3.481 \(\int x^3 \text{csch}(a+b x) \text{sech}^3(a+b x) \, dx\)
Optimal. Leaf size=240 \[ -\frac{3 x^2 \text{PolyLog}\left (2,-e^{2 a+2 b x}\right )}{2 b^2}+\frac{3 x^2 \text{PolyLog}\left (2,e^{2 a+2 b x}\right )}{2 b^2}+\frac{3 x \text{PolyLog}\left (3,-e^{2 a+2 b x}\right )}{2 b^3}-\frac{3 x \text{PolyLog}\left (3,e^{2 a+2 b x}\right )}{2 b^3}+\frac{3 \text{PolyLog}\left (2,-e^{2 (a+b x)}\right )}{2 b^4}-\frac{3 \text{PolyLog}\left (4,-e^{2 a+2 b x}\right )}{4 b^4}+\frac{3 \text{PolyLog}\left (4,e^{2 a+2 b x}\right )}{4 b^4}-\frac{3 x^2 \tanh (a+b x)}{2 b^2}+\frac{3 x \log \left (e^{2 (a+b x)}+1\right )}{b^3}-\frac{x^3 \tanh ^2(a+b x)}{2 b}-\frac{2 x^3 \tanh ^{-1}\left (e^{2 a+2 b x}\right )}{b}-\frac{3 x^2}{2 b^2}+\frac{x^3}{2 b} \]
[Out]
(-3*x^2)/(2*b^2) + x^3/(2*b) - (2*x^3*ArcTanh[E^(2*a + 2*b*x)])/b + (3*x*Log[1 + E^(2*(a + b*x))])/b^3 + (3*Po
lyLog[2, -E^(2*(a + b*x))])/(2*b^4) - (3*x^2*PolyLog[2, -E^(2*a + 2*b*x)])/(2*b^2) + (3*x^2*PolyLog[2, E^(2*a
+ 2*b*x)])/(2*b^2) + (3*x*PolyLog[3, -E^(2*a + 2*b*x)])/(2*b^3) - (3*x*PolyLog[3, E^(2*a + 2*b*x)])/(2*b^3) -
(3*PolyLog[4, -E^(2*a + 2*b*x)])/(4*b^4) + (3*PolyLog[4, E^(2*a + 2*b*x)])/(4*b^4) - (3*x^2*Tanh[a + b*x])/(2*
b^2) - (x^3*Tanh[a + b*x]^2)/(2*b)
________________________________________________________________________________________
Rubi [A] time = 0.43124, antiderivative size = 240, normalized size of antiderivative = 1.,
number of steps used = 20, number of rules used = 16, integrand size = 18, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.889, Rules used
= {2620, 14, 5462, 2551, 12, 4182, 2531, 6609, 2282, 6589, 3720, 3718, 2190, 2279, 2391, 30}
\[ -\frac{3 x^2 \text{PolyLog}\left (2,-e^{2 a+2 b x}\right )}{2 b^2}+\frac{3 x^2 \text{PolyLog}\left (2,e^{2 a+2 b x}\right )}{2 b^2}+\frac{3 x \text{PolyLog}\left (3,-e^{2 a+2 b x}\right )}{2 b^3}-\frac{3 x \text{PolyLog}\left (3,e^{2 a+2 b x}\right )}{2 b^3}+\frac{3 \text{PolyLog}\left (2,-e^{2 (a+b x)}\right )}{2 b^4}-\frac{3 \text{PolyLog}\left (4,-e^{2 a+2 b x}\right )}{4 b^4}+\frac{3 \text{PolyLog}\left (4,e^{2 a+2 b x}\right )}{4 b^4}-\frac{3 x^2 \tanh (a+b x)}{2 b^2}+\frac{3 x \log \left (e^{2 (a+b x)}+1\right )}{b^3}-\frac{x^3 \tanh ^2(a+b x)}{2 b}-\frac{2 x^3 \tanh ^{-1}\left (e^{2 a+2 b x}\right )}{b}-\frac{3 x^2}{2 b^2}+\frac{x^3}{2 b} \]
Antiderivative was successfully verified.
[In]
Int[x^3*Csch[a + b*x]*Sech[a + b*x]^3,x]
[Out]
(-3*x^2)/(2*b^2) + x^3/(2*b) - (2*x^3*ArcTanh[E^(2*a + 2*b*x)])/b + (3*x*Log[1 + E^(2*(a + b*x))])/b^3 + (3*Po
lyLog[2, -E^(2*(a + b*x))])/(2*b^4) - (3*x^2*PolyLog[2, -E^(2*a + 2*b*x)])/(2*b^2) + (3*x^2*PolyLog[2, E^(2*a
+ 2*b*x)])/(2*b^2) + (3*x*PolyLog[3, -E^(2*a + 2*b*x)])/(2*b^3) - (3*x*PolyLog[3, E^(2*a + 2*b*x)])/(2*b^3) -
(3*PolyLog[4, -E^(2*a + 2*b*x)])/(4*b^4) + (3*PolyLog[4, E^(2*a + 2*b*x)])/(4*b^4) - (3*x^2*Tanh[a + b*x])/(2*
b^2) - (x^3*Tanh[a + b*x]^2)/(2*b)
Rule 2620
Int[csc[(e_.) + (f_.)*(x_)]^(m_.)*sec[(e_.) + (f_.)*(x_)]^(n_.), x_Symbol] :> Dist[1/f, Subst[Int[(1 + x^2)^((
m + n)/2 - 1)/x^m, x], x, Tan[e + f*x]], x] /; FreeQ[{e, f}, x] && IntegersQ[m, n, (m + n)/2]
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 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 2551
Int[Log[u_]*((a_.) + (b_.)*(x_))^(m_.), x_Symbol] :> Simp[((a + b*x)^(m + 1)*Log[u])/(b*(m + 1)), x] - Dist[1/
(b*(m + 1)), Int[SimplifyIntegrand[((a + b*x)^(m + 1)*D[u, x])/u, x], x], x] /; FreeQ[{a, b, m}, x] && Inverse
FunctionFreeQ[u, x] && NeQ[m, -1]
Rule 12
Int[(a_)*(u_), x_Symbol] :> Dist[a, Int[u, x], x] /; FreeQ[a, x] && !MatchQ[u, (b_)*(v_) /; FreeQ[b, x]]
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 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 6609
Int[((e_.) + (f_.)*(x_))^(m_.)*PolyLog[n_, (d_.)*((F_)^((c_.)*((a_.) + (b_.)*(x_))))^(p_.)], x_Symbol] :> Simp
[((e + f*x)^m*PolyLog[n + 1, d*(F^(c*(a + b*x)))^p])/(b*c*p*Log[F]), x] - Dist[(f*m)/(b*c*p*Log[F]), Int[(e +
f*x)^(m - 1)*PolyLog[n + 1, d*(F^(c*(a + b*x)))^p], x], x] /; FreeQ[{F, a, b, c, d, e, f, n, p}, 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 3720
Int[((c_.) + (d_.)*(x_))^(m_.)*((b_.)*tan[(e_.) + (f_.)*(x_)])^(n_), x_Symbol] :> Simp[(b*(c + d*x)^m*(b*Tan[e
+ f*x])^(n - 1))/(f*(n - 1)), x] + (-Dist[(b*d*m)/(f*(n - 1)), Int[(c + d*x)^(m - 1)*(b*Tan[e + f*x])^(n - 1)
, x], x] - Dist[b^2, Int[(c + d*x)^m*(b*Tan[e + f*x])^(n - 2), x], x]) /; FreeQ[{b, c, d, e, f}, x] && GtQ[n,
1] && GtQ[m, 0]
Rule 3718
Int[((c_.) + (d_.)*(x_))^(m_.)*tan[(e_.) + (Complex[0, fz_])*(f_.)*(x_)], x_Symbol] :> -Simp[(I*(c + d*x)^(m +
1))/(d*(m + 1)), x] + Dist[2*I, Int[((c + d*x)^m*E^(2*(-(I*e) + f*fz*x)))/(1 + E^(2*(-(I*e) + f*fz*x))), x],
x] /; FreeQ[{c, d, e, f, fz}, x] && 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 30
Int[(x_)^(m_.), x_Symbol] :> Simp[x^(m + 1)/(m + 1), x] /; FreeQ[m, x] && NeQ[m, -1]
Rubi steps
\begin{align*} \int x^3 \text{csch}(a+b x) \text{sech}^3(a+b x) \, dx &=\frac{x^3 \log (\tanh (a+b x))}{b}-\frac{x^3 \tanh ^2(a+b x)}{2 b}-3 \int x^2 \left (\frac{\log (\tanh (a+b x))}{b}-\frac{\tanh ^2(a+b x)}{2 b}\right ) \, dx\\ &=\frac{x^3 \log (\tanh (a+b x))}{b}-\frac{x^3 \tanh ^2(a+b x)}{2 b}-3 \int \left (\frac{x^2 \log (\tanh (a+b x))}{b}-\frac{x^2 \tanh ^2(a+b x)}{2 b}\right ) \, dx\\ &=\frac{x^3 \log (\tanh (a+b x))}{b}-\frac{x^3 \tanh ^2(a+b x)}{2 b}+\frac{3 \int x^2 \tanh ^2(a+b x) \, dx}{2 b}-\frac{3 \int x^2 \log (\tanh (a+b x)) \, dx}{b}\\ &=-\frac{3 x^2 \tanh (a+b x)}{2 b^2}-\frac{x^3 \tanh ^2(a+b x)}{2 b}+\frac{3 \int x \tanh (a+b x) \, dx}{b^2}+\frac{\int 2 b x^3 \text{csch}(2 a+2 b x) \, dx}{b}+\frac{3 \int x^2 \, dx}{2 b}\\ &=-\frac{3 x^2}{2 b^2}+\frac{x^3}{2 b}-\frac{3 x^2 \tanh (a+b x)}{2 b^2}-\frac{x^3 \tanh ^2(a+b x)}{2 b}+2 \int x^3 \text{csch}(2 a+2 b x) \, dx+\frac{6 \int \frac{e^{2 (a+b x)} x}{1+e^{2 (a+b x)}} \, dx}{b^2}\\ &=-\frac{3 x^2}{2 b^2}+\frac{x^3}{2 b}-\frac{2 x^3 \tanh ^{-1}\left (e^{2 a+2 b x}\right )}{b}+\frac{3 x \log \left (1+e^{2 (a+b x)}\right )}{b^3}-\frac{3 x^2 \tanh (a+b x)}{2 b^2}-\frac{x^3 \tanh ^2(a+b x)}{2 b}-\frac{3 \int \log \left (1+e^{2 (a+b x)}\right ) \, dx}{b^3}-\frac{3 \int x^2 \log \left (1-e^{2 a+2 b x}\right ) \, dx}{b}+\frac{3 \int x^2 \log \left (1+e^{2 a+2 b x}\right ) \, dx}{b}\\ &=-\frac{3 x^2}{2 b^2}+\frac{x^3}{2 b}-\frac{2 x^3 \tanh ^{-1}\left (e^{2 a+2 b x}\right )}{b}+\frac{3 x \log \left (1+e^{2 (a+b x)}\right )}{b^3}-\frac{3 x^2 \text{Li}_2\left (-e^{2 a+2 b x}\right )}{2 b^2}+\frac{3 x^2 \text{Li}_2\left (e^{2 a+2 b x}\right )}{2 b^2}-\frac{3 x^2 \tanh (a+b x)}{2 b^2}-\frac{x^3 \tanh ^2(a+b x)}{2 b}-\frac{3 \operatorname{Subst}\left (\int \frac{\log (1+x)}{x} \, dx,x,e^{2 (a+b x)}\right )}{2 b^4}+\frac{3 \int x \text{Li}_2\left (-e^{2 a+2 b x}\right ) \, dx}{b^2}-\frac{3 \int x \text{Li}_2\left (e^{2 a+2 b x}\right ) \, dx}{b^2}\\ &=-\frac{3 x^2}{2 b^2}+\frac{x^3}{2 b}-\frac{2 x^3 \tanh ^{-1}\left (e^{2 a+2 b x}\right )}{b}+\frac{3 x \log \left (1+e^{2 (a+b x)}\right )}{b^3}+\frac{3 \text{Li}_2\left (-e^{2 (a+b x)}\right )}{2 b^4}-\frac{3 x^2 \text{Li}_2\left (-e^{2 a+2 b x}\right )}{2 b^2}+\frac{3 x^2 \text{Li}_2\left (e^{2 a+2 b x}\right )}{2 b^2}+\frac{3 x \text{Li}_3\left (-e^{2 a+2 b x}\right )}{2 b^3}-\frac{3 x \text{Li}_3\left (e^{2 a+2 b x}\right )}{2 b^3}-\frac{3 x^2 \tanh (a+b x)}{2 b^2}-\frac{x^3 \tanh ^2(a+b x)}{2 b}-\frac{3 \int \text{Li}_3\left (-e^{2 a+2 b x}\right ) \, dx}{2 b^3}+\frac{3 \int \text{Li}_3\left (e^{2 a+2 b x}\right ) \, dx}{2 b^3}\\ &=-\frac{3 x^2}{2 b^2}+\frac{x^3}{2 b}-\frac{2 x^3 \tanh ^{-1}\left (e^{2 a+2 b x}\right )}{b}+\frac{3 x \log \left (1+e^{2 (a+b x)}\right )}{b^3}+\frac{3 \text{Li}_2\left (-e^{2 (a+b x)}\right )}{2 b^4}-\frac{3 x^2 \text{Li}_2\left (-e^{2 a+2 b x}\right )}{2 b^2}+\frac{3 x^2 \text{Li}_2\left (e^{2 a+2 b x}\right )}{2 b^2}+\frac{3 x \text{Li}_3\left (-e^{2 a+2 b x}\right )}{2 b^3}-\frac{3 x \text{Li}_3\left (e^{2 a+2 b x}\right )}{2 b^3}-\frac{3 x^2 \tanh (a+b x)}{2 b^2}-\frac{x^3 \tanh ^2(a+b x)}{2 b}-\frac{3 \operatorname{Subst}\left (\int \frac{\text{Li}_3(-x)}{x} \, dx,x,e^{2 a+2 b x}\right )}{4 b^4}+\frac{3 \operatorname{Subst}\left (\int \frac{\text{Li}_3(x)}{x} \, dx,x,e^{2 a+2 b x}\right )}{4 b^4}\\ &=-\frac{3 x^2}{2 b^2}+\frac{x^3}{2 b}-\frac{2 x^3 \tanh ^{-1}\left (e^{2 a+2 b x}\right )}{b}+\frac{3 x \log \left (1+e^{2 (a+b x)}\right )}{b^3}+\frac{3 \text{Li}_2\left (-e^{2 (a+b x)}\right )}{2 b^4}-\frac{3 x^2 \text{Li}_2\left (-e^{2 a+2 b x}\right )}{2 b^2}+\frac{3 x^2 \text{Li}_2\left (e^{2 a+2 b x}\right )}{2 b^2}+\frac{3 x \text{Li}_3\left (-e^{2 a+2 b x}\right )}{2 b^3}-\frac{3 x \text{Li}_3\left (e^{2 a+2 b x}\right )}{2 b^3}-\frac{3 \text{Li}_4\left (-e^{2 a+2 b x}\right )}{4 b^4}+\frac{3 \text{Li}_4\left (e^{2 a+2 b x}\right )}{4 b^4}-\frac{3 x^2 \tanh (a+b x)}{2 b^2}-\frac{x^3 \tanh ^2(a+b x)}{2 b}\\ \end{align*}
Mathematica [A] time = 7.3167, size = 462, normalized size = 1.92 \[ -\frac{e^{2 a} \left (6 \left (1-e^{-2 a}\right ) \left (b^2 x^2 \text{PolyLog}\left (2,-e^{-a-b x}\right )+2 \left (b x \text{PolyLog}\left (3,-e^{-a-b x}\right )+\text{PolyLog}\left (4,-e^{-a-b x}\right )\right )\right )+6 \left (1-e^{-2 a}\right ) \left (b^2 x^2 \text{PolyLog}\left (2,e^{-a-b x}\right )+2 \left (b x \text{PolyLog}\left (3,e^{-a-b x}\right )+\text{PolyLog}\left (4,e^{-a-b x}\right )\right )\right )+e^{-2 a} b^4 x^4-2 \left (1-e^{-2 a}\right ) b^3 x^3 \log \left (1-e^{-a-b x}\right )-2 \left (1-e^{-2 a}\right ) b^3 x^3 \log \left (e^{-a-b x}+1\right )\right )}{2 \left (e^{2 a}-1\right ) b^4}-\frac{e^{2 a} \left (-3 e^{-2 a} \left (e^{2 a}+1\right ) \left (2 b^2 x^2 \text{PolyLog}\left (2,-e^{-2 (a+b x)}\right )+2 b x \text{PolyLog}\left (3,-e^{-2 (a+b x)}\right )+\text{PolyLog}\left (4,-e^{-2 (a+b x)}\right )\right )+6 \left (e^{-2 a}+1\right ) \text{PolyLog}\left (2,-e^{-2 (a+b x)}\right )+2 e^{-2 a} b^4 x^4-12 e^{-2 a} b^2 x^2+4 \left (e^{-2 a}+1\right ) b^3 x^3 \log \left (e^{-2 (a+b x)}+1\right )-12 \left (e^{-2 a}+1\right ) b x \log \left (e^{-2 (a+b x)}+1\right )\right )}{4 \left (e^{2 a}+1\right ) b^4}-\frac{3 x^2 \text{sech}(a) \sinh (b x) \text{sech}(a+b x)}{2 b^2}+\frac{x^3 \text{sech}^2(a+b x)}{2 b}+\frac{1}{4} x^4 \text{csch}(a) \text{sech}(a) \]
Antiderivative was successfully verified.
[In]
Integrate[x^3*Csch[a + b*x]*Sech[a + b*x]^3,x]
[Out]
-(E^(2*a)*((b^4*x^4)/E^(2*a) - 2*b^3*(1 - E^(-2*a))*x^3*Log[1 - E^(-a - b*x)] - 2*b^3*(1 - E^(-2*a))*x^3*Log[1
+ E^(-a - b*x)] + 6*(1 - E^(-2*a))*(b^2*x^2*PolyLog[2, -E^(-a - b*x)] + 2*(b*x*PolyLog[3, -E^(-a - b*x)] + Po
lyLog[4, -E^(-a - b*x)])) + 6*(1 - E^(-2*a))*(b^2*x^2*PolyLog[2, E^(-a - b*x)] + 2*(b*x*PolyLog[3, E^(-a - b*x
)] + PolyLog[4, E^(-a - b*x)]))))/(2*b^4*(-1 + E^(2*a))) - (E^(2*a)*((-12*b^2*x^2)/E^(2*a) + (2*b^4*x^4)/E^(2*
a) - 12*b*(1 + E^(-2*a))*x*Log[1 + E^(-2*(a + b*x))] + 4*b^3*(1 + E^(-2*a))*x^3*Log[1 + E^(-2*(a + b*x))] + 6*
(1 + E^(-2*a))*PolyLog[2, -E^(-2*(a + b*x))] - (3*(1 + E^(2*a))*(2*b^2*x^2*PolyLog[2, -E^(-2*(a + b*x))] + 2*b
*x*PolyLog[3, -E^(-2*(a + b*x))] + PolyLog[4, -E^(-2*(a + b*x))]))/E^(2*a)))/(4*b^4*(1 + E^(2*a))) + (x^4*Csch
[a]*Sech[a])/4 + (x^3*Sech[a + b*x]^2)/(2*b) - (3*x^2*Sech[a]*Sech[a + b*x]*Sinh[b*x])/(2*b^2)
________________________________________________________________________________________
Maple [A] time = 0.088, size = 359, normalized size = 1.5 \begin{align*}{\frac{{x}^{2} \left ( 2\,bx{{\rm e}^{2\,bx+2\,a}}+3\,{{\rm e}^{2\,bx+2\,a}}+3 \right ) }{{b}^{2} \left ( 1+{{\rm e}^{2\,bx+2\,a}} \right ) ^{2}}}-3\,{\frac{{x}^{2}}{{b}^{2}}}-3\,{\frac{{a}^{2}}{{b}^{4}}}+3\,{\frac{x\ln \left ( 1+{{\rm e}^{2\,bx+2\,a}} \right ) }{{b}^{3}}}-6\,{\frac{ax}{{b}^{3}}}+{\frac{\ln \left ( 1+{{\rm e}^{bx+a}} \right ){x}^{3}}{b}}-{\frac{{x}^{3}\ln \left ( 1+{{\rm e}^{2\,bx+2\,a}} \right ) }{b}}-{\frac{3\,{x}^{2}{\it polylog} \left ( 2,-{{\rm e}^{2\,bx+2\,a}} \right ) }{2\,{b}^{2}}}+{\frac{3\,x{\it polylog} \left ( 3,-{{\rm e}^{2\,bx+2\,a}} \right ) }{2\,{b}^{3}}}+3\,{\frac{{\it polylog} \left ( 2,-{{\rm e}^{bx+a}} \right ){x}^{2}}{{b}^{2}}}-6\,{\frac{{\it polylog} \left ( 3,-{{\rm e}^{bx+a}} \right ) x}{{b}^{3}}}+{\frac{\ln \left ( 1-{{\rm e}^{bx+a}} \right ){x}^{3}}{b}}+3\,{\frac{{\it polylog} \left ( 2,{{\rm e}^{bx+a}} \right ){x}^{2}}{{b}^{2}}}-6\,{\frac{{\it polylog} \left ( 3,{{\rm e}^{bx+a}} \right ) x}{{b}^{3}}}+{\frac{3\,{\it polylog} \left ( 2,-{{\rm e}^{2\,bx+2\,a}} \right ) }{2\,{b}^{4}}}+6\,{\frac{{\it polylog} \left ( 4,{{\rm e}^{bx+a}} \right ) }{{b}^{4}}}+6\,{\frac{{\it polylog} \left ( 4,-{{\rm e}^{bx+a}} \right ) }{{b}^{4}}}-{\frac{3\,{\it polylog} \left ( 4,-{{\rm e}^{2\,bx+2\,a}} \right ) }{4\,{b}^{4}}}-{\frac{{a}^{3}\ln \left ({{\rm e}^{bx+a}}-1 \right ) }{{b}^{4}}}+6\,{\frac{a\ln \left ({{\rm e}^{bx+a}} \right ) }{{b}^{4}}}+{\frac{\ln \left ( 1-{{\rm e}^{bx+a}} \right ){a}^{3}}{{b}^{4}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
int(x^3*csch(b*x+a)*sech(b*x+a)^3,x)
[Out]
x^2*(2*b*x*exp(2*b*x+2*a)+3*exp(2*b*x+2*a)+3)/b^2/(1+exp(2*b*x+2*a))^2-3*x^2/b^2-3/b^4*a^2+3*x*ln(1+exp(2*b*x+
2*a))/b^3-6/b^3*a*x+1/b*ln(1+exp(b*x+a))*x^3-x^3*ln(1+exp(2*b*x+2*a))/b-3/2*x^2*polylog(2,-exp(2*b*x+2*a))/b^2
+3/2*x*polylog(3,-exp(2*b*x+2*a))/b^3+3/b^2*polylog(2,-exp(b*x+a))*x^2-6/b^3*polylog(3,-exp(b*x+a))*x+1/b*ln(1
-exp(b*x+a))*x^3+3/b^2*polylog(2,exp(b*x+a))*x^2-6/b^3*polylog(3,exp(b*x+a))*x+3/2*polylog(2,-exp(2*b*x+2*a))/
b^4+6/b^4*polylog(4,exp(b*x+a))+6/b^4*polylog(4,-exp(b*x+a))-3/4*polylog(4,-exp(2*b*x+2*a))/b^4-1/b^4*a^3*ln(e
xp(b*x+a)-1)+6/b^4*a*ln(exp(b*x+a))+1/b^4*ln(1-exp(b*x+a))*a^3
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Maxima [A] time = 1.20206, size = 444, normalized size = 1.85 \begin{align*} -\frac{1}{2} \, x^{4} + \frac{3 \, x^{2} +{\left (2 \, b x^{3} e^{\left (2 \, a\right )} + 3 \, x^{2} e^{\left (2 \, a\right )}\right )} e^{\left (2 \, b x\right )}}{b^{2} e^{\left (4 \, b x + 4 \, a\right )} + 2 \, b^{2} e^{\left (2 \, b x + 2 \, a\right )} + b^{2}} + \frac{b^{4} x^{4} - 6 \, b^{2} x^{2}}{2 \, b^{4}} - \frac{4 \, b^{3} x^{3} \log \left (e^{\left (2 \, b x + 2 \, a\right )} + 1\right ) + 6 \, b^{2} x^{2}{\rm Li}_2\left (-e^{\left (2 \, b x + 2 \, a\right )}\right ) - 6 \, b x{\rm Li}_{3}(-e^{\left (2 \, b x + 2 \, a\right )}) + 3 \,{\rm Li}_{4}(-e^{\left (2 \, b x + 2 \, a\right )})}{3 \, b^{4}} + \frac{b^{3} x^{3} \log \left (e^{\left (b x + a\right )} + 1\right ) + 3 \, b^{2} x^{2}{\rm Li}_2\left (-e^{\left (b x + a\right )}\right ) - 6 \, b x{\rm Li}_{3}(-e^{\left (b x + a\right )}) + 6 \,{\rm Li}_{4}(-e^{\left (b x + a\right )})}{b^{4}} + \frac{b^{3} x^{3} \log \left (-e^{\left (b x + a\right )} + 1\right ) + 3 \, b^{2} x^{2}{\rm Li}_2\left (e^{\left (b x + a\right )}\right ) - 6 \, b x{\rm Li}_{3}(e^{\left (b x + a\right )}) + 6 \,{\rm Li}_{4}(e^{\left (b x + a\right )})}{b^{4}} + \frac{3 \,{\left (2 \, b x \log \left (e^{\left (2 \, b x + 2 \, a\right )} + 1\right ) +{\rm Li}_2\left (-e^{\left (2 \, b x + 2 \, a\right )}\right )\right )}}{2 \, b^{4}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(x^3*csch(b*x+a)*sech(b*x+a)^3,x, algorithm="maxima")
[Out]
-1/2*x^4 + (3*x^2 + (2*b*x^3*e^(2*a) + 3*x^2*e^(2*a))*e^(2*b*x))/(b^2*e^(4*b*x + 4*a) + 2*b^2*e^(2*b*x + 2*a)
+ b^2) + 1/2*(b^4*x^4 - 6*b^2*x^2)/b^4 - 1/3*(4*b^3*x^3*log(e^(2*b*x + 2*a) + 1) + 6*b^2*x^2*dilog(-e^(2*b*x +
2*a)) - 6*b*x*polylog(3, -e^(2*b*x + 2*a)) + 3*polylog(4, -e^(2*b*x + 2*a)))/b^4 + (b^3*x^3*log(e^(b*x + a) +
1) + 3*b^2*x^2*dilog(-e^(b*x + a)) - 6*b*x*polylog(3, -e^(b*x + a)) + 6*polylog(4, -e^(b*x + a)))/b^4 + (b^3*
x^3*log(-e^(b*x + a) + 1) + 3*b^2*x^2*dilog(e^(b*x + a)) - 6*b*x*polylog(3, e^(b*x + a)) + 6*polylog(4, e^(b*x
+ a)))/b^4 + 3/2*(2*b*x*log(e^(2*b*x + 2*a) + 1) + dilog(-e^(2*b*x + 2*a)))/b^4
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Fricas [C] time = 3.18665, size = 8806, normalized size = 36.69 \begin{align*} \text{result too large to display} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(x^3*csch(b*x+a)*sech(b*x+a)^3,x, algorithm="fricas")
[Out]
-(3*(b^2*x^2 - a^2)*cosh(b*x + a)^4 + 12*(b^2*x^2 - a^2)*cosh(b*x + a)*sinh(b*x + a)^3 + 3*(b^2*x^2 - a^2)*sin
h(b*x + a)^4 - (2*b^3*x^3 - 3*b^2*x^2 + 6*a^2)*cosh(b*x + a)^2 - (2*b^3*x^3 - 3*b^2*x^2 - 18*(b^2*x^2 - a^2)*c
osh(b*x + a)^2 + 6*a^2)*sinh(b*x + a)^2 - 3*a^2 - 3*(b^2*x^2*cosh(b*x + a)^4 + 4*b^2*x^2*cosh(b*x + a)*sinh(b*
x + a)^3 + b^2*x^2*sinh(b*x + a)^4 + 2*b^2*x^2*cosh(b*x + a)^2 + b^2*x^2 + 2*(3*b^2*x^2*cosh(b*x + a)^2 + b^2*
x^2)*sinh(b*x + a)^2 + 4*(b^2*x^2*cosh(b*x + a)^3 + b^2*x^2*cosh(b*x + a))*sinh(b*x + a))*dilog(cosh(b*x + a)
+ sinh(b*x + a)) + 3*((b^2*x^2 - 1)*cosh(b*x + a)^4 + 4*(b^2*x^2 - 1)*cosh(b*x + a)*sinh(b*x + a)^3 + (b^2*x^2
- 1)*sinh(b*x + a)^4 + b^2*x^2 + 2*(b^2*x^2 - 1)*cosh(b*x + a)^2 + 2*(b^2*x^2 + 3*(b^2*x^2 - 1)*cosh(b*x + a)
^2 - 1)*sinh(b*x + a)^2 + 4*((b^2*x^2 - 1)*cosh(b*x + a)^3 + (b^2*x^2 - 1)*cosh(b*x + a))*sinh(b*x + a) - 1)*d
ilog(I*cosh(b*x + a) + I*sinh(b*x + a)) + 3*((b^2*x^2 - 1)*cosh(b*x + a)^4 + 4*(b^2*x^2 - 1)*cosh(b*x + a)*sin
h(b*x + a)^3 + (b^2*x^2 - 1)*sinh(b*x + a)^4 + b^2*x^2 + 2*(b^2*x^2 - 1)*cosh(b*x + a)^2 + 2*(b^2*x^2 + 3*(b^2
*x^2 - 1)*cosh(b*x + a)^2 - 1)*sinh(b*x + a)^2 + 4*((b^2*x^2 - 1)*cosh(b*x + a)^3 + (b^2*x^2 - 1)*cosh(b*x + a
))*sinh(b*x + a) - 1)*dilog(-I*cosh(b*x + a) - I*sinh(b*x + a)) - 3*(b^2*x^2*cosh(b*x + a)^4 + 4*b^2*x^2*cosh(
b*x + a)*sinh(b*x + a)^3 + b^2*x^2*sinh(b*x + a)^4 + 2*b^2*x^2*cosh(b*x + a)^2 + b^2*x^2 + 2*(3*b^2*x^2*cosh(b
*x + a)^2 + b^2*x^2)*sinh(b*x + a)^2 + 4*(b^2*x^2*cosh(b*x + a)^3 + b^2*x^2*cosh(b*x + a))*sinh(b*x + a))*dilo
g(-cosh(b*x + a) - sinh(b*x + a)) - (b^3*x^3*cosh(b*x + a)^4 + 4*b^3*x^3*cosh(b*x + a)*sinh(b*x + a)^3 + b^3*x
^3*sinh(b*x + a)^4 + 2*b^3*x^3*cosh(b*x + a)^2 + b^3*x^3 + 2*(3*b^3*x^3*cosh(b*x + a)^2 + b^3*x^3)*sinh(b*x +
a)^2 + 4*(b^3*x^3*cosh(b*x + a)^3 + b^3*x^3*cosh(b*x + a))*sinh(b*x + a))*log(cosh(b*x + a) + sinh(b*x + a) +
1) - ((a^3 - 3*a)*cosh(b*x + a)^4 + 4*(a^3 - 3*a)*cosh(b*x + a)*sinh(b*x + a)^3 + (a^3 - 3*a)*sinh(b*x + a)^4
+ a^3 + 2*(a^3 - 3*a)*cosh(b*x + a)^2 + 2*(a^3 + 3*(a^3 - 3*a)*cosh(b*x + a)^2 - 3*a)*sinh(b*x + a)^2 + 4*((a^
3 - 3*a)*cosh(b*x + a)^3 + (a^3 - 3*a)*cosh(b*x + a))*sinh(b*x + a) - 3*a)*log(cosh(b*x + a) + sinh(b*x + a) +
I) - ((a^3 - 3*a)*cosh(b*x + a)^4 + 4*(a^3 - 3*a)*cosh(b*x + a)*sinh(b*x + a)^3 + (a^3 - 3*a)*sinh(b*x + a)^4
+ a^3 + 2*(a^3 - 3*a)*cosh(b*x + a)^2 + 2*(a^3 + 3*(a^3 - 3*a)*cosh(b*x + a)^2 - 3*a)*sinh(b*x + a)^2 + 4*((a
^3 - 3*a)*cosh(b*x + a)^3 + (a^3 - 3*a)*cosh(b*x + a))*sinh(b*x + a) - 3*a)*log(cosh(b*x + a) + sinh(b*x + a)
- I) + (a^3*cosh(b*x + a)^4 + 4*a^3*cosh(b*x + a)*sinh(b*x + a)^3 + a^3*sinh(b*x + a)^4 + 2*a^3*cosh(b*x + a)^
2 + a^3 + 2*(3*a^3*cosh(b*x + a)^2 + a^3)*sinh(b*x + a)^2 + 4*(a^3*cosh(b*x + a)^3 + a^3*cosh(b*x + a))*sinh(b
*x + a))*log(cosh(b*x + a) + sinh(b*x + a) - 1) + (b^3*x^3 + (b^3*x^3 + a^3 - 3*b*x - 3*a)*cosh(b*x + a)^4 + 4
*(b^3*x^3 + a^3 - 3*b*x - 3*a)*cosh(b*x + a)*sinh(b*x + a)^3 + (b^3*x^3 + a^3 - 3*b*x - 3*a)*sinh(b*x + a)^4 +
a^3 + 2*(b^3*x^3 + a^3 - 3*b*x - 3*a)*cosh(b*x + a)^2 + 2*(b^3*x^3 + a^3 + 3*(b^3*x^3 + a^3 - 3*b*x - 3*a)*co
sh(b*x + a)^2 - 3*b*x - 3*a)*sinh(b*x + a)^2 - 3*b*x + 4*((b^3*x^3 + a^3 - 3*b*x - 3*a)*cosh(b*x + a)^3 + (b^3
*x^3 + a^3 - 3*b*x - 3*a)*cosh(b*x + a))*sinh(b*x + a) - 3*a)*log(I*cosh(b*x + a) + I*sinh(b*x + a) + 1) + (b^
3*x^3 + (b^3*x^3 + a^3 - 3*b*x - 3*a)*cosh(b*x + a)^4 + 4*(b^3*x^3 + a^3 - 3*b*x - 3*a)*cosh(b*x + a)*sinh(b*x
+ a)^3 + (b^3*x^3 + a^3 - 3*b*x - 3*a)*sinh(b*x + a)^4 + a^3 + 2*(b^3*x^3 + a^3 - 3*b*x - 3*a)*cosh(b*x + a)^
2 + 2*(b^3*x^3 + a^3 + 3*(b^3*x^3 + a^3 - 3*b*x - 3*a)*cosh(b*x + a)^2 - 3*b*x - 3*a)*sinh(b*x + a)^2 - 3*b*x
+ 4*((b^3*x^3 + a^3 - 3*b*x - 3*a)*cosh(b*x + a)^3 + (b^3*x^3 + a^3 - 3*b*x - 3*a)*cosh(b*x + a))*sinh(b*x + a
) - 3*a)*log(-I*cosh(b*x + a) - I*sinh(b*x + a) + 1) - (b^3*x^3 + (b^3*x^3 + a^3)*cosh(b*x + a)^4 + 4*(b^3*x^3
+ a^3)*cosh(b*x + a)*sinh(b*x + a)^3 + (b^3*x^3 + a^3)*sinh(b*x + a)^4 + a^3 + 2*(b^3*x^3 + a^3)*cosh(b*x + a
)^2 + 2*(b^3*x^3 + a^3 + 3*(b^3*x^3 + a^3)*cosh(b*x + a)^2)*sinh(b*x + a)^2 + 4*((b^3*x^3 + a^3)*cosh(b*x + a)
^3 + (b^3*x^3 + a^3)*cosh(b*x + a))*sinh(b*x + a))*log(-cosh(b*x + a) - sinh(b*x + a) + 1) - 6*(cosh(b*x + a)^
4 + 4*cosh(b*x + a)*sinh(b*x + a)^3 + sinh(b*x + a)^4 + 2*(3*cosh(b*x + a)^2 + 1)*sinh(b*x + a)^2 + 2*cosh(b*x
+ a)^2 + 4*(cosh(b*x + a)^3 + cosh(b*x + a))*sinh(b*x + a) + 1)*polylog(4, cosh(b*x + a) + sinh(b*x + a)) + 6
*(cosh(b*x + a)^4 + 4*cosh(b*x + a)*sinh(b*x + a)^3 + sinh(b*x + a)^4 + 2*(3*cosh(b*x + a)^2 + 1)*sinh(b*x + a
)^2 + 2*cosh(b*x + a)^2 + 4*(cosh(b*x + a)^3 + cosh(b*x + a))*sinh(b*x + a) + 1)*polylog(4, I*cosh(b*x + a) +
I*sinh(b*x + a)) + 6*(cosh(b*x + a)^4 + 4*cosh(b*x + a)*sinh(b*x + a)^3 + sinh(b*x + a)^4 + 2*(3*cosh(b*x + a)
^2 + 1)*sinh(b*x + a)^2 + 2*cosh(b*x + a)^2 + 4*(cosh(b*x + a)^3 + cosh(b*x + a))*sinh(b*x + a) + 1)*polylog(4
, -I*cosh(b*x + a) - I*sinh(b*x + a)) - 6*(cosh(b*x + a)^4 + 4*cosh(b*x + a)*sinh(b*x + a)^3 + sinh(b*x + a)^4
+ 2*(3*cosh(b*x + a)^2 + 1)*sinh(b*x + a)^2 + 2*cosh(b*x + a)^2 + 4*(cosh(b*x + a)^3 + cosh(b*x + a))*sinh(b*
x + a) + 1)*polylog(4, -cosh(b*x + a) - sinh(b*x + a)) + 6*(b*x*cosh(b*x + a)^4 + 4*b*x*cosh(b*x + a)*sinh(b*x
+ a)^3 + b*x*sinh(b*x + a)^4 + 2*b*x*cosh(b*x + a)^2 + 2*(3*b*x*cosh(b*x + a)^2 + b*x)*sinh(b*x + a)^2 + b*x
+ 4*(b*x*cosh(b*x + a)^3 + b*x*cosh(b*x + a))*sinh(b*x + a))*polylog(3, cosh(b*x + a) + sinh(b*x + a)) - 6*(b*
x*cosh(b*x + a)^4 + 4*b*x*cosh(b*x + a)*sinh(b*x + a)^3 + b*x*sinh(b*x + a)^4 + 2*b*x*cosh(b*x + a)^2 + 2*(3*b
*x*cosh(b*x + a)^2 + b*x)*sinh(b*x + a)^2 + b*x + 4*(b*x*cosh(b*x + a)^3 + b*x*cosh(b*x + a))*sinh(b*x + a))*p
olylog(3, I*cosh(b*x + a) + I*sinh(b*x + a)) - 6*(b*x*cosh(b*x + a)^4 + 4*b*x*cosh(b*x + a)*sinh(b*x + a)^3 +
b*x*sinh(b*x + a)^4 + 2*b*x*cosh(b*x + a)^2 + 2*(3*b*x*cosh(b*x + a)^2 + b*x)*sinh(b*x + a)^2 + b*x + 4*(b*x*c
osh(b*x + a)^3 + b*x*cosh(b*x + a))*sinh(b*x + a))*polylog(3, -I*cosh(b*x + a) - I*sinh(b*x + a)) + 6*(b*x*cos
h(b*x + a)^4 + 4*b*x*cosh(b*x + a)*sinh(b*x + a)^3 + b*x*sinh(b*x + a)^4 + 2*b*x*cosh(b*x + a)^2 + 2*(3*b*x*co
sh(b*x + a)^2 + b*x)*sinh(b*x + a)^2 + b*x + 4*(b*x*cosh(b*x + a)^3 + b*x*cosh(b*x + a))*sinh(b*x + a))*polylo
g(3, -cosh(b*x + a) - sinh(b*x + a)) + 2*(6*(b^2*x^2 - a^2)*cosh(b*x + a)^3 - (2*b^3*x^3 - 3*b^2*x^2 + 6*a^2)*
cosh(b*x + a))*sinh(b*x + a))/(b^4*cosh(b*x + a)^4 + 4*b^4*cosh(b*x + a)*sinh(b*x + a)^3 + b^4*sinh(b*x + a)^4
+ 2*b^4*cosh(b*x + a)^2 + b^4 + 2*(3*b^4*cosh(b*x + a)^2 + b^4)*sinh(b*x + a)^2 + 4*(b^4*cosh(b*x + a)^3 + b^
4*cosh(b*x + a))*sinh(b*x + a))
________________________________________________________________________________________
Sympy [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int x^{3} \operatorname{csch}{\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**3*csch(b*x+a)*sech(b*x+a)**3,x)
[Out]
Integral(x**3*csch(a + b*x)*sech(a + b*x)**3, x)
________________________________________________________________________________________
Giac [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int x^{3} \operatorname{csch}\left (b x + a\right ) \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^3*csch(b*x+a)*sech(b*x+a)^3,x, algorithm="giac")
[Out]
integrate(x^3*csch(b*x + a)*sech(b*x + a)^3, x)