\(\int x^2 \text {sech}(a+b x) \tanh ^2(a+b x) \, dx\) [90]

Optimal result

Integrand size = 18, antiderivative size = 143 \[ \int x^2 \text {sech}(a+b x) \tanh ^2(a+b x) \, dx=\frac {x^2 \arctan \left (e^{a+b x}\right )}{b}+\frac {\arctan (\sinh (a+b x))}{b^3}-\frac {i x \operatorname {PolyLog}\left (2,-i e^{a+b x}\right )}{b^2}+\frac {i x \operatorname {PolyLog}\left (2,i e^{a+b x}\right )}{b^2}+\frac {i \operatorname {PolyLog}\left (3,-i e^{a+b x}\right )}{b^3}-\frac {i \operatorname {PolyLog}\left (3,i e^{a+b x}\right )}{b^3}-\frac {x \text {sech}(a+b x)}{b^2}-\frac {x^2 \text {sech}(a+b x) \tanh (a+b x)}{2 b} \] Output:

x^2*arctan(exp(b*x+a))/b+arctan(sinh(b*x+a))/b^3-I*x*polylog(2,-I*exp(b*x+ 
a))/b^2+I*x*polylog(2,I*exp(b*x+a))/b^2+I*polylog(3,-I*exp(b*x+a))/b^3-I*p 
olylog(3,I*exp(b*x+a))/b^3-x*sech(b*x+a)/b^2-1/2*x^2*sech(b*x+a)*tanh(b*x+ 
a)/b
 

Mathematica [A] (verified)

Time = 0.55 (sec) , antiderivative size = 180, normalized size of antiderivative = 1.26 \[ \int x^2 \text {sech}(a+b x) \tanh ^2(a+b x) \, dx=\frac {i \left (-4 i \arctan \left (e^{a+b x}\right )+b^2 x^2 \log \left (1-i e^{a+b x}\right )-b^2 x^2 \log \left (1+i e^{a+b x}\right )-2 b x \operatorname {PolyLog}\left (2,-i e^{a+b x}\right )+2 b x \operatorname {PolyLog}\left (2,i e^{a+b x}\right )+2 \operatorname {PolyLog}\left (3,-i e^{a+b x}\right )-2 \operatorname {PolyLog}\left (3,i e^{a+b x}\right )\right )}{2 b^3}-\frac {x \text {sech}(a) \text {sech}(a+b x) (2 \cosh (a)+b x \sinh (a))}{2 b^2}-\frac {x^2 \text {sech}(a) \text {sech}^2(a+b x) \sinh (b x)}{2 b} \] Input:

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

Output:

((I/2)*((-4*I)*ArcTan[E^(a + b*x)] + 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*Po 
lyLog[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*Sech[a]*Sech[a + b*x]*(2*Cosh[a] + b*x*Sinh[a]))/( 
2*b^2) - (x^2*Sech[a]*Sech[a + b*x]^2*Sinh[b*x])/(2*b)
 

Rubi [A] (verified)

Time = 1.59 (sec) , antiderivative size = 259, normalized size of antiderivative = 1.81, number of steps used = 13, number of rules used = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.667, Rules used = {5978, 3042, 4668, 3011, 2720, 4674, 3042, 4257, 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.

\(\displaystyle \int x^2 \tanh ^2(a+b x) \text {sech}(a+b x) \, dx\)

\(\Big \downarrow \) 5978

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 4668

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

\(\Big \downarrow \) 3011

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

\(\Big \downarrow \) 2720

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

\(\Big \downarrow \) 4674

\(\displaystyle \frac {2 i \left (\frac {\int e^{-a-b x} \operatorname {PolyLog}\left (2,-i e^{a+b x}\right )de^{a+b x}}{b^2}-\frac {x \operatorname {PolyLog}\left (2,-i e^{a+b x}\right )}{b}\right )}{b}-\frac {2 i \left (\frac {\int e^{-a-b x} \operatorname {PolyLog}\left (2,i e^{a+b x}\right )de^{a+b x}}{b^2}-\frac {x \operatorname {PolyLog}\left (2,i e^{a+b x}\right )}{b}\right )}{b}+\frac {\int \text {sech}(a+b x)dx}{b^2}-\frac {1}{2} \int x^2 \text {sech}(a+b x)dx+\frac {2 x^2 \arctan \left (e^{a+b x}\right )}{b}-\frac {x \text {sech}(a+b x)}{b^2}-\frac {x^2 \tanh (a+b x) \text {sech}(a+b x)}{2 b}\)

\(\Big \downarrow \) 3042

\(\displaystyle \frac {2 i \left (\frac {\int e^{-a-b x} \operatorname {PolyLog}\left (2,-i e^{a+b x}\right )de^{a+b x}}{b^2}-\frac {x \operatorname {PolyLog}\left (2,-i e^{a+b x}\right )}{b}\right )}{b}-\frac {2 i \left (\frac {\int e^{-a-b x} \operatorname {PolyLog}\left (2,i e^{a+b x}\right )de^{a+b x}}{b^2}-\frac {x \operatorname {PolyLog}\left (2,i e^{a+b x}\right )}{b}\right )}{b}+\frac {\int \csc \left (i a+i b x+\frac {\pi }{2}\right )dx}{b^2}-\frac {1}{2} \int x^2 \csc \left (i a+i b x+\frac {\pi }{2}\right )dx+\frac {2 x^2 \arctan \left (e^{a+b x}\right )}{b}-\frac {x \text {sech}(a+b x)}{b^2}-\frac {x^2 \tanh (a+b x) \text {sech}(a+b x)}{2 b}\)

\(\Big \downarrow \) 4257

\(\displaystyle \frac {2 i \left (\frac {\int e^{-a-b x} \operatorname {PolyLog}\left (2,-i e^{a+b x}\right )de^{a+b x}}{b^2}-\frac {x \operatorname {PolyLog}\left (2,-i e^{a+b x}\right )}{b}\right )}{b}-\frac {2 i \left (\frac {\int e^{-a-b x} \operatorname {PolyLog}\left (2,i e^{a+b x}\right )de^{a+b x}}{b^2}-\frac {x \operatorname {PolyLog}\left (2,i e^{a+b x}\right )}{b}\right )}{b}-\frac {1}{2} \int x^2 \csc \left (i a+i b x+\frac {\pi }{2}\right )dx+\frac {\arctan (\sinh (a+b x))}{b^3}+\frac {2 x^2 \arctan \left (e^{a+b x}\right )}{b}-\frac {x \text {sech}(a+b x)}{b^2}-\frac {x^2 \tanh (a+b x) \text {sech}(a+b x)}{2 b}\)

\(\Big \downarrow \) 4668

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

\(\Big \downarrow \) 3011

\(\displaystyle \frac {1}{2} \left (-\frac {2 i \left (\frac {\int \operatorname {PolyLog}\left (2,-i e^{a+b x}\right )dx}{b}-\frac {x \operatorname {PolyLog}\left (2,-i e^{a+b x}\right )}{b}\right )}{b}+\frac {2 i \left (\frac {\int \operatorname {PolyLog}\left (2,i e^{a+b x}\right )dx}{b}-\frac {x \operatorname {PolyLog}\left (2,i e^{a+b x}\right )}{b}\right )}{b}-\frac {2 x^2 \arctan \left (e^{a+b x}\right )}{b}\right )+\frac {2 i \left (\frac {\int e^{-a-b x} \operatorname {PolyLog}\left (2,-i e^{a+b x}\right )de^{a+b x}}{b^2}-\frac {x \operatorname {PolyLog}\left (2,-i e^{a+b x}\right )}{b}\right )}{b}-\frac {2 i \left (\frac {\int e^{-a-b x} \operatorname {PolyLog}\left (2,i e^{a+b x}\right )de^{a+b x}}{b^2}-\frac {x \operatorname {PolyLog}\left (2,i e^{a+b x}\right )}{b}\right )}{b}+\frac {\arctan (\sinh (a+b x))}{b^3}+\frac {2 x^2 \arctan \left (e^{a+b x}\right )}{b}-\frac {x \text {sech}(a+b x)}{b^2}-\frac {x^2 \tanh (a+b x) \text {sech}(a+b x)}{2 b}\)

\(\Big \downarrow \) 2720

\(\displaystyle \frac {1}{2} \left (-\frac {2 i \left (\frac {\int e^{-a-b x} \operatorname {PolyLog}\left (2,-i e^{a+b x}\right )de^{a+b x}}{b^2}-\frac {x \operatorname {PolyLog}\left (2,-i e^{a+b x}\right )}{b}\right )}{b}+\frac {2 i \left (\frac {\int e^{-a-b x} \operatorname {PolyLog}\left (2,i e^{a+b x}\right )de^{a+b x}}{b^2}-\frac {x \operatorname {PolyLog}\left (2,i e^{a+b x}\right )}{b}\right )}{b}-\frac {2 x^2 \arctan \left (e^{a+b x}\right )}{b}\right )+\frac {2 i \left (\frac {\int e^{-a-b x} \operatorname {PolyLog}\left (2,-i e^{a+b x}\right )de^{a+b x}}{b^2}-\frac {x \operatorname {PolyLog}\left (2,-i e^{a+b x}\right )}{b}\right )}{b}-\frac {2 i \left (\frac {\int e^{-a-b x} \operatorname {PolyLog}\left (2,i e^{a+b x}\right )de^{a+b x}}{b^2}-\frac {x \operatorname {PolyLog}\left (2,i e^{a+b x}\right )}{b}\right )}{b}+\frac {\arctan (\sinh (a+b x))}{b^3}+\frac {2 x^2 \arctan \left (e^{a+b x}\right )}{b}-\frac {x \text {sech}(a+b x)}{b^2}-\frac {x^2 \tanh (a+b x) \text {sech}(a+b x)}{2 b}\)

\(\Big \downarrow \) 7143

\(\displaystyle \frac {\arctan (\sinh (a+b x))}{b^3}+\frac {1}{2} \left (-\frac {2 x^2 \arctan \left (e^{a+b x}\right )}{b}-\frac {2 i \left (\frac {\operatorname {PolyLog}\left (3,-i e^{a+b x}\right )}{b^2}-\frac {x \operatorname {PolyLog}\left (2,-i e^{a+b x}\right )}{b}\right )}{b}+\frac {2 i \left (\frac {\operatorname {PolyLog}\left (3,i e^{a+b x}\right )}{b^2}-\frac {x \operatorname {PolyLog}\left (2,i e^{a+b x}\right )}{b}\right )}{b}\right )+\frac {2 x^2 \arctan \left (e^{a+b x}\right )}{b}+\frac {2 i \left (\frac {\operatorname {PolyLog}\left (3,-i e^{a+b x}\right )}{b^2}-\frac {x \operatorname {PolyLog}\left (2,-i e^{a+b x}\right )}{b}\right )}{b}-\frac {2 i \left (\frac {\operatorname {PolyLog}\left (3,i e^{a+b x}\right )}{b^2}-\frac {x \operatorname {PolyLog}\left (2,i e^{a+b x}\right )}{b}\right )}{b}-\frac {x \text {sech}(a+b x)}{b^2}-\frac {x^2 \tanh (a+b x) \text {sech}(a+b x)}{2 b}\)

Input:

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

Output:

(2*x^2*ArcTan[E^(a + b*x)])/b + ArcTan[Sinh[a + b*x]]/b^3 + ((2*I)*(-((x*P 
olyLog[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 + ((-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)/2 - (x*Sech[a + 
 b*x])/b^2 - (x^2*Sech[a + b*x]*Tanh[a + b*x])/(2*b)
 

Defintions of rubi rules used

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 4257

Int[csc[(c_.) + (d_.)*(x_)], x_Symbol] :> Simp[-ArcTanh[Cos[c + d*x]]/d, x] 
 /; FreeQ[{c, d}, x]
 

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 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 5978

Int[((c_.) + (d_.)*(x_))^(m_.)*Sech[(a_.) + (b_.)*(x_)]*Tanh[(a_.) + (b_.)* 
(x_)]^(p_), x_Symbol] :> Int[(c + d*x)^m*Sech[a + b*x]*Tanh[a + b*x]^(p - 2 
), x] - Int[(c + d*x)^m*Sech[a + b*x]^3*Tanh[a + b*x]^(p - 2), x] /; FreeQ[ 
{a, b, c, d, m}, x] && IGtQ[p/2, 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]
 

Maple [F]

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

Input:

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

Output:

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

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. 1577 vs. \(2 (118) = 236\).

Time = 0.15 (sec) , antiderivative size = 1577, normalized size of antiderivative = 11.03 \[ \int x^2 \text {sech}(a+b x) \tanh ^2(a+b x) \, dx=\text {Too large to display} \] Input:

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

Output:

-1/2*(2*(b^2*x^2 + 2*b*x)*cosh(b*x + a)^3 + 6*(b^2*x^2 + 2*b*x)*cosh(b*x + 
 a)*sinh(b*x + a)^2 + 2*(b^2*x^2 + 2*b*x)*sinh(b*x + a)^3 - 2*(b^2*x^2 - 2 
*b*x)*cosh(b*x + a) + 2*(-I*b*x*cosh(b*x + a)^4 - 4*I*b*x*cosh(b*x + a)*si 
nh(b*x + a)^3 - I*b*x*sinh(b*x + a)^4 - 2*I*b*x*cosh(b*x + a)^2 + 2*(-3*I* 
b*x*cosh(b*x + a)^2 - I*b*x)*sinh(b*x + a)^2 - I*b*x + 4*(-I*b*x*cosh(b*x 
+ a)^3 - I*b*x*cosh(b*x + a))*sinh(b*x + a))*dilog(I*cosh(b*x + a) + I*sin 
h(b*x + a)) + 2*(I*b*x*cosh(b*x + a)^4 + 4*I*b*x*cosh(b*x + a)*sinh(b*x + 
a)^3 + I*b*x*sinh(b*x + a)^4 + 2*I*b*x*cosh(b*x + a)^2 + 2*(3*I*b*x*cosh(b 
*x + a)^2 + I*b*x)*sinh(b*x + a)^2 + I*b*x + 4*(I*b*x*cosh(b*x + a)^3 + I* 
b*x*cosh(b*x + a))*sinh(b*x + a))*dilog(-I*cosh(b*x + a) - I*sinh(b*x + a) 
) - ((I*a^2 + 2*I)*cosh(b*x + a)^4 - 4*(-I*a^2 - 2*I)*cosh(b*x + a)*sinh(b 
*x + a)^3 + (I*a^2 + 2*I)*sinh(b*x + a)^4 - 2*(-I*a^2 - 2*I)*cosh(b*x + a) 
^2 - 2*(3*(-I*a^2 - 2*I)*cosh(b*x + a)^2 - I*a^2 - 2*I)*sinh(b*x + a)^2 + 
I*a^2 - 4*((-I*a^2 - 2*I)*cosh(b*x + a)^3 + (-I*a^2 - 2*I)*cosh(b*x + a))* 
sinh(b*x + a) + 2*I)*log(cosh(b*x + a) + sinh(b*x + a) + I) - ((-I*a^2 - 2 
*I)*cosh(b*x + a)^4 - 4*(I*a^2 + 2*I)*cosh(b*x + a)*sinh(b*x + a)^3 + (-I* 
a^2 - 2*I)*sinh(b*x + a)^4 - 2*(I*a^2 + 2*I)*cosh(b*x + a)^2 - 2*(3*(I*a^2 
 + 2*I)*cosh(b*x + a)^2 + I*a^2 + 2*I)*sinh(b*x + a)^2 - I*a^2 - 4*((I*a^2 
 + 2*I)*cosh(b*x + a)^3 + (I*a^2 + 2*I)*cosh(b*x + a))*sinh(b*x + a) - 2*I 
)*log(cosh(b*x + a) + sinh(b*x + a) - I) - ((-I*b^2*x^2 + I*a^2)*cosh(b...
 

Sympy [F]

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

integrate(x**2*sech(b*x+a)*tanh(b*x+a)**2,x)
 

Output:

Integral(x**2*tanh(a + b*x)**2*sech(a + b*x), x)
 

Maxima [F]

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

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

Output:

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

Giac [F]

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

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

Output:

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

\[ \int x^2 \text {sech}(a+b x) \tanh ^2(a+b x) \, dx=\frac {\frac {44 e^{4 b x +4 a} \mathit {atan} \left (e^{b x +a}\right )}{9}+\frac {88 e^{2 b x +2 a} \mathit {atan} \left (e^{b x +a}\right )}{9}+\frac {44 \mathit {atan} \left (e^{b x +a}\right )}{9}+\frac {8 e^{4 b x +5 a} \left (\int \frac {e^{b x} x^{2}}{e^{6 b x +6 a}+3 e^{4 b x +4 a}+3 e^{2 b x +2 a}+1}d x \right ) b^{3}}{3}+\frac {64 e^{4 b x +5 a} \left (\int \frac {e^{b x} x}{e^{6 b x +6 a}+3 e^{4 b x +4 a}+3 e^{2 b x +2 a}+1}d x \right ) b^{2}}{9}-2 e^{3 b x +3 a} b^{2} x^{2}-4 e^{3 b x +3 a} b x +\frac {8 e^{3 b x +3 a}}{9}+\frac {16 e^{2 b x +3 a} \left (\int \frac {e^{b x} x^{2}}{e^{6 b x +6 a}+3 e^{4 b x +4 a}+3 e^{2 b x +2 a}+1}d x \right ) b^{3}}{3}+\frac {128 e^{2 b x +3 a} \left (\int \frac {e^{b x} x}{e^{6 b x +6 a}+3 e^{4 b x +4 a}+3 e^{2 b x +2 a}+1}d x \right ) b^{2}}{9}-\frac {2 e^{b x +a} b^{2} x^{2}}{3}-\frac {52 e^{b x +a} b x}{9}+\frac {8 e^{b x +a}}{9}+\frac {8 e^{a} \left (\int \frac {e^{b x} x^{2}}{e^{6 b x +6 a}+3 e^{4 b x +4 a}+3 e^{2 b x +2 a}+1}d x \right ) b^{3}}{3}+\frac {64 e^{a} \left (\int \frac {e^{b x} x}{e^{6 b x +6 a}+3 e^{4 b x +4 a}+3 e^{2 b x +2 a}+1}d x \right ) b^{2}}{9}}{b^{3} \left (e^{4 b x +4 a}+2 e^{2 b x +2 a}+1\right )} \] Input:

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

Output:

(2*(22*e**(4*a + 4*b*x)*atan(e**(a + b*x)) + 44*e**(2*a + 2*b*x)*atan(e**( 
a + b*x)) + 22*atan(e**(a + b*x)) + 12*e**(5*a + 4*b*x)*int((e**(b*x)*x**2 
)/(e**(6*a + 6*b*x) + 3*e**(4*a + 4*b*x) + 3*e**(2*a + 2*b*x) + 1),x)*b**3 
 + 32*e**(5*a + 4*b*x)*int((e**(b*x)*x)/(e**(6*a + 6*b*x) + 3*e**(4*a + 4* 
b*x) + 3*e**(2*a + 2*b*x) + 1),x)*b**2 - 9*e**(3*a + 3*b*x)*b**2*x**2 - 18 
*e**(3*a + 3*b*x)*b*x + 4*e**(3*a + 3*b*x) + 24*e**(3*a + 2*b*x)*int((e**( 
b*x)*x**2)/(e**(6*a + 6*b*x) + 3*e**(4*a + 4*b*x) + 3*e**(2*a + 2*b*x) + 1 
),x)*b**3 + 64*e**(3*a + 2*b*x)*int((e**(b*x)*x)/(e**(6*a + 6*b*x) + 3*e** 
(4*a + 4*b*x) + 3*e**(2*a + 2*b*x) + 1),x)*b**2 - 3*e**(a + b*x)*b**2*x**2 
 - 26*e**(a + b*x)*b*x + 4*e**(a + b*x) + 12*e**a*int((e**(b*x)*x**2)/(e** 
(6*a + 6*b*x) + 3*e**(4*a + 4*b*x) + 3*e**(2*a + 2*b*x) + 1),x)*b**3 + 32* 
e**a*int((e**(b*x)*x)/(e**(6*a + 6*b*x) + 3*e**(4*a + 4*b*x) + 3*e**(2*a + 
 2*b*x) + 1),x)*b**2))/(9*b**3*(e**(4*a + 4*b*x) + 2*e**(2*a + 2*b*x) + 1) 
)