3.4.56 \(\int x^3 \sinh (a+b x) \tanh (a+b x) \, dx\) [356]

3.4.56.1 Optimal result

Integrand size = 16, antiderivative size = 195 \[ \int x^3 \sinh (a+b x) \tanh (a+b x) \, dx=-\frac {2 x^3 \arctan \left (e^{a+b x}\right )}{b}-\frac {6 \cosh (a+b x)}{b^4}-\frac {3 x^2 \cosh (a+b x)}{b^2}+\frac {3 i x^2 \operatorname {PolyLog}\left (2,-i e^{a+b x}\right )}{b^2}-\frac {3 i x^2 \operatorname {PolyLog}\left (2,i e^{a+b x}\right )}{b^2}-\frac {6 i x \operatorname {PolyLog}\left (3,-i e^{a+b x}\right )}{b^3}+\frac {6 i x \operatorname {PolyLog}\left (3,i e^{a+b x}\right )}{b^3}+\frac {6 i \operatorname {PolyLog}\left (4,-i e^{a+b x}\right )}{b^4}-\frac {6 i \operatorname {PolyLog}\left (4,i e^{a+b x}\right )}{b^4}+\frac {6 x \sinh (a+b x)}{b^3}+\frac {x^3 \sinh (a+b x)}{b} \]

output

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

3.4.56.2 Mathematica [A] (verified)

Time = 0.20 (sec) , antiderivative size = 211, normalized size of antiderivative = 1.08 \[ \int x^3 \sinh (a+b x) \tanh (a+b x) \, dx=-\frac {i \left (-6 i \cosh (a+b x)-3 i b^2 x^2 \cosh (a+b x)+b^3 x^3 \log \left (1-i e^{a+b x}\right )-b^3 x^3 \log \left (1+i e^{a+b x}\right )-3 b^2 x^2 \operatorname {PolyLog}\left (2,-i e^{a+b x}\right )+3 b^2 x^2 \operatorname {PolyLog}\left (2,i e^{a+b x}\right )+6 b x \operatorname {PolyLog}\left (3,-i e^{a+b x}\right )-6 b x \operatorname {PolyLog}\left (3,i e^{a+b x}\right )-6 \operatorname {PolyLog}\left (4,-i e^{a+b x}\right )+6 \operatorname {PolyLog}\left (4,i e^{a+b x}\right )+6 i b x \sinh (a+b x)+i b^3 x^3 \sinh (a+b x)\right )}{b^4} \]

input

Integrate[x^3*Sinh[a + b*x]*Tanh[a + b*x],x]
 

output

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

3.4.56.3 Rubi [A] (verified)

Time = 1.11 (sec) , antiderivative size = 226, normalized size of antiderivative = 1.16, number of steps used = 19, number of rules used = 18, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 1.125, Rules used = {5972, 3042, 3777, 26, 3042, 26, 3777, 3042, 3777, 26, 3042, 26, 3118, 4668, 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.

\(\displaystyle \int x^3 \sinh (a+b x) \tanh (a+b x) \, dx\)

\(\Big \downarrow \) 5972

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 3777

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

\(\Big \downarrow \) 26

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 26

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

\(\Big \downarrow \) 3777

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 3777

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

\(\Big \downarrow \) 26

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 26

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

\(\Big \downarrow \) 3118

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

\(\Big \downarrow \) 4668

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

\(\Big \downarrow \) 3011

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

\(\Big \downarrow \) 7163

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

\(\Big \downarrow \) 2720

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

\(\Big \downarrow \) 7143

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

input

Int[x^3*Sinh[a + b*x]*Tanh[a + b*x],x]
 

output

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

3.4.56.3.1 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 3118

Int[sin[(c_.) + (d_.)*(x_)], x_Symbol] :> Simp[-Cos[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 4668

Int[csc[(e_.) + Pi*(k_.) + (Complex[0, fz_])*(f_.)*(x_)]*((c_.) + (d_.)*(x_ 
))^(m_.), x_Symbol] :> Simp[-2*(c + d*x)^m*(ArcTanh[E^((-I)*e + f*fz*x)/E^( 
I*k*Pi)]/(f*fz*I)), x] + (-Simp[d*(m/(f*fz*I))   Int[(c + d*x)^(m - 1)*Log[ 
1 - E^((-I)*e + f*fz*x)/E^(I*k*Pi)], x], x] + Simp[d*(m/(f*fz*I))   Int[(c 
+ d*x)^(m - 1)*Log[1 + E^((-I)*e + f*fz*x)/E^(I*k*Pi)], x], x]) /; FreeQ[{c 
, d, e, f, fz}, x] && IntegerQ[2*k] && IGtQ[m, 0]
 

rule 5972

Int[((c_.) + (d_.)*(x_))^(m_.)*Sinh[(a_.) + (b_.)*(x_)]^(n_.)*Tanh[(a_.) + 
(b_.)*(x_)]^(p_.), x_Symbol] :> Int[(c + d*x)^m*Sinh[a + b*x]^n*Tanh[a + b* 
x]^(p - 2), x] - Int[(c + d*x)^m*Sinh[a + b*x]^(n - 2)*Tanh[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]
 

3.4.56.4 Maple [F]

\[\int x^{3} \operatorname {sech}\left (b x +a \right ) \sinh \left (b x +a \right )^{2}d x\]

input

int(x^3*sech(b*x+a)*sinh(b*x+a)^2,x)
 

output

int(x^3*sech(b*x+a)*sinh(b*x+a)^2,x)
 

3.4.56.5 Fricas [B] (verification not implemented)

Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 609 vs. \(2 (162) = 324\).

Time = 0.27 (sec) , antiderivative size = 609, normalized size of antiderivative = 3.12 \[ \int x^3 \sinh (a+b x) \tanh (a+b x) \, dx=-\frac {b^{3} x^{3} + 3 \, b^{2} x^{2} - {\left (b^{3} x^{3} - 3 \, b^{2} x^{2} + 6 \, b x - 6\right )} \cosh \left (b x + a\right )^{2} - 2 \, {\left (b^{3} x^{3} - 3 \, b^{2} x^{2} + 6 \, b x - 6\right )} \cosh \left (b x + a\right ) \sinh \left (b x + a\right ) - {\left (b^{3} x^{3} - 3 \, b^{2} x^{2} + 6 \, b x - 6\right )} \sinh \left (b x + a\right )^{2} + 6 \, b x + 6 \, {\left (i \, b^{2} x^{2} \cosh \left (b x + a\right ) + i \, b^{2} x^{2} \sinh \left (b x + a\right )\right )} {\rm Li}_2\left (i \, \cosh \left (b x + a\right ) + i \, \sinh \left (b x + a\right )\right ) + 6 \, {\left (-i \, b^{2} x^{2} \cosh \left (b x + a\right ) - i \, b^{2} x^{2} \sinh \left (b x + a\right )\right )} {\rm Li}_2\left (-i \, \cosh \left (b x + a\right ) - i \, \sinh \left (b x + a\right )\right ) + 2 \, {\left (-i \, a^{3} \cosh \left (b x + a\right ) - i \, a^{3} \sinh \left (b x + a\right )\right )} \log \left (\cosh \left (b x + a\right ) + \sinh \left (b x + a\right ) + i\right ) + 2 \, {\left (i \, a^{3} \cosh \left (b x + a\right ) + i \, a^{3} \sinh \left (b x + a\right )\right )} \log \left (\cosh \left (b x + a\right ) + \sinh \left (b x + a\right ) - i\right ) + 2 \, {\left ({\left (-i \, b^{3} x^{3} - i \, a^{3}\right )} \cosh \left (b x + a\right ) + {\left (-i \, b^{3} x^{3} - i \, a^{3}\right )} \sinh \left (b x + a\right )\right )} \log \left (i \, \cosh \left (b x + a\right ) + i \, \sinh \left (b x + a\right ) + 1\right ) + 2 \, {\left ({\left (i \, b^{3} x^{3} + i \, a^{3}\right )} \cosh \left (b x + a\right ) + {\left (i \, b^{3} x^{3} + i \, a^{3}\right )} \sinh \left (b x + a\right )\right )} \log \left (-i \, \cosh \left (b x + a\right ) - i \, \sinh \left (b x + a\right ) + 1\right ) + 12 \, {\left (i \, \cosh \left (b x + a\right ) + i \, \sinh \left (b x + a\right )\right )} {\rm polylog}\left (4, i \, \cosh \left (b x + a\right ) + i \, \sinh \left (b x + a\right )\right ) + 12 \, {\left (-i \, \cosh \left (b x + a\right ) - i \, \sinh \left (b x + a\right )\right )} {\rm polylog}\left (4, -i \, \cosh \left (b x + a\right ) - i \, \sinh \left (b x + a\right )\right ) + 12 \, {\left (-i \, b x \cosh \left (b x + a\right ) - i \, b x \sinh \left (b x + a\right )\right )} {\rm polylog}\left (3, i \, \cosh \left (b x + a\right ) + i \, \sinh \left (b x + a\right )\right ) + 12 \, {\left (i \, b x \cosh \left (b x + a\right ) + i \, b x \sinh \left (b x + a\right )\right )} {\rm polylog}\left (3, -i \, \cosh \left (b x + a\right ) - i \, \sinh \left (b x + a\right )\right ) + 6}{2 \, {\left (b^{4} \cosh \left (b x + a\right ) + b^{4} \sinh \left (b x + a\right )\right )}} \]

input

integrate(x^3*sech(b*x+a)*sinh(b*x+a)^2,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*(I*b^2*x^2*cos 
h(b*x + a) + I*b^2*x^2*sinh(b*x + a))*dilog(I*cosh(b*x + a) + I*sinh(b*x + 
 a)) + 6*(-I*b^2*x^2*cosh(b*x + a) - I*b^2*x^2*sinh(b*x + a))*dilog(-I*cos 
h(b*x + a) - I*sinh(b*x + a)) + 2*(-I*a^3*cosh(b*x + a) - I*a^3*sinh(b*x + 
 a))*log(cosh(b*x + a) + sinh(b*x + a) + I) + 2*(I*a^3*cosh(b*x + a) + I*a 
^3*sinh(b*x + a))*log(cosh(b*x + a) + sinh(b*x + a) - I) + 2*((-I*b^3*x^3 
- I*a^3)*cosh(b*x + a) + (-I*b^3*x^3 - I*a^3)*sinh(b*x + a))*log(I*cosh(b* 
x + a) + I*sinh(b*x + a) + 1) + 2*((I*b^3*x^3 + I*a^3)*cosh(b*x + a) + (I* 
b^3*x^3 + I*a^3)*sinh(b*x + a))*log(-I*cosh(b*x + a) - I*sinh(b*x + a) + 1 
) + 12*(I*cosh(b*x + a) + I*sinh(b*x + a))*polylog(4, I*cosh(b*x + a) + I* 
sinh(b*x + a)) + 12*(-I*cosh(b*x + a) - I*sinh(b*x + a))*polylog(4, -I*cos 
h(b*x + a) - I*sinh(b*x + a)) + 12*(-I*b*x*cosh(b*x + a) - I*b*x*sinh(b*x 
+ a))*polylog(3, I*cosh(b*x + a) + I*sinh(b*x + a)) + 12*(I*b*x*cosh(b*x + 
 a) + I*b*x*sinh(b*x + a))*polylog(3, -I*cosh(b*x + a) - I*sinh(b*x + a)) 
+ 6)/(b^4*cosh(b*x + a) + b^4*sinh(b*x + a))
 

3.4.56.6 Sympy [F]

\[ \int x^3 \sinh (a+b x) \tanh (a+b x) \, dx=\int x^{3} \sinh ^{2}{\left (a + b x \right )} \operatorname {sech}{\left (a + b x \right )}\, dx \]

input

integrate(x**3*sech(b*x+a)*sinh(b*x+a)**2,x)
 

output

Integral(x**3*sinh(a + b*x)**2*sech(a + b*x), x)
 

3.4.56.7 Maxima [F]

\[ \int x^3 \sinh (a+b x) \tanh (a+b x) \, dx=\int { x^{3} \operatorname {sech}\left (b x + a\right ) \sinh \left (b x + a\right )^{2} \,d x } \]

input

integrate(x^3*sech(b*x+a)*sinh(b*x+a)^2,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 - 2*integrat 
e(x^3*e^(b*x + a)/(e^(2*b*x + 2*a) + 1), x)
 

3.4.56.8 Giac [F]

\[ \int x^3 \sinh (a+b x) \tanh (a+b x) \, dx=\int { x^{3} \operatorname {sech}\left (b x + a\right ) \sinh \left (b x + a\right )^{2} \,d x } \]

input

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

output

integrate(x^3*sech(b*x + a)*sinh(b*x + a)^2, x)
 

3.4.56.9 Mupad [F(-1)]

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

input

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

output

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