3.4.84 \(\int x^3 \sinh (a+b x) \tanh ^2(a+b x) \, dx\) [384]

3.4.84.1 Optimal result

Integrand size = 18, antiderivative size = 162 \[ \int x^3 \sinh (a+b x) \tanh ^2(a+b x) \, dx=-\frac {6 x^2 \arctan \left (e^{a+b x}\right )}{b^2}+\frac {6 x \cosh (a+b x)}{b^3}+\frac {x^3 \cosh (a+b x)}{b}+\frac {6 i x \operatorname {PolyLog}\left (2,-i e^{a+b x}\right )}{b^3}-\frac {6 i x \operatorname {PolyLog}\left (2,i e^{a+b x}\right )}{b^3}-\frac {6 i \operatorname {PolyLog}\left (3,-i e^{a+b x}\right )}{b^4}+\frac {6 i \operatorname {PolyLog}\left (3,i e^{a+b x}\right )}{b^4}+\frac {x^3 \text {sech}(a+b x)}{b}-\frac {6 \sinh (a+b x)}{b^4}-\frac {3 x^2 \sinh (a+b x)}{b^2} \]

output

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

3.4.84.2 Mathematica [A] (verified)

Time = 0.88 (sec) , antiderivative size = 196, normalized size of antiderivative = 1.21 \[ \int x^3 \sinh (a+b x) \tanh ^2(a+b x) \, dx=\frac {-3 i \left (b^2 x^2 \log \left (1-i e^{a+b x}\right )-b^2 x^2 \log \left (1+i e^{a+b x}\right )-2 b x \operatorname {PolyLog}\left (2,-i e^{a+b x}\right )+2 b x \operatorname {PolyLog}\left (2,i e^{a+b x}\right )+2 \operatorname {PolyLog}\left (3,-i e^{a+b x}\right )-2 \operatorname {PolyLog}\left (3,i e^{a+b x}\right )\right )+b^3 x^3 \text {sech}(a+b x)+\cosh (b x) \left (b x \left (6+b^2 x^2\right ) \cosh (a)-3 \left (2+b^2 x^2\right ) \sinh (a)\right )+\left (-3 \left (2+b^2 x^2\right ) \cosh (a)+b x \left (6+b^2 x^2\right ) \sinh (a)\right ) \sinh (b x)}{b^4} \]

input

Integrate[x^3*Sinh[a + b*x]*Tanh[a + b*x]^2,x]
 

output

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

3.4.84.3 Rubi [A] (verified)

Time = 1.13 (sec) , antiderivative size = 201, normalized size of antiderivative = 1.24, number of steps used = 19, number of rules used = 18, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 1.000, Rules used = {5972, 3042, 26, 3777, 3042, 3777, 26, 3042, 26, 3777, 3042, 3117, 5941, 3042, 4668, 3011, 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.

\(\displaystyle \int x^3 \sinh (a+b x) \tanh ^2(a+b x) \, dx\)

\(\Big \downarrow \) 5972

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 26

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

\(\Big \downarrow \) 3777

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 3777

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

\(\Big \downarrow \) 26

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 26

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

\(\Big \downarrow \) 3777

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 3117

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

\(\Big \downarrow \) 5941

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 4668

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

\(\Big \downarrow \) 3011

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

\(\Big \downarrow \) 2720

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

\(\Big \downarrow \) 7143

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

input

Int[x^3*Sinh[a + b*x]*Tanh[a + b*x]^2,x]
 

output

(-3*((2*x^2*ArcTan[E^(a + b*x)])/b + ((2*I)*(-((x*PolyLog[2, (-I)*E^(a + b 
*x)])/b) + PolyLog[3, (-I)*E^(a + b*x)]/b^2))/b - ((2*I)*(-((x*PolyLog[2, 
I*E^(a + b*x)])/b) + PolyLog[3, I*E^(a + b*x)]/b^2))/b))/b + (x^3*Sech[a + 
 b*x])/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)
 

3.4.84.3.1 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 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 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 5941

Int[(x_)^(m_.)*Sech[(a_.) + (b_.)*(x_)^(n_.)]^(p_.)*Tanh[(a_.) + (b_.)*(x_) 
^(n_.)]^(q_.), x_Symbol] :> Simp[(-x^(m - n + 1))*(Sech[a + b*x^n]^p/(b*n*p 
)), x] + Simp[(m - n + 1)/(b*n*p)   Int[x^(m - n)*Sech[a + b*x^n]^p, x], x] 
 /; FreeQ[{a, b, p}, x] && RationalQ[m] && IntegerQ[n] && GeQ[m - n, 0] && 
EqQ[q, 1]
 

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]
 

3.4.84.4 Maple [F]

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

input

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

output

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

3.4.84.5 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. 1225 vs. \(2 (139) = 278\).

Time = 0.28 (sec) , antiderivative size = 1225, normalized size of antiderivative = 7.56 \[ \int x^3 \sinh (a+b x) \tanh ^2(a+b x) \, dx=\text {Too large to display} \]

input

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

output

1/2*(b^3*x^3 + (b^3*x^3 - 3*b^2*x^2 + 6*b*x - 6)*cosh(b*x + a)^4 + 4*(b^3* 
x^3 - 3*b^2*x^2 + 6*b*x - 6)*cosh(b*x + a)*sinh(b*x + a)^3 + (b^3*x^3 - 3* 
b^2*x^2 + 6*b*x - 6)*sinh(b*x + a)^4 + 3*b^2*x^2 + 6*(b^3*x^3 + 2*b*x)*cos 
h(b*x + a)^2 + 6*(b^3*x^3 + (b^3*x^3 - 3*b^2*x^2 + 6*b*x - 6)*cosh(b*x + a 
)^2 + 2*b*x)*sinh(b*x + a)^2 + 6*b*x - 12*(I*b*x*cosh(b*x + a)^3 + 3*I*b*x 
*cosh(b*x + a)*sinh(b*x + a)^2 + I*b*x*sinh(b*x + a)^3 + I*b*x*cosh(b*x + 
a) + (3*I*b*x*cosh(b*x + a)^2 + I*b*x)*sinh(b*x + a))*dilog(I*cosh(b*x + a 
) + I*sinh(b*x + a)) - 12*(-I*b*x*cosh(b*x + a)^3 - 3*I*b*x*cosh(b*x + a)* 
sinh(b*x + a)^2 - I*b*x*sinh(b*x + a)^3 - I*b*x*cosh(b*x + a) + (-3*I*b*x* 
cosh(b*x + a)^2 - I*b*x)*sinh(b*x + a))*dilog(-I*cosh(b*x + a) - I*sinh(b* 
x + a)) - 6*(I*a^2*cosh(b*x + a)^3 + 3*I*a^2*cosh(b*x + a)*sinh(b*x + a)^2 
 + I*a^2*sinh(b*x + a)^3 + I*a^2*cosh(b*x + a) + (3*I*a^2*cosh(b*x + a)^2 
+ I*a^2)*sinh(b*x + a))*log(cosh(b*x + a) + sinh(b*x + a) + I) - 6*(-I*a^2 
*cosh(b*x + a)^3 - 3*I*a^2*cosh(b*x + a)*sinh(b*x + a)^2 - I*a^2*sinh(b*x 
+ a)^3 - I*a^2*cosh(b*x + a) + (-3*I*a^2*cosh(b*x + a)^2 - I*a^2)*sinh(b*x 
 + a))*log(cosh(b*x + a) + sinh(b*x + a) - I) - 6*((-I*b^2*x^2 + I*a^2)*co 
sh(b*x + a)^3 + 3*(-I*b^2*x^2 + I*a^2)*cosh(b*x + a)*sinh(b*x + a)^2 + (-I 
*b^2*x^2 + I*a^2)*sinh(b*x + a)^3 + (-I*b^2*x^2 + I*a^2)*cosh(b*x + a) + ( 
-I*b^2*x^2 + 3*(-I*b^2*x^2 + I*a^2)*cosh(b*x + a)^2 + I*a^2)*sinh(b*x + a) 
)*log(I*cosh(b*x + a) + I*sinh(b*x + a) + 1) - 6*((I*b^2*x^2 - I*a^2)*c...
 

3.4.84.6 Sympy [F]

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

input

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

output

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

3.4.84.7 Maxima [F]

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

input

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

output

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

3.4.84.8 Giac [F]

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

input

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

output

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

3.4.84.9 Mupad [F(-1)]

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

input

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

output

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