ c*x]*Sqrt[1 + c*x]) + (I*b^2*c^2*Sqrt[d - c^2*d*x^2]*PolyLog[3, (-I)*E^ArcCosh[c*x]])/(Sqrt[-1 + c*x]*Sqrt[1
+ c*x]) - (I*b^2*c^2*Sqrt[d - c^2*d*x^2]*PolyLog[3, I*E^ArcCosh[c*x]])/(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 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 5662

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))/(Sqr
t[-1 + c*x]*Sqrt[1 + c*x]), x], x] /; FreeQ[{a, b, c, d, m}, x] && IGtQ[n, 0] && NeQ[m, -1]

Rule 92

Int[1/(Sqrt[(a_.) + (b_.)*(x_)]*Sqrt[(c_.) + (d_.)*(x_)]*((e_.) + (f_.)*(x_))), x_Symbol] :> Dist[b*f, Subst[I
nt[1/(d*(b*e - a*f)^2 + b*f^2*x^2), x], x, Sqrt[a + b*x]*Sqrt[c + d*x]], x] /; FreeQ[{a, b, c, d, e, f}, x] &&
EqQ[2*b*d*e - f*(b*c + a*d), 0]

Rule 205

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(Rt[a/b, 2]*ArcTan[x/Rt[a/b, 2]])/a, x] /; FreeQ[{a, b}, x]
&& PosQ[a/b]

Rule 5761

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

Rule 4180

Int[csc[(e_.) + Pi*(k_.) + (Complex[0, fz_])*(f_.)*(x_)]*((c_.) + (d_.)*(x_))^(m_.), x_Symbol] :> Simp[(-2*(c
+ d*x)^m*ArcTanh[E^(-(I*e) + f*fz*x)/E^(I*k*Pi)])/(f*fz*I), x] + (-Dist[(d*m)/(f*fz*I), Int[(c + d*x)^(m - 1)*
Log[1 - E^(-(I*e) + f*fz*x)/E^(I*k*Pi)], x], x] + Dist[(d*m)/(f*fz*I), Int[(c + d*x)^(m - 1)*Log[1 + E^(-(I*e)
+ f*fz*x)/E^(I*k*Pi)], x], x]) /; FreeQ[{c, d, e, f, fz}, x] && IntegerQ[2*k] && IGtQ[m, 0]

Rule 2531

Int[Log[1 + (e_.)*((F_)^((c_.)*((a_.) + (b_.)*(x_))))^(n_.)]*((f_.) + (g_.)*(x_))^(m_.), x_Symbol] :> -Simp[((
f + g*x)^m*PolyLog[2, -(e*(F^(c*(a + b*x)))^n)])/(b*c*n*Log[F]), x] + Dist[(g*m)/(b*c*n*Log[F]), Int[(f + g*x)
^(m - 1)*PolyLog[2, -(e*(F^(c*(a + b*x)))^n)], x], x] /; FreeQ[{F, a, b, c, e, f, g, n}, x] && GtQ[m, 0]

Rule 2282

Int[u_, x_Symbol] :> With[{v = FunctionOfExponential[u, x]}, Dist[v/D[v, x], Subst[Int[FunctionOfExponentialFu
nction[u, x]/x, x], x, v], x]] /; FunctionOfExponentialQ[u, x] &&  !MatchQ[u, (w_)*((a_.)*(v_)^(n_))^(m_) /; F
reeQ[{a, m, n}, x] && IntegerQ[m*n]] &&  !MatchQ[u, E^((c_.)*((a_.) + (b_.)*x))*(F_)[v_] /; FreeQ[{a, b, c}, x
] && InverseFunctionQ[F[x]]]

Rule 6589

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

Rubi steps

\begin{align*} \int \frac{\sqrt{d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )^2}{x^3} \, dx &=\frac{\sqrt{d-c^2 d x^2} \int \frac{\sqrt{-1+c x} \sqrt{1+c x} \left (a+b \cosh ^{-1}(c x)\right )^2}{x^3} \, dx}{\sqrt{-1+c x} \sqrt{1+c x}}\\ &=-\frac{\sqrt{d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )^2}{2 x^2}+\frac{\left (b c \sqrt{d-c^2 d x^2}\right ) \int \frac{a+b \cosh ^{-1}(c x)}{x^2} \, dx}{\sqrt{-1+c x} \sqrt{1+c x}}+\frac{\left (c^2 \sqrt{d-c^2 d x^2}\right ) \int \frac{\left (a+b \cosh ^{-1}(c x)\right )^2}{x \sqrt{-1+c x} \sqrt{1+c x}} \, dx}{2 \sqrt{-1+c x} \sqrt{1+c x}}\\ &=-\frac{b c \sqrt{d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )}{x \sqrt{-1+c x} \sqrt{1+c x}}-\frac{\sqrt{d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )^2}{2 x^2}+\frac{\left (c^2 \sqrt{d-c^2 d x^2}\right ) \operatorname{Subst}\left (\int (a+b x)^2 \text{sech}(x) \, dx,x,\cosh ^{-1}(c x)\right )}{2 \sqrt{-1+c x} \sqrt{1+c x}}+\frac{\left (b^2 c^2 \sqrt{d-c^2 d x^2}\right ) \int \frac{1}{x \sqrt{-1+c x} \sqrt{1+c x}} \, dx}{\sqrt{-1+c x} \sqrt{1+c x}}\\ &=-\frac{b c \sqrt{d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )}{x \sqrt{-1+c x} \sqrt{1+c x}}-\frac{\sqrt{d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )^2}{2 x^2}+\frac{c^2 \sqrt{d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )^2 \tan ^{-1}\left (e^{\cosh ^{-1}(c x)}\right )}{\sqrt{-1+c x} \sqrt{1+c x}}-\frac{\left (i b c^2 \sqrt{d-c^2 d x^2}\right ) \operatorname{Subst}\left (\int (a+b x) \log \left (1-i e^x\right ) \, dx,x,\cosh ^{-1}(c x)\right )}{\sqrt{-1+c x} \sqrt{1+c x}}+\frac{\left (i b c^2 \sqrt{d-c^2 d x^2}\right ) \operatorname{Subst}\left (\int (a+b x) \log \left (1+i e^x\right ) \, dx,x,\cosh ^{-1}(c x)\right )}{\sqrt{-1+c x} \sqrt{1+c x}}+\frac{\left (b^2 c^3 \sqrt{d-c^2 d x^2}\right ) \operatorname{Subst}\left (\int \frac{1}{c+c x^2} \, dx,x,\sqrt{-1+c x} \sqrt{1+c x}\right )}{\sqrt{-1+c x} \sqrt{1+c x}}\\ &=-\frac{b c \sqrt{d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )}{x \sqrt{-1+c x} \sqrt{1+c x}}-\frac{\sqrt{d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )^2}{2 x^2}+\frac{c^2 \sqrt{d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )^2 \tan ^{-1}\left (e^{\cosh ^{-1}(c x)}\right )}{\sqrt{-1+c x} \sqrt{1+c x}}+\frac{b^2 c^2 \sqrt{d-c^2 d x^2} \tan ^{-1}\left (\sqrt{-1+c x} \sqrt{1+c x}\right )}{\sqrt{-1+c x} \sqrt{1+c x}}-\frac{i b c^2 \sqrt{d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right ) \text{Li}_2\left (-i e^{\cosh ^{-1}(c x)}\right )}{\sqrt{-1+c x} \sqrt{1+c x}}+\frac{i b c^2 \sqrt{d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right ) \text{Li}_2\left (i e^{\cosh ^{-1}(c x)}\right )}{\sqrt{-1+c x} \sqrt{1+c x}}+\frac{\left (i b^2 c^2 \sqrt{d-c^2 d x^2}\right ) \operatorname{Subst}\left (\int \text{Li}_2\left (-i e^x\right ) \, dx,x,\cosh ^{-1}(c x)\right )}{\sqrt{-1+c x} \sqrt{1+c x}}-\frac{\left (i b^2 c^2 \sqrt{d-c^2 d x^2}\right ) \operatorname{Subst}\left (\int \text{Li}_2\left (i e^x\right ) \, dx,x,\cosh ^{-1}(c x)\right )}{\sqrt{-1+c x} \sqrt{1+c x}}\\ &=-\frac{b c \sqrt{d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )}{x \sqrt{-1+c x} \sqrt{1+c x}}-\frac{\sqrt{d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )^2}{2 x^2}+\frac{c^2 \sqrt{d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )^2 \tan ^{-1}\left (e^{\cosh ^{-1}(c x)}\right )}{\sqrt{-1+c x} \sqrt{1+c x}}+\frac{b^2 c^2 \sqrt{d-c^2 d x^2} \tan ^{-1}\left (\sqrt{-1+c x} \sqrt{1+c x}\right )}{\sqrt{-1+c x} \sqrt{1+c x}}-\frac{i b c^2 \sqrt{d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right ) \text{Li}_2\left (-i e^{\cosh ^{-1}(c x)}\right )}{\sqrt{-1+c x} \sqrt{1+c x}}+\frac{i b c^2 \sqrt{d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right ) \text{Li}_2\left (i e^{\cosh ^{-1}(c x)}\right )}{\sqrt{-1+c x} \sqrt{1+c x}}+\frac{\left (i b^2 c^2 \sqrt{d-c^2 d x^2}\right ) \operatorname{Subst}\left (\int \frac{\text{Li}_2(-i x)}{x} \, dx,x,e^{\cosh ^{-1}(c x)}\right )}{\sqrt{-1+c x} \sqrt{1+c x}}-\frac{\left (i b^2 c^2 \sqrt{d-c^2 d x^2}\right ) \operatorname{Subst}\left (\int \frac{\text{Li}_2(i x)}{x} \, dx,x,e^{\cosh ^{-1}(c x)}\right )}{\sqrt{-1+c x} \sqrt{1+c x}}\\ &=-\frac{b c \sqrt{d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )}{x \sqrt{-1+c x} \sqrt{1+c x}}-\frac{\sqrt{d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )^2}{2 x^2}+\frac{c^2 \sqrt{d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )^2 \tan ^{-1}\left (e^{\cosh ^{-1}(c x)}\right )}{\sqrt{-1+c x} \sqrt{1+c x}}+\frac{b^2 c^2 \sqrt{d-c^2 d x^2} \tan ^{-1}\left (\sqrt{-1+c x} \sqrt{1+c x}\right )}{\sqrt{-1+c x} \sqrt{1+c x}}-\frac{i b c^2 \sqrt{d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right ) \text{Li}_2\left (-i e^{\cosh ^{-1}(c x)}\right )}{\sqrt{-1+c x} \sqrt{1+c x}}+\frac{i b c^2 \sqrt{d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right ) \text{Li}_2\left (i e^{\cosh ^{-1}(c x)}\right )}{\sqrt{-1+c x} \sqrt{1+c x}}+\frac{i b^2 c^2 \sqrt{d-c^2 d x^2} \text{Li}_3\left (-i e^{\cosh ^{-1}(c x)}\right )}{\sqrt{-1+c x} \sqrt{1+c x}}-\frac{i b^2 c^2 \sqrt{d-c^2 d x^2} \text{Li}_3\left (i e^{\cosh ^{-1}(c x)}\right )}{\sqrt{-1+c x} \sqrt{1+c x}}\\ \end{align*}

Mathematica [A]  time = 79.7534, size = 547, normalized size = 1.28 \[ \frac{1}{2} a \left (\frac{2 b d (c x+1) \left (i c^2 x^2 \sqrt{\frac{c x-1}{c x+1}} \text{PolyLog}\left (2,-i e^{-\cosh ^{-1}(c x)}\right )-i c^2 x^2 \sqrt{\frac{c x-1}{c x+1}} \text{PolyLog}\left (2,i e^{-\cosh ^{-1}(c x)}\right )+i c^2 x^2 \sqrt{\frac{c x-1}{c x+1}} \cosh ^{-1}(c x) \log \left (1-i e^{-\cosh ^{-1}(c x)}\right )-i c^2 x^2 \sqrt{\frac{c x-1}{c x+1}} \cosh ^{-1}(c x) \log \left (1+i e^{-\cosh ^{-1}(c x)}\right )+c x \sqrt{\frac{c x-1}{c x+1}}+c x \cosh ^{-1}(c x)-\cosh ^{-1}(c x)\right )}{x^2 \sqrt{d-c^2 d x^2}}-\frac{a \sqrt{d-c^2 d x^2}}{x^2}+a c^2 \sqrt{d} \log \left (\sqrt{d} \sqrt{d-c^2 d x^2}+d\right )-a c^2 \sqrt{d} \log (x)\right )+\frac{1}{2} b^2 c^2 \sqrt{d-c^2 d x^2} \left (-\frac{i \left (2 \cosh ^{-1}(c x) \text{PolyLog}\left (2,-i e^{-\cosh ^{-1}(c x)}\right )-2 \cosh ^{-1}(c x) \text{PolyLog}\left (2,i e^{-\cosh ^{-1}(c x)}\right )+2 \text{PolyLog}\left (3,-i e^{-\cosh ^{-1}(c x)}\right )-2 \text{PolyLog}\left (3,i e^{-\cosh ^{-1}(c x)}\right )+\cosh ^{-1}(c x)^2 \log \left (1-i e^{-\cosh ^{-1}(c x)}\right )-\cosh ^{-1}(c x)^2 \log \left (1+i e^{-\cosh ^{-1}(c x)}\right )+4 i \tan ^{-1}\left (\tanh \left (\frac{1}{2} \cosh ^{-1}(c x)\right )\right )\right )}{\sqrt{\frac{c x-1}{c x+1}} (c x+1)}-\frac{\cosh ^{-1}(c x)^2}{c^2 x^2}+\frac{2 \sqrt{\frac{c x-1}{c x+1}} \cosh ^{-1}(c x)}{c x-c^2 x^2}\right ) \]

Warning: Unable to verify antiderivative.

[In]

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

[Out]

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

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Maple [F]  time = 0.348, size = 0, normalized size = 0. \begin{align*} \int{\frac{ \left ( a+b{\rm arccosh} \left (cx\right ) \right ) ^{2}}{{x}^{3}}\sqrt{-{c}^{2}d{x}^{2}+d}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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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((a+b*arccosh(c*x))^2*(-c^2*d*x^2+d)^(1/2)/x^3,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{\sqrt{-c^{2} d x^{2} + d}{\left (b^{2} \operatorname{arcosh}\left (c x\right )^{2} + 2 \, a b \operatorname{arcosh}\left (c x\right ) + a^{2}\right )}}{x^{3}}, x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \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^{3}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\sqrt{-c^{2} d x^{2} + d}{\left (b \operatorname{arcosh}\left (c x\right ) + a\right )}^{2}}{x^{3}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b*arccosh(c*x))^2*(-c^2*d*x^2+d)^(1/2)/x^3,x, algorithm="giac")

[Out]

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