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

Optimal. Leaf size=1395 \[ \text{result too large to display} \]

[Out]

(2*b^2*x^(3/2))/(a^2*(a^2 - b^2)*d) + x^2/(2*a^2) - (6*b^2*x*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^(3/2)*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^(3/2)*Log[1 + (a*E^(c + d*Sqrt[x]))/(b - Sqrt[-a^2 + b^2])])/(a^2*Sqrt[-a^2 + b^2]*d
) - (6*b^2*x*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^(3/2)*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^(3/2)*Log[1 + (a*E^(c
+ d*Sqrt[x]))/(b + Sqrt[-a^2 + b^2])])/(a^2*Sqrt[-a^2 + b^2]*d) - (12*b^2*Sqrt[x]*PolyLog[2, -((a*E^(c + d*Sqr
t[x]))/(b - Sqrt[-a^2 + b^2]))])/(a^2*(a^2 - b^2)*d^3) + (6*b^3*x*PolyLog[2, -((a*E^(c + d*Sqrt[x]))/(b - Sqrt
[-a^2 + b^2]))])/(a^2*(-a^2 + b^2)^(3/2)*d^2) - (12*b*x*PolyLog[2, -((a*E^(c + d*Sqrt[x]))/(b - Sqrt[-a^2 + b^
2]))])/(a^2*Sqrt[-a^2 + b^2]*d^2) - (12*b^2*Sqrt[x]*PolyLog[2, -((a*E^(c + d*Sqrt[x]))/(b + Sqrt[-a^2 + b^2]))
])/(a^2*(a^2 - b^2)*d^3) - (6*b^3*x*PolyLog[2, -((a*E^(c + d*Sqrt[x]))/(b + Sqrt[-a^2 + b^2]))])/(a^2*(-a^2 +
b^2)^(3/2)*d^2) + (12*b*x*PolyLog[2, -((a*E^(c + d*Sqrt[x]))/(b + Sqrt[-a^2 + b^2]))])/(a^2*Sqrt[-a^2 + b^2]*d
^2) + (12*b^2*PolyLog[3, -((a*E^(c + d*Sqrt[x]))/(b - Sqrt[-a^2 + b^2]))])/(a^2*(a^2 - b^2)*d^4) - (12*b^3*Sqr
t[x]*PolyLog[3, -((a*E^(c + d*Sqrt[x]))/(b - Sqrt[-a^2 + b^2]))])/(a^2*(-a^2 + b^2)^(3/2)*d^3) + (24*b*Sqrt[x]
*PolyLog[3, -((a*E^(c + d*Sqrt[x]))/(b - Sqrt[-a^2 + b^2]))])/(a^2*Sqrt[-a^2 + b^2]*d^3) + (12*b^2*PolyLog[3,
-((a*E^(c + d*Sqrt[x]))/(b + Sqrt[-a^2 + b^2]))])/(a^2*(a^2 - b^2)*d^4) + (12*b^3*Sqrt[x]*PolyLog[3, -((a*E^(c
+ d*Sqrt[x]))/(b + Sqrt[-a^2 + b^2]))])/(a^2*(-a^2 + b^2)^(3/2)*d^3) - (24*b*Sqrt[x]*PolyLog[3, -((a*E^(c + d
*Sqrt[x]))/(b + Sqrt[-a^2 + b^2]))])/(a^2*Sqrt[-a^2 + b^2]*d^3) + (12*b^3*PolyLog[4, -((a*E^(c + d*Sqrt[x]))/(
b - Sqrt[-a^2 + b^2]))])/(a^2*(-a^2 + b^2)^(3/2)*d^4) - (24*b*PolyLog[4, -((a*E^(c + d*Sqrt[x]))/(b - Sqrt[-a^
2 + b^2]))])/(a^2*Sqrt[-a^2 + b^2]*d^4) - (12*b^3*PolyLog[4, -((a*E^(c + d*Sqrt[x]))/(b + Sqrt[-a^2 + b^2]))])
/(a^2*(-a^2 + b^2)^(3/2)*d^4) + (24*b*PolyLog[4, -((a*E^(c + d*Sqrt[x]))/(b + Sqrt[-a^2 + b^2]))])/(a^2*Sqrt[-
a^2 + b^2]*d^4) + (2*b^2*x^(3/2)*Sinh[c + d*Sqrt[x]])/(a*(a^2 - b^2)*d*(b + a*Cosh[c + d*Sqrt[x]]))

________________________________________________________________________________________

Rubi [A]  time = 2.31258, antiderivative size = 1395, normalized size of antiderivative = 1., number of steps used = 37, number of rules used = 11, integrand size = 18, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.611, Rules used = {5436, 4191, 3324, 3320, 2264, 2190, 2531, 6609, 2282, 6589, 5562} \[ \frac{2 x^{3/2} \log \left (\frac{e^{c+d \sqrt{x}} a}{b-\sqrt{b^2-a^2}}+1\right ) b^3}{a^2 \left (b^2-a^2\right )^{3/2} d}-\frac{2 x^{3/2} \log \left (\frac{e^{c+d \sqrt{x}} a}{b+\sqrt{b^2-a^2}}+1\right ) b^3}{a^2 \left (b^2-a^2\right )^{3/2} d}+\frac{6 x \text{PolyLog}\left (2,-\frac{a e^{c+d \sqrt{x}}}{b-\sqrt{b^2-a^2}}\right ) b^3}{a^2 \left (b^2-a^2\right )^{3/2} d^2}-\frac{6 x \text{PolyLog}\left (2,-\frac{a e^{c+d \sqrt{x}}}{b+\sqrt{b^2-a^2}}\right ) b^3}{a^2 \left (b^2-a^2\right )^{3/2} d^2}-\frac{12 \sqrt{x} \text{PolyLog}\left (3,-\frac{a e^{c+d \sqrt{x}}}{b-\sqrt{b^2-a^2}}\right ) b^3}{a^2 \left (b^2-a^2\right )^{3/2} d^3}+\frac{12 \sqrt{x} \text{PolyLog}\left (3,-\frac{a e^{c+d \sqrt{x}}}{b+\sqrt{b^2-a^2}}\right ) b^3}{a^2 \left (b^2-a^2\right )^{3/2} d^3}+\frac{12 \text{PolyLog}\left (4,-\frac{a e^{c+d \sqrt{x}}}{b-\sqrt{b^2-a^2}}\right ) b^3}{a^2 \left (b^2-a^2\right )^{3/2} d^4}-\frac{12 \text{PolyLog}\left (4,-\frac{a e^{c+d \sqrt{x}}}{b+\sqrt{b^2-a^2}}\right ) b^3}{a^2 \left (b^2-a^2\right )^{3/2} d^4}+\frac{2 x^{3/2} b^2}{a^2 \left (a^2-b^2\right ) d}-\frac{6 x \log \left (\frac{e^{c+d \sqrt{x}} a}{b-\sqrt{b^2-a^2}}+1\right ) b^2}{a^2 \left (a^2-b^2\right ) d^2}-\frac{6 x \log \left (\frac{e^{c+d \sqrt{x}} a}{b+\sqrt{b^2-a^2}}+1\right ) b^2}{a^2 \left (a^2-b^2\right ) d^2}-\frac{12 \sqrt{x} \text{PolyLog}\left (2,-\frac{a e^{c+d \sqrt{x}}}{b-\sqrt{b^2-a^2}}\right ) b^2}{a^2 \left (a^2-b^2\right ) d^3}-\frac{12 \sqrt{x} \text{PolyLog}\left (2,-\frac{a e^{c+d \sqrt{x}}}{b+\sqrt{b^2-a^2}}\right ) b^2}{a^2 \left (a^2-b^2\right ) d^3}+\frac{12 \text{PolyLog}\left (3,-\frac{a e^{c+d \sqrt{x}}}{b-\sqrt{b^2-a^2}}\right ) b^2}{a^2 \left (a^2-b^2\right ) d^4}+\frac{12 \text{PolyLog}\left (3,-\frac{a e^{c+d \sqrt{x}}}{b+\sqrt{b^2-a^2}}\right ) b^2}{a^2 \left (a^2-b^2\right ) d^4}+\frac{2 x^{3/2} \sinh \left (c+d \sqrt{x}\right ) b^2}{a \left (a^2-b^2\right ) d \left (b+a \cosh \left (c+d \sqrt{x}\right )\right )}-\frac{4 x^{3/2} \log \left (\frac{e^{c+d \sqrt{x}} a}{b-\sqrt{b^2-a^2}}+1\right ) b}{a^2 \sqrt{b^2-a^2} d}+\frac{4 x^{3/2} \log \left (\frac{e^{c+d \sqrt{x}} a}{b+\sqrt{b^2-a^2}}+1\right ) b}{a^2 \sqrt{b^2-a^2} d}-\frac{12 x \text{PolyLog}\left (2,-\frac{a e^{c+d \sqrt{x}}}{b-\sqrt{b^2-a^2}}\right ) b}{a^2 \sqrt{b^2-a^2} d^2}+\frac{12 x \text{PolyLog}\left (2,-\frac{a e^{c+d \sqrt{x}}}{b+\sqrt{b^2-a^2}}\right ) b}{a^2 \sqrt{b^2-a^2} d^2}+\frac{24 \sqrt{x} \text{PolyLog}\left (3,-\frac{a e^{c+d \sqrt{x}}}{b-\sqrt{b^2-a^2}}\right ) b}{a^2 \sqrt{b^2-a^2} d^3}-\frac{24 \sqrt{x} \text{PolyLog}\left (3,-\frac{a e^{c+d \sqrt{x}}}{b+\sqrt{b^2-a^2}}\right ) b}{a^2 \sqrt{b^2-a^2} d^3}-\frac{24 \text{PolyLog}\left (4,-\frac{a e^{c+d \sqrt{x}}}{b-\sqrt{b^2-a^2}}\right ) b}{a^2 \sqrt{b^2-a^2} d^4}+\frac{24 \text{PolyLog}\left (4,-\frac{a e^{c+d \sqrt{x}}}{b+\sqrt{b^2-a^2}}\right ) b}{a^2 \sqrt{b^2-a^2} d^4}+\frac{x^2}{2 a^2} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(2*b^2*x^(3/2))/(a^2*(a^2 - b^2)*d) + x^2/(2*a^2) - (6*b^2*x*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^(3/2)*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^(3/2)*Log[1 + (a*E^(c + d*Sqrt[x]))/(b - Sqrt[-a^2 + b^2])])/(a^2*Sqrt[-a^2 + b^2]*d
) - (6*b^2*x*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^(3/2)*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^(3/2)*Log[1 + (a*E^(c
+ d*Sqrt[x]))/(b + Sqrt[-a^2 + b^2])])/(a^2*Sqrt[-a^2 + b^2]*d) - (12*b^2*Sqrt[x]*PolyLog[2, -((a*E^(c + d*Sqr
t[x]))/(b - Sqrt[-a^2 + b^2]))])/(a^2*(a^2 - b^2)*d^3) + (6*b^3*x*PolyLog[2, -((a*E^(c + d*Sqrt[x]))/(b - Sqrt
[-a^2 + b^2]))])/(a^2*(-a^2 + b^2)^(3/2)*d^2) - (12*b*x*PolyLog[2, -((a*E^(c + d*Sqrt[x]))/(b - Sqrt[-a^2 + b^
2]))])/(a^2*Sqrt[-a^2 + b^2]*d^2) - (12*b^2*Sqrt[x]*PolyLog[2, -((a*E^(c + d*Sqrt[x]))/(b + Sqrt[-a^2 + b^2]))
])/(a^2*(a^2 - b^2)*d^3) - (6*b^3*x*PolyLog[2, -((a*E^(c + d*Sqrt[x]))/(b + Sqrt[-a^2 + b^2]))])/(a^2*(-a^2 +
b^2)^(3/2)*d^2) + (12*b*x*PolyLog[2, -((a*E^(c + d*Sqrt[x]))/(b + Sqrt[-a^2 + b^2]))])/(a^2*Sqrt[-a^2 + b^2]*d
^2) + (12*b^2*PolyLog[3, -((a*E^(c + d*Sqrt[x]))/(b - Sqrt[-a^2 + b^2]))])/(a^2*(a^2 - b^2)*d^4) - (12*b^3*Sqr
t[x]*PolyLog[3, -((a*E^(c + d*Sqrt[x]))/(b - Sqrt[-a^2 + b^2]))])/(a^2*(-a^2 + b^2)^(3/2)*d^3) + (24*b*Sqrt[x]
*PolyLog[3, -((a*E^(c + d*Sqrt[x]))/(b - Sqrt[-a^2 + b^2]))])/(a^2*Sqrt[-a^2 + b^2]*d^3) + (12*b^2*PolyLog[3,
-((a*E^(c + d*Sqrt[x]))/(b + Sqrt[-a^2 + b^2]))])/(a^2*(a^2 - b^2)*d^4) + (12*b^3*Sqrt[x]*PolyLog[3, -((a*E^(c
+ d*Sqrt[x]))/(b + Sqrt[-a^2 + b^2]))])/(a^2*(-a^2 + b^2)^(3/2)*d^3) - (24*b*Sqrt[x]*PolyLog[3, -((a*E^(c + d
*Sqrt[x]))/(b + Sqrt[-a^2 + b^2]))])/(a^2*Sqrt[-a^2 + b^2]*d^3) + (12*b^3*PolyLog[4, -((a*E^(c + d*Sqrt[x]))/(
b - Sqrt[-a^2 + b^2]))])/(a^2*(-a^2 + b^2)^(3/2)*d^4) - (24*b*PolyLog[4, -((a*E^(c + d*Sqrt[x]))/(b - Sqrt[-a^
2 + b^2]))])/(a^2*Sqrt[-a^2 + b^2]*d^4) - (12*b^3*PolyLog[4, -((a*E^(c + d*Sqrt[x]))/(b + Sqrt[-a^2 + b^2]))])
/(a^2*(-a^2 + b^2)^(3/2)*d^4) + (24*b*PolyLog[4, -((a*E^(c + d*Sqrt[x]))/(b + Sqrt[-a^2 + b^2]))])/(a^2*Sqrt[-
a^2 + b^2]*d^4) + (2*b^2*x^(3/2)*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}{\left (a+b \text{sech}\left (c+d \sqrt{x}\right )\right )^2} \, dx &=2 \operatorname{Subst}\left (\int \frac{x^3}{(a+b \text{sech}(c+d x))^2} \, dx,x,\sqrt{x}\right )\\ &=2 \operatorname{Subst}\left (\int \left (\frac{x^3}{a^2}+\frac{b^2 x^3}{a^2 (b+a \cosh (c+d x))^2}-\frac{2 b x^3}{a^2 (b+a \cosh (c+d x))}\right ) \, dx,x,\sqrt{x}\right )\\ &=\frac{x^2}{2 a^2}-\frac{(4 b) \operatorname{Subst}\left (\int \frac{x^3}{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^3}{(b+a \cosh (c+d x))^2} \, dx,x,\sqrt{x}\right )}{a^2}\\ &=\frac{x^2}{2 a^2}+\frac{2 b^2 x^{3/2} \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^3}{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^3}{b+a \cosh (c+d x)} \, dx,x,\sqrt{x}\right )}{a^2 \left (a^2-b^2\right )}-\frac{\left (6 b^2\right ) \operatorname{Subst}\left (\int \frac{x^2 \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^{3/2}}{a^2 \left (a^2-b^2\right ) d}+\frac{x^2}{2 a^2}+\frac{2 b^2 x^{3/2} \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^3}{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^3}{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^3}{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 (6 b^2\right ) \operatorname{Subst}\left (\int \frac{e^{c+d x} x^2}{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 (6 b^2\right ) \operatorname{Subst}\left (\int \frac{e^{c+d x} x^2}{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^{3/2}}{a^2 \left (a^2-b^2\right ) d}+\frac{x^2}{2 a^2}-\frac{6 b^2 x \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^{3/2} \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{6 b^2 x \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^{3/2} \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^{3/2} \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^3}{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^3}{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 (12 b^2\right ) \operatorname{Subst}\left (\int x \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 (12 b^2\right ) \operatorname{Subst}\left (\int x \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{(12 b) \operatorname{Subst}\left (\int x^2 \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{(12 b) \operatorname{Subst}\left (\int x^2 \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^{3/2}}{a^2 \left (a^2-b^2\right ) d}+\frac{x^2}{2 a^2}-\frac{6 b^2 x \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^{3/2} \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^{3/2} \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{6 b^2 x \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^{3/2} \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^{3/2} \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{12 b^2 \sqrt{x} \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{12 b x \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{12 b^2 \sqrt{x} \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{12 b x \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^{3/2} \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 (12 b^2\right ) \operatorname{Subst}\left (\int \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 (12 b^2\right ) \operatorname{Subst}\left (\int \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{(24 b) \operatorname{Subst}\left (\int x \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{(24 b) \operatorname{Subst}\left (\int x \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 (6 b^3\right ) \operatorname{Subst}\left (\int x^2 \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 (6 b^3\right ) \operatorname{Subst}\left (\int x^2 \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^{3/2}}{a^2 \left (a^2-b^2\right ) d}+\frac{x^2}{2 a^2}-\frac{6 b^2 x \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^{3/2} \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^{3/2} \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{6 b^2 x \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^{3/2} \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^{3/2} \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{12 b^2 \sqrt{x} \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{6 b^3 x \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{12 b x \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{12 b^2 \sqrt{x} \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{6 b^3 x \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{12 b x \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{24 b \sqrt{x} \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{24 b \sqrt{x} \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^{3/2} \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 (12 b^2\right ) \operatorname{Subst}\left (\int \frac{\text{Li}_2\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^4}+\frac{\left (12 b^2\right ) \operatorname{Subst}\left (\int \frac{\text{Li}_2\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^4}-\frac{(24 b) \operatorname{Subst}\left (\int \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{(24 b) \operatorname{Subst}\left (\int \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 (12 b^3\right ) \operatorname{Subst}\left (\int x \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 (12 b^3\right ) \operatorname{Subst}\left (\int x \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^{3/2}}{a^2 \left (a^2-b^2\right ) d}+\frac{x^2}{2 a^2}-\frac{6 b^2 x \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^{3/2} \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^{3/2} \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{6 b^2 x \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^{3/2} \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^{3/2} \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{12 b^2 \sqrt{x} \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{6 b^3 x \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{12 b x \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{12 b^2 \sqrt{x} \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{6 b^3 x \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{12 b x \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{12 b^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{12 b^3 \sqrt{x} \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{24 b \sqrt{x} \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{12 b^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{12 b^3 \sqrt{x} \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{24 b \sqrt{x} \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^{3/2} \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{(24 b) \operatorname{Subst}\left (\int \frac{\text{Li}_3\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^4}+\frac{(24 b) \operatorname{Subst}\left (\int \frac{\text{Li}_3\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^4}+\frac{\left (12 b^3\right ) \operatorname{Subst}\left (\int \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 (12 b^3\right ) \operatorname{Subst}\left (\int \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^{3/2}}{a^2 \left (a^2-b^2\right ) d}+\frac{x^2}{2 a^2}-\frac{6 b^2 x \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^{3/2} \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^{3/2} \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{6 b^2 x \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^{3/2} \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^{3/2} \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{12 b^2 \sqrt{x} \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{6 b^3 x \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{12 b x \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{12 b^2 \sqrt{x} \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{6 b^3 x \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{12 b x \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{12 b^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{12 b^3 \sqrt{x} \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{24 b \sqrt{x} \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{12 b^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{12 b^3 \sqrt{x} \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{24 b \sqrt{x} \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{24 b \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{24 b \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^{3/2} \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 (12 b^3\right ) \operatorname{Subst}\left (\int \frac{\text{Li}_3\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^4}-\frac{\left (12 b^3\right ) \operatorname{Subst}\left (\int \frac{\text{Li}_3\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^4}\\ &=\frac{2 b^2 x^{3/2}}{a^2 \left (a^2-b^2\right ) d}+\frac{x^2}{2 a^2}-\frac{6 b^2 x \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^{3/2} \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^{3/2} \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{6 b^2 x \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^{3/2} \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^{3/2} \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{12 b^2 \sqrt{x} \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{6 b^3 x \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{12 b x \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{12 b^2 \sqrt{x} \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{6 b^3 x \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{12 b x \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{12 b^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{12 b^3 \sqrt{x} \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{24 b \sqrt{x} \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{12 b^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{12 b^3 \sqrt{x} \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{24 b \sqrt{x} \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{12 b^3 \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{24 b \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{12 b^3 \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{24 b \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^{3/2} \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 = 16.514, size = 1393, normalized size = 1. \[ \frac{\left (b+a \cosh \left (c+d \sqrt{x}\right )\right ) \text{sech}^2\left (c+d \sqrt{x}\right ) \left (\frac{4 x^{3/2} \text{sech}(c) \left (a \sinh \left (d \sqrt{x}\right )-b \sinh (c)\right ) b^2}{(a-b) (a+b) d}+\frac{4 e^c \left (b+a \cosh \left (c+d \sqrt{x}\right )\right ) \left (2 b e^c x^{3/2}+\frac{e^{-c} \left (1+e^{2 c}\right ) \left (-2 a^2 e^c x^{3/2} \log \left (\frac{e^{2 c+d \sqrt{x}} a}{b e^c-\sqrt{\left (b^2-a^2\right ) e^{2 c}}}+1\right ) d^3+b^2 e^c x^{3/2} \log \left (\frac{e^{2 c+d \sqrt{x}} a}{b e^c-\sqrt{\left (b^2-a^2\right ) e^{2 c}}}+1\right ) d^3+2 a^2 e^c x^{3/2} \log \left (\frac{e^{2 c+d \sqrt{x}} a}{e^c b+\sqrt{\left (b^2-a^2\right ) e^{2 c}}}+1\right ) d^3-b^2 e^c x^{3/2} \log \left (\frac{e^{2 c+d \sqrt{x}} a}{e^c b+\sqrt{\left (b^2-a^2\right ) e^{2 c}}}+1\right ) d^3-3 b \sqrt{\left (b^2-a^2\right ) e^{2 c}} x \log \left (\frac{e^{2 c+d \sqrt{x}} a}{b e^c-\sqrt{\left (b^2-a^2\right ) e^{2 c}}}+1\right ) d^2-3 b \sqrt{\left (b^2-a^2\right ) e^{2 c}} x \log \left (\frac{e^{2 c+d \sqrt{x}} a}{e^c b+\sqrt{\left (b^2-a^2\right ) e^{2 c}}}+1\right ) d^2+3 \left (-2 d e^c \sqrt{x} a^2-2 b \sqrt{\left (b^2-a^2\right ) e^{2 c}}+b^2 d e^c \sqrt{x}\right ) \sqrt{x} \text{PolyLog}\left (2,-\frac{a e^{2 c+d \sqrt{x}}}{b e^c-\sqrt{\left (b^2-a^2\right ) e^{2 c}}}\right ) d-3 \left (-2 d e^c \sqrt{x} a^2+2 b \sqrt{\left (b^2-a^2\right ) e^{2 c}}+b^2 d e^c \sqrt{x}\right ) \sqrt{x} \text{PolyLog}\left (2,-\frac{a e^{2 c+d \sqrt{x}}}{e^c b+\sqrt{\left (b^2-a^2\right ) e^{2 c}}}\right ) d+12 a^2 e^c \sqrt{x} \text{PolyLog}\left (3,-\frac{a e^{2 c+d \sqrt{x}}}{b e^c-\sqrt{\left (b^2-a^2\right ) e^{2 c}}}\right ) d-6 b^2 e^c \sqrt{x} \text{PolyLog}\left (3,-\frac{a e^{2 c+d \sqrt{x}}}{b e^c-\sqrt{\left (b^2-a^2\right ) e^{2 c}}}\right ) d-12 a^2 e^c \sqrt{x} \text{PolyLog}\left (3,-\frac{a e^{2 c+d \sqrt{x}}}{e^c b+\sqrt{\left (b^2-a^2\right ) e^{2 c}}}\right ) d+6 b^2 e^c \sqrt{x} \text{PolyLog}\left (3,-\frac{a e^{2 c+d \sqrt{x}}}{e^c b+\sqrt{\left (b^2-a^2\right ) e^{2 c}}}\right ) d+6 b \sqrt{\left (b^2-a^2\right ) e^{2 c}} \text{PolyLog}\left (3,-\frac{a e^{2 c+d \sqrt{x}}}{b e^c-\sqrt{\left (b^2-a^2\right ) e^{2 c}}}\right )+6 b \sqrt{\left (b^2-a^2\right ) e^{2 c}} \text{PolyLog}\left (3,-\frac{a e^{2 c+d \sqrt{x}}}{e^c b+\sqrt{\left (b^2-a^2\right ) e^{2 c}}}\right )-12 a^2 e^c \text{PolyLog}\left (4,-\frac{a e^{2 c+d \sqrt{x}}}{b e^c-\sqrt{\left (b^2-a^2\right ) e^{2 c}}}\right )+6 b^2 e^c \text{PolyLog}\left (4,-\frac{a e^{2 c+d \sqrt{x}}}{b e^c-\sqrt{\left (b^2-a^2\right ) e^{2 c}}}\right )+12 a^2 e^c \text{PolyLog}\left (4,-\frac{a e^{2 c+d \sqrt{x}}}{e^c b+\sqrt{\left (b^2-a^2\right ) e^{2 c}}}\right )-6 b^2 e^c \text{PolyLog}\left (4,-\frac{a e^{2 c+d \sqrt{x}}}{e^c b+\sqrt{\left (b^2-a^2\right ) e^{2 c}}}\right )\right )}{d^3 \sqrt{\left (b^2-a^2\right ) e^{2 c}}}\right ) b}{\left (a^2-b^2\right ) d \left (1+e^{2 c}\right )}+x^2 \left (b+a \cosh \left (c+d \sqrt{x}\right )\right )\right )}{2 a^2 \left (a+b \text{sech}\left (c+d \sqrt{x}\right )\right )^2} \]

Antiderivative was successfully verified.

[In]

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

[Out]

((b + a*Cosh[c + d*Sqrt[x]])*Sech[c + d*Sqrt[x]]^2*(x^2*(b + a*Cosh[c + d*Sqrt[x]]) + (4*b*E^c*(b + a*Cosh[c +
d*Sqrt[x]])*(2*b*E^c*x^(3/2) + ((1 + E^(2*c))*(-3*b*d^2*Sqrt[(-a^2 + b^2)*E^(2*c)]*x*Log[1 + (a*E^(2*c + d*Sq
rt[x]))/(b*E^c - Sqrt[(-a^2 + b^2)*E^(2*c)])] - 2*a^2*d^3*E^c*x^(3/2)*Log[1 + (a*E^(2*c + d*Sqrt[x]))/(b*E^c -
Sqrt[(-a^2 + b^2)*E^(2*c)])] + b^2*d^3*E^c*x^(3/2)*Log[1 + (a*E^(2*c + d*Sqrt[x]))/(b*E^c - Sqrt[(-a^2 + b^2)
*E^(2*c)])] - 3*b*d^2*Sqrt[(-a^2 + b^2)*E^(2*c)]*x*Log[1 + (a*E^(2*c + d*Sqrt[x]))/(b*E^c + Sqrt[(-a^2 + b^2)*
E^(2*c)])] + 2*a^2*d^3*E^c*x^(3/2)*Log[1 + (a*E^(2*c + d*Sqrt[x]))/(b*E^c + Sqrt[(-a^2 + b^2)*E^(2*c)])] - b^2
*d^3*E^c*x^(3/2)*Log[1 + (a*E^(2*c + d*Sqrt[x]))/(b*E^c + Sqrt[(-a^2 + b^2)*E^(2*c)])] + 3*d*(-2*b*Sqrt[(-a^2
+ b^2)*E^(2*c)] - 2*a^2*d*E^c*Sqrt[x] + b^2*d*E^c*Sqrt[x])*Sqrt[x]*PolyLog[2, -((a*E^(2*c + d*Sqrt[x]))/(b*E^c
- Sqrt[(-a^2 + b^2)*E^(2*c)]))] - 3*d*(2*b*Sqrt[(-a^2 + b^2)*E^(2*c)] - 2*a^2*d*E^c*Sqrt[x] + b^2*d*E^c*Sqrt[
x])*Sqrt[x]*PolyLog[2, -((a*E^(2*c + d*Sqrt[x]))/(b*E^c + Sqrt[(-a^2 + b^2)*E^(2*c)]))] + 6*b*Sqrt[(-a^2 + b^2
)*E^(2*c)]*PolyLog[3, -((a*E^(2*c + d*Sqrt[x]))/(b*E^c - Sqrt[(-a^2 + b^2)*E^(2*c)]))] + 12*a^2*d*E^c*Sqrt[x]*
PolyLog[3, -((a*E^(2*c + d*Sqrt[x]))/(b*E^c - Sqrt[(-a^2 + b^2)*E^(2*c)]))] - 6*b^2*d*E^c*Sqrt[x]*PolyLog[3, -
((a*E^(2*c + d*Sqrt[x]))/(b*E^c - Sqrt[(-a^2 + b^2)*E^(2*c)]))] + 6*b*Sqrt[(-a^2 + b^2)*E^(2*c)]*PolyLog[3, -(
(a*E^(2*c + d*Sqrt[x]))/(b*E^c + Sqrt[(-a^2 + b^2)*E^(2*c)]))] - 12*a^2*d*E^c*Sqrt[x]*PolyLog[3, -((a*E^(2*c +
d*Sqrt[x]))/(b*E^c + Sqrt[(-a^2 + b^2)*E^(2*c)]))] + 6*b^2*d*E^c*Sqrt[x]*PolyLog[3, -((a*E^(2*c + d*Sqrt[x]))
/(b*E^c + Sqrt[(-a^2 + b^2)*E^(2*c)]))] - 12*a^2*E^c*PolyLog[4, -((a*E^(2*c + d*Sqrt[x]))/(b*E^c - Sqrt[(-a^2
+ b^2)*E^(2*c)]))] + 6*b^2*E^c*PolyLog[4, -((a*E^(2*c + d*Sqrt[x]))/(b*E^c - Sqrt[(-a^2 + b^2)*E^(2*c)]))] + 1
2*a^2*E^c*PolyLog[4, -((a*E^(2*c + d*Sqrt[x]))/(b*E^c + Sqrt[(-a^2 + b^2)*E^(2*c)]))] - 6*b^2*E^c*PolyLog[4, -
((a*E^(2*c + d*Sqrt[x]))/(b*E^c + Sqrt[(-a^2 + b^2)*E^(2*c)]))]))/(d^3*E^c*Sqrt[(-a^2 + b^2)*E^(2*c)])))/((a^2
- b^2)*d*(1 + E^(2*c))) + (4*b^2*x^(3/2)*Sech[c]*(-(b*Sinh[c]) + a*Sinh[d*Sqrt[x]]))/((a - b)*(a + b)*d)))/(2
*a^2*(a + b*Sech[c + d*Sqrt[x]])^2)

________________________________________________________________________________________

Maple [F]  time = 0.096, size = 0, normalized size = 0. \begin{align*} \int{x \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/(a+b*sech(c+d*x^(1/2)))^2,x)

[Out]

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

________________________________________________________________________________________

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

[Out]

Exception raised: ValueError

________________________________________________________________________________________

Fricas [F]  time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (\frac{x}{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/(a+b*sech(c+d*x^(1/2)))^2,x, algorithm="fricas")

[Out]

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

________________________________________________________________________________________

Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{x}{\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/(a+b*sech(c+d*x**(1/2)))**2,x)

[Out]

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

________________________________________________________________________________________

Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{x}{{\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/(a+b*sech(c+d*x^(1/2)))^2,x, algorithm="giac")

[Out]

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