\(\int x^3 \cosh (a+b x) \coth (a+b x) \, dx\) [143]

Optimal result

Integrand size = 16, antiderivative size = 165 \[ \int x^3 \cosh (a+b x) \coth (a+b x) \, dx=-\frac {2 x^3 \text {arctanh}\left (e^{a+b x}\right )}{b}+\frac {6 x \cosh (a+b x)}{b^3}+\frac {x^3 \cosh (a+b x)}{b}-\frac {3 x^2 \operatorname {PolyLog}\left (2,-e^{a+b x}\right )}{b^2}+\frac {3 x^2 \operatorname {PolyLog}\left (2,e^{a+b x}\right )}{b^2}+\frac {6 x \operatorname {PolyLog}\left (3,-e^{a+b x}\right )}{b^3}-\frac {6 x \operatorname {PolyLog}\left (3,e^{a+b x}\right )}{b^3}-\frac {6 \operatorname {PolyLog}\left (4,-e^{a+b x}\right )}{b^4}+\frac {6 \operatorname {PolyLog}\left (4,e^{a+b x}\right )}{b^4}-\frac {6 \sinh (a+b x)}{b^4}-\frac {3 x^2 \sinh (a+b x)}{b^2} \] Output:

-2*x^3*arctanh(exp(b*x+a))/b+6*x*cosh(b*x+a)/b^3+x^3*cosh(b*x+a)/b-3*x^2*p 
olylog(2,-exp(b*x+a))/b^2+3*x^2*polylog(2,exp(b*x+a))/b^2+6*x*polylog(3,-e 
xp(b*x+a))/b^3-6*x*polylog(3,exp(b*x+a))/b^3-6*polylog(4,-exp(b*x+a))/b^4+ 
6*polylog(4,exp(b*x+a))/b^4-6*sinh(b*x+a)/b^4-3*x^2*sinh(b*x+a)/b^2
 

Mathematica [A] (verified)

Time = 0.14 (sec) , antiderivative size = 175, normalized size of antiderivative = 1.06 \[ \int x^3 \cosh (a+b x) \coth (a+b x) \, dx=\frac {6 b x \cosh (a+b x)+b^3 x^3 \cosh (a+b x)+b^3 x^3 \log \left (1-e^{a+b x}\right )-b^3 x^3 \log \left (1+e^{a+b x}\right )-3 b^2 x^2 \operatorname {PolyLog}\left (2,-e^{a+b x}\right )+3 b^2 x^2 \operatorname {PolyLog}\left (2,e^{a+b x}\right )+6 b x \operatorname {PolyLog}\left (3,-e^{a+b x}\right )-6 b x \operatorname {PolyLog}\left (3,e^{a+b x}\right )-6 \operatorname {PolyLog}\left (4,-e^{a+b x}\right )+6 \operatorname {PolyLog}\left (4,e^{a+b x}\right )-6 \sinh (a+b x)-3 b^2 x^2 \sinh (a+b x)}{b^4} \] Input:

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

Output:

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

Rubi [C] (verified)

Result contains complex when optimal does not.

Time = 1.21 (sec) , antiderivative size = 225, normalized size of antiderivative = 1.36, number of steps used = 18, number of rules used = 17, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 1.062, Rules used = {5973, 3042, 26, 3777, 3042, 3777, 26, 3042, 26, 3777, 3042, 3117, 4670, 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.

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

\(\Big \downarrow \) 5973

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 26

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

\(\Big \downarrow \) 3777

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 3777

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

\(\Big \downarrow \) 26

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 26

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

\(\Big \downarrow \) 3777

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 3117

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

\(\Big \downarrow \) 4670

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

\(\Big \downarrow \) 3011

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

\(\Big \downarrow \) 7163

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

\(\Big \downarrow \) 2720

\(\displaystyle i \left (-\frac {3 i \left (\frac {2 \left (\frac {x \operatorname {PolyLog}\left (3,-e^{a+b x}\right )}{b}-\frac {\int e^{-a-b x} \operatorname {PolyLog}\left (3,-e^{a+b x}\right )de^{a+b x}}{b^2}\right )}{b}-\frac {x^2 \operatorname {PolyLog}\left (2,-e^{a+b x}\right )}{b}\right )}{b}+\frac {3 i \left (\frac {2 \left (\frac {x \operatorname {PolyLog}\left (3,e^{a+b x}\right )}{b}-\frac {\int e^{-a-b x} \operatorname {PolyLog}\left (3,e^{a+b x}\right )de^{a+b x}}{b^2}\right )}{b}-\frac {x^2 \operatorname {PolyLog}\left (2,e^{a+b x}\right )}{b}\right )}{b}+\frac {2 i x^3 \text {arctanh}\left (e^{a+b x}\right )}{b}\right )-i \left (\frac {i x^3 \cosh (a+b x)}{b}-\frac {3 i \left (\frac {x^2 \sinh (a+b x)}{b}+\frac {2 i \left (\frac {i x \cosh (a+b x)}{b}-\frac {i \sinh (a+b x)}{b^2}\right )}{b}\right )}{b}\right )\)

\(\Big \downarrow \) 7143

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

Input:

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

Output:

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

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 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 3117

Int[sin[Pi/2 + (c_.) + (d_.)*(x_)], x_Symbol] :> Simp[Sin[c + d*x]/d, x] /; 
 FreeQ[{c, d}, x]
 

rule 3777

Int[((c_.) + (d_.)*(x_))^(m_.)*sin[(e_.) + (f_.)*(x_)], x_Symbol] :> Simp[( 
-(c + d*x)^m)*(Cos[e + f*x]/f), x] + Simp[d*(m/f)   Int[(c + d*x)^(m - 1)*C 
os[e + f*x], x], x] /; FreeQ[{c, d, e, f}, x] && GtQ[m, 0]
 

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 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 = 1.02 (sec) , antiderivative size = 246, normalized size of antiderivative = 1.49

method	result	size
risch	\(\frac {\left (b^{3} x^{3}-3 b^{2} x^{2}+6 b x -6\right ) {\mathrm e}^{b x +a}}{2 b^{4}}+\frac {\left (b^{3} x^{3}+3 b^{2} x^{2}+6 b x +6\right ) {\mathrm e}^{-b x -a}}{2 b^{4}}-\frac {6 \operatorname {polylog}\left (4, -{\mathrm e}^{b x +a}\right )}{b^{4}}+\frac {6 \operatorname {polylog}\left (4, {\mathrm e}^{b x +a}\right )}{b^{4}}-\frac {6 x \operatorname {polylog}\left (3, {\mathrm e}^{b x +a}\right )}{b^{3}}-\frac {\ln \left ({\mathrm e}^{b x +a}+1\right ) a^{3}}{b^{4}}+\frac {\ln \left (1-{\mathrm e}^{b x +a}\right ) a^{3}}{b^{4}}+\frac {2 a^{3} \operatorname {arctanh}\left ({\mathrm e}^{b x +a}\right )}{b^{4}}-\frac {\ln \left ({\mathrm e}^{b x +a}+1\right ) x^{3}}{b}-\frac {3 x^{2} \operatorname {polylog}\left (2, -{\mathrm e}^{b x +a}\right )}{b^{2}}+\frac {6 x \operatorname {polylog}\left (3, -{\mathrm e}^{b x +a}\right )}{b^{3}}+\frac {\ln \left (1-{\mathrm e}^{b x +a}\right ) x^{3}}{b}+\frac {3 x^{2} \operatorname {polylog}\left (2, {\mathrm e}^{b x +a}\right )}{b^{2}}\)	\(246\)

Input:

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

Output:

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

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 511 vs. \(2 (156) = 312\).

Time = 0.10 (sec) , antiderivative size = 511, normalized size of antiderivative = 3.10 \[ \int x^3 \cosh (a+b x) \coth (a+b x) \, dx =\text {Too large to display} \] Input:

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

Output:

1/2*(b^3*x^3 + 3*b^2*x^2 + (b^3*x^3 - 3*b^2*x^2 + 6*b*x - 6)*cosh(b*x + a) 
^2 + 2*(b^3*x^3 - 3*b^2*x^2 + 6*b*x - 6)*cosh(b*x + a)*sinh(b*x + a) + (b^ 
3*x^3 - 3*b^2*x^2 + 6*b*x - 6)*sinh(b*x + a)^2 + 6*b*x + 6*(b^2*x^2*cosh(b 
*x + a) + b^2*x^2*sinh(b*x + a))*dilog(cosh(b*x + a) + sinh(b*x + a)) - 6* 
(b^2*x^2*cosh(b*x + a) + b^2*x^2*sinh(b*x + a))*dilog(-cosh(b*x + a) - sin 
h(b*x + a)) - 2*(b^3*x^3*cosh(b*x + a) + b^3*x^3*sinh(b*x + a))*log(cosh(b 
*x + a) + sinh(b*x + a) + 1) - 2*(a^3*cosh(b*x + a) + a^3*sinh(b*x + a))*l 
og(cosh(b*x + a) + sinh(b*x + a) - 1) + 2*((b^3*x^3 + a^3)*cosh(b*x + a) + 
 (b^3*x^3 + a^3)*sinh(b*x + a))*log(-cosh(b*x + a) - sinh(b*x + a) + 1) + 
12*(cosh(b*x + a) + sinh(b*x + a))*polylog(4, cosh(b*x + a) + sinh(b*x + a 
)) - 12*(cosh(b*x + a) + sinh(b*x + a))*polylog(4, -cosh(b*x + a) - sinh(b 
*x + a)) - 12*(b*x*cosh(b*x + a) + b*x*sinh(b*x + a))*polylog(3, cosh(b*x 
+ a) + sinh(b*x + a)) + 12*(b*x*cosh(b*x + a) + b*x*sinh(b*x + a))*polylog 
(3, -cosh(b*x + a) - sinh(b*x + a)) + 6)/(b^4*cosh(b*x + a) + b^4*sinh(b*x 
 + a))
 

Sympy [F]

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

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

Output:

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

Maxima [A] (verification not implemented)

Time = 0.12 (sec) , antiderivative size = 206, normalized size of antiderivative = 1.25 \[ \int x^3 \cosh (a+b x) \coth (a+b x) \, dx=\frac {{\left ({\left (b^{3} x^{3} e^{\left (2 \, a\right )} - 3 \, b^{2} x^{2} e^{\left (2 \, a\right )} + 6 \, b x e^{\left (2 \, a\right )} - 6 \, e^{\left (2 \, a\right )}\right )} e^{\left (b x\right )} + {\left (b^{3} x^{3} + 3 \, b^{2} x^{2} + 6 \, b x + 6\right )} e^{\left (-b x\right )}\right )} e^{\left (-a\right )}}{2 \, b^{4}} - \frac {b^{3} x^{3} \log \left (e^{\left (b x + a\right )} + 1\right ) + 3 \, b^{2} x^{2} {\rm Li}_2\left (-e^{\left (b x + a\right )}\right ) - 6 \, b x {\rm Li}_{3}(-e^{\left (b x + a\right )}) + 6 \, {\rm Li}_{4}(-e^{\left (b x + a\right )})}{b^{4}} + \frac {b^{3} x^{3} \log \left (-e^{\left (b x + a\right )} + 1\right ) + 3 \, b^{2} x^{2} {\rm Li}_2\left (e^{\left (b x + a\right )}\right ) - 6 \, b x {\rm Li}_{3}(e^{\left (b x + a\right )}) + 6 \, {\rm Li}_{4}(e^{\left (b x + a\right )})}{b^{4}} \] Input:

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

Output:

1/2*((b^3*x^3*e^(2*a) - 3*b^2*x^2*e^(2*a) + 6*b*x*e^(2*a) - 6*e^(2*a))*e^( 
b*x) + (b^3*x^3 + 3*b^2*x^2 + 6*b*x + 6)*e^(-b*x))*e^(-a)/b^4 - (b^3*x^3*l 
og(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*pol 
ylog(4, e^(b*x + a)))/b^4
 

Giac [F]

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

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

Output:

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

\[ \int x^3 \cosh (a+b x) \coth (a+b x) \, dx=\frac {e^{2 b x +2 a} b^{3} x^{3}-3 e^{2 b x +2 a} b^{2} x^{2}+6 e^{2 b x +2 a} b x -6 e^{2 b x +2 a}+4 e^{b x} \left (\int \frac {x^{3}}{e^{3 b x +2 a}-e^{b x}}d x \right ) b^{4}-3 b^{3} x^{3}-9 b^{2} x^{2}-18 b x -18}{2 e^{b x +a} b^{4}} \] Input:

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

Output:

(e**(2*a + 2*b*x)*b**3*x**3 - 3*e**(2*a + 2*b*x)*b**2*x**2 + 6*e**(2*a + 2 
*b*x)*b*x - 6*e**(2*a + 2*b*x) + 4*e**(b*x)*int(x**3/(e**(2*a + 3*b*x) - e 
**(b*x)),x)*b**4 - 3*b**3*x**3 - 9*b**2*x**2 - 18*b*x - 18)/(2*e**(a + b*x 
)*b**4)