3.132 \(\int \frac{a+b \cosh ^{-1}(c x)}{x^3 (d-c^2 d x^2)^{5/2}} \, dx\)

Optimal. Leaf size=479 \[ -\frac{5 i b c^2 \sqrt{c x-1} \sqrt{c x+1} \text{PolyLog}\left (2,-i e^{\cosh ^{-1}(c x)}\right )}{2 d^2 \sqrt{d-c^2 d x^2}}+\frac{5 i b c^2 \sqrt{c x-1} \sqrt{c x+1} \text{PolyLog}\left (2,i e^{\cosh ^{-1}(c x)}\right )}{2 d^2 \sqrt{d-c^2 d x^2}}+\frac{5 c^2 \left (a+b \cosh ^{-1}(c x)\right )}{2 d^2 \sqrt{d-c^2 d x^2}}+\frac{5 c^2 \sqrt{c x-1} \sqrt{c x+1} \tan ^{-1}\left (e^{\cosh ^{-1}(c x)}\right ) \left (a+b \cosh ^{-1}(c x)\right )}{d^2 \sqrt{d-c^2 d x^2}}+\frac{5 c^2 \left (a+b \cosh ^{-1}(c x)\right )}{6 d \left (d-c^2 d x^2\right )^{3/2}}-\frac{a+b \cosh ^{-1}(c x)}{2 d x^2 \left (d-c^2 d x^2\right )^{3/2}}+\frac{5 b c^3 x \sqrt{c x-1} \sqrt{c x+1}}{12 d^2 \left (1-c^2 x^2\right ) \sqrt{d-c^2 d x^2}}+\frac{3 b c \sqrt{c x-1} \sqrt{c x+1}}{4 d^2 x \sqrt{d-c^2 d x^2}}-\frac{b c \sqrt{c x-1} \sqrt{c x+1}}{4 d^2 x \left (1-c^2 x^2\right ) \sqrt{d-c^2 d x^2}}+\frac{13 b c^2 \sqrt{c x-1} \sqrt{c x+1} \tanh ^{-1}(c x)}{6 d^2 \sqrt{d-c^2 d x^2}} \]

[Out]

(3*b*c*Sqrt[-1 + c*x]*Sqrt[1 + c*x])/(4*d^2*x*Sqrt[d - c^2*d*x^2]) - (b*c*Sqrt[-1 + c*x]*Sqrt[1 + c*x])/(4*d^2
*x*(1 - c^2*x^2)*Sqrt[d - c^2*d*x^2]) + (5*b*c^3*x*Sqrt[-1 + c*x]*Sqrt[1 + c*x])/(12*d^2*(1 - c^2*x^2)*Sqrt[d
- c^2*d*x^2]) + (5*c^2*(a + b*ArcCosh[c*x]))/(6*d*(d - c^2*d*x^2)^(3/2)) - (a + b*ArcCosh[c*x])/(2*d*x^2*(d -
c^2*d*x^2)^(3/2)) + (5*c^2*(a + b*ArcCosh[c*x]))/(2*d^2*Sqrt[d - c^2*d*x^2]) + (5*c^2*Sqrt[-1 + c*x]*Sqrt[1 +
c*x]*(a + b*ArcCosh[c*x])*ArcTan[E^ArcCosh[c*x]])/(d^2*Sqrt[d - c^2*d*x^2]) + (13*b*c^2*Sqrt[-1 + c*x]*Sqrt[1
+ c*x]*ArcTanh[c*x])/(6*d^2*Sqrt[d - c^2*d*x^2]) - (((5*I)/2)*b*c^2*Sqrt[-1 + c*x]*Sqrt[1 + c*x]*PolyLog[2, (-
I)*E^ArcCosh[c*x]])/(d^2*Sqrt[d - c^2*d*x^2]) + (((5*I)/2)*b*c^2*Sqrt[-1 + c*x]*Sqrt[1 + c*x]*PolyLog[2, I*E^A
rcCosh[c*x]])/(d^2*Sqrt[d - c^2*d*x^2])

________________________________________________________________________________________

Rubi [A]  time = 1.14279, antiderivative size = 509, normalized size of antiderivative = 1.06, number of steps used = 16, number of rules used = 11, integrand size = 27, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.407, Rules used = {5798, 5748, 5756, 5761, 4180, 2279, 2391, 207, 199, 290, 325} \[ -\frac{5 i b c^2 \sqrt{c x-1} \sqrt{c x+1} \text{PolyLog}\left (2,-i e^{\cosh ^{-1}(c x)}\right )}{2 d^2 \sqrt{d-c^2 d x^2}}+\frac{5 i b c^2 \sqrt{c x-1} \sqrt{c x+1} \text{PolyLog}\left (2,i e^{\cosh ^{-1}(c x)}\right )}{2 d^2 \sqrt{d-c^2 d x^2}}+\frac{5 c^2 \left (a+b \cosh ^{-1}(c x)\right )}{6 d^2 (1-c x) (c x+1) \sqrt{d-c^2 d x^2}}+\frac{5 c^2 \left (a+b \cosh ^{-1}(c x)\right )}{2 d^2 \sqrt{d-c^2 d x^2}}-\frac{a+b \cosh ^{-1}(c x)}{2 d^2 x^2 (1-c x) (c x+1) \sqrt{d-c^2 d x^2}}+\frac{5 c^2 \sqrt{c x-1} \sqrt{c x+1} \tan ^{-1}\left (e^{\cosh ^{-1}(c x)}\right ) \left (a+b \cosh ^{-1}(c x)\right )}{d^2 \sqrt{d-c^2 d x^2}}+\frac{5 b c^3 x \sqrt{c x-1} \sqrt{c x+1}}{12 d^2 \left (1-c^2 x^2\right ) \sqrt{d-c^2 d x^2}}+\frac{3 b c \sqrt{c x-1} \sqrt{c x+1}}{4 d^2 x \sqrt{d-c^2 d x^2}}-\frac{b c \sqrt{c x-1} \sqrt{c x+1}}{4 d^2 x \left (1-c^2 x^2\right ) \sqrt{d-c^2 d x^2}}+\frac{13 b c^2 \sqrt{c x-1} \sqrt{c x+1} \tanh ^{-1}(c x)}{6 d^2 \sqrt{d-c^2 d x^2}} \]

Antiderivative was successfully verified.

[In]

Int[(a + b*ArcCosh[c*x])/(x^3*(d - c^2*d*x^2)^(5/2)),x]

[Out]

(3*b*c*Sqrt[-1 + c*x]*Sqrt[1 + c*x])/(4*d^2*x*Sqrt[d - c^2*d*x^2]) - (b*c*Sqrt[-1 + c*x]*Sqrt[1 + c*x])/(4*d^2
*x*(1 - c^2*x^2)*Sqrt[d - c^2*d*x^2]) + (5*b*c^3*x*Sqrt[-1 + c*x]*Sqrt[1 + c*x])/(12*d^2*(1 - c^2*x^2)*Sqrt[d
- c^2*d*x^2]) + (5*c^2*(a + b*ArcCosh[c*x]))/(2*d^2*Sqrt[d - c^2*d*x^2]) + (5*c^2*(a + b*ArcCosh[c*x]))/(6*d^2
*(1 - c*x)*(1 + c*x)*Sqrt[d - c^2*d*x^2]) - (a + b*ArcCosh[c*x])/(2*d^2*x^2*(1 - c*x)*(1 + c*x)*Sqrt[d - c^2*d
*x^2]) + (5*c^2*Sqrt[-1 + c*x]*Sqrt[1 + c*x]*(a + b*ArcCosh[c*x])*ArcTan[E^ArcCosh[c*x]])/(d^2*Sqrt[d - c^2*d*
x^2]) + (13*b*c^2*Sqrt[-1 + c*x]*Sqrt[1 + c*x]*ArcTanh[c*x])/(6*d^2*Sqrt[d - c^2*d*x^2]) - (((5*I)/2)*b*c^2*Sq
rt[-1 + c*x]*Sqrt[1 + c*x]*PolyLog[2, (-I)*E^ArcCosh[c*x]])/(d^2*Sqrt[d - c^2*d*x^2]) + (((5*I)/2)*b*c^2*Sqrt[
-1 + c*x]*Sqrt[1 + c*x]*PolyLog[2, I*E^ArcCosh[c*x]])/(d^2*Sqrt[d - c^2*d*x^2])

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 5748

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 + 1)*(d2 + e2*x)^(p + 1)*(a + b*ArcCosh[c*x])^n)/(d1*
d2*f*(m + 1)), x] + (Dist[(c^2*(m + 2*p + 3))/(f^2*(m + 1)), Int[(f*x)^(m + 2)*(d1 + e1*x)^p*(d2 + e2*x)^p*(a
+ b*ArcCosh[c*x])^n, x], x] + Dist[(b*c*n*(-(d1*d2))^IntPart[p]*(d1 + e1*x)^FracPart[p]*(d2 + e2*x)^FracPart[p
])/(f*(m + 1)*(1 + c*x)^FracPart[p]*(-1 + c*x)^FracPart[p]), 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, p}, x] && EqQ[e1 - c*d1, 0] && EqQ[e2 +
c*d2, 0] && GtQ[n, 0] && LtQ[m, -1] && IntegerQ[m] && IntegerQ[p + 1/2]

Rule 5756

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 + 1)*(d2 + e2*x)^(p + 1)*(a + b*ArcCosh[c*x])^n)/(2*
d1*d2*f*(p + 1)), x] + (Dist[(m + 2*p + 3)/(2*d1*d2*(p + 1)), Int[(f*x)^m*(d1 + e1*x)^(p + 1)*(d2 + e2*x)^(p +
1)*(a + b*ArcCosh[c*x])^n, x], x] - Dist[(b*c*n*(-(d1*d2))^IntPart[p]*(d1 + e1*x)^FracPart[p]*(d2 + e2*x)^Fra
cPart[p])/(2*f*(p + 1)*(1 + c*x)^FracPart[p]*(-1 + c*x)^FracPart[p]), 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, m}, x] && EqQ[e1 - c*d1, 0] &&
EqQ[e2 + c*d2, 0] && GtQ[n, 0] && LtQ[p, -1] &&  !GtQ[m, 1] && (IntegerQ[m] || EqQ[n, 1]) && IntegerQ[p + 1/2]

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 207

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> -Simp[ArcTanh[(Rt[b, 2]*x)/Rt[-a, 2]]/(Rt[-a, 2]*Rt[b, 2]), x] /;
FreeQ[{a, b}, x] && NegQ[a/b] && (LtQ[a, 0] || GtQ[b, 0])

Rule 199

Int[((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> -Simp[(x*(a + b*x^n)^(p + 1))/(a*n*(p + 1)), x] + Dist[(n*(p +
1) + 1)/(a*n*(p + 1)), Int[(a + b*x^n)^(p + 1), x], x] /; FreeQ[{a, b}, x] && IGtQ[n, 0] && LtQ[p, -1] && (In
tegerQ[2*p] || (n == 2 && IntegerQ[4*p]) || (n == 2 && IntegerQ[3*p]) || Denominator[p + 1/n] < Denominator[p]
)

Rule 290

Int[((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> -Simp[((c*x)^(m + 1)*(a + b*x^n)^(p + 1))/(
a*c*n*(p + 1)), x] + Dist[(m + n*(p + 1) + 1)/(a*n*(p + 1)), Int[(c*x)^m*(a + b*x^n)^(p + 1), x], x] /; FreeQ[
{a, b, c, m}, x] && IGtQ[n, 0] && LtQ[p, -1] && IntBinomialQ[a, b, c, n, m, p, x]

Rule 325

Int[((c_.)*(x_))^(m_)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[((c*x)^(m + 1)*(a + b*x^n)^(p + 1))/(a*
c*(m + 1)), x] - Dist[(b*(m + n*(p + 1) + 1))/(a*c^n*(m + 1)), Int[(c*x)^(m + n)*(a + b*x^n)^p, x], x] /; Free
Q[{a, b, c, p}, x] && IGtQ[n, 0] && LtQ[m, -1] && IntBinomialQ[a, b, c, n, m, p, x]

Rubi steps

\begin{align*} \int \frac{a+b \cosh ^{-1}(c x)}{x^3 \left (d-c^2 d x^2\right )^{5/2}} \, dx &=\frac{\left (\sqrt{-1+c x} \sqrt{1+c x}\right ) \int \frac{a+b \cosh ^{-1}(c x)}{x^3 (-1+c x)^{5/2} (1+c x)^{5/2}} \, dx}{d^2 \sqrt{d-c^2 d x^2}}\\ &=-\frac{a+b \cosh ^{-1}(c x)}{2 d^2 x^2 (1-c x) (1+c x) \sqrt{d-c^2 d x^2}}-\frac{\left (b c \sqrt{-1+c x} \sqrt{1+c x}\right ) \int \frac{1}{x^2 \left (-1+c^2 x^2\right )^2} \, dx}{2 d^2 \sqrt{d-c^2 d x^2}}+\frac{\left (5 c^2 \sqrt{-1+c x} \sqrt{1+c x}\right ) \int \frac{a+b \cosh ^{-1}(c x)}{x (-1+c x)^{5/2} (1+c x)^{5/2}} \, dx}{2 d^2 \sqrt{d-c^2 d x^2}}\\ &=-\frac{b c \sqrt{-1+c x} \sqrt{1+c x}}{4 d^2 x \left (1-c^2 x^2\right ) \sqrt{d-c^2 d x^2}}+\frac{5 c^2 \left (a+b \cosh ^{-1}(c x)\right )}{6 d^2 (1-c x) (1+c x) \sqrt{d-c^2 d x^2}}-\frac{a+b \cosh ^{-1}(c x)}{2 d^2 x^2 (1-c x) (1+c x) \sqrt{d-c^2 d x^2}}+\frac{\left (3 b c \sqrt{-1+c x} \sqrt{1+c x}\right ) \int \frac{1}{x^2 \left (-1+c^2 x^2\right )} \, dx}{4 d^2 \sqrt{d-c^2 d x^2}}-\frac{\left (5 c^2 \sqrt{-1+c x} \sqrt{1+c x}\right ) \int \frac{a+b \cosh ^{-1}(c x)}{x (-1+c x)^{3/2} (1+c x)^{3/2}} \, dx}{2 d^2 \sqrt{d-c^2 d x^2}}+\frac{\left (5 b c^3 \sqrt{-1+c x} \sqrt{1+c x}\right ) \int \frac{1}{\left (-1+c^2 x^2\right )^2} \, dx}{6 d^2 \sqrt{d-c^2 d x^2}}\\ &=\frac{3 b c \sqrt{-1+c x} \sqrt{1+c x}}{4 d^2 x \sqrt{d-c^2 d x^2}}-\frac{b c \sqrt{-1+c x} \sqrt{1+c x}}{4 d^2 x \left (1-c^2 x^2\right ) \sqrt{d-c^2 d x^2}}+\frac{5 b c^3 x \sqrt{-1+c x} \sqrt{1+c x}}{12 d^2 \left (1-c^2 x^2\right ) \sqrt{d-c^2 d x^2}}+\frac{5 c^2 \left (a+b \cosh ^{-1}(c x)\right )}{2 d^2 \sqrt{d-c^2 d x^2}}+\frac{5 c^2 \left (a+b \cosh ^{-1}(c x)\right )}{6 d^2 (1-c x) (1+c x) \sqrt{d-c^2 d x^2}}-\frac{a+b \cosh ^{-1}(c x)}{2 d^2 x^2 (1-c x) (1+c x) \sqrt{d-c^2 d x^2}}+\frac{\left (5 c^2 \sqrt{-1+c x} \sqrt{1+c x}\right ) \int \frac{a+b \cosh ^{-1}(c x)}{x \sqrt{-1+c x} \sqrt{1+c x}} \, dx}{2 d^2 \sqrt{d-c^2 d x^2}}-\frac{\left (5 b c^3 \sqrt{-1+c x} \sqrt{1+c x}\right ) \int \frac{1}{-1+c^2 x^2} \, dx}{12 d^2 \sqrt{d-c^2 d x^2}}+\frac{\left (3 b c^3 \sqrt{-1+c x} \sqrt{1+c x}\right ) \int \frac{1}{-1+c^2 x^2} \, dx}{4 d^2 \sqrt{d-c^2 d x^2}}-\frac{\left (5 b c^3 \sqrt{-1+c x} \sqrt{1+c x}\right ) \int \frac{1}{-1+c^2 x^2} \, dx}{2 d^2 \sqrt{d-c^2 d x^2}}\\ &=\frac{3 b c \sqrt{-1+c x} \sqrt{1+c x}}{4 d^2 x \sqrt{d-c^2 d x^2}}-\frac{b c \sqrt{-1+c x} \sqrt{1+c x}}{4 d^2 x \left (1-c^2 x^2\right ) \sqrt{d-c^2 d x^2}}+\frac{5 b c^3 x \sqrt{-1+c x} \sqrt{1+c x}}{12 d^2 \left (1-c^2 x^2\right ) \sqrt{d-c^2 d x^2}}+\frac{5 c^2 \left (a+b \cosh ^{-1}(c x)\right )}{2 d^2 \sqrt{d-c^2 d x^2}}+\frac{5 c^2 \left (a+b \cosh ^{-1}(c x)\right )}{6 d^2 (1-c x) (1+c x) \sqrt{d-c^2 d x^2}}-\frac{a+b \cosh ^{-1}(c x)}{2 d^2 x^2 (1-c x) (1+c x) \sqrt{d-c^2 d x^2}}+\frac{13 b c^2 \sqrt{-1+c x} \sqrt{1+c x} \tanh ^{-1}(c x)}{6 d^2 \sqrt{d-c^2 d x^2}}+\frac{\left (5 c^2 \sqrt{-1+c x} \sqrt{1+c x}\right ) \operatorname{Subst}\left (\int (a+b x) \text{sech}(x) \, dx,x,\cosh ^{-1}(c x)\right )}{2 d^2 \sqrt{d-c^2 d x^2}}\\ &=\frac{3 b c \sqrt{-1+c x} \sqrt{1+c x}}{4 d^2 x \sqrt{d-c^2 d x^2}}-\frac{b c \sqrt{-1+c x} \sqrt{1+c x}}{4 d^2 x \left (1-c^2 x^2\right ) \sqrt{d-c^2 d x^2}}+\frac{5 b c^3 x \sqrt{-1+c x} \sqrt{1+c x}}{12 d^2 \left (1-c^2 x^2\right ) \sqrt{d-c^2 d x^2}}+\frac{5 c^2 \left (a+b \cosh ^{-1}(c x)\right )}{2 d^2 \sqrt{d-c^2 d x^2}}+\frac{5 c^2 \left (a+b \cosh ^{-1}(c x)\right )}{6 d^2 (1-c x) (1+c x) \sqrt{d-c^2 d x^2}}-\frac{a+b \cosh ^{-1}(c x)}{2 d^2 x^2 (1-c x) (1+c x) \sqrt{d-c^2 d x^2}}+\frac{5 c^2 \sqrt{-1+c x} \sqrt{1+c x} \left (a+b \cosh ^{-1}(c x)\right ) \tan ^{-1}\left (e^{\cosh ^{-1}(c x)}\right )}{d^2 \sqrt{d-c^2 d x^2}}+\frac{13 b c^2 \sqrt{-1+c x} \sqrt{1+c x} \tanh ^{-1}(c x)}{6 d^2 \sqrt{d-c^2 d x^2}}-\frac{\left (5 i b c^2 \sqrt{-1+c x} \sqrt{1+c x}\right ) \operatorname{Subst}\left (\int \log \left (1-i e^x\right ) \, dx,x,\cosh ^{-1}(c x)\right )}{2 d^2 \sqrt{d-c^2 d x^2}}+\frac{\left (5 i b c^2 \sqrt{-1+c x} \sqrt{1+c x}\right ) \operatorname{Subst}\left (\int \log \left (1+i e^x\right ) \, dx,x,\cosh ^{-1}(c x)\right )}{2 d^2 \sqrt{d-c^2 d x^2}}\\ &=\frac{3 b c \sqrt{-1+c x} \sqrt{1+c x}}{4 d^2 x \sqrt{d-c^2 d x^2}}-\frac{b c \sqrt{-1+c x} \sqrt{1+c x}}{4 d^2 x \left (1-c^2 x^2\right ) \sqrt{d-c^2 d x^2}}+\frac{5 b c^3 x \sqrt{-1+c x} \sqrt{1+c x}}{12 d^2 \left (1-c^2 x^2\right ) \sqrt{d-c^2 d x^2}}+\frac{5 c^2 \left (a+b \cosh ^{-1}(c x)\right )}{2 d^2 \sqrt{d-c^2 d x^2}}+\frac{5 c^2 \left (a+b \cosh ^{-1}(c x)\right )}{6 d^2 (1-c x) (1+c x) \sqrt{d-c^2 d x^2}}-\frac{a+b \cosh ^{-1}(c x)}{2 d^2 x^2 (1-c x) (1+c x) \sqrt{d-c^2 d x^2}}+\frac{5 c^2 \sqrt{-1+c x} \sqrt{1+c x} \left (a+b \cosh ^{-1}(c x)\right ) \tan ^{-1}\left (e^{\cosh ^{-1}(c x)}\right )}{d^2 \sqrt{d-c^2 d x^2}}+\frac{13 b c^2 \sqrt{-1+c x} \sqrt{1+c x} \tanh ^{-1}(c x)}{6 d^2 \sqrt{d-c^2 d x^2}}-\frac{\left (5 i b c^2 \sqrt{-1+c x} \sqrt{1+c x}\right ) \operatorname{Subst}\left (\int \frac{\log (1-i x)}{x} \, dx,x,e^{\cosh ^{-1}(c x)}\right )}{2 d^2 \sqrt{d-c^2 d x^2}}+\frac{\left (5 i b c^2 \sqrt{-1+c x} \sqrt{1+c x}\right ) \operatorname{Subst}\left (\int \frac{\log (1+i x)}{x} \, dx,x,e^{\cosh ^{-1}(c x)}\right )}{2 d^2 \sqrt{d-c^2 d x^2}}\\ &=\frac{3 b c \sqrt{-1+c x} \sqrt{1+c x}}{4 d^2 x \sqrt{d-c^2 d x^2}}-\frac{b c \sqrt{-1+c x} \sqrt{1+c x}}{4 d^2 x \left (1-c^2 x^2\right ) \sqrt{d-c^2 d x^2}}+\frac{5 b c^3 x \sqrt{-1+c x} \sqrt{1+c x}}{12 d^2 \left (1-c^2 x^2\right ) \sqrt{d-c^2 d x^2}}+\frac{5 c^2 \left (a+b \cosh ^{-1}(c x)\right )}{2 d^2 \sqrt{d-c^2 d x^2}}+\frac{5 c^2 \left (a+b \cosh ^{-1}(c x)\right )}{6 d^2 (1-c x) (1+c x) \sqrt{d-c^2 d x^2}}-\frac{a+b \cosh ^{-1}(c x)}{2 d^2 x^2 (1-c x) (1+c x) \sqrt{d-c^2 d x^2}}+\frac{5 c^2 \sqrt{-1+c x} \sqrt{1+c x} \left (a+b \cosh ^{-1}(c x)\right ) \tan ^{-1}\left (e^{\cosh ^{-1}(c x)}\right )}{d^2 \sqrt{d-c^2 d x^2}}+\frac{13 b c^2 \sqrt{-1+c x} \sqrt{1+c x} \tanh ^{-1}(c x)}{6 d^2 \sqrt{d-c^2 d x^2}}-\frac{5 i b c^2 \sqrt{-1+c x} \sqrt{1+c x} \text{Li}_2\left (-i e^{\cosh ^{-1}(c x)}\right )}{2 d^2 \sqrt{d-c^2 d x^2}}+\frac{5 i b c^2 \sqrt{-1+c x} \sqrt{1+c x} \text{Li}_2\left (i e^{\cosh ^{-1}(c x)}\right )}{2 d^2 \sqrt{d-c^2 d x^2}}\\ \end{align*}

Mathematica [A]  time = 7.20668, size = 500, normalized size = 1.04 \[ \frac{b c^2 \left (-30 i \sqrt{\frac{c x-1}{c x+1}} (c x+1) \text{PolyLog}\left (2,-i e^{-\cosh ^{-1}(c x)}\right )+30 i \sqrt{\frac{c x-1}{c x+1}} (c x+1) \text{PolyLog}\left (2,i e^{-\cosh ^{-1}(c x)}\right )+\frac{6 (c x-1) (c x+1) \cosh ^{-1}(c x)}{c^2 x^2}+\frac{6 \sqrt{\frac{c x-1}{c x+1}} (c x+1)}{c x}+26 \cosh ^{-1}(c x) \cosh ^2\left (\frac{1}{2} \cosh ^{-1}(c x)\right )-30 i \sqrt{\frac{c x-1}{c x+1}} (c x+1) \cosh ^{-1}(c x) \log \left (1-i e^{-\cosh ^{-1}(c x)}\right )+30 i \sqrt{\frac{c x-1}{c x+1}} (c x+1) \cosh ^{-1}(c x) \log \left (1+i e^{-\cosh ^{-1}(c x)}\right )-26 \cosh ^{-1}(c x) \sinh ^2\left (\frac{1}{2} \cosh ^{-1}(c x)\right )-\cosh ^{-1}(c x) \tanh ^2\left (\frac{1}{2} \cosh ^{-1}(c x)\right )-\tanh \left (\frac{1}{2} \cosh ^{-1}(c x)\right )-\cosh ^{-1}(c x) \coth ^2\left (\frac{1}{2} \cosh ^{-1}(c x)\right )-\coth \left (\frac{1}{2} \cosh ^{-1}(c x)\right )-26 \sqrt{\frac{c x-1}{c x+1}} (c x+1) \log \left (\tanh \left (\frac{1}{2} \cosh ^{-1}(c x)\right )\right )\right )}{12 d^2 \sqrt{-d (c x-1) (c x+1)}}+\sqrt{-d \left (c^2 x^2-1\right )} \left (-\frac{2 a c^2}{d^3 \left (c^2 x^2-1\right )}+\frac{a c^2}{3 d^3 \left (c^2 x^2-1\right )^2}-\frac{a}{2 d^3 x^2}\right )-\frac{5 a c^2 \log \left (\sqrt{d} \sqrt{-d \left (c^2 x^2-1\right )}+d\right )}{2 d^{5/2}}+\frac{5 a c^2 \log (x)}{2 d^{5/2}} \]

Warning: Unable to verify antiderivative.

[In]

Integrate[(a + b*ArcCosh[c*x])/(x^3*(d - c^2*d*x^2)^(5/2)),x]

[Out]

Sqrt[-(d*(-1 + c^2*x^2))]*(-a/(2*d^3*x^2) + (a*c^2)/(3*d^3*(-1 + c^2*x^2)^2) - (2*a*c^2)/(d^3*(-1 + c^2*x^2)))
+ (5*a*c^2*Log[x])/(2*d^(5/2)) - (5*a*c^2*Log[d + Sqrt[d]*Sqrt[-(d*(-1 + c^2*x^2))]])/(2*d^(5/2)) + (b*c^2*((
6*Sqrt[(-1 + c*x)/(1 + c*x)]*(1 + c*x))/(c*x) + (6*(-1 + c*x)*(1 + c*x)*ArcCosh[c*x])/(c^2*x^2) + 26*ArcCosh[c
*x]*Cosh[ArcCosh[c*x]/2]^2 - Coth[ArcCosh[c*x]/2] - ArcCosh[c*x]*Coth[ArcCosh[c*x]/2]^2 - (30*I)*Sqrt[(-1 + c*
x)/(1 + c*x)]*(1 + c*x)*ArcCosh[c*x]*Log[1 - I/E^ArcCosh[c*x]] + (30*I)*Sqrt[(-1 + c*x)/(1 + c*x)]*(1 + c*x)*A
rcCosh[c*x]*Log[1 + I/E^ArcCosh[c*x]] - 26*Sqrt[(-1 + c*x)/(1 + c*x)]*(1 + c*x)*Log[Tanh[ArcCosh[c*x]/2]] - (3
0*I)*Sqrt[(-1 + c*x)/(1 + c*x)]*(1 + c*x)*PolyLog[2, (-I)/E^ArcCosh[c*x]] + (30*I)*Sqrt[(-1 + c*x)/(1 + c*x)]*
(1 + c*x)*PolyLog[2, I/E^ArcCosh[c*x]] - 26*ArcCosh[c*x]*Sinh[ArcCosh[c*x]/2]^2 - Tanh[ArcCosh[c*x]/2] - ArcCo
sh[c*x]*Tanh[ArcCosh[c*x]/2]^2))/(12*d^2*Sqrt[-(d*(-1 + c*x)*(1 + c*x))])

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Maple [A]  time = 0.285, size = 801, 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((a+b*arccosh(c*x))/x^3/(-c^2*d*x^2+d)^(5/2),x)

[Out]

-1/2*a/d/x^2/(-c^2*d*x^2+d)^(3/2)+5/6*a*c^2/d/(-c^2*d*x^2+d)^(3/2)+5/2*a*c^2/d^2/(-c^2*d*x^2+d)^(1/2)-5/2*a*c^
2/d^(5/2)*ln((2*d+2*d^(1/2)*(-c^2*d*x^2+d)^(1/2))/x)-5/2*b*(-d*(c^2*x^2-1))^(1/2)/d^3/(c^4*x^4-2*c^2*x^2+1)*x^
2*arccosh(c*x)*c^4-1/3*b*(-d*(c^2*x^2-1))^(1/2)/d^3/(c^4*x^4-2*c^2*x^2+1)*x*(c*x+1)^(1/2)*(c*x-1)^(1/2)*c^3+10
/3*b*(-d*(c^2*x^2-1))^(1/2)/d^3/(c^4*x^4-2*c^2*x^2+1)*arccosh(c*x)*c^2+1/2*b*(-d*(c^2*x^2-1))^(1/2)/d^3/(c^4*x
^4-2*c^2*x^2+1)/x*(c*x+1)^(1/2)*(c*x-1)^(1/2)*c-1/2*b*(-d*(c^2*x^2-1))^(1/2)/d^3/(c^4*x^4-2*c^2*x^2+1)/x^2*arc
cosh(c*x)+13/6*b*(-d*(c^2*x^2-1))^(1/2)*(c*x-1)^(1/2)*(c*x+1)^(1/2)/d^3/(c^2*x^2-1)*ln(c*x+(c*x-1)^(1/2)*(c*x+
1)^(1/2)-1)*c^2-13/6*b*(-d*(c^2*x^2-1))^(1/2)*(c*x-1)^(1/2)*(c*x+1)^(1/2)/d^3/(c^2*x^2-1)*ln(1+c*x+(c*x-1)^(1/
2)*(c*x+1)^(1/2))*c^2-5/2*I*b*(-d*(c^2*x^2-1))^(1/2)*(c*x-1)^(1/2)*(c*x+1)^(1/2)/d^3/(c^2*x^2-1)*arccosh(c*x)*
ln(1-I*(c*x+(c*x-1)^(1/2)*(c*x+1)^(1/2)))*c^2+5/2*I*b*(-d*(c^2*x^2-1))^(1/2)*(c*x-1)^(1/2)*(c*x+1)^(1/2)/d^3/(
c^2*x^2-1)*arccosh(c*x)*ln(1+I*(c*x+(c*x-1)^(1/2)*(c*x+1)^(1/2)))*c^2+5/2*I*b*(-d*(c^2*x^2-1))^(1/2)*(c*x-1)^(
1/2)*(c*x+1)^(1/2)/d^3/(c^2*x^2-1)*dilog(1+I*(c*x+(c*x-1)^(1/2)*(c*x+1)^(1/2)))*c^2-5/2*I*b*(-d*(c^2*x^2-1))^(
1/2)*(c*x-1)^(1/2)*(c*x+1)^(1/2)/d^3/(c^2*x^2-1)*dilog(1-I*(c*x+(c*x-1)^(1/2)*(c*x+1)^(1/2)))*c^2

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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))/x^3/(-c^2*d*x^2+d)^(5/2),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 \operatorname{arcosh}\left (c x\right ) + a\right )}}{c^{6} d^{3} x^{9} - 3 \, c^{4} d^{3} x^{7} + 3 \, c^{2} d^{3} x^{5} - d^{3} x^{3}}, x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b*arccosh(c*x))/x^3/(-c^2*d*x^2+d)^(5/2),x, algorithm="fricas")

[Out]

integral(-sqrt(-c^2*d*x^2 + d)*(b*arccosh(c*x) + a)/(c^6*d^3*x^9 - 3*c^4*d^3*x^7 + 3*c^2*d^3*x^5 - d^3*x^3), x
)

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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((a+b*acosh(c*x))/x**3/(-c**2*d*x**2+d)**(5/2),x)

[Out]

Timed out

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{b \operatorname{arcosh}\left (c x\right ) + a}{{\left (-c^{2} d x^{2} + d\right )}^{\frac{5}{2}} x^{3}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b*arccosh(c*x))/x^3/(-c^2*d*x^2+d)^(5/2),x, algorithm="giac")

[Out]

integrate((b*arccosh(c*x) + a)/((-c^2*d*x^2 + d)^(5/2)*x^3), x)