3.23 \(\int \frac{x^5}{(a+b \text{sech}(c+d x^2))^2} \, dx\)

Optimal. Leaf size=994 \[ \frac{x^6}{6 a^2}-\frac{b \log \left (\frac{e^{d x^2+c} a}{b-\sqrt{b^2-a^2}}+1\right ) x^4}{a^2 \sqrt{b^2-a^2} d}+\frac{b^3 \log \left (\frac{e^{d x^2+c} a}{b-\sqrt{b^2-a^2}}+1\right ) x^4}{2 a^2 \left (b^2-a^2\right )^{3/2} d}+\frac{b \log \left (\frac{e^{d x^2+c} a}{b+\sqrt{b^2-a^2}}+1\right ) x^4}{a^2 \sqrt{b^2-a^2} d}-\frac{b^3 \log \left (\frac{e^{d x^2+c} a}{b+\sqrt{b^2-a^2}}+1\right ) x^4}{2 a^2 \left (b^2-a^2\right )^{3/2} d}+\frac{b^2 \sinh \left (d x^2+c\right ) x^4}{2 a \left (a^2-b^2\right ) d \left (b+a \cosh \left (d x^2+c\right )\right )}+\frac{b^2 x^4}{2 a^2 \left (a^2-b^2\right ) d}-\frac{b^2 \log \left (\frac{e^{d x^2+c} a}{b-\sqrt{b^2-a^2}}+1\right ) x^2}{a^2 \left (a^2-b^2\right ) d^2}-\frac{b^2 \log \left (\frac{e^{d x^2+c} a}{b+\sqrt{b^2-a^2}}+1\right ) x^2}{a^2 \left (a^2-b^2\right ) d^2}-\frac{2 b \text{PolyLog}\left (2,-\frac{a e^{d x^2+c}}{b-\sqrt{b^2-a^2}}\right ) x^2}{a^2 \sqrt{b^2-a^2} d^2}+\frac{b^3 \text{PolyLog}\left (2,-\frac{a e^{d x^2+c}}{b-\sqrt{b^2-a^2}}\right ) x^2}{a^2 \left (b^2-a^2\right )^{3/2} d^2}+\frac{2 b \text{PolyLog}\left (2,-\frac{a e^{d x^2+c}}{b+\sqrt{b^2-a^2}}\right ) x^2}{a^2 \sqrt{b^2-a^2} d^2}-\frac{b^3 \text{PolyLog}\left (2,-\frac{a e^{d x^2+c}}{b+\sqrt{b^2-a^2}}\right ) x^2}{a^2 \left (b^2-a^2\right )^{3/2} d^2}-\frac{b^2 \text{PolyLog}\left (2,-\frac{a e^{d x^2+c}}{b-\sqrt{b^2-a^2}}\right )}{a^2 \left (a^2-b^2\right ) d^3}-\frac{b^2 \text{PolyLog}\left (2,-\frac{a e^{d x^2+c}}{b+\sqrt{b^2-a^2}}\right )}{a^2 \left (a^2-b^2\right ) d^3}+\frac{2 b \text{PolyLog}\left (3,-\frac{a e^{d x^2+c}}{b-\sqrt{b^2-a^2}}\right )}{a^2 \sqrt{b^2-a^2} d^3}-\frac{b^3 \text{PolyLog}\left (3,-\frac{a e^{d x^2+c}}{b-\sqrt{b^2-a^2}}\right )}{a^2 \left (b^2-a^2\right )^{3/2} d^3}-\frac{2 b \text{PolyLog}\left (3,-\frac{a e^{d x^2+c}}{b+\sqrt{b^2-a^2}}\right )}{a^2 \sqrt{b^2-a^2} d^3}+\frac{b^3 \text{PolyLog}\left (3,-\frac{a e^{d x^2+c}}{b+\sqrt{b^2-a^2}}\right )}{a^2 \left (b^2-a^2\right )^{3/2} d^3} \]

[Out]

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

________________________________________________________________________________________

Rubi [A]  time = 2.11466, antiderivative size = 994, normalized size of antiderivative = 1., number of steps used = 31, number of rules used = 12, integrand size = 18, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.667, Rules used = {5436, 4191, 3324, 3320, 2264, 2190, 2531, 2282, 6589, 5562, 2279, 2391} \[ \frac{x^6}{6 a^2}-\frac{b \log \left (\frac{e^{d x^2+c} a}{b-\sqrt{b^2-a^2}}+1\right ) x^4}{a^2 \sqrt{b^2-a^2} d}+\frac{b^3 \log \left (\frac{e^{d x^2+c} a}{b-\sqrt{b^2-a^2}}+1\right ) x^4}{2 a^2 \left (b^2-a^2\right )^{3/2} d}+\frac{b \log \left (\frac{e^{d x^2+c} a}{b+\sqrt{b^2-a^2}}+1\right ) x^4}{a^2 \sqrt{b^2-a^2} d}-\frac{b^3 \log \left (\frac{e^{d x^2+c} a}{b+\sqrt{b^2-a^2}}+1\right ) x^4}{2 a^2 \left (b^2-a^2\right )^{3/2} d}+\frac{b^2 \sinh \left (d x^2+c\right ) x^4}{2 a \left (a^2-b^2\right ) d \left (b+a \cosh \left (d x^2+c\right )\right )}+\frac{b^2 x^4}{2 a^2 \left (a^2-b^2\right ) d}-\frac{b^2 \log \left (\frac{e^{d x^2+c} a}{b-\sqrt{b^2-a^2}}+1\right ) x^2}{a^2 \left (a^2-b^2\right ) d^2}-\frac{b^2 \log \left (\frac{e^{d x^2+c} a}{b+\sqrt{b^2-a^2}}+1\right ) x^2}{a^2 \left (a^2-b^2\right ) d^2}-\frac{2 b \text{PolyLog}\left (2,-\frac{a e^{d x^2+c}}{b-\sqrt{b^2-a^2}}\right ) x^2}{a^2 \sqrt{b^2-a^2} d^2}+\frac{b^3 \text{PolyLog}\left (2,-\frac{a e^{d x^2+c}}{b-\sqrt{b^2-a^2}}\right ) x^2}{a^2 \left (b^2-a^2\right )^{3/2} d^2}+\frac{2 b \text{PolyLog}\left (2,-\frac{a e^{d x^2+c}}{b+\sqrt{b^2-a^2}}\right ) x^2}{a^2 \sqrt{b^2-a^2} d^2}-\frac{b^3 \text{PolyLog}\left (2,-\frac{a e^{d x^2+c}}{b+\sqrt{b^2-a^2}}\right ) x^2}{a^2 \left (b^2-a^2\right )^{3/2} d^2}-\frac{b^2 \text{PolyLog}\left (2,-\frac{a e^{d x^2+c}}{b-\sqrt{b^2-a^2}}\right )}{a^2 \left (a^2-b^2\right ) d^3}-\frac{b^2 \text{PolyLog}\left (2,-\frac{a e^{d x^2+c}}{b+\sqrt{b^2-a^2}}\right )}{a^2 \left (a^2-b^2\right ) d^3}+\frac{2 b \text{PolyLog}\left (3,-\frac{a e^{d x^2+c}}{b-\sqrt{b^2-a^2}}\right )}{a^2 \sqrt{b^2-a^2} d^3}-\frac{b^3 \text{PolyLog}\left (3,-\frac{a e^{d x^2+c}}{b-\sqrt{b^2-a^2}}\right )}{a^2 \left (b^2-a^2\right )^{3/2} d^3}-\frac{2 b \text{PolyLog}\left (3,-\frac{a e^{d x^2+c}}{b+\sqrt{b^2-a^2}}\right )}{a^2 \sqrt{b^2-a^2} d^3}+\frac{b^3 \text{PolyLog}\left (3,-\frac{a e^{d x^2+c}}{b+\sqrt{b^2-a^2}}\right )}{a^2 \left (b^2-a^2\right )^{3/2} d^3} \]

Antiderivative was successfully verified.

[In]

Int[x^5/(a + b*Sech[c + d*x^2])^2,x]

[Out]

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

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 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]

Rule 2279

Int[Log[(a_) + (b_.)*((F_)^((e_.)*((c_.) + (d_.)*(x_))))^(n_.)], x_Symbol] :> Dist[1/(d*e*n*Log[F]), Subst[Int
[Log[a + b*x]/x, x], x, (F^(e*(c + d*x)))^n], x] /; FreeQ[{F, a, b, c, d, e, n}, x] && GtQ[a, 0]

Rule 2391

Int[Log[(c_.)*((d_) + (e_.)*(x_)^(n_.))]/(x_), x_Symbol] :> -Simp[PolyLog[2, -(c*e*x^n)]/n, x] /; FreeQ[{c, d,
e, n}, x] && EqQ[c*d, 1]

Rubi steps

\begin{align*} \int \frac{x^5}{\left (a+b \text{sech}\left (c+d x^2\right )\right )^2} \, dx &=\frac{1}{2} \operatorname{Subst}\left (\int \frac{x^2}{(a+b \text{sech}(c+d x))^2} \, dx,x,x^2\right )\\ &=\frac{1}{2} \operatorname{Subst}\left (\int \left (\frac{x^2}{a^2}+\frac{b^2 x^2}{a^2 (b+a \cosh (c+d x))^2}-\frac{2 b x^2}{a^2 (b+a \cosh (c+d x))}\right ) \, dx,x,x^2\right )\\ &=\frac{x^6}{6 a^2}-\frac{b \operatorname{Subst}\left (\int \frac{x^2}{b+a \cosh (c+d x)} \, dx,x,x^2\right )}{a^2}+\frac{b^2 \operatorname{Subst}\left (\int \frac{x^2}{(b+a \cosh (c+d x))^2} \, dx,x,x^2\right )}{2 a^2}\\ &=\frac{x^6}{6 a^2}+\frac{b^2 x^4 \sinh \left (c+d x^2\right )}{2 a \left (a^2-b^2\right ) d \left (b+a \cosh \left (c+d x^2\right )\right )}-\frac{(2 b) \operatorname{Subst}\left (\int \frac{e^{c+d x} x^2}{a+2 b e^{c+d x}+a e^{2 (c+d x)}} \, dx,x,x^2\right )}{a^2}-\frac{b^3 \operatorname{Subst}\left (\int \frac{x^2}{b+a \cosh (c+d x)} \, dx,x,x^2\right )}{2 a^2 \left (a^2-b^2\right )}-\frac{b^2 \operatorname{Subst}\left (\int \frac{x \sinh (c+d x)}{b+a \cosh (c+d x)} \, dx,x,x^2\right )}{a \left (a^2-b^2\right ) d}\\ &=\frac{b^2 x^4}{2 a^2 \left (a^2-b^2\right ) d}+\frac{x^6}{6 a^2}+\frac{b^2 x^4 \sinh \left (c+d x^2\right )}{2 a \left (a^2-b^2\right ) d \left (b+a \cosh \left (c+d x^2\right )\right )}-\frac{b^3 \operatorname{Subst}\left (\int \frac{e^{c+d x} x^2}{a+2 b e^{c+d x}+a e^{2 (c+d x)}} \, dx,x,x^2\right )}{a^2 \left (a^2-b^2\right )}-\frac{(2 b) \operatorname{Subst}\left (\int \frac{e^{c+d x} x^2}{2 b-2 \sqrt{-a^2+b^2}+2 a e^{c+d x}} \, dx,x,x^2\right )}{a \sqrt{-a^2+b^2}}+\frac{(2 b) \operatorname{Subst}\left (\int \frac{e^{c+d x} x^2}{2 b+2 \sqrt{-a^2+b^2}+2 a e^{c+d x}} \, dx,x,x^2\right )}{a \sqrt{-a^2+b^2}}-\frac{b^2 \operatorname{Subst}\left (\int \frac{e^{c+d x} x}{b-\sqrt{-a^2+b^2}+a e^{c+d x}} \, dx,x,x^2\right )}{a \left (a^2-b^2\right ) d}-\frac{b^2 \operatorname{Subst}\left (\int \frac{e^{c+d x} x}{b+\sqrt{-a^2+b^2}+a e^{c+d x}} \, dx,x,x^2\right )}{a \left (a^2-b^2\right ) d}\\ &=\frac{b^2 x^4}{2 a^2 \left (a^2-b^2\right ) d}+\frac{x^6}{6 a^2}-\frac{b^2 x^2 \log \left (1+\frac{a e^{c+d x^2}}{b-\sqrt{-a^2+b^2}}\right )}{a^2 \left (a^2-b^2\right ) d^2}-\frac{b x^4 \log \left (1+\frac{a e^{c+d x^2}}{b-\sqrt{-a^2+b^2}}\right )}{a^2 \sqrt{-a^2+b^2} d}-\frac{b^2 x^2 \log \left (1+\frac{a e^{c+d x^2}}{b+\sqrt{-a^2+b^2}}\right )}{a^2 \left (a^2-b^2\right ) d^2}+\frac{b x^4 \log \left (1+\frac{a e^{c+d x^2}}{b+\sqrt{-a^2+b^2}}\right )}{a^2 \sqrt{-a^2+b^2} d}+\frac{b^2 x^4 \sinh \left (c+d x^2\right )}{2 a \left (a^2-b^2\right ) d \left (b+a \cosh \left (c+d x^2\right )\right )}+\frac{b^3 \operatorname{Subst}\left (\int \frac{e^{c+d x} x^2}{2 b-2 \sqrt{-a^2+b^2}+2 a e^{c+d x}} \, dx,x,x^2\right )}{a \left (-a^2+b^2\right )^{3/2}}-\frac{b^3 \operatorname{Subst}\left (\int \frac{e^{c+d x} x^2}{2 b+2 \sqrt{-a^2+b^2}+2 a e^{c+d x}} \, dx,x,x^2\right )}{a \left (-a^2+b^2\right )^{3/2}}+\frac{b^2 \operatorname{Subst}\left (\int \log \left (1+\frac{a e^{c+d x}}{b-\sqrt{-a^2+b^2}}\right ) \, dx,x,x^2\right )}{a^2 \left (a^2-b^2\right ) d^2}+\frac{b^2 \operatorname{Subst}\left (\int \log \left (1+\frac{a e^{c+d x}}{b+\sqrt{-a^2+b^2}}\right ) \, dx,x,x^2\right )}{a^2 \left (a^2-b^2\right ) d^2}+\frac{(2 b) \operatorname{Subst}\left (\int x \log \left (1+\frac{2 a e^{c+d x}}{2 b-2 \sqrt{-a^2+b^2}}\right ) \, dx,x,x^2\right )}{a^2 \sqrt{-a^2+b^2} d}-\frac{(2 b) \operatorname{Subst}\left (\int x \log \left (1+\frac{2 a e^{c+d x}}{2 b+2 \sqrt{-a^2+b^2}}\right ) \, dx,x,x^2\right )}{a^2 \sqrt{-a^2+b^2} d}\\ &=\frac{b^2 x^4}{2 a^2 \left (a^2-b^2\right ) d}+\frac{x^6}{6 a^2}-\frac{b^2 x^2 \log \left (1+\frac{a e^{c+d x^2}}{b-\sqrt{-a^2+b^2}}\right )}{a^2 \left (a^2-b^2\right ) d^2}+\frac{b^3 x^4 \log \left (1+\frac{a e^{c+d x^2}}{b-\sqrt{-a^2+b^2}}\right )}{2 a^2 \left (-a^2+b^2\right )^{3/2} d}-\frac{b x^4 \log \left (1+\frac{a e^{c+d x^2}}{b-\sqrt{-a^2+b^2}}\right )}{a^2 \sqrt{-a^2+b^2} d}-\frac{b^2 x^2 \log \left (1+\frac{a e^{c+d x^2}}{b+\sqrt{-a^2+b^2}}\right )}{a^2 \left (a^2-b^2\right ) d^2}-\frac{b^3 x^4 \log \left (1+\frac{a e^{c+d x^2}}{b+\sqrt{-a^2+b^2}}\right )}{2 a^2 \left (-a^2+b^2\right )^{3/2} d}+\frac{b x^4 \log \left (1+\frac{a e^{c+d x^2}}{b+\sqrt{-a^2+b^2}}\right )}{a^2 \sqrt{-a^2+b^2} d}-\frac{2 b x^2 \text{Li}_2\left (-\frac{a e^{c+d x^2}}{b-\sqrt{-a^2+b^2}}\right )}{a^2 \sqrt{-a^2+b^2} d^2}+\frac{2 b x^2 \text{Li}_2\left (-\frac{a e^{c+d x^2}}{b+\sqrt{-a^2+b^2}}\right )}{a^2 \sqrt{-a^2+b^2} d^2}+\frac{b^2 x^4 \sinh \left (c+d x^2\right )}{2 a \left (a^2-b^2\right ) d \left (b+a \cosh \left (c+d x^2\right )\right )}+\frac{b^2 \operatorname{Subst}\left (\int \frac{\log \left (1+\frac{a x}{b-\sqrt{-a^2+b^2}}\right )}{x} \, dx,x,e^{c+d x^2}\right )}{a^2 \left (a^2-b^2\right ) d^3}+\frac{b^2 \operatorname{Subst}\left (\int \frac{\log \left (1+\frac{a x}{b+\sqrt{-a^2+b^2}}\right )}{x} \, dx,x,e^{c+d x^2}\right )}{a^2 \left (a^2-b^2\right ) d^3}+\frac{(2 b) \operatorname{Subst}\left (\int \text{Li}_2\left (-\frac{2 a e^{c+d x}}{2 b-2 \sqrt{-a^2+b^2}}\right ) \, dx,x,x^2\right )}{a^2 \sqrt{-a^2+b^2} d^2}-\frac{(2 b) \operatorname{Subst}\left (\int \text{Li}_2\left (-\frac{2 a e^{c+d x}}{2 b+2 \sqrt{-a^2+b^2}}\right ) \, dx,x,x^2\right )}{a^2 \sqrt{-a^2+b^2} d^2}-\frac{b^3 \operatorname{Subst}\left (\int x \log \left (1+\frac{2 a e^{c+d x}}{2 b-2 \sqrt{-a^2+b^2}}\right ) \, dx,x,x^2\right )}{a^2 \left (-a^2+b^2\right )^{3/2} d}+\frac{b^3 \operatorname{Subst}\left (\int x \log \left (1+\frac{2 a e^{c+d x}}{2 b+2 \sqrt{-a^2+b^2}}\right ) \, dx,x,x^2\right )}{a^2 \left (-a^2+b^2\right )^{3/2} d}\\ &=\frac{b^2 x^4}{2 a^2 \left (a^2-b^2\right ) d}+\frac{x^6}{6 a^2}-\frac{b^2 x^2 \log \left (1+\frac{a e^{c+d x^2}}{b-\sqrt{-a^2+b^2}}\right )}{a^2 \left (a^2-b^2\right ) d^2}+\frac{b^3 x^4 \log \left (1+\frac{a e^{c+d x^2}}{b-\sqrt{-a^2+b^2}}\right )}{2 a^2 \left (-a^2+b^2\right )^{3/2} d}-\frac{b x^4 \log \left (1+\frac{a e^{c+d x^2}}{b-\sqrt{-a^2+b^2}}\right )}{a^2 \sqrt{-a^2+b^2} d}-\frac{b^2 x^2 \log \left (1+\frac{a e^{c+d x^2}}{b+\sqrt{-a^2+b^2}}\right )}{a^2 \left (a^2-b^2\right ) d^2}-\frac{b^3 x^4 \log \left (1+\frac{a e^{c+d x^2}}{b+\sqrt{-a^2+b^2}}\right )}{2 a^2 \left (-a^2+b^2\right )^{3/2} d}+\frac{b x^4 \log \left (1+\frac{a e^{c+d x^2}}{b+\sqrt{-a^2+b^2}}\right )}{a^2 \sqrt{-a^2+b^2} d}-\frac{b^2 \text{Li}_2\left (-\frac{a e^{c+d x^2}}{b-\sqrt{-a^2+b^2}}\right )}{a^2 \left (a^2-b^2\right ) d^3}+\frac{b^3 x^2 \text{Li}_2\left (-\frac{a e^{c+d x^2}}{b-\sqrt{-a^2+b^2}}\right )}{a^2 \left (-a^2+b^2\right )^{3/2} d^2}-\frac{2 b x^2 \text{Li}_2\left (-\frac{a e^{c+d x^2}}{b-\sqrt{-a^2+b^2}}\right )}{a^2 \sqrt{-a^2+b^2} d^2}-\frac{b^2 \text{Li}_2\left (-\frac{a e^{c+d x^2}}{b+\sqrt{-a^2+b^2}}\right )}{a^2 \left (a^2-b^2\right ) d^3}-\frac{b^3 x^2 \text{Li}_2\left (-\frac{a e^{c+d x^2}}{b+\sqrt{-a^2+b^2}}\right )}{a^2 \left (-a^2+b^2\right )^{3/2} d^2}+\frac{2 b x^2 \text{Li}_2\left (-\frac{a e^{c+d x^2}}{b+\sqrt{-a^2+b^2}}\right )}{a^2 \sqrt{-a^2+b^2} d^2}+\frac{b^2 x^4 \sinh \left (c+d x^2\right )}{2 a \left (a^2-b^2\right ) d \left (b+a \cosh \left (c+d x^2\right )\right )}+\frac{(2 b) \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 x^2}\right )}{a^2 \sqrt{-a^2+b^2} d^3}-\frac{(2 b) \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 x^2}\right )}{a^2 \sqrt{-a^2+b^2} d^3}-\frac{b^3 \operatorname{Subst}\left (\int \text{Li}_2\left (-\frac{2 a e^{c+d x}}{2 b-2 \sqrt{-a^2+b^2}}\right ) \, dx,x,x^2\right )}{a^2 \left (-a^2+b^2\right )^{3/2} d^2}+\frac{b^3 \operatorname{Subst}\left (\int \text{Li}_2\left (-\frac{2 a e^{c+d x}}{2 b+2 \sqrt{-a^2+b^2}}\right ) \, dx,x,x^2\right )}{a^2 \left (-a^2+b^2\right )^{3/2} d^2}\\ &=\frac{b^2 x^4}{2 a^2 \left (a^2-b^2\right ) d}+\frac{x^6}{6 a^2}-\frac{b^2 x^2 \log \left (1+\frac{a e^{c+d x^2}}{b-\sqrt{-a^2+b^2}}\right )}{a^2 \left (a^2-b^2\right ) d^2}+\frac{b^3 x^4 \log \left (1+\frac{a e^{c+d x^2}}{b-\sqrt{-a^2+b^2}}\right )}{2 a^2 \left (-a^2+b^2\right )^{3/2} d}-\frac{b x^4 \log \left (1+\frac{a e^{c+d x^2}}{b-\sqrt{-a^2+b^2}}\right )}{a^2 \sqrt{-a^2+b^2} d}-\frac{b^2 x^2 \log \left (1+\frac{a e^{c+d x^2}}{b+\sqrt{-a^2+b^2}}\right )}{a^2 \left (a^2-b^2\right ) d^2}-\frac{b^3 x^4 \log \left (1+\frac{a e^{c+d x^2}}{b+\sqrt{-a^2+b^2}}\right )}{2 a^2 \left (-a^2+b^2\right )^{3/2} d}+\frac{b x^4 \log \left (1+\frac{a e^{c+d x^2}}{b+\sqrt{-a^2+b^2}}\right )}{a^2 \sqrt{-a^2+b^2} d}-\frac{b^2 \text{Li}_2\left (-\frac{a e^{c+d x^2}}{b-\sqrt{-a^2+b^2}}\right )}{a^2 \left (a^2-b^2\right ) d^3}+\frac{b^3 x^2 \text{Li}_2\left (-\frac{a e^{c+d x^2}}{b-\sqrt{-a^2+b^2}}\right )}{a^2 \left (-a^2+b^2\right )^{3/2} d^2}-\frac{2 b x^2 \text{Li}_2\left (-\frac{a e^{c+d x^2}}{b-\sqrt{-a^2+b^2}}\right )}{a^2 \sqrt{-a^2+b^2} d^2}-\frac{b^2 \text{Li}_2\left (-\frac{a e^{c+d x^2}}{b+\sqrt{-a^2+b^2}}\right )}{a^2 \left (a^2-b^2\right ) d^3}-\frac{b^3 x^2 \text{Li}_2\left (-\frac{a e^{c+d x^2}}{b+\sqrt{-a^2+b^2}}\right )}{a^2 \left (-a^2+b^2\right )^{3/2} d^2}+\frac{2 b x^2 \text{Li}_2\left (-\frac{a e^{c+d x^2}}{b+\sqrt{-a^2+b^2}}\right )}{a^2 \sqrt{-a^2+b^2} d^2}+\frac{2 b \text{Li}_3\left (-\frac{a e^{c+d x^2}}{b-\sqrt{-a^2+b^2}}\right )}{a^2 \sqrt{-a^2+b^2} d^3}-\frac{2 b \text{Li}_3\left (-\frac{a e^{c+d x^2}}{b+\sqrt{-a^2+b^2}}\right )}{a^2 \sqrt{-a^2+b^2} d^3}+\frac{b^2 x^4 \sinh \left (c+d x^2\right )}{2 a \left (a^2-b^2\right ) d \left (b+a \cosh \left (c+d x^2\right )\right )}-\frac{b^3 \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 x^2}\right )}{a^2 \left (-a^2+b^2\right )^{3/2} d^3}+\frac{b^3 \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 x^2}\right )}{a^2 \left (-a^2+b^2\right )^{3/2} d^3}\\ &=\frac{b^2 x^4}{2 a^2 \left (a^2-b^2\right ) d}+\frac{x^6}{6 a^2}-\frac{b^2 x^2 \log \left (1+\frac{a e^{c+d x^2}}{b-\sqrt{-a^2+b^2}}\right )}{a^2 \left (a^2-b^2\right ) d^2}+\frac{b^3 x^4 \log \left (1+\frac{a e^{c+d x^2}}{b-\sqrt{-a^2+b^2}}\right )}{2 a^2 \left (-a^2+b^2\right )^{3/2} d}-\frac{b x^4 \log \left (1+\frac{a e^{c+d x^2}}{b-\sqrt{-a^2+b^2}}\right )}{a^2 \sqrt{-a^2+b^2} d}-\frac{b^2 x^2 \log \left (1+\frac{a e^{c+d x^2}}{b+\sqrt{-a^2+b^2}}\right )}{a^2 \left (a^2-b^2\right ) d^2}-\frac{b^3 x^4 \log \left (1+\frac{a e^{c+d x^2}}{b+\sqrt{-a^2+b^2}}\right )}{2 a^2 \left (-a^2+b^2\right )^{3/2} d}+\frac{b x^4 \log \left (1+\frac{a e^{c+d x^2}}{b+\sqrt{-a^2+b^2}}\right )}{a^2 \sqrt{-a^2+b^2} d}-\frac{b^2 \text{Li}_2\left (-\frac{a e^{c+d x^2}}{b-\sqrt{-a^2+b^2}}\right )}{a^2 \left (a^2-b^2\right ) d^3}+\frac{b^3 x^2 \text{Li}_2\left (-\frac{a e^{c+d x^2}}{b-\sqrt{-a^2+b^2}}\right )}{a^2 \left (-a^2+b^2\right )^{3/2} d^2}-\frac{2 b x^2 \text{Li}_2\left (-\frac{a e^{c+d x^2}}{b-\sqrt{-a^2+b^2}}\right )}{a^2 \sqrt{-a^2+b^2} d^2}-\frac{b^2 \text{Li}_2\left (-\frac{a e^{c+d x^2}}{b+\sqrt{-a^2+b^2}}\right )}{a^2 \left (a^2-b^2\right ) d^3}-\frac{b^3 x^2 \text{Li}_2\left (-\frac{a e^{c+d x^2}}{b+\sqrt{-a^2+b^2}}\right )}{a^2 \left (-a^2+b^2\right )^{3/2} d^2}+\frac{2 b x^2 \text{Li}_2\left (-\frac{a e^{c+d x^2}}{b+\sqrt{-a^2+b^2}}\right )}{a^2 \sqrt{-a^2+b^2} d^2}-\frac{b^3 \text{Li}_3\left (-\frac{a e^{c+d x^2}}{b-\sqrt{-a^2+b^2}}\right )}{a^2 \left (-a^2+b^2\right )^{3/2} d^3}+\frac{2 b \text{Li}_3\left (-\frac{a e^{c+d x^2}}{b-\sqrt{-a^2+b^2}}\right )}{a^2 \sqrt{-a^2+b^2} d^3}+\frac{b^3 \text{Li}_3\left (-\frac{a e^{c+d x^2}}{b+\sqrt{-a^2+b^2}}\right )}{a^2 \left (-a^2+b^2\right )^{3/2} d^3}-\frac{2 b \text{Li}_3\left (-\frac{a e^{c+d x^2}}{b+\sqrt{-a^2+b^2}}\right )}{a^2 \sqrt{-a^2+b^2} d^3}+\frac{b^2 x^4 \sinh \left (c+d x^2\right )}{2 a \left (a^2-b^2\right ) d \left (b+a \cosh \left (c+d x^2\right )\right )}\\ \end{align*}

Mathematica [A]  time = 12.6476, size = 1565, normalized size = 1.57 \[ \text{result too large to display} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [F]  time = 0.116, size = 0, normalized size = 0. \begin{align*} \int{\frac{{x}^{5}}{ \left ( a+b{\rm sech} \left (d{x}^{2}+c\right ) \right ) ^{2}}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^5/(a+b*sech(d*x^2+c))^2,x)

[Out]

int(x^5/(a+b*sech(d*x^2+c))^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^5/(a+b*sech(d*x^2+c))^2,x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [C]  time = 2.80036, size = 8047, normalized size = 8.1 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/6*((a^5 - 2*a^3*b^2 + a*b^4)*d^3*x^6 - 6*(a^3*b^2 - a*b^4)*c^2 + ((a^5 - 2*a^3*b^2 + a*b^4)*d^3*x^6 + 6*(a^3
*b^2 - a*b^4)*d^2*x^4 - 6*(a^3*b^2 - a*b^4)*c^2)*cosh(d*x^2 + c)^2 + ((a^5 - 2*a^3*b^2 + a*b^4)*d^3*x^6 + 6*(a
^3*b^2 - a*b^4)*d^2*x^4 - 6*(a^3*b^2 - a*b^4)*c^2)*sinh(d*x^2 + c)^2 - 6*(2*a^4*b - a^2*b^3 + (2*a^4*b - a^2*b
^3)*cosh(d*x^2 + c)^2 + (2*a^4*b - a^2*b^3)*sinh(d*x^2 + c)^2 + 2*(2*a^3*b^2 - a*b^4)*cosh(d*x^2 + c) + 2*(2*a
^3*b^2 - a*b^4 + (2*a^4*b - a^2*b^3)*cosh(d*x^2 + c))*sinh(d*x^2 + c))*sqrt(-(a^2 - b^2)/a^2)*polylog(3, -(b*c
osh(d*x^2 + c) + b*sinh(d*x^2 + c) + (a*cosh(d*x^2 + c) + a*sinh(d*x^2 + c))*sqrt(-(a^2 - b^2)/a^2))/a) + 6*(2
*a^4*b - a^2*b^3 + (2*a^4*b - a^2*b^3)*cosh(d*x^2 + c)^2 + (2*a^4*b - a^2*b^3)*sinh(d*x^2 + c)^2 + 2*(2*a^3*b^
2 - a*b^4)*cosh(d*x^2 + c) + 2*(2*a^3*b^2 - a*b^4 + (2*a^4*b - a^2*b^3)*cosh(d*x^2 + c))*sinh(d*x^2 + c))*sqrt
(-(a^2 - b^2)/a^2)*polylog(3, -(b*cosh(d*x^2 + c) + b*sinh(d*x^2 + c) - (a*cosh(d*x^2 + c) + a*sinh(d*x^2 + c)
)*sqrt(-(a^2 - b^2)/a^2))/a) + 2*((a^4*b - 2*a^2*b^3 + b^5)*d^3*x^6 + 3*(a^2*b^3 - b^5)*d^2*x^4 - 6*(a^2*b^3 -
b^5)*c^2)*cosh(d*x^2 + c) - 6*(a^3*b^2 - a*b^4 + (a^3*b^2 - a*b^4)*cosh(d*x^2 + c)^2 + (a^3*b^2 - a*b^4)*sinh
(d*x^2 + c)^2 + 2*(a^2*b^3 - b^5)*cosh(d*x^2 + c) + 2*(a^2*b^3 - b^5 + (a^3*b^2 - a*b^4)*cosh(d*x^2 + c))*sinh
(d*x^2 + c) - ((2*a^4*b - a^2*b^3)*d*x^2*cosh(d*x^2 + c)^2 + (2*a^4*b - a^2*b^3)*d*x^2*sinh(d*x^2 + c)^2 + 2*(
2*a^3*b^2 - a*b^4)*d*x^2*cosh(d*x^2 + c) + (2*a^4*b - a^2*b^3)*d*x^2 + 2*((2*a^4*b - a^2*b^3)*d*x^2*cosh(d*x^2
+ c) + (2*a^3*b^2 - a*b^4)*d*x^2)*sinh(d*x^2 + c))*sqrt(-(a^2 - b^2)/a^2))*dilog(-(b*cosh(d*x^2 + c) + b*sinh
(d*x^2 + c) + (a*cosh(d*x^2 + c) + a*sinh(d*x^2 + c))*sqrt(-(a^2 - b^2)/a^2) + a)/a + 1) - 6*(a^3*b^2 - a*b^4
+ (a^3*b^2 - a*b^4)*cosh(d*x^2 + c)^2 + (a^3*b^2 - a*b^4)*sinh(d*x^2 + c)^2 + 2*(a^2*b^3 - b^5)*cosh(d*x^2 + c
) + 2*(a^2*b^3 - b^5 + (a^3*b^2 - a*b^4)*cosh(d*x^2 + c))*sinh(d*x^2 + c) + ((2*a^4*b - a^2*b^3)*d*x^2*cosh(d*
x^2 + c)^2 + (2*a^4*b - a^2*b^3)*d*x^2*sinh(d*x^2 + c)^2 + 2*(2*a^3*b^2 - a*b^4)*d*x^2*cosh(d*x^2 + c) + (2*a^
4*b - a^2*b^3)*d*x^2 + 2*((2*a^4*b - a^2*b^3)*d*x^2*cosh(d*x^2 + c) + (2*a^3*b^2 - a*b^4)*d*x^2)*sinh(d*x^2 +
c))*sqrt(-(a^2 - b^2)/a^2))*dilog(-(b*cosh(d*x^2 + c) + b*sinh(d*x^2 + c) - (a*cosh(d*x^2 + c) + a*sinh(d*x^2
+ c))*sqrt(-(a^2 - b^2)/a^2) + a)/a + 1) + 3*(2*(a^3*b^2 - a*b^4)*c*cosh(d*x^2 + c)^2 + 2*(a^3*b^2 - a*b^4)*c*
sinh(d*x^2 + c)^2 + 4*(a^2*b^3 - b^5)*c*cosh(d*x^2 + c) + 2*(a^3*b^2 - a*b^4)*c + 4*((a^3*b^2 - a*b^4)*c*cosh(
d*x^2 + c) + (a^2*b^3 - b^5)*c)*sinh(d*x^2 + c) - ((2*a^4*b - a^2*b^3)*c^2*cosh(d*x^2 + c)^2 + (2*a^4*b - a^2*
b^3)*c^2*sinh(d*x^2 + c)^2 + 2*(2*a^3*b^2 - a*b^4)*c^2*cosh(d*x^2 + c) + (2*a^4*b - a^2*b^3)*c^2 + 2*((2*a^4*b
- a^2*b^3)*c^2*cosh(d*x^2 + c) + (2*a^3*b^2 - a*b^4)*c^2)*sinh(d*x^2 + c))*sqrt(-(a^2 - b^2)/a^2))*log(2*a*co
sh(d*x^2 + c) + 2*a*sinh(d*x^2 + c) + 2*a*sqrt(-(a^2 - b^2)/a^2) + 2*b) + 3*(2*(a^3*b^2 - a*b^4)*c*cosh(d*x^2
+ c)^2 + 2*(a^3*b^2 - a*b^4)*c*sinh(d*x^2 + c)^2 + 4*(a^2*b^3 - b^5)*c*cosh(d*x^2 + c) + 2*(a^3*b^2 - a*b^4)*c
+ 4*((a^3*b^2 - a*b^4)*c*cosh(d*x^2 + c) + (a^2*b^3 - b^5)*c)*sinh(d*x^2 + c) + ((2*a^4*b - a^2*b^3)*c^2*cosh
(d*x^2 + c)^2 + (2*a^4*b - a^2*b^3)*c^2*sinh(d*x^2 + c)^2 + 2*(2*a^3*b^2 - a*b^4)*c^2*cosh(d*x^2 + c) + (2*a^4
*b - a^2*b^3)*c^2 + 2*((2*a^4*b - a^2*b^3)*c^2*cosh(d*x^2 + c) + (2*a^3*b^2 - a*b^4)*c^2)*sinh(d*x^2 + c))*sqr
t(-(a^2 - b^2)/a^2))*log(2*a*cosh(d*x^2 + c) + 2*a*sinh(d*x^2 + c) - 2*a*sqrt(-(a^2 - b^2)/a^2) + 2*b) - 3*(2*
(a^3*b^2 - a*b^4)*d*x^2 + 2*((a^3*b^2 - a*b^4)*d*x^2 + (a^3*b^2 - a*b^4)*c)*cosh(d*x^2 + c)^2 + 2*((a^3*b^2 -
a*b^4)*d*x^2 + (a^3*b^2 - a*b^4)*c)*sinh(d*x^2 + c)^2 + 2*(a^3*b^2 - a*b^4)*c + 4*((a^2*b^3 - b^5)*d*x^2 + (a^
2*b^3 - b^5)*c)*cosh(d*x^2 + c) + 4*((a^2*b^3 - b^5)*d*x^2 + (a^2*b^3 - b^5)*c + ((a^3*b^2 - a*b^4)*d*x^2 + (a
^3*b^2 - a*b^4)*c)*cosh(d*x^2 + c))*sinh(d*x^2 + c) - ((2*a^4*b - a^2*b^3)*d^2*x^4 - (2*a^4*b - a^2*b^3)*c^2 +
((2*a^4*b - a^2*b^3)*d^2*x^4 - (2*a^4*b - a^2*b^3)*c^2)*cosh(d*x^2 + c)^2 + ((2*a^4*b - a^2*b^3)*d^2*x^4 - (2
*a^4*b - a^2*b^3)*c^2)*sinh(d*x^2 + c)^2 + 2*((2*a^3*b^2 - a*b^4)*d^2*x^4 - (2*a^3*b^2 - a*b^4)*c^2)*cosh(d*x^
2 + c) + 2*((2*a^3*b^2 - a*b^4)*d^2*x^4 - (2*a^3*b^2 - a*b^4)*c^2 + ((2*a^4*b - a^2*b^3)*d^2*x^4 - (2*a^4*b -
a^2*b^3)*c^2)*cosh(d*x^2 + c))*sinh(d*x^2 + c))*sqrt(-(a^2 - b^2)/a^2))*log((b*cosh(d*x^2 + c) + b*sinh(d*x^2
+ c) + (a*cosh(d*x^2 + c) + a*sinh(d*x^2 + c))*sqrt(-(a^2 - b^2)/a^2) + a)/a) - 3*(2*(a^3*b^2 - a*b^4)*d*x^2 +
2*((a^3*b^2 - a*b^4)*d*x^2 + (a^3*b^2 - a*b^4)*c)*cosh(d*x^2 + c)^2 + 2*((a^3*b^2 - a*b^4)*d*x^2 + (a^3*b^2 -
a*b^4)*c)*sinh(d*x^2 + c)^2 + 2*(a^3*b^2 - a*b^4)*c + 4*((a^2*b^3 - b^5)*d*x^2 + (a^2*b^3 - b^5)*c)*cosh(d*x^
2 + c) + 4*((a^2*b^3 - b^5)*d*x^2 + (a^2*b^3 - b^5)*c + ((a^3*b^2 - a*b^4)*d*x^2 + (a^3*b^2 - a*b^4)*c)*cosh(d
*x^2 + c))*sinh(d*x^2 + c) + ((2*a^4*b - a^2*b^3)*d^2*x^4 - (2*a^4*b - a^2*b^3)*c^2 + ((2*a^4*b - a^2*b^3)*d^2
*x^4 - (2*a^4*b - a^2*b^3)*c^2)*cosh(d*x^2 + c)^2 + ((2*a^4*b - a^2*b^3)*d^2*x^4 - (2*a^4*b - a^2*b^3)*c^2)*si
nh(d*x^2 + c)^2 + 2*((2*a^3*b^2 - a*b^4)*d^2*x^4 - (2*a^3*b^2 - a*b^4)*c^2)*cosh(d*x^2 + c) + 2*((2*a^3*b^2 -
a*b^4)*d^2*x^4 - (2*a^3*b^2 - a*b^4)*c^2 + ((2*a^4*b - a^2*b^3)*d^2*x^4 - (2*a^4*b - a^2*b^3)*c^2)*cosh(d*x^2
+ c))*sinh(d*x^2 + c))*sqrt(-(a^2 - b^2)/a^2))*log((b*cosh(d*x^2 + c) + b*sinh(d*x^2 + c) - (a*cosh(d*x^2 + c)
+ a*sinh(d*x^2 + c))*sqrt(-(a^2 - b^2)/a^2) + a)/a) + 2*((a^4*b - 2*a^2*b^3 + b^5)*d^3*x^6 + 3*(a^2*b^3 - b^5
)*d^2*x^4 - 6*(a^2*b^3 - b^5)*c^2 + ((a^5 - 2*a^3*b^2 + a*b^4)*d^3*x^6 + 6*(a^3*b^2 - a*b^4)*d^2*x^4 - 6*(a^3*
b^2 - a*b^4)*c^2)*cosh(d*x^2 + c))*sinh(d*x^2 + c))/((a^7 - 2*a^5*b^2 + a^3*b^4)*d^3*cosh(d*x^2 + c)^2 + (a^7
- 2*a^5*b^2 + a^3*b^4)*d^3*sinh(d*x^2 + c)^2 + 2*(a^6*b - 2*a^4*b^3 + a^2*b^5)*d^3*cosh(d*x^2 + c) + (a^7 - 2*
a^5*b^2 + a^3*b^4)*d^3 + 2*((a^7 - 2*a^5*b^2 + a^3*b^4)*d^3*cosh(d*x^2 + c) + (a^6*b - 2*a^4*b^3 + a^2*b^5)*d^
3)*sinh(d*x^2 + c))

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x**5/(a+b*sech(d*x**2+c))**2,x)

[Out]

Integral(x**5/(a + b*sech(c + d*x**2))**2, x)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate(x^5/(b*sech(d*x^2 + c) + a)^2, x)