\(\int x^3 \text {csch}^3(a+b x) \text {sech}^3(a+b x) \, dx\) [244]

Optimal result

Integrand size = 20, antiderivative size = 240 \[ \int x^3 \text {csch}^3(a+b x) \text {sech}^3(a+b x) \, dx=-\frac {6 x \text {arctanh}\left (e^{2 a+2 b x}\right )}{b^3}+\frac {4 x^3 \text {arctanh}\left (e^{2 a+2 b x}\right )}{b}-\frac {3 x^2 \text {csch}(2 a+2 b x)}{b^2}-\frac {2 x^3 \coth (2 a+2 b x) \text {csch}(2 a+2 b x)}{b}-\frac {3 \operatorname {PolyLog}\left (2,-e^{2 a+2 b x}\right )}{2 b^4}+\frac {3 x^2 \operatorname {PolyLog}\left (2,-e^{2 a+2 b x}\right )}{b^2}+\frac {3 \operatorname {PolyLog}\left (2,e^{2 a+2 b x}\right )}{2 b^4}-\frac {3 x^2 \operatorname {PolyLog}\left (2,e^{2 a+2 b x}\right )}{b^2}-\frac {3 x \operatorname {PolyLog}\left (3,-e^{2 a+2 b x}\right )}{b^3}+\frac {3 x \operatorname {PolyLog}\left (3,e^{2 a+2 b x}\right )}{b^3}+\frac {3 \operatorname {PolyLog}\left (4,-e^{2 a+2 b x}\right )}{2 b^4}-\frac {3 \operatorname {PolyLog}\left (4,e^{2 a+2 b x}\right )}{2 b^4} \] Output:

-6*x*arctanh(exp(2*b*x+2*a))/b^3+4*x^3*arctanh(exp(2*b*x+2*a))/b-3*x^2*csc 
h(2*b*x+2*a)/b^2-2*x^3*coth(2*b*x+2*a)*csch(2*b*x+2*a)/b-3/2*polylog(2,-ex 
p(2*b*x+2*a))/b^4+3*x^2*polylog(2,-exp(2*b*x+2*a))/b^2+3/2*polylog(2,exp(2 
*b*x+2*a))/b^4-3*x^2*polylog(2,exp(2*b*x+2*a))/b^2-3*x*polylog(3,-exp(2*b* 
x+2*a))/b^3+3*x*polylog(3,exp(2*b*x+2*a))/b^3+3/2*polylog(4,-exp(2*b*x+2*a 
))/b^4-3/2*polylog(4,exp(2*b*x+2*a))/b^4
 

Mathematica [A] (verified)

Time = 3.20 (sec) , antiderivative size = 274, normalized size of antiderivative = 1.14 \[ \int x^3 \text {csch}^3(a+b x) \text {sech}^3(a+b x) \, dx=\frac {-b^3 x^3 \text {csch}^2(a+b x)+6 b x \log \left (1-e^{2 (a+b x)}\right )-4 b^3 x^3 \log \left (1-e^{2 (a+b x)}\right )-6 b x \log \left (1+e^{2 (a+b x)}\right )+4 b^3 x^3 \log \left (1+e^{2 (a+b x)}\right )+\left (-3+6 b^2 x^2\right ) \operatorname {PolyLog}\left (2,-e^{2 (a+b x)}\right )+\left (3-6 b^2 x^2\right ) \operatorname {PolyLog}\left (2,e^{2 (a+b x)}\right )-6 b x \operatorname {PolyLog}\left (3,-e^{2 (a+b x)}\right )+6 b x \operatorname {PolyLog}\left (3,e^{2 (a+b x)}\right )+3 \operatorname {PolyLog}\left (4,-e^{2 (a+b x)}\right )-3 \operatorname {PolyLog}\left (4,e^{2 (a+b x)}\right )-3 b^2 x^2 \text {csch}(a) \text {sech}(a)-b^3 x^3 \text {sech}^2(a+b x)+3 b^2 x^2 \text {csch}(a) \text {csch}(a+b x) \sinh (b x)+3 b^2 x^2 \text {sech}(a) \text {sech}(a+b x) \sinh (b x)}{2 b^4} \] Input:

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

Output:

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

Rubi [C] (verified)

Result contains complex when optimal does not.

Time = 1.39 (sec) , antiderivative size = 311, normalized size of antiderivative = 1.30, number of steps used = 15, number of rules used = 14, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.700, Rules used = {5984, 3042, 26, 4674, 26, 3042, 26, 4670, 2715, 2838, 3011, 7163, 2720, 7143}

Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.

\(\displaystyle \int x^3 \text {csch}^3(a+b x) \text {sech}^3(a+b x) \, dx\)

\(\Big \downarrow \) 5984

\(\displaystyle 8 \int x^3 \text {csch}^3(2 a+2 b x)dx\)

\(\Big \downarrow \) 3042

\(\displaystyle 8 \int -i x^3 \csc (2 i a+2 i b x)^3dx\)

\(\Big \downarrow \) 26

\(\displaystyle -8 i \int x^3 \csc (2 i a+2 i b x)^3dx\)

\(\Big \downarrow \) 4674

\(\displaystyle -8 i \left (-\frac {3 \int -i x \text {csch}(2 a+2 b x)dx}{4 b^2}+\frac {1}{2} \int -i x^3 \text {csch}(2 a+2 b x)dx-\frac {3 i x^2 \text {csch}(2 a+2 b x)}{8 b^2}-\frac {i x^3 \coth (2 a+2 b x) \text {csch}(2 a+2 b x)}{4 b}\right )\)

\(\Big \downarrow \) 26

\(\displaystyle -8 i \left (\frac {3 i \int x \text {csch}(2 a+2 b x)dx}{4 b^2}-\frac {1}{2} i \int x^3 \text {csch}(2 a+2 b x)dx-\frac {3 i x^2 \text {csch}(2 a+2 b x)}{8 b^2}-\frac {i x^3 \coth (2 a+2 b x) \text {csch}(2 a+2 b x)}{4 b}\right )\)

\(\Big \downarrow \) 3042

\(\displaystyle -8 i \left (\frac {3 i \int i x \csc (2 i a+2 i b x)dx}{4 b^2}-\frac {1}{2} i \int i x^3 \csc (2 i a+2 i b x)dx-\frac {3 i x^2 \text {csch}(2 a+2 b x)}{8 b^2}-\frac {i x^3 \coth (2 a+2 b x) \text {csch}(2 a+2 b x)}{4 b}\right )\)

\(\Big \downarrow \) 26

\(\displaystyle -8 i \left (-\frac {3 \int x \csc (2 i a+2 i b x)dx}{4 b^2}+\frac {1}{2} \int x^3 \csc (2 i a+2 i b x)dx-\frac {3 i x^2 \text {csch}(2 a+2 b x)}{8 b^2}-\frac {i x^3 \coth (2 a+2 b x) \text {csch}(2 a+2 b x)}{4 b}\right )\)

\(\Big \downarrow \) 4670

\(\displaystyle -8 i \left (-\frac {3 \left (\frac {i \int \log \left (1-e^{2 a+2 b x}\right )dx}{2 b}-\frac {i \int \log \left (1+e^{2 a+2 b x}\right )dx}{2 b}+\frac {i x \text {arctanh}\left (e^{2 a+2 b x}\right )}{b}\right )}{4 b^2}+\frac {1}{2} \left (\frac {3 i \int x^2 \log \left (1-e^{2 a+2 b x}\right )dx}{2 b}-\frac {3 i \int x^2 \log \left (1+e^{2 a+2 b x}\right )dx}{2 b}+\frac {i x^3 \text {arctanh}\left (e^{2 a+2 b x}\right )}{b}\right )-\frac {3 i x^2 \text {csch}(2 a+2 b x)}{8 b^2}-\frac {i x^3 \coth (2 a+2 b x) \text {csch}(2 a+2 b x)}{4 b}\right )\)

\(\Big \downarrow \) 2715

\(\displaystyle -8 i \left (-\frac {3 \left (\frac {i \int e^{-2 a-2 b x} \log \left (1-e^{2 a+2 b x}\right )de^{2 a+2 b x}}{4 b^2}-\frac {i \int e^{-2 a-2 b x} \log \left (1+e^{2 a+2 b x}\right )de^{2 a+2 b x}}{4 b^2}+\frac {i x \text {arctanh}\left (e^{2 a+2 b x}\right )}{b}\right )}{4 b^2}+\frac {1}{2} \left (\frac {3 i \int x^2 \log \left (1-e^{2 a+2 b x}\right )dx}{2 b}-\frac {3 i \int x^2 \log \left (1+e^{2 a+2 b x}\right )dx}{2 b}+\frac {i x^3 \text {arctanh}\left (e^{2 a+2 b x}\right )}{b}\right )-\frac {3 i x^2 \text {csch}(2 a+2 b x)}{8 b^2}-\frac {i x^3 \coth (2 a+2 b x) \text {csch}(2 a+2 b x)}{4 b}\right )\)

\(\Big \downarrow \) 2838

\(\displaystyle -8 i \left (\frac {1}{2} \left (\frac {3 i \int x^2 \log \left (1-e^{2 a+2 b x}\right )dx}{2 b}-\frac {3 i \int x^2 \log \left (1+e^{2 a+2 b x}\right )dx}{2 b}+\frac {i x^3 \text {arctanh}\left (e^{2 a+2 b x}\right )}{b}\right )-\frac {3 \left (\frac {i x \text {arctanh}\left (e^{2 a+2 b x}\right )}{b}+\frac {i \operatorname {PolyLog}\left (2,-e^{2 a+2 b x}\right )}{4 b^2}-\frac {i \operatorname {PolyLog}\left (2,e^{2 a+2 b x}\right )}{4 b^2}\right )}{4 b^2}-\frac {3 i x^2 \text {csch}(2 a+2 b x)}{8 b^2}-\frac {i x^3 \coth (2 a+2 b x) \text {csch}(2 a+2 b x)}{4 b}\right )\)

\(\Big \downarrow \) 3011

\(\displaystyle -8 i \left (\frac {1}{2} \left (-\frac {3 i \left (\frac {\int x \operatorname {PolyLog}\left (2,-e^{2 a+2 b x}\right )dx}{b}-\frac {x^2 \operatorname {PolyLog}\left (2,-e^{2 a+2 b x}\right )}{2 b}\right )}{2 b}+\frac {3 i \left (\frac {\int x \operatorname {PolyLog}\left (2,e^{2 a+2 b x}\right )dx}{b}-\frac {x^2 \operatorname {PolyLog}\left (2,e^{2 a+2 b x}\right )}{2 b}\right )}{2 b}+\frac {i x^3 \text {arctanh}\left (e^{2 a+2 b x}\right )}{b}\right )-\frac {3 \left (\frac {i x \text {arctanh}\left (e^{2 a+2 b x}\right )}{b}+\frac {i \operatorname {PolyLog}\left (2,-e^{2 a+2 b x}\right )}{4 b^2}-\frac {i \operatorname {PolyLog}\left (2,e^{2 a+2 b x}\right )}{4 b^2}\right )}{4 b^2}-\frac {3 i x^2 \text {csch}(2 a+2 b x)}{8 b^2}-\frac {i x^3 \coth (2 a+2 b x) \text {csch}(2 a+2 b x)}{4 b}\right )\)

\(\Big \downarrow \) 7163

\(\displaystyle -8 i \left (\frac {1}{2} \left (-\frac {3 i \left (\frac {\frac {x \operatorname {PolyLog}\left (3,-e^{2 a+2 b x}\right )}{2 b}-\frac {\int \operatorname {PolyLog}\left (3,-e^{2 a+2 b x}\right )dx}{2 b}}{b}-\frac {x^2 \operatorname {PolyLog}\left (2,-e^{2 a+2 b x}\right )}{2 b}\right )}{2 b}+\frac {3 i \left (\frac {\frac {x \operatorname {PolyLog}\left (3,e^{2 a+2 b x}\right )}{2 b}-\frac {\int \operatorname {PolyLog}\left (3,e^{2 a+2 b x}\right )dx}{2 b}}{b}-\frac {x^2 \operatorname {PolyLog}\left (2,e^{2 a+2 b x}\right )}{2 b}\right )}{2 b}+\frac {i x^3 \text {arctanh}\left (e^{2 a+2 b x}\right )}{b}\right )-\frac {3 \left (\frac {i x \text {arctanh}\left (e^{2 a+2 b x}\right )}{b}+\frac {i \operatorname {PolyLog}\left (2,-e^{2 a+2 b x}\right )}{4 b^2}-\frac {i \operatorname {PolyLog}\left (2,e^{2 a+2 b x}\right )}{4 b^2}\right )}{4 b^2}-\frac {3 i x^2 \text {csch}(2 a+2 b x)}{8 b^2}-\frac {i x^3 \coth (2 a+2 b x) \text {csch}(2 a+2 b x)}{4 b}\right )\)

\(\Big \downarrow \) 2720

\(\displaystyle -8 i \left (\frac {1}{2} \left (-\frac {3 i \left (\frac {\frac {x \operatorname {PolyLog}\left (3,-e^{2 a+2 b x}\right )}{2 b}-\frac {\int e^{-2 a-2 b x} \operatorname {PolyLog}\left (3,-e^{2 a+2 b x}\right )de^{2 a+2 b x}}{4 b^2}}{b}-\frac {x^2 \operatorname {PolyLog}\left (2,-e^{2 a+2 b x}\right )}{2 b}\right )}{2 b}+\frac {3 i \left (\frac {\frac {x \operatorname {PolyLog}\left (3,e^{2 a+2 b x}\right )}{2 b}-\frac {\int e^{-2 a-2 b x} \operatorname {PolyLog}\left (3,e^{2 a+2 b x}\right )de^{2 a+2 b x}}{4 b^2}}{b}-\frac {x^2 \operatorname {PolyLog}\left (2,e^{2 a+2 b x}\right )}{2 b}\right )}{2 b}+\frac {i x^3 \text {arctanh}\left (e^{2 a+2 b x}\right )}{b}\right )-\frac {3 \left (\frac {i x \text {arctanh}\left (e^{2 a+2 b x}\right )}{b}+\frac {i \operatorname {PolyLog}\left (2,-e^{2 a+2 b x}\right )}{4 b^2}-\frac {i \operatorname {PolyLog}\left (2,e^{2 a+2 b x}\right )}{4 b^2}\right )}{4 b^2}-\frac {3 i x^2 \text {csch}(2 a+2 b x)}{8 b^2}-\frac {i x^3 \coth (2 a+2 b x) \text {csch}(2 a+2 b x)}{4 b}\right )\)

\(\Big \downarrow \) 7143

\(\displaystyle -8 i \left (\frac {1}{2} \left (\frac {i x^3 \text {arctanh}\left (e^{2 a+2 b x}\right )}{b}-\frac {3 i \left (\frac {\frac {x \operatorname {PolyLog}\left (3,-e^{2 a+2 b x}\right )}{2 b}-\frac {\operatorname {PolyLog}\left (4,-e^{2 a+2 b x}\right )}{4 b^2}}{b}-\frac {x^2 \operatorname {PolyLog}\left (2,-e^{2 a+2 b x}\right )}{2 b}\right )}{2 b}+\frac {3 i \left (\frac {\frac {x \operatorname {PolyLog}\left (3,e^{2 a+2 b x}\right )}{2 b}-\frac {\operatorname {PolyLog}\left (4,e^{2 a+2 b x}\right )}{4 b^2}}{b}-\frac {x^2 \operatorname {PolyLog}\left (2,e^{2 a+2 b x}\right )}{2 b}\right )}{2 b}\right )-\frac {3 \left (\frac {i x \text {arctanh}\left (e^{2 a+2 b x}\right )}{b}+\frac {i \operatorname {PolyLog}\left (2,-e^{2 a+2 b x}\right )}{4 b^2}-\frac {i \operatorname {PolyLog}\left (2,e^{2 a+2 b x}\right )}{4 b^2}\right )}{4 b^2}-\frac {3 i x^2 \text {csch}(2 a+2 b x)}{8 b^2}-\frac {i x^3 \coth (2 a+2 b x) \text {csch}(2 a+2 b x)}{4 b}\right )\)

Input:

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

Output:

(-8*I)*((((-3*I)/8)*x^2*Csch[2*a + 2*b*x])/b^2 - ((I/4)*x^3*Coth[2*a + 2*b 
*x]*Csch[2*a + 2*b*x])/b - (3*((I*x*ArcTanh[E^(2*a + 2*b*x)])/b + ((I/4)*P 
olyLog[2, -E^(2*a + 2*b*x)])/b^2 - ((I/4)*PolyLog[2, E^(2*a + 2*b*x)])/b^2 
))/(4*b^2) + ((I*x^3*ArcTanh[E^(2*a + 2*b*x)])/b - (((3*I)/2)*(-1/2*(x^2*P 
olyLog[2, -E^(2*a + 2*b*x)])/b + ((x*PolyLog[3, -E^(2*a + 2*b*x)])/(2*b) - 
 PolyLog[4, -E^(2*a + 2*b*x)]/(4*b^2))/b))/b + (((3*I)/2)*(-1/2*(x^2*PolyL 
og[2, E^(2*a + 2*b*x)])/b + ((x*PolyLog[3, E^(2*a + 2*b*x)])/(2*b) - PolyL 
og[4, E^(2*a + 2*b*x)]/(4*b^2))/b))/b)/2)
 

Defintions of rubi rules used

rule 26

Int[(Complex[0, a_])*(Fx_), x_Symbol] :> Simp[(Complex[Identity[0], a])   I 
nt[Fx, x], x] /; FreeQ[a, x] && EqQ[a^2, 1]
 

rule 2715

Int[Log[(a_) + (b_.)*((F_)^((e_.)*((c_.) + (d_.)*(x_))))^(n_.)], x_Symbol] 
:> Simp[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 2720

Int[u_, x_Symbol] :> With[{v = FunctionOfExponential[u, x]}, Simp[v/D[v, x] 
   Subst[Int[FunctionOfExponentialFunction[u, x]/x, x], x, v], x]] /; Funct 
ionOfExponentialQ[u, x] &&  !MatchQ[u, (w_)*((a_.)*(v_)^(n_))^(m_) /; FreeQ 
[{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 2838

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 3011

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] + Simp[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 3042

Int[u_, x_Symbol] :> Int[DeactivateTrig[u, x], x] /; FunctionOfTrigOfLinear 
Q[u, x]
 

rule 4670

Int[csc[(e_.) + (Complex[0, fz_])*(f_.)*(x_)]*((c_.) + (d_.)*(x_))^(m_.), x 
_Symbol] :> Simp[-2*(c + d*x)^m*(ArcTanh[E^((-I)*e + f*fz*x)]/(f*fz*I)), x] 
 + (-Simp[d*(m/(f*fz*I))   Int[(c + d*x)^(m - 1)*Log[1 - E^((-I)*e + f*fz*x 
)], x], x] + Simp[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 4674

Int[(csc[(e_.) + (f_.)*(x_)]*(b_.))^(n_)*((c_.) + (d_.)*(x_))^(m_), x_Symbo 
l] :> Simp[(-b^2)*(c + d*x)^m*Cot[e + f*x]*((b*Csc[e + f*x])^(n - 2)/(f*(n 
- 1))), x] + (-Simp[b^2*d*m*(c + d*x)^(m - 1)*((b*Csc[e + f*x])^(n - 2)/(f^ 
2*(n - 1)*(n - 2))), x] + Simp[b^2*d^2*m*((m - 1)/(f^2*(n - 1)*(n - 2))) 
Int[(c + d*x)^(m - 2)*(b*Csc[e + f*x])^(n - 2), x], x] + Simp[b^2*((n - 2)/ 
(n - 1))   Int[(c + d*x)^m*(b*Csc[e + f*x])^(n - 2), x], x]) /; FreeQ[{b, c 
, d, e, f}, x] && GtQ[n, 1] && NeQ[n, 2] && GtQ[m, 1]
 

rule 5984

Int[Csch[(a_.) + (b_.)*(x_)]^(n_.)*((c_.) + (d_.)*(x_))^(m_.)*Sech[(a_.) + 
(b_.)*(x_)]^(n_.), x_Symbol] :> Simp[2^n   Int[(c + d*x)^m*Csch[2*a + 2*b*x 
]^n, x], x] /; FreeQ[{a, b, c, d}, x] && RationalQ[m] && IntegerQ[n]
 

rule 7143

Int[PolyLog[n_, (c_.)*((a_.) + (b_.)*(x_))^(p_.)]/((d_.) + (e_.)*(x_)), x_S 
ymbol] :> 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 7163

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] - Simp[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]
 

Maple [A] (verified)

Time = 27.36 (sec) , antiderivative size = 445, normalized size of antiderivative = 1.85

method	result	size
risch	\(-\frac {2 x^{2} {\mathrm e}^{2 b x +2 a} \left (2 b x \,{\mathrm e}^{4 b x +4 a}+3 \,{\mathrm e}^{4 b x +4 a}+2 b x -3\right )}{b^{2} \left ({\mathrm e}^{2 b x +2 a}-1\right )^{2} \left ({\mathrm e}^{2 b x +2 a}+1\right )^{2}}-\frac {12 \operatorname {polylog}\left (4, {\mathrm e}^{b x +a}\right )}{b^{4}}+\frac {3 \operatorname {polylog}\left (4, -{\mathrm e}^{2 b x +2 a}\right )}{2 b^{4}}-\frac {12 \operatorname {polylog}\left (4, -{\mathrm e}^{b x +a}\right )}{b^{4}}+\frac {3 \operatorname {polylog}\left (2, {\mathrm e}^{b x +a}\right )}{b^{4}}-\frac {3 \operatorname {polylog}\left (2, -{\mathrm e}^{2 b x +2 a}\right )}{2 b^{4}}+\frac {3 \operatorname {polylog}\left (2, -{\mathrm e}^{b x +a}\right )}{b^{4}}-\frac {3 a \ln \left ({\mathrm e}^{b x +a}-1\right )}{b^{4}}+\frac {2 a^{3} \ln \left ({\mathrm e}^{b x +a}-1\right )}{b^{4}}-\frac {2 \ln \left (1-{\mathrm e}^{b x +a}\right ) a^{3}}{b^{4}}+\frac {3 \ln \left (1-{\mathrm e}^{b x +a}\right ) a}{b^{4}}+\frac {3 \ln \left (1-{\mathrm e}^{b x +a}\right ) x}{b^{3}}-\frac {2 \ln \left (1-{\mathrm e}^{b x +a}\right ) x^{3}}{b}+\frac {3 \ln \left ({\mathrm e}^{b x +a}+1\right ) x}{b^{3}}-\frac {3 x \ln \left ({\mathrm e}^{2 b x +2 a}+1\right )}{b^{3}}-\frac {6 x^{2} \operatorname {polylog}\left (2, {\mathrm e}^{b x +a}\right )}{b^{2}}+\frac {12 x \operatorname {polylog}\left (3, {\mathrm e}^{b x +a}\right )}{b^{3}}+\frac {2 x^{3} \ln \left ({\mathrm e}^{2 b x +2 a}+1\right )}{b}+\frac {3 x^{2} \operatorname {polylog}\left (2, -{\mathrm e}^{2 b x +2 a}\right )}{b^{2}}-\frac {3 x \operatorname {polylog}\left (3, -{\mathrm e}^{2 b x +2 a}\right )}{b^{3}}-\frac {2 \ln \left ({\mathrm e}^{b x +a}+1\right ) x^{3}}{b}-\frac {6 x^{2} \operatorname {polylog}\left (2, -{\mathrm e}^{b x +a}\right )}{b^{2}}+\frac {12 x \operatorname {polylog}\left (3, -{\mathrm e}^{b x +a}\right )}{b^{3}}\)	\(445\)

Input:

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

Output:

-2*x^2*exp(2*b*x+2*a)*(2*b*x*exp(4*b*x+4*a)+3*exp(4*b*x+4*a)+2*b*x-3)/b^2/ 
(exp(2*b*x+2*a)-1)^2/(exp(2*b*x+2*a)+1)^2-12*polylog(4,exp(b*x+a))/b^4+3/2 
*polylog(4,-exp(2*b*x+2*a))/b^4-12*polylog(4,-exp(b*x+a))/b^4+3*polylog(2, 
exp(b*x+a))/b^4-3/2*polylog(2,-exp(2*b*x+2*a))/b^4+3*polylog(2,-exp(b*x+a) 
)/b^4-3/b^4*a*ln(exp(b*x+a)-1)+2/b^4*a^3*ln(exp(b*x+a)-1)-2/b^4*ln(1-exp(b 
*x+a))*a^3+3/b^4*ln(1-exp(b*x+a))*a+3/b^3*ln(1-exp(b*x+a))*x-2/b*ln(1-exp( 
b*x+a))*x^3+3/b^3*ln(exp(b*x+a)+1)*x-3*x*ln(exp(2*b*x+2*a)+1)/b^3-6*x^2*po 
lylog(2,exp(b*x+a))/b^2+12*x*polylog(3,exp(b*x+a))/b^3+2*x^3*ln(exp(2*b*x+ 
2*a)+1)/b+3*x^2*polylog(2,-exp(2*b*x+2*a))/b^2-3*x*polylog(3,-exp(2*b*x+2* 
a))/b^3-2/b*ln(exp(b*x+a)+1)*x^3-6*x^2*polylog(2,-exp(b*x+a))/b^2+12*x*pol 
ylog(3,-exp(b*x+a))/b^3
 

Fricas [C] (verification not implemented)

Result contains complex when optimal does not.

Time = 0.17 (sec) , antiderivative size = 6764, normalized size of antiderivative = 28.18 \[ \int x^3 \text {csch}^3(a+b x) \text {sech}^3(a+b x) \, dx=\text {Too large to display} \] Input:

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

Output:

Too large to include
 

Sympy [F]

\[ \int x^3 \text {csch}^3(a+b x) \text {sech}^3(a+b x) \, dx=\int x^{3} \operatorname {csch}^{3}{\left (a + b x \right )} \operatorname {sech}^{3}{\left (a + b x \right )}\, dx \] Input:

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

Output:

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

Maxima [A] (verification not implemented)

Time = 0.07 (sec) , antiderivative size = 381, normalized size of antiderivative = 1.59 \[ \int x^3 \text {csch}^3(a+b x) \text {sech}^3(a+b x) \, dx=-\frac {2 \, {\left ({\left (2 \, b x^{3} e^{\left (6 \, a\right )} + 3 \, x^{2} e^{\left (6 \, a\right )}\right )} e^{\left (6 \, b x\right )} + {\left (2 \, b x^{3} e^{\left (2 \, a\right )} - 3 \, x^{2} e^{\left (2 \, a\right )}\right )} e^{\left (2 \, b x\right )}\right )}}{b^{2} e^{\left (8 \, b x + 8 \, a\right )} - 2 \, b^{2} e^{\left (4 \, b x + 4 \, a\right )} + b^{2}} + \frac {2 \, {\left (4 \, b^{3} x^{3} \log \left (e^{\left (2 \, b x + 2 \, a\right )} + 1\right ) + 6 \, b^{2} x^{2} {\rm Li}_2\left (-e^{\left (2 \, b x + 2 \, a\right )}\right ) - 6 \, b x {\rm Li}_{3}(-e^{\left (2 \, b x + 2 \, a\right )}) + 3 \, {\rm Li}_{4}(-e^{\left (2 \, b x + 2 \, a\right )})\right )}}{3 \, b^{4}} - \frac {2 \, {\left (b^{3} x^{3} \log \left (e^{\left (b x + a\right )} + 1\right ) + 3 \, b^{2} x^{2} {\rm Li}_2\left (-e^{\left (b x + a\right )}\right ) - 6 \, b x {\rm Li}_{3}(-e^{\left (b x + a\right )}) + 6 \, {\rm Li}_{4}(-e^{\left (b x + a\right )})\right )}}{b^{4}} - \frac {2 \, {\left (b^{3} x^{3} \log \left (-e^{\left (b x + a\right )} + 1\right ) + 3 \, b^{2} x^{2} {\rm Li}_2\left (e^{\left (b x + a\right )}\right ) - 6 \, b x {\rm Li}_{3}(e^{\left (b x + a\right )}) + 6 \, {\rm Li}_{4}(e^{\left (b x + a\right )})\right )}}{b^{4}} - \frac {3 \, {\left (2 \, b x \log \left (e^{\left (2 \, b x + 2 \, a\right )} + 1\right ) + {\rm Li}_2\left (-e^{\left (2 \, b x + 2 \, a\right )}\right )\right )}}{2 \, b^{4}} + \frac {3 \, {\left (b x \log \left (e^{\left (b x + a\right )} + 1\right ) + {\rm Li}_2\left (-e^{\left (b x + a\right )}\right )\right )}}{b^{4}} + \frac {3 \, {\left (b x \log \left (-e^{\left (b x + a\right )} + 1\right ) + {\rm Li}_2\left (e^{\left (b x + a\right )}\right )\right )}}{b^{4}} \] Input:

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

Output:

-2*((2*b*x^3*e^(6*a) + 3*x^2*e^(6*a))*e^(6*b*x) + (2*b*x^3*e^(2*a) - 3*x^2 
*e^(2*a))*e^(2*b*x))/(b^2*e^(8*b*x + 8*a) - 2*b^2*e^(4*b*x + 4*a) + b^2) + 
 2/3*(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 - 2*(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 - 2*(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 + 3*(b*x*log(e^(b*x + a) + 1) + d 
ilog(-e^(b*x + a)))/b^4 + 3*(b*x*log(-e^(b*x + a) + 1) + dilog(e^(b*x + a) 
))/b^4
 

Giac [F]

\[ \int x^3 \text {csch}^3(a+b x) \text {sech}^3(a+b x) \, dx=\int { x^{3} \operatorname {csch}\left (b x + a\right )^{3} \operatorname {sech}\left (b x + a\right )^{3} \,d x } \] Input:

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

Output:

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

Mupad [F(-1)]

Timed out. \[ \int x^3 \text {csch}^3(a+b x) \text {sech}^3(a+b x) \, dx=\int \frac {x^3}{{\mathrm {cosh}\left (a+b\,x\right )}^3\,{\mathrm {sinh}\left (a+b\,x\right )}^3} \,d x \] Input:

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

Output:

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

Reduce [F]

\[ \int x^3 \text {csch}^3(a+b x) \text {sech}^3(a+b x) \, dx=\frac {-\frac {64 e^{8 b x +10 a} \left (\int \frac {e^{2 b x} x^{3}}{e^{12 b x +12 a}-3 e^{8 b x +8 a}+3 e^{4 b x +4 a}-1}d x \right ) b^{4}}{3}-\frac {128 e^{8 b x +10 a} \left (\int \frac {e^{2 b x} x^{2}}{e^{12 b x +12 a}-3 e^{8 b x +8 a}+3 e^{4 b x +4 a}-1}d x \right ) b^{3}}{3}-\frac {128 e^{8 b x +10 a} \left (\int \frac {e^{2 b x} x}{e^{12 b x +12 a}-3 e^{8 b x +8 a}+3 e^{4 b x +4 a}-1}d x \right ) b^{2}}{9}+\frac {2 e^{8 b x +8 a} \mathrm {log}\left (e^{2 b x +2 a}+1\right )}{9}-\frac {2 e^{8 b x +8 a} \mathrm {log}\left (e^{b x +a}-1\right )}{9}-\frac {2 e^{8 b x +8 a} \mathrm {log}\left (e^{b x +a}+1\right )}{9}-\frac {4 e^{6 b x +6 a}}{9}+\frac {128 e^{4 b x +6 a} \left (\int \frac {e^{2 b x} x^{3}}{e^{12 b x +12 a}-3 e^{8 b x +8 a}+3 e^{4 b x +4 a}-1}d x \right ) b^{4}}{3}+\frac {256 e^{4 b x +6 a} \left (\int \frac {e^{2 b x} x^{2}}{e^{12 b x +12 a}-3 e^{8 b x +8 a}+3 e^{4 b x +4 a}-1}d x \right ) b^{3}}{3}+\frac {256 e^{4 b x +6 a} \left (\int \frac {e^{2 b x} x}{e^{12 b x +12 a}-3 e^{8 b x +8 a}+3 e^{4 b x +4 a}-1}d x \right ) b^{2}}{9}-\frac {4 e^{4 b x +4 a} \mathrm {log}\left (e^{2 b x +2 a}+1\right )}{9}+\frac {4 e^{4 b x +4 a} \mathrm {log}\left (e^{b x +a}-1\right )}{9}+\frac {4 e^{4 b x +4 a} \mathrm {log}\left (e^{b x +a}+1\right )}{9}-\frac {32 e^{2 b x +2 a} b^{3} x^{3}}{3}-\frac {16 e^{2 b x +2 a} b^{2} x^{2}}{3}-\frac {16 e^{2 b x +2 a} b x}{9}+\frac {4 e^{2 b x +2 a}}{9}-\frac {64 e^{2 a} \left (\int \frac {e^{2 b x} x^{3}}{e^{12 b x +12 a}-3 e^{8 b x +8 a}+3 e^{4 b x +4 a}-1}d x \right ) b^{4}}{3}-\frac {128 e^{2 a} \left (\int \frac {e^{2 b x} x^{2}}{e^{12 b x +12 a}-3 e^{8 b x +8 a}+3 e^{4 b x +4 a}-1}d x \right ) b^{3}}{3}-\frac {128 e^{2 a} \left (\int \frac {e^{2 b x} x}{e^{12 b x +12 a}-3 e^{8 b x +8 a}+3 e^{4 b x +4 a}-1}d x \right ) b^{2}}{9}+\frac {2 \,\mathrm {log}\left (e^{2 b x +2 a}+1\right )}{9}-\frac {2 \,\mathrm {log}\left (e^{b x +a}-1\right )}{9}-\frac {2 \,\mathrm {log}\left (e^{b x +a}+1\right )}{9}}{b^{4} \left (e^{8 b x +8 a}-2 e^{4 b x +4 a}+1\right )} \] Input:

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

Output:

(2*( - 96*e**(10*a + 8*b*x)*int((e**(2*b*x)*x**3)/(e**(12*a + 12*b*x) - 3* 
e**(8*a + 8*b*x) + 3*e**(4*a + 4*b*x) - 1),x)*b**4 - 192*e**(10*a + 8*b*x) 
*int((e**(2*b*x)*x**2)/(e**(12*a + 12*b*x) - 3*e**(8*a + 8*b*x) + 3*e**(4* 
a + 4*b*x) - 1),x)*b**3 - 64*e**(10*a + 8*b*x)*int((e**(2*b*x)*x)/(e**(12* 
a + 12*b*x) - 3*e**(8*a + 8*b*x) + 3*e**(4*a + 4*b*x) - 1),x)*b**2 + e**(8 
*a + 8*b*x)*log(e**(2*a + 2*b*x) + 1) - e**(8*a + 8*b*x)*log(e**(a + b*x) 
- 1) - e**(8*a + 8*b*x)*log(e**(a + b*x) + 1) - 2*e**(6*a + 6*b*x) + 192*e 
**(6*a + 4*b*x)*int((e**(2*b*x)*x**3)/(e**(12*a + 12*b*x) - 3*e**(8*a + 8* 
b*x) + 3*e**(4*a + 4*b*x) - 1),x)*b**4 + 384*e**(6*a + 4*b*x)*int((e**(2*b 
*x)*x**2)/(e**(12*a + 12*b*x) - 3*e**(8*a + 8*b*x) + 3*e**(4*a + 4*b*x) - 
1),x)*b**3 + 128*e**(6*a + 4*b*x)*int((e**(2*b*x)*x)/(e**(12*a + 12*b*x) - 
 3*e**(8*a + 8*b*x) + 3*e**(4*a + 4*b*x) - 1),x)*b**2 - 2*e**(4*a + 4*b*x) 
*log(e**(2*a + 2*b*x) + 1) + 2*e**(4*a + 4*b*x)*log(e**(a + b*x) - 1) + 2* 
e**(4*a + 4*b*x)*log(e**(a + b*x) + 1) - 48*e**(2*a + 2*b*x)*b**3*x**3 - 2 
4*e**(2*a + 2*b*x)*b**2*x**2 - 8*e**(2*a + 2*b*x)*b*x + 2*e**(2*a + 2*b*x) 
 - 96*e**(2*a)*int((e**(2*b*x)*x**3)/(e**(12*a + 12*b*x) - 3*e**(8*a + 8*b 
*x) + 3*e**(4*a + 4*b*x) - 1),x)*b**4 - 192*e**(2*a)*int((e**(2*b*x)*x**2) 
/(e**(12*a + 12*b*x) - 3*e**(8*a + 8*b*x) + 3*e**(4*a + 4*b*x) - 1),x)*b** 
3 - 64*e**(2*a)*int((e**(2*b*x)*x)/(e**(12*a + 12*b*x) - 3*e**(8*a + 8*b*x 
) + 3*e**(4*a + 4*b*x) - 1),x)*b**2 + log(e**(2*a + 2*b*x) + 1) - log(e...