3.1.46 \(\int \frac {x^3}{(a+b \text {csch}(c+d \sqrt {x}))^2} \, dx\) [46]

Optimal. Leaf size=2663 \[ \text {result too large to display} \]

[Out]

-2*b^2*x^(7/2)/a^2/(a^2+b^2)/d-20160*b*polylog(8,-a*exp(c+d*x^(1/2))/(b-(a^2+b^2)^(1/2)))/a^2/d^8/(a^2+b^2)^(1
/2)+20160*b*polylog(8,-a*exp(c+d*x^(1/2))/(b+(a^2+b^2)^(1/2)))/a^2/d^8/(a^2+b^2)^(1/2)+10080*b^3*polylog(8,-a*
exp(c+d*x^(1/2))/(b-(a^2+b^2)^(1/2)))/a^2/(a^2+b^2)^(3/2)/d^8-10080*b^3*polylog(8,-a*exp(c+d*x^(1/2))/(b+(a^2+
b^2)^(1/2)))/a^2/(a^2+b^2)^(3/2)/d^8-10080*b^2*polylog(7,-a*exp(c+d*x^(1/2))/(b-(a^2+b^2)^(1/2)))/a^2/(a^2+b^2
)/d^8-10080*b^2*polylog(7,-a*exp(c+d*x^(1/2))/(b+(a^2+b^2)^(1/2)))/a^2/(a^2+b^2)/d^8+10080*b*x*polylog(6,-a*ex
p(c+d*x^(1/2))/(b+(a^2+b^2)^(1/2)))/a^2/d^6/(a^2+b^2)^(1/2)+10080*b^2*polylog(6,-a*exp(c+d*x^(1/2))/(b-(a^2+b^
2)^(1/2)))*x^(1/2)/a^2/(a^2+b^2)/d^7+10080*b^2*polylog(6,-a*exp(c+d*x^(1/2))/(b+(a^2+b^2)^(1/2)))*x^(1/2)/a^2/
(a^2+b^2)/d^7-10080*b^3*polylog(7,-a*exp(c+d*x^(1/2))/(b-(a^2+b^2)^(1/2)))*x^(1/2)/a^2/(a^2+b^2)^(3/2)/d^7+100
80*b^3*polylog(7,-a*exp(c+d*x^(1/2))/(b+(a^2+b^2)^(1/2)))*x^(1/2)/a^2/(a^2+b^2)^(3/2)/d^7+20160*b*polylog(7,-a
*exp(c+d*x^(1/2))/(b-(a^2+b^2)^(1/2)))*x^(1/2)/a^2/d^7/(a^2+b^2)^(1/2)-20160*b*polylog(7,-a*exp(c+d*x^(1/2))/(
b+(a^2+b^2)^(1/2)))*x^(1/2)/a^2/d^7/(a^2+b^2)^(1/2)+1/4*x^4/a^2-2*b^2*x^(7/2)*cosh(c+d*x^(1/2))/a/(a^2+b^2)/d/
(b+a*sinh(c+d*x^(1/2)))+14*b^2*x^3*ln(1+a*exp(c+d*x^(1/2))/(b-(a^2+b^2)^(1/2)))/a^2/(a^2+b^2)/d^2+2*b^3*x^(7/2
)*ln(1+a*exp(c+d*x^(1/2))/(b-(a^2+b^2)^(1/2)))/a^2/(a^2+b^2)^(3/2)/d+14*b^2*x^3*ln(1+a*exp(c+d*x^(1/2))/(b+(a^
2+b^2)^(1/2)))/a^2/(a^2+b^2)/d^2-2*b^3*x^(7/2)*ln(1+a*exp(c+d*x^(1/2))/(b+(a^2+b^2)^(1/2)))/a^2/(a^2+b^2)^(3/2
)/d+84*b^2*x^(5/2)*polylog(2,-a*exp(c+d*x^(1/2))/(b-(a^2+b^2)^(1/2)))/a^2/(a^2+b^2)/d^3+14*b^3*x^3*polylog(2,-
a*exp(c+d*x^(1/2))/(b-(a^2+b^2)^(1/2)))/a^2/(a^2+b^2)^(3/2)/d^2+84*b^2*x^(5/2)*polylog(2,-a*exp(c+d*x^(1/2))/(
b+(a^2+b^2)^(1/2)))/a^2/(a^2+b^2)/d^3-14*b^3*x^3*polylog(2,-a*exp(c+d*x^(1/2))/(b+(a^2+b^2)^(1/2)))/a^2/(a^2+b
^2)^(3/2)/d^2-420*b^2*x^2*polylog(3,-a*exp(c+d*x^(1/2))/(b-(a^2+b^2)^(1/2)))/a^2/(a^2+b^2)/d^4-84*b^3*x^(5/2)*
polylog(3,-a*exp(c+d*x^(1/2))/(b-(a^2+b^2)^(1/2)))/a^2/(a^2+b^2)^(3/2)/d^3-420*b^2*x^2*polylog(3,-a*exp(c+d*x^
(1/2))/(b+(a^2+b^2)^(1/2)))/a^2/(a^2+b^2)/d^4+84*b^3*x^(5/2)*polylog(3,-a*exp(c+d*x^(1/2))/(b+(a^2+b^2)^(1/2))
)/a^2/(a^2+b^2)^(3/2)/d^3+1680*b^2*x^(3/2)*polylog(4,-a*exp(c+d*x^(1/2))/(b-(a^2+b^2)^(1/2)))/a^2/(a^2+b^2)/d^
5+420*b^3*x^2*polylog(4,-a*exp(c+d*x^(1/2))/(b-(a^2+b^2)^(1/2)))/a^2/(a^2+b^2)^(3/2)/d^4+1680*b^2*x^(3/2)*poly
log(4,-a*exp(c+d*x^(1/2))/(b+(a^2+b^2)^(1/2)))/a^2/(a^2+b^2)/d^5-420*b^3*x^2*polylog(4,-a*exp(c+d*x^(1/2))/(b+
(a^2+b^2)^(1/2)))/a^2/(a^2+b^2)^(3/2)/d^4-5040*b^2*x*polylog(5,-a*exp(c+d*x^(1/2))/(b-(a^2+b^2)^(1/2)))/a^2/(a
^2+b^2)/d^6-1680*b^3*x^(3/2)*polylog(5,-a*exp(c+d*x^(1/2))/(b-(a^2+b^2)^(1/2)))/a^2/(a^2+b^2)^(3/2)/d^5-5040*b
^2*x*polylog(5,-a*exp(c+d*x^(1/2))/(b+(a^2+b^2)^(1/2)))/a^2/(a^2+b^2)/d^6+1680*b^3*x^(3/2)*polylog(5,-a*exp(c+
d*x^(1/2))/(b+(a^2+b^2)^(1/2)))/a^2/(a^2+b^2)^(3/2)/d^5+5040*b^3*x*polylog(6,-a*exp(c+d*x^(1/2))/(b-(a^2+b^2)^
(1/2)))/a^2/(a^2+b^2)^(3/2)/d^6-5040*b^3*x*polylog(6,-a*exp(c+d*x^(1/2))/(b+(a^2+b^2)^(1/2)))/a^2/(a^2+b^2)^(3
/2)/d^6-4*b*x^(7/2)*ln(1+a*exp(c+d*x^(1/2))/(b-(a^2+b^2)^(1/2)))/a^2/d/(a^2+b^2)^(1/2)+4*b*x^(7/2)*ln(1+a*exp(
c+d*x^(1/2))/(b+(a^2+b^2)^(1/2)))/a^2/d/(a^2+b^2)^(1/2)-28*b*x^3*polylog(2,-a*exp(c+d*x^(1/2))/(b-(a^2+b^2)^(1
/2)))/a^2/d^2/(a^2+b^2)^(1/2)+28*b*x^3*polylog(2,-a*exp(c+d*x^(1/2))/(b+(a^2+b^2)^(1/2)))/a^2/d^2/(a^2+b^2)^(1
/2)+168*b*x^(5/2)*polylog(3,-a*exp(c+d*x^(1/2))/(b-(a^2+b^2)^(1/2)))/a^2/d^3/(a^2+b^2)^(1/2)-168*b*x^(5/2)*pol
ylog(3,-a*exp(c+d*x^(1/2))/(b+(a^2+b^2)^(1/2)))/a^2/d^3/(a^2+b^2)^(1/2)-840*b*x^2*polylog(4,-a*exp(c+d*x^(1/2)
)/(b-(a^2+b^2)^(1/2)))/a^2/d^4/(a^2+b^2)^(1/2)+840*b*x^2*polylog(4,-a*exp(c+d*x^(1/2))/(b+(a^2+b^2)^(1/2)))/a^
2/d^4/(a^2+b^2)^(1/2)+3360*b*x^(3/2)*polylog(5,-a*exp(c+d*x^(1/2))/(b-(a^2+b^2)^(1/2)))/a^2/d^5/(a^2+b^2)^(1/2
)-3360*b*x^(3/2)*polylog(5,-a*exp(c+d*x^(1/2))/(b+(a^2+b^2)^(1/2)))/a^2/d^5/(a^2+b^2)^(1/2)-10080*b*x*polylog(
6,-a*exp(c+d*x^(1/2))/(b-(a^2+b^2)^(1/2)))/a^2/d^6/(a^2+b^2)^(1/2)

________________________________________________________________________________________

Rubi [A]
time = 2.52, antiderivative size = 2663, normalized size of antiderivative = 1.00, number of steps used = 61, number of rules used = 11, integrand size = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.550, Rules used = {5545, 4276, 3405, 3403, 2296, 2221, 2611, 6744, 2320, 6724, 5680} \begin {gather*} \text {Too large to display} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[x^3/(a + b*Csch[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)*Lo
g[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 - Sqrt[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*Sqrt[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*Sq
rt[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) + (33
60*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)*PolyL
og[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*S
qrt[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) + (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) - (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*S
qrt[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*Poly
Log[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*S
qrt[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)*Cosh[c + d*Sqrt[x]])/(a*(a^2 + b^2)*d*(b + a
*Sinh[c + d*Sqrt[x]]))

Rule 2221

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/(b*f*g*n*Log[F]))*Log[1 + b*((F^(g*(e + f*x)))^n/a)], 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 2296

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 2320

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 2611

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 3403

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

Rule 3405

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 4276

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 5545

Int[((a_.) + Csch[(c_.) + (d_.)*(x_)^(n_)]*(b_.))^(p_.)*(x_)^(m_.), x_Symbol] :> Dist[1/n, Subst[Int[x^(Simpli
fy[(m + 1)/n] - 1)*(a + b*Csch[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 5680

Int[(Cosh[(c_.) + (d_.)*(x_)]*((e_.) + (f_.)*(x_))^(m_.))/((a_) + (b_.)*Sinh[(c_.) + (d_.)*(x_)]), 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]

Rule 6724

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 6744

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]

Rubi steps

\begin {align*} \int \frac {x^3}{\left (a+b \text {csch}\left (c+d \sqrt {x}\right )\right )^2} \, dx &=2 \text {Subst}\left (\int \frac {x^7}{(a+b \text {csch}(c+d x))^2} \, dx,x,\sqrt {x}\right )\\ &=2 \text {Subst}\left (\int \left (\frac {x^7}{a^2}+\frac {b^2 x^7}{a^2 (b+a \sinh (c+d x))^2}-\frac {2 b x^7}{a^2 (b+a \sinh (c+d x))}\right ) \, dx,x,\sqrt {x}\right )\\ &=\frac {x^4}{4 a^2}-\frac {(4 b) \text {Subst}\left (\int \frac {x^7}{b+a \sinh (c+d x)} \, dx,x,\sqrt {x}\right )}{a^2}+\frac {\left (2 b^2\right ) \text {Subst}\left (\int \frac {x^7}{(b+a \sinh (c+d x))^2} \, dx,x,\sqrt {x}\right )}{a^2}\\ &=\frac {x^4}{4 a^2}-\frac {2 b^2 x^{7/2} \cosh \left (c+d \sqrt {x}\right )}{a \left (a^2+b^2\right ) d \left (b+a \sinh \left (c+d \sqrt {x}\right )\right )}-\frac {(8 b) \text {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 ) \text {Subst}\left (\int \frac {x^7}{b+a \sinh (c+d x)} \, dx,x,\sqrt {x}\right )}{a^2 \left (a^2+b^2\right )}+\frac {\left (14 b^2\right ) \text {Subst}\left (\int \frac {x^6 \cosh (c+d x)}{b+a \sinh (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} \cosh \left (c+d \sqrt {x}\right )}{a \left (a^2+b^2\right ) d \left (b+a \sinh \left (c+d \sqrt {x}\right )\right )}+\frac {\left (4 b^3\right ) \text {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) \text {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) \text {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 ) \text {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 ) \text {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} \cosh \left (c+d \sqrt {x}\right )}{a \left (a^2+b^2\right ) d \left (b+a \sinh \left (c+d \sqrt {x}\right )\right )}+\frac {\left (4 b^3\right ) \text {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 ) \text {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 ) \text {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 ) \text {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) \text {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) \text {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} \cosh \left (c+d \sqrt {x}\right )}{a \left (a^2+b^2\right ) d \left (b+a \sinh \left (c+d \sqrt {x}\right )\right )}-\frac {\left (420 b^2\right ) \text {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 ) \text {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) \text {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) \text {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 ) \text {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 ) \text {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} \cosh \left (c+d \sqrt {x}\right )}{a \left (a^2+b^2\right ) d \left (b+a \sinh \left (c+d \sqrt {x}\right )\right )}+\frac {\left (1680 b^2\right ) \text {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 ) \text {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) \text {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) \text {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 ) \text {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 ) \text {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} \cosh \left (c+d \sqrt {x}\right )}{a \left (a^2+b^2\right ) d \left (b+a \sinh \left (c+d \sqrt {x}\right )\right )}-\frac {\left (5040 b^2\right ) \text {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 ) \text {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) \text {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) \text {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 ) \text {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 ) \text {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} \cosh \left (c+d \sqrt {x}\right )}{a \left (a^2+b^2\right ) d \left (b+a \sinh \left (c+d \sqrt {x}\right )\right )}+\frac {\left (10080 b^2\right ) \text {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 ) \text {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) \text {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) \text {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 ) \text {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 ) \text {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} \cosh \left (c+d \sqrt {x}\right )}{a \left (a^2+b^2\right ) d \left (b+a \sinh \left (c+d \sqrt {x}\right )\right )}-\frac {\left (10080 b^2\right ) \text {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 ) \text {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) \text {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) \text {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 ) \text {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 ) \text {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} \cosh \left (c+d \sqrt {x}\right )}{a \left (a^2+b^2\right ) d \left (b+a \sinh \left (c+d \sqrt {x}\right )\right )}-\frac {\left (10080 b^2\right ) \text {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 ) \text {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) \text {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) \text {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 ) \text {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 ) \text {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} \cosh \left (c+d \sqrt {x}\right )}{a \left (a^2+b^2\right ) d \left (b+a \sinh \left (c+d \sqrt {x}\right )\right )}-\frac {(20160 b) \text {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) \text {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 ) \text {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 ) \text {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} \cosh \left (c+d \sqrt {x}\right )}{a \left (a^2+b^2\right ) d \left (b+a \sinh \left (c+d \sqrt {x}\right )\right )}+\frac {\left (10080 b^3\right ) \text {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 ) \text {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} \cosh \left (c+d \sqrt {x}\right )}{a \left (a^2+b^2\right ) d \left (b+a \sinh \left (c+d \sqrt {x}\right )\right )}\\ \end {align*}

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Mathematica [A]
time = 15.67, size = 2909, normalized size = 1.09 \begin {gather*} \text {Result too large to show} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(x^4*Csch[c + d*Sqrt[x]]^2*(b + a*Sinh[c + d*Sqrt[x]])^2)/(4*a^2*(a + b*Csch[c + d*Sqrt[x]])^2) + (2*b*Csch[c
+ d*Sqrt[x]]^2*((-2*b*d^7*E^(2*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*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*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*S
qrt[(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*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)]))] - 210*b*d^4*Sq
rt[(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)*PolyL
og[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*PolyLog[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*Sq
rt[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)]))] - 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)*PolyLog[5, -((a*E^(2*c + d*Sqrt[x]))/(b*E^c + Sq
rt[(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 - S
qrt[(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)]))] + 25
20*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*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]*Po
lyLog[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)]*Po
lyLog[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*PolyLog[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)]))] + 50
40*b^2*E^c*PolyLog[8, -((a*E^(2*c + d*Sqrt[x]))/(b*E^c + Sqrt[(a^2 + b^2)*E^(2*c)]))])/Sqrt[(a^2 + b^2)*E^(2*c
)])*(b + a*Sinh[c + d*Sqrt[x]])^2)/(a^2*(a^2 + b^2)*d^8*(a + b*Csch[c + d*Sqrt[x]])^2) + (Csch[c/2]*Csch[c + d
*Sqrt[x]]^2*Sech[c/2]*(b + a*Sinh[c + d*Sqrt[x]])*(b^3*x^(7/2)*Cosh[c] + a*b^2*x^(7/2)*Sinh[d*Sqrt[x]]))/(a^2*
(a^2 + b^2)*d*(a + b*Csch[c + d*Sqrt[x]])^2)

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Maple [F]
time = 6.02, size = 0, normalized size = 0.00 \[\int \frac {x^{3}}{\left (a +b \,\mathrm {csch}\left (c +d \sqrt {x}\right )\right )^{2}}\, dx\]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/4*(16*a*b^2*x^(7/2) - (a^3*d*e^(2*c) + a*b^2*d*e^(2*c))*x^4*e^(2*d*sqrt(x)) + (a^3*d + a*b^2*d)*x^4 - 2*(8*b
^3*x^(7/2)*e^c + (a^2*b*d*e^c + b^3*d*e^c)*x^4)*e^(d*sqrt(x)))/(a^5*d + a^3*b^2*d - (a^5*d*e^(2*c) + a^3*b^2*d
*e^(2*c))*e^(2*d*sqrt(x)) - 2*(a^4*b*d*e^c + a^2*b^3*d*e^c)*e^(d*sqrt(x))) - integrate(2*(7*a*b^2*x^(5/2) - (7
*b^3*x^(5/2)*e^c + (2*a^2*b*d*e^c + b^3*d*e^c)*x^3)*e^(d*sqrt(x)))/(a^5*d + a^3*b^2*d - (a^5*d*e^(2*c) + a^3*b
^2*d*e^(2*c))*e^(2*d*sqrt(x)) - 2*(a^4*b*d*e^c + a^2*b^3*d*e^c)*e^(d*sqrt(x))), x)

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Fricas [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {x^{3}}{\left (a + b \operatorname {csch}{\left (c + d \sqrt {x} \right )}\right )^{2}}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Giac [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

sage0*x

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Mupad [F]
time = 0.00, size = -1, normalized size = -0.00 \begin {gather*} \int \frac {x^3}{{\left (a+\frac {b}{\mathrm {sinh}\left (c+d\,\sqrt {x}\right )}\right )}^2} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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