3.47 \(\int \frac{x^3}{(a+b \text{sech}(c+d \sqrt{x}))^2} \, dx\)

Optimal. Leaf size=2851 \[ \text{result too large to display} \]

[Out]

(2*b^2*x^(7/2))/(a^2*(a^2 - b^2)*d) + x^4/(4*a^2) - (14*b^2*x^3*Log[1 + (a*E^(c + d*Sqrt[x]))/(b - Sqrt[-a^2 +
b^2])])/(a^2*(a^2 - b^2)*d^2) + (2*b^3*x^(7/2)*Log[1 + (a*E^(c + d*Sqrt[x]))/(b - Sqrt[-a^2 + b^2])])/(a^2*(-
a^2 + b^2)^(3/2)*d) - (4*b*x^(7/2)*Log[1 + (a*E^(c + d*Sqrt[x]))/(b - Sqrt[-a^2 + b^2])])/(a^2*Sqrt[-a^2 + b^2
]*d) - (14*b^2*x^3*Log[1 + (a*E^(c + d*Sqrt[x]))/(b + Sqrt[-a^2 + b^2])])/(a^2*(a^2 - b^2)*d^2) - (2*b^3*x^(7/
2)*Log[1 + (a*E^(c + d*Sqrt[x]))/(b + Sqrt[-a^2 + b^2])])/(a^2*(-a^2 + b^2)^(3/2)*d) + (4*b*x^(7/2)*Log[1 + (a
*E^(c + d*Sqrt[x]))/(b + Sqrt[-a^2 + b^2])])/(a^2*Sqrt[-a^2 + b^2]*d) - (84*b^2*x^(5/2)*PolyLog[2, -((a*E^(c +
d*Sqrt[x]))/(b - Sqrt[-a^2 + b^2]))])/(a^2*(a^2 - b^2)*d^3) + (14*b^3*x^3*PolyLog[2, -((a*E^(c + d*Sqrt[x]))/
(b - Sqrt[-a^2 + b^2]))])/(a^2*(-a^2 + b^2)^(3/2)*d^2) - (28*b*x^3*PolyLog[2, -((a*E^(c + d*Sqrt[x]))/(b - Sqr
t[-a^2 + b^2]))])/(a^2*Sqrt[-a^2 + b^2]*d^2) - (84*b^2*x^(5/2)*PolyLog[2, -((a*E^(c + d*Sqrt[x]))/(b + Sqrt[-a
^2 + b^2]))])/(a^2*(a^2 - b^2)*d^3) - (14*b^3*x^3*PolyLog[2, -((a*E^(c + d*Sqrt[x]))/(b + Sqrt[-a^2 + b^2]))])
/(a^2*(-a^2 + b^2)^(3/2)*d^2) + (28*b*x^3*PolyLog[2, -((a*E^(c + d*Sqrt[x]))/(b + Sqrt[-a^2 + b^2]))])/(a^2*Sq
rt[-a^2 + b^2]*d^2) + (420*b^2*x^2*PolyLog[3, -((a*E^(c + d*Sqrt[x]))/(b - Sqrt[-a^2 + b^2]))])/(a^2*(a^2 - b^
2)*d^4) - (84*b^3*x^(5/2)*PolyLog[3, -((a*E^(c + d*Sqrt[x]))/(b - Sqrt[-a^2 + b^2]))])/(a^2*(-a^2 + b^2)^(3/2)
*d^3) + (168*b*x^(5/2)*PolyLog[3, -((a*E^(c + d*Sqrt[x]))/(b - Sqrt[-a^2 + b^2]))])/(a^2*Sqrt[-a^2 + b^2]*d^3)
+ (420*b^2*x^2*PolyLog[3, -((a*E^(c + d*Sqrt[x]))/(b + Sqrt[-a^2 + b^2]))])/(a^2*(a^2 - b^2)*d^4) + (84*b^3*x
^(5/2)*PolyLog[3, -((a*E^(c + d*Sqrt[x]))/(b + Sqrt[-a^2 + b^2]))])/(a^2*(-a^2 + b^2)^(3/2)*d^3) - (168*b*x^(5
/2)*PolyLog[3, -((a*E^(c + d*Sqrt[x]))/(b + Sqrt[-a^2 + b^2]))])/(a^2*Sqrt[-a^2 + b^2]*d^3) - (1680*b^2*x^(3/2
)*PolyLog[4, -((a*E^(c + d*Sqrt[x]))/(b - Sqrt[-a^2 + b^2]))])/(a^2*(a^2 - b^2)*d^5) + (420*b^3*x^2*PolyLog[4,
-((a*E^(c + d*Sqrt[x]))/(b - Sqrt[-a^2 + b^2]))])/(a^2*(-a^2 + b^2)^(3/2)*d^4) - (840*b*x^2*PolyLog[4, -((a*E
^(c + d*Sqrt[x]))/(b - Sqrt[-a^2 + b^2]))])/(a^2*Sqrt[-a^2 + b^2]*d^4) - (1680*b^2*x^(3/2)*PolyLog[4, -((a*E^(
c + d*Sqrt[x]))/(b + Sqrt[-a^2 + b^2]))])/(a^2*(a^2 - b^2)*d^5) - (420*b^3*x^2*PolyLog[4, -((a*E^(c + d*Sqrt[x
]))/(b + Sqrt[-a^2 + b^2]))])/(a^2*(-a^2 + b^2)^(3/2)*d^4) + (840*b*x^2*PolyLog[4, -((a*E^(c + d*Sqrt[x]))/(b
+ Sqrt[-a^2 + b^2]))])/(a^2*Sqrt[-a^2 + b^2]*d^4) + (5040*b^2*x*PolyLog[5, -((a*E^(c + d*Sqrt[x]))/(b - Sqrt[-
a^2 + b^2]))])/(a^2*(a^2 - b^2)*d^6) - (1680*b^3*x^(3/2)*PolyLog[5, -((a*E^(c + d*Sqrt[x]))/(b - Sqrt[-a^2 + b
^2]))])/(a^2*(-a^2 + b^2)^(3/2)*d^5) + (3360*b*x^(3/2)*PolyLog[5, -((a*E^(c + d*Sqrt[x]))/(b - Sqrt[-a^2 + b^2
]))])/(a^2*Sqrt[-a^2 + b^2]*d^5) + (5040*b^2*x*PolyLog[5, -((a*E^(c + d*Sqrt[x]))/(b + Sqrt[-a^2 + b^2]))])/(a
^2*(a^2 - b^2)*d^6) + (1680*b^3*x^(3/2)*PolyLog[5, -((a*E^(c + d*Sqrt[x]))/(b + Sqrt[-a^2 + b^2]))])/(a^2*(-a^
2 + b^2)^(3/2)*d^5) - (3360*b*x^(3/2)*PolyLog[5, -((a*E^(c + d*Sqrt[x]))/(b + Sqrt[-a^2 + b^2]))])/(a^2*Sqrt[-
a^2 + b^2]*d^5) - (10080*b^2*Sqrt[x]*PolyLog[6, -((a*E^(c + d*Sqrt[x]))/(b - Sqrt[-a^2 + b^2]))])/(a^2*(a^2 -
b^2)*d^7) + (5040*b^3*x*PolyLog[6, -((a*E^(c + d*Sqrt[x]))/(b - Sqrt[-a^2 + b^2]))])/(a^2*(-a^2 + b^2)^(3/2)*d
^6) - (10080*b*x*PolyLog[6, -((a*E^(c + d*Sqrt[x]))/(b - Sqrt[-a^2 + b^2]))])/(a^2*Sqrt[-a^2 + b^2]*d^6) - (10
080*b^2*Sqrt[x]*PolyLog[6, -((a*E^(c + d*Sqrt[x]))/(b + Sqrt[-a^2 + b^2]))])/(a^2*(a^2 - b^2)*d^7) - (5040*b^3
*x*PolyLog[6, -((a*E^(c + d*Sqrt[x]))/(b + Sqrt[-a^2 + b^2]))])/(a^2*(-a^2 + b^2)^(3/2)*d^6) + (10080*b*x*Poly
Log[6, -((a*E^(c + d*Sqrt[x]))/(b + Sqrt[-a^2 + b^2]))])/(a^2*Sqrt[-a^2 + b^2]*d^6) + (10080*b^2*PolyLog[7, -(
(a*E^(c + d*Sqrt[x]))/(b - Sqrt[-a^2 + b^2]))])/(a^2*(a^2 - b^2)*d^8) - (10080*b^3*Sqrt[x]*PolyLog[7, -((a*E^(
c + d*Sqrt[x]))/(b - Sqrt[-a^2 + b^2]))])/(a^2*(-a^2 + b^2)^(3/2)*d^7) + (20160*b*Sqrt[x]*PolyLog[7, -((a*E^(c
+ d*Sqrt[x]))/(b - Sqrt[-a^2 + b^2]))])/(a^2*Sqrt[-a^2 + b^2]*d^7) + (10080*b^2*PolyLog[7, -((a*E^(c + d*Sqrt
[x]))/(b + Sqrt[-a^2 + b^2]))])/(a^2*(a^2 - b^2)*d^8) + (10080*b^3*Sqrt[x]*PolyLog[7, -((a*E^(c + d*Sqrt[x]))/
(b + Sqrt[-a^2 + b^2]))])/(a^2*(-a^2 + b^2)^(3/2)*d^7) - (20160*b*Sqrt[x]*PolyLog[7, -((a*E^(c + d*Sqrt[x]))/(
b + Sqrt[-a^2 + b^2]))])/(a^2*Sqrt[-a^2 + b^2]*d^7) + (10080*b^3*PolyLog[8, -((a*E^(c + d*Sqrt[x]))/(b - Sqrt[
-a^2 + b^2]))])/(a^2*(-a^2 + b^2)^(3/2)*d^8) - (20160*b*PolyLog[8, -((a*E^(c + d*Sqrt[x]))/(b - Sqrt[-a^2 + b^
2]))])/(a^2*Sqrt[-a^2 + b^2]*d^8) - (10080*b^3*PolyLog[8, -((a*E^(c + d*Sqrt[x]))/(b + Sqrt[-a^2 + b^2]))])/(a
^2*(-a^2 + b^2)^(3/2)*d^8) + (20160*b*PolyLog[8, -((a*E^(c + d*Sqrt[x]))/(b + Sqrt[-a^2 + b^2]))])/(a^2*Sqrt[-
a^2 + b^2]*d^8) + (2*b^2*x^(7/2)*Sinh[c + d*Sqrt[x]])/(a*(a^2 - b^2)*d*(b + a*Cosh[c + d*Sqrt[x]]))

________________________________________________________________________________________

Rubi [A]  time = 3.78249, antiderivative size = 2851, normalized size of antiderivative = 1., number of steps used = 61, number of rules used = 11, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.55, Rules used = {5436, 4191, 3324, 3320, 2264, 2190, 2531, 6609, 2282, 6589, 5562} \[ \text{result too large to display} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(2*b^2*x^(7/2))/(a^2*(a^2 - b^2)*d) + x^4/(4*a^2) - (14*b^2*x^3*Log[1 + (a*E^(c + d*Sqrt[x]))/(b - Sqrt[-a^2 +
b^2])])/(a^2*(a^2 - b^2)*d^2) + (2*b^3*x^(7/2)*Log[1 + (a*E^(c + d*Sqrt[x]))/(b - Sqrt[-a^2 + b^2])])/(a^2*(-
a^2 + b^2)^(3/2)*d) - (4*b*x^(7/2)*Log[1 + (a*E^(c + d*Sqrt[x]))/(b - Sqrt[-a^2 + b^2])])/(a^2*Sqrt[-a^2 + b^2
]*d) - (14*b^2*x^3*Log[1 + (a*E^(c + d*Sqrt[x]))/(b + Sqrt[-a^2 + b^2])])/(a^2*(a^2 - b^2)*d^2) - (2*b^3*x^(7/
2)*Log[1 + (a*E^(c + d*Sqrt[x]))/(b + Sqrt[-a^2 + b^2])])/(a^2*(-a^2 + b^2)^(3/2)*d) + (4*b*x^(7/2)*Log[1 + (a
*E^(c + d*Sqrt[x]))/(b + Sqrt[-a^2 + b^2])])/(a^2*Sqrt[-a^2 + b^2]*d) - (84*b^2*x^(5/2)*PolyLog[2, -((a*E^(c +
d*Sqrt[x]))/(b - Sqrt[-a^2 + b^2]))])/(a^2*(a^2 - b^2)*d^3) + (14*b^3*x^3*PolyLog[2, -((a*E^(c + d*Sqrt[x]))/
(b - Sqrt[-a^2 + b^2]))])/(a^2*(-a^2 + b^2)^(3/2)*d^2) - (28*b*x^3*PolyLog[2, -((a*E^(c + d*Sqrt[x]))/(b - Sqr
t[-a^2 + b^2]))])/(a^2*Sqrt[-a^2 + b^2]*d^2) - (84*b^2*x^(5/2)*PolyLog[2, -((a*E^(c + d*Sqrt[x]))/(b + Sqrt[-a
^2 + b^2]))])/(a^2*(a^2 - b^2)*d^3) - (14*b^3*x^3*PolyLog[2, -((a*E^(c + d*Sqrt[x]))/(b + Sqrt[-a^2 + b^2]))])
/(a^2*(-a^2 + b^2)^(3/2)*d^2) + (28*b*x^3*PolyLog[2, -((a*E^(c + d*Sqrt[x]))/(b + Sqrt[-a^2 + b^2]))])/(a^2*Sq
rt[-a^2 + b^2]*d^2) + (420*b^2*x^2*PolyLog[3, -((a*E^(c + d*Sqrt[x]))/(b - Sqrt[-a^2 + b^2]))])/(a^2*(a^2 - b^
2)*d^4) - (84*b^3*x^(5/2)*PolyLog[3, -((a*E^(c + d*Sqrt[x]))/(b - Sqrt[-a^2 + b^2]))])/(a^2*(-a^2 + b^2)^(3/2)
*d^3) + (168*b*x^(5/2)*PolyLog[3, -((a*E^(c + d*Sqrt[x]))/(b - Sqrt[-a^2 + b^2]))])/(a^2*Sqrt[-a^2 + b^2]*d^3)
+ (420*b^2*x^2*PolyLog[3, -((a*E^(c + d*Sqrt[x]))/(b + Sqrt[-a^2 + b^2]))])/(a^2*(a^2 - b^2)*d^4) + (84*b^3*x
^(5/2)*PolyLog[3, -((a*E^(c + d*Sqrt[x]))/(b + Sqrt[-a^2 + b^2]))])/(a^2*(-a^2 + b^2)^(3/2)*d^3) - (168*b*x^(5
/2)*PolyLog[3, -((a*E^(c + d*Sqrt[x]))/(b + Sqrt[-a^2 + b^2]))])/(a^2*Sqrt[-a^2 + b^2]*d^3) - (1680*b^2*x^(3/2
)*PolyLog[4, -((a*E^(c + d*Sqrt[x]))/(b - Sqrt[-a^2 + b^2]))])/(a^2*(a^2 - b^2)*d^5) + (420*b^3*x^2*PolyLog[4,
-((a*E^(c + d*Sqrt[x]))/(b - Sqrt[-a^2 + b^2]))])/(a^2*(-a^2 + b^2)^(3/2)*d^4) - (840*b*x^2*PolyLog[4, -((a*E
^(c + d*Sqrt[x]))/(b - Sqrt[-a^2 + b^2]))])/(a^2*Sqrt[-a^2 + b^2]*d^4) - (1680*b^2*x^(3/2)*PolyLog[4, -((a*E^(
c + d*Sqrt[x]))/(b + Sqrt[-a^2 + b^2]))])/(a^2*(a^2 - b^2)*d^5) - (420*b^3*x^2*PolyLog[4, -((a*E^(c + d*Sqrt[x
]))/(b + Sqrt[-a^2 + b^2]))])/(a^2*(-a^2 + b^2)^(3/2)*d^4) + (840*b*x^2*PolyLog[4, -((a*E^(c + d*Sqrt[x]))/(b
+ Sqrt[-a^2 + b^2]))])/(a^2*Sqrt[-a^2 + b^2]*d^4) + (5040*b^2*x*PolyLog[5, -((a*E^(c + d*Sqrt[x]))/(b - Sqrt[-
a^2 + b^2]))])/(a^2*(a^2 - b^2)*d^6) - (1680*b^3*x^(3/2)*PolyLog[5, -((a*E^(c + d*Sqrt[x]))/(b - Sqrt[-a^2 + b
^2]))])/(a^2*(-a^2 + b^2)^(3/2)*d^5) + (3360*b*x^(3/2)*PolyLog[5, -((a*E^(c + d*Sqrt[x]))/(b - Sqrt[-a^2 + b^2
]))])/(a^2*Sqrt[-a^2 + b^2]*d^5) + (5040*b^2*x*PolyLog[5, -((a*E^(c + d*Sqrt[x]))/(b + Sqrt[-a^2 + b^2]))])/(a
^2*(a^2 - b^2)*d^6) + (1680*b^3*x^(3/2)*PolyLog[5, -((a*E^(c + d*Sqrt[x]))/(b + Sqrt[-a^2 + b^2]))])/(a^2*(-a^
2 + b^2)^(3/2)*d^5) - (3360*b*x^(3/2)*PolyLog[5, -((a*E^(c + d*Sqrt[x]))/(b + Sqrt[-a^2 + b^2]))])/(a^2*Sqrt[-
a^2 + b^2]*d^5) - (10080*b^2*Sqrt[x]*PolyLog[6, -((a*E^(c + d*Sqrt[x]))/(b - Sqrt[-a^2 + b^2]))])/(a^2*(a^2 -
b^2)*d^7) + (5040*b^3*x*PolyLog[6, -((a*E^(c + d*Sqrt[x]))/(b - Sqrt[-a^2 + b^2]))])/(a^2*(-a^2 + b^2)^(3/2)*d
^6) - (10080*b*x*PolyLog[6, -((a*E^(c + d*Sqrt[x]))/(b - Sqrt[-a^2 + b^2]))])/(a^2*Sqrt[-a^2 + b^2]*d^6) - (10
080*b^2*Sqrt[x]*PolyLog[6, -((a*E^(c + d*Sqrt[x]))/(b + Sqrt[-a^2 + b^2]))])/(a^2*(a^2 - b^2)*d^7) - (5040*b^3
*x*PolyLog[6, -((a*E^(c + d*Sqrt[x]))/(b + Sqrt[-a^2 + b^2]))])/(a^2*(-a^2 + b^2)^(3/2)*d^6) + (10080*b*x*Poly
Log[6, -((a*E^(c + d*Sqrt[x]))/(b + Sqrt[-a^2 + b^2]))])/(a^2*Sqrt[-a^2 + b^2]*d^6) + (10080*b^2*PolyLog[7, -(
(a*E^(c + d*Sqrt[x]))/(b - Sqrt[-a^2 + b^2]))])/(a^2*(a^2 - b^2)*d^8) - (10080*b^3*Sqrt[x]*PolyLog[7, -((a*E^(
c + d*Sqrt[x]))/(b - Sqrt[-a^2 + b^2]))])/(a^2*(-a^2 + b^2)^(3/2)*d^7) + (20160*b*Sqrt[x]*PolyLog[7, -((a*E^(c
+ d*Sqrt[x]))/(b - Sqrt[-a^2 + b^2]))])/(a^2*Sqrt[-a^2 + b^2]*d^7) + (10080*b^2*PolyLog[7, -((a*E^(c + d*Sqrt
[x]))/(b + Sqrt[-a^2 + b^2]))])/(a^2*(a^2 - b^2)*d^8) + (10080*b^3*Sqrt[x]*PolyLog[7, -((a*E^(c + d*Sqrt[x]))/
(b + Sqrt[-a^2 + b^2]))])/(a^2*(-a^2 + b^2)^(3/2)*d^7) - (20160*b*Sqrt[x]*PolyLog[7, -((a*E^(c + d*Sqrt[x]))/(
b + Sqrt[-a^2 + b^2]))])/(a^2*Sqrt[-a^2 + b^2]*d^7) + (10080*b^3*PolyLog[8, -((a*E^(c + d*Sqrt[x]))/(b - Sqrt[
-a^2 + b^2]))])/(a^2*(-a^2 + b^2)^(3/2)*d^8) - (20160*b*PolyLog[8, -((a*E^(c + d*Sqrt[x]))/(b - Sqrt[-a^2 + b^
2]))])/(a^2*Sqrt[-a^2 + b^2]*d^8) - (10080*b^3*PolyLog[8, -((a*E^(c + d*Sqrt[x]))/(b + Sqrt[-a^2 + b^2]))])/(a
^2*(-a^2 + b^2)^(3/2)*d^8) + (20160*b*PolyLog[8, -((a*E^(c + d*Sqrt[x]))/(b + Sqrt[-a^2 + b^2]))])/(a^2*Sqrt[-
a^2 + b^2]*d^8) + (2*b^2*x^(7/2)*Sinh[c + d*Sqrt[x]])/(a*(a^2 - b^2)*d*(b + a*Cosh[c + d*Sqrt[x]]))

Rule 5436

Int[(x_)^(m_.)*((a_.) + (b_.)*Sech[(c_.) + (d_.)*(x_)^(n_)])^(p_.), x_Symbol] :> Dist[1/n, Subst[Int[x^(Simpli
fy[(m + 1)/n] - 1)*(a + b*Sech[c + d*x])^p, x], x, x^n], x] /; FreeQ[{a, b, c, d, m, n, p}, x] && IGtQ[Simplif
y[(m + 1)/n], 0] && IntegerQ[p]

Rule 4191

Int[(csc[(e_.) + (f_.)*(x_)]*(b_.) + (a_))^(n_.)*((c_.) + (d_.)*(x_))^(m_.), x_Symbol] :> Int[ExpandIntegrand[
(c + d*x)^m, 1/(Sin[e + f*x]^n/(b + a*Sin[e + f*x])^n), x], x] /; FreeQ[{a, b, c, d, e, f}, x] && ILtQ[n, 0] &
& IGtQ[m, 0]

Rule 3324

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

Rule 3320

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

Rule 2264

Int[((F_)^(u_)*((f_.) + (g_.)*(x_))^(m_.))/((a_.) + (b_.)*(F_)^(u_) + (c_.)*(F_)^(v_)), x_Symbol] :> With[{q =
Rt[b^2 - 4*a*c, 2]}, Dist[(2*c)/q, Int[((f + g*x)^m*F^u)/(b - q + 2*c*F^u), x], x] - Dist[(2*c)/q, Int[((f +
g*x)^m*F^u)/(b + q + 2*c*F^u), x], x]] /; FreeQ[{F, a, b, c, f, g}, x] && EqQ[v, 2*u] && LinearQ[u, x] && NeQ[
b^2 - 4*a*c, 0] && 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 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 6609

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

Rule 5562

Int[(((e_.) + (f_.)*(x_))^(m_.)*Sinh[(c_.) + (d_.)*(x_)])/(Cosh[(c_.) + (d_.)*(x_)]*(b_.) + (a_)), x_Symbol] :
> -Simp[(e + f*x)^(m + 1)/(b*f*(m + 1)), x] + (Int[((e + f*x)^m*E^(c + d*x))/(a - Rt[a^2 - b^2, 2] + b*E^(c +
d*x)), x] + Int[((e + f*x)^m*E^(c + d*x))/(a + Rt[a^2 - b^2, 2] + b*E^(c + d*x)), x]) /; FreeQ[{a, b, c, d, e,
f}, x] && IGtQ[m, 0] && NeQ[a^2 - b^2, 0]

Rubi steps

\begin{align*} \int \frac{x^3}{\left (a+b \text{sech}\left (c+d \sqrt{x}\right )\right )^2} \, dx &=2 \operatorname{Subst}\left (\int \frac{x^7}{(a+b \text{sech}(c+d x))^2} \, dx,x,\sqrt{x}\right )\\ &=2 \operatorname{Subst}\left (\int \left (\frac{x^7}{a^2}+\frac{b^2 x^7}{a^2 (b+a \cosh (c+d x))^2}-\frac{2 b x^7}{a^2 (b+a \cosh (c+d x))}\right ) \, dx,x,\sqrt{x}\right )\\ &=\frac{x^4}{4 a^2}-\frac{(4 b) \operatorname{Subst}\left (\int \frac{x^7}{b+a \cosh (c+d x)} \, dx,x,\sqrt{x}\right )}{a^2}+\frac{\left (2 b^2\right ) \operatorname{Subst}\left (\int \frac{x^7}{(b+a \cosh (c+d x))^2} \, dx,x,\sqrt{x}\right )}{a^2}\\ &=\frac{x^4}{4 a^2}+\frac{2 b^2 x^{7/2} \sinh \left (c+d \sqrt{x}\right )}{a \left (a^2-b^2\right ) d \left (b+a \cosh \left (c+d \sqrt{x}\right )\right )}-\frac{(8 b) \operatorname{Subst}\left (\int \frac{e^{c+d x} x^7}{a+2 b e^{c+d x}+a e^{2 (c+d x)}} \, dx,x,\sqrt{x}\right )}{a^2}-\frac{\left (2 b^3\right ) \operatorname{Subst}\left (\int \frac{x^7}{b+a \cosh (c+d x)} \, dx,x,\sqrt{x}\right )}{a^2 \left (a^2-b^2\right )}-\frac{\left (14 b^2\right ) \operatorname{Subst}\left (\int \frac{x^6 \sinh (c+d x)}{b+a \cosh (c+d x)} \, dx,x,\sqrt{x}\right )}{a \left (a^2-b^2\right ) d}\\ &=\frac{2 b^2 x^{7/2}}{a^2 \left (a^2-b^2\right ) d}+\frac{x^4}{4 a^2}+\frac{2 b^2 x^{7/2} \sinh \left (c+d \sqrt{x}\right )}{a \left (a^2-b^2\right ) d \left (b+a \cosh \left (c+d \sqrt{x}\right )\right )}-\frac{\left (4 b^3\right ) \operatorname{Subst}\left (\int \frac{e^{c+d x} x^7}{a+2 b e^{c+d x}+a e^{2 (c+d x)}} \, dx,x,\sqrt{x}\right )}{a^2 \left (a^2-b^2\right )}-\frac{(8 b) \operatorname{Subst}\left (\int \frac{e^{c+d x} x^7}{2 b-2 \sqrt{-a^2+b^2}+2 a e^{c+d x}} \, dx,x,\sqrt{x}\right )}{a \sqrt{-a^2+b^2}}+\frac{(8 b) \operatorname{Subst}\left (\int \frac{e^{c+d x} x^7}{2 b+2 \sqrt{-a^2+b^2}+2 a e^{c+d x}} \, dx,x,\sqrt{x}\right )}{a \sqrt{-a^2+b^2}}-\frac{\left (14 b^2\right ) \operatorname{Subst}\left (\int \frac{e^{c+d x} x^6}{b-\sqrt{-a^2+b^2}+a e^{c+d x}} \, dx,x,\sqrt{x}\right )}{a \left (a^2-b^2\right ) d}-\frac{\left (14 b^2\right ) \operatorname{Subst}\left (\int \frac{e^{c+d x} x^6}{b+\sqrt{-a^2+b^2}+a e^{c+d x}} \, dx,x,\sqrt{x}\right )}{a \left (a^2-b^2\right ) d}\\ &=\frac{2 b^2 x^{7/2}}{a^2 \left (a^2-b^2\right ) d}+\frac{x^4}{4 a^2}-\frac{14 b^2 x^3 \log \left (1+\frac{a e^{c+d \sqrt{x}}}{b-\sqrt{-a^2+b^2}}\right )}{a^2 \left (a^2-b^2\right ) d^2}-\frac{4 b x^{7/2} \log \left (1+\frac{a e^{c+d \sqrt{x}}}{b-\sqrt{-a^2+b^2}}\right )}{a^2 \sqrt{-a^2+b^2} d}-\frac{14 b^2 x^3 \log \left (1+\frac{a e^{c+d \sqrt{x}}}{b+\sqrt{-a^2+b^2}}\right )}{a^2 \left (a^2-b^2\right ) d^2}+\frac{4 b x^{7/2} \log \left (1+\frac{a e^{c+d \sqrt{x}}}{b+\sqrt{-a^2+b^2}}\right )}{a^2 \sqrt{-a^2+b^2} d}+\frac{2 b^2 x^{7/2} \sinh \left (c+d \sqrt{x}\right )}{a \left (a^2-b^2\right ) d \left (b+a \cosh \left (c+d \sqrt{x}\right )\right )}+\frac{\left (4 b^3\right ) \operatorname{Subst}\left (\int \frac{e^{c+d x} x^7}{2 b-2 \sqrt{-a^2+b^2}+2 a e^{c+d x}} \, dx,x,\sqrt{x}\right )}{a \left (-a^2+b^2\right )^{3/2}}-\frac{\left (4 b^3\right ) \operatorname{Subst}\left (\int \frac{e^{c+d x} x^7}{2 b+2 \sqrt{-a^2+b^2}+2 a e^{c+d x}} \, dx,x,\sqrt{x}\right )}{a \left (-a^2+b^2\right )^{3/2}}+\frac{\left (84 b^2\right ) \operatorname{Subst}\left (\int x^5 \log \left (1+\frac{a e^{c+d x}}{b-\sqrt{-a^2+b^2}}\right ) \, dx,x,\sqrt{x}\right )}{a^2 \left (a^2-b^2\right ) d^2}+\frac{\left (84 b^2\right ) \operatorname{Subst}\left (\int x^5 \log \left (1+\frac{a e^{c+d x}}{b+\sqrt{-a^2+b^2}}\right ) \, dx,x,\sqrt{x}\right )}{a^2 \left (a^2-b^2\right ) d^2}+\frac{(28 b) \operatorname{Subst}\left (\int x^6 \log \left (1+\frac{2 a e^{c+d x}}{2 b-2 \sqrt{-a^2+b^2}}\right ) \, dx,x,\sqrt{x}\right )}{a^2 \sqrt{-a^2+b^2} d}-\frac{(28 b) \operatorname{Subst}\left (\int x^6 \log \left (1+\frac{2 a e^{c+d x}}{2 b+2 \sqrt{-a^2+b^2}}\right ) \, dx,x,\sqrt{x}\right )}{a^2 \sqrt{-a^2+b^2} d}\\ &=\frac{2 b^2 x^{7/2}}{a^2 \left (a^2-b^2\right ) d}+\frac{x^4}{4 a^2}-\frac{14 b^2 x^3 \log \left (1+\frac{a e^{c+d \sqrt{x}}}{b-\sqrt{-a^2+b^2}}\right )}{a^2 \left (a^2-b^2\right ) d^2}+\frac{2 b^3 x^{7/2} \log \left (1+\frac{a e^{c+d \sqrt{x}}}{b-\sqrt{-a^2+b^2}}\right )}{a^2 \left (-a^2+b^2\right )^{3/2} d}-\frac{4 b x^{7/2} \log \left (1+\frac{a e^{c+d \sqrt{x}}}{b-\sqrt{-a^2+b^2}}\right )}{a^2 \sqrt{-a^2+b^2} d}-\frac{14 b^2 x^3 \log \left (1+\frac{a e^{c+d \sqrt{x}}}{b+\sqrt{-a^2+b^2}}\right )}{a^2 \left (a^2-b^2\right ) d^2}-\frac{2 b^3 x^{7/2} \log \left (1+\frac{a e^{c+d \sqrt{x}}}{b+\sqrt{-a^2+b^2}}\right )}{a^2 \left (-a^2+b^2\right )^{3/2} d}+\frac{4 b x^{7/2} \log \left (1+\frac{a e^{c+d \sqrt{x}}}{b+\sqrt{-a^2+b^2}}\right )}{a^2 \sqrt{-a^2+b^2} d}-\frac{84 b^2 x^{5/2} \text{Li}_2\left (-\frac{a e^{c+d \sqrt{x}}}{b-\sqrt{-a^2+b^2}}\right )}{a^2 \left (a^2-b^2\right ) d^3}-\frac{28 b x^3 \text{Li}_2\left (-\frac{a e^{c+d \sqrt{x}}}{b-\sqrt{-a^2+b^2}}\right )}{a^2 \sqrt{-a^2+b^2} d^2}-\frac{84 b^2 x^{5/2} \text{Li}_2\left (-\frac{a e^{c+d \sqrt{x}}}{b+\sqrt{-a^2+b^2}}\right )}{a^2 \left (a^2-b^2\right ) d^3}+\frac{28 b x^3 \text{Li}_2\left (-\frac{a e^{c+d \sqrt{x}}}{b+\sqrt{-a^2+b^2}}\right )}{a^2 \sqrt{-a^2+b^2} d^2}+\frac{2 b^2 x^{7/2} \sinh \left (c+d \sqrt{x}\right )}{a \left (a^2-b^2\right ) d \left (b+a \cosh \left (c+d \sqrt{x}\right )\right )}+\frac{\left (420 b^2\right ) \operatorname{Subst}\left (\int x^4 \text{Li}_2\left (-\frac{a e^{c+d x}}{b-\sqrt{-a^2+b^2}}\right ) \, dx,x,\sqrt{x}\right )}{a^2 \left (a^2-b^2\right ) d^3}+\frac{\left (420 b^2\right ) \operatorname{Subst}\left (\int x^4 \text{Li}_2\left (-\frac{a e^{c+d x}}{b+\sqrt{-a^2+b^2}}\right ) \, dx,x,\sqrt{x}\right )}{a^2 \left (a^2-b^2\right ) d^3}+\frac{(168 b) \operatorname{Subst}\left (\int x^5 \text{Li}_2\left (-\frac{2 a e^{c+d x}}{2 b-2 \sqrt{-a^2+b^2}}\right ) \, dx,x,\sqrt{x}\right )}{a^2 \sqrt{-a^2+b^2} d^2}-\frac{(168 b) \operatorname{Subst}\left (\int x^5 \text{Li}_2\left (-\frac{2 a e^{c+d x}}{2 b+2 \sqrt{-a^2+b^2}}\right ) \, dx,x,\sqrt{x}\right )}{a^2 \sqrt{-a^2+b^2} d^2}-\frac{\left (14 b^3\right ) \operatorname{Subst}\left (\int x^6 \log \left (1+\frac{2 a e^{c+d x}}{2 b-2 \sqrt{-a^2+b^2}}\right ) \, dx,x,\sqrt{x}\right )}{a^2 \left (-a^2+b^2\right )^{3/2} d}+\frac{\left (14 b^3\right ) \operatorname{Subst}\left (\int x^6 \log \left (1+\frac{2 a e^{c+d x}}{2 b+2 \sqrt{-a^2+b^2}}\right ) \, dx,x,\sqrt{x}\right )}{a^2 \left (-a^2+b^2\right )^{3/2} d}\\ &=\frac{2 b^2 x^{7/2}}{a^2 \left (a^2-b^2\right ) d}+\frac{x^4}{4 a^2}-\frac{14 b^2 x^3 \log \left (1+\frac{a e^{c+d \sqrt{x}}}{b-\sqrt{-a^2+b^2}}\right )}{a^2 \left (a^2-b^2\right ) d^2}+\frac{2 b^3 x^{7/2} \log \left (1+\frac{a e^{c+d \sqrt{x}}}{b-\sqrt{-a^2+b^2}}\right )}{a^2 \left (-a^2+b^2\right )^{3/2} d}-\frac{4 b x^{7/2} \log \left (1+\frac{a e^{c+d \sqrt{x}}}{b-\sqrt{-a^2+b^2}}\right )}{a^2 \sqrt{-a^2+b^2} d}-\frac{14 b^2 x^3 \log \left (1+\frac{a e^{c+d \sqrt{x}}}{b+\sqrt{-a^2+b^2}}\right )}{a^2 \left (a^2-b^2\right ) d^2}-\frac{2 b^3 x^{7/2} \log \left (1+\frac{a e^{c+d \sqrt{x}}}{b+\sqrt{-a^2+b^2}}\right )}{a^2 \left (-a^2+b^2\right )^{3/2} d}+\frac{4 b x^{7/2} \log \left (1+\frac{a e^{c+d \sqrt{x}}}{b+\sqrt{-a^2+b^2}}\right )}{a^2 \sqrt{-a^2+b^2} d}-\frac{84 b^2 x^{5/2} \text{Li}_2\left (-\frac{a e^{c+d \sqrt{x}}}{b-\sqrt{-a^2+b^2}}\right )}{a^2 \left (a^2-b^2\right ) d^3}+\frac{14 b^3 x^3 \text{Li}_2\left (-\frac{a e^{c+d \sqrt{x}}}{b-\sqrt{-a^2+b^2}}\right )}{a^2 \left (-a^2+b^2\right )^{3/2} d^2}-\frac{28 b x^3 \text{Li}_2\left (-\frac{a e^{c+d \sqrt{x}}}{b-\sqrt{-a^2+b^2}}\right )}{a^2 \sqrt{-a^2+b^2} d^2}-\frac{84 b^2 x^{5/2} \text{Li}_2\left (-\frac{a e^{c+d \sqrt{x}}}{b+\sqrt{-a^2+b^2}}\right )}{a^2 \left (a^2-b^2\right ) d^3}-\frac{14 b^3 x^3 \text{Li}_2\left (-\frac{a e^{c+d \sqrt{x}}}{b+\sqrt{-a^2+b^2}}\right )}{a^2 \left (-a^2+b^2\right )^{3/2} d^2}+\frac{28 b x^3 \text{Li}_2\left (-\frac{a e^{c+d \sqrt{x}}}{b+\sqrt{-a^2+b^2}}\right )}{a^2 \sqrt{-a^2+b^2} d^2}+\frac{420 b^2 x^2 \text{Li}_3\left (-\frac{a e^{c+d \sqrt{x}}}{b-\sqrt{-a^2+b^2}}\right )}{a^2 \left (a^2-b^2\right ) d^4}+\frac{168 b x^{5/2} \text{Li}_3\left (-\frac{a e^{c+d \sqrt{x}}}{b-\sqrt{-a^2+b^2}}\right )}{a^2 \sqrt{-a^2+b^2} d^3}+\frac{420 b^2 x^2 \text{Li}_3\left (-\frac{a e^{c+d \sqrt{x}}}{b+\sqrt{-a^2+b^2}}\right )}{a^2 \left (a^2-b^2\right ) d^4}-\frac{168 b x^{5/2} \text{Li}_3\left (-\frac{a e^{c+d \sqrt{x}}}{b+\sqrt{-a^2+b^2}}\right )}{a^2 \sqrt{-a^2+b^2} d^3}+\frac{2 b^2 x^{7/2} \sinh \left (c+d \sqrt{x}\right )}{a \left (a^2-b^2\right ) d \left (b+a \cosh \left (c+d \sqrt{x}\right )\right )}-\frac{\left (1680 b^2\right ) \operatorname{Subst}\left (\int x^3 \text{Li}_3\left (-\frac{a e^{c+d x}}{b-\sqrt{-a^2+b^2}}\right ) \, dx,x,\sqrt{x}\right )}{a^2 \left (a^2-b^2\right ) d^4}-\frac{\left (1680 b^2\right ) \operatorname{Subst}\left (\int x^3 \text{Li}_3\left (-\frac{a e^{c+d x}}{b+\sqrt{-a^2+b^2}}\right ) \, dx,x,\sqrt{x}\right )}{a^2 \left (a^2-b^2\right ) d^4}-\frac{(840 b) \operatorname{Subst}\left (\int x^4 \text{Li}_3\left (-\frac{2 a e^{c+d x}}{2 b-2 \sqrt{-a^2+b^2}}\right ) \, dx,x,\sqrt{x}\right )}{a^2 \sqrt{-a^2+b^2} d^3}+\frac{(840 b) \operatorname{Subst}\left (\int x^4 \text{Li}_3\left (-\frac{2 a e^{c+d x}}{2 b+2 \sqrt{-a^2+b^2}}\right ) \, dx,x,\sqrt{x}\right )}{a^2 \sqrt{-a^2+b^2} d^3}-\frac{\left (84 b^3\right ) \operatorname{Subst}\left (\int x^5 \text{Li}_2\left (-\frac{2 a e^{c+d x}}{2 b-2 \sqrt{-a^2+b^2}}\right ) \, dx,x,\sqrt{x}\right )}{a^2 \left (-a^2+b^2\right )^{3/2} d^2}+\frac{\left (84 b^3\right ) \operatorname{Subst}\left (\int x^5 \text{Li}_2\left (-\frac{2 a e^{c+d x}}{2 b+2 \sqrt{-a^2+b^2}}\right ) \, dx,x,\sqrt{x}\right )}{a^2 \left (-a^2+b^2\right )^{3/2} d^2}\\ &=\frac{2 b^2 x^{7/2}}{a^2 \left (a^2-b^2\right ) d}+\frac{x^4}{4 a^2}-\frac{14 b^2 x^3 \log \left (1+\frac{a e^{c+d \sqrt{x}}}{b-\sqrt{-a^2+b^2}}\right )}{a^2 \left (a^2-b^2\right ) d^2}+\frac{2 b^3 x^{7/2} \log \left (1+\frac{a e^{c+d \sqrt{x}}}{b-\sqrt{-a^2+b^2}}\right )}{a^2 \left (-a^2+b^2\right )^{3/2} d}-\frac{4 b x^{7/2} \log \left (1+\frac{a e^{c+d \sqrt{x}}}{b-\sqrt{-a^2+b^2}}\right )}{a^2 \sqrt{-a^2+b^2} d}-\frac{14 b^2 x^3 \log \left (1+\frac{a e^{c+d \sqrt{x}}}{b+\sqrt{-a^2+b^2}}\right )}{a^2 \left (a^2-b^2\right ) d^2}-\frac{2 b^3 x^{7/2} \log \left (1+\frac{a e^{c+d \sqrt{x}}}{b+\sqrt{-a^2+b^2}}\right )}{a^2 \left (-a^2+b^2\right )^{3/2} d}+\frac{4 b x^{7/2} \log \left (1+\frac{a e^{c+d \sqrt{x}}}{b+\sqrt{-a^2+b^2}}\right )}{a^2 \sqrt{-a^2+b^2} d}-\frac{84 b^2 x^{5/2} \text{Li}_2\left (-\frac{a e^{c+d \sqrt{x}}}{b-\sqrt{-a^2+b^2}}\right )}{a^2 \left (a^2-b^2\right ) d^3}+\frac{14 b^3 x^3 \text{Li}_2\left (-\frac{a e^{c+d \sqrt{x}}}{b-\sqrt{-a^2+b^2}}\right )}{a^2 \left (-a^2+b^2\right )^{3/2} d^2}-\frac{28 b x^3 \text{Li}_2\left (-\frac{a e^{c+d \sqrt{x}}}{b-\sqrt{-a^2+b^2}}\right )}{a^2 \sqrt{-a^2+b^2} d^2}-\frac{84 b^2 x^{5/2} \text{Li}_2\left (-\frac{a e^{c+d \sqrt{x}}}{b+\sqrt{-a^2+b^2}}\right )}{a^2 \left (a^2-b^2\right ) d^3}-\frac{14 b^3 x^3 \text{Li}_2\left (-\frac{a e^{c+d \sqrt{x}}}{b+\sqrt{-a^2+b^2}}\right )}{a^2 \left (-a^2+b^2\right )^{3/2} d^2}+\frac{28 b x^3 \text{Li}_2\left (-\frac{a e^{c+d \sqrt{x}}}{b+\sqrt{-a^2+b^2}}\right )}{a^2 \sqrt{-a^2+b^2} d^2}+\frac{420 b^2 x^2 \text{Li}_3\left (-\frac{a e^{c+d \sqrt{x}}}{b-\sqrt{-a^2+b^2}}\right )}{a^2 \left (a^2-b^2\right ) d^4}-\frac{84 b^3 x^{5/2} \text{Li}_3\left (-\frac{a e^{c+d \sqrt{x}}}{b-\sqrt{-a^2+b^2}}\right )}{a^2 \left (-a^2+b^2\right )^{3/2} d^3}+\frac{168 b x^{5/2} \text{Li}_3\left (-\frac{a e^{c+d \sqrt{x}}}{b-\sqrt{-a^2+b^2}}\right )}{a^2 \sqrt{-a^2+b^2} d^3}+\frac{420 b^2 x^2 \text{Li}_3\left (-\frac{a e^{c+d \sqrt{x}}}{b+\sqrt{-a^2+b^2}}\right )}{a^2 \left (a^2-b^2\right ) d^4}+\frac{84 b^3 x^{5/2} \text{Li}_3\left (-\frac{a e^{c+d \sqrt{x}}}{b+\sqrt{-a^2+b^2}}\right )}{a^2 \left (-a^2+b^2\right )^{3/2} d^3}-\frac{168 b x^{5/2} \text{Li}_3\left (-\frac{a e^{c+d \sqrt{x}}}{b+\sqrt{-a^2+b^2}}\right )}{a^2 \sqrt{-a^2+b^2} d^3}-\frac{1680 b^2 x^{3/2} \text{Li}_4\left (-\frac{a e^{c+d \sqrt{x}}}{b-\sqrt{-a^2+b^2}}\right )}{a^2 \left (a^2-b^2\right ) d^5}-\frac{840 b x^2 \text{Li}_4\left (-\frac{a e^{c+d \sqrt{x}}}{b-\sqrt{-a^2+b^2}}\right )}{a^2 \sqrt{-a^2+b^2} d^4}-\frac{1680 b^2 x^{3/2} \text{Li}_4\left (-\frac{a e^{c+d \sqrt{x}}}{b+\sqrt{-a^2+b^2}}\right )}{a^2 \left (a^2-b^2\right ) d^5}+\frac{840 b x^2 \text{Li}_4\left (-\frac{a e^{c+d \sqrt{x}}}{b+\sqrt{-a^2+b^2}}\right )}{a^2 \sqrt{-a^2+b^2} d^4}+\frac{2 b^2 x^{7/2} \sinh \left (c+d \sqrt{x}\right )}{a \left (a^2-b^2\right ) d \left (b+a \cosh \left (c+d \sqrt{x}\right )\right )}+\frac{\left (5040 b^2\right ) \operatorname{Subst}\left (\int x^2 \text{Li}_4\left (-\frac{a e^{c+d x}}{b-\sqrt{-a^2+b^2}}\right ) \, dx,x,\sqrt{x}\right )}{a^2 \left (a^2-b^2\right ) d^5}+\frac{\left (5040 b^2\right ) \operatorname{Subst}\left (\int x^2 \text{Li}_4\left (-\frac{a e^{c+d x}}{b+\sqrt{-a^2+b^2}}\right ) \, dx,x,\sqrt{x}\right )}{a^2 \left (a^2-b^2\right ) d^5}+\frac{(3360 b) \operatorname{Subst}\left (\int x^3 \text{Li}_4\left (-\frac{2 a e^{c+d x}}{2 b-2 \sqrt{-a^2+b^2}}\right ) \, dx,x,\sqrt{x}\right )}{a^2 \sqrt{-a^2+b^2} d^4}-\frac{(3360 b) \operatorname{Subst}\left (\int x^3 \text{Li}_4\left (-\frac{2 a e^{c+d x}}{2 b+2 \sqrt{-a^2+b^2}}\right ) \, dx,x,\sqrt{x}\right )}{a^2 \sqrt{-a^2+b^2} d^4}+\frac{\left (420 b^3\right ) \operatorname{Subst}\left (\int x^4 \text{Li}_3\left (-\frac{2 a e^{c+d x}}{2 b-2 \sqrt{-a^2+b^2}}\right ) \, dx,x,\sqrt{x}\right )}{a^2 \left (-a^2+b^2\right )^{3/2} d^3}-\frac{\left (420 b^3\right ) \operatorname{Subst}\left (\int x^4 \text{Li}_3\left (-\frac{2 a e^{c+d x}}{2 b+2 \sqrt{-a^2+b^2}}\right ) \, dx,x,\sqrt{x}\right )}{a^2 \left (-a^2+b^2\right )^{3/2} d^3}\\ &=\frac{2 b^2 x^{7/2}}{a^2 \left (a^2-b^2\right ) d}+\frac{x^4}{4 a^2}-\frac{14 b^2 x^3 \log \left (1+\frac{a e^{c+d \sqrt{x}}}{b-\sqrt{-a^2+b^2}}\right )}{a^2 \left (a^2-b^2\right ) d^2}+\frac{2 b^3 x^{7/2} \log \left (1+\frac{a e^{c+d \sqrt{x}}}{b-\sqrt{-a^2+b^2}}\right )}{a^2 \left (-a^2+b^2\right )^{3/2} d}-\frac{4 b x^{7/2} \log \left (1+\frac{a e^{c+d \sqrt{x}}}{b-\sqrt{-a^2+b^2}}\right )}{a^2 \sqrt{-a^2+b^2} d}-\frac{14 b^2 x^3 \log \left (1+\frac{a e^{c+d \sqrt{x}}}{b+\sqrt{-a^2+b^2}}\right )}{a^2 \left (a^2-b^2\right ) d^2}-\frac{2 b^3 x^{7/2} \log \left (1+\frac{a e^{c+d \sqrt{x}}}{b+\sqrt{-a^2+b^2}}\right )}{a^2 \left (-a^2+b^2\right )^{3/2} d}+\frac{4 b x^{7/2} \log \left (1+\frac{a e^{c+d \sqrt{x}}}{b+\sqrt{-a^2+b^2}}\right )}{a^2 \sqrt{-a^2+b^2} d}-\frac{84 b^2 x^{5/2} \text{Li}_2\left (-\frac{a e^{c+d \sqrt{x}}}{b-\sqrt{-a^2+b^2}}\right )}{a^2 \left (a^2-b^2\right ) d^3}+\frac{14 b^3 x^3 \text{Li}_2\left (-\frac{a e^{c+d \sqrt{x}}}{b-\sqrt{-a^2+b^2}}\right )}{a^2 \left (-a^2+b^2\right )^{3/2} d^2}-\frac{28 b x^3 \text{Li}_2\left (-\frac{a e^{c+d \sqrt{x}}}{b-\sqrt{-a^2+b^2}}\right )}{a^2 \sqrt{-a^2+b^2} d^2}-\frac{84 b^2 x^{5/2} \text{Li}_2\left (-\frac{a e^{c+d \sqrt{x}}}{b+\sqrt{-a^2+b^2}}\right )}{a^2 \left (a^2-b^2\right ) d^3}-\frac{14 b^3 x^3 \text{Li}_2\left (-\frac{a e^{c+d \sqrt{x}}}{b+\sqrt{-a^2+b^2}}\right )}{a^2 \left (-a^2+b^2\right )^{3/2} d^2}+\frac{28 b x^3 \text{Li}_2\left (-\frac{a e^{c+d \sqrt{x}}}{b+\sqrt{-a^2+b^2}}\right )}{a^2 \sqrt{-a^2+b^2} d^2}+\frac{420 b^2 x^2 \text{Li}_3\left (-\frac{a e^{c+d \sqrt{x}}}{b-\sqrt{-a^2+b^2}}\right )}{a^2 \left (a^2-b^2\right ) d^4}-\frac{84 b^3 x^{5/2} \text{Li}_3\left (-\frac{a e^{c+d \sqrt{x}}}{b-\sqrt{-a^2+b^2}}\right )}{a^2 \left (-a^2+b^2\right )^{3/2} d^3}+\frac{168 b x^{5/2} \text{Li}_3\left (-\frac{a e^{c+d \sqrt{x}}}{b-\sqrt{-a^2+b^2}}\right )}{a^2 \sqrt{-a^2+b^2} d^3}+\frac{420 b^2 x^2 \text{Li}_3\left (-\frac{a e^{c+d \sqrt{x}}}{b+\sqrt{-a^2+b^2}}\right )}{a^2 \left (a^2-b^2\right ) d^4}+\frac{84 b^3 x^{5/2} \text{Li}_3\left (-\frac{a e^{c+d \sqrt{x}}}{b+\sqrt{-a^2+b^2}}\right )}{a^2 \left (-a^2+b^2\right )^{3/2} d^3}-\frac{168 b x^{5/2} \text{Li}_3\left (-\frac{a e^{c+d \sqrt{x}}}{b+\sqrt{-a^2+b^2}}\right )}{a^2 \sqrt{-a^2+b^2} d^3}-\frac{1680 b^2 x^{3/2} \text{Li}_4\left (-\frac{a e^{c+d \sqrt{x}}}{b-\sqrt{-a^2+b^2}}\right )}{a^2 \left (a^2-b^2\right ) d^5}+\frac{420 b^3 x^2 \text{Li}_4\left (-\frac{a e^{c+d \sqrt{x}}}{b-\sqrt{-a^2+b^2}}\right )}{a^2 \left (-a^2+b^2\right )^{3/2} d^4}-\frac{840 b x^2 \text{Li}_4\left (-\frac{a e^{c+d \sqrt{x}}}{b-\sqrt{-a^2+b^2}}\right )}{a^2 \sqrt{-a^2+b^2} d^4}-\frac{1680 b^2 x^{3/2} \text{Li}_4\left (-\frac{a e^{c+d \sqrt{x}}}{b+\sqrt{-a^2+b^2}}\right )}{a^2 \left (a^2-b^2\right ) d^5}-\frac{420 b^3 x^2 \text{Li}_4\left (-\frac{a e^{c+d \sqrt{x}}}{b+\sqrt{-a^2+b^2}}\right )}{a^2 \left (-a^2+b^2\right )^{3/2} d^4}+\frac{840 b x^2 \text{Li}_4\left (-\frac{a e^{c+d \sqrt{x}}}{b+\sqrt{-a^2+b^2}}\right )}{a^2 \sqrt{-a^2+b^2} d^4}+\frac{5040 b^2 x \text{Li}_5\left (-\frac{a e^{c+d \sqrt{x}}}{b-\sqrt{-a^2+b^2}}\right )}{a^2 \left (a^2-b^2\right ) d^6}+\frac{3360 b x^{3/2} \text{Li}_5\left (-\frac{a e^{c+d \sqrt{x}}}{b-\sqrt{-a^2+b^2}}\right )}{a^2 \sqrt{-a^2+b^2} d^5}+\frac{5040 b^2 x \text{Li}_5\left (-\frac{a e^{c+d \sqrt{x}}}{b+\sqrt{-a^2+b^2}}\right )}{a^2 \left (a^2-b^2\right ) d^6}-\frac{3360 b x^{3/2} \text{Li}_5\left (-\frac{a e^{c+d \sqrt{x}}}{b+\sqrt{-a^2+b^2}}\right )}{a^2 \sqrt{-a^2+b^2} d^5}+\frac{2 b^2 x^{7/2} \sinh \left (c+d \sqrt{x}\right )}{a \left (a^2-b^2\right ) d \left (b+a \cosh \left (c+d \sqrt{x}\right )\right )}-\frac{\left (10080 b^2\right ) \operatorname{Subst}\left (\int x \text{Li}_5\left (-\frac{a e^{c+d x}}{b-\sqrt{-a^2+b^2}}\right ) \, dx,x,\sqrt{x}\right )}{a^2 \left (a^2-b^2\right ) d^6}-\frac{\left (10080 b^2\right ) \operatorname{Subst}\left (\int x \text{Li}_5\left (-\frac{a e^{c+d x}}{b+\sqrt{-a^2+b^2}}\right ) \, dx,x,\sqrt{x}\right )}{a^2 \left (a^2-b^2\right ) d^6}-\frac{(10080 b) \operatorname{Subst}\left (\int x^2 \text{Li}_5\left (-\frac{2 a e^{c+d x}}{2 b-2 \sqrt{-a^2+b^2}}\right ) \, dx,x,\sqrt{x}\right )}{a^2 \sqrt{-a^2+b^2} d^5}+\frac{(10080 b) \operatorname{Subst}\left (\int x^2 \text{Li}_5\left (-\frac{2 a e^{c+d x}}{2 b+2 \sqrt{-a^2+b^2}}\right ) \, dx,x,\sqrt{x}\right )}{a^2 \sqrt{-a^2+b^2} d^5}-\frac{\left (1680 b^3\right ) \operatorname{Subst}\left (\int x^3 \text{Li}_4\left (-\frac{2 a e^{c+d x}}{2 b-2 \sqrt{-a^2+b^2}}\right ) \, dx,x,\sqrt{x}\right )}{a^2 \left (-a^2+b^2\right )^{3/2} d^4}+\frac{\left (1680 b^3\right ) \operatorname{Subst}\left (\int x^3 \text{Li}_4\left (-\frac{2 a e^{c+d x}}{2 b+2 \sqrt{-a^2+b^2}}\right ) \, dx,x,\sqrt{x}\right )}{a^2 \left (-a^2+b^2\right )^{3/2} d^4}\\ &=\frac{2 b^2 x^{7/2}}{a^2 \left (a^2-b^2\right ) d}+\frac{x^4}{4 a^2}-\frac{14 b^2 x^3 \log \left (1+\frac{a e^{c+d \sqrt{x}}}{b-\sqrt{-a^2+b^2}}\right )}{a^2 \left (a^2-b^2\right ) d^2}+\frac{2 b^3 x^{7/2} \log \left (1+\frac{a e^{c+d \sqrt{x}}}{b-\sqrt{-a^2+b^2}}\right )}{a^2 \left (-a^2+b^2\right )^{3/2} d}-\frac{4 b x^{7/2} \log \left (1+\frac{a e^{c+d \sqrt{x}}}{b-\sqrt{-a^2+b^2}}\right )}{a^2 \sqrt{-a^2+b^2} d}-\frac{14 b^2 x^3 \log \left (1+\frac{a e^{c+d \sqrt{x}}}{b+\sqrt{-a^2+b^2}}\right )}{a^2 \left (a^2-b^2\right ) d^2}-\frac{2 b^3 x^{7/2} \log \left (1+\frac{a e^{c+d \sqrt{x}}}{b+\sqrt{-a^2+b^2}}\right )}{a^2 \left (-a^2+b^2\right )^{3/2} d}+\frac{4 b x^{7/2} \log \left (1+\frac{a e^{c+d \sqrt{x}}}{b+\sqrt{-a^2+b^2}}\right )}{a^2 \sqrt{-a^2+b^2} d}-\frac{84 b^2 x^{5/2} \text{Li}_2\left (-\frac{a e^{c+d \sqrt{x}}}{b-\sqrt{-a^2+b^2}}\right )}{a^2 \left (a^2-b^2\right ) d^3}+\frac{14 b^3 x^3 \text{Li}_2\left (-\frac{a e^{c+d \sqrt{x}}}{b-\sqrt{-a^2+b^2}}\right )}{a^2 \left (-a^2+b^2\right )^{3/2} d^2}-\frac{28 b x^3 \text{Li}_2\left (-\frac{a e^{c+d \sqrt{x}}}{b-\sqrt{-a^2+b^2}}\right )}{a^2 \sqrt{-a^2+b^2} d^2}-\frac{84 b^2 x^{5/2} \text{Li}_2\left (-\frac{a e^{c+d \sqrt{x}}}{b+\sqrt{-a^2+b^2}}\right )}{a^2 \left (a^2-b^2\right ) d^3}-\frac{14 b^3 x^3 \text{Li}_2\left (-\frac{a e^{c+d \sqrt{x}}}{b+\sqrt{-a^2+b^2}}\right )}{a^2 \left (-a^2+b^2\right )^{3/2} d^2}+\frac{28 b x^3 \text{Li}_2\left (-\frac{a e^{c+d \sqrt{x}}}{b+\sqrt{-a^2+b^2}}\right )}{a^2 \sqrt{-a^2+b^2} d^2}+\frac{420 b^2 x^2 \text{Li}_3\left (-\frac{a e^{c+d \sqrt{x}}}{b-\sqrt{-a^2+b^2}}\right )}{a^2 \left (a^2-b^2\right ) d^4}-\frac{84 b^3 x^{5/2} \text{Li}_3\left (-\frac{a e^{c+d \sqrt{x}}}{b-\sqrt{-a^2+b^2}}\right )}{a^2 \left (-a^2+b^2\right )^{3/2} d^3}+\frac{168 b x^{5/2} \text{Li}_3\left (-\frac{a e^{c+d \sqrt{x}}}{b-\sqrt{-a^2+b^2}}\right )}{a^2 \sqrt{-a^2+b^2} d^3}+\frac{420 b^2 x^2 \text{Li}_3\left (-\frac{a e^{c+d \sqrt{x}}}{b+\sqrt{-a^2+b^2}}\right )}{a^2 \left (a^2-b^2\right ) d^4}+\frac{84 b^3 x^{5/2} \text{Li}_3\left (-\frac{a e^{c+d \sqrt{x}}}{b+\sqrt{-a^2+b^2}}\right )}{a^2 \left (-a^2+b^2\right )^{3/2} d^3}-\frac{168 b x^{5/2} \text{Li}_3\left (-\frac{a e^{c+d \sqrt{x}}}{b+\sqrt{-a^2+b^2}}\right )}{a^2 \sqrt{-a^2+b^2} d^3}-\frac{1680 b^2 x^{3/2} \text{Li}_4\left (-\frac{a e^{c+d \sqrt{x}}}{b-\sqrt{-a^2+b^2}}\right )}{a^2 \left (a^2-b^2\right ) d^5}+\frac{420 b^3 x^2 \text{Li}_4\left (-\frac{a e^{c+d \sqrt{x}}}{b-\sqrt{-a^2+b^2}}\right )}{a^2 \left (-a^2+b^2\right )^{3/2} d^4}-\frac{840 b x^2 \text{Li}_4\left (-\frac{a e^{c+d \sqrt{x}}}{b-\sqrt{-a^2+b^2}}\right )}{a^2 \sqrt{-a^2+b^2} d^4}-\frac{1680 b^2 x^{3/2} \text{Li}_4\left (-\frac{a e^{c+d \sqrt{x}}}{b+\sqrt{-a^2+b^2}}\right )}{a^2 \left (a^2-b^2\right ) d^5}-\frac{420 b^3 x^2 \text{Li}_4\left (-\frac{a e^{c+d \sqrt{x}}}{b+\sqrt{-a^2+b^2}}\right )}{a^2 \left (-a^2+b^2\right )^{3/2} d^4}+\frac{840 b x^2 \text{Li}_4\left (-\frac{a e^{c+d \sqrt{x}}}{b+\sqrt{-a^2+b^2}}\right )}{a^2 \sqrt{-a^2+b^2} d^4}+\frac{5040 b^2 x \text{Li}_5\left (-\frac{a e^{c+d \sqrt{x}}}{b-\sqrt{-a^2+b^2}}\right )}{a^2 \left (a^2-b^2\right ) d^6}-\frac{1680 b^3 x^{3/2} \text{Li}_5\left (-\frac{a e^{c+d \sqrt{x}}}{b-\sqrt{-a^2+b^2}}\right )}{a^2 \left (-a^2+b^2\right )^{3/2} d^5}+\frac{3360 b x^{3/2} \text{Li}_5\left (-\frac{a e^{c+d \sqrt{x}}}{b-\sqrt{-a^2+b^2}}\right )}{a^2 \sqrt{-a^2+b^2} d^5}+\frac{5040 b^2 x \text{Li}_5\left (-\frac{a e^{c+d \sqrt{x}}}{b+\sqrt{-a^2+b^2}}\right )}{a^2 \left (a^2-b^2\right ) d^6}+\frac{1680 b^3 x^{3/2} \text{Li}_5\left (-\frac{a e^{c+d \sqrt{x}}}{b+\sqrt{-a^2+b^2}}\right )}{a^2 \left (-a^2+b^2\right )^{3/2} d^5}-\frac{3360 b x^{3/2} \text{Li}_5\left (-\frac{a e^{c+d \sqrt{x}}}{b+\sqrt{-a^2+b^2}}\right )}{a^2 \sqrt{-a^2+b^2} d^5}-\frac{10080 b^2 \sqrt{x} \text{Li}_6\left (-\frac{a e^{c+d \sqrt{x}}}{b-\sqrt{-a^2+b^2}}\right )}{a^2 \left (a^2-b^2\right ) d^7}-\frac{10080 b x \text{Li}_6\left (-\frac{a e^{c+d \sqrt{x}}}{b-\sqrt{-a^2+b^2}}\right )}{a^2 \sqrt{-a^2+b^2} d^6}-\frac{10080 b^2 \sqrt{x} \text{Li}_6\left (-\frac{a e^{c+d \sqrt{x}}}{b+\sqrt{-a^2+b^2}}\right )}{a^2 \left (a^2-b^2\right ) d^7}+\frac{10080 b x \text{Li}_6\left (-\frac{a e^{c+d \sqrt{x}}}{b+\sqrt{-a^2+b^2}}\right )}{a^2 \sqrt{-a^2+b^2} d^6}+\frac{2 b^2 x^{7/2} \sinh \left (c+d \sqrt{x}\right )}{a \left (a^2-b^2\right ) d \left (b+a \cosh \left (c+d \sqrt{x}\right )\right )}+\frac{\left (10080 b^2\right ) \operatorname{Subst}\left (\int \text{Li}_6\left (-\frac{a e^{c+d x}}{b-\sqrt{-a^2+b^2}}\right ) \, dx,x,\sqrt{x}\right )}{a^2 \left (a^2-b^2\right ) d^7}+\frac{\left (10080 b^2\right ) \operatorname{Subst}\left (\int \text{Li}_6\left (-\frac{a e^{c+d x}}{b+\sqrt{-a^2+b^2}}\right ) \, dx,x,\sqrt{x}\right )}{a^2 \left (a^2-b^2\right ) d^7}+\frac{(20160 b) \operatorname{Subst}\left (\int x \text{Li}_6\left (-\frac{2 a e^{c+d x}}{2 b-2 \sqrt{-a^2+b^2}}\right ) \, dx,x,\sqrt{x}\right )}{a^2 \sqrt{-a^2+b^2} d^6}-\frac{(20160 b) \operatorname{Subst}\left (\int x \text{Li}_6\left (-\frac{2 a e^{c+d x}}{2 b+2 \sqrt{-a^2+b^2}}\right ) \, dx,x,\sqrt{x}\right )}{a^2 \sqrt{-a^2+b^2} d^6}+\frac{\left (5040 b^3\right ) \operatorname{Subst}\left (\int x^2 \text{Li}_5\left (-\frac{2 a e^{c+d x}}{2 b-2 \sqrt{-a^2+b^2}}\right ) \, dx,x,\sqrt{x}\right )}{a^2 \left (-a^2+b^2\right )^{3/2} d^5}-\frac{\left (5040 b^3\right ) \operatorname{Subst}\left (\int x^2 \text{Li}_5\left (-\frac{2 a e^{c+d x}}{2 b+2 \sqrt{-a^2+b^2}}\right ) \, dx,x,\sqrt{x}\right )}{a^2 \left (-a^2+b^2\right )^{3/2} d^5}\\ &=\frac{2 b^2 x^{7/2}}{a^2 \left (a^2-b^2\right ) d}+\frac{x^4}{4 a^2}-\frac{14 b^2 x^3 \log \left (1+\frac{a e^{c+d \sqrt{x}}}{b-\sqrt{-a^2+b^2}}\right )}{a^2 \left (a^2-b^2\right ) d^2}+\frac{2 b^3 x^{7/2} \log \left (1+\frac{a e^{c+d \sqrt{x}}}{b-\sqrt{-a^2+b^2}}\right )}{a^2 \left (-a^2+b^2\right )^{3/2} d}-\frac{4 b x^{7/2} \log \left (1+\frac{a e^{c+d \sqrt{x}}}{b-\sqrt{-a^2+b^2}}\right )}{a^2 \sqrt{-a^2+b^2} d}-\frac{14 b^2 x^3 \log \left (1+\frac{a e^{c+d \sqrt{x}}}{b+\sqrt{-a^2+b^2}}\right )}{a^2 \left (a^2-b^2\right ) d^2}-\frac{2 b^3 x^{7/2} \log \left (1+\frac{a e^{c+d \sqrt{x}}}{b+\sqrt{-a^2+b^2}}\right )}{a^2 \left (-a^2+b^2\right )^{3/2} d}+\frac{4 b x^{7/2} \log \left (1+\frac{a e^{c+d \sqrt{x}}}{b+\sqrt{-a^2+b^2}}\right )}{a^2 \sqrt{-a^2+b^2} d}-\frac{84 b^2 x^{5/2} \text{Li}_2\left (-\frac{a e^{c+d \sqrt{x}}}{b-\sqrt{-a^2+b^2}}\right )}{a^2 \left (a^2-b^2\right ) d^3}+\frac{14 b^3 x^3 \text{Li}_2\left (-\frac{a e^{c+d \sqrt{x}}}{b-\sqrt{-a^2+b^2}}\right )}{a^2 \left (-a^2+b^2\right )^{3/2} d^2}-\frac{28 b x^3 \text{Li}_2\left (-\frac{a e^{c+d \sqrt{x}}}{b-\sqrt{-a^2+b^2}}\right )}{a^2 \sqrt{-a^2+b^2} d^2}-\frac{84 b^2 x^{5/2} \text{Li}_2\left (-\frac{a e^{c+d \sqrt{x}}}{b+\sqrt{-a^2+b^2}}\right )}{a^2 \left (a^2-b^2\right ) d^3}-\frac{14 b^3 x^3 \text{Li}_2\left (-\frac{a e^{c+d \sqrt{x}}}{b+\sqrt{-a^2+b^2}}\right )}{a^2 \left (-a^2+b^2\right )^{3/2} d^2}+\frac{28 b x^3 \text{Li}_2\left (-\frac{a e^{c+d \sqrt{x}}}{b+\sqrt{-a^2+b^2}}\right )}{a^2 \sqrt{-a^2+b^2} d^2}+\frac{420 b^2 x^2 \text{Li}_3\left (-\frac{a e^{c+d \sqrt{x}}}{b-\sqrt{-a^2+b^2}}\right )}{a^2 \left (a^2-b^2\right ) d^4}-\frac{84 b^3 x^{5/2} \text{Li}_3\left (-\frac{a e^{c+d \sqrt{x}}}{b-\sqrt{-a^2+b^2}}\right )}{a^2 \left (-a^2+b^2\right )^{3/2} d^3}+\frac{168 b x^{5/2} \text{Li}_3\left (-\frac{a e^{c+d \sqrt{x}}}{b-\sqrt{-a^2+b^2}}\right )}{a^2 \sqrt{-a^2+b^2} d^3}+\frac{420 b^2 x^2 \text{Li}_3\left (-\frac{a e^{c+d \sqrt{x}}}{b+\sqrt{-a^2+b^2}}\right )}{a^2 \left (a^2-b^2\right ) d^4}+\frac{84 b^3 x^{5/2} \text{Li}_3\left (-\frac{a e^{c+d \sqrt{x}}}{b+\sqrt{-a^2+b^2}}\right )}{a^2 \left (-a^2+b^2\right )^{3/2} d^3}-\frac{168 b x^{5/2} \text{Li}_3\left (-\frac{a e^{c+d \sqrt{x}}}{b+\sqrt{-a^2+b^2}}\right )}{a^2 \sqrt{-a^2+b^2} d^3}-\frac{1680 b^2 x^{3/2} \text{Li}_4\left (-\frac{a e^{c+d \sqrt{x}}}{b-\sqrt{-a^2+b^2}}\right )}{a^2 \left (a^2-b^2\right ) d^5}+\frac{420 b^3 x^2 \text{Li}_4\left (-\frac{a e^{c+d \sqrt{x}}}{b-\sqrt{-a^2+b^2}}\right )}{a^2 \left (-a^2+b^2\right )^{3/2} d^4}-\frac{840 b x^2 \text{Li}_4\left (-\frac{a e^{c+d \sqrt{x}}}{b-\sqrt{-a^2+b^2}}\right )}{a^2 \sqrt{-a^2+b^2} d^4}-\frac{1680 b^2 x^{3/2} \text{Li}_4\left (-\frac{a e^{c+d \sqrt{x}}}{b+\sqrt{-a^2+b^2}}\right )}{a^2 \left (a^2-b^2\right ) d^5}-\frac{420 b^3 x^2 \text{Li}_4\left (-\frac{a e^{c+d \sqrt{x}}}{b+\sqrt{-a^2+b^2}}\right )}{a^2 \left (-a^2+b^2\right )^{3/2} d^4}+\frac{840 b x^2 \text{Li}_4\left (-\frac{a e^{c+d \sqrt{x}}}{b+\sqrt{-a^2+b^2}}\right )}{a^2 \sqrt{-a^2+b^2} d^4}+\frac{5040 b^2 x \text{Li}_5\left (-\frac{a e^{c+d \sqrt{x}}}{b-\sqrt{-a^2+b^2}}\right )}{a^2 \left (a^2-b^2\right ) d^6}-\frac{1680 b^3 x^{3/2} \text{Li}_5\left (-\frac{a e^{c+d \sqrt{x}}}{b-\sqrt{-a^2+b^2}}\right )}{a^2 \left (-a^2+b^2\right )^{3/2} d^5}+\frac{3360 b x^{3/2} \text{Li}_5\left (-\frac{a e^{c+d \sqrt{x}}}{b-\sqrt{-a^2+b^2}}\right )}{a^2 \sqrt{-a^2+b^2} d^5}+\frac{5040 b^2 x \text{Li}_5\left (-\frac{a e^{c+d \sqrt{x}}}{b+\sqrt{-a^2+b^2}}\right )}{a^2 \left (a^2-b^2\right ) d^6}+\frac{1680 b^3 x^{3/2} \text{Li}_5\left (-\frac{a e^{c+d \sqrt{x}}}{b+\sqrt{-a^2+b^2}}\right )}{a^2 \left (-a^2+b^2\right )^{3/2} d^5}-\frac{3360 b x^{3/2} \text{Li}_5\left (-\frac{a e^{c+d \sqrt{x}}}{b+\sqrt{-a^2+b^2}}\right )}{a^2 \sqrt{-a^2+b^2} d^5}-\frac{10080 b^2 \sqrt{x} \text{Li}_6\left (-\frac{a e^{c+d \sqrt{x}}}{b-\sqrt{-a^2+b^2}}\right )}{a^2 \left (a^2-b^2\right ) d^7}+\frac{5040 b^3 x \text{Li}_6\left (-\frac{a e^{c+d \sqrt{x}}}{b-\sqrt{-a^2+b^2}}\right )}{a^2 \left (-a^2+b^2\right )^{3/2} d^6}-\frac{10080 b x \text{Li}_6\left (-\frac{a e^{c+d \sqrt{x}}}{b-\sqrt{-a^2+b^2}}\right )}{a^2 \sqrt{-a^2+b^2} d^6}-\frac{10080 b^2 \sqrt{x} \text{Li}_6\left (-\frac{a e^{c+d \sqrt{x}}}{b+\sqrt{-a^2+b^2}}\right )}{a^2 \left (a^2-b^2\right ) d^7}-\frac{5040 b^3 x \text{Li}_6\left (-\frac{a e^{c+d \sqrt{x}}}{b+\sqrt{-a^2+b^2}}\right )}{a^2 \left (-a^2+b^2\right )^{3/2} d^6}+\frac{10080 b x \text{Li}_6\left (-\frac{a e^{c+d \sqrt{x}}}{b+\sqrt{-a^2+b^2}}\right )}{a^2 \sqrt{-a^2+b^2} d^6}+\frac{20160 b \sqrt{x} \text{Li}_7\left (-\frac{a e^{c+d \sqrt{x}}}{b-\sqrt{-a^2+b^2}}\right )}{a^2 \sqrt{-a^2+b^2} d^7}-\frac{20160 b \sqrt{x} \text{Li}_7\left (-\frac{a e^{c+d \sqrt{x}}}{b+\sqrt{-a^2+b^2}}\right )}{a^2 \sqrt{-a^2+b^2} d^7}+\frac{2 b^2 x^{7/2} \sinh \left (c+d \sqrt{x}\right )}{a \left (a^2-b^2\right ) d \left (b+a \cosh \left (c+d \sqrt{x}\right )\right )}+\frac{\left (10080 b^2\right ) \operatorname{Subst}\left (\int \frac{\text{Li}_6\left (\frac{a x}{-b+\sqrt{-a^2+b^2}}\right )}{x} \, dx,x,e^{c+d \sqrt{x}}\right )}{a^2 \left (a^2-b^2\right ) d^8}+\frac{\left (10080 b^2\right ) \operatorname{Subst}\left (\int \frac{\text{Li}_6\left (-\frac{a x}{b+\sqrt{-a^2+b^2}}\right )}{x} \, dx,x,e^{c+d \sqrt{x}}\right )}{a^2 \left (a^2-b^2\right ) d^8}-\frac{(20160 b) \operatorname{Subst}\left (\int \text{Li}_7\left (-\frac{2 a e^{c+d x}}{2 b-2 \sqrt{-a^2+b^2}}\right ) \, dx,x,\sqrt{x}\right )}{a^2 \sqrt{-a^2+b^2} d^7}+\frac{(20160 b) \operatorname{Subst}\left (\int \text{Li}_7\left (-\frac{2 a e^{c+d x}}{2 b+2 \sqrt{-a^2+b^2}}\right ) \, dx,x,\sqrt{x}\right )}{a^2 \sqrt{-a^2+b^2} d^7}-\frac{\left (10080 b^3\right ) \operatorname{Subst}\left (\int x \text{Li}_6\left (-\frac{2 a e^{c+d x}}{2 b-2 \sqrt{-a^2+b^2}}\right ) \, dx,x,\sqrt{x}\right )}{a^2 \left (-a^2+b^2\right )^{3/2} d^6}+\frac{\left (10080 b^3\right ) \operatorname{Subst}\left (\int x \text{Li}_6\left (-\frac{2 a e^{c+d x}}{2 b+2 \sqrt{-a^2+b^2}}\right ) \, dx,x,\sqrt{x}\right )}{a^2 \left (-a^2+b^2\right )^{3/2} d^6}\\ &=\frac{2 b^2 x^{7/2}}{a^2 \left (a^2-b^2\right ) d}+\frac{x^4}{4 a^2}-\frac{14 b^2 x^3 \log \left (1+\frac{a e^{c+d \sqrt{x}}}{b-\sqrt{-a^2+b^2}}\right )}{a^2 \left (a^2-b^2\right ) d^2}+\frac{2 b^3 x^{7/2} \log \left (1+\frac{a e^{c+d \sqrt{x}}}{b-\sqrt{-a^2+b^2}}\right )}{a^2 \left (-a^2+b^2\right )^{3/2} d}-\frac{4 b x^{7/2} \log \left (1+\frac{a e^{c+d \sqrt{x}}}{b-\sqrt{-a^2+b^2}}\right )}{a^2 \sqrt{-a^2+b^2} d}-\frac{14 b^2 x^3 \log \left (1+\frac{a e^{c+d \sqrt{x}}}{b+\sqrt{-a^2+b^2}}\right )}{a^2 \left (a^2-b^2\right ) d^2}-\frac{2 b^3 x^{7/2} \log \left (1+\frac{a e^{c+d \sqrt{x}}}{b+\sqrt{-a^2+b^2}}\right )}{a^2 \left (-a^2+b^2\right )^{3/2} d}+\frac{4 b x^{7/2} \log \left (1+\frac{a e^{c+d \sqrt{x}}}{b+\sqrt{-a^2+b^2}}\right )}{a^2 \sqrt{-a^2+b^2} d}-\frac{84 b^2 x^{5/2} \text{Li}_2\left (-\frac{a e^{c+d \sqrt{x}}}{b-\sqrt{-a^2+b^2}}\right )}{a^2 \left (a^2-b^2\right ) d^3}+\frac{14 b^3 x^3 \text{Li}_2\left (-\frac{a e^{c+d \sqrt{x}}}{b-\sqrt{-a^2+b^2}}\right )}{a^2 \left (-a^2+b^2\right )^{3/2} d^2}-\frac{28 b x^3 \text{Li}_2\left (-\frac{a e^{c+d \sqrt{x}}}{b-\sqrt{-a^2+b^2}}\right )}{a^2 \sqrt{-a^2+b^2} d^2}-\frac{84 b^2 x^{5/2} \text{Li}_2\left (-\frac{a e^{c+d \sqrt{x}}}{b+\sqrt{-a^2+b^2}}\right )}{a^2 \left (a^2-b^2\right ) d^3}-\frac{14 b^3 x^3 \text{Li}_2\left (-\frac{a e^{c+d \sqrt{x}}}{b+\sqrt{-a^2+b^2}}\right )}{a^2 \left (-a^2+b^2\right )^{3/2} d^2}+\frac{28 b x^3 \text{Li}_2\left (-\frac{a e^{c+d \sqrt{x}}}{b+\sqrt{-a^2+b^2}}\right )}{a^2 \sqrt{-a^2+b^2} d^2}+\frac{420 b^2 x^2 \text{Li}_3\left (-\frac{a e^{c+d \sqrt{x}}}{b-\sqrt{-a^2+b^2}}\right )}{a^2 \left (a^2-b^2\right ) d^4}-\frac{84 b^3 x^{5/2} \text{Li}_3\left (-\frac{a e^{c+d \sqrt{x}}}{b-\sqrt{-a^2+b^2}}\right )}{a^2 \left (-a^2+b^2\right )^{3/2} d^3}+\frac{168 b x^{5/2} \text{Li}_3\left (-\frac{a e^{c+d \sqrt{x}}}{b-\sqrt{-a^2+b^2}}\right )}{a^2 \sqrt{-a^2+b^2} d^3}+\frac{420 b^2 x^2 \text{Li}_3\left (-\frac{a e^{c+d \sqrt{x}}}{b+\sqrt{-a^2+b^2}}\right )}{a^2 \left (a^2-b^2\right ) d^4}+\frac{84 b^3 x^{5/2} \text{Li}_3\left (-\frac{a e^{c+d \sqrt{x}}}{b+\sqrt{-a^2+b^2}}\right )}{a^2 \left (-a^2+b^2\right )^{3/2} d^3}-\frac{168 b x^{5/2} \text{Li}_3\left (-\frac{a e^{c+d \sqrt{x}}}{b+\sqrt{-a^2+b^2}}\right )}{a^2 \sqrt{-a^2+b^2} d^3}-\frac{1680 b^2 x^{3/2} \text{Li}_4\left (-\frac{a e^{c+d \sqrt{x}}}{b-\sqrt{-a^2+b^2}}\right )}{a^2 \left (a^2-b^2\right ) d^5}+\frac{420 b^3 x^2 \text{Li}_4\left (-\frac{a e^{c+d \sqrt{x}}}{b-\sqrt{-a^2+b^2}}\right )}{a^2 \left (-a^2+b^2\right )^{3/2} d^4}-\frac{840 b x^2 \text{Li}_4\left (-\frac{a e^{c+d \sqrt{x}}}{b-\sqrt{-a^2+b^2}}\right )}{a^2 \sqrt{-a^2+b^2} d^4}-\frac{1680 b^2 x^{3/2} \text{Li}_4\left (-\frac{a e^{c+d \sqrt{x}}}{b+\sqrt{-a^2+b^2}}\right )}{a^2 \left (a^2-b^2\right ) d^5}-\frac{420 b^3 x^2 \text{Li}_4\left (-\frac{a e^{c+d \sqrt{x}}}{b+\sqrt{-a^2+b^2}}\right )}{a^2 \left (-a^2+b^2\right )^{3/2} d^4}+\frac{840 b x^2 \text{Li}_4\left (-\frac{a e^{c+d \sqrt{x}}}{b+\sqrt{-a^2+b^2}}\right )}{a^2 \sqrt{-a^2+b^2} d^4}+\frac{5040 b^2 x \text{Li}_5\left (-\frac{a e^{c+d \sqrt{x}}}{b-\sqrt{-a^2+b^2}}\right )}{a^2 \left (a^2-b^2\right ) d^6}-\frac{1680 b^3 x^{3/2} \text{Li}_5\left (-\frac{a e^{c+d \sqrt{x}}}{b-\sqrt{-a^2+b^2}}\right )}{a^2 \left (-a^2+b^2\right )^{3/2} d^5}+\frac{3360 b x^{3/2} \text{Li}_5\left (-\frac{a e^{c+d \sqrt{x}}}{b-\sqrt{-a^2+b^2}}\right )}{a^2 \sqrt{-a^2+b^2} d^5}+\frac{5040 b^2 x \text{Li}_5\left (-\frac{a e^{c+d \sqrt{x}}}{b+\sqrt{-a^2+b^2}}\right )}{a^2 \left (a^2-b^2\right ) d^6}+\frac{1680 b^3 x^{3/2} \text{Li}_5\left (-\frac{a e^{c+d \sqrt{x}}}{b+\sqrt{-a^2+b^2}}\right )}{a^2 \left (-a^2+b^2\right )^{3/2} d^5}-\frac{3360 b x^{3/2} \text{Li}_5\left (-\frac{a e^{c+d \sqrt{x}}}{b+\sqrt{-a^2+b^2}}\right )}{a^2 \sqrt{-a^2+b^2} d^5}-\frac{10080 b^2 \sqrt{x} \text{Li}_6\left (-\frac{a e^{c+d \sqrt{x}}}{b-\sqrt{-a^2+b^2}}\right )}{a^2 \left (a^2-b^2\right ) d^7}+\frac{5040 b^3 x \text{Li}_6\left (-\frac{a e^{c+d \sqrt{x}}}{b-\sqrt{-a^2+b^2}}\right )}{a^2 \left (-a^2+b^2\right )^{3/2} d^6}-\frac{10080 b x \text{Li}_6\left (-\frac{a e^{c+d \sqrt{x}}}{b-\sqrt{-a^2+b^2}}\right )}{a^2 \sqrt{-a^2+b^2} d^6}-\frac{10080 b^2 \sqrt{x} \text{Li}_6\left (-\frac{a e^{c+d \sqrt{x}}}{b+\sqrt{-a^2+b^2}}\right )}{a^2 \left (a^2-b^2\right ) d^7}-\frac{5040 b^3 x \text{Li}_6\left (-\frac{a e^{c+d \sqrt{x}}}{b+\sqrt{-a^2+b^2}}\right )}{a^2 \left (-a^2+b^2\right )^{3/2} d^6}+\frac{10080 b x \text{Li}_6\left (-\frac{a e^{c+d \sqrt{x}}}{b+\sqrt{-a^2+b^2}}\right )}{a^2 \sqrt{-a^2+b^2} d^6}+\frac{10080 b^2 \text{Li}_7\left (-\frac{a e^{c+d \sqrt{x}}}{b-\sqrt{-a^2+b^2}}\right )}{a^2 \left (a^2-b^2\right ) d^8}-\frac{10080 b^3 \sqrt{x} \text{Li}_7\left (-\frac{a e^{c+d \sqrt{x}}}{b-\sqrt{-a^2+b^2}}\right )}{a^2 \left (-a^2+b^2\right )^{3/2} d^7}+\frac{20160 b \sqrt{x} \text{Li}_7\left (-\frac{a e^{c+d \sqrt{x}}}{b-\sqrt{-a^2+b^2}}\right )}{a^2 \sqrt{-a^2+b^2} d^7}+\frac{10080 b^2 \text{Li}_7\left (-\frac{a e^{c+d \sqrt{x}}}{b+\sqrt{-a^2+b^2}}\right )}{a^2 \left (a^2-b^2\right ) d^8}+\frac{10080 b^3 \sqrt{x} \text{Li}_7\left (-\frac{a e^{c+d \sqrt{x}}}{b+\sqrt{-a^2+b^2}}\right )}{a^2 \left (-a^2+b^2\right )^{3/2} d^7}-\frac{20160 b \sqrt{x} \text{Li}_7\left (-\frac{a e^{c+d \sqrt{x}}}{b+\sqrt{-a^2+b^2}}\right )}{a^2 \sqrt{-a^2+b^2} d^7}+\frac{2 b^2 x^{7/2} \sinh \left (c+d \sqrt{x}\right )}{a \left (a^2-b^2\right ) d \left (b+a \cosh \left (c+d \sqrt{x}\right )\right )}-\frac{(20160 b) \operatorname{Subst}\left (\int \frac{\text{Li}_7\left (\frac{a x}{-b+\sqrt{-a^2+b^2}}\right )}{x} \, dx,x,e^{c+d \sqrt{x}}\right )}{a^2 \sqrt{-a^2+b^2} d^8}+\frac{(20160 b) \operatorname{Subst}\left (\int \frac{\text{Li}_7\left (-\frac{a x}{b+\sqrt{-a^2+b^2}}\right )}{x} \, dx,x,e^{c+d \sqrt{x}}\right )}{a^2 \sqrt{-a^2+b^2} d^8}+\frac{\left (10080 b^3\right ) \operatorname{Subst}\left (\int \text{Li}_7\left (-\frac{2 a e^{c+d x}}{2 b-2 \sqrt{-a^2+b^2}}\right ) \, dx,x,\sqrt{x}\right )}{a^2 \left (-a^2+b^2\right )^{3/2} d^7}-\frac{\left (10080 b^3\right ) \operatorname{Subst}\left (\int \text{Li}_7\left (-\frac{2 a e^{c+d x}}{2 b+2 \sqrt{-a^2+b^2}}\right ) \, dx,x,\sqrt{x}\right )}{a^2 \left (-a^2+b^2\right )^{3/2} d^7}\\ &=\frac{2 b^2 x^{7/2}}{a^2 \left (a^2-b^2\right ) d}+\frac{x^4}{4 a^2}-\frac{14 b^2 x^3 \log \left (1+\frac{a e^{c+d \sqrt{x}}}{b-\sqrt{-a^2+b^2}}\right )}{a^2 \left (a^2-b^2\right ) d^2}+\frac{2 b^3 x^{7/2} \log \left (1+\frac{a e^{c+d \sqrt{x}}}{b-\sqrt{-a^2+b^2}}\right )}{a^2 \left (-a^2+b^2\right )^{3/2} d}-\frac{4 b x^{7/2} \log \left (1+\frac{a e^{c+d \sqrt{x}}}{b-\sqrt{-a^2+b^2}}\right )}{a^2 \sqrt{-a^2+b^2} d}-\frac{14 b^2 x^3 \log \left (1+\frac{a e^{c+d \sqrt{x}}}{b+\sqrt{-a^2+b^2}}\right )}{a^2 \left (a^2-b^2\right ) d^2}-\frac{2 b^3 x^{7/2} \log \left (1+\frac{a e^{c+d \sqrt{x}}}{b+\sqrt{-a^2+b^2}}\right )}{a^2 \left (-a^2+b^2\right )^{3/2} d}+\frac{4 b x^{7/2} \log \left (1+\frac{a e^{c+d \sqrt{x}}}{b+\sqrt{-a^2+b^2}}\right )}{a^2 \sqrt{-a^2+b^2} d}-\frac{84 b^2 x^{5/2} \text{Li}_2\left (-\frac{a e^{c+d \sqrt{x}}}{b-\sqrt{-a^2+b^2}}\right )}{a^2 \left (a^2-b^2\right ) d^3}+\frac{14 b^3 x^3 \text{Li}_2\left (-\frac{a e^{c+d \sqrt{x}}}{b-\sqrt{-a^2+b^2}}\right )}{a^2 \left (-a^2+b^2\right )^{3/2} d^2}-\frac{28 b x^3 \text{Li}_2\left (-\frac{a e^{c+d \sqrt{x}}}{b-\sqrt{-a^2+b^2}}\right )}{a^2 \sqrt{-a^2+b^2} d^2}-\frac{84 b^2 x^{5/2} \text{Li}_2\left (-\frac{a e^{c+d \sqrt{x}}}{b+\sqrt{-a^2+b^2}}\right )}{a^2 \left (a^2-b^2\right ) d^3}-\frac{14 b^3 x^3 \text{Li}_2\left (-\frac{a e^{c+d \sqrt{x}}}{b+\sqrt{-a^2+b^2}}\right )}{a^2 \left (-a^2+b^2\right )^{3/2} d^2}+\frac{28 b x^3 \text{Li}_2\left (-\frac{a e^{c+d \sqrt{x}}}{b+\sqrt{-a^2+b^2}}\right )}{a^2 \sqrt{-a^2+b^2} d^2}+\frac{420 b^2 x^2 \text{Li}_3\left (-\frac{a e^{c+d \sqrt{x}}}{b-\sqrt{-a^2+b^2}}\right )}{a^2 \left (a^2-b^2\right ) d^4}-\frac{84 b^3 x^{5/2} \text{Li}_3\left (-\frac{a e^{c+d \sqrt{x}}}{b-\sqrt{-a^2+b^2}}\right )}{a^2 \left (-a^2+b^2\right )^{3/2} d^3}+\frac{168 b x^{5/2} \text{Li}_3\left (-\frac{a e^{c+d \sqrt{x}}}{b-\sqrt{-a^2+b^2}}\right )}{a^2 \sqrt{-a^2+b^2} d^3}+\frac{420 b^2 x^2 \text{Li}_3\left (-\frac{a e^{c+d \sqrt{x}}}{b+\sqrt{-a^2+b^2}}\right )}{a^2 \left (a^2-b^2\right ) d^4}+\frac{84 b^3 x^{5/2} \text{Li}_3\left (-\frac{a e^{c+d \sqrt{x}}}{b+\sqrt{-a^2+b^2}}\right )}{a^2 \left (-a^2+b^2\right )^{3/2} d^3}-\frac{168 b x^{5/2} \text{Li}_3\left (-\frac{a e^{c+d \sqrt{x}}}{b+\sqrt{-a^2+b^2}}\right )}{a^2 \sqrt{-a^2+b^2} d^3}-\frac{1680 b^2 x^{3/2} \text{Li}_4\left (-\frac{a e^{c+d \sqrt{x}}}{b-\sqrt{-a^2+b^2}}\right )}{a^2 \left (a^2-b^2\right ) d^5}+\frac{420 b^3 x^2 \text{Li}_4\left (-\frac{a e^{c+d \sqrt{x}}}{b-\sqrt{-a^2+b^2}}\right )}{a^2 \left (-a^2+b^2\right )^{3/2} d^4}-\frac{840 b x^2 \text{Li}_4\left (-\frac{a e^{c+d \sqrt{x}}}{b-\sqrt{-a^2+b^2}}\right )}{a^2 \sqrt{-a^2+b^2} d^4}-\frac{1680 b^2 x^{3/2} \text{Li}_4\left (-\frac{a e^{c+d \sqrt{x}}}{b+\sqrt{-a^2+b^2}}\right )}{a^2 \left (a^2-b^2\right ) d^5}-\frac{420 b^3 x^2 \text{Li}_4\left (-\frac{a e^{c+d \sqrt{x}}}{b+\sqrt{-a^2+b^2}}\right )}{a^2 \left (-a^2+b^2\right )^{3/2} d^4}+\frac{840 b x^2 \text{Li}_4\left (-\frac{a e^{c+d \sqrt{x}}}{b+\sqrt{-a^2+b^2}}\right )}{a^2 \sqrt{-a^2+b^2} d^4}+\frac{5040 b^2 x \text{Li}_5\left (-\frac{a e^{c+d \sqrt{x}}}{b-\sqrt{-a^2+b^2}}\right )}{a^2 \left (a^2-b^2\right ) d^6}-\frac{1680 b^3 x^{3/2} \text{Li}_5\left (-\frac{a e^{c+d \sqrt{x}}}{b-\sqrt{-a^2+b^2}}\right )}{a^2 \left (-a^2+b^2\right )^{3/2} d^5}+\frac{3360 b x^{3/2} \text{Li}_5\left (-\frac{a e^{c+d \sqrt{x}}}{b-\sqrt{-a^2+b^2}}\right )}{a^2 \sqrt{-a^2+b^2} d^5}+\frac{5040 b^2 x \text{Li}_5\left (-\frac{a e^{c+d \sqrt{x}}}{b+\sqrt{-a^2+b^2}}\right )}{a^2 \left (a^2-b^2\right ) d^6}+\frac{1680 b^3 x^{3/2} \text{Li}_5\left (-\frac{a e^{c+d \sqrt{x}}}{b+\sqrt{-a^2+b^2}}\right )}{a^2 \left (-a^2+b^2\right )^{3/2} d^5}-\frac{3360 b x^{3/2} \text{Li}_5\left (-\frac{a e^{c+d \sqrt{x}}}{b+\sqrt{-a^2+b^2}}\right )}{a^2 \sqrt{-a^2+b^2} d^5}-\frac{10080 b^2 \sqrt{x} \text{Li}_6\left (-\frac{a e^{c+d \sqrt{x}}}{b-\sqrt{-a^2+b^2}}\right )}{a^2 \left (a^2-b^2\right ) d^7}+\frac{5040 b^3 x \text{Li}_6\left (-\frac{a e^{c+d \sqrt{x}}}{b-\sqrt{-a^2+b^2}}\right )}{a^2 \left (-a^2+b^2\right )^{3/2} d^6}-\frac{10080 b x \text{Li}_6\left (-\frac{a e^{c+d \sqrt{x}}}{b-\sqrt{-a^2+b^2}}\right )}{a^2 \sqrt{-a^2+b^2} d^6}-\frac{10080 b^2 \sqrt{x} \text{Li}_6\left (-\frac{a e^{c+d \sqrt{x}}}{b+\sqrt{-a^2+b^2}}\right )}{a^2 \left (a^2-b^2\right ) d^7}-\frac{5040 b^3 x \text{Li}_6\left (-\frac{a e^{c+d \sqrt{x}}}{b+\sqrt{-a^2+b^2}}\right )}{a^2 \left (-a^2+b^2\right )^{3/2} d^6}+\frac{10080 b x \text{Li}_6\left (-\frac{a e^{c+d \sqrt{x}}}{b+\sqrt{-a^2+b^2}}\right )}{a^2 \sqrt{-a^2+b^2} d^6}+\frac{10080 b^2 \text{Li}_7\left (-\frac{a e^{c+d \sqrt{x}}}{b-\sqrt{-a^2+b^2}}\right )}{a^2 \left (a^2-b^2\right ) d^8}-\frac{10080 b^3 \sqrt{x} \text{Li}_7\left (-\frac{a e^{c+d \sqrt{x}}}{b-\sqrt{-a^2+b^2}}\right )}{a^2 \left (-a^2+b^2\right )^{3/2} d^7}+\frac{20160 b \sqrt{x} \text{Li}_7\left (-\frac{a e^{c+d \sqrt{x}}}{b-\sqrt{-a^2+b^2}}\right )}{a^2 \sqrt{-a^2+b^2} d^7}+\frac{10080 b^2 \text{Li}_7\left (-\frac{a e^{c+d \sqrt{x}}}{b+\sqrt{-a^2+b^2}}\right )}{a^2 \left (a^2-b^2\right ) d^8}+\frac{10080 b^3 \sqrt{x} \text{Li}_7\left (-\frac{a e^{c+d \sqrt{x}}}{b+\sqrt{-a^2+b^2}}\right )}{a^2 \left (-a^2+b^2\right )^{3/2} d^7}-\frac{20160 b \sqrt{x} \text{Li}_7\left (-\frac{a e^{c+d \sqrt{x}}}{b+\sqrt{-a^2+b^2}}\right )}{a^2 \sqrt{-a^2+b^2} d^7}-\frac{20160 b \text{Li}_8\left (-\frac{a e^{c+d \sqrt{x}}}{b-\sqrt{-a^2+b^2}}\right )}{a^2 \sqrt{-a^2+b^2} d^8}+\frac{20160 b \text{Li}_8\left (-\frac{a e^{c+d \sqrt{x}}}{b+\sqrt{-a^2+b^2}}\right )}{a^2 \sqrt{-a^2+b^2} d^8}+\frac{2 b^2 x^{7/2} \sinh \left (c+d \sqrt{x}\right )}{a \left (a^2-b^2\right ) d \left (b+a \cosh \left (c+d \sqrt{x}\right )\right )}+\frac{\left (10080 b^3\right ) \operatorname{Subst}\left (\int \frac{\text{Li}_7\left (\frac{a x}{-b+\sqrt{-a^2+b^2}}\right )}{x} \, dx,x,e^{c+d \sqrt{x}}\right )}{a^2 \left (-a^2+b^2\right )^{3/2} d^8}-\frac{\left (10080 b^3\right ) \operatorname{Subst}\left (\int \frac{\text{Li}_7\left (-\frac{a x}{b+\sqrt{-a^2+b^2}}\right )}{x} \, dx,x,e^{c+d \sqrt{x}}\right )}{a^2 \left (-a^2+b^2\right )^{3/2} d^8}\\ &=\frac{2 b^2 x^{7/2}}{a^2 \left (a^2-b^2\right ) d}+\frac{x^4}{4 a^2}-\frac{14 b^2 x^3 \log \left (1+\frac{a e^{c+d \sqrt{x}}}{b-\sqrt{-a^2+b^2}}\right )}{a^2 \left (a^2-b^2\right ) d^2}+\frac{2 b^3 x^{7/2} \log \left (1+\frac{a e^{c+d \sqrt{x}}}{b-\sqrt{-a^2+b^2}}\right )}{a^2 \left (-a^2+b^2\right )^{3/2} d}-\frac{4 b x^{7/2} \log \left (1+\frac{a e^{c+d \sqrt{x}}}{b-\sqrt{-a^2+b^2}}\right )}{a^2 \sqrt{-a^2+b^2} d}-\frac{14 b^2 x^3 \log \left (1+\frac{a e^{c+d \sqrt{x}}}{b+\sqrt{-a^2+b^2}}\right )}{a^2 \left (a^2-b^2\right ) d^2}-\frac{2 b^3 x^{7/2} \log \left (1+\frac{a e^{c+d \sqrt{x}}}{b+\sqrt{-a^2+b^2}}\right )}{a^2 \left (-a^2+b^2\right )^{3/2} d}+\frac{4 b x^{7/2} \log \left (1+\frac{a e^{c+d \sqrt{x}}}{b+\sqrt{-a^2+b^2}}\right )}{a^2 \sqrt{-a^2+b^2} d}-\frac{84 b^2 x^{5/2} \text{Li}_2\left (-\frac{a e^{c+d \sqrt{x}}}{b-\sqrt{-a^2+b^2}}\right )}{a^2 \left (a^2-b^2\right ) d^3}+\frac{14 b^3 x^3 \text{Li}_2\left (-\frac{a e^{c+d \sqrt{x}}}{b-\sqrt{-a^2+b^2}}\right )}{a^2 \left (-a^2+b^2\right )^{3/2} d^2}-\frac{28 b x^3 \text{Li}_2\left (-\frac{a e^{c+d \sqrt{x}}}{b-\sqrt{-a^2+b^2}}\right )}{a^2 \sqrt{-a^2+b^2} d^2}-\frac{84 b^2 x^{5/2} \text{Li}_2\left (-\frac{a e^{c+d \sqrt{x}}}{b+\sqrt{-a^2+b^2}}\right )}{a^2 \left (a^2-b^2\right ) d^3}-\frac{14 b^3 x^3 \text{Li}_2\left (-\frac{a e^{c+d \sqrt{x}}}{b+\sqrt{-a^2+b^2}}\right )}{a^2 \left (-a^2+b^2\right )^{3/2} d^2}+\frac{28 b x^3 \text{Li}_2\left (-\frac{a e^{c+d \sqrt{x}}}{b+\sqrt{-a^2+b^2}}\right )}{a^2 \sqrt{-a^2+b^2} d^2}+\frac{420 b^2 x^2 \text{Li}_3\left (-\frac{a e^{c+d \sqrt{x}}}{b-\sqrt{-a^2+b^2}}\right )}{a^2 \left (a^2-b^2\right ) d^4}-\frac{84 b^3 x^{5/2} \text{Li}_3\left (-\frac{a e^{c+d \sqrt{x}}}{b-\sqrt{-a^2+b^2}}\right )}{a^2 \left (-a^2+b^2\right )^{3/2} d^3}+\frac{168 b x^{5/2} \text{Li}_3\left (-\frac{a e^{c+d \sqrt{x}}}{b-\sqrt{-a^2+b^2}}\right )}{a^2 \sqrt{-a^2+b^2} d^3}+\frac{420 b^2 x^2 \text{Li}_3\left (-\frac{a e^{c+d \sqrt{x}}}{b+\sqrt{-a^2+b^2}}\right )}{a^2 \left (a^2-b^2\right ) d^4}+\frac{84 b^3 x^{5/2} \text{Li}_3\left (-\frac{a e^{c+d \sqrt{x}}}{b+\sqrt{-a^2+b^2}}\right )}{a^2 \left (-a^2+b^2\right )^{3/2} d^3}-\frac{168 b x^{5/2} \text{Li}_3\left (-\frac{a e^{c+d \sqrt{x}}}{b+\sqrt{-a^2+b^2}}\right )}{a^2 \sqrt{-a^2+b^2} d^3}-\frac{1680 b^2 x^{3/2} \text{Li}_4\left (-\frac{a e^{c+d \sqrt{x}}}{b-\sqrt{-a^2+b^2}}\right )}{a^2 \left (a^2-b^2\right ) d^5}+\frac{420 b^3 x^2 \text{Li}_4\left (-\frac{a e^{c+d \sqrt{x}}}{b-\sqrt{-a^2+b^2}}\right )}{a^2 \left (-a^2+b^2\right )^{3/2} d^4}-\frac{840 b x^2 \text{Li}_4\left (-\frac{a e^{c+d \sqrt{x}}}{b-\sqrt{-a^2+b^2}}\right )}{a^2 \sqrt{-a^2+b^2} d^4}-\frac{1680 b^2 x^{3/2} \text{Li}_4\left (-\frac{a e^{c+d \sqrt{x}}}{b+\sqrt{-a^2+b^2}}\right )}{a^2 \left (a^2-b^2\right ) d^5}-\frac{420 b^3 x^2 \text{Li}_4\left (-\frac{a e^{c+d \sqrt{x}}}{b+\sqrt{-a^2+b^2}}\right )}{a^2 \left (-a^2+b^2\right )^{3/2} d^4}+\frac{840 b x^2 \text{Li}_4\left (-\frac{a e^{c+d \sqrt{x}}}{b+\sqrt{-a^2+b^2}}\right )}{a^2 \sqrt{-a^2+b^2} d^4}+\frac{5040 b^2 x \text{Li}_5\left (-\frac{a e^{c+d \sqrt{x}}}{b-\sqrt{-a^2+b^2}}\right )}{a^2 \left (a^2-b^2\right ) d^6}-\frac{1680 b^3 x^{3/2} \text{Li}_5\left (-\frac{a e^{c+d \sqrt{x}}}{b-\sqrt{-a^2+b^2}}\right )}{a^2 \left (-a^2+b^2\right )^{3/2} d^5}+\frac{3360 b x^{3/2} \text{Li}_5\left (-\frac{a e^{c+d \sqrt{x}}}{b-\sqrt{-a^2+b^2}}\right )}{a^2 \sqrt{-a^2+b^2} d^5}+\frac{5040 b^2 x \text{Li}_5\left (-\frac{a e^{c+d \sqrt{x}}}{b+\sqrt{-a^2+b^2}}\right )}{a^2 \left (a^2-b^2\right ) d^6}+\frac{1680 b^3 x^{3/2} \text{Li}_5\left (-\frac{a e^{c+d \sqrt{x}}}{b+\sqrt{-a^2+b^2}}\right )}{a^2 \left (-a^2+b^2\right )^{3/2} d^5}-\frac{3360 b x^{3/2} \text{Li}_5\left (-\frac{a e^{c+d \sqrt{x}}}{b+\sqrt{-a^2+b^2}}\right )}{a^2 \sqrt{-a^2+b^2} d^5}-\frac{10080 b^2 \sqrt{x} \text{Li}_6\left (-\frac{a e^{c+d \sqrt{x}}}{b-\sqrt{-a^2+b^2}}\right )}{a^2 \left (a^2-b^2\right ) d^7}+\frac{5040 b^3 x \text{Li}_6\left (-\frac{a e^{c+d \sqrt{x}}}{b-\sqrt{-a^2+b^2}}\right )}{a^2 \left (-a^2+b^2\right )^{3/2} d^6}-\frac{10080 b x \text{Li}_6\left (-\frac{a e^{c+d \sqrt{x}}}{b-\sqrt{-a^2+b^2}}\right )}{a^2 \sqrt{-a^2+b^2} d^6}-\frac{10080 b^2 \sqrt{x} \text{Li}_6\left (-\frac{a e^{c+d \sqrt{x}}}{b+\sqrt{-a^2+b^2}}\right )}{a^2 \left (a^2-b^2\right ) d^7}-\frac{5040 b^3 x \text{Li}_6\left (-\frac{a e^{c+d \sqrt{x}}}{b+\sqrt{-a^2+b^2}}\right )}{a^2 \left (-a^2+b^2\right )^{3/2} d^6}+\frac{10080 b x \text{Li}_6\left (-\frac{a e^{c+d \sqrt{x}}}{b+\sqrt{-a^2+b^2}}\right )}{a^2 \sqrt{-a^2+b^2} d^6}+\frac{10080 b^2 \text{Li}_7\left (-\frac{a e^{c+d \sqrt{x}}}{b-\sqrt{-a^2+b^2}}\right )}{a^2 \left (a^2-b^2\right ) d^8}-\frac{10080 b^3 \sqrt{x} \text{Li}_7\left (-\frac{a e^{c+d \sqrt{x}}}{b-\sqrt{-a^2+b^2}}\right )}{a^2 \left (-a^2+b^2\right )^{3/2} d^7}+\frac{20160 b \sqrt{x} \text{Li}_7\left (-\frac{a e^{c+d \sqrt{x}}}{b-\sqrt{-a^2+b^2}}\right )}{a^2 \sqrt{-a^2+b^2} d^7}+\frac{10080 b^2 \text{Li}_7\left (-\frac{a e^{c+d \sqrt{x}}}{b+\sqrt{-a^2+b^2}}\right )}{a^2 \left (a^2-b^2\right ) d^8}+\frac{10080 b^3 \sqrt{x} \text{Li}_7\left (-\frac{a e^{c+d \sqrt{x}}}{b+\sqrt{-a^2+b^2}}\right )}{a^2 \left (-a^2+b^2\right )^{3/2} d^7}-\frac{20160 b \sqrt{x} \text{Li}_7\left (-\frac{a e^{c+d \sqrt{x}}}{b+\sqrt{-a^2+b^2}}\right )}{a^2 \sqrt{-a^2+b^2} d^7}+\frac{10080 b^3 \text{Li}_8\left (-\frac{a e^{c+d \sqrt{x}}}{b-\sqrt{-a^2+b^2}}\right )}{a^2 \left (-a^2+b^2\right )^{3/2} d^8}-\frac{20160 b \text{Li}_8\left (-\frac{a e^{c+d \sqrt{x}}}{b-\sqrt{-a^2+b^2}}\right )}{a^2 \sqrt{-a^2+b^2} d^8}-\frac{10080 b^3 \text{Li}_8\left (-\frac{a e^{c+d \sqrt{x}}}{b+\sqrt{-a^2+b^2}}\right )}{a^2 \left (-a^2+b^2\right )^{3/2} d^8}+\frac{20160 b \text{Li}_8\left (-\frac{a e^{c+d \sqrt{x}}}{b+\sqrt{-a^2+b^2}}\right )}{a^2 \sqrt{-a^2+b^2} d^8}+\frac{2 b^2 x^{7/2} \sinh \left (c+d \sqrt{x}\right )}{a \left (a^2-b^2\right ) d \left (b+a \cosh \left (c+d \sqrt{x}\right )\right )}\\ \end{align*}

Mathematica [A]  time = 18.4928, size = 3033, normalized size = 1.06 \[ \text{Result too large to show} \]

Warning: Unable to verify antiderivative.

[In]

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

[Out]

(x^4*(b + a*Cosh[c + d*Sqrt[x]])^2*Sech[c + d*Sqrt[x]]^2)/(4*a^2*(a + b*Sech[c + d*Sqrt[x]])^2) + (2*b*E^c*(b
+ a*Cosh[c + d*Sqrt[x]])^2*(2*b*E^c*x^(7/2) + ((1 + E^(2*c))*(-7*b*d^6*Sqrt[(-a^2 + b^2)*E^(2*c)]*x^3*Log[1 +
(a*E^(2*c + d*Sqrt[x]))/(b*E^c - Sqrt[(-a^2 + b^2)*E^(2*c)])] - 2*a^2*d^7*E^c*x^(7/2)*Log[1 + (a*E^(2*c + d*Sq
rt[x]))/(b*E^c - Sqrt[(-a^2 + b^2)*E^(2*c)])] + b^2*d^7*E^c*x^(7/2)*Log[1 + (a*E^(2*c + d*Sqrt[x]))/(b*E^c - S
qrt[(-a^2 + b^2)*E^(2*c)])] - 7*b*d^6*Sqrt[(-a^2 + b^2)*E^(2*c)]*x^3*Log[1 + (a*E^(2*c + d*Sqrt[x]))/(b*E^c +
Sqrt[(-a^2 + b^2)*E^(2*c)])] + 2*a^2*d^7*E^c*x^(7/2)*Log[1 + (a*E^(2*c + d*Sqrt[x]))/(b*E^c + Sqrt[(-a^2 + b^2
)*E^(2*c)])] - b^2*d^7*E^c*x^(7/2)*Log[1 + (a*E^(2*c + d*Sqrt[x]))/(b*E^c + Sqrt[(-a^2 + b^2)*E^(2*c)])] + 7*d
^5*(-6*b*Sqrt[(-a^2 + b^2)*E^(2*c)] - 2*a^2*d*E^c*Sqrt[x] + b^2*d*E^c*Sqrt[x])*x^(5/2)*PolyLog[2, -((a*E^(2*c
+ d*Sqrt[x]))/(b*E^c - Sqrt[(-a^2 + b^2)*E^(2*c)]))] - 7*d^5*(6*b*Sqrt[(-a^2 + b^2)*E^(2*c)] - 2*a^2*d*E^c*Sqr
t[x] + b^2*d*E^c*Sqrt[x])*x^(5/2)*PolyLog[2, -((a*E^(2*c + d*Sqrt[x]))/(b*E^c + Sqrt[(-a^2 + b^2)*E^(2*c)]))]
+ 210*b*d^4*Sqrt[(-a^2 + b^2)*E^(2*c)]*x^2*PolyLog[3, -((a*E^(2*c + d*Sqrt[x]))/(b*E^c - Sqrt[(-a^2 + b^2)*E^(
2*c)]))] + 84*a^2*d^5*E^c*x^(5/2)*PolyLog[3, -((a*E^(2*c + d*Sqrt[x]))/(b*E^c - Sqrt[(-a^2 + b^2)*E^(2*c)]))]
- 42*b^2*d^5*E^c*x^(5/2)*PolyLog[3, -((a*E^(2*c + d*Sqrt[x]))/(b*E^c - Sqrt[(-a^2 + b^2)*E^(2*c)]))] + 210*b*d
^4*Sqrt[(-a^2 + b^2)*E^(2*c)]*x^2*PolyLog[3, -((a*E^(2*c + d*Sqrt[x]))/(b*E^c + Sqrt[(-a^2 + b^2)*E^(2*c)]))]
- 84*a^2*d^5*E^c*x^(5/2)*PolyLog[3, -((a*E^(2*c + d*Sqrt[x]))/(b*E^c + Sqrt[(-a^2 + b^2)*E^(2*c)]))] + 42*b^2*
d^5*E^c*x^(5/2)*PolyLog[3, -((a*E^(2*c + d*Sqrt[x]))/(b*E^c + Sqrt[(-a^2 + b^2)*E^(2*c)]))] - 840*b*d^3*Sqrt[(
-a^2 + b^2)*E^(2*c)]*x^(3/2)*PolyLog[4, -((a*E^(2*c + d*Sqrt[x]))/(b*E^c - Sqrt[(-a^2 + b^2)*E^(2*c)]))] - 420
*a^2*d^4*E^c*x^2*PolyLog[4, -((a*E^(2*c + d*Sqrt[x]))/(b*E^c - Sqrt[(-a^2 + b^2)*E^(2*c)]))] + 210*b^2*d^4*E^c
*x^2*PolyLog[4, -((a*E^(2*c + d*Sqrt[x]))/(b*E^c - Sqrt[(-a^2 + b^2)*E^(2*c)]))] - 840*b*d^3*Sqrt[(-a^2 + b^2)
*E^(2*c)]*x^(3/2)*PolyLog[4, -((a*E^(2*c + d*Sqrt[x]))/(b*E^c + Sqrt[(-a^2 + b^2)*E^(2*c)]))] + 420*a^2*d^4*E^
c*x^2*PolyLog[4, -((a*E^(2*c + d*Sqrt[x]))/(b*E^c + Sqrt[(-a^2 + b^2)*E^(2*c)]))] - 210*b^2*d^4*E^c*x^2*PolyLo
g[4, -((a*E^(2*c + d*Sqrt[x]))/(b*E^c + Sqrt[(-a^2 + b^2)*E^(2*c)]))] + 2520*b*d^2*Sqrt[(-a^2 + b^2)*E^(2*c)]*
x*PolyLog[5, -((a*E^(2*c + d*Sqrt[x]))/(b*E^c - Sqrt[(-a^2 + b^2)*E^(2*c)]))] + 1680*a^2*d^3*E^c*x^(3/2)*PolyL
og[5, -((a*E^(2*c + d*Sqrt[x]))/(b*E^c - Sqrt[(-a^2 + b^2)*E^(2*c)]))] - 840*b^2*d^3*E^c*x^(3/2)*PolyLog[5, -(
(a*E^(2*c + d*Sqrt[x]))/(b*E^c - Sqrt[(-a^2 + b^2)*E^(2*c)]))] + 2520*b*d^2*Sqrt[(-a^2 + b^2)*E^(2*c)]*x*PolyL
og[5, -((a*E^(2*c + d*Sqrt[x]))/(b*E^c + Sqrt[(-a^2 + b^2)*E^(2*c)]))] - 1680*a^2*d^3*E^c*x^(3/2)*PolyLog[5, -
((a*E^(2*c + d*Sqrt[x]))/(b*E^c + Sqrt[(-a^2 + b^2)*E^(2*c)]))] + 840*b^2*d^3*E^c*x^(3/2)*PolyLog[5, -((a*E^(2
*c + d*Sqrt[x]))/(b*E^c + Sqrt[(-a^2 + b^2)*E^(2*c)]))] - 5040*b*d*Sqrt[(-a^2 + b^2)*E^(2*c)]*Sqrt[x]*PolyLog[
6, -((a*E^(2*c + d*Sqrt[x]))/(b*E^c - Sqrt[(-a^2 + b^2)*E^(2*c)]))] - 5040*a^2*d^2*E^c*x*PolyLog[6, -((a*E^(2*
c + d*Sqrt[x]))/(b*E^c - Sqrt[(-a^2 + b^2)*E^(2*c)]))] + 2520*b^2*d^2*E^c*x*PolyLog[6, -((a*E^(2*c + d*Sqrt[x]
))/(b*E^c - Sqrt[(-a^2 + b^2)*E^(2*c)]))] - 5040*b*d*Sqrt[(-a^2 + b^2)*E^(2*c)]*Sqrt[x]*PolyLog[6, -((a*E^(2*c
+ d*Sqrt[x]))/(b*E^c + Sqrt[(-a^2 + b^2)*E^(2*c)]))] + 5040*a^2*d^2*E^c*x*PolyLog[6, -((a*E^(2*c + d*Sqrt[x])
)/(b*E^c + Sqrt[(-a^2 + b^2)*E^(2*c)]))] - 2520*b^2*d^2*E^c*x*PolyLog[6, -((a*E^(2*c + d*Sqrt[x]))/(b*E^c + Sq
rt[(-a^2 + b^2)*E^(2*c)]))] + 5040*b*Sqrt[(-a^2 + b^2)*E^(2*c)]*PolyLog[7, -((a*E^(2*c + d*Sqrt[x]))/(b*E^c -
Sqrt[(-a^2 + b^2)*E^(2*c)]))] + 10080*a^2*d*E^c*Sqrt[x]*PolyLog[7, -((a*E^(2*c + d*Sqrt[x]))/(b*E^c - Sqrt[(-a
^2 + b^2)*E^(2*c)]))] - 5040*b^2*d*E^c*Sqrt[x]*PolyLog[7, -((a*E^(2*c + d*Sqrt[x]))/(b*E^c - Sqrt[(-a^2 + b^2)
*E^(2*c)]))] + 5040*b*Sqrt[(-a^2 + b^2)*E^(2*c)]*PolyLog[7, -((a*E^(2*c + d*Sqrt[x]))/(b*E^c + Sqrt[(-a^2 + b^
2)*E^(2*c)]))] - 10080*a^2*d*E^c*Sqrt[x]*PolyLog[7, -((a*E^(2*c + d*Sqrt[x]))/(b*E^c + Sqrt[(-a^2 + b^2)*E^(2*
c)]))] + 5040*b^2*d*E^c*Sqrt[x]*PolyLog[7, -((a*E^(2*c + d*Sqrt[x]))/(b*E^c + Sqrt[(-a^2 + b^2)*E^(2*c)]))] -
10080*a^2*E^c*PolyLog[8, -((a*E^(2*c + d*Sqrt[x]))/(b*E^c - Sqrt[(-a^2 + b^2)*E^(2*c)]))] + 5040*b^2*E^c*PolyL
og[8, -((a*E^(2*c + d*Sqrt[x]))/(b*E^c - Sqrt[(-a^2 + b^2)*E^(2*c)]))] + 10080*a^2*E^c*PolyLog[8, -((a*E^(2*c
+ d*Sqrt[x]))/(b*E^c + Sqrt[(-a^2 + b^2)*E^(2*c)]))] - 5040*b^2*E^c*PolyLog[8, -((a*E^(2*c + d*Sqrt[x]))/(b*E^
c + Sqrt[(-a^2 + b^2)*E^(2*c)]))]))/(d^7*E^c*Sqrt[(-a^2 + b^2)*E^(2*c)]))*Sech[c + d*Sqrt[x]]^2)/(a^2*(a^2 - b
^2)*d*(1 + E^(2*c))*(a + b*Sech[c + d*Sqrt[x]])^2) + (2*(b + a*Cosh[c + d*Sqrt[x]])*Sech[c]*Sech[c + d*Sqrt[x]
]^2*(b^3*x^(7/2)*Sinh[c] - a*b^2*x^(7/2)*Sinh[d*Sqrt[x]]))/(a^2*(-a + b)*(a + b)*d*(a + b*Sech[c + d*Sqrt[x]])
^2)

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Maple [F]  time = 0.105, size = 0, normalized size = 0. \begin{align*} \int{{x}^{3} \left ( a+b{\rm sech} \left (c+d\sqrt{x}\right ) \right ) ^{-2}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^3/(a+b*sech(c+d*x^(1/2)))^2,x, algorithm="fricas")

[Out]

integral(x^3/(b^2*sech(d*sqrt(x) + c)^2 + 2*a*b*sech(d*sqrt(x) + c) + a^2), x)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Integral(x**3/(a + b*sech(c + d*sqrt(x)))**2, x)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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