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

Optimal. Leaf size=555 \[ -\frac{b \text{PolyLog}\left (2,-\frac{a e^{c+d x^2}}{b-\sqrt{b^2-a^2}}\right )}{a^2 d^2 \sqrt{b^2-a^2}}+\frac{b^3 \text{PolyLog}\left (2,-\frac{a e^{c+d x^2}}{b-\sqrt{b^2-a^2}}\right )}{2 a^2 d^2 \left (b^2-a^2\right )^{3/2}}+\frac{b \text{PolyLog}\left (2,-\frac{a e^{c+d x^2}}{\sqrt{b^2-a^2}+b}\right )}{a^2 d^2 \sqrt{b^2-a^2}}-\frac{b^3 \text{PolyLog}\left (2,-\frac{a e^{c+d x^2}}{\sqrt{b^2-a^2}+b}\right )}{2 a^2 d^2 \left (b^2-a^2\right )^{3/2}}-\frac{b^2 \log \left (a \cosh \left (c+d x^2\right )+b\right )}{2 a^2 d^2 \left (a^2-b^2\right )}-\frac{b x^2 \log \left (\frac{a e^{c+d x^2}}{b-\sqrt{b^2-a^2}}+1\right )}{a^2 d \sqrt{b^2-a^2}}+\frac{b^3 x^2 \log \left (\frac{a e^{c+d x^2}}{b-\sqrt{b^2-a^2}}+1\right )}{2 a^2 d \left (b^2-a^2\right )^{3/2}}+\frac{b x^2 \log \left (\frac{a e^{c+d x^2}}{\sqrt{b^2-a^2}+b}+1\right )}{a^2 d \sqrt{b^2-a^2}}-\frac{b^3 x^2 \log \left (\frac{a e^{c+d x^2}}{\sqrt{b^2-a^2}+b}+1\right )}{2 a^2 d \left (b^2-a^2\right )^{3/2}}+\frac{b^2 x^2 \sinh \left (c+d x^2\right )}{2 a d \left (a^2-b^2\right ) \left (a \cosh \left (c+d x^2\right )+b\right )}+\frac{x^4}{4 a^2} \]

[Out]

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

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Rubi [A]  time = 1.10232, antiderivative size = 555, normalized size of antiderivative = 1., number of steps used = 22, number of rules used = 10, integrand size = 18, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.556, Rules used = {5436, 4191, 3324, 3320, 2264, 2190, 2279, 2391, 2668, 31} \[ -\frac{b \text{PolyLog}\left (2,-\frac{a e^{c+d x^2}}{b-\sqrt{b^2-a^2}}\right )}{a^2 d^2 \sqrt{b^2-a^2}}+\frac{b^3 \text{PolyLog}\left (2,-\frac{a e^{c+d x^2}}{b-\sqrt{b^2-a^2}}\right )}{2 a^2 d^2 \left (b^2-a^2\right )^{3/2}}+\frac{b \text{PolyLog}\left (2,-\frac{a e^{c+d x^2}}{\sqrt{b^2-a^2}+b}\right )}{a^2 d^2 \sqrt{b^2-a^2}}-\frac{b^3 \text{PolyLog}\left (2,-\frac{a e^{c+d x^2}}{\sqrt{b^2-a^2}+b}\right )}{2 a^2 d^2 \left (b^2-a^2\right )^{3/2}}-\frac{b^2 \log \left (a \cosh \left (c+d x^2\right )+b\right )}{2 a^2 d^2 \left (a^2-b^2\right )}-\frac{b x^2 \log \left (\frac{a e^{c+d x^2}}{b-\sqrt{b^2-a^2}}+1\right )}{a^2 d \sqrt{b^2-a^2}}+\frac{b^3 x^2 \log \left (\frac{a e^{c+d x^2}}{b-\sqrt{b^2-a^2}}+1\right )}{2 a^2 d \left (b^2-a^2\right )^{3/2}}+\frac{b x^2 \log \left (\frac{a e^{c+d x^2}}{\sqrt{b^2-a^2}+b}+1\right )}{a^2 d \sqrt{b^2-a^2}}-\frac{b^3 x^2 \log \left (\frac{a e^{c+d x^2}}{\sqrt{b^2-a^2}+b}+1\right )}{2 a^2 d \left (b^2-a^2\right )^{3/2}}+\frac{b^2 x^2 \sinh \left (c+d x^2\right )}{2 a d \left (a^2-b^2\right ) \left (a \cosh \left (c+d x^2\right )+b\right )}+\frac{x^4}{4 a^2} \]

Antiderivative was successfully verified.

[In]

Int[x^3/(a + b*Sech[c + d*x^2])^2,x]

[Out]

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

Rule 2668

Int[cos[(e_.) + (f_.)*(x_)]^(p_.)*((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_.), x_Symbol] :> Dist[1/(b^p*f), S
ubst[Int[(a + x)^m*(b^2 - x^2)^((p - 1)/2), x], x, b*Sin[e + f*x]], x] /; FreeQ[{a, b, e, f, m}, x] && Integer
Q[(p - 1)/2] && NeQ[a^2 - b^2, 0]

Rule 31

Int[((a_) + (b_.)*(x_))^(-1), x_Symbol] :> Simp[Log[RemoveContent[a + b*x, x]]/b, x] /; FreeQ[{a, b}, x]

Rubi steps

\begin{align*} \int \frac{x^3}{\left (a+b \text{sech}\left (c+d x^2\right )\right )^2} \, dx &=\frac{1}{2} \operatorname{Subst}\left (\int \frac{x}{(a+b \text{sech}(c+d x))^2} \, dx,x,x^2\right )\\ &=\frac{1}{2} \operatorname{Subst}\left (\int \left (\frac{x}{a^2}+\frac{b^2 x}{a^2 (b+a \cosh (c+d x))^2}-\frac{2 b x}{a^2 (b+a \cosh (c+d x))}\right ) \, dx,x,x^2\right )\\ &=\frac{x^4}{4 a^2}-\frac{b \operatorname{Subst}\left (\int \frac{x}{b+a \cosh (c+d x)} \, dx,x,x^2\right )}{a^2}+\frac{b^2 \operatorname{Subst}\left (\int \frac{x}{(b+a \cosh (c+d x))^2} \, dx,x,x^2\right )}{2 a^2}\\ &=\frac{x^4}{4 a^2}+\frac{b^2 x^2 \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}{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}{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{\sinh (c+d x)}{b+a \cosh (c+d x)} \, dx,x,x^2\right )}{2 a \left (a^2-b^2\right ) d}\\ &=\frac{x^4}{4 a^2}+\frac{b^2 x^2 \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}{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 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 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{1}{b+x} \, dx,x,a \cosh \left (c+d x^2\right )\right )}{2 a^2 \left (a^2-b^2\right ) d^2}\\ &=\frac{x^4}{4 a^2}-\frac{b x^2 \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 x^2 \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 \log \left (b+a \cosh \left (c+d x^2\right )\right )}{2 a^2 \left (a^2-b^2\right ) d^2}+\frac{b^2 x^2 \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 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 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 \operatorname{Subst}\left (\int \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 \operatorname{Subst}\left (\int \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{x^4}{4 a^2}+\frac{b^3 x^2 \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^2 \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^3 x^2 \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^2 \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 \log \left (b+a \cosh \left (c+d x^2\right )\right )}{2 a^2 \left (a^2-b^2\right ) d^2}+\frac{b^2 x^2 \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 \operatorname{Subst}\left (\int \frac{\log \left (1+\frac{2 a x}{2 b-2 \sqrt{-a^2+b^2}}\right )}{x} \, dx,x,e^{c+d x^2}\right )}{a^2 \sqrt{-a^2+b^2} d^2}-\frac{b \operatorname{Subst}\left (\int \frac{\log \left (1+\frac{2 a x}{2 b+2 \sqrt{-a^2+b^2}}\right )}{x} \, dx,x,e^{c+d x^2}\right )}{a^2 \sqrt{-a^2+b^2} d^2}-\frac{b^3 \operatorname{Subst}\left (\int \log \left (1+\frac{2 a e^{c+d x}}{2 b-2 \sqrt{-a^2+b^2}}\right ) \, dx,x,x^2\right )}{2 a^2 \left (-a^2+b^2\right )^{3/2} d}+\frac{b^3 \operatorname{Subst}\left (\int \log \left (1+\frac{2 a e^{c+d x}}{2 b+2 \sqrt{-a^2+b^2}}\right ) \, dx,x,x^2\right )}{2 a^2 \left (-a^2+b^2\right )^{3/2} d}\\ &=\frac{x^4}{4 a^2}+\frac{b^3 x^2 \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^2 \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^3 x^2 \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^2 \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 \log \left (b+a \cosh \left (c+d x^2\right )\right )}{2 a^2 \left (a^2-b^2\right ) d^2}-\frac{b \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 \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^2 \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{\log \left (1+\frac{2 a x}{2 b-2 \sqrt{-a^2+b^2}}\right )}{x} \, dx,x,e^{c+d x^2}\right )}{2 a^2 \left (-a^2+b^2\right )^{3/2} d^2}+\frac{b^3 \operatorname{Subst}\left (\int \frac{\log \left (1+\frac{2 a x}{2 b+2 \sqrt{-a^2+b^2}}\right )}{x} \, dx,x,e^{c+d x^2}\right )}{2 a^2 \left (-a^2+b^2\right )^{3/2} d^2}\\ &=\frac{x^4}{4 a^2}+\frac{b^3 x^2 \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^2 \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^3 x^2 \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^2 \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 \log \left (b+a \cosh \left (c+d x^2\right )\right )}{2 a^2 \left (a^2-b^2\right ) d^2}+\frac{b^3 \text{Li}_2\left (-\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^2}-\frac{b \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}_2\left (-\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^2}+\frac{b \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^2 \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 = 6.36479, size = 755, normalized size = 1.36 \[ \frac{\text{sech}^2\left (c+d x^2\right ) \left (a \cosh \left (c+d x^2\right )+b\right ) \left (-\frac{2 b \left (a^2-b^2\right ) \left (a \cosh \left (c+d x^2\right )+b\right ) \left (\sqrt{a^2-b^2} \left (b^2-2 a^2\right ) \text{PolyLog}\left (2,\frac{a e^{c+d x^2}}{\sqrt{b^2-a^2}-b}\right )+\sqrt{a^2-b^2} \left (2 a^2-b^2\right ) \text{PolyLog}\left (2,-\frac{a e^{c+d x^2}}{\sqrt{b^2-a^2}+b}\right )+b \sqrt{-\left (a^2-b^2\right )^2} \left (c+d x^2\right )-2 a^2 \sqrt{a^2-b^2} \left (c+d x^2\right ) \log \left (\frac{a e^{c+d x^2}}{b-\sqrt{b^2-a^2}}+1\right )+2 a^2 \sqrt{a^2-b^2} \left (c+d x^2\right ) \log \left (\frac{a e^{c+d x^2}}{\sqrt{b^2-a^2}+b}+1\right )+b^2 \sqrt{a^2-b^2} \left (c+d x^2\right ) \log \left (\frac{a e^{c+d x^2}}{b-\sqrt{b^2-a^2}}+1\right )-b^2 \sqrt{a^2-b^2} \left (c+d x^2\right ) \log \left (\frac{a e^{c+d x^2}}{\sqrt{b^2-a^2}+b}+1\right )-b \sqrt{-\left (a^2-b^2\right )^2} \log \left (a e^{2 \left (c+d x^2\right )}+a+2 b e^{c+d x^2}\right )-4 a^2 c \sqrt{b^2-a^2} \tan ^{-1}\left (\frac{a e^{-c-d x^2}+b}{\sqrt{a^2-b^2}}\right )+2 b^2 c \sqrt{b^2-a^2} \tan ^{-1}\left (\frac{a e^{-c-d x^2}+b}{\sqrt{a^2-b^2}}\right )-2 b^2 \sqrt{b^2-a^2} \tan ^{-1}\left (\frac{a e^{-c-d x^2}+b}{\sqrt{a^2-b^2}}\right )-2 b^2 \sqrt{b^2-a^2} \tan ^{-1}\left (\frac{a e^{c+d x^2}+b}{\sqrt{a^2-b^2}}\right )\right )}{\left (-\left (a^2-b^2\right )^2\right )^{3/2}}+\frac{2 a b^2 d x^2 \sinh \left (c+d x^2\right )}{(a-b) (a+b)}+\left (d x^2-c\right ) \left (c+d x^2\right ) \left (a \cosh \left (c+d x^2\right )+b\right )\right )}{4 a^2 d^2 \left (a+b \text{sech}\left (c+d x^2\right )\right )^2} \]

Warning: Unable to verify antiderivative.

[In]

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

[Out]

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

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Maple [F]  time = 0.128, size = 0, normalized size = 0. \begin{align*} \int{\frac{{x}^{3}}{ \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^3/(a+b*sech(d*x^2+c))^2,x)

[Out]

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

[Out]

Exception raised: ValueError

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Fricas [B]  time = 2.53235, size = 5169, normalized size = 9.31 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/4*((a^5 - 2*a^3*b^2 + a*b^4)*d^2*x^4 + ((a^5 - 2*a^3*b^2 + a*b^4)*d^2*x^4 + 4*(a^3*b^2 - a*b^4)*d*x^2 + 4*(a
^3*b^2 - a*b^4)*c)*cosh(d*x^2 + c)^2 + ((a^5 - 2*a^3*b^2 + a*b^4)*d^2*x^4 + 4*(a^3*b^2 - a*b^4)*d*x^2 + 4*(a^3
*b^2 - a*b^4)*c)*sinh(d*x^2 + c)^2 + 2*(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)*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) - 2*(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)*dilog(-(b*co
sh(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)
+ 2*((2*a^4*b - a^2*b^3)*d*x^2 + ((2*a^4*b - a^2*b^3)*d*x^2 + (2*a^4*b - a^2*b^3)*c)*cosh(d*x^2 + c)^2 + ((2*
a^4*b - a^2*b^3)*d*x^2 + (2*a^4*b - a^2*b^3)*c)*sinh(d*x^2 + c)^2 + (2*a^4*b - a^2*b^3)*c + 2*((2*a^3*b^2 - a*
b^4)*d*x^2 + (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 + (2*a^3*b^2 - a*b^4)*c + (
(2*a^4*b - a^2*b^3)*d*x^2 + (2*a^4*b - a^2*b^3)*c)*cosh(d*x^2 + c))*sinh(d*x^2 + c))*sqrt(-(a^2 - b^2)/a^2)*lo
g((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*((2*a^4*b - a^2*b^3)*d*x^2 + ((2*a^4*b - a^2*b^3)*d*x^2 + (2*a^4*b - a^2*b^3)*c)*cosh(d*x^2 + c)^2 + (
(2*a^4*b - a^2*b^3)*d*x^2 + (2*a^4*b - a^2*b^3)*c)*sinh(d*x^2 + c)^2 + (2*a^4*b - a^2*b^3)*c + 2*((2*a^3*b^2 -
a*b^4)*d*x^2 + (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 + (2*a^3*b^2 - a*b^4)*c
+ ((2*a^4*b - a^2*b^3)*d*x^2 + (2*a^4*b - a^2*b^3)*c)*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) + 4*(a^3*b^2 - a*b^4)*c + 2*((a^4*b - 2*a^2*b^3 + b^5)*d^2*x^4 + 2*(a^2*b^3 - b^5)*d*x^2 + 4*(a^2*b^3 -
b^5)*c)*cosh(d*x^2 + c) - 2*(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)*c*cosh(d*x^2 + c)^2 + (2*a^4*b - a^2*b^3)*c*sinh(d*x^2 + c)^2 + 2*(2*a^3*b^2
- a*b^4)*c*cosh(d*x^2 + c) + (2*a^4*b - a^2*b^3)*c + 2*((2*a^4*b - a^2*b^3)*c*cosh(d*x^2 + c) + (2*a^3*b^2 - a
*b^4)*c)*sinh(d*x^2 + c))*sqrt(-(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) - 2*(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)*c*cosh(d*x^2 + c)^2 + (2*a^4*b - a^2*b^3)*c*sinh(d*x^2 + c)^2 + 2*(2*a^3*b^2 - a
*b^4)*c*cosh(d*x^2 + c) + (2*a^4*b - a^2*b^3)*c + 2*((2*a^4*b - a^2*b^3)*c*cosh(d*x^2 + c) + (2*a^3*b^2 - a*b^
4)*c)*sinh(d*x^2 + c))*sqrt(-(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) + 2*((a^4*b - 2*a^2*b^3 + b^5)*d^2*x^4 + 2*(a^2*b^3 - b^5)*d*x^2 + 4*(a^2*b^3 - b^5)*c + ((
a^5 - 2*a^3*b^2 + a*b^4)*d^2*x^4 + 4*(a^3*b^2 - a*b^4)*d*x^2 + 4*(a^3*b^2 - a*b^4)*c)*cosh(d*x^2 + c))*sinh(d*
x^2 + c))/((a^7 - 2*a^5*b^2 + a^3*b^4)*d^2*cosh(d*x^2 + c)^2 + (a^7 - 2*a^5*b^2 + a^3*b^4)*d^2*sinh(d*x^2 + c)
^2 + 2*(a^6*b - 2*a^4*b^3 + a^2*b^5)*d^2*cosh(d*x^2 + c) + (a^7 - 2*a^5*b^2 + a^3*b^4)*d^2 + 2*((a^7 - 2*a^5*b
^2 + a^3*b^4)*d^2*cosh(d*x^2 + c) + (a^6*b - 2*a^4*b^3 + a^2*b^5)*d^2)*sinh(d*x^2 + c))

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Integral(x**3/(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^{3}}{{\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^3/(a+b*sech(d*x^2+c))^2,x, algorithm="giac")

[Out]

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