3.5.40 \(\int x^2 \cosh (a+b x) \coth ^2(a+b x) \, dx\) [440]

3.5.40.1 Optimal result

Integrand size = 18, antiderivative size = 95 \[ \int x^2 \cosh (a+b x) \coth ^2(a+b x) \, dx=-\frac {4 x \text {arctanh}\left (e^{a+b x}\right )}{b^2}-\frac {2 x \cosh (a+b x)}{b^2}-\frac {x^2 \text {csch}(a+b x)}{b}-\frac {2 \operatorname {PolyLog}\left (2,-e^{a+b x}\right )}{b^3}+\frac {2 \operatorname {PolyLog}\left (2,e^{a+b x}\right )}{b^3}+\frac {2 \sinh (a+b x)}{b^3}+\frac {x^2 \sinh (a+b x)}{b} \]

output

-4*x*arctanh(exp(b*x+a))/b^2-2*x*cosh(b*x+a)/b^2-x^2*csch(b*x+a)/b-2*polyl 
og(2,-exp(b*x+a))/b^3+2*polylog(2,exp(b*x+a))/b^3+2*sinh(b*x+a)/b^3+x^2*si 
nh(b*x+a)/b
 

3.5.40.2 Mathematica [A] (verified)

Time = 0.22 (sec) , antiderivative size = 152, normalized size of antiderivative = 1.60 \[ \int x^2 \cosh (a+b x) \coth ^2(a+b x) \, dx=\frac {\text {csch}\left (\frac {1}{2} (a+b x)\right ) \text {sech}\left (\frac {1}{2} (a+b x)\right ) \left (-2-3 b^2 x^2+2 \cosh (2 (a+b x))+b^2 x^2 \cosh (2 (a+b x))+4 b x \log \left (1-e^{a+b x}\right ) \sinh (a+b x)-4 b x \log \left (1+e^{a+b x}\right ) \sinh (a+b x)-4 \operatorname {PolyLog}\left (2,-e^{a+b x}\right ) \sinh (a+b x)+4 \operatorname {PolyLog}\left (2,e^{a+b x}\right ) \sinh (a+b x)-2 b x \sinh (2 (a+b x))\right )}{4 b^3} \]

input

Integrate[x^2*Cosh[a + b*x]*Coth[a + b*x]^2,x]
 

output

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

3.5.40.3 Rubi [C] (verified)

Result contains complex when optimal does not.

Time = 0.77 (sec) , antiderivative size = 121, normalized size of antiderivative = 1.27, number of steps used = 16, number of rules used = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.833, Rules used = {5973, 3042, 3777, 26, 3042, 26, 3777, 3042, 3117, 5942, 3042, 26, 4670, 2715, 2838}

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.

\(\displaystyle \int x^2 \cosh (a+b x) \coth ^2(a+b x) \, dx\)

\(\Big \downarrow \) 5973

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

\(\Big \downarrow \) 3042

\(\displaystyle \int x^2 \coth (a+b x) \text {csch}(a+b x)dx+\int x^2 \sin \left (i a+i b x+\frac {\pi }{2}\right )dx\)

\(\Big \downarrow \) 3777

\(\displaystyle \int x^2 \coth (a+b x) \text {csch}(a+b x)dx-\frac {2 i \int -i x \sinh (a+b x)dx}{b}+\frac {x^2 \sinh (a+b x)}{b}\)

\(\Big \downarrow \) 26

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

\(\Big \downarrow \) 3042

\(\displaystyle \int x^2 \coth (a+b x) \text {csch}(a+b x)dx-\frac {2 \int -i x \sin (i a+i b x)dx}{b}+\frac {x^2 \sinh (a+b x)}{b}\)

\(\Big \downarrow \) 26

\(\displaystyle \int x^2 \coth (a+b x) \text {csch}(a+b x)dx+\frac {2 i \int x \sin (i a+i b x)dx}{b}+\frac {x^2 \sinh (a+b x)}{b}\)

\(\Big \downarrow \) 3777

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 3117

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

\(\Big \downarrow \) 5942

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 26

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

\(\Big \downarrow \) 4670

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

\(\Big \downarrow \) 2715

\(\displaystyle \frac {2 i \left (\frac {i \int e^{-a-b x} \log \left (1-e^{a+b x}\right )de^{a+b x}}{b^2}-\frac {i \int e^{-a-b x} \log \left (1+e^{a+b x}\right )de^{a+b x}}{b^2}+\frac {2 i x \text {arctanh}\left (e^{a+b x}\right )}{b}\right )}{b}+\frac {2 i \left (\frac {i x \cosh (a+b x)}{b}-\frac {i \sinh (a+b x)}{b^2}\right )}{b}+\frac {x^2 \sinh (a+b x)}{b}-\frac {x^2 \text {csch}(a+b x)}{b}\)

\(\Big \downarrow \) 2838

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

input

Int[x^2*Cosh[a + b*x]*Coth[a + b*x]^2,x]
 

output

-((x^2*Csch[a + b*x])/b) + ((2*I)*(((2*I)*x*ArcTanh[E^(a + b*x)])/b + (I*P 
olyLog[2, -E^(a + b*x)])/b^2 - (I*PolyLog[2, E^(a + b*x)])/b^2))/b + (x^2* 
Sinh[a + b*x])/b + ((2*I)*((I*x*Cosh[a + b*x])/b - (I*Sinh[a + b*x])/b^2)) 
/b
 

3.5.40.3.1 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 2715

Int[Log[(a_) + (b_.)*((F_)^((e_.)*((c_.) + (d_.)*(x_))))^(n_.)], x_Symbol] 
:> Simp[1/(d*e*n*Log[F])   Subst[Int[Log[a + b*x]/x, x], x, (F^(e*(c + d*x) 
))^n], x] /; FreeQ[{F, a, b, c, d, e, n}, x] && GtQ[a, 0]
 

rule 2838

Int[Log[(c_.)*((d_) + (e_.)*(x_)^(n_.))]/(x_), x_Symbol] :> Simp[-PolyLog[2 
, (-c)*e*x^n]/n, x] /; FreeQ[{c, d, e, n}, x] && EqQ[c*d, 1]
 

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 4670

Int[csc[(e_.) + (Complex[0, fz_])*(f_.)*(x_)]*((c_.) + (d_.)*(x_))^(m_.), x 
_Symbol] :> Simp[-2*(c + d*x)^m*(ArcTanh[E^((-I)*e + f*fz*x)]/(f*fz*I)), x] 
 + (-Simp[d*(m/(f*fz*I))   Int[(c + d*x)^(m - 1)*Log[1 - E^((-I)*e + f*fz*x 
)], x], x] + Simp[d*(m/(f*fz*I))   Int[(c + d*x)^(m - 1)*Log[1 + E^((-I)*e 
+ f*fz*x)], x], x]) /; FreeQ[{c, d, e, f, fz}, x] && IGtQ[m, 0]
 

rule 5942

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

rule 5973

Int[Cosh[(a_.) + (b_.)*(x_)]^(n_.)*Coth[(a_.) + (b_.)*(x_)]^(p_.)*((c_.) + 
(d_.)*(x_))^(m_.), x_Symbol] :> Int[(c + d*x)^m*Cosh[a + b*x]^n*Coth[a + b* 
x]^(p - 2), x] + Int[(c + d*x)^m*Cosh[a + b*x]^(n - 2)*Coth[a + b*x]^p, x] 
/; FreeQ[{a, b, c, d, m}, x] && IGtQ[n, 0] && IGtQ[p, 0]
 

3.5.40.4 Maple [A] (verified)

Time = 0.59 (sec) , antiderivative size = 185, normalized size of antiderivative = 1.95

method	result	size
risch	\(\frac {\left (x^{2} b^{2}-2 b x +2\right ) {\mathrm e}^{b x +a}}{2 b^{3}}-\frac {\left (x^{2} b^{2}+2 b x +2\right ) {\mathrm e}^{-b x -a}}{2 b^{3}}-\frac {2 x^{2} {\mathrm e}^{b x +a}}{b \left ({\mathrm e}^{2 b x +2 a}-1\right )}+\frac {2 \ln \left (1-{\mathrm e}^{b x +a}\right ) x}{b^{2}}+\frac {2 \ln \left (1-{\mathrm e}^{b x +a}\right ) a}{b^{3}}+\frac {2 \operatorname {polylog}\left (2, {\mathrm e}^{b x +a}\right )}{b^{3}}-\frac {2 \ln \left ({\mathrm e}^{b x +a}+1\right ) x}{b^{2}}-\frac {2 \ln \left ({\mathrm e}^{b x +a}+1\right ) a}{b^{3}}-\frac {2 \operatorname {polylog}\left (2, -{\mathrm e}^{b x +a}\right )}{b^{3}}+\frac {4 a \,\operatorname {arctanh}\left ({\mathrm e}^{b x +a}\right )}{b^{3}}\)	\(185\)

input

int(x^2*cosh(b*x+a)^3*csch(b*x+a)^2,x,method=_RETURNVERBOSE)
 

output

1/2*(b^2*x^2-2*b*x+2)/b^3*exp(b*x+a)-1/2*(b^2*x^2+2*b*x+2)/b^3*exp(-b*x-a) 
-2/b*x^2*exp(b*x+a)/(exp(2*b*x+2*a)-1)+2/b^2*ln(1-exp(b*x+a))*x+2/b^3*ln(1 
-exp(b*x+a))*a+2*polylog(2,exp(b*x+a))/b^3-2/b^2*ln(exp(b*x+a)+1)*x-2/b^3* 
ln(exp(b*x+a)+1)*a-2*polylog(2,-exp(b*x+a))/b^3+4/b^3*a*arctanh(exp(b*x+a) 
)
 

3.5.40.5 Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 731 vs. \(2 (90) = 180\).

Time = 0.27 (sec) , antiderivative size = 731, normalized size of antiderivative = 7.69 \[ \int x^2 \cosh (a+b x) \coth ^2(a+b x) \, dx=\frac {{\left (b^{2} x^{2} - 2 \, b x + 2\right )} \cosh \left (b x + a\right )^{4} + 4 \, {\left (b^{2} x^{2} - 2 \, b x + 2\right )} \cosh \left (b x + a\right ) \sinh \left (b x + a\right )^{3} + {\left (b^{2} x^{2} - 2 \, b x + 2\right )} \sinh \left (b x + a\right )^{4} + b^{2} x^{2} - 2 \, {\left (3 \, b^{2} x^{2} + 2\right )} \cosh \left (b x + a\right )^{2} - 2 \, {\left (3 \, b^{2} x^{2} - 3 \, {\left (b^{2} x^{2} - 2 \, b x + 2\right )} \cosh \left (b x + a\right )^{2} + 2\right )} \sinh \left (b x + a\right )^{2} + 2 \, b x + 4 \, {\left (\cosh \left (b x + a\right )^{3} + 3 \, \cosh \left (b x + a\right ) \sinh \left (b x + a\right )^{2} + \sinh \left (b x + a\right )^{3} + {\left (3 \, \cosh \left (b x + a\right )^{2} - 1\right )} \sinh \left (b x + a\right ) - \cosh \left (b x + a\right )\right )} {\rm Li}_2\left (\cosh \left (b x + a\right ) + \sinh \left (b x + a\right )\right ) - 4 \, {\left (\cosh \left (b x + a\right )^{3} + 3 \, \cosh \left (b x + a\right ) \sinh \left (b x + a\right )^{2} + \sinh \left (b x + a\right )^{3} + {\left (3 \, \cosh \left (b x + a\right )^{2} - 1\right )} \sinh \left (b x + a\right ) - \cosh \left (b x + a\right )\right )} {\rm Li}_2\left (-\cosh \left (b x + a\right ) - \sinh \left (b x + a\right )\right ) - 4 \, {\left (b x \cosh \left (b x + a\right )^{3} + 3 \, b x \cosh \left (b x + a\right ) \sinh \left (b x + a\right )^{2} + b x \sinh \left (b x + a\right )^{3} - b x \cosh \left (b x + a\right ) + {\left (3 \, b x \cosh \left (b x + a\right )^{2} - b x\right )} \sinh \left (b x + a\right )\right )} \log \left (\cosh \left (b x + a\right ) + \sinh \left (b x + a\right ) + 1\right ) - 4 \, {\left (a \cosh \left (b x + a\right )^{3} + 3 \, a \cosh \left (b x + a\right ) \sinh \left (b x + a\right )^{2} + a \sinh \left (b x + a\right )^{3} - a \cosh \left (b x + a\right ) + {\left (3 \, a \cosh \left (b x + a\right )^{2} - a\right )} \sinh \left (b x + a\right )\right )} \log \left (\cosh \left (b x + a\right ) + \sinh \left (b x + a\right ) - 1\right ) + 4 \, {\left ({\left (b x + a\right )} \cosh \left (b x + a\right )^{3} + 3 \, {\left (b x + a\right )} \cosh \left (b x + a\right ) \sinh \left (b x + a\right )^{2} + {\left (b x + a\right )} \sinh \left (b x + a\right )^{3} - {\left (b x + a\right )} \cosh \left (b x + a\right ) + {\left (3 \, {\left (b x + a\right )} \cosh \left (b x + a\right )^{2} - b x - a\right )} \sinh \left (b x + a\right )\right )} \log \left (-\cosh \left (b x + a\right ) - \sinh \left (b x + a\right ) + 1\right ) + 4 \, {\left ({\left (b^{2} x^{2} - 2 \, b x + 2\right )} \cosh \left (b x + a\right )^{3} - {\left (3 \, b^{2} x^{2} + 2\right )} \cosh \left (b x + a\right )\right )} \sinh \left (b x + a\right ) + 2}{2 \, {\left (b^{3} \cosh \left (b x + a\right )^{3} + 3 \, b^{3} \cosh \left (b x + a\right ) \sinh \left (b x + a\right )^{2} + b^{3} \sinh \left (b x + a\right )^{3} - b^{3} \cosh \left (b x + a\right ) + {\left (3 \, b^{3} \cosh \left (b x + a\right )^{2} - b^{3}\right )} \sinh \left (b x + a\right )\right )}} \]

input

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

output

1/2*((b^2*x^2 - 2*b*x + 2)*cosh(b*x + a)^4 + 4*(b^2*x^2 - 2*b*x + 2)*cosh( 
b*x + a)*sinh(b*x + a)^3 + (b^2*x^2 - 2*b*x + 2)*sinh(b*x + a)^4 + b^2*x^2 
 - 2*(3*b^2*x^2 + 2)*cosh(b*x + a)^2 - 2*(3*b^2*x^2 - 3*(b^2*x^2 - 2*b*x + 
 2)*cosh(b*x + a)^2 + 2)*sinh(b*x + a)^2 + 2*b*x + 4*(cosh(b*x + a)^3 + 3* 
cosh(b*x + a)*sinh(b*x + a)^2 + sinh(b*x + a)^3 + (3*cosh(b*x + a)^2 - 1)* 
sinh(b*x + a) - cosh(b*x + a))*dilog(cosh(b*x + a) + sinh(b*x + a)) - 4*(c 
osh(b*x + a)^3 + 3*cosh(b*x + a)*sinh(b*x + a)^2 + sinh(b*x + a)^3 + (3*co 
sh(b*x + a)^2 - 1)*sinh(b*x + a) - cosh(b*x + a))*dilog(-cosh(b*x + a) - s 
inh(b*x + a)) - 4*(b*x*cosh(b*x + a)^3 + 3*b*x*cosh(b*x + a)*sinh(b*x + a) 
^2 + b*x*sinh(b*x + a)^3 - b*x*cosh(b*x + a) + (3*b*x*cosh(b*x + a)^2 - b* 
x)*sinh(b*x + a))*log(cosh(b*x + a) + sinh(b*x + a) + 1) - 4*(a*cosh(b*x + 
 a)^3 + 3*a*cosh(b*x + a)*sinh(b*x + a)^2 + a*sinh(b*x + a)^3 - a*cosh(b*x 
 + a) + (3*a*cosh(b*x + a)^2 - a)*sinh(b*x + a))*log(cosh(b*x + a) + sinh( 
b*x + a) - 1) + 4*((b*x + a)*cosh(b*x + a)^3 + 3*(b*x + a)*cosh(b*x + a)*s 
inh(b*x + a)^2 + (b*x + a)*sinh(b*x + a)^3 - (b*x + a)*cosh(b*x + a) + (3* 
(b*x + a)*cosh(b*x + a)^2 - b*x - a)*sinh(b*x + a))*log(-cosh(b*x + a) - s 
inh(b*x + a) + 1) + 4*((b^2*x^2 - 2*b*x + 2)*cosh(b*x + a)^3 - (3*b^2*x^2 
+ 2)*cosh(b*x + a))*sinh(b*x + a) + 2)/(b^3*cosh(b*x + a)^3 + 3*b^3*cosh(b 
*x + a)*sinh(b*x + a)^2 + b^3*sinh(b*x + a)^3 - b^3*cosh(b*x + a) + (3*b^3 
*cosh(b*x + a)^2 - b^3)*sinh(b*x + a))
 

3.5.40.6 Sympy [F(-1)]

Timed out. \[ \int x^2 \cosh (a+b x) \coth ^2(a+b x) \, dx=\text {Timed out} \]

input

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

output

Timed out
 

3.5.40.7 Maxima [A] (verification not implemented)

Time = 0.28 (sec) , antiderivative size = 157, normalized size of antiderivative = 1.65 \[ \int x^2 \cosh (a+b x) \coth ^2(a+b x) \, dx=\frac {{\left (b^{2} x^{2} e^{\left (4 \, a\right )} - 2 \, b x e^{\left (4 \, a\right )} + 2 \, e^{\left (4 \, a\right )}\right )} e^{\left (3 \, b x\right )} - 2 \, {\left (3 \, b^{2} x^{2} e^{\left (2 \, a\right )} + 2 \, e^{\left (2 \, a\right )}\right )} e^{\left (b x\right )} + {\left (b^{2} x^{2} + 2 \, b x + 2\right )} e^{\left (-b x\right )}}{2 \, {\left (b^{3} e^{\left (2 \, b x + 3 \, a\right )} - b^{3} e^{a}\right )}} - \frac {2 \, {\left (b x \log \left (e^{\left (b x + a\right )} + 1\right ) + {\rm Li}_2\left (-e^{\left (b x + a\right )}\right )\right )}}{b^{3}} + \frac {2 \, {\left (b x \log \left (-e^{\left (b x + a\right )} + 1\right ) + {\rm Li}_2\left (e^{\left (b x + a\right )}\right )\right )}}{b^{3}} \]

input

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

output

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

3.5.40.8 Giac [F]

\[ \int x^2 \cosh (a+b x) \coth ^2(a+b x) \, dx=\int { x^{2} \cosh \left (b x + a\right )^{3} \operatorname {csch}\left (b x + a\right )^{2} \,d x } \]

input

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

output

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

3.5.40.9 Mupad [F(-1)]

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

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)