3.489 \(\int x^2 \text{csch}^2(a+b x) \text{sech}(a+b x) \, dx\)

Optimal. Leaf size=157 \[ \frac{2 i x \text{PolyLog}\left (2,-i e^{a+b x}\right )}{b^2}-\frac{2 i x \text{PolyLog}\left (2,i e^{a+b x}\right )}{b^2}-\frac{2 \text{PolyLog}\left (2,-e^{a+b x}\right )}{b^3}+\frac{2 \text{PolyLog}\left (2,e^{a+b x}\right )}{b^3}-\frac{2 i \text{PolyLog}\left (3,-i e^{a+b x}\right )}{b^3}+\frac{2 i \text{PolyLog}\left (3,i e^{a+b x}\right )}{b^3}-\frac{4 x \tanh ^{-1}\left (e^{a+b x}\right )}{b^2}-\frac{2 x^2 \tan ^{-1}\left (e^{a+b x}\right )}{b}-\frac{x^2 \text{csch}(a+b x)}{b} \]

[Out]

(-2*x^2*ArcTan[E^(a + b*x)])/b - (4*x*ArcTanh[E^(a + b*x)])/b^2 - (x^2*Csch[a + b*x])/b - (2*PolyLog[2, -E^(a
+ b*x)])/b^3 + ((2*I)*x*PolyLog[2, (-I)*E^(a + b*x)])/b^2 - ((2*I)*x*PolyLog[2, I*E^(a + b*x)])/b^2 + (2*PolyL
og[2, E^(a + b*x)])/b^3 - ((2*I)*PolyLog[3, (-I)*E^(a + b*x)])/b^3 + ((2*I)*PolyLog[3, I*E^(a + b*x)])/b^3

________________________________________________________________________________________

Rubi [A]  time = 0.233922, antiderivative size = 157, normalized size of antiderivative = 1., number of steps used = 17, number of rules used = 14, integrand size = 18, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.778, Rules used = {2621, 321, 207, 5462, 14, 5205, 12, 4180, 2531, 2282, 6589, 4182, 2279, 2391} \[ \frac{2 i x \text{PolyLog}\left (2,-i e^{a+b x}\right )}{b^2}-\frac{2 i x \text{PolyLog}\left (2,i e^{a+b x}\right )}{b^2}-\frac{2 \text{PolyLog}\left (2,-e^{a+b x}\right )}{b^3}+\frac{2 \text{PolyLog}\left (2,e^{a+b x}\right )}{b^3}-\frac{2 i \text{PolyLog}\left (3,-i e^{a+b x}\right )}{b^3}+\frac{2 i \text{PolyLog}\left (3,i e^{a+b x}\right )}{b^3}-\frac{4 x \tanh ^{-1}\left (e^{a+b x}\right )}{b^2}-\frac{2 x^2 \tan ^{-1}\left (e^{a+b x}\right )}{b}-\frac{x^2 \text{csch}(a+b x)}{b} \]

Antiderivative was successfully verified.

[In]

Int[x^2*Csch[a + b*x]^2*Sech[a + b*x],x]

[Out]

(-2*x^2*ArcTan[E^(a + b*x)])/b - (4*x*ArcTanh[E^(a + b*x)])/b^2 - (x^2*Csch[a + b*x])/b - (2*PolyLog[2, -E^(a
+ b*x)])/b^3 + ((2*I)*x*PolyLog[2, (-I)*E^(a + b*x)])/b^2 - ((2*I)*x*PolyLog[2, I*E^(a + b*x)])/b^2 + (2*PolyL
og[2, E^(a + b*x)])/b^3 - ((2*I)*PolyLog[3, (-I)*E^(a + b*x)])/b^3 + ((2*I)*PolyLog[3, I*E^(a + b*x)])/b^3

Rule 2621

Int[(csc[(e_.) + (f_.)*(x_)]*(a_.))^(m_)*sec[(e_.) + (f_.)*(x_)]^(n_.), x_Symbol] :> -Dist[(f*a^n)^(-1), Subst
[Int[x^(m + n - 1)/(-1 + x^2/a^2)^((n + 1)/2), x], x, a*Csc[e + f*x]], x] /; FreeQ[{a, e, f, m}, x] && Integer
Q[(n + 1)/2] &&  !(IntegerQ[(m + 1)/2] && LtQ[0, m, n])

Rule 321

Int[((c_.)*(x_))^(m_)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[(c^(n - 1)*(c*x)^(m - n + 1)*(a + b*x^n
)^(p + 1))/(b*(m + n*p + 1)), x] - Dist[(a*c^n*(m - n + 1))/(b*(m + n*p + 1)), Int[(c*x)^(m - n)*(a + b*x^n)^p
, x], x] /; FreeQ[{a, b, c, p}, x] && IGtQ[n, 0] && GtQ[m, n - 1] && NeQ[m + n*p + 1, 0] && IntBinomialQ[a, b,
c, n, m, p, x]

Rule 207

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> -Simp[ArcTanh[(Rt[b, 2]*x)/Rt[-a, 2]]/(Rt[-a, 2]*Rt[b, 2]), x] /;
FreeQ[{a, b}, x] && NegQ[a/b] && (LtQ[a, 0] || GtQ[b, 0])

Rule 5462

Int[Csch[(a_.) + (b_.)*(x_)]^(n_.)*((c_.) + (d_.)*(x_))^(m_.)*Sech[(a_.) + (b_.)*(x_)]^(p_.), x_Symbol] :> Wit
h[{u = IntHide[Csch[a + b*x]^n*Sech[a + b*x]^p, x]}, Dist[(c + d*x)^m, u, x] - Dist[d*m, Int[(c + d*x)^(m - 1)
*u, x], x]] /; FreeQ[{a, b, c, d}, x] && IntegersQ[n, p] && GtQ[m, 0] && NeQ[n, p]

Rule 14

Int[(u_)*((c_.)*(x_))^(m_.), x_Symbol] :> Int[ExpandIntegrand[(c*x)^m*u, x], x] /; FreeQ[{c, m}, x] && SumQ[u]
&&  !LinearQ[u, x] &&  !MatchQ[u, (a_) + (b_.)*(v_) /; FreeQ[{a, b}, x] && InverseFunctionQ[v]]

Rule 5205

Int[((a_.) + ArcTan[u_]*(b_.))*((c_.) + (d_.)*(x_))^(m_.), x_Symbol] :> Simp[((c + d*x)^(m + 1)*(a + b*ArcTan[
u]))/(d*(m + 1)), x] - Dist[b/(d*(m + 1)), Int[SimplifyIntegrand[((c + d*x)^(m + 1)*D[u, x])/(1 + u^2), x], x]
, x] /; FreeQ[{a, b, c, d, m}, x] && NeQ[m, -1] && InverseFunctionFreeQ[u, x] &&  !FunctionOfQ[(c + d*x)^(m +
1), u, x] && FalseQ[PowerVariableExpn[u, m + 1, x]]

Rule 12

Int[(a_)*(u_), x_Symbol] :> Dist[a, Int[u, x], x] /; FreeQ[a, x] &&  !MatchQ[u, (b_)*(v_) /; FreeQ[b, x]]

Rule 4180

Int[csc[(e_.) + Pi*(k_.) + (Complex[0, fz_])*(f_.)*(x_)]*((c_.) + (d_.)*(x_))^(m_.), x_Symbol] :> Simp[(-2*(c
+ d*x)^m*ArcTanh[E^(-(I*e) + f*fz*x)/E^(I*k*Pi)])/(f*fz*I), x] + (-Dist[(d*m)/(f*fz*I), Int[(c + d*x)^(m - 1)*
Log[1 - E^(-(I*e) + f*fz*x)/E^(I*k*Pi)], x], x] + Dist[(d*m)/(f*fz*I), Int[(c + d*x)^(m - 1)*Log[1 + E^(-(I*e)
+ f*fz*x)/E^(I*k*Pi)], x], x]) /; FreeQ[{c, d, e, f, fz}, x] && IntegerQ[2*k] && 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 4182

Int[csc[(e_.) + (Complex[0, fz_])*(f_.)*(x_)]*((c_.) + (d_.)*(x_))^(m_.), x_Symbol] :> Simp[(-2*(c + d*x)^m*Ar
cTanh[E^(-(I*e) + f*fz*x)])/(f*fz*I), x] + (-Dist[(d*m)/(f*fz*I), Int[(c + d*x)^(m - 1)*Log[1 - E^(-(I*e) + f*
fz*x)], x], x] + Dist[(d*m)/(f*fz*I), Int[(c + d*x)^(m - 1)*Log[1 + E^(-(I*e) + f*fz*x)], x], x]) /; FreeQ[{c,
d, e, f, fz}, 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]

Rubi steps

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

Mathematica [A]  time = 1.85044, size = 312, normalized size = 1.99 \[ \frac{4 i b x \text{PolyLog}\left (2,-i e^{a+b x}\right )-4 i b x \text{PolyLog}\left (2,i e^{a+b x}\right )+4 \text{PolyLog}\left (2,-e^{-a-b x}\right )-4 \text{PolyLog}\left (2,e^{-a-b x}\right )-4 i \text{PolyLog}\left (3,-i e^{a+b x}\right )+4 i \text{PolyLog}\left (3,i e^{a+b x}\right )-2 i b^2 x^2 \log \left (1-i e^{a+b x}\right )+2 i b^2 x^2 \log \left (1+i e^{a+b x}\right )-2 b^2 x^2 \text{csch}(a)+b^2 x^2 \text{csch}\left (\frac{a}{2}\right ) \sinh \left (\frac{b x}{2}\right ) \text{csch}\left (\frac{1}{2} (a+b x)\right )+b^2 x^2 \text{sech}\left (\frac{a}{2}\right ) \sinh \left (\frac{b x}{2}\right ) \text{sech}\left (\frac{1}{2} (a+b x)\right )+4 b x \log \left (1-e^{-a-b x}\right )-4 b x \log \left (e^{-a-b x}+1\right )+4 a \log \left (1-e^{-a-b x}\right )-4 a \log \left (e^{-a-b x}+1\right )-4 a \log \left (\tanh \left (\frac{1}{2} (a+b x)\right )\right )}{2 b^3} \]

Antiderivative was successfully verified.

[In]

Integrate[x^2*Csch[a + b*x]^2*Sech[a + b*x],x]

[Out]

(-2*b^2*x^2*Csch[a] + 4*a*Log[1 - E^(-a - b*x)] + 4*b*x*Log[1 - E^(-a - b*x)] - 4*a*Log[1 + E^(-a - b*x)] - 4*
b*x*Log[1 + E^(-a - b*x)] - (2*I)*b^2*x^2*Log[1 - I*E^(a + b*x)] + (2*I)*b^2*x^2*Log[1 + I*E^(a + b*x)] - 4*a*
Log[Tanh[(a + b*x)/2]] + 4*PolyLog[2, -E^(-a - b*x)] - 4*PolyLog[2, E^(-a - b*x)] + (4*I)*b*x*PolyLog[2, (-I)*
E^(a + b*x)] - (4*I)*b*x*PolyLog[2, I*E^(a + b*x)] - (4*I)*PolyLog[3, (-I)*E^(a + b*x)] + (4*I)*PolyLog[3, I*E
^(a + b*x)] + b^2*x^2*Csch[a/2]*Csch[(a + b*x)/2]*Sinh[(b*x)/2] + b^2*x^2*Sech[a/2]*Sech[(a + b*x)/2]*Sinh[(b*
x)/2])/(2*b^3)

________________________________________________________________________________________

Maple [F]  time = 0.371, size = 0, normalized size = 0. \begin{align*} \int{x}^{2} \left ({\rm csch} \left (bx+a\right ) \right ) ^{2}{\rm sech} \left (bx+a\right )\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^2*csch(b*x+a)^2*sech(b*x+a),x)

[Out]

int(x^2*csch(b*x+a)^2*sech(b*x+a),x)

________________________________________________________________________________________

Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} -\frac{2 \, x^{2} e^{\left (b x + a\right )}}{b e^{\left (2 \, b x + 2 \, a\right )} - b} - \frac{2 \,{\left (b x \log \left (e^{\left (b x + a\right )} + 1\right ) +{\rm Li}_2\left (-e^{\left (b x + a\right )}\right )\right )}}{b^{3}} + \frac{2 \,{\left (b x \log \left (-e^{\left (b x + a\right )} + 1\right ) +{\rm Li}_2\left (e^{\left (b x + a\right )}\right )\right )}}{b^{3}} - 8 \, \int \frac{x^{2} e^{\left (b x + a\right )}}{4 \,{\left (e^{\left (2 \, b x + 2 \, a\right )} + 1\right )}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^2*csch(b*x+a)^2*sech(b*x+a),x, algorithm="maxima")

[Out]

-2*x^2*e^(b*x + a)/(b*e^(2*b*x + 2*a) - b) - 2*(b*x*log(e^(b*x + a) + 1) + dilog(-e^(b*x + a)))/b^3 + 2*(b*x*l
og(-e^(b*x + a) + 1) + dilog(e^(b*x + a)))/b^3 - 8*integrate(1/4*x^2*e^(b*x + a)/(e^(2*b*x + 2*a) + 1), x)

________________________________________________________________________________________

Fricas [C]  time = 2.76475, size = 2685, normalized size = 17.1 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^2*csch(b*x+a)^2*sech(b*x+a),x, algorithm="fricas")

[Out]

-(2*b^2*x^2*cosh(b*x + a) + 2*b^2*x^2*sinh(b*x + a) - 2*(cosh(b*x + a)^2 + 2*cosh(b*x + a)*sinh(b*x + a) + sin
h(b*x + a)^2 - 1)*dilog(cosh(b*x + a) + sinh(b*x + a)) - (-2*I*b*x*cosh(b*x + a)^2 - 4*I*b*x*cosh(b*x + a)*sin
h(b*x + a) - 2*I*b*x*sinh(b*x + a)^2 + 2*I*b*x)*dilog(I*cosh(b*x + a) + I*sinh(b*x + a)) - (2*I*b*x*cosh(b*x +
a)^2 + 4*I*b*x*cosh(b*x + a)*sinh(b*x + a) + 2*I*b*x*sinh(b*x + a)^2 - 2*I*b*x)*dilog(-I*cosh(b*x + a) - I*si
nh(b*x + a)) + 2*(cosh(b*x + a)^2 + 2*cosh(b*x + a)*sinh(b*x + a) + sinh(b*x + a)^2 - 1)*dilog(-cosh(b*x + a)
- sinh(b*x + a)) + 2*(b*x*cosh(b*x + a)^2 + 2*b*x*cosh(b*x + a)*sinh(b*x + a) + b*x*sinh(b*x + a)^2 - b*x)*log
(cosh(b*x + a) + sinh(b*x + a) + 1) - (-I*a^2*cosh(b*x + a)^2 - 2*I*a^2*cosh(b*x + a)*sinh(b*x + a) - I*a^2*si
nh(b*x + a)^2 + I*a^2)*log(cosh(b*x + a) + sinh(b*x + a) + I) - (I*a^2*cosh(b*x + a)^2 + 2*I*a^2*cosh(b*x + a)
*sinh(b*x + a) + I*a^2*sinh(b*x + a)^2 - I*a^2)*log(cosh(b*x + a) + sinh(b*x + a) - I) + 2*(a*cosh(b*x + a)^2
+ 2*a*cosh(b*x + a)*sinh(b*x + a) + a*sinh(b*x + a)^2 - a)*log(cosh(b*x + a) + sinh(b*x + a) - 1) - (-I*b^2*x^
2 + (I*b^2*x^2 - I*a^2)*cosh(b*x + a)^2 + (2*I*b^2*x^2 - 2*I*a^2)*cosh(b*x + a)*sinh(b*x + a) + (I*b^2*x^2 - I
*a^2)*sinh(b*x + a)^2 + I*a^2)*log(I*cosh(b*x + a) + I*sinh(b*x + a) + 1) - (I*b^2*x^2 + (-I*b^2*x^2 + I*a^2)*
cosh(b*x + a)^2 + (-2*I*b^2*x^2 + 2*I*a^2)*cosh(b*x + a)*sinh(b*x + a) + (-I*b^2*x^2 + I*a^2)*sinh(b*x + a)^2
- I*a^2)*log(-I*cosh(b*x + a) - I*sinh(b*x + a) + 1) - 2*((b*x + a)*cosh(b*x + a)^2 + 2*(b*x + a)*cosh(b*x + a
)*sinh(b*x + a) + (b*x + a)*sinh(b*x + a)^2 - b*x - a)*log(-cosh(b*x + a) - sinh(b*x + a) + 1) - (2*I*cosh(b*x
+ a)^2 + 4*I*cosh(b*x + a)*sinh(b*x + a) + 2*I*sinh(b*x + a)^2 - 2*I)*polylog(3, I*cosh(b*x + a) + I*sinh(b*x
+ a)) - (-2*I*cosh(b*x + a)^2 - 4*I*cosh(b*x + a)*sinh(b*x + a) - 2*I*sinh(b*x + a)^2 + 2*I)*polylog(3, -I*co
sh(b*x + a) - I*sinh(b*x + a)))/(b^3*cosh(b*x + a)^2 + 2*b^3*cosh(b*x + a)*sinh(b*x + a) + b^3*sinh(b*x + a)^2
- b^3)

________________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x**2*csch(b*x+a)**2*sech(b*x+a),x)

[Out]

Integral(x**2*csch(a + b*x)**2*sech(a + b*x), x)

________________________________________________________________________________________

Giac [F(-1)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^2*csch(b*x+a)^2*sech(b*x+a),x, algorithm="giac")

[Out]

Timed out