\(\int x^3 \cosh ^2(a+b x) \coth (a+b x) \, dx\) [149]

Optimal result

Integrand size = 18, antiderivative size = 180 \[ \int x^3 \cosh ^2(a+b x) \coth (a+b x) \, dx=\frac {3 x}{8 b^3}+\frac {x^3}{4 b}-\frac {x^4}{4}+\frac {x^3 \log \left (1-e^{2 (a+b x)}\right )}{b}+\frac {3 x^2 \operatorname {PolyLog}\left (2,e^{2 (a+b x)}\right )}{2 b^2}-\frac {3 x \operatorname {PolyLog}\left (3,e^{2 (a+b x)}\right )}{2 b^3}+\frac {3 \operatorname {PolyLog}\left (4,e^{2 (a+b x)}\right )}{4 b^4}-\frac {3 \cosh (a+b x) \sinh (a+b x)}{8 b^4}-\frac {3 x^2 \cosh (a+b x) \sinh (a+b x)}{4 b^2}+\frac {3 x \sinh ^2(a+b x)}{4 b^3}+\frac {x^3 \sinh ^2(a+b x)}{2 b} \] Output:

3/8*x/b^3+1/4*x^3/b-1/4*x^4+x^3*ln(1-exp(2*b*x+2*a))/b+3/2*x^2*polylog(2,e 
xp(2*b*x+2*a))/b^2-3/2*x*polylog(3,exp(2*b*x+2*a))/b^3+3/4*polylog(4,exp(2 
*b*x+2*a))/b^4-3/8*cosh(b*x+a)*sinh(b*x+a)/b^4-3/4*x^2*cosh(b*x+a)*sinh(b* 
x+a)/b^2+3/4*x*sinh(b*x+a)^2/b^3+1/2*x^3*sinh(b*x+a)^2/b
 

Mathematica [A] (verified)

Time = 0.45 (sec) , antiderivative size = 236, normalized size of antiderivative = 1.31 \[ \int x^3 \cosh ^2(a+b x) \coth (a+b x) \, dx=\frac {\sinh (a) (\cosh (a)+\sinh (a)) \left (4 b^4 x^4+6 b x \cosh (2 (a+b x))+4 b^3 x^3 \cosh (2 (a+b x))+16 b^3 x^3 \log \left (1-e^{-a-b x}\right )+16 b^3 x^3 \log \left (1+e^{-a-b x}\right )-48 b^2 x^2 \operatorname {PolyLog}\left (2,-e^{-a-b x}\right )-48 b^2 x^2 \operatorname {PolyLog}\left (2,e^{-a-b x}\right )-96 b x \operatorname {PolyLog}\left (3,-e^{-a-b x}\right )-96 b x \operatorname {PolyLog}\left (3,e^{-a-b x}\right )-96 \operatorname {PolyLog}\left (4,-e^{-a-b x}\right )-96 \operatorname {PolyLog}\left (4,e^{-a-b x}\right )-3 \sinh (2 (a+b x))-6 b^2 x^2 \sinh (2 (a+b x))\right )}{8 b^4 \left (-1+e^{2 a}\right )} \] Input:

Integrate[x^3*Cosh[a + b*x]^2*Coth[a + b*x],x]
 

Output:

(Sinh[a]*(Cosh[a] + Sinh[a])*(4*b^4*x^4 + 6*b*x*Cosh[2*(a + b*x)] + 4*b^3* 
x^3*Cosh[2*(a + b*x)] + 16*b^3*x^3*Log[1 - E^(-a - b*x)] + 16*b^3*x^3*Log[ 
1 + E^(-a - b*x)] - 48*b^2*x^2*PolyLog[2, -E^(-a - b*x)] - 48*b^2*x^2*Poly 
Log[2, E^(-a - b*x)] - 96*b*x*PolyLog[3, -E^(-a - b*x)] - 96*b*x*PolyLog[3 
, E^(-a - b*x)] - 96*PolyLog[4, -E^(-a - b*x)] - 96*PolyLog[4, E^(-a - b*x 
)] - 3*Sinh[2*(a + b*x)] - 6*b^2*x^2*Sinh[2*(a + b*x)]))/(8*b^4*(-1 + E^(2 
*a)))
 

Rubi [C] (verified)

Result contains complex when optimal does not.

Time = 1.82 (sec) , antiderivative size = 246, normalized size of antiderivative = 1.37, number of steps used = 20, number of rules used = 19, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 1.056, Rules used = {5973, 3042, 26, 4201, 2620, 3011, 5895, 3042, 25, 3792, 15, 25, 3042, 25, 3115, 24, 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.

\(\displaystyle \int x^3 \cosh ^2(a+b x) \coth (a+b x) \, dx\)

\(\Big \downarrow \) 5973

\(\displaystyle \int x^3 \coth (a+b x)dx+\int x^3 \cosh (a+b x) \sinh (a+b x)dx\)

\(\Big \downarrow \) 3042

\(\displaystyle \int x^3 \cosh (a+b x) \sinh (a+b x)dx+\int -i x^3 \tan \left (i a+i b x+\frac {\pi }{2}\right )dx\)

\(\Big \downarrow \) 26

\(\displaystyle \int x^3 \cosh (a+b x) \sinh (a+b x)dx-i \int x^3 \tan \left (\frac {1}{2} (2 i a+\pi )+i b x\right )dx\)

\(\Big \downarrow \) 4201

\(\displaystyle \int x^3 \cosh (a+b x) \sinh (a+b x)dx-i \left (2 i \int \frac {e^{2 a+2 b x-i \pi } x^3}{1+e^{2 a+2 b x-i \pi }}dx-\frac {i x^4}{4}\right )\)

\(\Big \downarrow \) 2620

\(\displaystyle \int x^3 \cosh (a+b x) \sinh (a+b x)dx-i \left (2 i \left (\frac {x^3 \log \left (1+e^{2 a+2 b x-i \pi }\right )}{2 b}-\frac {3 \int x^2 \log \left (1+e^{2 a+2 b x-i \pi }\right )dx}{2 b}\right )-\frac {i x^4}{4}\right )\)

\(\Big \downarrow \) 3011

\(\displaystyle \int x^3 \cosh (a+b x) \sinh (a+b x)dx-i \left (2 i \left (\frac {x^3 \log \left (1+e^{2 a+2 b x-i \pi }\right )}{2 b}-\frac {3 \left (\frac {\int x \operatorname {PolyLog}\left (2,-e^{2 a+2 b x-i \pi }\right )dx}{b}-\frac {x^2 \operatorname {PolyLog}\left (2,-e^{2 a+2 b x-i \pi }\right )}{2 b}\right )}{2 b}\right )-\frac {i x^4}{4}\right )\)

\(\Big \downarrow \) 5895

\(\displaystyle -i \left (2 i \left (\frac {x^3 \log \left (1+e^{2 a+2 b x-i \pi }\right )}{2 b}-\frac {3 \left (\frac {\int x \operatorname {PolyLog}\left (2,-e^{2 a+2 b x-i \pi }\right )dx}{b}-\frac {x^2 \operatorname {PolyLog}\left (2,-e^{2 a+2 b x-i \pi }\right )}{2 b}\right )}{2 b}\right )-\frac {i x^4}{4}\right )-\frac {3 \int x^2 \sinh ^2(a+b x)dx}{2 b}+\frac {x^3 \sinh ^2(a+b x)}{2 b}\)

\(\Big \downarrow \) 3042

\(\displaystyle -i \left (2 i \left (\frac {x^3 \log \left (1+e^{2 a+2 b x-i \pi }\right )}{2 b}-\frac {3 \left (\frac {\int x \operatorname {PolyLog}\left (2,-e^{2 a+2 b x-i \pi }\right )dx}{b}-\frac {x^2 \operatorname {PolyLog}\left (2,-e^{2 a+2 b x-i \pi }\right )}{2 b}\right )}{2 b}\right )-\frac {i x^4}{4}\right )-\frac {3 \int -x^2 \sin (i a+i b x)^2dx}{2 b}+\frac {x^3 \sinh ^2(a+b x)}{2 b}\)

\(\Big \downarrow \) 25

\(\displaystyle -i \left (2 i \left (\frac {x^3 \log \left (1+e^{2 a+2 b x-i \pi }\right )}{2 b}-\frac {3 \left (\frac {\int x \operatorname {PolyLog}\left (2,-e^{2 a+2 b x-i \pi }\right )dx}{b}-\frac {x^2 \operatorname {PolyLog}\left (2,-e^{2 a+2 b x-i \pi }\right )}{2 b}\right )}{2 b}\right )-\frac {i x^4}{4}\right )+\frac {3 \int x^2 \sin (i a+i b x)^2dx}{2 b}+\frac {x^3 \sinh ^2(a+b x)}{2 b}\)

\(\Big \downarrow \) 3792

\(\displaystyle \frac {3 \left (\frac {\int -\sinh ^2(a+b x)dx}{2 b^2}+\frac {\int x^2dx}{2}+\frac {x \sinh ^2(a+b x)}{2 b^2}-\frac {x^2 \sinh (a+b x) \cosh (a+b x)}{2 b}\right )}{2 b}-i \left (2 i \left (\frac {x^3 \log \left (1+e^{2 a+2 b x-i \pi }\right )}{2 b}-\frac {3 \left (\frac {\int x \operatorname {PolyLog}\left (2,-e^{2 a+2 b x-i \pi }\right )dx}{b}-\frac {x^2 \operatorname {PolyLog}\left (2,-e^{2 a+2 b x-i \pi }\right )}{2 b}\right )}{2 b}\right )-\frac {i x^4}{4}\right )+\frac {x^3 \sinh ^2(a+b x)}{2 b}\)

\(\Big \downarrow \) 15

\(\displaystyle \frac {3 \left (\frac {\int -\sinh ^2(a+b x)dx}{2 b^2}+\frac {x \sinh ^2(a+b x)}{2 b^2}-\frac {x^2 \sinh (a+b x) \cosh (a+b x)}{2 b}+\frac {x^3}{6}\right )}{2 b}-i \left (2 i \left (\frac {x^3 \log \left (1+e^{2 a+2 b x-i \pi }\right )}{2 b}-\frac {3 \left (\frac {\int x \operatorname {PolyLog}\left (2,-e^{2 a+2 b x-i \pi }\right )dx}{b}-\frac {x^2 \operatorname {PolyLog}\left (2,-e^{2 a+2 b x-i \pi }\right )}{2 b}\right )}{2 b}\right )-\frac {i x^4}{4}\right )+\frac {x^3 \sinh ^2(a+b x)}{2 b}\)

\(\Big \downarrow \) 25

\(\displaystyle \frac {3 \left (-\frac {\int \sinh ^2(a+b x)dx}{2 b^2}+\frac {x \sinh ^2(a+b x)}{2 b^2}-\frac {x^2 \sinh (a+b x) \cosh (a+b x)}{2 b}+\frac {x^3}{6}\right )}{2 b}-i \left (2 i \left (\frac {x^3 \log \left (1+e^{2 a+2 b x-i \pi }\right )}{2 b}-\frac {3 \left (\frac {\int x \operatorname {PolyLog}\left (2,-e^{2 a+2 b x-i \pi }\right )dx}{b}-\frac {x^2 \operatorname {PolyLog}\left (2,-e^{2 a+2 b x-i \pi }\right )}{2 b}\right )}{2 b}\right )-\frac {i x^4}{4}\right )+\frac {x^3 \sinh ^2(a+b x)}{2 b}\)

\(\Big \downarrow \) 3042

\(\displaystyle \frac {3 \left (-\frac {\int -\sin (i a+i b x)^2dx}{2 b^2}+\frac {x \sinh ^2(a+b x)}{2 b^2}-\frac {x^2 \sinh (a+b x) \cosh (a+b x)}{2 b}+\frac {x^3}{6}\right )}{2 b}-i \left (2 i \left (\frac {x^3 \log \left (1+e^{2 a+2 b x-i \pi }\right )}{2 b}-\frac {3 \left (\frac {\int x \operatorname {PolyLog}\left (2,-e^{2 a+2 b x-i \pi }\right )dx}{b}-\frac {x^2 \operatorname {PolyLog}\left (2,-e^{2 a+2 b x-i \pi }\right )}{2 b}\right )}{2 b}\right )-\frac {i x^4}{4}\right )+\frac {x^3 \sinh ^2(a+b x)}{2 b}\)

\(\Big \downarrow \) 25

\(\displaystyle \frac {3 \left (\frac {\int \sin (i a+i b x)^2dx}{2 b^2}+\frac {x \sinh ^2(a+b x)}{2 b^2}-\frac {x^2 \sinh (a+b x) \cosh (a+b x)}{2 b}+\frac {x^3}{6}\right )}{2 b}-i \left (2 i \left (\frac {x^3 \log \left (1+e^{2 a+2 b x-i \pi }\right )}{2 b}-\frac {3 \left (\frac {\int x \operatorname {PolyLog}\left (2,-e^{2 a+2 b x-i \pi }\right )dx}{b}-\frac {x^2 \operatorname {PolyLog}\left (2,-e^{2 a+2 b x-i \pi }\right )}{2 b}\right )}{2 b}\right )-\frac {i x^4}{4}\right )+\frac {x^3 \sinh ^2(a+b x)}{2 b}\)

\(\Big \downarrow \) 3115

\(\displaystyle \frac {3 \left (\frac {\frac {\int 1dx}{2}-\frac {\sinh (a+b x) \cosh (a+b x)}{2 b}}{2 b^2}+\frac {x \sinh ^2(a+b x)}{2 b^2}-\frac {x^2 \sinh (a+b x) \cosh (a+b x)}{2 b}+\frac {x^3}{6}\right )}{2 b}-i \left (2 i \left (\frac {x^3 \log \left (1+e^{2 a+2 b x-i \pi }\right )}{2 b}-\frac {3 \left (\frac {\int x \operatorname {PolyLog}\left (2,-e^{2 a+2 b x-i \pi }\right )dx}{b}-\frac {x^2 \operatorname {PolyLog}\left (2,-e^{2 a+2 b x-i \pi }\right )}{2 b}\right )}{2 b}\right )-\frac {i x^4}{4}\right )+\frac {x^3 \sinh ^2(a+b x)}{2 b}\)

\(\Big \downarrow \) 24

\(\displaystyle -i \left (2 i \left (\frac {x^3 \log \left (1+e^{2 a+2 b x-i \pi }\right )}{2 b}-\frac {3 \left (\frac {\int x \operatorname {PolyLog}\left (2,-e^{2 a+2 b x-i \pi }\right )dx}{b}-\frac {x^2 \operatorname {PolyLog}\left (2,-e^{2 a+2 b x-i \pi }\right )}{2 b}\right )}{2 b}\right )-\frac {i x^4}{4}\right )+\frac {3 \left (\frac {x \sinh ^2(a+b x)}{2 b^2}+\frac {\frac {x}{2}-\frac {\sinh (a+b x) \cosh (a+b x)}{2 b}}{2 b^2}-\frac {x^2 \sinh (a+b x) \cosh (a+b x)}{2 b}+\frac {x^3}{6}\right )}{2 b}+\frac {x^3 \sinh ^2(a+b x)}{2 b}\)

\(\Big \downarrow \) 7163

\(\displaystyle -i \left (2 i \left (\frac {x^3 \log \left (1+e^{2 a+2 b x-i \pi }\right )}{2 b}-\frac {3 \left (\frac {\frac {x \operatorname {PolyLog}\left (3,-e^{2 a+2 b x-i \pi }\right )}{2 b}-\frac {\int \operatorname {PolyLog}\left (3,-e^{2 a+2 b x-i \pi }\right )dx}{2 b}}{b}-\frac {x^2 \operatorname {PolyLog}\left (2,-e^{2 a+2 b x-i \pi }\right )}{2 b}\right )}{2 b}\right )-\frac {i x^4}{4}\right )+\frac {3 \left (\frac {x \sinh ^2(a+b x)}{2 b^2}+\frac {\frac {x}{2}-\frac {\sinh (a+b x) \cosh (a+b x)}{2 b}}{2 b^2}-\frac {x^2 \sinh (a+b x) \cosh (a+b x)}{2 b}+\frac {x^3}{6}\right )}{2 b}+\frac {x^3 \sinh ^2(a+b x)}{2 b}\)

\(\Big \downarrow \) 2720

\(\displaystyle -i \left (2 i \left (\frac {x^3 \log \left (1+e^{2 a+2 b x-i \pi }\right )}{2 b}-\frac {3 \left (\frac {\frac {x \operatorname {PolyLog}\left (3,-e^{2 a+2 b x-i \pi }\right )}{2 b}-\frac {\int e^{-2 a-2 b x+i \pi } \operatorname {PolyLog}\left (3,-e^{2 a+2 b x-i \pi }\right )de^{2 a+2 b x-i \pi }}{4 b^2}}{b}-\frac {x^2 \operatorname {PolyLog}\left (2,-e^{2 a+2 b x-i \pi }\right )}{2 b}\right )}{2 b}\right )-\frac {i x^4}{4}\right )+\frac {3 \left (\frac {x \sinh ^2(a+b x)}{2 b^2}+\frac {\frac {x}{2}-\frac {\sinh (a+b x) \cosh (a+b x)}{2 b}}{2 b^2}-\frac {x^2 \sinh (a+b x) \cosh (a+b x)}{2 b}+\frac {x^3}{6}\right )}{2 b}+\frac {x^3 \sinh ^2(a+b x)}{2 b}\)

\(\Big \downarrow \) 7143

\(\displaystyle -i \left (2 i \left (\frac {x^3 \log \left (1+e^{2 a+2 b x-i \pi }\right )}{2 b}-\frac {3 \left (\frac {\frac {x \operatorname {PolyLog}\left (3,-e^{2 a+2 b x-i \pi }\right )}{2 b}-\frac {\operatorname {PolyLog}\left (4,-e^{2 a+2 b x-i \pi }\right )}{4 b^2}}{b}-\frac {x^2 \operatorname {PolyLog}\left (2,-e^{2 a+2 b x-i \pi }\right )}{2 b}\right )}{2 b}\right )-\frac {i x^4}{4}\right )+\frac {3 \left (\frac {x \sinh ^2(a+b x)}{2 b^2}+\frac {\frac {x}{2}-\frac {\sinh (a+b x) \cosh (a+b x)}{2 b}}{2 b^2}-\frac {x^2 \sinh (a+b x) \cosh (a+b x)}{2 b}+\frac {x^3}{6}\right )}{2 b}+\frac {x^3 \sinh ^2(a+b x)}{2 b}\)

Input:

Int[x^3*Cosh[a + b*x]^2*Coth[a + b*x],x]
 

Output:

(-I)*((-1/4*I)*x^4 + (2*I)*((x^3*Log[1 + E^(2*a - I*Pi + 2*b*x)])/(2*b) - 
(3*(-1/2*(x^2*PolyLog[2, -E^(2*a - I*Pi + 2*b*x)])/b + ((x*PolyLog[3, -E^( 
2*a - I*Pi + 2*b*x)])/(2*b) - PolyLog[4, -E^(2*a - I*Pi + 2*b*x)]/(4*b^2)) 
/b))/(2*b))) + (x^3*Sinh[a + b*x]^2)/(2*b) + (3*(x^3/6 - (x^2*Cosh[a + b*x 
]*Sinh[a + b*x])/(2*b) + (x*Sinh[a + b*x]^2)/(2*b^2) + (x/2 - (Cosh[a + b* 
x]*Sinh[a + b*x])/(2*b))/(2*b^2)))/(2*b)
 

Defintions of rubi rules used

rule 15

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

rule 24

Int[a_, x_Symbol] :> Simp[a*x, x] /; FreeQ[a, x]
 

rule 25

Int[-(Fx_), x_Symbol] :> Simp[Identity[-1]   Int[Fx, x], x]
 

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 2620

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] - Si 
mp[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 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 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 3115

Int[((b_.)*sin[(c_.) + (d_.)*(x_)])^(n_), x_Symbol] :> Simp[(-b)*Cos[c + d* 
x]*((b*Sin[c + d*x])^(n - 1)/(d*n)), x] + Simp[b^2*((n - 1)/n)   Int[(b*Sin 
[c + d*x])^(n - 2), x], x] /; FreeQ[{b, c, d}, x] && GtQ[n, 1] && IntegerQ[ 
2*n]
 

rule 3792

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

rule 4201

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] + Simp[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 5895

Int[Cosh[(a_.) + (b_.)*(x_)^(n_.)]*(x_)^(m_.)*Sinh[(a_.) + (b_.)*(x_)^(n_.) 
]^(p_.), x_Symbol] :> Simp[x^(m - n + 1)*(Sinh[a + b*x^n]^(p + 1)/(b*n*(p + 
 1))), x] - Simp[(m - n + 1)/(b*n*(p + 1))   Int[x^(m - n)*Sinh[a + b*x^n]^ 
(p + 1), x], x] /; FreeQ[{a, b, p}, x] && LtQ[0, n, m + 1] && NeQ[p, -1]
 

rule 5973

Int[Cosh[(a_.) + (b_.)*(x_)]^(n_.)*Coth[(a_.) + (b_.)*(x_)]^(p_.)*((c_.) + 
(d_.)*(x_))^(m_.), x_Symbol] :> Int[(c + d*x)^m*Cosh[a + b*x]^n*Coth[a + b* 
x]^(p - 2), x] + Int[(c + d*x)^m*Cosh[a + b*x]^(n - 2)*Coth[a + b*x]^p, x] 
/; FreeQ[{a, b, c, d, m}, x] && IGtQ[n, 0] && IGtQ[p, 0]
 

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 = 1.49 (sec) , antiderivative size = 272, normalized size of antiderivative = 1.51

method	result	size
risch	\(-\frac {x^{4}}{4}+\frac {\left (4 b^{3} x^{3}-6 b^{2} x^{2}+6 b x -3\right ) {\mathrm e}^{2 b x +2 a}}{32 b^{4}}+\frac {\left (4 b^{3} x^{3}+6 b^{2} x^{2}+6 b x +3\right ) {\mathrm e}^{-2 b x -2 a}}{32 b^{4}}-\frac {3 a^{4}}{2 b^{4}}+\frac {6 \operatorname {polylog}\left (4, -{\mathrm e}^{b x +a}\right )}{b^{4}}+\frac {\ln \left ({\mathrm e}^{b x +a}+1\right ) x^{3}}{b}+\frac {6 \operatorname {polylog}\left (4, {\mathrm e}^{b x +a}\right )}{b^{4}}+\frac {\ln \left (1-{\mathrm e}^{b x +a}\right ) a^{3}}{b^{4}}-\frac {a^{3} \ln \left ({\mathrm e}^{b x +a}-1\right )}{b^{4}}+\frac {2 a^{3} \ln \left ({\mathrm e}^{b x +a}\right )}{b^{4}}-\frac {2 a^{3} x}{b^{3}}+\frac {3 x^{2} \operatorname {polylog}\left (2, -{\mathrm e}^{b x +a}\right )}{b^{2}}-\frac {6 x \operatorname {polylog}\left (3, -{\mathrm e}^{b x +a}\right )}{b^{3}}+\frac {\ln \left (1-{\mathrm e}^{b x +a}\right ) x^{3}}{b}+\frac {3 x^{2} \operatorname {polylog}\left (2, {\mathrm e}^{b x +a}\right )}{b^{2}}-\frac {6 x \operatorname {polylog}\left (3, {\mathrm e}^{b x +a}\right )}{b^{3}}\)	\(272\)

Input:

int(x^3*cosh(b*x+a)^2*coth(b*x+a),x,method=_RETURNVERBOSE)
 

Output:

-1/4*x^4+1/32*(4*b^3*x^3-6*b^2*x^2+6*b*x-3)/b^4*exp(2*b*x+2*a)+1/32*(4*b^3 
*x^3+6*b^2*x^2+6*b*x+3)/b^4*exp(-2*b*x-2*a)-3/2/b^4*a^4+6*polylog(4,-exp(b 
*x+a))/b^4+1/b*ln(exp(b*x+a)+1)*x^3+6*polylog(4,exp(b*x+a))/b^4+1/b^4*ln(1 
-exp(b*x+a))*a^3-1/b^4*a^3*ln(exp(b*x+a)-1)+2/b^4*a^3*ln(exp(b*x+a))-2/b^3 
*a^3*x+3*x^2*polylog(2,-exp(b*x+a))/b^2-6*x*polylog(3,-exp(b*x+a))/b^3+1/b 
*ln(1-exp(b*x+a))*x^3+3*x^2*polylog(2,exp(b*x+a))/b^2-6*x*polylog(3,exp(b* 
x+a))/b^3
 

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 876 vs. \(2 (159) = 318\).

Time = 0.11 (sec) , antiderivative size = 876, normalized size of antiderivative = 4.87 \[ \int x^3 \cosh ^2(a+b x) \coth (a+b x) \, dx=\text {Too large to display} \] Input:

integrate(x^3*cosh(b*x+a)^2*coth(b*x+a),x, algorithm="fricas")
 

Output:

1/32*(4*b^3*x^3 + (4*b^3*x^3 - 6*b^2*x^2 + 6*b*x - 3)*cosh(b*x + a)^4 + 4* 
(4*b^3*x^3 - 6*b^2*x^2 + 6*b*x - 3)*cosh(b*x + a)*sinh(b*x + a)^3 + (4*b^3 
*x^3 - 6*b^2*x^2 + 6*b*x - 3)*sinh(b*x + a)^4 + 6*b^2*x^2 - 8*(b^4*x^4 - 2 
*a^4)*cosh(b*x + a)^2 - 2*(4*b^4*x^4 - 8*a^4 - 3*(4*b^3*x^3 - 6*b^2*x^2 + 
6*b*x - 3)*cosh(b*x + a)^2)*sinh(b*x + a)^2 + 6*b*x + 96*(b^2*x^2*cosh(b*x 
 + a)^2 + 2*b^2*x^2*cosh(b*x + a)*sinh(b*x + a) + b^2*x^2*sinh(b*x + a)^2) 
*dilog(cosh(b*x + a) + sinh(b*x + a)) + 96*(b^2*x^2*cosh(b*x + a)^2 + 2*b^ 
2*x^2*cosh(b*x + a)*sinh(b*x + a) + b^2*x^2*sinh(b*x + a)^2)*dilog(-cosh(b 
*x + a) - sinh(b*x + a)) + 32*(b^3*x^3*cosh(b*x + a)^2 + 2*b^3*x^3*cosh(b* 
x + a)*sinh(b*x + a) + b^3*x^3*sinh(b*x + a)^2)*log(cosh(b*x + a) + sinh(b 
*x + a) + 1) - 32*(a^3*cosh(b*x + a)^2 + 2*a^3*cosh(b*x + a)*sinh(b*x + a) 
 + a^3*sinh(b*x + a)^2)*log(cosh(b*x + a) + sinh(b*x + a) - 1) + 32*((b^3* 
x^3 + a^3)*cosh(b*x + a)^2 + 2*(b^3*x^3 + a^3)*cosh(b*x + a)*sinh(b*x + a) 
 + (b^3*x^3 + a^3)*sinh(b*x + a)^2)*log(-cosh(b*x + a) - sinh(b*x + a) + 1 
) + 192*(cosh(b*x + a)^2 + 2*cosh(b*x + a)*sinh(b*x + a) + sinh(b*x + a)^2 
)*polylog(4, cosh(b*x + a) + sinh(b*x + a)) + 192*(cosh(b*x + a)^2 + 2*cos 
h(b*x + a)*sinh(b*x + a) + sinh(b*x + a)^2)*polylog(4, -cosh(b*x + a) - si 
nh(b*x + a)) - 192*(b*x*cosh(b*x + a)^2 + 2*b*x*cosh(b*x + a)*sinh(b*x + a 
) + b*x*sinh(b*x + a)^2)*polylog(3, cosh(b*x + a) + sinh(b*x + a)) - 192*( 
b*x*cosh(b*x + a)^2 + 2*b*x*cosh(b*x + a)*sinh(b*x + a) + b*x*sinh(b*x ...
 

Sympy [F]

\[ \int x^3 \cosh ^2(a+b x) \coth (a+b x) \, dx=\int x^{3} \cosh ^{2}{\left (a + b x \right )} \coth {\left (a + b x \right )}\, dx \] Input:

integrate(x**3*cosh(b*x+a)**2*coth(b*x+a),x)
 

Output:

Integral(x**3*cosh(a + b*x)**2*coth(a + b*x), x)
 

Maxima [A] (verification not implemented)

Time = 0.15 (sec) , antiderivative size = 225, normalized size of antiderivative = 1.25 \[ \int x^3 \cosh ^2(a+b x) \coth (a+b x) \, dx=-\frac {1}{2} \, x^{4} + \frac {{\left (8 \, b^{4} x^{4} e^{\left (2 \, a\right )} + {\left (4 \, b^{3} x^{3} e^{\left (4 \, a\right )} - 6 \, b^{2} x^{2} e^{\left (4 \, a\right )} + 6 \, b x e^{\left (4 \, a\right )} - 3 \, e^{\left (4 \, a\right )}\right )} e^{\left (2 \, b x\right )} + {\left (4 \, b^{3} x^{3} + 6 \, b^{2} x^{2} + 6 \, b x + 3\right )} e^{\left (-2 \, b x\right )}\right )} e^{\left (-2 \, a\right )}}{32 \, b^{4}} + \frac {b^{3} x^{3} \log \left (e^{\left (b x + a\right )} + 1\right ) + 3 \, b^{2} x^{2} {\rm Li}_2\left (-e^{\left (b x + a\right )}\right ) - 6 \, b x {\rm Li}_{3}(-e^{\left (b x + a\right )}) + 6 \, {\rm Li}_{4}(-e^{\left (b x + a\right )})}{b^{4}} + \frac {b^{3} x^{3} \log \left (-e^{\left (b x + a\right )} + 1\right ) + 3 \, b^{2} x^{2} {\rm Li}_2\left (e^{\left (b x + a\right )}\right ) - 6 \, b x {\rm Li}_{3}(e^{\left (b x + a\right )}) + 6 \, {\rm Li}_{4}(e^{\left (b x + a\right )})}{b^{4}} \] Input:

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

Output:

-1/2*x^4 + 1/32*(8*b^4*x^4*e^(2*a) + (4*b^3*x^3*e^(4*a) - 6*b^2*x^2*e^(4*a 
) + 6*b*x*e^(4*a) - 3*e^(4*a))*e^(2*b*x) + (4*b^3*x^3 + 6*b^2*x^2 + 6*b*x 
+ 3)*e^(-2*b*x))*e^(-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
 

Giac [F]

\[ \int x^3 \cosh ^2(a+b x) \coth (a+b x) \, dx=\int { x^{3} \cosh \left (b x + a\right )^{2} \coth \left (b x + a\right ) \,d x } \] Input:

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

Output:

integrate(x^3*cosh(b*x + a)^2*coth(b*x + a), x)
 

Mupad [F(-1)]

Timed out. \[ \int x^3 \cosh ^2(a+b x) \coth (a+b x) \, dx=\int x^3\,{\mathrm {cosh}\left (a+b\,x\right )}^2\,\mathrm {coth}\left (a+b\,x\right ) \,d x \] Input:

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

Output:

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

Reduce [F]

\[ \int x^3 \cosh ^2(a+b x) \coth (a+b x) \, dx=\frac {4 e^{4 b x +4 a} b^{3} x^{3}-6 e^{4 b x +4 a} b^{2} x^{2}+6 e^{4 b x +4 a} b x -3 e^{4 b x +4 a}+64 e^{2 b x +2 a} \left (\int \frac {x^{3}}{e^{4 b x +4 a}-e^{2 b x +2 a}}d x \right ) b^{4}+8 e^{2 b x +2 a} b^{4} x^{4}-28 b^{3} x^{3}-42 b^{2} x^{2}-42 b x -21}{32 e^{2 b x +2 a} b^{4}} \] Input:

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

Output:

(4*e**(4*a + 4*b*x)*b**3*x**3 - 6*e**(4*a + 4*b*x)*b**2*x**2 + 6*e**(4*a + 
 4*b*x)*b*x - 3*e**(4*a + 4*b*x) + 64*e**(2*a + 2*b*x)*int(x**3/(e**(4*a + 
 4*b*x) - e**(2*a + 2*b*x)),x)*b**4 + 8*e**(2*a + 2*b*x)*b**4*x**4 - 28*b* 
*3*x**3 - 42*b**2*x**2 - 42*b*x - 21)/(32*e**(2*a + 2*b*x)*b**4)