\(\int x^2 \text {csch}^2(a+b x) \text {sech}(a+b x) \, dx\) [198]

Optimal result

Integrand size = 18, antiderivative size = 157 \[ \int x^2 \text {csch}^2(a+b x) \text {sech}(a+b x) \, dx=-\frac {2 x^2 \arctan \left (e^{a+b x}\right )}{b}-\frac {4 x \text {arctanh}\left (e^{a+b x}\right )}{b^2}-\frac {x^2 \text {csch}(a+b x)}{b}-\frac {2 \operatorname {PolyLog}\left (2,-e^{a+b x}\right )}{b^3}+\frac {2 i x \operatorname {PolyLog}\left (2,-i e^{a+b x}\right )}{b^2}-\frac {2 i x \operatorname {PolyLog}\left (2,i e^{a+b x}\right )}{b^2}+\frac {2 \operatorname {PolyLog}\left (2,e^{a+b x}\right )}{b^3}-\frac {2 i \operatorname {PolyLog}\left (3,-i e^{a+b x}\right )}{b^3}+\frac {2 i \operatorname {PolyLog}\left (3,i e^{a+b x}\right )}{b^3} \] Output:

-2*x^2*arctan(exp(b*x+a))/b-4*x*arctanh(exp(b*x+a))/b^2-x^2*csch(b*x+a)/b- 
2*polylog(2,-exp(b*x+a))/b^3+2*I*x*polylog(2,-I*exp(b*x+a))/b^2-2*I*x*poly 
log(2,I*exp(b*x+a))/b^2+2*polylog(2,exp(b*x+a))/b^3-2*I*polylog(3,-I*exp(b 
*x+a))/b^3+2*I*polylog(3,I*exp(b*x+a))/b^3
 

Mathematica [A] (verified)

Time = 0.98 (sec) , antiderivative size = 252, normalized size of antiderivative = 1.61 \[ \int x^2 \text {csch}^2(a+b x) \text {sech}(a+b x) \, dx=\frac {-2 b^2 x^2 \text {csch}(a)+4 b x \log \left (1-e^{a+b x}\right )-2 i b^2 x^2 \log \left (1-i e^{a+b x}\right )+2 i b^2 x^2 \log \left (1+i e^{a+b x}\right )-4 b x \log \left (1+e^{a+b x}\right )-4 \operatorname {PolyLog}\left (2,-e^{a+b x}\right )+4 i b x \operatorname {PolyLog}\left (2,-i e^{a+b x}\right )-4 i b x \operatorname {PolyLog}\left (2,i e^{a+b x}\right )+4 \operatorname {PolyLog}\left (2,e^{a+b x}\right )-4 i \operatorname {PolyLog}\left (3,-i e^{a+b x}\right )+4 i \operatorname {PolyLog}\left (3,i e^{a+b x}\right )+b^2 x^2 \text {csch}\left (\frac {a}{2}\right ) \text {csch}\left (\frac {1}{2} (a+b x)\right ) \sinh \left (\frac {b x}{2}\right )+b^2 x^2 \text {sech}\left (\frac {a}{2}\right ) \text {sech}\left (\frac {1}{2} (a+b x)\right ) \sinh \left (\frac {b x}{2}\right )}{2 b^3} \] Input:

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

Output:

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

Rubi [A] (verified)

Time = 0.81 (sec) , antiderivative size = 191, normalized size of antiderivative = 1.22, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.222, Rules used = {5985, 25, 2010, 2009}

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.

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

\(\Big \downarrow \) 5985

\(\displaystyle -2 \int -x \left (\frac {\arctan (\sinh (a+b x))}{b}+\frac {\text {csch}(a+b x)}{b}\right )dx-\frac {x^2 \arctan (\sinh (a+b x))}{b}-\frac {x^2 \text {csch}(a+b x)}{b}\)

\(\Big \downarrow \) 25

\(\displaystyle 2 \int x \left (\frac {\arctan (\sinh (a+b x))}{b}+\frac {\text {csch}(a+b x)}{b}\right )dx-\frac {x^2 \arctan (\sinh (a+b x))}{b}-\frac {x^2 \text {csch}(a+b x)}{b}\)

\(\Big \downarrow \) 2010

\(\displaystyle 2 \int \left (\frac {x \arctan (\sinh (a+b x))}{b}+\frac {x \text {csch}(a+b x)}{b}\right )dx-\frac {x^2 \arctan (\sinh (a+b x))}{b}-\frac {x^2 \text {csch}(a+b x)}{b}\)

\(\Big \downarrow \) 2009

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

Input:

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

Output:

-((x^2*ArcTan[Sinh[a + b*x]])/b) - (x^2*Csch[a + b*x])/b + 2*(-((x^2*ArcTa 
n[E^(a + b*x)])/b) + (x^2*ArcTan[Sinh[a + b*x]])/(2*b) - (2*x*ArcTanh[E^(a 
 + b*x)])/b^2 - PolyLog[2, -E^(a + b*x)]/b^3 + (I*x*PolyLog[2, (-I)*E^(a + 
 b*x)])/b^2 - (I*x*PolyLog[2, I*E^(a + b*x)])/b^2 + PolyLog[2, E^(a + b*x) 
]/b^3 - (I*PolyLog[3, (-I)*E^(a + b*x)])/b^3 + (I*PolyLog[3, I*E^(a + b*x) 
])/b^3)
 

Defintions of rubi rules used

rule 25

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

rule 2009

Int[u_, x_Symbol] :> Simp[IntSum[u, x], x] /; SumQ[u]
 

rule 2010

Int[(u_)*((c_.)*(x_))^(m_.), x_Symbol] :> Int[ExpandIntegrand[(c*x)^m*u, x] 
, x] /; FreeQ[{c, m}, x] && SumQ[u] &&  !LinearQ[u, x] &&  !MatchQ[u, (a_) 
+ (b_.)*(v_) /; FreeQ[{a, b}, x] && InverseFunctionQ[v]]
 

rule 5985

Int[Csch[(a_.) + (b_.)*(x_)]^(n_.)*((c_.) + (d_.)*(x_))^(m_.)*Sech[(a_.) + 
(b_.)*(x_)]^(p_.), x_Symbol] :> With[{u = IntHide[Csch[a + b*x]^n*Sech[a + 
b*x]^p, x]}, Simp[(c + d*x)^m   u, x] - Simp[d*m   Int[(c + d*x)^(m - 1)*u, 
 x], x]] /; FreeQ[{a, b, c, d}, x] && IntegersQ[n, p] && GtQ[m, 0] && NeQ[n 
, p]
 

Maple [F]

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

Input:

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

Output:

int(x^2*csch(b*x+a)^2*sech(b*x+a),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. 966 vs. \(2 (129) = 258\).

Time = 0.10 (sec) , antiderivative size = 966, normalized size of antiderivative = 6.15 \[ \int x^2 \text {csch}^2(a+b x) \text {sech}(a+b x) \, dx=\text {Too large to display} \] Input:

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

Output:

-(2*b^2*x^2*cosh(b*x + a) + 2*b^2*x^2*sinh(b*x + a) - 2*(cosh(b*x + a)^2 + 
 2*cosh(b*x + a)*sinh(b*x + a) + sinh(b*x + a)^2 - 1)*dilog(cosh(b*x + a) 
+ sinh(b*x + a)) + 2*(I*b*x*cosh(b*x + a)^2 + 2*I*b*x*cosh(b*x + a)*sinh(b 
*x + a) + I*b*x*sinh(b*x + a)^2 - I*b*x)*dilog(I*cosh(b*x + a) + I*sinh(b* 
x + a)) + 2*(-I*b*x*cosh(b*x + a)^2 - 2*I*b*x*cosh(b*x + a)*sinh(b*x + a) 
- I*b*x*sinh(b*x + a)^2 + I*b*x)*dilog(-I*cosh(b*x + a) - I*sinh(b*x + a)) 
 + 2*(cosh(b*x + a)^2 + 2*cosh(b*x + a)*sinh(b*x + a) + sinh(b*x + a)^2 - 
1)*dilog(-cosh(b*x + a) - sinh(b*x + a)) + 2*(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 - b*x)*log(cosh(b*x + a) 
 + sinh(b*x + a) + 1) - (-I*a^2*cosh(b*x + a)^2 - 2*I*a^2*cosh(b*x + a)*si 
nh(b*x + a) - I*a^2*sinh(b*x + a)^2 + I*a^2)*log(cosh(b*x + a) + sinh(b*x 
+ a) + I) - (I*a^2*cosh(b*x + a)^2 + 2*I*a^2*cosh(b*x + a)*sinh(b*x + a) + 
 I*a^2*sinh(b*x + a)^2 - I*a^2)*log(cosh(b*x + a) + sinh(b*x + a) - I) + 2 
*(a*cosh(b*x + a)^2 + 2*a*cosh(b*x + a)*sinh(b*x + a) + a*sinh(b*x + a)^2 
- a)*log(cosh(b*x + a) + sinh(b*x + a) - 1) - (-I*b^2*x^2 + (I*b^2*x^2 - I 
*a^2)*cosh(b*x + a)^2 - 2*(-I*b^2*x^2 + I*a^2)*cosh(b*x + a)*sinh(b*x + a) 
 + (I*b^2*x^2 - I*a^2)*sinh(b*x + a)^2 + I*a^2)*log(I*cosh(b*x + a) + I*si 
nh(b*x + a) + 1) - (I*b^2*x^2 + (-I*b^2*x^2 + I*a^2)*cosh(b*x + a)^2 - 2*( 
I*b^2*x^2 - I*a^2)*cosh(b*x + a)*sinh(b*x + a) + (-I*b^2*x^2 + I*a^2)*sinh 
(b*x + a)^2 - I*a^2)*log(-I*cosh(b*x + a) - I*sinh(b*x + a) + 1) - 2*((...
 

Sympy [F]

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

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

Output:

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

Maxima [F]

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

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

Output:

-2*x^2*e^(b*x + a)/(b*e^(2*b*x + 2*a) - b) - 2*(b*x*log(e^(b*x + a) + 1) + 
 dilog(-e^(b*x + a)))/b^3 + 2*(b*x*log(-e^(b*x + a) + 1) + dilog(e^(b*x + 
a)))/b^3 - 8*integrate(1/4*x^2*e^(b*x + a)/(e^(2*b*x + 2*a) + 1), x)
 

Giac [F(-1)]

Timed out. \[ \int x^2 \text {csch}^2(a+b x) \text {sech}(a+b x) \, dx=\text {Timed out} \] Input:

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

Output:

Timed out
 

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

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

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

Output:

int(csch(a + b*x)**2*sech(a + b*x)*x**2,x)