3.5.6 \(\int x^2 \cosh (a+b x) \coth (a+b x) \, dx\) [406]

3.5.6.1 Optimal result

Integrand size = 16, antiderivative size = 115 \[ \int x^2 \cosh (a+b x) \coth (a+b x) \, dx=-\frac {2 x^2 \text {arctanh}\left (e^{a+b x}\right )}{b}+\frac {2 \cosh (a+b x)}{b^3}+\frac {x^2 \cosh (a+b x)}{b}-\frac {2 x \operatorname {PolyLog}\left (2,-e^{a+b x}\right )}{b^2}+\frac {2 x \operatorname {PolyLog}\left (2,e^{a+b x}\right )}{b^2}+\frac {2 \operatorname {PolyLog}\left (3,-e^{a+b x}\right )}{b^3}-\frac {2 \operatorname {PolyLog}\left (3,e^{a+b x}\right )}{b^3}-\frac {2 x \sinh (a+b x)}{b^2} \]

output

-2*x^2*arctanh(exp(b*x+a))/b+2*cosh(b*x+a)/b^3+x^2*cosh(b*x+a)/b-2*x*polyl 
og(2,-exp(b*x+a))/b^2+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^3-2*x*sinh(b*x+a)/b^2
 

3.5.6.2 Mathematica [A] (verified)

Time = 0.11 (sec) , antiderivative size = 125, normalized size of antiderivative = 1.09 \[ \int x^2 \cosh (a+b x) \coth (a+b x) \, dx=\frac {2 \cosh (a+b x)+b^2 x^2 \cosh (a+b x)+b^2 x^2 \log \left (1-e^{a+b x}\right )-b^2 x^2 \log \left (1+e^{a+b x}\right )-2 b x \operatorname {PolyLog}\left (2,-e^{a+b x}\right )+2 b x \operatorname {PolyLog}\left (2,e^{a+b x}\right )+2 \operatorname {PolyLog}\left (3,-e^{a+b x}\right )-2 \operatorname {PolyLog}\left (3,e^{a+b x}\right )-2 b x \sinh (a+b x)}{b^3} \]

input

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

output

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

3.5.6.3 Rubi [C] (verified)

Result contains complex when optimal does not.

Time = 0.78 (sec) , antiderivative size = 151, normalized size of antiderivative = 1.31, number of steps used = 15, number of rules used = 14, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.875, Rules used = {5973, 3042, 26, 3777, 3042, 3777, 26, 3042, 26, 3118, 4670, 3011, 2720, 7143}

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.

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

\(\Big \downarrow \) 5973

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

\(\Big \downarrow \) 3042

\(\displaystyle \int -i x^2 \sin (i a+i b x)dx+\int i x^2 \csc (i a+i b x)dx\)

\(\Big \downarrow \) 26

\(\displaystyle i \int x^2 \csc (i a+i b x)dx-i \int x^2 \sin (i a+i b x)dx\)

\(\Big \downarrow \) 3777

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 3777

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

\(\Big \downarrow \) 26

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 26

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

\(\Big \downarrow \) 3118

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

\(\Big \downarrow \) 4670

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

\(\Big \downarrow \) 3011

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

\(\Big \downarrow \) 2720

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

\(\Big \downarrow \) 7143

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

input

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

output

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

3.5.6.3.1 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 2720

Int[u_, x_Symbol] :> With[{v = FunctionOfExponential[u, x]}, Simp[v/D[v, x] 
   Subst[Int[FunctionOfExponentialFunction[u, x]/x, x], x, v], x]] /; Funct 
ionOfExponentialQ[u, x] &&  !MatchQ[u, (w_)*((a_.)*(v_)^(n_))^(m_) /; FreeQ 
[{a, m, n}, x] && IntegerQ[m*n]] &&  !MatchQ[u, E^((c_.)*((a_.) + (b_.)*x)) 
*(F_)[v_] /; FreeQ[{a, b, c}, x] && InverseFunctionQ[F[x]]]
 

rule 3011

Int[Log[1 + (e_.)*((F_)^((c_.)*((a_.) + (b_.)*(x_))))^(n_.)]*((f_.) + (g_.) 
*(x_))^(m_.), x_Symbol] :> Simp[(-(f + g*x)^m)*(PolyLog[2, (-e)*(F^(c*(a + 
b*x)))^n]/(b*c*n*Log[F])), x] + Simp[g*(m/(b*c*n*Log[F]))   Int[(f + g*x)^( 
m - 1)*PolyLog[2, (-e)*(F^(c*(a + b*x)))^n], x], x] /; FreeQ[{F, a, b, c, e 
, f, g, n}, x] && GtQ[m, 0]
 

rule 3042

Int[u_, x_Symbol] :> Int[DeactivateTrig[u, x], x] /; FunctionOfTrigOfLinear 
Q[u, x]
 

rule 3118

Int[sin[(c_.) + (d_.)*(x_)], x_Symbol] :> Simp[-Cos[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 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]
 

rule 7143

Int[PolyLog[n_, (c_.)*((a_.) + (b_.)*(x_))^(p_.)]/((d_.) + (e_.)*(x_)), x_S 
ymbol] :> Simp[PolyLog[n + 1, c*(a + b*x)^p]/(e*p), x] /; FreeQ[{a, b, c, d 
, e, n, p}, x] && EqQ[b*d, a*e]
 

3.5.6.4 Maple [A] (verified)

Time = 0.62 (sec) , antiderivative size = 196, normalized size of antiderivative = 1.70

method	result	size
risch	\(\frac {\left (x^{2} b^{2}-2 b x +2\right ) {\mathrm e}^{b x +a}}{2 b^{3}}+\frac {\left (x^{2} b^{2}+2 b x +2\right ) {\mathrm e}^{-b x -a}}{2 b^{3}}-\frac {2 a^{2} \operatorname {arctanh}\left ({\mathrm e}^{b x +a}\right )}{b^{3}}+\frac {\ln \left (1-{\mathrm e}^{b x +a}\right ) x^{2}}{b}-\frac {\ln \left (1-{\mathrm e}^{b x +a}\right ) a^{2}}{b^{3}}+\frac {2 x \operatorname {polylog}\left (2, {\mathrm e}^{b x +a}\right )}{b^{2}}-\frac {2 \operatorname {polylog}\left (3, {\mathrm e}^{b x +a}\right )}{b^{3}}-\frac {\ln \left ({\mathrm e}^{b x +a}+1\right ) x^{2}}{b}+\frac {\ln \left ({\mathrm e}^{b x +a}+1\right ) a^{2}}{b^{3}}-\frac {2 x \operatorname {polylog}\left (2, -{\mathrm e}^{b x +a}\right )}{b^{2}}+\frac {2 \operatorname {polylog}\left (3, -{\mathrm e}^{b x +a}\right )}{b^{3}}\)	\(196\)

input

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

3.5.6.5 Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 391 vs. \(2 (108) = 216\).

Time = 0.26 (sec) , antiderivative size = 391, normalized size of antiderivative = 3.40 \[ \int x^2 \cosh (a+b x) \coth (a+b x) \, dx=\frac {b^{2} x^{2} + {\left (b^{2} x^{2} - 2 \, b x + 2\right )} \cosh \left (b x + a\right )^{2} + 2 \, {\left (b^{2} x^{2} - 2 \, b x + 2\right )} \cosh \left (b x + a\right ) \sinh \left (b x + a\right ) + {\left (b^{2} x^{2} - 2 \, b x + 2\right )} \sinh \left (b x + a\right )^{2} + 2 \, b x + 4 \, {\left (b x \cosh \left (b x + a\right ) + b x \sinh \left (b x + a\right )\right )} {\rm Li}_2\left (\cosh \left (b x + a\right ) + \sinh \left (b x + a\right )\right ) - 4 \, {\left (b x \cosh \left (b x + a\right ) + b x \sinh \left (b x + a\right )\right )} {\rm Li}_2\left (-\cosh \left (b x + a\right ) - \sinh \left (b x + a\right )\right ) - 2 \, {\left (b^{2} x^{2} \cosh \left (b x + a\right ) + b^{2} x^{2} \sinh \left (b x + a\right )\right )} \log \left (\cosh \left (b x + a\right ) + \sinh \left (b x + a\right ) + 1\right ) + 2 \, {\left (a^{2} \cosh \left (b x + a\right ) + a^{2} \sinh \left (b x + a\right )\right )} \log \left (\cosh \left (b x + a\right ) + \sinh \left (b x + a\right ) - 1\right ) + 2 \, {\left ({\left (b^{2} x^{2} - a^{2}\right )} \cosh \left (b x + a\right ) + {\left (b^{2} x^{2} - a^{2}\right )} \sinh \left (b x + a\right )\right )} \log \left (-\cosh \left (b x + a\right ) - \sinh \left (b x + a\right ) + 1\right ) - 4 \, {\left (\cosh \left (b x + a\right ) + \sinh \left (b x + a\right )\right )} {\rm polylog}\left (3, \cosh \left (b x + a\right ) + \sinh \left (b x + a\right )\right ) + 4 \, {\left (\cosh \left (b x + a\right ) + \sinh \left (b x + a\right )\right )} {\rm polylog}\left (3, -\cosh \left (b x + a\right ) - \sinh \left (b x + a\right )\right ) + 2}{2 \, {\left (b^{3} \cosh \left (b x + a\right ) + b^{3} \sinh \left (b x + a\right )\right )}} \]

input

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

output

1/2*(b^2*x^2 + (b^2*x^2 - 2*b*x + 2)*cosh(b*x + a)^2 + 2*(b^2*x^2 - 2*b*x 
+ 2)*cosh(b*x + a)*sinh(b*x + a) + (b^2*x^2 - 2*b*x + 2)*sinh(b*x + a)^2 + 
 2*b*x + 4*(b*x*cosh(b*x + a) + b*x*sinh(b*x + a))*dilog(cosh(b*x + a) + s 
inh(b*x + a)) - 4*(b*x*cosh(b*x + a) + b*x*sinh(b*x + a))*dilog(-cosh(b*x 
+ a) - sinh(b*x + a)) - 2*(b^2*x^2*cosh(b*x + a) + b^2*x^2*sinh(b*x + a))* 
log(cosh(b*x + a) + sinh(b*x + a) + 1) + 2*(a^2*cosh(b*x + a) + a^2*sinh(b 
*x + a))*log(cosh(b*x + a) + sinh(b*x + a) - 1) + 2*((b^2*x^2 - a^2)*cosh( 
b*x + a) + (b^2*x^2 - a^2)*sinh(b*x + a))*log(-cosh(b*x + a) - sinh(b*x + 
a) + 1) - 4*(cosh(b*x + a) + sinh(b*x + a))*polylog(3, cosh(b*x + a) + sin 
h(b*x + a)) + 4*(cosh(b*x + a) + sinh(b*x + a))*polylog(3, -cosh(b*x + a) 
- sinh(b*x + a)) + 2)/(b^3*cosh(b*x + a) + b^3*sinh(b*x + a))
 

3.5.6.6 Sympy [F]

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

input

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

output

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

3.5.6.7 Maxima [A] (verification not implemented)

Time = 0.26 (sec) , antiderivative size = 152, normalized size of antiderivative = 1.32 \[ \int x^2 \cosh (a+b x) \coth (a+b x) \, dx=\frac {{\left ({\left (b^{2} x^{2} e^{\left (2 \, a\right )} - 2 \, b x e^{\left (2 \, a\right )} + 2 \, e^{\left (2 \, a\right )}\right )} e^{\left (b x\right )} + {\left (b^{2} x^{2} + 2 \, b x + 2\right )} e^{\left (-b x\right )}\right )} e^{\left (-a\right )}}{2 \, b^{3}} - \frac {b^{2} x^{2} \log \left (e^{\left (b x + a\right )} + 1\right ) + 2 \, b x {\rm Li}_2\left (-e^{\left (b x + a\right )}\right ) - 2 \, {\rm Li}_{3}(-e^{\left (b x + a\right )})}{b^{3}} + \frac {b^{2} x^{2} \log \left (-e^{\left (b x + a\right )} + 1\right ) + 2 \, b x {\rm Li}_2\left (e^{\left (b x + a\right )}\right ) - 2 \, {\rm Li}_{3}(e^{\left (b x + a\right )})}{b^{3}} \]

input

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

output

1/2*((b^2*x^2*e^(2*a) - 2*b*x*e^(2*a) + 2*e^(2*a))*e^(b*x) + (b^2*x^2 + 2* 
b*x + 2)*e^(-b*x))*e^(-a)/b^3 - (b^2*x^2*log(e^(b*x + a) + 1) + 2*b*x*dilo 
g(-e^(b*x + a)) - 2*polylog(3, -e^(b*x + a)))/b^3 + (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
 

3.5.6.8 Giac [F]

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

input

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

output

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

3.5.6.9 Mupad [F(-1)]

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

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)