3.6.8 \(\int x^3 \text {csch}^3(a+b x) \text {sech}(a+b x) \, dx\) [508]

Optimal. Leaf size=240 \[ -\frac {3 x^2}{2 b^2}+\frac {x^3}{2 b}+\frac {2 x^3 \tanh ^{-1}\left (e^{2 a+2 b x}\right )}{b}-\frac {3 x^2 \coth (a+b x)}{2 b^2}-\frac {x^3 \coth ^2(a+b x)}{2 b}+\frac {3 x \log \left (1-e^{2 (a+b x)}\right )}{b^3}+\frac {3 \text {PolyLog}\left (2,e^{2 (a+b x)}\right )}{2 b^4}+\frac {3 x^2 \text {PolyLog}\left (2,-e^{2 a+2 b x}\right )}{2 b^2}-\frac {3 x^2 \text {PolyLog}\left (2,e^{2 a+2 b x}\right )}{2 b^2}-\frac {3 x \text {PolyLog}\left (3,-e^{2 a+2 b x}\right )}{2 b^3}+\frac {3 x \text {PolyLog}\left (3,e^{2 a+2 b x}\right )}{2 b^3}+\frac {3 \text {PolyLog}\left (4,-e^{2 a+2 b x}\right )}{4 b^4}-\frac {3 \text {PolyLog}\left (4,e^{2 a+2 b x}\right )}{4 b^4} \]

[Out]

-3/2*x^2/b^2+1/2*x^3/b+2*x^3*arctanh(exp(2*b*x+2*a))/b-3/2*x^2*coth(b*x+a)/b^2-1/2*x^3*coth(b*x+a)^2/b+3*x*ln(
1-exp(2*b*x+2*a))/b^3+3/2*polylog(2,exp(2*b*x+2*a))/b^4+3/2*x^2*polylog(2,-exp(2*b*x+2*a))/b^2-3/2*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+3/2*x*polylog(3,exp(2*b*x+2*a))/b^3+3/4*polylog(4,
-exp(2*b*x+2*a))/b^4-3/4*polylog(4,exp(2*b*x+2*a))/b^4

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Rubi [A]
time = 0.31, antiderivative size = 240, normalized size of antiderivative = 1.00, 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 = {2700, 14, 5570, 3801, 3797, 2221, 2317, 2438, 30, 2631, 12, 4267, 2611, 6744, 2320, 6724} \begin {gather*} \frac {3 \text {Li}_2\left (e^{2 (a+b x)}\right )}{2 b^4}+\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 \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 \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 \coth (a+b x)}{2 b^2}+\frac {2 x^3 \tanh ^{-1}\left (e^{2 a+2 b x}\right )}{b}-\frac {x^3 \coth ^2(a+b x)}{2 b}-\frac {3 x^2}{2 b^2}+\frac {x^3}{2 b} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[x^3*Csch[a + b*x]^3*Sech[a + b*x],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^2*Coth[a + b*x])/(2*b^2) - (x^3*Coth[
a + b*x]^2)/(2*b) + (3*x*Log[1 - E^(2*(a + b*x))])/b^3 + (3*PolyLog[2, E^(2*(a + b*x))])/(2*b^4) + (3*x^2*Poly
Log[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*Pol
yLog[4, E^(2*a + 2*b*x)])/(4*b^4)

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 30

Int[(x_)^(m_.), x_Symbol] :> Simp[x^(m + 1)/(m + 1), x] /; FreeQ[m, x] && NeQ[m, -1]

Rule 2221

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/(b*f*g*n*Log[F]))*Log[1 + b*((F^(g*(e + f*x)))^n/a)], 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 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 2631

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 2700

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 3797

Int[((c_.) + (d_.)*(x_))^(m_.)*tan[(e_.) + Pi*(k_.) + (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))/E^(2*I*k*Pi))))/E^(2*I*k*Pi), x], x] /; FreeQ[{c, d, e, f, fz}, x] && IntegerQ[4*k] && IGtQ[m, 0]

Rule 3801

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 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 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 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 6744

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]

Rubi steps

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

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Mathematica [A]
time = 4.05, size = 281, normalized size = 1.17 \begin {gather*} \frac {1}{4} \left (-\frac {12 x^2}{b^2}-\frac {12 x^2}{b^2 \left (-1+e^{2 a}\right )}+\frac {2 x^4}{-1+e^{2 a}}+\frac {2 x^4}{1+e^{2 a}}-\frac {2 x^3 \text {csch}^2(a+b x)}{b}+\frac {12 x \log \left (1-e^{2 (a+b x)}\right )}{b^3}-\frac {4 x^3 \log \left (1-e^{2 (a+b x)}\right )}{b}+\frac {4 x^3 \log \left (1+e^{2 (a+b x)}\right )}{b}+\frac {6 x^2 \text {PolyLog}\left (2,-e^{2 (a+b x)}\right )}{b^2}-\frac {6 \left (-1+b^2 x^2\right ) \text {PolyLog}\left (2,e^{2 (a+b x)}\right )}{b^4}-\frac {6 x \text {PolyLog}\left (3,-e^{2 (a+b x)}\right )}{b^3}+\frac {6 x \text {PolyLog}\left (3,e^{2 (a+b x)}\right )}{b^3}+\frac {3 \text {PolyLog}\left (4,-e^{2 (a+b x)}\right )}{b^4}-\frac {3 \text {PolyLog}\left (4,e^{2 (a+b x)}\right )}{b^4}-x^4 \text {csch}(a) \text {sech}(a)+\frac {6 x^2 \text {csch}(a) \text {csch}(a+b x) \sinh (b x)}{b^2}\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

((-12*x^2)/b^2 - (12*x^2)/(b^2*(-1 + E^(2*a))) + (2*x^4)/(-1 + E^(2*a)) + (2*x^4)/(1 + E^(2*a)) - (2*x^3*Csch[
a + b*x]^2)/b + (12*x*Log[1 - E^(2*(a + b*x))])/b^3 - (4*x^3*Log[1 - E^(2*(a + b*x))])/b + (4*x^3*Log[1 + E^(2
*(a + b*x))])/b + (6*x^2*PolyLog[2, -E^(2*(a + b*x))])/b^2 - (6*(-1 + b^2*x^2)*PolyLog[2, E^(2*(a + b*x))])/b^
4 - (6*x*PolyLog[3, -E^(2*(a + b*x))])/b^3 + (6*x*PolyLog[3, E^(2*(a + b*x))])/b^3 + (3*PolyLog[4, -E^(2*(a +
b*x))])/b^4 - (3*PolyLog[4, E^(2*(a + b*x))])/b^4 - x^4*Csch[a]*Sech[a] + (6*x^2*Csch[a]*Csch[a + b*x]*Sinh[b*
x])/b^2)/4

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Maple [A]
time = 2.04, size = 417, normalized size = 1.74

method	result	size
risch	\(\frac {3 a \ln \left (1-{\mathrm e}^{b x +a}\right )}{b^{4}}+\frac {3 \ln \left (1-{\mathrm e}^{b x +a}\right ) x}{b^{3}}+\frac {x^{3} \ln \left ({\mathrm e}^{2 b x +2 a}+1\right )}{b}-\frac {x^{2} \left (2 b x \,{\mathrm e}^{2 b x +2 a}+3 \,{\mathrm e}^{2 b x +2 a}-3\right )}{b^{2} \left ({\mathrm e}^{2 b x +2 a}-1\right )^{2}}-\frac {3 x^{2}}{b^{2}}-\frac {6 a x}{b^{3}}+\frac {3 \ln \left ({\mathrm e}^{b x +a}+1\right ) x}{b^{3}}+\frac {6 x \polylog \left (3, -{\mathrm e}^{b x +a}\right )}{b^{3}}-\frac {3 x^{2} \polylog \left (2, {\mathrm e}^{b x +a}\right )}{b^{2}}+\frac {6 x \polylog \left (3, {\mathrm e}^{b x +a}\right )}{b^{3}}-\frac {3 x^{2} \polylog \left (2, -{\mathrm e}^{b x +a}\right )}{b^{2}}-\frac {\ln \left (1-{\mathrm e}^{b x +a}\right ) x^{3}}{b}-\frac {\ln \left ({\mathrm e}^{b x +a}+1\right ) x^{3}}{b}-\frac {\ln \left (1-{\mathrm e}^{b x +a}\right ) a^{3}}{b^{4}}+\frac {6 a \ln \left ({\mathrm e}^{b x +a}\right )}{b^{4}}+\frac {3 \polylog \left (4, -{\mathrm e}^{2 b x +2 a}\right )}{4 b^{4}}-\frac {3 a \ln \left ({\mathrm e}^{b x +a}-1\right )}{b^{4}}+\frac {a^{3} \ln \left ({\mathrm e}^{b x +a}-1\right )}{b^{4}}+\frac {3 \polylog \left (2, -{\mathrm e}^{b x +a}\right )}{b^{4}}+\frac {3 x^{2} \polylog \left (2, -{\mathrm e}^{2 b x +2 a}\right )}{2 b^{2}}-\frac {3 x \polylog \left (3, -{\mathrm e}^{2 b x +2 a}\right )}{2 b^{3}}-\frac {6 \polylog \left (4, {\mathrm e}^{b x +a}\right )}{b^{4}}+\frac {3 \polylog \left (2, {\mathrm e}^{b x +a}\right )}{b^{4}}-\frac {3 a^{2}}{b^{4}}-\frac {6 \polylog \left (4, -{\mathrm e}^{b x +a}\right )}{b^{4}}\)	\(417\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^3*csch(b*x+a)^3*sech(b*x+a),x,method=_RETURNVERBOSE)

[Out]

-3*x^2*polylog(2,exp(b*x+a))/b^2+6*x*polylog(3,-exp(b*x+a))/b^3+6*x*polylog(3,exp(b*x+a))/b^3+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^4*a*ln(1-exp(b*x+a))+3/b^3*ln(1-exp(b*x+a))*x+
x^3*ln(exp(2*b*x+2*a)+1)/b-x^2*(2*b*x*exp(2*b*x+2*a)+3*exp(2*b*x+2*a)-3)/b^2/(exp(2*b*x+2*a)-1)^2-3/b^2*x^2-6*
a*x/b^3+3/b^3*ln(exp(b*x+a)+1)*x-1/b*ln(1-exp(b*x+a))*x^3-1/b*ln(exp(b*x+a)+1)*x^3-1/b^4*ln(1-exp(b*x+a))*a^3-
3*x^2*polylog(2,-exp(b*x+a))/b^2+6/b^4*a*ln(exp(b*x+a))-3/b^4*a*ln(exp(b*x+a)-1)+1/b^4*a^3*ln(exp(b*x+a)-1)-3/
b^4*a^2+3*polylog(2,-exp(b*x+a))/b^4+3*polylog(2,exp(b*x+a))/b^4+3/4*polylog(4,-exp(2*b*x+2*a))/b^4-6*polylog(
4,-exp(b*x+a))/b^4-6*polylog(4,exp(b*x+a))/b^4

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Maxima [A]
time = 0.29, size = 352, normalized size = 1.47 \begin {gather*} -\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 (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^{4}} + \frac {3 \, {\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^{4}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^3*csch(b*x+a)^3*sech(b*x+a),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*(b*x*log(e^(b*x + a) + 1) + dilog(-e^(b*x + a)))/b^4 + 3*(b*x*log(-e^(b*x + a) + 1) + dilog(e^
(b*x + a)))/b^4

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Fricas [C] Result contains complex when optimal does not.
time = 0.45, size = 3394, normalized size = 14.14 \begin {gather*} \text {Too large to display} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^3*csch(b*x+a)^3*sech(b*x+a),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 - 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)*dilog(cosh(b*x + a) + sinh(b*x + a)) - 3*(b^2*x^2*cosh(b*x + a)^4 + 4*b^2*x^2*co
sh(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*cos
h(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))*d
ilog(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)*sin
h(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(-I*cosh(b*x + a) - I*s
inh(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)*dilo
g(-cosh(b*x + a) - sinh(b*x + a)) + (b^3*x^3 + (b^3*x^3 - 3*b*x)*cosh(b*x + a)^4 + 4*(b^3*x^3 - 3*b*x)*cosh(b*
x + a)*sinh(b*x + a)^3 + (b^3*x^3 - 3*b*x)*sinh(b*x + a)^4 - 2*(b^3*x^3 - 3*b*x)*cosh(b*x + a)^2 - 2*(b^3*x^3
- 3*(b^3*x^3 - 3*b*x)*cosh(b*x + a)^2 - 3*b*x)*sinh(b*x + a)^2 - 3*b*x + 4*((b^3*x^3 - 3*b*x)*cosh(b*x + a)^3
- (b^3*x^3 - 3*b*x)*cosh(b*x + a))*sinh(b*x + a))*log(cosh(b*x + a) + sinh(b*x + a) + 1) + (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) + 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) - 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) - 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(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(-
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(-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(...

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Giac [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

[Out]

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

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