\(\int x^2 \text {csch}^2(a+b x) \text {sech}^3(a+b x) \, dx\) [209]

Optimal result

Integrand size = 20, antiderivative size = 202 \[ \int x^2 \text {csch}^2(a+b x) \text {sech}^3(a+b x) \, dx=-\frac {3 x^2 \arctan \left (e^{a+b x}\right )}{b}+\frac {\arctan (\sinh (a+b x))}{b^3}-\frac {4 x \text {arctanh}\left (e^{a+b x}\right )}{b^2}-\frac {x^2 \text {csch}(a+b x)}{b}-\frac {2 \operatorname {PolyLog}\left (2,-e^{a+b x}\right )}{b^3}+\frac {3 i x \operatorname {PolyLog}\left (2,-i e^{a+b x}\right )}{b^2}-\frac {3 i x \operatorname {PolyLog}\left (2,i e^{a+b x}\right )}{b^2}+\frac {2 \operatorname {PolyLog}\left (2,e^{a+b x}\right )}{b^3}-\frac {3 i \operatorname {PolyLog}\left (3,-i e^{a+b x}\right )}{b^3}+\frac {3 i \operatorname {PolyLog}\left (3,i e^{a+b x}\right )}{b^3}-\frac {x \text {sech}(a+b x)}{b^2}-\frac {x^2 \text {sech}(a+b x) \tanh (a+b x)}{2 b} \] Output:

-3*x^2*arctan(exp(b*x+a))/b+arctan(sinh(b*x+a))/b^3-4*x*arctanh(exp(b*x+a) 
)/b^2-x^2*csch(b*x+a)/b-2*polylog(2,-exp(b*x+a))/b^3+3*I*x*polylog(2,-I*ex 
p(b*x+a))/b^2-3*I*x*polylog(2,I*exp(b*x+a))/b^2+2*polylog(2,exp(b*x+a))/b^ 
3-3*I*polylog(3,-I*exp(b*x+a))/b^3+3*I*polylog(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 = 4.53 (sec) , antiderivative size = 310, normalized size of antiderivative = 1.53 \[ \int x^2 \text {csch}^2(a+b x) \text {sech}^3(a+b x) \, dx=\frac {4 \arctan \left (e^{a+b x}\right )-2 b^2 x^2 \text {csch}(a)+4 b x \log \left (1-e^{a+b x}\right )-3 i b^2 x^2 \log \left (1-i e^{a+b x}\right )+3 i b^2 x^2 \log \left (1+i e^{a+b x}\right )-4 b x \log \left (1+e^{a+b x}\right )-4 \operatorname {PolyLog}\left (2,-e^{a+b x}\right )+6 i b x \operatorname {PolyLog}\left (2,-i e^{a+b x}\right )-6 i b x \operatorname {PolyLog}\left (2,i e^{a+b x}\right )+4 \operatorname {PolyLog}\left (2,e^{a+b x}\right )-6 i \operatorname {PolyLog}\left (3,-i e^{a+b x}\right )+6 i \operatorname {PolyLog}\left (3,i e^{a+b x}\right )-2 b x \text {sech}(a+b x)+b^2 x^2 \text {csch}\left (\frac {a}{2}\right ) \text {csch}\left (\frac {1}{2} (a+b x)\right ) \sinh \left (\frac {b x}{2}\right )+b^2 x^2 \text {sech}\left (\frac {a}{2}\right ) \text {sech}\left (\frac {1}{2} (a+b x)\right ) \sinh \left (\frac {b x}{2}\right )-b^2 x^2 \text {sech}(a) \text {sech}^2(a+b x) \sinh (b x)-b^2 x^2 \text {sech}(a+b x) \tanh (a)}{2 b^3} \] Input:

Integrate[x^2*Csch[a + b*x]^2*Sech[a + b*x]^3,x]
 

Output:

(4*ArcTan[E^(a + b*x)] - 2*b^2*x^2*Csch[a] + 4*b*x*Log[1 - E^(a + b*x)] - 
(3*I)*b^2*x^2*Log[1 - I*E^(a + b*x)] + (3*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)] + (6*I)*b*x*Po 
lyLog[2, (-I)*E^(a + b*x)] - (6*I)*b*x*PolyLog[2, I*E^(a + b*x)] + 4*PolyL 
og[2, E^(a + b*x)] - (6*I)*PolyLog[3, (-I)*E^(a + b*x)] + (6*I)*PolyLog[3, 
 I*E^(a + b*x)] - 2*b*x*Sech[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] - b^2* 
x^2*Sech[a]*Sech[a + b*x]^2*Sinh[b*x] - b^2*x^2*Sech[a + b*x]*Tanh[a])/(2* 
b^3)
 

Rubi [A] (verified)

Time = 0.71 (sec) , antiderivative size = 206, normalized size of antiderivative = 1.02, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.200, Rules used = {5985, 27, 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.

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

\(\Big \downarrow \) 5985

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 2010

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

\(\Big \downarrow \) 2009

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

Input:

Int[x^2*Csch[a + b*x]^2*Sech[a + b*x]^3,x]
 

Output:

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

Defintions of rubi rules used

rule 27

Int[(a_)*(Fx_), x_Symbol] :> Simp[a   Int[Fx, x], x] /; FreeQ[a, x] &&  !Ma 
tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, 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]

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

Input:

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

Output:

int(x^2*csch(b*x+a)^2*sech(b*x+a)^3,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. 3825 vs. \(2 (172) = 344\).

Time = 0.17 (sec) , antiderivative size = 3825, normalized size of antiderivative = 18.94 \[ \int x^2 \text {csch}^2(a+b x) \text {sech}^3(a+b x) \, dx=\text {Too large to display} \] Input:

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

Output:

Too large to include
 

Sympy [F]

\[ \int x^2 \text {csch}^2(a+b x) \text {sech}^3(a+b x) \, dx=\int x^{2} \operatorname {csch}^{2}{\left (a + b x \right )} \operatorname {sech}^{3}{\left (a + b x \right )}\, dx \] Input:

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

Output:

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

Maxima [F]

\[ \int x^2 \text {csch}^2(a+b x) \text {sech}^3(a+b x) \, dx=\int { x^{2} \operatorname {csch}\left (b x + a\right )^{2} \operatorname {sech}\left (b x + a\right )^{3} \,d x } \] Input:

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

Output:

-96*b^2*integrate(1/32*x^2*e^(b*x + a)/(b^2*e^(2*b*x + 2*a) + b^2), x) - ( 
2*b*x^2*e^(3*b*x + 3*a) + (3*b*x^2*e^(5*a) + 2*x*e^(5*a))*e^(5*b*x) + (3*b 
*x^2*e^a - 2*x*e^a)*e^(b*x))/(b^2*e^(6*b*x + 6*a) + b^2*e^(4*b*x + 4*a) - 
b^2*e^(2*b*x + 2*a) - b^2) - 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 + 2*arc 
tan(e^(b*x + a))/b^3
 

Giac [F]

\[ \int x^2 \text {csch}^2(a+b x) \text {sech}^3(a+b x) \, dx=\int { x^{2} \operatorname {csch}\left (b x + a\right )^{2} \operatorname {sech}\left (b x + a\right )^{3} \,d x } \] Input:

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

Output:

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

\[ \int x^2 \text {csch}^2(a+b x) \text {sech}^3(a+b x) \, dx=\text {too large to display} \] Input:

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

Output:

(8*( - 4*e**(6*a + 6*b*x)*atan(e**(a + b*x)) - 4*e**(4*a + 4*b*x)*atan(e** 
(a + b*x)) + 4*e**(2*a + 2*b*x)*atan(e**(a + b*x)) + 4*atan(e**(a + b*x)) 
+ 40*e**(9*a + 6*b*x)*int((e**(3*b*x)*x**2)/(e**(10*a + 10*b*x) + e**(8*a 
+ 8*b*x) - 2*e**(6*a + 6*b*x) - 2*e**(4*a + 4*b*x) + e**(2*a + 2*b*x) + 1) 
,x)*b**3 + 16*e**(9*a + 6*b*x)*int((e**(3*b*x)*x)/(e**(10*a + 10*b*x) + e* 
*(8*a + 8*b*x) - 2*e**(6*a + 6*b*x) - 2*e**(4*a + 4*b*x) + e**(2*a + 2*b*x 
) + 1),x)*b**2 - 20*e**(7*a + 6*b*x)*int((e**(b*x)*x**2)/(e**(10*a + 10*b* 
x) + e**(8*a + 8*b*x) - 2*e**(6*a + 6*b*x) - 2*e**(4*a + 4*b*x) + e**(2*a 
+ 2*b*x) + 1),x)*b**3 - 48*e**(7*a + 6*b*x)*int((e**(b*x)*x)/(e**(10*a + 1 
0*b*x) + e**(8*a + 8*b*x) - 2*e**(6*a + 6*b*x) - 2*e**(4*a + 4*b*x) + e**( 
2*a + 2*b*x) + 1),x)*b**2 + e**(6*a + 6*b*x)*log(e**(a + b*x) - 1) - e**(6 
*a + 6*b*x)*log(e**(a + b*x) + 1) - 2*e**(5*a + 5*b*x) + 40*e**(7*a + 4*b* 
x)*int((e**(3*b*x)*x**2)/(e**(10*a + 10*b*x) + e**(8*a + 8*b*x) - 2*e**(6* 
a + 6*b*x) - 2*e**(4*a + 4*b*x) + e**(2*a + 2*b*x) + 1),x)*b**3 + 16*e**(7 
*a + 4*b*x)*int((e**(3*b*x)*x)/(e**(10*a + 10*b*x) + e**(8*a + 8*b*x) - 2* 
e**(6*a + 6*b*x) - 2*e**(4*a + 4*b*x) + e**(2*a + 2*b*x) + 1),x)*b**2 - 20 
*e**(5*a + 4*b*x)*int((e**(b*x)*x**2)/(e**(10*a + 10*b*x) + e**(8*a + 8*b* 
x) - 2*e**(6*a + 6*b*x) - 2*e**(4*a + 4*b*x) + e**(2*a + 2*b*x) + 1),x)*b* 
*3 - 48*e**(5*a + 4*b*x)*int((e**(b*x)*x)/(e**(10*a + 10*b*x) + e**(8*a + 
8*b*x) - 2*e**(6*a + 6*b*x) - 2*e**(4*a + 4*b*x) + e**(2*a + 2*b*x) + 1...