3.6.16 \(\int x^2 \text {csch}^3(a+b x) \text {sech}^2(a+b x) \, dx\) [516]

Optimal. Leaf size=197 \[ \frac {4 x \text {ArcTan}\left (e^{a+b x}\right )}{b^2}+\frac {3 x^2 \tanh ^{-1}\left (e^{a+b x}\right )}{b}-\frac {\tanh ^{-1}(\cosh (a+b x))}{b^3}-\frac {x \text {csch}(a+b x)}{b^2}+\frac {3 x \text {PolyLog}\left (2,-e^{a+b x}\right )}{b^2}-\frac {2 i \text {PolyLog}\left (2,-i e^{a+b x}\right )}{b^3}+\frac {2 i \text {PolyLog}\left (2,i e^{a+b x}\right )}{b^3}-\frac {3 x \text {PolyLog}\left (2,e^{a+b x}\right )}{b^2}-\frac {3 \text {PolyLog}\left (3,-e^{a+b x}\right )}{b^3}+\frac {3 \text {PolyLog}\left (3,e^{a+b x}\right )}{b^3}-\frac {3 x^2 \text {sech}(a+b x)}{2 b}-\frac {x^2 \text {csch}^2(a+b x) \text {sech}(a+b x)}{2 b} \]

[Out]

4*x*arctan(exp(b*x+a))/b^2+3*x^2*arctanh(exp(b*x+a))/b-arctanh(cosh(b*x+a))/b^3-x*csch(b*x+a)/b^2+3*x*polylog(
2,-exp(b*x+a))/b^2-2*I*polylog(2,-I*exp(b*x+a))/b^3+2*I*polylog(2,I*exp(b*x+a))/b^3-3*x*polylog(2,exp(b*x+a))/
b^2-3*polylog(3,-exp(b*x+a))/b^3+3*polylog(3,exp(b*x+a))/b^3-3/2*x^2*sech(b*x+a)/b-1/2*x^2*csch(b*x+a)^2*sech(
b*x+a)/b

________________________________________________________________________________________

Rubi [A]
time = 0.37, antiderivative size = 197, normalized size of antiderivative = 1.00, number of steps used = 29, number of rules used = 19, integrand size = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.950, Rules used = {2702, 294, 327, 213, 5570, 14, 6408, 12, 4267, 2611, 2320, 6724, 6874, 4265, 2317, 2438, 2701, 5311, 3855} \begin {gather*} \frac {4 x \text {ArcTan}\left (e^{a+b x}\right )}{b^2}-\frac {2 i \text {Li}_2\left (-i e^{a+b x}\right )}{b^3}+\frac {2 i \text {Li}_2\left (i e^{a+b x}\right )}{b^3}-\frac {3 \text {Li}_3\left (-e^{a+b x}\right )}{b^3}+\frac {3 \text {Li}_3\left (e^{a+b x}\right )}{b^3}-\frac {\tanh ^{-1}(\cosh (a+b x))}{b^3}+\frac {3 x \text {Li}_2\left (-e^{a+b x}\right )}{b^2}-\frac {3 x \text {Li}_2\left (e^{a+b x}\right )}{b^2}-\frac {x \text {csch}(a+b x)}{b^2}+\frac {3 x^2 \tanh ^{-1}\left (e^{a+b x}\right )}{b}-\frac {3 x^2 \text {sech}(a+b x)}{2 b}-\frac {x^2 \text {csch}^2(a+b x) \text {sech}(a+b x)}{2 b} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[x^2*Csch[a + b*x]^3*Sech[a + b*x]^2,x]

[Out]

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

Rule 12

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

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 213

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

Rule 294

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 327

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 2317

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 2320

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 2438

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 2611

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 2701

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 2702

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 3855

Int[csc[(c_.) + (d_.)*(x_)], x_Symbol] :> Simp[-ArcTanh[Cos[c + d*x]]/d, x] /; FreeQ[{c, d}, x]

Rule 4265

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 4267

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 5311

Int[ArcTan[u_], x_Symbol] :> Simp[x*ArcTan[u], x] - Int[SimplifyIntegrand[x*(D[u, x]/(1 + u^2)), x], x] /; Inv
erseFunctionFreeQ[u, x]

Rule 5570

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 6408

Int[((a_.) + ArcTanh[u_]*(b_.))*((c_.) + (d_.)*(x_))^(m_.), x_Symbol] :> Simp[(c + d*x)^(m + 1)*((a + b*ArcTan
h[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 6724

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 6874

Int[u_, x_Symbol] :> With[{v = ExpandIntegrand[u, x]}, Int[v, x] /; SumQ[v]]

Rubi steps

\begin {align*} \int x^2 \text {csch}^3(a+b x) \text {sech}^2(a+b x) \, dx &=\frac {3 x^2 \tanh ^{-1}(\cosh (a+b x))}{2 b}-\frac {3 x^2 \text {sech}(a+b x)}{2 b}-\frac {x^2 \text {csch}^2(a+b x) \text {sech}(a+b x)}{2 b}-2 \int x \left (\frac {3 \tanh ^{-1}(\cosh (a+b x))}{2 b}-\frac {3 \text {sech}(a+b x)}{2 b}-\frac {\text {csch}^2(a+b x) \text {sech}(a+b x)}{2 b}\right ) \, dx\\ &=\frac {3 x^2 \tanh ^{-1}(\cosh (a+b x))}{2 b}-\frac {3 x^2 \text {sech}(a+b x)}{2 b}-\frac {x^2 \text {csch}^2(a+b x) \text {sech}(a+b x)}{2 b}-2 \int \left (\frac {3 x \tanh ^{-1}(\cosh (a+b x))}{2 b}-\frac {x \left (3+\text {csch}^2(a+b x)\right ) \text {sech}(a+b x)}{2 b}\right ) \, dx\\ &=\frac {3 x^2 \tanh ^{-1}(\cosh (a+b x))}{2 b}-\frac {3 x^2 \text {sech}(a+b x)}{2 b}-\frac {x^2 \text {csch}^2(a+b x) \text {sech}(a+b x)}{2 b}+\frac {\int x \left (3+\text {csch}^2(a+b x)\right ) \text {sech}(a+b x) \, dx}{b}-\frac {3 \int x \tanh ^{-1}(\cosh (a+b x)) \, dx}{b}\\ &=-\frac {3 x^2 \text {sech}(a+b x)}{2 b}-\frac {x^2 \text {csch}^2(a+b x) \text {sech}(a+b x)}{2 b}+\frac {\int \left (3 x \text {sech}(a+b x)+x \text {csch}^2(a+b x) \text {sech}(a+b x)\right ) \, dx}{b}-\frac {3 \int b x^2 \text {csch}(a+b x) \, dx}{2 b}\\ &=-\frac {3 x^2 \text {sech}(a+b x)}{2 b}-\frac {x^2 \text {csch}^2(a+b x) \text {sech}(a+b x)}{2 b}-\frac {3}{2} \int x^2 \text {csch}(a+b x) \, dx+\frac {\int x \text {csch}^2(a+b x) \text {sech}(a+b x) \, dx}{b}+\frac {3 \int x \text {sech}(a+b x) \, dx}{b}\\ &=\frac {6 x \tan ^{-1}\left (e^{a+b x}\right )}{b^2}-\frac {x \tan ^{-1}(\sinh (a+b x))}{b^2}+\frac {3 x^2 \tanh ^{-1}\left (e^{a+b x}\right )}{b}-\frac {x \text {csch}(a+b x)}{b^2}-\frac {3 x^2 \text {sech}(a+b x)}{2 b}-\frac {x^2 \text {csch}^2(a+b x) \text {sech}(a+b x)}{2 b}-\frac {(3 i) \int \log \left (1-i e^{a+b x}\right ) \, dx}{b^2}+\frac {(3 i) \int \log \left (1+i e^{a+b x}\right ) \, dx}{b^2}-\frac {\int \left (-\frac {\tan ^{-1}(\sinh (a+b x))}{b}-\frac {\text {csch}(a+b x)}{b}\right ) \, dx}{b}+\frac {3 \int x \log \left (1-e^{a+b x}\right ) \, dx}{b}-\frac {3 \int x \log \left (1+e^{a+b x}\right ) \, dx}{b}\\ &=\frac {6 x \tan ^{-1}\left (e^{a+b x}\right )}{b^2}-\frac {x \tan ^{-1}(\sinh (a+b x))}{b^2}+\frac {3 x^2 \tanh ^{-1}\left (e^{a+b x}\right )}{b}-\frac {x \text {csch}(a+b x)}{b^2}+\frac {3 x \text {Li}_2\left (-e^{a+b x}\right )}{b^2}-\frac {3 x \text {Li}_2\left (e^{a+b x}\right )}{b^2}-\frac {3 x^2 \text {sech}(a+b x)}{2 b}-\frac {x^2 \text {csch}^2(a+b x) \text {sech}(a+b x)}{2 b}-\frac {(3 i) \text {Subst}\left (\int \frac {\log (1-i x)}{x} \, dx,x,e^{a+b x}\right )}{b^3}+\frac {(3 i) \text {Subst}\left (\int \frac {\log (1+i x)}{x} \, dx,x,e^{a+b x}\right )}{b^3}+\frac {\int \tan ^{-1}(\sinh (a+b x)) \, dx}{b^2}+\frac {\int \text {csch}(a+b x) \, dx}{b^2}-\frac {3 \int \text {Li}_2\left (-e^{a+b x}\right ) \, dx}{b^2}+\frac {3 \int \text {Li}_2\left (e^{a+b x}\right ) \, dx}{b^2}\\ &=\frac {6 x \tan ^{-1}\left (e^{a+b x}\right )}{b^2}+\frac {3 x^2 \tanh ^{-1}\left (e^{a+b x}\right )}{b}-\frac {\tanh ^{-1}(\cosh (a+b x))}{b^3}-\frac {x \text {csch}(a+b x)}{b^2}+\frac {3 x \text {Li}_2\left (-e^{a+b x}\right )}{b^2}-\frac {3 i \text {Li}_2\left (-i e^{a+b x}\right )}{b^3}+\frac {3 i \text {Li}_2\left (i e^{a+b x}\right )}{b^3}-\frac {3 x \text {Li}_2\left (e^{a+b x}\right )}{b^2}-\frac {3 x^2 \text {sech}(a+b x)}{2 b}-\frac {x^2 \text {csch}^2(a+b x) \text {sech}(a+b x)}{2 b}-\frac {3 \text {Subst}\left (\int \frac {\text {Li}_2(-x)}{x} \, dx,x,e^{a+b x}\right )}{b^3}+\frac {3 \text {Subst}\left (\int \frac {\text {Li}_2(x)}{x} \, dx,x,e^{a+b x}\right )}{b^3}-\frac {\int b x \text {sech}(a+b x) \, dx}{b^2}\\ &=\frac {6 x \tan ^{-1}\left (e^{a+b x}\right )}{b^2}+\frac {3 x^2 \tanh ^{-1}\left (e^{a+b x}\right )}{b}-\frac {\tanh ^{-1}(\cosh (a+b x))}{b^3}-\frac {x \text {csch}(a+b x)}{b^2}+\frac {3 x \text {Li}_2\left (-e^{a+b x}\right )}{b^2}-\frac {3 i \text {Li}_2\left (-i e^{a+b x}\right )}{b^3}+\frac {3 i \text {Li}_2\left (i e^{a+b x}\right )}{b^3}-\frac {3 x \text {Li}_2\left (e^{a+b x}\right )}{b^2}-\frac {3 \text {Li}_3\left (-e^{a+b x}\right )}{b^3}+\frac {3 \text {Li}_3\left (e^{a+b x}\right )}{b^3}-\frac {3 x^2 \text {sech}(a+b x)}{2 b}-\frac {x^2 \text {csch}^2(a+b x) \text {sech}(a+b x)}{2 b}-\frac {\int x \text {sech}(a+b x) \, dx}{b}\\ &=\frac {4 x \tan ^{-1}\left (e^{a+b x}\right )}{b^2}+\frac {3 x^2 \tanh ^{-1}\left (e^{a+b x}\right )}{b}-\frac {\tanh ^{-1}(\cosh (a+b x))}{b^3}-\frac {x \text {csch}(a+b x)}{b^2}+\frac {3 x \text {Li}_2\left (-e^{a+b x}\right )}{b^2}-\frac {3 i \text {Li}_2\left (-i e^{a+b x}\right )}{b^3}+\frac {3 i \text {Li}_2\left (i e^{a+b x}\right )}{b^3}-\frac {3 x \text {Li}_2\left (e^{a+b x}\right )}{b^2}-\frac {3 \text {Li}_3\left (-e^{a+b x}\right )}{b^3}+\frac {3 \text {Li}_3\left (e^{a+b x}\right )}{b^3}-\frac {3 x^2 \text {sech}(a+b x)}{2 b}-\frac {x^2 \text {csch}^2(a+b x) \text {sech}(a+b x)}{2 b}+\frac {i \int \log \left (1-i e^{a+b x}\right ) \, dx}{b^2}-\frac {i \int \log \left (1+i e^{a+b x}\right ) \, dx}{b^2}\\ &=\frac {4 x \tan ^{-1}\left (e^{a+b x}\right )}{b^2}+\frac {3 x^2 \tanh ^{-1}\left (e^{a+b x}\right )}{b}-\frac {\tanh ^{-1}(\cosh (a+b x))}{b^3}-\frac {x \text {csch}(a+b x)}{b^2}+\frac {3 x \text {Li}_2\left (-e^{a+b x}\right )}{b^2}-\frac {3 i \text {Li}_2\left (-i e^{a+b x}\right )}{b^3}+\frac {3 i \text {Li}_2\left (i e^{a+b x}\right )}{b^3}-\frac {3 x \text {Li}_2\left (e^{a+b x}\right )}{b^2}-\frac {3 \text {Li}_3\left (-e^{a+b x}\right )}{b^3}+\frac {3 \text {Li}_3\left (e^{a+b x}\right )}{b^3}-\frac {3 x^2 \text {sech}(a+b x)}{2 b}-\frac {x^2 \text {csch}^2(a+b x) \text {sech}(a+b x)}{2 b}+\frac {i \text {Subst}\left (\int \frac {\log (1-i x)}{x} \, dx,x,e^{a+b x}\right )}{b^3}-\frac {i \text {Subst}\left (\int \frac {\log (1+i x)}{x} \, dx,x,e^{a+b x}\right )}{b^3}\\ &=\frac {4 x \tan ^{-1}\left (e^{a+b x}\right )}{b^2}+\frac {3 x^2 \tanh ^{-1}\left (e^{a+b x}\right )}{b}-\frac {\tanh ^{-1}(\cosh (a+b x))}{b^3}-\frac {x \text {csch}(a+b x)}{b^2}+\frac {3 x \text {Li}_2\left (-e^{a+b x}\right )}{b^2}-\frac {2 i \text {Li}_2\left (-i e^{a+b x}\right )}{b^3}+\frac {2 i \text {Li}_2\left (i e^{a+b x}\right )}{b^3}-\frac {3 x \text {Li}_2\left (e^{a+b x}\right )}{b^2}-\frac {3 \text {Li}_3\left (-e^{a+b x}\right )}{b^3}+\frac {3 \text {Li}_3\left (e^{a+b x}\right )}{b^3}-\frac {3 x^2 \text {sech}(a+b x)}{2 b}-\frac {x^2 \text {csch}^2(a+b x) \text {sech}(a+b x)}{2 b}\\ \end {align*}

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Mathematica [B] Both result and optimal contain complex but leaf count is larger than twice the leaf count of optimal. \(425\) vs. \(2(197)=394\).
time = 7.54, size = 425, normalized size = 2.16 \begin {gather*} -\frac {x \text {csch}(a)}{b^2}-\frac {x^2 \text {csch}^2\left (\frac {a}{2}+\frac {b x}{2}\right )}{8 b}+\frac {2 \left (-\left (\left (-i a+\frac {\pi }{2}-i b x\right ) \left (\log \left (1-e^{i \left (-i a+\frac {\pi }{2}-i b x\right )}\right )-\log \left (1+e^{i \left (-i a+\frac {\pi }{2}-i b x\right )}\right )\right )\right )+\left (-i a+\frac {\pi }{2}\right ) \log \left (\tan \left (\frac {1}{2} \left (-i a+\frac {\pi }{2}-i b x\right )\right )\right )-i \left (\text {PolyLog}\left (2,-e^{i \left (-i a+\frac {\pi }{2}-i b x\right )}\right )-\text {PolyLog}\left (2,e^{i \left (-i a+\frac {\pi }{2}-i b x\right )}\right )\right )\right )}{b^3}-\frac {4 \tanh ^{-1}\left (e^{a+b x}\right )+3 b^2 x^2 \log \left (1-e^{a+b x}\right )-3 b^2 x^2 \log \left (1+e^{a+b x}\right )-6 b x \text {PolyLog}\left (2,-e^{a+b x}\right )+6 b x \text {PolyLog}\left (2,e^{a+b x}\right )+6 \text {PolyLog}\left (3,-e^{a+b x}\right )-6 \text {PolyLog}\left (3,e^{a+b x}\right )}{2 b^3}-\frac {x^2 \text {sech}^2\left (\frac {a}{2}+\frac {b x}{2}\right )}{8 b}-\frac {x^2 \text {sech}(a+b x)}{b}+\frac {x \text {csch}\left (\frac {a}{2}\right ) \text {csch}\left (\frac {a}{2}+\frac {b x}{2}\right ) \sinh \left (\frac {b x}{2}\right )}{2 b^2}+\frac {x \text {sech}\left (\frac {a}{2}\right ) \text {sech}\left (\frac {a}{2}+\frac {b x}{2}\right ) \sinh \left (\frac {b x}{2}\right )}{2 b^2} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[x^2*Csch[a + b*x]^3*Sech[a + b*x]^2,x]

[Out]

-((x*Csch[a])/b^2) - (x^2*Csch[a/2 + (b*x)/2]^2)/(8*b) + (2*(-(((-I)*a + Pi/2 - I*b*x)*(Log[1 - E^(I*((-I)*a +
Pi/2 - I*b*x))] - Log[1 + E^(I*((-I)*a + Pi/2 - I*b*x))])) + ((-I)*a + Pi/2)*Log[Tan[((-I)*a + Pi/2 - I*b*x)/
2]] - I*(PolyLog[2, -E^(I*((-I)*a + Pi/2 - I*b*x))] - PolyLog[2, E^(I*((-I)*a + Pi/2 - I*b*x))])))/b^3 - (4*Ar
cTanh[E^(a + b*x)] + 3*b^2*x^2*Log[1 - E^(a + b*x)] - 3*b^2*x^2*Log[1 + E^(a + b*x)] - 6*b*x*PolyLog[2, -E^(a
+ b*x)] + 6*b*x*PolyLog[2, E^(a + b*x)] + 6*PolyLog[3, -E^(a + b*x)] - 6*PolyLog[3, E^(a + b*x)])/(2*b^3) - (x
^2*Sech[a/2 + (b*x)/2]^2)/(8*b) - (x^2*Sech[a + b*x])/b + (x*Csch[a/2]*Csch[a/2 + (b*x)/2]*Sinh[(b*x)/2])/(2*b
^2) + (x*Sech[a/2]*Sech[a/2 + (b*x)/2]*Sinh[(b*x)/2])/(2*b^2)

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Maple [F]
time = 1.58, size = 0, normalized size = 0.00 \[\int x^{2} \mathrm {csch}\left (b x +a \right )^{3} \mathrm {sech}\left (b x +a \right )^{2}\, dx\]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^2*csch(b*x+a)^3*sech(b*x+a)^2,x)

[Out]

int(x^2*csch(b*x+a)^3*sech(b*x+a)^2,x)

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Maxima [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^2*csch(b*x+a)^3*sech(b*x+a)^2,x, algorithm="maxima")

[Out]

(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) + 3/2*(b^2*x^2*log(e^(b*x + a) + 1) + 2*b*x*
dilog(-e^(b*x + a)) - 2*polylog(3, -e^(b*x + a)))/b^3 - 3/2*(b^2*x^2*log(-e^(b*x + a) + 1) + 2*b*x*dilog(e^(b*
x + a)) - 2*polylog(3, e^(b*x + a)))/b^3 - log(e^(b*x + a) + 1)/b^3 + log(e^(b*x + a) - 1)/b^3 + 32*integrate(
1/8*x*e^(b*x + a)/(b*e^(2*b*x + 2*a) + b), x)

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Fricas [B] Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 3804 vs. \(2 (173) = 346\).
time = 0.44, size = 3804, normalized size = 19.31 \begin {gather*} \text {Too large to display} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^2*csch(b*x+a)^3*sech(b*x+a)^2,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) - 6*(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*c
osh(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)*s
inh(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))*di
log(cosh(b*x + a) + sinh(b*x + a)) - 4*(-I*cosh(b*x + a)^6 - 6*I*cosh(b*x + a)*sinh(b*x + a)^5 - I*sinh(b*x +
a)^6 + (-15*I*cosh(b*x + a)^2 + I)*sinh(b*x + a)^4 + I*cosh(b*x + a)^4 + 4*(-5*I*cosh(b*x + a)^3 + I*cosh(b*x
+ a))*sinh(b*x + a)^3 + (-15*I*cosh(b*x + a)^4 + 6*I*cosh(b*x + a)^2 + I)*sinh(b*x + a)^2 + I*cosh(b*x + a)^2
+ 2*(-3*I*cosh(b*x + a)^5 + 2*I*cosh(b*x + a)^3 + I*cosh(b*x + a))*sinh(b*x + a) - I)*dilog(I*cosh(b*x + a) +
I*sinh(b*x + a)) - 4*(I*cosh(b*x + a)^6 + 6*I*cosh(b*x + a)*sinh(b*x + a)^5 + I*sinh(b*x + a)^6 + (15*I*cosh(b
*x + a)^2 - I)*sinh(b*x + a)^4 - I*cosh(b*x + a)^4 + 4*(5*I*cosh(b*x + a)^3 - I*cosh(b*x + a))*sinh(b*x + a)^3
+ (15*I*cosh(b*x + a)^4 - 6*I*cosh(b*x + a)^2 - I)*sinh(b*x + a)^2 - I*cosh(b*x + a)^2 + 2*(3*I*cosh(b*x + a)
^5 - 2*I*cosh(b*x + a)^3 - I*cosh(b*x + a))*sinh(b*x + a) + I)*dilog(-I*cosh(b*x + a) - I*sinh(b*x + a)) + 6*(
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*cos
h(b*x + a)^5 - 2*b*x*cosh(b*x + a)^3 - b*x*cosh(b*x + a))*sinh(b*x + a))*dilog(-cosh(b*x + a) - sinh(b*x + a))
+ ((3*b^2*x^2 - 2)*cosh(b*x + a)^6 + 6*(3*b^2*x^2 - 2)*cosh(b*x + a)*sinh(b*x + a)^5 + (3*b^2*x^2 - 2)*sinh(b
*x + a)^6 - (3*b^2*x^2 - 2)*cosh(b*x + a)^4 - (3*b^2*x^2 - 15*(3*b^2*x^2 - 2)*cosh(b*x + a)^2 - 2)*sinh(b*x +
a)^4 + 3*b^2*x^2 + 4*(5*(3*b^2*x^2 - 2)*cosh(b*x + a)^3 - (3*b^2*x^2 - 2)*cosh(b*x + a))*sinh(b*x + a)^3 - (3*
b^2*x^2 - 2)*cosh(b*x + a)^2 + (15*(3*b^2*x^2 - 2)*cosh(b*x + a)^4 - 3*b^2*x^2 - 6*(3*b^2*x^2 - 2)*cosh(b*x +
a)^2 + 2)*sinh(b*x + a)^2 + 2*(3*(3*b^2*x^2 - 2)*cosh(b*x + a)^5 - 2*(3*b^2*x^2 - 2)*cosh(b*x + a)^3 - (3*b^2*
x^2 - 2)*cosh(b*x + a))*sinh(b*x + a) - 2)*log(cosh(b*x + a) + sinh(b*x + a) + 1) - 4*(I*a*cosh(b*x + a)^6 + 6
*I*a*cosh(b*x + a)*sinh(b*x + a)^5 + I*a*sinh(b*x + a)^6 - I*a*cosh(b*x + a)^4 + (15*I*a*cosh(b*x + a)^2 - I*a
)*sinh(b*x + a)^4 + 4*(5*I*a*cosh(b*x + a)^3 - I*a*cosh(b*x + a))*sinh(b*x + a)^3 - I*a*cosh(b*x + a)^2 + (15*
I*a*cosh(b*x + a)^4 - 6*I*a*cosh(b*x + a)^2 - I*a)*sinh(b*x + a)^2 + 2*(3*I*a*cosh(b*x + a)^5 - 2*I*a*cosh(b*x
+ a)^3 - I*a*cosh(b*x + a))*sinh(b*x + a) + I*a)*log(cosh(b*x + a) + sinh(b*x + a) + I) - 4*(-I*a*cosh(b*x +
a)^6 - 6*I*a*cosh(b*x + a)*sinh(b*x + a)^5 - I*a*sinh(b*x + a)^6 + I*a*cosh(b*x + a)^4 + (-15*I*a*cosh(b*x + a
)^2 + I*a)*sinh(b*x + a)^4 + 4*(-5*I*a*cosh(b*x + a)^3 + I*a*cosh(b*x + a))*sinh(b*x + a)^3 + I*a*cosh(b*x + a
)^2 + (-15*I*a*cosh(b*x + a)^4 + 6*I*a*cosh(b*x + a)^2 + I*a)*sinh(b*x + a)^2 + 2*(-3*I*a*cosh(b*x + a)^5 + 2*
I*a*cosh(b*x + a)^3 + I*a*cosh(b*x + a))*sinh(b*x + a) - I*a)*log(cosh(b*x + a) + sinh(b*x + a) - I) - ((3*a^2
- 2)*cosh(b*x + a)^6 + 6*(3*a^2 - 2)*cosh(b*x + a)*sinh(b*x + a)^5 + (3*a^2 - 2)*sinh(b*x + a)^6 - (3*a^2 - 2
)*cosh(b*x + a)^4 + (15*(3*a^2 - 2)*cosh(b*x + a)^2 - 3*a^2 + 2)*sinh(b*x + a)^4 + 4*(5*(3*a^2 - 2)*cosh(b*x +
a)^3 - (3*a^2 - 2)*cosh(b*x + a))*sinh(b*x + a)^3 - (3*a^2 - 2)*cosh(b*x + a)^2 + (15*(3*a^2 - 2)*cosh(b*x +
a)^4 - 6*(3*a^2 - 2)*cosh(b*x + a)^2 - 3*a^2 + 2)*sinh(b*x + a)^2 + 3*a^2 + 2*(3*(3*a^2 - 2)*cosh(b*x + a)^5 -
2*(3*a^2 - 2)*cosh(b*x + a)^3 - (3*a^2 - 2)*cosh(b*x + a))*sinh(b*x + a) - 2)*log(cosh(b*x + a) + sinh(b*x +
a) - 1) - 4*((I*b*x + I*a)*cosh(b*x + a)^6 + 6*(I*b*x + I*a)*cosh(b*x + a)*sinh(b*x + a)^5 + (I*b*x + I*a)*sin
h(b*x + a)^6 + (-I*b*x - I*a)*cosh(b*x + a)^4 + (15*(I*b*x + I*a)*cosh(b*x + a)^2 - I*b*x - I*a)*sinh(b*x + a)
^4 + 4*(5*(I*b*x + I*a)*cosh(b*x + a)^3 + (-I*b*x - I*a)*cosh(b*x + a))*sinh(b*x + a)^3 + (-I*b*x - I*a)*cosh(
b*x + a)^2 + (15*(I*b*x + I*a)*cosh(b*x + a)^4 + 6*(-I*b*x - I*a)*cosh(b*x + a)^2 - I*b*x - I*a)*sinh(b*x + a)
^2 + I*b*x + 2*(3*(I*b*x + I*a)*cosh(b*x + a)^5 + 2*(-I*b*x - I*a)*cosh(b*x + a)^3 + (-I*b*x - I*a)*cosh(b*x +
a))*sinh(b*x + a) + I*a)*log(I*cosh(b*x + a) + I*sinh(b*x + a) + 1) - 4*((-I*b*x - I*a)*cosh(b*x + a)^6 + 6*(
-I*b*x - I*a)*cosh(b*x + a)*sinh(b*x + a)^5 + (-I*b*x - I*a)*sinh(b*x + a)^6 + (I*b*x + I*a)*cosh(b*x + a)^4 +
(15*(-I*b*x - I*a)*cosh(b*x + a)^2 + I*b*x + I...

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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int x^{2} \operatorname {csch}^{3}{\left (a + b x \right )} \operatorname {sech}^{2}{\left (a + b x \right )}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x**2*csch(b*x+a)**3*sech(b*x+a)**2,x)

[Out]

Integral(x**2*csch(a + b*x)**3*sech(a + b*x)**2, x)

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Giac [F(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: AttributeError} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^2*csch(b*x+a)^3*sech(b*x+a)^2,x, algorithm="giac")

[Out]

Exception raised: AttributeError >> type

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Mupad [F]
time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \int \frac {x^2}{{\mathrm {cosh}\left (a+b\,x\right )}^2\,{\mathrm {sinh}\left (a+b\,x\right )}^3} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^2/(cosh(a + b*x)^2*sinh(a + b*x)^3),x)

[Out]

int(x^2/(cosh(a + b*x)^2*sinh(a + b*x)^3), x)

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