\(\int x^2 \coth (a+b x) \text {csch}(a+b x) \, dx\) [180]

Optimal result

Integrand size = 16, antiderivative size = 59 \[ \int x^2 \coth (a+b x) \text {csch}(a+b x) \, dx=-\frac {4 x \text {arctanh}\left (e^{a+b x}\right )}{b^2}-\frac {x^2 \text {csch}(a+b x)}{b}-\frac {2 \operatorname {PolyLog}\left (2,-e^{a+b x}\right )}{b^3}+\frac {2 \operatorname {PolyLog}\left (2,e^{a+b x}\right )}{b^3} \] Output:

-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*polylog(2,exp(b*x+a))/b^3
 

Mathematica [A] (verified)

Time = 0.21 (sec) , antiderivative size = 69, normalized size of antiderivative = 1.17 \[ \int x^2 \coth (a+b x) \text {csch}(a+b x) \, dx=\frac {-b x \left (b x \text {csch}(a+b x)-2 \log \left (1-e^{a+b x}\right )+2 \log \left (1+e^{a+b x}\right )\right )-2 \operatorname {PolyLog}\left (2,-e^{a+b x}\right )+2 \operatorname {PolyLog}\left (2,e^{a+b x}\right )}{b^3} \] Input:

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

Output:

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

Rubi [C] (verified)

Result contains complex when optimal does not.

Time = 0.38 (sec) , antiderivative size = 73, normalized size of antiderivative = 1.24, number of steps used = 7, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.375, Rules used = {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.

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

\(\Big \downarrow \) 5942

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 26

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

\(\Big \downarrow \) 4670

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

\(\Big \downarrow \) 2715

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

\(\Big \downarrow \) 2838

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

Input:

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

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 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]
 

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(133\) vs. \(2(56)=112\).

Time = 0.57 (sec) , antiderivative size = 134, normalized size of antiderivative = 2.27

method	result	size
risch	\(-\frac {2 x^{2} {\mathrm e}^{b x +a}}{b \left ({\mathrm e}^{2 b x +2 a}-1\right )}-\frac {2 \ln \left ({\mathrm e}^{b x +a}+1\right ) x}{b^{2}}-\frac {2 \ln \left ({\mathrm e}^{b x +a}+1\right ) a}{b^{3}}-\frac {2 \operatorname {polylog}\left (2, -{\mathrm e}^{b x +a}\right )}{b^{3}}+\frac {2 \ln \left (1-{\mathrm e}^{b x +a}\right ) x}{b^{2}}+\frac {2 \ln \left (1-{\mathrm e}^{b x +a}\right ) a}{b^{3}}+\frac {2 \operatorname {polylog}\left (2, {\mathrm e}^{b x +a}\right )}{b^{3}}+\frac {4 a \,\operatorname {arctanh}\left ({\mathrm e}^{b x +a}\right )}{b^{3}}\)	\(134\)

Input:

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

Output:

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

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 367 vs. \(2 (54) = 108\).

Time = 0.10 (sec) , antiderivative size = 367, normalized size of antiderivative = 6.22 \[ \int x^2 \coth (a+b x) \text {csch}(a+b x) \, dx=-\frac {2 \, {\left (b^{2} x^{2} \cosh \left (b x + a\right ) + b^{2} x^{2} \sinh \left (b x + a\right ) - {\left (\cosh \left (b x + a\right )^{2} + 2 \, \cosh \left (b x + a\right ) \sinh \left (b x + a\right ) + \sinh \left (b x + a\right )^{2} - 1\right )} {\rm Li}_2\left (\cosh \left (b x + a\right ) + \sinh \left (b x + a\right )\right ) + {\left (\cosh \left (b x + a\right )^{2} + 2 \, \cosh \left (b x + a\right ) \sinh \left (b x + a\right ) + \sinh \left (b x + a\right )^{2} - 1\right )} {\rm Li}_2\left (-\cosh \left (b x + a\right ) - \sinh \left (b x + a\right )\right ) + {\left (b x \cosh \left (b x + a\right )^{2} + 2 \, b x \cosh \left (b x + a\right ) \sinh \left (b x + a\right ) + b x \sinh \left (b x + a\right )^{2} - b x\right )} \log \left (\cosh \left (b x + a\right ) + \sinh \left (b x + a\right ) + 1\right ) + {\left (a \cosh \left (b x + a\right )^{2} + 2 \, a \cosh \left (b x + a\right ) \sinh \left (b x + a\right ) + a \sinh \left (b x + a\right )^{2} - a\right )} \log \left (\cosh \left (b x + a\right ) + \sinh \left (b x + a\right ) - 1\right ) - {\left ({\left (b x + a\right )} \cosh \left (b x + a\right )^{2} + 2 \, {\left (b x + a\right )} \cosh \left (b x + a\right ) \sinh \left (b x + a\right ) + {\left (b x + a\right )} \sinh \left (b x + a\right )^{2} - b x - a\right )} \log \left (-\cosh \left (b x + a\right ) - \sinh \left (b x + a\right ) + 1\right )\right )}}{b^{3} \cosh \left (b x + a\right )^{2} + 2 \, b^{3} \cosh \left (b x + a\right ) \sinh \left (b x + a\right ) + b^{3} \sinh \left (b x + a\right )^{2} - b^{3}} \] Input:

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

Output:

-2*(b^2*x^2*cosh(b*x + a) + b^2*x^2*sinh(b*x + a) - (cosh(b*x + a)^2 + 2*c 
osh(b*x + a)*sinh(b*x + a) + sinh(b*x + a)^2 - 1)*dilog(cosh(b*x + a) + si 
nh(b*x + a)) + (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)) + (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) + (a*cosh(b*x + a)^2 + 2*a*cosh(b*x + a)*sin 
h(b*x + a) + a*sinh(b*x + a)^2 - a)*log(cosh(b*x + a) + sinh(b*x + a) - 1) 
 - ((b*x + a)*cosh(b*x + a)^2 + 2*(b*x + a)*cosh(b*x + a)*sinh(b*x + a) + 
(b*x + a)*sinh(b*x + a)^2 - b*x - a)*log(-cosh(b*x + a) - sinh(b*x + a) + 
1))/(b^3*cosh(b*x + a)^2 + 2*b^3*cosh(b*x + a)*sinh(b*x + a) + b^3*sinh(b* 
x + a)^2 - b^3)
 

Sympy [F]

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

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

Output:

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

Maxima [A] (verification not implemented)

Time = 0.13 (sec) , antiderivative size = 83, normalized size of antiderivative = 1.41 \[ \int x^2 \coth (a+b x) \text {csch}(a+b x) \, dx=-\frac {2 \, x^{2} e^{\left (b x + a\right )}}{b e^{\left (2 \, b x + 2 \, a\right )} - b} - \frac {2 \, {\left (b x \log \left (e^{\left (b x + a\right )} + 1\right ) + {\rm Li}_2\left (-e^{\left (b x + a\right )}\right )\right )}}{b^{3}} + \frac {2 \, {\left (b x \log \left (-e^{\left (b x + a\right )} + 1\right ) + {\rm Li}_2\left (e^{\left (b x + a\right )}\right )\right )}}{b^{3}} \] Input:

integrate(x^2*coth(b*x+a)*csch(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
 

Giac [F]

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

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

Output:

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

\[ \int x^2 \coth (a+b x) \text {csch}(a+b x) \, dx=\frac {-8 e^{2 b x +3 a} \left (\int \frac {e^{b x} x}{e^{4 b x +4 a}-2 e^{2 b x +2 a}+1}d x \right ) b^{2}+2 e^{2 b x +2 a} \mathrm {log}\left (e^{b x +a}-1\right )-2 e^{2 b x +2 a} \mathrm {log}\left (e^{b x +a}+1\right )-2 e^{b x +a} b^{2} x^{2}-4 e^{b x +a} b x +8 e^{a} \left (\int \frac {e^{b x} x}{e^{4 b x +4 a}-2 e^{2 b x +2 a}+1}d x \right ) b^{2}-2 \,\mathrm {log}\left (e^{b x +a}-1\right )+2 \,\mathrm {log}\left (e^{b x +a}+1\right )}{b^{3} \left (e^{2 b x +2 a}-1\right )} \] Input:

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

Output:

(2*( - 4*e**(3*a + 2*b*x)*int((e**(b*x)*x)/(e**(4*a + 4*b*x) - 2*e**(2*a + 
 2*b*x) + 1),x)*b**2 + e**(2*a + 2*b*x)*log(e**(a + b*x) - 1) - e**(2*a + 
2*b*x)*log(e**(a + b*x) + 1) - e**(a + b*x)*b**2*x**2 - 2*e**(a + b*x)*b*x 
 + 4*e**a*int((e**(b*x)*x)/(e**(4*a + 4*b*x) - 2*e**(2*a + 2*b*x) + 1),x)* 
b**2 - log(e**(a + b*x) - 1) + log(e**(a + b*x) + 1)))/(b**3*(e**(2*a + 2* 
b*x) - 1))