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

Optimal. Leaf size=351 \[ \frac{x \text{PolyLog}\left (2,-\frac{(a+b) e^{2 c+2 d x}}{-2 \sqrt{-a} \sqrt{b}+a-b}\right )}{2 \sqrt{-a} \sqrt{b} d^2}-\frac{x \text{PolyLog}\left (2,-\frac{(a+b) e^{2 c+2 d x}}{2 \sqrt{-a} \sqrt{b}+a-b}\right )}{2 \sqrt{-a} \sqrt{b} d^2}-\frac{\text{PolyLog}\left (3,-\frac{(a+b) e^{2 c+2 d x}}{-2 \sqrt{-a} \sqrt{b}+a-b}\right )}{4 \sqrt{-a} \sqrt{b} d^3}+\frac{\text{PolyLog}\left (3,-\frac{(a+b) e^{2 c+2 d x}}{2 \sqrt{-a} \sqrt{b}+a-b}\right )}{4 \sqrt{-a} \sqrt{b} d^3}+\frac{x^2 \log \left (\frac{(a+b) e^{2 c+2 d x}}{-2 \sqrt{-a} \sqrt{b}+a-b}+1\right )}{2 \sqrt{-a} \sqrt{b} d}-\frac{x^2 \log \left (\frac{(a+b) e^{2 c+2 d x}}{2 \sqrt{-a} \sqrt{b}+a-b}+1\right )}{2 \sqrt{-a} \sqrt{b} d} \]

[Out]

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

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Rubi [A]  time = 0.866671, antiderivative size = 351, normalized size of antiderivative = 1., number of steps used = 11, number of rules used = 7, integrand size = 26, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.269, Rules used = {5632, 3320, 2264, 2190, 2531, 2282, 6589} \[ \frac{x \text{PolyLog}\left (2,-\frac{(a+b) e^{2 c+2 d x}}{-2 \sqrt{-a} \sqrt{b}+a-b}\right )}{2 \sqrt{-a} \sqrt{b} d^2}-\frac{x \text{PolyLog}\left (2,-\frac{(a+b) e^{2 c+2 d x}}{2 \sqrt{-a} \sqrt{b}+a-b}\right )}{2 \sqrt{-a} \sqrt{b} d^2}-\frac{\text{PolyLog}\left (3,-\frac{(a+b) e^{2 c+2 d x}}{-2 \sqrt{-a} \sqrt{b}+a-b}\right )}{4 \sqrt{-a} \sqrt{b} d^3}+\frac{\text{PolyLog}\left (3,-\frac{(a+b) e^{2 c+2 d x}}{2 \sqrt{-a} \sqrt{b}+a-b}\right )}{4 \sqrt{-a} \sqrt{b} d^3}+\frac{x^2 \log \left (\frac{(a+b) e^{2 c+2 d x}}{-2 \sqrt{-a} \sqrt{b}+a-b}+1\right )}{2 \sqrt{-a} \sqrt{b} d}-\frac{x^2 \log \left (\frac{(a+b) e^{2 c+2 d x}}{2 \sqrt{-a} \sqrt{b}+a-b}+1\right )}{2 \sqrt{-a} \sqrt{b} d} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

Rule 5632

Int[(((f_.) + (g_.)*(x_))^(m_.)*Sech[(d_.) + (e_.)*(x_)]^2)/((b_) + (c_.)*Tanh[(d_.) + (e_.)*(x_)]^2), x_Symbo
l] :> Dist[2, Int[(f + g*x)^m/(b - c + (b + c)*Cosh[2*d + 2*e*x]), x], x] /; FreeQ[{b, c, d, e, f, g}, x] && I
GtQ[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]

Rubi steps

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

Mathematica [C]  time = 2.30237, size = 316, normalized size = 0.9 \[ \frac{i \left (2 d x \text{PolyLog}\left (2,-\frac{\left (\sqrt{a}-i \sqrt{b}\right ) e^{2 (c+d x)}}{\sqrt{a}+i \sqrt{b}}\right )-2 d x \text{PolyLog}\left (2,-\frac{\left (\sqrt{a}+i \sqrt{b}\right ) e^{2 (c+d x)}}{\sqrt{a}-i \sqrt{b}}\right )-\text{PolyLog}\left (3,-\frac{\left (\sqrt{a}-i \sqrt{b}\right ) e^{2 (c+d x)}}{\sqrt{a}+i \sqrt{b}}\right )+\text{PolyLog}\left (3,-\frac{\left (\sqrt{a}+i \sqrt{b}\right ) e^{2 (c+d x)}}{\sqrt{a}-i \sqrt{b}}\right )+2 d^2 x^2 \log \left (1+\frac{\left (\sqrt{a}-i \sqrt{b}\right ) e^{2 (c+d x)}}{\sqrt{a}+i \sqrt{b}}\right )-2 d^2 x^2 \log \left (1+\frac{\left (\sqrt{a}+i \sqrt{b}\right ) e^{2 (c+d x)}}{\sqrt{a}-i \sqrt{b}}\right )\right )}{4 \sqrt{a} \sqrt{b} d^3} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [B]  time = 0.134, size = 1186, normalized size = 3.4 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-2/3/d^3*c^3/(-a*b)^(1/2)/(-2*(-a*b)^(1/2)-a+b)*b-1/4/d^3/(-a*b)^(1/2)/(-2*(-a*b)^(1/2)-a+b)*a*polylog(3,(a+b)
*exp(2*d*x+2*c)/(-2*(-a*b)^(1/2)-a+b))+1/4/d^3/(-a*b)^(1/2)/(-2*(-a*b)^(1/2)-a+b)*b*polylog(3,(a+b)*exp(2*d*x+
2*c)/(-2*(-a*b)^(1/2)-a+b))+1/2/d^3/(-a*b)^(1/2)/(-2*(-a*b)^(1/2)-a+b)*b*ln(1-(a+b)*exp(2*d*x+2*c)/(-2*(-a*b)^
(1/2)-a+b))*c^2-1/2/d^3/(-a*b)^(1/2)/(-2*(-a*b)^(1/2)-a+b)*a*ln(1-(a+b)*exp(2*d*x+2*c)/(-2*(-a*b)^(1/2)-a+b))*
c^2-1/2/d^2/(-a*b)^(1/2)/(-2*(-a*b)^(1/2)-a+b)*b*polylog(2,(a+b)*exp(2*d*x+2*c)/(-2*(-a*b)^(1/2)-a+b))*x+1/2/d
/(-a*b)^(1/2)/(-2*(-a*b)^(1/2)-a+b)*a*ln(1-(a+b)*exp(2*d*x+2*c)/(-2*(-a*b)^(1/2)-a+b))*x^2+1/d^2*c^2/(-a*b)^(1
/2)/(-2*(-a*b)^(1/2)-a+b)*a*x+1/2/d^2/(-a*b)^(1/2)/(-2*(-a*b)^(1/2)-a+b)*a*polylog(2,(a+b)*exp(2*d*x+2*c)/(-2*
(-a*b)^(1/2)-a+b))*x-1/2/d/(-a*b)^(1/2)/(-2*(-a*b)^(1/2)-a+b)*b*ln(1-(a+b)*exp(2*d*x+2*c)/(-2*(-a*b)^(1/2)-a+b
))*x^2-1/d^2*c^2/(-a*b)^(1/2)/(-2*(-a*b)^(1/2)-a+b)*b*x-1/3/(-a*b)^(1/2)*x^3-2/3/(-2*(-a*b)^(1/2)-a+b)*x^3+2/3
/d^3*c^3/(-a*b)^(1/2)/(-2*(-a*b)^(1/2)-a+b)*a-1/2/d^3/(-2*(-a*b)^(1/2)-a+b)*polylog(3,(a+b)*exp(2*d*x+2*c)/(-2
*(-a*b)^(1/2)-a+b))+2/3/d^3*c^3/(-a*b)^(1/2)+4/3/d^3*c^3/(-2*(-a*b)^(1/2)-a+b)-1/4/d^3/(-a*b)^(1/2)*polylog(3,
(a+b)*exp(2*d*x+2*c)/(2*(-a*b)^(1/2)-a+b))-1/3/(-a*b)^(1/2)/(-2*(-a*b)^(1/2)-a+b)*a*x^3+1/d^3*c^2/(a*b)^(1/2)*
arctan(1/4*(2*(a+b)*exp(2*d*x+2*c)+2*a-2*b)/(a*b)^(1/2))+1/d^2*c^2/(-a*b)^(1/2)*x+2/d^2*c^2/(-2*(-a*b)^(1/2)-a
+b)*x-1/d^3/(-2*(-a*b)^(1/2)-a+b)*ln(1-(a+b)*exp(2*d*x+2*c)/(-2*(-a*b)^(1/2)-a+b))*c^2-1/2/d^3/(-a*b)^(1/2)*ln
(1-(a+b)*exp(2*d*x+2*c)/(2*(-a*b)^(1/2)-a+b))*c^2+1/2/d^2/(-a*b)^(1/2)*polylog(2,(a+b)*exp(2*d*x+2*c)/(2*(-a*b
)^(1/2)-a+b))*x+1/d/(-2*(-a*b)^(1/2)-a+b)*ln(1-(a+b)*exp(2*d*x+2*c)/(-2*(-a*b)^(1/2)-a+b))*x^2+1/d^2/(-2*(-a*b
)^(1/2)-a+b)*polylog(2,(a+b)*exp(2*d*x+2*c)/(-2*(-a*b)^(1/2)-a+b))*x+1/2/d/(-a*b)^(1/2)*ln(1-(a+b)*exp(2*d*x+2
*c)/(2*(-a*b)^(1/2)-a+b))*x^2+1/3/(-a*b)^(1/2)/(-2*(-a*b)^(1/2)-a+b)*b*x^3

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [C]  time = 3.12261, size = 5269, normalized size = 15.01 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/2*(2*(a + b)*sqrt(-a*b/(a^2 + 2*a*b + b^2))*d*x*dilog(-(((a - b)*cosh(d*x + c) + (a - b)*sinh(d*x + c) - 2*(
(a + b)*cosh(d*x + c) + (a + b)*sinh(d*x + c))*sqrt(-a*b/(a^2 + 2*a*b + b^2)))*sqrt(-(2*(a + b)*sqrt(-a*b/(a^2
+ 2*a*b + b^2)) + a - b)/(a + b)) + a + b)/(a + b) + 1) + 2*(a + b)*sqrt(-a*b/(a^2 + 2*a*b + b^2))*d*x*dilog(
(((a - b)*cosh(d*x + c) + (a - b)*sinh(d*x + c) - 2*((a + b)*cosh(d*x + c) + (a + b)*sinh(d*x + c))*sqrt(-a*b/
(a^2 + 2*a*b + b^2)))*sqrt(-(2*(a + b)*sqrt(-a*b/(a^2 + 2*a*b + b^2)) + a - b)/(a + b)) - a - b)/(a + b) + 1)
- 2*(a + b)*sqrt(-a*b/(a^2 + 2*a*b + b^2))*d*x*dilog(-(((a - b)*cosh(d*x + c) + (a - b)*sinh(d*x + c) + 2*((a
+ b)*cosh(d*x + c) + (a + b)*sinh(d*x + c))*sqrt(-a*b/(a^2 + 2*a*b + b^2)))*sqrt((2*(a + b)*sqrt(-a*b/(a^2 + 2
*a*b + b^2)) - a + b)/(a + b)) + a + b)/(a + b) + 1) - 2*(a + b)*sqrt(-a*b/(a^2 + 2*a*b + b^2))*d*x*dilog((((a
- b)*cosh(d*x + c) + (a - b)*sinh(d*x + c) + 2*((a + b)*cosh(d*x + c) + (a + b)*sinh(d*x + c))*sqrt(-a*b/(a^2
+ 2*a*b + b^2)))*sqrt((2*(a + b)*sqrt(-a*b/(a^2 + 2*a*b + b^2)) - a + b)/(a + b)) - a - b)/(a + b) + 1) + (a
+ b)*sqrt(-a*b/(a^2 + 2*a*b + b^2))*c^2*log(2*sqrt(-(2*(a + b)*sqrt(-a*b/(a^2 + 2*a*b + b^2)) + a - b)/(a + b)
) + 2*cosh(d*x + c) + 2*sinh(d*x + c)) + (a + b)*sqrt(-a*b/(a^2 + 2*a*b + b^2))*c^2*log(-2*sqrt(-(2*(a + b)*sq
rt(-a*b/(a^2 + 2*a*b + b^2)) + a - b)/(a + b)) + 2*cosh(d*x + c) + 2*sinh(d*x + c)) - (a + b)*sqrt(-a*b/(a^2 +
2*a*b + b^2))*c^2*log(2*sqrt((2*(a + b)*sqrt(-a*b/(a^2 + 2*a*b + b^2)) - a + b)/(a + b)) + 2*cosh(d*x + c) +
2*sinh(d*x + c)) - (a + b)*sqrt(-a*b/(a^2 + 2*a*b + b^2))*c^2*log(-2*sqrt((2*(a + b)*sqrt(-a*b/(a^2 + 2*a*b +
b^2)) - a + b)/(a + b)) + 2*cosh(d*x + c) + 2*sinh(d*x + c)) + ((a + b)*d^2*x^2 - (a + b)*c^2)*sqrt(-a*b/(a^2
+ 2*a*b + b^2))*log((((a - b)*cosh(d*x + c) + (a - b)*sinh(d*x + c) - 2*((a + b)*cosh(d*x + c) + (a + b)*sinh(
d*x + c))*sqrt(-a*b/(a^2 + 2*a*b + b^2)))*sqrt(-(2*(a + b)*sqrt(-a*b/(a^2 + 2*a*b + b^2)) + a - b)/(a + b)) +
a + b)/(a + b)) + ((a + b)*d^2*x^2 - (a + b)*c^2)*sqrt(-a*b/(a^2 + 2*a*b + b^2))*log(-(((a - b)*cosh(d*x + c)
+ (a - b)*sinh(d*x + c) - 2*((a + b)*cosh(d*x + c) + (a + b)*sinh(d*x + c))*sqrt(-a*b/(a^2 + 2*a*b + b^2)))*sq
rt(-(2*(a + b)*sqrt(-a*b/(a^2 + 2*a*b + b^2)) + a - b)/(a + b)) - a - b)/(a + b)) - ((a + b)*d^2*x^2 - (a + b)
*c^2)*sqrt(-a*b/(a^2 + 2*a*b + b^2))*log((((a - b)*cosh(d*x + c) + (a - b)*sinh(d*x + c) + 2*((a + b)*cosh(d*x
+ c) + (a + b)*sinh(d*x + c))*sqrt(-a*b/(a^2 + 2*a*b + b^2)))*sqrt((2*(a + b)*sqrt(-a*b/(a^2 + 2*a*b + b^2))
- a + b)/(a + b)) + a + b)/(a + b)) - ((a + b)*d^2*x^2 - (a + b)*c^2)*sqrt(-a*b/(a^2 + 2*a*b + b^2))*log(-(((a
- b)*cosh(d*x + c) + (a - b)*sinh(d*x + c) + 2*((a + b)*cosh(d*x + c) + (a + b)*sinh(d*x + c))*sqrt(-a*b/(a^2
+ 2*a*b + b^2)))*sqrt((2*(a + b)*sqrt(-a*b/(a^2 + 2*a*b + b^2)) - a + b)/(a + b)) - a - b)/(a + b)) - 2*(a +
b)*sqrt(-a*b/(a^2 + 2*a*b + b^2))*polylog(3, ((a - b)*cosh(d*x + c) + (a - b)*sinh(d*x + c) - 2*((a + b)*cosh(
d*x + c) + (a + b)*sinh(d*x + c))*sqrt(-a*b/(a^2 + 2*a*b + b^2)))*sqrt(-(2*(a + b)*sqrt(-a*b/(a^2 + 2*a*b + b^
2)) + a - b)/(a + b))/(a + b)) - 2*(a + b)*sqrt(-a*b/(a^2 + 2*a*b + b^2))*polylog(3, -((a - b)*cosh(d*x + c) +
(a - b)*sinh(d*x + c) - 2*((a + b)*cosh(d*x + c) + (a + b)*sinh(d*x + c))*sqrt(-a*b/(a^2 + 2*a*b + b^2)))*sqr
t(-(2*(a + b)*sqrt(-a*b/(a^2 + 2*a*b + b^2)) + a - b)/(a + b))/(a + b)) + 2*(a + b)*sqrt(-a*b/(a^2 + 2*a*b + b
^2))*polylog(3, ((a - b)*cosh(d*x + c) + (a - b)*sinh(d*x + c) + 2*((a + b)*cosh(d*x + c) + (a + b)*sinh(d*x +
c))*sqrt(-a*b/(a^2 + 2*a*b + b^2)))*sqrt((2*(a + b)*sqrt(-a*b/(a^2 + 2*a*b + b^2)) - a + b)/(a + b))/(a + b))
+ 2*(a + b)*sqrt(-a*b/(a^2 + 2*a*b + b^2))*polylog(3, -((a - b)*cosh(d*x + c) + (a - b)*sinh(d*x + c) + 2*((a
+ b)*cosh(d*x + c) + (a + b)*sinh(d*x + c))*sqrt(-a*b/(a^2 + 2*a*b + b^2)))*sqrt((2*(a + b)*sqrt(-a*b/(a^2 +
2*a*b + b^2)) - a + b)/(a + b))/(a + b)))/(a*b*d^3)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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