3.102 \(\int \frac{(d-c^2 d x^2)^{5/2} (a+b \cosh ^{-1}(c x))}{x^5} \, dx\)

Optimal. Leaf size=407 \[ \frac{15 i b c^4 d^2 \sqrt{d-c^2 d x^2} \text{PolyLog}\left (2,-i e^{\cosh ^{-1}(c x)}\right )}{8 \sqrt{c x-1} \sqrt{c x+1}}-\frac{15 i b c^4 d^2 \sqrt{d-c^2 d x^2} \text{PolyLog}\left (2,i e^{\cosh ^{-1}(c x)}\right )}{8 \sqrt{c x-1} \sqrt{c x+1}}+\frac{15}{8} c^4 d^2 \sqrt{d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )-\frac{15 c^4 d^2 \sqrt{d-c^2 d x^2} \tan ^{-1}\left (e^{\cosh ^{-1}(c x)}\right ) \left (a+b \cosh ^{-1}(c x)\right )}{4 \sqrt{c x-1} \sqrt{c x+1}}+\frac{5 c^2 d \left (d-c^2 d x^2\right )^{3/2} \left (a+b \cosh ^{-1}(c x)\right )}{8 x^2}-\frac{\left (d-c^2 d x^2\right )^{5/2} \left (a+b \cosh ^{-1}(c x)\right )}{4 x^4}-\frac{b c^5 d^2 x \sqrt{d-c^2 d x^2}}{\sqrt{c x-1} \sqrt{c x+1}}+\frac{9 b c^3 d^2 \sqrt{d-c^2 d x^2}}{8 x \sqrt{c x-1} \sqrt{c x+1}}-\frac{b c d^2 \sqrt{d-c^2 d x^2}}{12 x^3 \sqrt{c x-1} \sqrt{c x+1}} \]

[Out]

-(b*c*d^2*Sqrt[d - c^2*d*x^2])/(12*x^3*Sqrt[-1 + c*x]*Sqrt[1 + c*x]) + (9*b*c^3*d^2*Sqrt[d - c^2*d*x^2])/(8*x*
Sqrt[-1 + c*x]*Sqrt[1 + c*x]) - (b*c^5*d^2*x*Sqrt[d - c^2*d*x^2])/(Sqrt[-1 + c*x]*Sqrt[1 + c*x]) + (15*c^4*d^2
*Sqrt[d - c^2*d*x^2]*(a + b*ArcCosh[c*x]))/8 + (5*c^2*d*(d - c^2*d*x^2)^(3/2)*(a + b*ArcCosh[c*x]))/(8*x^2) -
((d - c^2*d*x^2)^(5/2)*(a + b*ArcCosh[c*x]))/(4*x^4) - (15*c^4*d^2*Sqrt[d - c^2*d*x^2]*(a + b*ArcCosh[c*x])*Ar
cTan[E^ArcCosh[c*x]])/(4*Sqrt[-1 + c*x]*Sqrt[1 + c*x]) + (((15*I)/8)*b*c^4*d^2*Sqrt[d - c^2*d*x^2]*PolyLog[2,
(-I)*E^ArcCosh[c*x]])/(Sqrt[-1 + c*x]*Sqrt[1 + c*x]) - (((15*I)/8)*b*c^4*d^2*Sqrt[d - c^2*d*x^2]*PolyLog[2, I*
E^ArcCosh[c*x]])/(Sqrt[-1 + c*x]*Sqrt[1 + c*x])

________________________________________________________________________________________

Rubi [A]  time = 1.08268, antiderivative size = 438, normalized size of antiderivative = 1.08, number of steps used = 15, number of rules used = 10, integrand size = 27, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.37, Rules used = {5798, 5740, 5743, 5761, 4180, 2279, 2391, 8, 14, 270} \[ \frac{15 i b c^4 d^2 \sqrt{d-c^2 d x^2} \text{PolyLog}\left (2,-i e^{\cosh ^{-1}(c x)}\right )}{8 \sqrt{c x-1} \sqrt{c x+1}}-\frac{15 i b c^4 d^2 \sqrt{d-c^2 d x^2} \text{PolyLog}\left (2,i e^{\cosh ^{-1}(c x)}\right )}{8 \sqrt{c x-1} \sqrt{c x+1}}+\frac{15}{8} c^4 d^2 \sqrt{d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )+\frac{5 c^2 d^2 (1-c x) (c x+1) \sqrt{d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )}{8 x^2}-\frac{d^2 (1-c x)^2 (c x+1)^2 \sqrt{d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )}{4 x^4}-\frac{15 c^4 d^2 \sqrt{d-c^2 d x^2} \tan ^{-1}\left (e^{\cosh ^{-1}(c x)}\right ) \left (a+b \cosh ^{-1}(c x)\right )}{4 \sqrt{c x-1} \sqrt{c x+1}}-\frac{b c^5 d^2 x \sqrt{d-c^2 d x^2}}{\sqrt{c x-1} \sqrt{c x+1}}+\frac{9 b c^3 d^2 \sqrt{d-c^2 d x^2}}{8 x \sqrt{c x-1} \sqrt{c x+1}}-\frac{b c d^2 \sqrt{d-c^2 d x^2}}{12 x^3 \sqrt{c x-1} \sqrt{c x+1}} \]

Antiderivative was successfully verified.

[In]

Int[((d - c^2*d*x^2)^(5/2)*(a + b*ArcCosh[c*x]))/x^5,x]

[Out]

-(b*c*d^2*Sqrt[d - c^2*d*x^2])/(12*x^3*Sqrt[-1 + c*x]*Sqrt[1 + c*x]) + (9*b*c^3*d^2*Sqrt[d - c^2*d*x^2])/(8*x*
Sqrt[-1 + c*x]*Sqrt[1 + c*x]) - (b*c^5*d^2*x*Sqrt[d - c^2*d*x^2])/(Sqrt[-1 + c*x]*Sqrt[1 + c*x]) + (15*c^4*d^2
*Sqrt[d - c^2*d*x^2]*(a + b*ArcCosh[c*x]))/8 + (5*c^2*d^2*(1 - c*x)*(1 + c*x)*Sqrt[d - c^2*d*x^2]*(a + b*ArcCo
sh[c*x]))/(8*x^2) - (d^2*(1 - c*x)^2*(1 + c*x)^2*Sqrt[d - c^2*d*x^2]*(a + b*ArcCosh[c*x]))/(4*x^4) - (15*c^4*d
^2*Sqrt[d - c^2*d*x^2]*(a + b*ArcCosh[c*x])*ArcTan[E^ArcCosh[c*x]])/(4*Sqrt[-1 + c*x]*Sqrt[1 + c*x]) + (((15*I
)/8)*b*c^4*d^2*Sqrt[d - c^2*d*x^2]*PolyLog[2, (-I)*E^ArcCosh[c*x]])/(Sqrt[-1 + c*x]*Sqrt[1 + c*x]) - (((15*I)/
8)*b*c^4*d^2*Sqrt[d - c^2*d*x^2]*PolyLog[2, 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 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 5743

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 + 2)), x
] + (-Dist[(Sqrt[d1 + e1*x]*Sqrt[d2 + e2*x])/((m + 2)*Sqrt[1 + c*x]*Sqrt[-1 + c*x]), Int[((f*x)^m*(a + b*ArcCo
sh[c*x])^n)/(Sqrt[1 + c*x]*Sqrt[-1 + c*x]), x], x] - Dist[(b*c*n*Sqrt[d1 + e1*x]*Sqrt[d2 + e2*x])/(f*(m + 2)*S
qrt[1 + c*x]*Sqrt[-1 + c*x]), Int[(f*x)^(m + 1)*(a + b*ArcCosh[c*x])^(n - 1), x], x]) /; FreeQ[{a, b, c, d1, e
1, d2, e2, f, m}, x] && EqQ[e1 - c*d1, 0] && EqQ[e2 + c*d2, 0] && GtQ[n, 0] &&  !LtQ[m, -1] && (RationalQ[m] |
| EqQ[n, 1])

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 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 8

Int[a_, x_Symbol] :> Simp[a*x, x] /; FreeQ[a, x]

Rule 14

Int[(u_)*((c_.)*(x_))^(m_.), x_Symbol] :> Int[ExpandIntegrand[(c*x)^m*u, x], x] /; FreeQ[{c, m}, x] && SumQ[u]
&&  !LinearQ[u, x] &&  !MatchQ[u, (a_) + (b_.)*(v_) /; FreeQ[{a, b}, x] && InverseFunctionQ[v]]

Rule 270

Int[((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_.), x_Symbol] :> Int[ExpandIntegrand[(c*x)^m*(a + b*x^n)^p,
x], x] /; FreeQ[{a, b, c, m, n}, x] && IGtQ[p, 0]

Rubi steps

\begin{align*} \int \frac{\left (d-c^2 d x^2\right )^{5/2} \left (a+b \cosh ^{-1}(c x)\right )}{x^5} \, dx &=\frac{\left (d^2 \sqrt{d-c^2 d x^2}\right ) \int \frac{(-1+c x)^{5/2} (1+c x)^{5/2} \left (a+b \cosh ^{-1}(c x)\right )}{x^5} \, dx}{\sqrt{-1+c x} \sqrt{1+c x}}\\ &=-\frac{d^2 (1-c x)^2 (1+c x)^2 \sqrt{d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )}{4 x^4}+\frac{\left (b c d^2 \sqrt{d-c^2 d x^2}\right ) \int \frac{\left (-1+c^2 x^2\right )^2}{x^4} \, dx}{4 \sqrt{-1+c x} \sqrt{1+c x}}+\frac{\left (5 c^2 d^2 \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 )}{x^3} \, dx}{4 \sqrt{-1+c x} \sqrt{1+c x}}\\ &=\frac{5 c^2 d^2 (1-c x) (1+c x) \sqrt{d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )}{8 x^2}-\frac{d^2 (1-c x)^2 (1+c x)^2 \sqrt{d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )}{4 x^4}+\frac{\left (b c d^2 \sqrt{d-c^2 d x^2}\right ) \int \left (c^4+\frac{1}{x^4}-\frac{2 c^2}{x^2}\right ) \, dx}{4 \sqrt{-1+c x} \sqrt{1+c x}}+\frac{\left (5 b c^3 d^2 \sqrt{d-c^2 d x^2}\right ) \int \frac{-1+c^2 x^2}{x^2} \, dx}{8 \sqrt{-1+c x} \sqrt{1+c x}}+\frac{\left (15 c^4 d^2 \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 )}{x} \, dx}{8 \sqrt{-1+c x} \sqrt{1+c x}}\\ &=-\frac{b c d^2 \sqrt{d-c^2 d x^2}}{12 x^3 \sqrt{-1+c x} \sqrt{1+c x}}+\frac{b c^3 d^2 \sqrt{d-c^2 d x^2}}{2 x \sqrt{-1+c x} \sqrt{1+c x}}+\frac{b c^5 d^2 x \sqrt{d-c^2 d x^2}}{4 \sqrt{-1+c x} \sqrt{1+c x}}+\frac{15}{8} c^4 d^2 \sqrt{d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )+\frac{5 c^2 d^2 (1-c x) (1+c x) \sqrt{d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )}{8 x^2}-\frac{d^2 (1-c x)^2 (1+c x)^2 \sqrt{d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )}{4 x^4}+\frac{\left (5 b c^3 d^2 \sqrt{d-c^2 d x^2}\right ) \int \left (c^2-\frac{1}{x^2}\right ) \, dx}{8 \sqrt{-1+c x} \sqrt{1+c x}}-\frac{\left (15 c^4 d^2 \sqrt{d-c^2 d x^2}\right ) \int \frac{a+b \cosh ^{-1}(c x)}{x \sqrt{-1+c x} \sqrt{1+c x}} \, dx}{8 \sqrt{-1+c x} \sqrt{1+c x}}-\frac{\left (15 b c^5 d^2 \sqrt{d-c^2 d x^2}\right ) \int 1 \, dx}{8 \sqrt{-1+c x} \sqrt{1+c x}}\\ &=-\frac{b c d^2 \sqrt{d-c^2 d x^2}}{12 x^3 \sqrt{-1+c x} \sqrt{1+c x}}+\frac{9 b c^3 d^2 \sqrt{d-c^2 d x^2}}{8 x \sqrt{-1+c x} \sqrt{1+c x}}-\frac{b c^5 d^2 x \sqrt{d-c^2 d x^2}}{\sqrt{-1+c x} \sqrt{1+c x}}+\frac{15}{8} c^4 d^2 \sqrt{d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )+\frac{5 c^2 d^2 (1-c x) (1+c x) \sqrt{d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )}{8 x^2}-\frac{d^2 (1-c x)^2 (1+c x)^2 \sqrt{d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )}{4 x^4}-\frac{\left (15 c^4 d^2 \sqrt{d-c^2 d x^2}\right ) \operatorname{Subst}\left (\int (a+b x) \text{sech}(x) \, dx,x,\cosh ^{-1}(c x)\right )}{8 \sqrt{-1+c x} \sqrt{1+c x}}\\ &=-\frac{b c d^2 \sqrt{d-c^2 d x^2}}{12 x^3 \sqrt{-1+c x} \sqrt{1+c x}}+\frac{9 b c^3 d^2 \sqrt{d-c^2 d x^2}}{8 x \sqrt{-1+c x} \sqrt{1+c x}}-\frac{b c^5 d^2 x \sqrt{d-c^2 d x^2}}{\sqrt{-1+c x} \sqrt{1+c x}}+\frac{15}{8} c^4 d^2 \sqrt{d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )+\frac{5 c^2 d^2 (1-c x) (1+c x) \sqrt{d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )}{8 x^2}-\frac{d^2 (1-c x)^2 (1+c x)^2 \sqrt{d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )}{4 x^4}-\frac{15 c^4 d^2 \sqrt{d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right ) \tan ^{-1}\left (e^{\cosh ^{-1}(c x)}\right )}{4 \sqrt{-1+c x} \sqrt{1+c x}}+\frac{\left (15 i b c^4 d^2 \sqrt{d-c^2 d x^2}\right ) \operatorname{Subst}\left (\int \log \left (1-i e^x\right ) \, dx,x,\cosh ^{-1}(c x)\right )}{8 \sqrt{-1+c x} \sqrt{1+c x}}-\frac{\left (15 i b c^4 d^2 \sqrt{d-c^2 d x^2}\right ) \operatorname{Subst}\left (\int \log \left (1+i e^x\right ) \, dx,x,\cosh ^{-1}(c x)\right )}{8 \sqrt{-1+c x} \sqrt{1+c x}}\\ &=-\frac{b c d^2 \sqrt{d-c^2 d x^2}}{12 x^3 \sqrt{-1+c x} \sqrt{1+c x}}+\frac{9 b c^3 d^2 \sqrt{d-c^2 d x^2}}{8 x \sqrt{-1+c x} \sqrt{1+c x}}-\frac{b c^5 d^2 x \sqrt{d-c^2 d x^2}}{\sqrt{-1+c x} \sqrt{1+c x}}+\frac{15}{8} c^4 d^2 \sqrt{d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )+\frac{5 c^2 d^2 (1-c x) (1+c x) \sqrt{d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )}{8 x^2}-\frac{d^2 (1-c x)^2 (1+c x)^2 \sqrt{d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )}{4 x^4}-\frac{15 c^4 d^2 \sqrt{d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right ) \tan ^{-1}\left (e^{\cosh ^{-1}(c x)}\right )}{4 \sqrt{-1+c x} \sqrt{1+c x}}+\frac{\left (15 i b c^4 d^2 \sqrt{d-c^2 d x^2}\right ) \operatorname{Subst}\left (\int \frac{\log (1-i x)}{x} \, dx,x,e^{\cosh ^{-1}(c x)}\right )}{8 \sqrt{-1+c x} \sqrt{1+c x}}-\frac{\left (15 i b c^4 d^2 \sqrt{d-c^2 d x^2}\right ) \operatorname{Subst}\left (\int \frac{\log (1+i x)}{x} \, dx,x,e^{\cosh ^{-1}(c x)}\right )}{8 \sqrt{-1+c x} \sqrt{1+c x}}\\ &=-\frac{b c d^2 \sqrt{d-c^2 d x^2}}{12 x^3 \sqrt{-1+c x} \sqrt{1+c x}}+\frac{9 b c^3 d^2 \sqrt{d-c^2 d x^2}}{8 x \sqrt{-1+c x} \sqrt{1+c x}}-\frac{b c^5 d^2 x \sqrt{d-c^2 d x^2}}{\sqrt{-1+c x} \sqrt{1+c x}}+\frac{15}{8} c^4 d^2 \sqrt{d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )+\frac{5 c^2 d^2 (1-c x) (1+c x) \sqrt{d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )}{8 x^2}-\frac{d^2 (1-c x)^2 (1+c x)^2 \sqrt{d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )}{4 x^4}-\frac{15 c^4 d^2 \sqrt{d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right ) \tan ^{-1}\left (e^{\cosh ^{-1}(c x)}\right )}{4 \sqrt{-1+c x} \sqrt{1+c x}}+\frac{15 i b c^4 d^2 \sqrt{d-c^2 d x^2} \text{Li}_2\left (-i e^{\cosh ^{-1}(c x)}\right )}{8 \sqrt{-1+c x} \sqrt{1+c x}}-\frac{15 i b c^4 d^2 \sqrt{d-c^2 d x^2} \text{Li}_2\left (i e^{\cosh ^{-1}(c x)}\right )}{8 \sqrt{-1+c x} \sqrt{1+c x}}\\ \end{align*}

Mathematica [A]  time = 1.39051, size = 660, normalized size = 1.62 \[ \frac{-45 i b c^4 d^3 x^4 (c x-1) \text{PolyLog}\left (2,-i e^{-\cosh ^{-1}(c x)}\right )+45 i b c^4 d^3 x^4 (c x-1) \text{PolyLog}\left (2,i e^{-\cosh ^{-1}(c x)}\right )-24 a c^6 d^3 x^6 \sqrt{\frac{c x-1}{c x+1}}-3 a c^4 d^3 x^4 \sqrt{\frac{c x-1}{c x+1}}+33 a c^2 d^3 x^2 \sqrt{\frac{c x-1}{c x+1}}+45 a c^4 d^{5/2} x^4 \sqrt{\frac{c x-1}{c x+1}} \log (x) \sqrt{d-c^2 d x^2}-45 a c^4 d^{5/2} x^4 \sqrt{\frac{c x-1}{c x+1}} \sqrt{d-c^2 d x^2} \log \left (\sqrt{d} \sqrt{d-c^2 d x^2}+d\right )-6 a d^3 \sqrt{\frac{c x-1}{c x+1}}+24 b c^6 d^3 x^6-24 b c^5 d^3 x^5-27 b c^4 d^3 x^4+27 b c^3 d^3 x^3+2 b c^2 d^3 x^2-24 b c^6 d^3 x^6 \sqrt{\frac{c x-1}{c x+1}} \cosh ^{-1}(c x)-3 b c^4 d^3 x^4 \sqrt{\frac{c x-1}{c x+1}} \cosh ^{-1}(c x)+33 b c^2 d^3 x^2 \sqrt{\frac{c x-1}{c x+1}} \cosh ^{-1}(c x)-45 i b c^5 d^3 x^5 \cosh ^{-1}(c x) \log \left (1-i e^{-\cosh ^{-1}(c x)}\right )+45 i b c^5 d^3 x^5 \cosh ^{-1}(c x) \log \left (1+i e^{-\cosh ^{-1}(c x)}\right )+45 i b c^4 d^3 x^4 \cosh ^{-1}(c x) \log \left (1-i e^{-\cosh ^{-1}(c x)}\right )-45 i b c^4 d^3 x^4 \cosh ^{-1}(c x) \log \left (1+i e^{-\cosh ^{-1}(c x)}\right )-2 b c d^3 x-6 b d^3 \sqrt{\frac{c x-1}{c x+1}} \cosh ^{-1}(c x)}{24 x^4 \sqrt{\frac{c x-1}{c x+1}} \sqrt{d-c^2 d x^2}} \]

Warning: Unable to verify antiderivative.

[In]

Integrate[((d - c^2*d*x^2)^(5/2)*(a + b*ArcCosh[c*x]))/x^5,x]

[Out]

(-2*b*c*d^3*x + 2*b*c^2*d^3*x^2 + 27*b*c^3*d^3*x^3 - 27*b*c^4*d^3*x^4 - 24*b*c^5*d^3*x^5 + 24*b*c^6*d^3*x^6 -
6*a*d^3*Sqrt[(-1 + c*x)/(1 + c*x)] + 33*a*c^2*d^3*x^2*Sqrt[(-1 + c*x)/(1 + c*x)] - 3*a*c^4*d^3*x^4*Sqrt[(-1 +
c*x)/(1 + c*x)] - 24*a*c^6*d^3*x^6*Sqrt[(-1 + c*x)/(1 + c*x)] - 6*b*d^3*Sqrt[(-1 + c*x)/(1 + c*x)]*ArcCosh[c*x
] + 33*b*c^2*d^3*x^2*Sqrt[(-1 + c*x)/(1 + c*x)]*ArcCosh[c*x] - 3*b*c^4*d^3*x^4*Sqrt[(-1 + c*x)/(1 + c*x)]*ArcC
osh[c*x] - 24*b*c^6*d^3*x^6*Sqrt[(-1 + c*x)/(1 + c*x)]*ArcCosh[c*x] + (45*I)*b*c^4*d^3*x^4*ArcCosh[c*x]*Log[1
- I/E^ArcCosh[c*x]] - (45*I)*b*c^5*d^3*x^5*ArcCosh[c*x]*Log[1 - I/E^ArcCosh[c*x]] - (45*I)*b*c^4*d^3*x^4*ArcCo
sh[c*x]*Log[1 + I/E^ArcCosh[c*x]] + (45*I)*b*c^5*d^3*x^5*ArcCosh[c*x]*Log[1 + I/E^ArcCosh[c*x]] + 45*a*c^4*d^(
5/2)*x^4*Sqrt[(-1 + c*x)/(1 + c*x)]*Sqrt[d - c^2*d*x^2]*Log[x] - 45*a*c^4*d^(5/2)*x^4*Sqrt[(-1 + c*x)/(1 + c*x
)]*Sqrt[d - c^2*d*x^2]*Log[d + Sqrt[d]*Sqrt[d - c^2*d*x^2]] - (45*I)*b*c^4*d^3*x^4*(-1 + c*x)*PolyLog[2, (-I)/
E^ArcCosh[c*x]] + (45*I)*b*c^4*d^3*x^4*(-1 + c*x)*PolyLog[2, I/E^ArcCosh[c*x]])/(24*x^4*Sqrt[(-1 + c*x)/(1 + c
*x)]*Sqrt[d - c^2*d*x^2])

________________________________________________________________________________________

Maple [A]  time = 0.283, size = 691, normalized size = 1.7 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((-c^2*d*x^2+d)^(5/2)*(a+b*arccosh(c*x))/x^5,x)

[Out]

-1/4*a/d/x^4*(-c^2*d*x^2+d)^(7/2)+3/8*a*c^2/d/x^2*(-c^2*d*x^2+d)^(7/2)+3/8*a*c^4*(-c^2*d*x^2+d)^(5/2)+5/8*a*c^
4*d*(-c^2*d*x^2+d)^(3/2)-15/8*a*c^4*d^(5/2)*ln((2*d+2*d^(1/2)*(-c^2*d*x^2+d)^(1/2))/x)+15/8*a*c^4*(-c^2*d*x^2+
d)^(1/2)*d^2+b*(-d*(c^2*x^2-1))^(1/2)*d^2*c^6/(c*x+1)/(c*x-1)*arccosh(c*x)*x^2-b*(-d*(c^2*x^2-1))^(1/2)*d^2*c^
5/(c*x+1)^(1/2)/(c*x-1)^(1/2)*x+1/8*b*(-d*(c^2*x^2-1))^(1/2)*d^2*c^4/(c*x+1)/(c*x-1)*arccosh(c*x)+9/8*b*d^2*(-
d*(c^2*x^2-1))^(1/2)/(c*x+1)^(1/2)/x/(c*x-1)^(1/2)*c^3-11/8*b*d^2*(-d*(c^2*x^2-1))^(1/2)/(c*x+1)/x^2/(c*x-1)*a
rccosh(c*x)*c^2-1/12*b*d^2*(-d*(c^2*x^2-1))^(1/2)/(c*x+1)^(1/2)/x^3/(c*x-1)^(1/2)*c+1/4*b*d^2*(-d*(c^2*x^2-1))
^(1/2)/(c*x+1)/x^4/(c*x-1)*arccosh(c*x)-15/8*I*b*(-d*(c^2*x^2-1))^(1/2)/(c*x-1)^(1/2)/(c*x+1)^(1/2)*dilog(1-I*
(c*x+(c*x-1)^(1/2)*(c*x+1)^(1/2)))*d^2*c^4-15/8*I*b*(-d*(c^2*x^2-1))^(1/2)/(c*x-1)^(1/2)/(c*x+1)^(1/2)*arccosh
(c*x)*ln(1-I*(c*x+(c*x-1)^(1/2)*(c*x+1)^(1/2)))*d^2*c^4+15/8*I*b*(-d*(c^2*x^2-1))^(1/2)/(c*x-1)^(1/2)/(c*x+1)^
(1/2)*dilog(1+I*(c*x+(c*x-1)^(1/2)*(c*x+1)^(1/2)))*d^2*c^4+15/8*I*b*(-d*(c^2*x^2-1))^(1/2)/(c*x-1)^(1/2)/(c*x+
1)^(1/2)*arccosh(c*x)*ln(1+I*(c*x+(c*x-1)^(1/2)*(c*x+1)^(1/2)))*d^2*c^4

________________________________________________________________________________________

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)^(5/2)*(a+b*arccosh(c*x))/x^5,x, algorithm="maxima")

[Out]

Exception raised: ValueError

________________________________________________________________________________________

Fricas [F]  time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (\frac{{\left (a c^{4} d^{2} x^{4} - 2 \, a c^{2} d^{2} x^{2} + a d^{2} +{\left (b c^{4} d^{2} x^{4} - 2 \, b c^{2} d^{2} x^{2} + b d^{2}\right )} \operatorname{arcosh}\left (c x\right )\right )} \sqrt{-c^{2} d x^{2} + d}}{x^{5}}, x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-c^2*d*x^2+d)^(5/2)*(a+b*arccosh(c*x))/x^5,x, algorithm="fricas")

[Out]

integral((a*c^4*d^2*x^4 - 2*a*c^2*d^2*x^2 + a*d^2 + (b*c^4*d^2*x^4 - 2*b*c^2*d^2*x^2 + b*d^2)*arccosh(c*x))*sq
rt(-c^2*d*x^2 + d)/x^5, x)

________________________________________________________________________________________

Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-c**2*d*x**2+d)**(5/2)*(a+b*acosh(c*x))/x**5,x)

[Out]

Timed out

________________________________________________________________________________________

Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{{\left (-c^{2} d x^{2} + d\right )}^{\frac{5}{2}}{\left (b \operatorname{arcosh}\left (c x\right ) + a\right )}}{x^{5}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-c^2*d*x^2+d)^(5/2)*(a+b*arccosh(c*x))/x^5,x, algorithm="giac")

[Out]

integrate((-c^2*d*x^2 + d)^(5/2)*(b*arccosh(c*x) + a)/x^5, x)