3.502 \(\int x^2 \text{csch}^2(a+b x) \text{sech}^3(a+b x) \, dx\)
Optimal. Leaf size=206 \[ \frac{3 i x \text{PolyLog}\left (2,-i e^{a+b x}\right )}{b^2}-\frac{3 i x \text{PolyLog}\left (2,i e^{a+b x}\right )}{b^2}-\frac{2 \text{PolyLog}\left (2,-e^{a+b x}\right )}{b^3}+\frac{2 \text{PolyLog}\left (2,e^{a+b x}\right )}{b^3}-\frac{3 i \text{PolyLog}\left (3,-i e^{a+b x}\right )}{b^3}+\frac{3 i \text{PolyLog}\left (3,i e^{a+b x}\right )}{b^3}-\frac{4 x \tanh ^{-1}\left (e^{a+b x}\right )}{b^2}-\frac{x \text{sech}(a+b x)}{b^2}+\frac{\tan ^{-1}(\sinh (a+b x))}{b^3}-\frac{3 x^2 \tan ^{-1}\left (e^{a+b x}\right )}{b}-\frac{3 x^2 \text{csch}(a+b x)}{2 b}+\frac{x^2 \text{csch}(a+b x) \text{sech}^2(a+b x)}{2 b} \]
[Out]
(-3*x^2*ArcTan[E^(a + b*x)])/b + ArcTan[Sinh[a + b*x]]/b^3 - (4*x*ArcTanh[E^(a + b*x)])/b^2 - (3*x^2*Csch[a +
b*x])/(2*b) - (2*PolyLog[2, -E^(a + b*x)])/b^3 + ((3*I)*x*PolyLog[2, (-I)*E^(a + b*x)])/b^2 - ((3*I)*x*PolyLog
[2, I*E^(a + b*x)])/b^2 + (2*PolyLog[2, E^(a + b*x)])/b^3 - ((3*I)*PolyLog[3, (-I)*E^(a + b*x)])/b^3 + ((3*I)*
PolyLog[3, I*E^(a + b*x)])/b^3 - (x*Sech[a + b*x])/b^2 + (x^2*Csch[a + b*x]*Sech[a + b*x]^2)/(2*b)
________________________________________________________________________________________
Rubi [A] time = 0.434258, antiderivative size = 206, normalized size of antiderivative = 1.,
number of steps used = 29, number of rules used = 18, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.9, Rules used =
{2621, 288, 321, 207, 5462, 14, 5205, 12, 4180, 2531, 2282, 6589, 4182, 2279, 2391, 2622, 6271,
3770} \[ \frac{3 i x \text{PolyLog}\left (2,-i e^{a+b x}\right )}{b^2}-\frac{3 i x \text{PolyLog}\left (2,i e^{a+b x}\right )}{b^2}-\frac{2 \text{PolyLog}\left (2,-e^{a+b x}\right )}{b^3}+\frac{2 \text{PolyLog}\left (2,e^{a+b x}\right )}{b^3}-\frac{3 i \text{PolyLog}\left (3,-i e^{a+b x}\right )}{b^3}+\frac{3 i \text{PolyLog}\left (3,i e^{a+b x}\right )}{b^3}-\frac{4 x \tanh ^{-1}\left (e^{a+b x}\right )}{b^2}-\frac{x \text{sech}(a+b x)}{b^2}+\frac{\tan ^{-1}(\sinh (a+b x))}{b^3}-\frac{3 x^2 \tan ^{-1}\left (e^{a+b x}\right )}{b}-\frac{3 x^2 \text{csch}(a+b x)}{2 b}+\frac{x^2 \text{csch}(a+b x) \text{sech}^2(a+b x)}{2 b} \]
Antiderivative was successfully verified.
[In]
Int[x^2*Csch[a + b*x]^2*Sech[a + b*x]^3,x]
[Out]
(-3*x^2*ArcTan[E^(a + b*x)])/b + ArcTan[Sinh[a + b*x]]/b^3 - (4*x*ArcTanh[E^(a + b*x)])/b^2 - (3*x^2*Csch[a +
b*x])/(2*b) - (2*PolyLog[2, -E^(a + b*x)])/b^3 + ((3*I)*x*PolyLog[2, (-I)*E^(a + b*x)])/b^2 - ((3*I)*x*PolyLog
[2, I*E^(a + b*x)])/b^2 + (2*PolyLog[2, E^(a + b*x)])/b^3 - ((3*I)*PolyLog[3, (-I)*E^(a + b*x)])/b^3 + ((3*I)*
PolyLog[3, I*E^(a + b*x)])/b^3 - (x*Sech[a + b*x])/b^2 + (x^2*Csch[a + b*x]*Sech[a + b*x]^2)/(2*b)
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 288
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*n*(p + 1)), x] - Dist[(c^n*(m - n + 1))/(b*n*(p + 1)), Int[(c*x)^(m - n)*(a + b*x^n)^(p + 1), x
], x] /; FreeQ[{a, b, c}, x] && IGtQ[n, 0] && LtQ[p, -1] && GtQ[m + 1, n] && !ILtQ[(m + n*(p + 1) + 1)/n, 0]
&& IntBinomialQ[a, b, c, n, m, p, x]
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]
Rule 2622
Int[csc[(e_.) + (f_.)*(x_)]^(n_.)*((a_.)*sec[(e_.) + (f_.)*(x_)])^(m_), x_Symbol] :> Dist[1/(f*a^n), Subst[Int
[x^(m + n - 1)/(-1 + x^2/a^2)^((n + 1)/2), x], x, a*Sec[e + f*x]], x] /; FreeQ[{a, e, f, m}, x] && IntegerQ[(n
+ 1)/2] && !(IntegerQ[(m + 1)/2] && LtQ[0, m, n])
Rule 6271
Int[ArcTanh[u_], x_Symbol] :> Simp[x*ArcTanh[u], x] - Int[SimplifyIntegrand[(x*D[u, x])/(1 - u^2), x], x] /; I
nverseFunctionFreeQ[u, x]
Rule 3770
Int[csc[(c_.) + (d_.)*(x_)], x_Symbol] :> -Simp[ArcTanh[Cos[c + d*x]]/d, x] /; FreeQ[{c, d}, x]
Rubi steps
\begin{align*} \int x^2 \text{csch}^2(a+b x) \text{sech}^3(a+b x) \, dx &=-\frac{3 x^2 \tan ^{-1}(\sinh (a+b x))}{2 b}-\frac{3 x^2 \text{csch}(a+b x)}{2 b}+\frac{x^2 \text{csch}(a+b x) \text{sech}^2(a+b x)}{2 b}-2 \int x \left (-\frac{3 \tan ^{-1}(\sinh (a+b x))}{2 b}-\frac{3 \text{csch}(a+b x)}{2 b}+\frac{\text{csch}(a+b x) \text{sech}^2(a+b x)}{2 b}\right ) \, dx\\ &=-\frac{3 x^2 \tan ^{-1}(\sinh (a+b x))}{2 b}-\frac{3 x^2 \text{csch}(a+b x)}{2 b}+\frac{x^2 \text{csch}(a+b x) \text{sech}^2(a+b x)}{2 b}-2 \int \left (-\frac{3 x \left (\tan ^{-1}(\sinh (a+b x))+\text{csch}(a+b x)\right )}{2 b}+\frac{x \text{csch}(a+b x) \text{sech}^2(a+b x)}{2 b}\right ) \, dx\\ &=-\frac{3 x^2 \tan ^{-1}(\sinh (a+b x))}{2 b}-\frac{3 x^2 \text{csch}(a+b x)}{2 b}+\frac{x^2 \text{csch}(a+b x) \text{sech}^2(a+b x)}{2 b}-\frac{\int x \text{csch}(a+b x) \text{sech}^2(a+b x) \, dx}{b}+\frac{3 \int x \left (\tan ^{-1}(\sinh (a+b x))+\text{csch}(a+b x)\right ) \, dx}{b}\\ &=-\frac{3 x^2 \tan ^{-1}(\sinh (a+b x))}{2 b}+\frac{x \tanh ^{-1}(\cosh (a+b x))}{b^2}-\frac{3 x^2 \text{csch}(a+b x)}{2 b}-\frac{x \text{sech}(a+b x)}{b^2}+\frac{x^2 \text{csch}(a+b x) \text{sech}^2(a+b x)}{2 b}+\frac{\int \left (-\frac{\tanh ^{-1}(\cosh (a+b x))}{b}+\frac{\text{sech}(a+b x)}{b}\right ) \, dx}{b}+\frac{3 \int \left (x \tan ^{-1}(\sinh (a+b x))+x \text{csch}(a+b x)\right ) \, dx}{b}\\ &=-\frac{3 x^2 \tan ^{-1}(\sinh (a+b x))}{2 b}+\frac{x \tanh ^{-1}(\cosh (a+b x))}{b^2}-\frac{3 x^2 \text{csch}(a+b x)}{2 b}-\frac{x \text{sech}(a+b x)}{b^2}+\frac{x^2 \text{csch}(a+b x) \text{sech}^2(a+b x)}{2 b}-\frac{\int \tanh ^{-1}(\cosh (a+b x)) \, dx}{b^2}+\frac{\int \text{sech}(a+b x) \, dx}{b^2}+\frac{3 \int x \tan ^{-1}(\sinh (a+b x)) \, dx}{b}+\frac{3 \int x \text{csch}(a+b x) \, dx}{b}\\ &=\frac{\tan ^{-1}(\sinh (a+b x))}{b^3}-\frac{6 x \tanh ^{-1}\left (e^{a+b x}\right )}{b^2}-\frac{3 x^2 \text{csch}(a+b x)}{2 b}-\frac{x \text{sech}(a+b x)}{b^2}+\frac{x^2 \text{csch}(a+b x) \text{sech}^2(a+b x)}{2 b}-\frac{\int b x \text{csch}(a+b x) \, dx}{b^2}-\frac{3 \int \log \left (1-e^{a+b x}\right ) \, dx}{b^2}+\frac{3 \int \log \left (1+e^{a+b x}\right ) \, dx}{b^2}-\frac{3 \int b x^2 \text{sech}(a+b x) \, dx}{2 b}\\ &=\frac{\tan ^{-1}(\sinh (a+b x))}{b^3}-\frac{6 x \tanh ^{-1}\left (e^{a+b x}\right )}{b^2}-\frac{3 x^2 \text{csch}(a+b x)}{2 b}-\frac{x \text{sech}(a+b x)}{b^2}+\frac{x^2 \text{csch}(a+b x) \text{sech}^2(a+b x)}{2 b}-\frac{3}{2} \int x^2 \text{sech}(a+b x) \, dx-\frac{3 \operatorname{Subst}\left (\int \frac{\log (1-x)}{x} \, dx,x,e^{a+b x}\right )}{b^3}+\frac{3 \operatorname{Subst}\left (\int \frac{\log (1+x)}{x} \, dx,x,e^{a+b x}\right )}{b^3}-\frac{\int x \text{csch}(a+b x) \, dx}{b}\\ &=-\frac{3 x^2 \tan ^{-1}\left (e^{a+b x}\right )}{b}+\frac{\tan ^{-1}(\sinh (a+b x))}{b^3}-\frac{4 x \tanh ^{-1}\left (e^{a+b x}\right )}{b^2}-\frac{3 x^2 \text{csch}(a+b x)}{2 b}-\frac{3 \text{Li}_2\left (-e^{a+b x}\right )}{b^3}+\frac{3 \text{Li}_2\left (e^{a+b x}\right )}{b^3}-\frac{x \text{sech}(a+b x)}{b^2}+\frac{x^2 \text{csch}(a+b x) \text{sech}^2(a+b x)}{2 b}+\frac{\int \log \left (1-e^{a+b x}\right ) \, dx}{b^2}-\frac{\int \log \left (1+e^{a+b x}\right ) \, dx}{b^2}+\frac{(3 i) \int x \log \left (1-i e^{a+b x}\right ) \, dx}{b}-\frac{(3 i) \int x \log \left (1+i e^{a+b x}\right ) \, dx}{b}\\ &=-\frac{3 x^2 \tan ^{-1}\left (e^{a+b x}\right )}{b}+\frac{\tan ^{-1}(\sinh (a+b x))}{b^3}-\frac{4 x \tanh ^{-1}\left (e^{a+b x}\right )}{b^2}-\frac{3 x^2 \text{csch}(a+b x)}{2 b}-\frac{3 \text{Li}_2\left (-e^{a+b x}\right )}{b^3}+\frac{3 i x \text{Li}_2\left (-i e^{a+b x}\right )}{b^2}-\frac{3 i x \text{Li}_2\left (i e^{a+b x}\right )}{b^2}+\frac{3 \text{Li}_2\left (e^{a+b x}\right )}{b^3}-\frac{x \text{sech}(a+b x)}{b^2}+\frac{x^2 \text{csch}(a+b x) \text{sech}^2(a+b x)}{2 b}+\frac{\operatorname{Subst}\left (\int \frac{\log (1-x)}{x} \, dx,x,e^{a+b x}\right )}{b^3}-\frac{\operatorname{Subst}\left (\int \frac{\log (1+x)}{x} \, dx,x,e^{a+b x}\right )}{b^3}-\frac{(3 i) \int \text{Li}_2\left (-i e^{a+b x}\right ) \, dx}{b^2}+\frac{(3 i) \int \text{Li}_2\left (i e^{a+b x}\right ) \, dx}{b^2}\\ &=-\frac{3 x^2 \tan ^{-1}\left (e^{a+b x}\right )}{b}+\frac{\tan ^{-1}(\sinh (a+b x))}{b^3}-\frac{4 x \tanh ^{-1}\left (e^{a+b x}\right )}{b^2}-\frac{3 x^2 \text{csch}(a+b x)}{2 b}-\frac{2 \text{Li}_2\left (-e^{a+b x}\right )}{b^3}+\frac{3 i x \text{Li}_2\left (-i e^{a+b x}\right )}{b^2}-\frac{3 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{x \text{sech}(a+b x)}{b^2}+\frac{x^2 \text{csch}(a+b x) \text{sech}^2(a+b x)}{2 b}-\frac{(3 i) \operatorname{Subst}\left (\int \frac{\text{Li}_2(-i x)}{x} \, dx,x,e^{a+b x}\right )}{b^3}+\frac{(3 i) \operatorname{Subst}\left (\int \frac{\text{Li}_2(i x)}{x} \, dx,x,e^{a+b x}\right )}{b^3}\\ &=-\frac{3 x^2 \tan ^{-1}\left (e^{a+b x}\right )}{b}+\frac{\tan ^{-1}(\sinh (a+b x))}{b^3}-\frac{4 x \tanh ^{-1}\left (e^{a+b x}\right )}{b^2}-\frac{3 x^2 \text{csch}(a+b x)}{2 b}-\frac{2 \text{Li}_2\left (-e^{a+b x}\right )}{b^3}+\frac{3 i x \text{Li}_2\left (-i e^{a+b x}\right )}{b^2}-\frac{3 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{3 i \text{Li}_3\left (-i e^{a+b x}\right )}{b^3}+\frac{3 i \text{Li}_3\left (i e^{a+b x}\right )}{b^3}-\frac{x \text{sech}(a+b x)}{b^2}+\frac{x^2 \text{csch}(a+b x) \text{sech}^2(a+b x)}{2 b}\\ \end{align*}
Mathematica [A] time = 7.50837, size = 397, normalized size = 1.93 \[ -\frac{i \left (-6 b x \text{PolyLog}\left (2,-i e^{a+b x}\right )+6 b x \text{PolyLog}\left (2,i e^{a+b x}\right )+6 \text{PolyLog}\left (3,-i e^{a+b x}\right )-6 \text{PolyLog}\left (3,i e^{a+b x}\right )+3 b^2 x^2 \log \left (1-i e^{a+b x}\right )-3 b^2 x^2 \log \left (1+i e^{a+b x}\right )+4 i \tan ^{-1}\left (e^{a+b x}\right )\right )}{2 b^3}+\frac{2 \left (-a \log \left (\tanh \left (\frac{1}{2} (a+b x)\right )\right )-i \left (i \left (\text{PolyLog}\left (2,-e^{i (i a+i b x)}\right )-\text{PolyLog}\left (2,e^{i (i a+i b x)}\right )\right )+(i a+i b x) \left (\log \left (1-e^{i (i a+i b x)}\right )-\log \left (1+e^{i (i a+i b x)}\right )\right )\right )\right )}{b^3}-\frac{x \text{sech}(a) \text{sech}(a+b x) (b x \sinh (a)+2 \cosh (a))}{2 b^2}-\frac{x^2 \text{csch}(a)}{b}+\frac{x^2 \text{csch}\left (\frac{a}{2}\right ) \sinh \left (\frac{b x}{2}\right ) \text{csch}\left (\frac{a}{2}+\frac{b x}{2}\right )}{2 b}-\frac{x^2 \text{sech}(a) \sinh (b x) \text{sech}^2(a+b x)}{2 b}+\frac{x^2 \text{sech}\left (\frac{a}{2}\right ) \sinh \left (\frac{b x}{2}\right ) \text{sech}\left (\frac{a}{2}+\frac{b x}{2}\right )}{2 b} \]
Antiderivative was successfully verified.
[In]
Integrate[x^2*Csch[a + b*x]^2*Sech[a + b*x]^3,x]
[Out]
-((x^2*Csch[a])/b) + (2*(-(a*Log[Tanh[(a + b*x)/2]]) - I*((I*a + I*b*x)*(Log[1 - E^(I*(I*a + I*b*x))] - Log[1
+ E^(I*(I*a + I*b*x))]) + I*(PolyLog[2, -E^(I*(I*a + I*b*x))] - PolyLog[2, E^(I*(I*a + I*b*x))]))))/b^3 - ((I/
2)*((4*I)*ArcTan[E^(a + b*x)] + 3*b^2*x^2*Log[1 - I*E^(a + b*x)] - 3*b^2*x^2*Log[1 + I*E^(a + b*x)] - 6*b*x*Po
lyLog[2, (-I)*E^(a + b*x)] + 6*b*x*PolyLog[2, I*E^(a + b*x)] + 6*PolyLog[3, (-I)*E^(a + b*x)] - 6*PolyLog[3, I
*E^(a + b*x)]))/b^3 - (x*Sech[a]*Sech[a + b*x]*(2*Cosh[a] + b*x*Sinh[a]))/(2*b^2) + (x^2*Csch[a/2]*Csch[a/2 +
(b*x)/2]*Sinh[(b*x)/2])/(2*b) + (x^2*Sech[a/2]*Sech[a/2 + (b*x)/2]*Sinh[(b*x)/2])/(2*b) - (x^2*Sech[a]*Sech[a
+ b*x]^2*Sinh[b*x])/(2*b)
________________________________________________________________________________________
Maple [F] time = 0.441, size = 0, normalized size = 0. \begin{align*} \int{x}^{2} \left ({\rm csch} \left (bx+a\right ) \right ) ^{2} \left ({\rm sech} \left (bx+a\right ) \right ) ^{3}\, 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)^3,x)
[Out]
int(x^2*csch(b*x+a)^2*sech(b*x+a)^3,x)
________________________________________________________________________________________
Maxima [F] time = 0., size = 0, normalized size = 0. \begin{align*} -96 \, b^{2} \int \frac{x^{2} e^{\left (b x + a\right )}}{32 \,{\left (b^{2} e^{\left (2 \, b x + 2 \, a\right )} + b^{2}\right )}}\,{d x} - \frac{2 \, b x^{2} e^{\left (3 \, b x + 3 \, a\right )} +{\left (3 \, b x^{2} e^{\left (5 \, a\right )} + 2 \, x e^{\left (5 \, a\right )}\right )} e^{\left (5 \, b x\right )} +{\left (3 \, b x^{2} e^{a} - 2 \, x e^{a}\right )} e^{\left (b x\right )}}{b^{2} e^{\left (6 \, b x + 6 \, a\right )} + b^{2} e^{\left (4 \, b x + 4 \, a\right )} - b^{2} e^{\left (2 \, b x + 2 \, a\right )} - b^{2}} - \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}} + \frac{2 \, \arctan \left (e^{\left (b x + a\right )}\right )}{b^{3}} \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)^3,x, algorithm="maxima")
[Out]
-96*b^2*integrate(1/32*x^2*e^(b*x + a)/(b^2*e^(2*b*x + 2*a) + b^2), x) - (2*b*x^2*e^(3*b*x + 3*a) + (3*b*x^2*e
^(5*a) + 2*x*e^(5*a))*e^(5*b*x) + (3*b*x^2*e^a - 2*x*e^a)*e^(b*x))/(b^2*e^(6*b*x + 6*a) + b^2*e^(4*b*x + 4*a)
- b^2*e^(2*b*x + 2*a) - b^2) - 2*(b*x*log(e^(b*x + a) + 1) + dilog(-e^(b*x + a)))/b^3 + 2*(b*x*log(-e^(b*x + a
) + 1) + dilog(e^(b*x + a)))/b^3 + 2*arctan(e^(b*x + a))/b^3
________________________________________________________________________________________
Fricas [C] time = 3.08021, size = 10452, normalized size = 50.74 \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)^3,x, algorithm="fricas")
[Out]
-1/2*(4*b^2*x^2*cosh(b*x + a)^3 + 2*(3*b^2*x^2 + 2*b*x)*cosh(b*x + a)^5 + 10*(3*b^2*x^2 + 2*b*x)*cosh(b*x + a)
*sinh(b*x + a)^4 + 2*(3*b^2*x^2 + 2*b*x)*sinh(b*x + a)^5 + 4*(b^2*x^2 + 5*(3*b^2*x^2 + 2*b*x)*cosh(b*x + a)^2)
*sinh(b*x + a)^3 + 4*(3*b^2*x^2*cosh(b*x + a) + 5*(3*b^2*x^2 + 2*b*x)*cosh(b*x + a)^3)*sinh(b*x + a)^2 + 2*(3*
b^2*x^2 - 2*b*x)*cosh(b*x + a) - 4*(cosh(b*x + a)^6 + 6*cosh(b*x + a)*sinh(b*x + a)^5 + sinh(b*x + a)^6 + (15*
cosh(b*x + a)^2 + 1)*sinh(b*x + a)^4 + cosh(b*x + a)^4 + 4*(5*cosh(b*x + a)^3 + cosh(b*x + a))*sinh(b*x + a)^3
+ (15*cosh(b*x + a)^4 + 6*cosh(b*x + a)^2 - 1)*sinh(b*x + a)^2 - cosh(b*x + a)^2 + 2*(3*cosh(b*x + a)^5 + 2*c
osh(b*x + a)^3 - cosh(b*x + a))*sinh(b*x + a) - 1)*dilog(cosh(b*x + a) + sinh(b*x + a)) - (-6*I*b*x*cosh(b*x +
a)^6 - 36*I*b*x*cosh(b*x + a)*sinh(b*x + a)^5 - 6*I*b*x*sinh(b*x + a)^6 - 6*I*b*x*cosh(b*x + a)^4 + (-90*I*b*
x*cosh(b*x + a)^2 - 6*I*b*x)*sinh(b*x + a)^4 + 6*I*b*x*cosh(b*x + a)^2 + (-120*I*b*x*cosh(b*x + a)^3 - 24*I*b*
x*cosh(b*x + a))*sinh(b*x + a)^3 + (-90*I*b*x*cosh(b*x + a)^4 - 36*I*b*x*cosh(b*x + a)^2 + 6*I*b*x)*sinh(b*x +
a)^2 + 6*I*b*x + (-36*I*b*x*cosh(b*x + a)^5 - 24*I*b*x*cosh(b*x + a)^3 + 12*I*b*x*cosh(b*x + a))*sinh(b*x + a
))*dilog(I*cosh(b*x + a) + I*sinh(b*x + a)) - (6*I*b*x*cosh(b*x + a)^6 + 36*I*b*x*cosh(b*x + a)*sinh(b*x + a)^
5 + 6*I*b*x*sinh(b*x + a)^6 + 6*I*b*x*cosh(b*x + a)^4 + (90*I*b*x*cosh(b*x + a)^2 + 6*I*b*x)*sinh(b*x + a)^4 -
6*I*b*x*cosh(b*x + a)^2 + (120*I*b*x*cosh(b*x + a)^3 + 24*I*b*x*cosh(b*x + a))*sinh(b*x + a)^3 + (90*I*b*x*co
sh(b*x + a)^4 + 36*I*b*x*cosh(b*x + a)^2 - 6*I*b*x)*sinh(b*x + a)^2 - 6*I*b*x + (36*I*b*x*cosh(b*x + a)^5 + 24
*I*b*x*cosh(b*x + a)^3 - 12*I*b*x*cosh(b*x + a))*sinh(b*x + a))*dilog(-I*cosh(b*x + a) - I*sinh(b*x + a)) + 4*
(cosh(b*x + a)^6 + 6*cosh(b*x + a)*sinh(b*x + a)^5 + sinh(b*x + a)^6 + (15*cosh(b*x + a)^2 + 1)*sinh(b*x + a)^
4 + cosh(b*x + a)^4 + 4*(5*cosh(b*x + a)^3 + cosh(b*x + a))*sinh(b*x + a)^3 + (15*cosh(b*x + a)^4 + 6*cosh(b*x
+ a)^2 - 1)*sinh(b*x + a)^2 - cosh(b*x + a)^2 + 2*(3*cosh(b*x + a)^5 + 2*cosh(b*x + a)^3 - cosh(b*x + a))*sin
h(b*x + a) - 1)*dilog(-cosh(b*x + a) - sinh(b*x + a)) + 4*(b*x*cosh(b*x + a)^6 + 6*b*x*cosh(b*x + a)*sinh(b*x
+ a)^5 + b*x*sinh(b*x + a)^6 + b*x*cosh(b*x + a)^4 + (15*b*x*cosh(b*x + a)^2 + b*x)*sinh(b*x + a)^4 - b*x*cosh
(b*x + a)^2 + 4*(5*b*x*cosh(b*x + a)^3 + b*x*cosh(b*x + a))*sinh(b*x + a)^3 + (15*b*x*cosh(b*x + a)^4 + 6*b*x*
cosh(b*x + a)^2 - b*x)*sinh(b*x + a)^2 - b*x + 2*(3*b*x*cosh(b*x + a)^5 + 2*b*x*cosh(b*x + a)^3 - b*x*cosh(b*x
+ a))*sinh(b*x + a))*log(cosh(b*x + a) + sinh(b*x + a) + 1) - ((-3*I*a^2 + 2*I)*cosh(b*x + a)^6 + (-18*I*a^2
+ 12*I)*cosh(b*x + a)*sinh(b*x + a)^5 + (-3*I*a^2 + 2*I)*sinh(b*x + a)^6 + (-3*I*a^2 + 2*I)*cosh(b*x + a)^4 +
((-45*I*a^2 + 30*I)*cosh(b*x + a)^2 - 3*I*a^2 + 2*I)*sinh(b*x + a)^4 + ((-60*I*a^2 + 40*I)*cosh(b*x + a)^3 + (
-12*I*a^2 + 8*I)*cosh(b*x + a))*sinh(b*x + a)^3 + (3*I*a^2 - 2*I)*cosh(b*x + a)^2 + ((-45*I*a^2 + 30*I)*cosh(b
*x + a)^4 + (-18*I*a^2 + 12*I)*cosh(b*x + a)^2 + 3*I*a^2 - 2*I)*sinh(b*x + a)^2 + 3*I*a^2 + ((-18*I*a^2 + 12*I
)*cosh(b*x + a)^5 + (-12*I*a^2 + 8*I)*cosh(b*x + a)^3 + (6*I*a^2 - 4*I)*cosh(b*x + a))*sinh(b*x + a) - 2*I)*lo
g(cosh(b*x + a) + sinh(b*x + a) + I) - ((3*I*a^2 - 2*I)*cosh(b*x + a)^6 + (18*I*a^2 - 12*I)*cosh(b*x + a)*sinh
(b*x + a)^5 + (3*I*a^2 - 2*I)*sinh(b*x + a)^6 + (3*I*a^2 - 2*I)*cosh(b*x + a)^4 + ((45*I*a^2 - 30*I)*cosh(b*x
+ a)^2 + 3*I*a^2 - 2*I)*sinh(b*x + a)^4 + ((60*I*a^2 - 40*I)*cosh(b*x + a)^3 + (12*I*a^2 - 8*I)*cosh(b*x + a))
*sinh(b*x + a)^3 + (-3*I*a^2 + 2*I)*cosh(b*x + a)^2 + ((45*I*a^2 - 30*I)*cosh(b*x + a)^4 + (18*I*a^2 - 12*I)*c
osh(b*x + a)^2 - 3*I*a^2 + 2*I)*sinh(b*x + a)^2 - 3*I*a^2 + ((18*I*a^2 - 12*I)*cosh(b*x + a)^5 + (12*I*a^2 - 8
*I)*cosh(b*x + a)^3 + (-6*I*a^2 + 4*I)*cosh(b*x + a))*sinh(b*x + a) + 2*I)*log(cosh(b*x + a) + sinh(b*x + a) -
I) + 4*(a*cosh(b*x + a)^6 + 6*a*cosh(b*x + a)*sinh(b*x + a)^5 + a*sinh(b*x + a)^6 + a*cosh(b*x + a)^4 + (15*a
*cosh(b*x + a)^2 + a)*sinh(b*x + a)^4 + 4*(5*a*cosh(b*x + a)^3 + a*cosh(b*x + a))*sinh(b*x + a)^3 - a*cosh(b*x
+ a)^2 + (15*a*cosh(b*x + a)^4 + 6*a*cosh(b*x + a)^2 - a)*sinh(b*x + a)^2 + 2*(3*a*cosh(b*x + a)^5 + 2*a*cosh
(b*x + a)^3 - a*cosh(b*x + a))*sinh(b*x + a) - a)*log(cosh(b*x + a) + sinh(b*x + a) - 1) - ((3*I*b^2*x^2 - 3*I
*a^2)*cosh(b*x + a)^6 + (18*I*b^2*x^2 - 18*I*a^2)*cosh(b*x + a)*sinh(b*x + a)^5 + (3*I*b^2*x^2 - 3*I*a^2)*sinh
(b*x + a)^6 + (3*I*b^2*x^2 - 3*I*a^2)*cosh(b*x + a)^4 + (3*I*b^2*x^2 + (45*I*b^2*x^2 - 45*I*a^2)*cosh(b*x + a)
^2 - 3*I*a^2)*sinh(b*x + a)^4 - 3*I*b^2*x^2 + ((60*I*b^2*x^2 - 60*I*a^2)*cosh(b*x + a)^3 + (12*I*b^2*x^2 - 12*
I*a^2)*cosh(b*x + a))*sinh(b*x + a)^3 + (-3*I*b^2*x^2 + 3*I*a^2)*cosh(b*x + a)^2 + ((45*I*b^2*x^2 - 45*I*a^2)*
cosh(b*x + a)^4 - 3*I*b^2*x^2 + (18*I*b^2*x^2 - 18*I*a^2)*cosh(b*x + a)^2 + 3*I*a^2)*sinh(b*x + a)^2 + 3*I*a^2
+ ((18*I*b^2*x^2 - 18*I*a^2)*cosh(b*x + a)^5 + (12*I*b^2*x^2 - 12*I*a^2)*cosh(b*x + a)^3 + (-6*I*b^2*x^2 + 6*
I*a^2)*cosh(b*x + a))*sinh(b*x + a))*log(I*cosh(b*x + a) + I*sinh(b*x + a) + 1) - ((-3*I*b^2*x^2 + 3*I*a^2)*co
sh(b*x + a)^6 + (-18*I*b^2*x^2 + 18*I*a^2)*cosh(b*x + a)*sinh(b*x + a)^5 + (-3*I*b^2*x^2 + 3*I*a^2)*sinh(b*x +
a)^6 + (-3*I*b^2*x^2 + 3*I*a^2)*cosh(b*x + a)^4 + (-3*I*b^2*x^2 + (-45*I*b^2*x^2 + 45*I*a^2)*cosh(b*x + a)^2
+ 3*I*a^2)*sinh(b*x + a)^4 + 3*I*b^2*x^2 + ((-60*I*b^2*x^2 + 60*I*a^2)*cosh(b*x + a)^3 + (-12*I*b^2*x^2 + 12*I
*a^2)*cosh(b*x + a))*sinh(b*x + a)^3 + (3*I*b^2*x^2 - 3*I*a^2)*cosh(b*x + a)^2 + ((-45*I*b^2*x^2 + 45*I*a^2)*c
osh(b*x + a)^4 + 3*I*b^2*x^2 + (-18*I*b^2*x^2 + 18*I*a^2)*cosh(b*x + a)^2 - 3*I*a^2)*sinh(b*x + a)^2 - 3*I*a^2
+ ((-18*I*b^2*x^2 + 18*I*a^2)*cosh(b*x + a)^5 + (-12*I*b^2*x^2 + 12*I*a^2)*cosh(b*x + a)^3 + (6*I*b^2*x^2 - 6
*I*a^2)*cosh(b*x + a))*sinh(b*x + a))*log(-I*cosh(b*x + a) - I*sinh(b*x + a) + 1) - 4*((b*x + a)*cosh(b*x + a)
^6 + 6*(b*x + a)*cosh(b*x + a)*sinh(b*x + a)^5 + (b*x + a)*sinh(b*x + a)^6 + (b*x + a)*cosh(b*x + a)^4 + (15*(
b*x + a)*cosh(b*x + a)^2 + b*x + a)*sinh(b*x + a)^4 + 4*(5*(b*x + a)*cosh(b*x + a)^3 + (b*x + a)*cosh(b*x + a)
)*sinh(b*x + a)^3 - (b*x + a)*cosh(b*x + a)^2 + (15*(b*x + a)*cosh(b*x + a)^4 + 6*(b*x + a)*cosh(b*x + a)^2 -
b*x - a)*sinh(b*x + a)^2 - b*x + 2*(3*(b*x + a)*cosh(b*x + a)^5 + 2*(b*x + a)*cosh(b*x + a)^3 - (b*x + a)*cosh
(b*x + a))*sinh(b*x + a) - a)*log(-cosh(b*x + a) - sinh(b*x + a) + 1) - (6*I*cosh(b*x + a)^6 + 36*I*cosh(b*x +
a)*sinh(b*x + a)^5 + 6*I*sinh(b*x + a)^6 + (90*I*cosh(b*x + a)^2 + 6*I)*sinh(b*x + a)^4 + 6*I*cosh(b*x + a)^4
+ (120*I*cosh(b*x + a)^3 + 24*I*cosh(b*x + a))*sinh(b*x + a)^3 + (90*I*cosh(b*x + a)^4 + 36*I*cosh(b*x + a)^2
- 6*I)*sinh(b*x + a)^2 - 6*I*cosh(b*x + a)^2 + (36*I*cosh(b*x + a)^5 + 24*I*cosh(b*x + a)^3 - 12*I*cosh(b*x +
a))*sinh(b*x + a) - 6*I)*polylog(3, I*cosh(b*x + a) + I*sinh(b*x + a)) - (-6*I*cosh(b*x + a)^6 - 36*I*cosh(b*
x + a)*sinh(b*x + a)^5 - 6*I*sinh(b*x + a)^6 + (-90*I*cosh(b*x + a)^2 - 6*I)*sinh(b*x + a)^4 - 6*I*cosh(b*x +
a)^4 + (-120*I*cosh(b*x + a)^3 - 24*I*cosh(b*x + a))*sinh(b*x + a)^3 + (-90*I*cosh(b*x + a)^4 - 36*I*cosh(b*x
+ a)^2 + 6*I)*sinh(b*x + a)^2 + 6*I*cosh(b*x + a)^2 + (-36*I*cosh(b*x + a)^5 - 24*I*cosh(b*x + a)^3 + 12*I*cos
h(b*x + a))*sinh(b*x + a) + 6*I)*polylog(3, -I*cosh(b*x + a) - I*sinh(b*x + a)) + 2*(6*b^2*x^2*cosh(b*x + a)^2
+ 5*(3*b^2*x^2 + 2*b*x)*cosh(b*x + a)^4 + 3*b^2*x^2 - 2*b*x)*sinh(b*x + a))/(b^3*cosh(b*x + a)^6 + 6*b^3*cosh
(b*x + a)*sinh(b*x + a)^5 + b^3*sinh(b*x + a)^6 + b^3*cosh(b*x + a)^4 - b^3*cosh(b*x + a)^2 + (15*b^3*cosh(b*x
+ a)^2 + b^3)*sinh(b*x + a)^4 + 4*(5*b^3*cosh(b*x + a)^3 + b^3*cosh(b*x + a))*sinh(b*x + a)^3 - b^3 + (15*b^3
*cosh(b*x + a)^4 + 6*b^3*cosh(b*x + a)^2 - b^3)*sinh(b*x + a)^2 + 2*(3*b^3*cosh(b*x + a)^5 + 2*b^3*cosh(b*x +
a)^3 - b^3*cosh(b*x + a))*sinh(b*x + a))
________________________________________________________________________________________
Sympy [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int x^{2} \operatorname{csch}^{2}{\left (a + b x \right )} \operatorname{sech}^{3}{\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)**3,x)
[Out]
Integral(x**2*csch(a + b*x)**2*sech(a + b*x)**3, x)
________________________________________________________________________________________
Giac [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int x^{2} \operatorname{csch}\left (b x + a\right )^{2} \operatorname{sech}\left (b x + a\right )^{3}\,{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)^3,x, algorithm="giac")
[Out]
integrate(x^2*csch(b*x + a)^2*sech(b*x + a)^3, x)