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

Optimal. Leaf size=961 \[ \frac{x^4}{4 a}-\frac{2 b \log \left (\frac{e^{c+d \sqrt{x}} a}{b-\sqrt{b^2-a^2}}+1\right ) x^{7/2}}{a \sqrt{b^2-a^2} d}+\frac{2 b \log \left (\frac{e^{c+d \sqrt{x}} a}{b+\sqrt{b^2-a^2}}+1\right ) x^{7/2}}{a \sqrt{b^2-a^2} d}-\frac{14 b \text{PolyLog}\left (2,-\frac{a e^{c+d \sqrt{x}}}{b-\sqrt{b^2-a^2}}\right ) x^3}{a \sqrt{b^2-a^2} d^2}+\frac{14 b \text{PolyLog}\left (2,-\frac{a e^{c+d \sqrt{x}}}{b+\sqrt{b^2-a^2}}\right ) x^3}{a \sqrt{b^2-a^2} d^2}+\frac{84 b \text{PolyLog}\left (3,-\frac{a e^{c+d \sqrt{x}}}{b-\sqrt{b^2-a^2}}\right ) x^{5/2}}{a \sqrt{b^2-a^2} d^3}-\frac{84 b \text{PolyLog}\left (3,-\frac{a e^{c+d \sqrt{x}}}{b+\sqrt{b^2-a^2}}\right ) x^{5/2}}{a \sqrt{b^2-a^2} d^3}-\frac{420 b \text{PolyLog}\left (4,-\frac{a e^{c+d \sqrt{x}}}{b-\sqrt{b^2-a^2}}\right ) x^2}{a \sqrt{b^2-a^2} d^4}+\frac{420 b \text{PolyLog}\left (4,-\frac{a e^{c+d \sqrt{x}}}{b+\sqrt{b^2-a^2}}\right ) x^2}{a \sqrt{b^2-a^2} d^4}+\frac{1680 b \text{PolyLog}\left (5,-\frac{a e^{c+d \sqrt{x}}}{b-\sqrt{b^2-a^2}}\right ) x^{3/2}}{a \sqrt{b^2-a^2} d^5}-\frac{1680 b \text{PolyLog}\left (5,-\frac{a e^{c+d \sqrt{x}}}{b+\sqrt{b^2-a^2}}\right ) x^{3/2}}{a \sqrt{b^2-a^2} d^5}-\frac{5040 b \text{PolyLog}\left (6,-\frac{a e^{c+d \sqrt{x}}}{b-\sqrt{b^2-a^2}}\right ) x}{a \sqrt{b^2-a^2} d^6}+\frac{5040 b \text{PolyLog}\left (6,-\frac{a e^{c+d \sqrt{x}}}{b+\sqrt{b^2-a^2}}\right ) x}{a \sqrt{b^2-a^2} d^6}+\frac{10080 b \text{PolyLog}\left (7,-\frac{a e^{c+d \sqrt{x}}}{b-\sqrt{b^2-a^2}}\right ) \sqrt{x}}{a \sqrt{b^2-a^2} d^7}-\frac{10080 b \text{PolyLog}\left (7,-\frac{a e^{c+d \sqrt{x}}}{b+\sqrt{b^2-a^2}}\right ) \sqrt{x}}{a \sqrt{b^2-a^2} d^7}-\frac{10080 b \text{PolyLog}\left (8,-\frac{a e^{c+d \sqrt{x}}}{b-\sqrt{b^2-a^2}}\right )}{a \sqrt{b^2-a^2} d^8}+\frac{10080 b \text{PolyLog}\left (8,-\frac{a e^{c+d \sqrt{x}}}{b+\sqrt{b^2-a^2}}\right )}{a \sqrt{b^2-a^2} d^8} \]

[Out]

x^4/(4*a) - (2*b*x^(7/2)*Log[1 + (a*E^(c + d*Sqrt[x]))/(b - Sqrt[-a^2 + b^2])])/(a*Sqrt[-a^2 + b^2]*d) + (2*b*
x^(7/2)*Log[1 + (a*E^(c + d*Sqrt[x]))/(b + Sqrt[-a^2 + b^2])])/(a*Sqrt[-a^2 + b^2]*d) - (14*b*x^3*PolyLog[2, -
((a*E^(c + d*Sqrt[x]))/(b - Sqrt[-a^2 + b^2]))])/(a*Sqrt[-a^2 + b^2]*d^2) + (14*b*x^3*PolyLog[2, -((a*E^(c + d
*Sqrt[x]))/(b + Sqrt[-a^2 + b^2]))])/(a*Sqrt[-a^2 + b^2]*d^2) + (84*b*x^(5/2)*PolyLog[3, -((a*E^(c + d*Sqrt[x]
))/(b - Sqrt[-a^2 + b^2]))])/(a*Sqrt[-a^2 + b^2]*d^3) - (84*b*x^(5/2)*PolyLog[3, -((a*E^(c + d*Sqrt[x]))/(b +
Sqrt[-a^2 + b^2]))])/(a*Sqrt[-a^2 + b^2]*d^3) - (420*b*x^2*PolyLog[4, -((a*E^(c + d*Sqrt[x]))/(b - Sqrt[-a^2 +
b^2]))])/(a*Sqrt[-a^2 + b^2]*d^4) + (420*b*x^2*PolyLog[4, -((a*E^(c + d*Sqrt[x]))/(b + Sqrt[-a^2 + b^2]))])/(
a*Sqrt[-a^2 + b^2]*d^4) + (1680*b*x^(3/2)*PolyLog[5, -((a*E^(c + d*Sqrt[x]))/(b - Sqrt[-a^2 + b^2]))])/(a*Sqrt
[-a^2 + b^2]*d^5) - (1680*b*x^(3/2)*PolyLog[5, -((a*E^(c + d*Sqrt[x]))/(b + Sqrt[-a^2 + b^2]))])/(a*Sqrt[-a^2
+ b^2]*d^5) - (5040*b*x*PolyLog[6, -((a*E^(c + d*Sqrt[x]))/(b - Sqrt[-a^2 + b^2]))])/(a*Sqrt[-a^2 + b^2]*d^6)
+ (5040*b*x*PolyLog[6, -((a*E^(c + d*Sqrt[x]))/(b + Sqrt[-a^2 + b^2]))])/(a*Sqrt[-a^2 + b^2]*d^6) + (10080*b*S
qrt[x]*PolyLog[7, -((a*E^(c + d*Sqrt[x]))/(b - Sqrt[-a^2 + b^2]))])/(a*Sqrt[-a^2 + b^2]*d^7) - (10080*b*Sqrt[x
]*PolyLog[7, -((a*E^(c + d*Sqrt[x]))/(b + Sqrt[-a^2 + b^2]))])/(a*Sqrt[-a^2 + b^2]*d^7) - (10080*b*PolyLog[8,
-((a*E^(c + d*Sqrt[x]))/(b - Sqrt[-a^2 + b^2]))])/(a*Sqrt[-a^2 + b^2]*d^8) + (10080*b*PolyLog[8, -((a*E^(c + d
*Sqrt[x]))/(b + Sqrt[-a^2 + b^2]))])/(a*Sqrt[-a^2 + b^2]*d^8)

________________________________________________________________________________________

Rubi [A]  time = 1.33087, antiderivative size = 961, normalized size of antiderivative = 1., number of steps used = 23, number of rules used = 9, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.45, Rules used = {5436, 4191, 3320, 2264, 2190, 2531, 6609, 2282, 6589} \[ \frac{x^4}{4 a}-\frac{2 b \log \left (\frac{e^{c+d \sqrt{x}} a}{b-\sqrt{b^2-a^2}}+1\right ) x^{7/2}}{a \sqrt{b^2-a^2} d}+\frac{2 b \log \left (\frac{e^{c+d \sqrt{x}} a}{b+\sqrt{b^2-a^2}}+1\right ) x^{7/2}}{a \sqrt{b^2-a^2} d}-\frac{14 b \text{PolyLog}\left (2,-\frac{a e^{c+d \sqrt{x}}}{b-\sqrt{b^2-a^2}}\right ) x^3}{a \sqrt{b^2-a^2} d^2}+\frac{14 b \text{PolyLog}\left (2,-\frac{a e^{c+d \sqrt{x}}}{b+\sqrt{b^2-a^2}}\right ) x^3}{a \sqrt{b^2-a^2} d^2}+\frac{84 b \text{PolyLog}\left (3,-\frac{a e^{c+d \sqrt{x}}}{b-\sqrt{b^2-a^2}}\right ) x^{5/2}}{a \sqrt{b^2-a^2} d^3}-\frac{84 b \text{PolyLog}\left (3,-\frac{a e^{c+d \sqrt{x}}}{b+\sqrt{b^2-a^2}}\right ) x^{5/2}}{a \sqrt{b^2-a^2} d^3}-\frac{420 b \text{PolyLog}\left (4,-\frac{a e^{c+d \sqrt{x}}}{b-\sqrt{b^2-a^2}}\right ) x^2}{a \sqrt{b^2-a^2} d^4}+\frac{420 b \text{PolyLog}\left (4,-\frac{a e^{c+d \sqrt{x}}}{b+\sqrt{b^2-a^2}}\right ) x^2}{a \sqrt{b^2-a^2} d^4}+\frac{1680 b \text{PolyLog}\left (5,-\frac{a e^{c+d \sqrt{x}}}{b-\sqrt{b^2-a^2}}\right ) x^{3/2}}{a \sqrt{b^2-a^2} d^5}-\frac{1680 b \text{PolyLog}\left (5,-\frac{a e^{c+d \sqrt{x}}}{b+\sqrt{b^2-a^2}}\right ) x^{3/2}}{a \sqrt{b^2-a^2} d^5}-\frac{5040 b \text{PolyLog}\left (6,-\frac{a e^{c+d \sqrt{x}}}{b-\sqrt{b^2-a^2}}\right ) x}{a \sqrt{b^2-a^2} d^6}+\frac{5040 b \text{PolyLog}\left (6,-\frac{a e^{c+d \sqrt{x}}}{b+\sqrt{b^2-a^2}}\right ) x}{a \sqrt{b^2-a^2} d^6}+\frac{10080 b \text{PolyLog}\left (7,-\frac{a e^{c+d \sqrt{x}}}{b-\sqrt{b^2-a^2}}\right ) \sqrt{x}}{a \sqrt{b^2-a^2} d^7}-\frac{10080 b \text{PolyLog}\left (7,-\frac{a e^{c+d \sqrt{x}}}{b+\sqrt{b^2-a^2}}\right ) \sqrt{x}}{a \sqrt{b^2-a^2} d^7}-\frac{10080 b \text{PolyLog}\left (8,-\frac{a e^{c+d \sqrt{x}}}{b-\sqrt{b^2-a^2}}\right )}{a \sqrt{b^2-a^2} d^8}+\frac{10080 b \text{PolyLog}\left (8,-\frac{a e^{c+d \sqrt{x}}}{b+\sqrt{b^2-a^2}}\right )}{a \sqrt{b^2-a^2} d^8} \]

Antiderivative was successfully verified.

[In]

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

[Out]

x^4/(4*a) - (2*b*x^(7/2)*Log[1 + (a*E^(c + d*Sqrt[x]))/(b - Sqrt[-a^2 + b^2])])/(a*Sqrt[-a^2 + b^2]*d) + (2*b*
x^(7/2)*Log[1 + (a*E^(c + d*Sqrt[x]))/(b + Sqrt[-a^2 + b^2])])/(a*Sqrt[-a^2 + b^2]*d) - (14*b*x^3*PolyLog[2, -
((a*E^(c + d*Sqrt[x]))/(b - Sqrt[-a^2 + b^2]))])/(a*Sqrt[-a^2 + b^2]*d^2) + (14*b*x^3*PolyLog[2, -((a*E^(c + d
*Sqrt[x]))/(b + Sqrt[-a^2 + b^2]))])/(a*Sqrt[-a^2 + b^2]*d^2) + (84*b*x^(5/2)*PolyLog[3, -((a*E^(c + d*Sqrt[x]
))/(b - Sqrt[-a^2 + b^2]))])/(a*Sqrt[-a^2 + b^2]*d^3) - (84*b*x^(5/2)*PolyLog[3, -((a*E^(c + d*Sqrt[x]))/(b +
Sqrt[-a^2 + b^2]))])/(a*Sqrt[-a^2 + b^2]*d^3) - (420*b*x^2*PolyLog[4, -((a*E^(c + d*Sqrt[x]))/(b - Sqrt[-a^2 +
b^2]))])/(a*Sqrt[-a^2 + b^2]*d^4) + (420*b*x^2*PolyLog[4, -((a*E^(c + d*Sqrt[x]))/(b + Sqrt[-a^2 + b^2]))])/(
a*Sqrt[-a^2 + b^2]*d^4) + (1680*b*x^(3/2)*PolyLog[5, -((a*E^(c + d*Sqrt[x]))/(b - Sqrt[-a^2 + b^2]))])/(a*Sqrt
[-a^2 + b^2]*d^5) - (1680*b*x^(3/2)*PolyLog[5, -((a*E^(c + d*Sqrt[x]))/(b + Sqrt[-a^2 + b^2]))])/(a*Sqrt[-a^2
+ b^2]*d^5) - (5040*b*x*PolyLog[6, -((a*E^(c + d*Sqrt[x]))/(b - Sqrt[-a^2 + b^2]))])/(a*Sqrt[-a^2 + b^2]*d^6)
+ (5040*b*x*PolyLog[6, -((a*E^(c + d*Sqrt[x]))/(b + Sqrt[-a^2 + b^2]))])/(a*Sqrt[-a^2 + b^2]*d^6) + (10080*b*S
qrt[x]*PolyLog[7, -((a*E^(c + d*Sqrt[x]))/(b - Sqrt[-a^2 + b^2]))])/(a*Sqrt[-a^2 + b^2]*d^7) - (10080*b*Sqrt[x
]*PolyLog[7, -((a*E^(c + d*Sqrt[x]))/(b + Sqrt[-a^2 + b^2]))])/(a*Sqrt[-a^2 + b^2]*d^7) - (10080*b*PolyLog[8,
-((a*E^(c + d*Sqrt[x]))/(b - Sqrt[-a^2 + b^2]))])/(a*Sqrt[-a^2 + b^2]*d^8) + (10080*b*PolyLog[8, -((a*E^(c + d
*Sqrt[x]))/(b + Sqrt[-a^2 + b^2]))])/(a*Sqrt[-a^2 + b^2]*d^8)

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 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]

Rubi steps

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

Mathematica [A]  time = 3.02526, size = 939, normalized size = 0.98 \[ \frac{\left (b+a \cosh \left (c+d \sqrt{x}\right )\right ) \left (x^4-\frac{8 b e^c \left (x^{7/2} \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^7-x^{7/2} \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^7+7 x^3 \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^6-7 x^3 \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^6-42 x^{5/2} \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^5+42 x^{5/2} \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^5+210 x^2 \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 ) d^4-210 x^2 \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 ) d^4-840 x^{3/2} \text{PolyLog}\left (5,-\frac{a e^{2 c+d \sqrt{x}}}{b e^c-\sqrt{\left (b^2-a^2\right ) e^{2 c}}}\right ) d^3+840 x^{3/2} \text{PolyLog}\left (5,-\frac{a e^{2 c+d \sqrt{x}}}{e^c b+\sqrt{\left (b^2-a^2\right ) e^{2 c}}}\right ) d^3+2520 x \text{PolyLog}\left (6,-\frac{a e^{2 c+d \sqrt{x}}}{b e^c-\sqrt{\left (b^2-a^2\right ) e^{2 c}}}\right ) d^2-2520 x \text{PolyLog}\left (6,-\frac{a e^{2 c+d \sqrt{x}}}{e^c b+\sqrt{\left (b^2-a^2\right ) e^{2 c}}}\right ) d^2-5040 \sqrt{x} \text{PolyLog}\left (7,-\frac{a e^{2 c+d \sqrt{x}}}{b e^c-\sqrt{\left (b^2-a^2\right ) e^{2 c}}}\right ) d+5040 \sqrt{x} \text{PolyLog}\left (7,-\frac{a e^{2 c+d \sqrt{x}}}{e^c b+\sqrt{\left (b^2-a^2\right ) e^{2 c}}}\right ) d+5040 \text{PolyLog}\left (8,-\frac{a e^{2 c+d \sqrt{x}}}{b e^c-\sqrt{\left (b^2-a^2\right ) e^{2 c}}}\right )-5040 \text{PolyLog}\left (8,-\frac{a e^{2 c+d \sqrt{x}}}{e^c b+\sqrt{\left (b^2-a^2\right ) e^{2 c}}}\right )\right )}{d^8 \sqrt{\left (b^2-a^2\right ) e^{2 c}}}\right ) \text{sech}\left (c+d \sqrt{x}\right )}{4 a \left (a+b \text{sech}\left (c+d \sqrt{x}\right )\right )} \]

Antiderivative was successfully verified.

[In]

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

[Out]

((b + a*Cosh[c + d*Sqrt[x]])*(x^4 - (8*b*E^c*(d^7*x^(7/2)*Log[1 + (a*E^(2*c + d*Sqrt[x]))/(b*E^c - Sqrt[(-a^2
+ b^2)*E^(2*c)])] - d^7*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^6*
x^3*PolyLog[2, -((a*E^(2*c + d*Sqrt[x]))/(b*E^c - Sqrt[(-a^2 + b^2)*E^(2*c)]))] - 7*d^6*x^3*PolyLog[2, -((a*E^
(2*c + d*Sqrt[x]))/(b*E^c + Sqrt[(-a^2 + b^2)*E^(2*c)]))] - 42*d^5*x^(5/2)*PolyLog[3, -((a*E^(2*c + d*Sqrt[x])
)/(b*E^c - Sqrt[(-a^2 + b^2)*E^(2*c)]))] + 42*d^5*x^(5/2)*PolyLog[3, -((a*E^(2*c + d*Sqrt[x]))/(b*E^c + Sqrt[(
-a^2 + b^2)*E^(2*c)]))] + 210*d^4*x^2*PolyLog[4, -((a*E^(2*c + d*Sqrt[x]))/(b*E^c - Sqrt[(-a^2 + b^2)*E^(2*c)]
))] - 210*d^4*x^2*PolyLog[4, -((a*E^(2*c + d*Sqrt[x]))/(b*E^c + Sqrt[(-a^2 + b^2)*E^(2*c)]))] - 840*d^3*x^(3/2
)*PolyLog[5, -((a*E^(2*c + d*Sqrt[x]))/(b*E^c - Sqrt[(-a^2 + b^2)*E^(2*c)]))] + 840*d^3*x^(3/2)*PolyLog[5, -((
a*E^(2*c + d*Sqrt[x]))/(b*E^c + Sqrt[(-a^2 + b^2)*E^(2*c)]))] + 2520*d^2*x*PolyLog[6, -((a*E^(2*c + d*Sqrt[x])
)/(b*E^c - Sqrt[(-a^2 + b^2)*E^(2*c)]))] - 2520*d^2*x*PolyLog[6, -((a*E^(2*c + d*Sqrt[x]))/(b*E^c + Sqrt[(-a^2
+ b^2)*E^(2*c)]))] - 5040*d*Sqrt[x]*PolyLog[7, -((a*E^(2*c + d*Sqrt[x]))/(b*E^c - Sqrt[(-a^2 + b^2)*E^(2*c)])
)] + 5040*d*Sqrt[x]*PolyLog[7, -((a*E^(2*c + d*Sqrt[x]))/(b*E^c + Sqrt[(-a^2 + b^2)*E^(2*c)]))] + 5040*PolyLog
[8, -((a*E^(2*c + d*Sqrt[x]))/(b*E^c - Sqrt[(-a^2 + b^2)*E^(2*c)]))] - 5040*PolyLog[8, -((a*E^(2*c + d*Sqrt[x]
))/(b*E^c + Sqrt[(-a^2 + b^2)*E^(2*c)]))]))/(d^8*Sqrt[(-a^2 + b^2)*E^(2*c)]))*Sech[c + d*Sqrt[x]])/(4*a*(a + b
*Sech[c + d*Sqrt[x]]))

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Maple [F]  time = 0.088, size = 0, normalized size = 0. \begin{align*} \int{{x}^{3} \left ( a+b{\rm sech} \left (c+d\sqrt{x}\right ) \right ) ^{-1}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [F(-2)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: ValueError} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Exception raised: ValueError

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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