\(\int x^2 \text {csch}(a+b x) \text {sech}^2(a+b x) \, dx\) [168]

Optimal result

Integrand size = 18, antiderivative size = 146 \[ \int x^2 \text {csch}(a+b x) \text {sech}^2(a+b x) \, dx=-\frac {4 x \arctan \left (e^{a+b x}\right )}{b^2}-\frac {2 x^2 \text {arctanh}\left (e^{a+b x}\right )}{b}-\frac {2 x \operatorname {PolyLog}\left (2,-e^{a+b x}\right )}{b^2}+\frac {2 i \operatorname {PolyLog}\left (2,-i e^{a+b x}\right )}{b^3}-\frac {2 i \operatorname {PolyLog}\left (2,i e^{a+b x}\right )}{b^3}+\frac {2 x \operatorname {PolyLog}\left (2,e^{a+b x}\right )}{b^2}+\frac {2 \operatorname {PolyLog}\left (3,-e^{a+b x}\right )}{b^3}-\frac {2 \operatorname {PolyLog}\left (3,e^{a+b x}\right )}{b^3}+\frac {x^2 \text {sech}(a+b x)}{b} \] Output:

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

Mathematica [A] (verified)

Time = 0.18 (sec) , antiderivative size = 181, normalized size of antiderivative = 1.24 \[ \int x^2 \text {csch}(a+b x) \text {sech}^2(a+b x) \, dx=\frac {b^2 x^2 \log \left (1-e^{a+b x}\right )-2 i b x \log \left (1-i e^{a+b x}\right )+2 i b x \log \left (1+i e^{a+b x}\right )-b^2 x^2 \log \left (1+e^{a+b x}\right )-2 b x \operatorname {PolyLog}\left (2,-e^{a+b x}\right )+2 i \operatorname {PolyLog}\left (2,-i e^{a+b x}\right )-2 i \operatorname {PolyLog}\left (2,i e^{a+b x}\right )+2 b x \operatorname {PolyLog}\left (2,e^{a+b x}\right )+2 \operatorname {PolyLog}\left (3,-e^{a+b x}\right )-2 \operatorname {PolyLog}\left (3,e^{a+b x}\right )+b^2 x^2 \text {sech}(a+b x)}{b^3} \] Input:

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

Output:

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

Rubi [A] (verified)

Time = 0.52 (sec) , antiderivative size = 179, normalized size of antiderivative = 1.23, 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.

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

\(\Big \downarrow \) 5985

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

\(\Big \downarrow \) 25

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

\(\Big \downarrow \) 2010

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

\(\Big \downarrow \) 2009

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

Input:

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

Output:

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

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]

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

Input:

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

Output:

int(x^2*csch(b*x+a)*sech(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. 937 vs. \(2 (126) = 252\).

Time = 0.10 (sec) , antiderivative size = 937, normalized size of antiderivative = 6.42 \[ \int x^2 \text {csch}(a+b x) \text {sech}^2(a+b x) \, dx=\text {Too large to display} \] Input:

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

Output:

(2*b^2*x^2*cosh(b*x + a) + 2*b^2*x^2*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)*dilog(c 
osh(b*x + a) + sinh(b*x + a)) - 2*(I*cosh(b*x + a)^2 + 2*I*cosh(b*x + a)*s 
inh(b*x + a) + I*sinh(b*x + a)^2 + I)*dilog(I*cosh(b*x + a) + I*sinh(b*x + 
 a)) - 2*(-I*cosh(b*x + a)^2 - 2*I*cosh(b*x + a)*sinh(b*x + a) - I*sinh(b* 
x + a)^2 - I)*dilog(-I*cosh(b*x + a) - I*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)*di 
log(-cosh(b*x + a) - sinh(b*x + a)) - (b^2*x^2*cosh(b*x + a)^2 + 2*b^2*x^2 
*cosh(b*x + a)*sinh(b*x + a) + b^2*x^2*sinh(b*x + a)^2 + b^2*x^2)*log(cosh 
(b*x + a) + sinh(b*x + a) + 1) - 2*(-I*a*cosh(b*x + a)^2 - 2*I*a*cosh(b*x 
+ a)*sinh(b*x + a) - I*a*sinh(b*x + a)^2 - I*a)*log(cosh(b*x + a) + sinh(b 
*x + a) + I) - 2*(I*a*cosh(b*x + a)^2 + 2*I*a*cosh(b*x + a)*sinh(b*x + a) 
+ I*a*sinh(b*x + a)^2 + I*a)*log(cosh(b*x + a) + sinh(b*x + a) - I) + (a^2 
*cosh(b*x + a)^2 + 2*a^2*cosh(b*x + a)*sinh(b*x + a) + a^2*sinh(b*x + a)^2 
 + a^2)*log(cosh(b*x + a) + sinh(b*x + a) - 1) - 2*((-I*b*x - I*a)*cosh(b* 
x + a)^2 + 2*(-I*b*x - I*a)*cosh(b*x + a)*sinh(b*x + a) + (-I*b*x - I*a)*s 
inh(b*x + a)^2 - I*b*x - I*a)*log(I*cosh(b*x + a) + I*sinh(b*x + a) + 1) - 
 2*((I*b*x + I*a)*cosh(b*x + a)^2 + 2*(I*b*x + I*a)*cosh(b*x + a)*sinh(b*x 
 + a) + (I*b*x + I*a)*sinh(b*x + a)^2 + I*b*x + I*a)*log(-I*cosh(b*x + a) 
- I*sinh(b*x + a) + 1) + (b^2*x^2 + (b^2*x^2 - a^2)*cosh(b*x + a)^2 + 2...
 

Sympy [F]

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

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

Output:

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

Maxima [F]

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

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

Output:

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

Giac [F(-1)]

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

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

Output:

Timed out
 

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

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

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

Output:

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