\(\int \frac {x^3 \text {csch}(x) \text {sech}(x)}{\sqrt {a \text {sech}^2(x)}} \, dx\) [604]

Optimal result

Integrand size = 18, antiderivative size = 150 \[ \int \frac {x^3 \text {csch}(x) \text {sech}(x)}{\sqrt {a \text {sech}^2(x)}} \, dx=-\frac {2 x^3 \text {arctanh}\left (e^x\right ) \text {sech}(x)}{\sqrt {a \text {sech}^2(x)}}-\frac {3 x^2 \operatorname {PolyLog}\left (2,-e^x\right ) \text {sech}(x)}{\sqrt {a \text {sech}^2(x)}}+\frac {3 x^2 \operatorname {PolyLog}\left (2,e^x\right ) \text {sech}(x)}{\sqrt {a \text {sech}^2(x)}}+\frac {6 x \operatorname {PolyLog}\left (3,-e^x\right ) \text {sech}(x)}{\sqrt {a \text {sech}^2(x)}}-\frac {6 x \operatorname {PolyLog}\left (3,e^x\right ) \text {sech}(x)}{\sqrt {a \text {sech}^2(x)}}-\frac {6 \operatorname {PolyLog}\left (4,-e^x\right ) \text {sech}(x)}{\sqrt {a \text {sech}^2(x)}}+\frac {6 \operatorname {PolyLog}\left (4,e^x\right ) \text {sech}(x)}{\sqrt {a \text {sech}^2(x)}} \] Output:

-2*x^3*arctanh(exp(x))*sech(x)/(a*sech(x)^2)^(1/2)-3*x^2*polylog(2,-exp(x) 
)*sech(x)/(a*sech(x)^2)^(1/2)+3*x^2*polylog(2,exp(x))*sech(x)/(a*sech(x)^2 
)^(1/2)+6*x*polylog(3,-exp(x))*sech(x)/(a*sech(x)^2)^(1/2)-6*x*polylog(3,e 
xp(x))*sech(x)/(a*sech(x)^2)^(1/2)-6*polylog(4,-exp(x))*sech(x)/(a*sech(x) 
^2)^(1/2)+6*polylog(4,exp(x))*sech(x)/(a*sech(x)^2)^(1/2)
                                                                                    
                                                                                    
 

Mathematica [A] (verified)

Time = 0.11 (sec) , antiderivative size = 93, normalized size of antiderivative = 0.62 \[ \int \frac {x^3 \text {csch}(x) \text {sech}(x)}{\sqrt {a \text {sech}^2(x)}} \, dx=\frac {\left (x^3 \log \left (1-e^x\right )-x^3 \log \left (1+e^x\right )-3 x^2 \operatorname {PolyLog}\left (2,-e^x\right )+3 x^2 \operatorname {PolyLog}\left (2,e^x\right )+6 x \operatorname {PolyLog}\left (3,-e^x\right )-6 x \operatorname {PolyLog}\left (3,e^x\right )-6 \operatorname {PolyLog}\left (4,-e^x\right )+6 \operatorname {PolyLog}\left (4,e^x\right )\right ) \text {sech}(x)}{\sqrt {a \text {sech}^2(x)}} \] Input:

Integrate[(x^3*Csch[x]*Sech[x])/Sqrt[a*Sech[x]^2],x]
 

Output:

((x^3*Log[1 - E^x] - x^3*Log[1 + E^x] - 3*x^2*PolyLog[2, -E^x] + 3*x^2*Pol 
yLog[2, E^x] + 6*x*PolyLog[3, -E^x] - 6*x*PolyLog[3, E^x] - 6*PolyLog[4, - 
E^x] + 6*PolyLog[4, E^x])*Sech[x])/Sqrt[a*Sech[x]^2]
 

Rubi [C] (verified)

Result contains complex when optimal does not.

Time = 0.94 (sec) , antiderivative size = 98, normalized size of antiderivative = 0.65, number of steps used = 9, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.444, Rules used = {7271, 3042, 26, 4670, 3011, 7163, 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.

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

\(\Big \downarrow \) 7271

\(\displaystyle \frac {\text {sech}(x) \int x^3 \text {csch}(x)dx}{\sqrt {a \text {sech}^2(x)}}\)

\(\Big \downarrow \) 3042

\(\displaystyle \frac {\text {sech}(x) \int i x^3 \csc (i x)dx}{\sqrt {a \text {sech}^2(x)}}\)

\(\Big \downarrow \) 26

\(\displaystyle \frac {i \text {sech}(x) \int x^3 \csc (i x)dx}{\sqrt {a \text {sech}^2(x)}}\)

\(\Big \downarrow \) 4670

\(\displaystyle \frac {i \text {sech}(x) \left (3 i \int x^2 \log \left (1-e^x\right )dx-3 i \int x^2 \log \left (1+e^x\right )dx+2 i x^3 \text {arctanh}\left (e^x\right )\right )}{\sqrt {a \text {sech}^2(x)}}\)

\(\Big \downarrow \) 3011

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

\(\Big \downarrow \) 7163

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

\(\Big \downarrow \) 2720

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

\(\Big \downarrow \) 7143

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

Input:

Int[(x^3*Csch[x]*Sech[x])/Sqrt[a*Sech[x]^2],x]
 

Output:

(I*((2*I)*x^3*ArcTanh[E^x] - (3*I)*(-(x^2*PolyLog[2, -E^x]) + 2*(x*PolyLog 
[3, -E^x] - PolyLog[4, -E^x])) + (3*I)*(-(x^2*PolyLog[2, E^x]) + 2*(x*Poly 
Log[3, E^x] - PolyLog[4, E^x])))*Sech[x])/Sqrt[a*Sech[x]^2]
 

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 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 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]
 

rule 7163

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

rule 7271

Int[(u_.)*((a_.)*(v_)^(m_.))^(p_), x_Symbol] :> Simp[a^IntPart[p]*((a*v^m)^ 
FracPart[p]/v^(m*FracPart[p]))   Int[u*v^(m*p), x], x] /; FreeQ[{a, m, p}, 
x] &&  !IntegerQ[p] &&  !FreeQ[v, x] &&  !(EqQ[a, 1] && EqQ[m, 1]) &&  !(Eq 
Q[v, x] && EqQ[m, 1])
 

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(280\) vs. \(2(129)=258\).

Time = 0.10 (sec) , antiderivative size = 281, normalized size of antiderivative = 1.87

method	result	size
risch	\(-\frac {{\mathrm e}^{x} x^{3} \ln \left (1+{\mathrm e}^{x}\right )}{\sqrt {\frac {a \,{\mathrm e}^{2 x}}{\left (1+{\mathrm e}^{2 x}\right )^{2}}}\, \left (1+{\mathrm e}^{2 x}\right )}-\frac {3 \,{\mathrm e}^{x} x^{2} \operatorname {polylog}\left (2, -{\mathrm e}^{x}\right )}{\sqrt {\frac {a \,{\mathrm e}^{2 x}}{\left (1+{\mathrm e}^{2 x}\right )^{2}}}\, \left (1+{\mathrm e}^{2 x}\right )}+\frac {6 \,{\mathrm e}^{x} x \operatorname {polylog}\left (3, -{\mathrm e}^{x}\right )}{\sqrt {\frac {a \,{\mathrm e}^{2 x}}{\left (1+{\mathrm e}^{2 x}\right )^{2}}}\, \left (1+{\mathrm e}^{2 x}\right )}-\frac {6 \,{\mathrm e}^{x} \operatorname {polylog}\left (4, -{\mathrm e}^{x}\right )}{\sqrt {\frac {a \,{\mathrm e}^{2 x}}{\left (1+{\mathrm e}^{2 x}\right )^{2}}}\, \left (1+{\mathrm e}^{2 x}\right )}+\frac {{\mathrm e}^{x} x^{3} \ln \left (1-{\mathrm e}^{x}\right )}{\sqrt {\frac {a \,{\mathrm e}^{2 x}}{\left (1+{\mathrm e}^{2 x}\right )^{2}}}\, \left (1+{\mathrm e}^{2 x}\right )}+\frac {3 \,{\mathrm e}^{x} x^{2} \operatorname {polylog}\left (2, {\mathrm e}^{x}\right )}{\sqrt {\frac {a \,{\mathrm e}^{2 x}}{\left (1+{\mathrm e}^{2 x}\right )^{2}}}\, \left (1+{\mathrm e}^{2 x}\right )}-\frac {6 \,{\mathrm e}^{x} x \operatorname {polylog}\left (3, {\mathrm e}^{x}\right )}{\sqrt {\frac {a \,{\mathrm e}^{2 x}}{\left (1+{\mathrm e}^{2 x}\right )^{2}}}\, \left (1+{\mathrm e}^{2 x}\right )}+\frac {6 \,{\mathrm e}^{x} \operatorname {polylog}\left (4, {\mathrm e}^{x}\right )}{\sqrt {\frac {a \,{\mathrm e}^{2 x}}{\left (1+{\mathrm e}^{2 x}\right )^{2}}}\, \left (1+{\mathrm e}^{2 x}\right )}\)	\(281\)

Input:

int(x^3*csch(x)*sech(x)/(a*sech(x)^2)^(1/2),x,method=_RETURNVERBOSE)
 

Output:

-1/(a*exp(2*x)/(1+exp(2*x))^2)^(1/2)/(1+exp(2*x))*exp(x)*x^3*ln(1+exp(x))- 
3/(a*exp(2*x)/(1+exp(2*x))^2)^(1/2)/(1+exp(2*x))*exp(x)*x^2*polylog(2,-exp 
(x))+6/(a*exp(2*x)/(1+exp(2*x))^2)^(1/2)/(1+exp(2*x))*exp(x)*x*polylog(3,- 
exp(x))-6/(a*exp(2*x)/(1+exp(2*x))^2)^(1/2)/(1+exp(2*x))*exp(x)*polylog(4, 
-exp(x))+1/(a*exp(2*x)/(1+exp(2*x))^2)^(1/2)/(1+exp(2*x))*exp(x)*x^3*ln(1- 
exp(x))+3/(a*exp(2*x)/(1+exp(2*x))^2)^(1/2)/(1+exp(2*x))*exp(x)*x^2*polylo 
g(2,exp(x))-6/(a*exp(2*x)/(1+exp(2*x))^2)^(1/2)/(1+exp(2*x))*exp(x)*x*poly 
log(3,exp(x))+6/(a*exp(2*x)/(1+exp(2*x))^2)^(1/2)/(1+exp(2*x))*exp(x)*poly 
log(4,exp(x))
 

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 272 vs. \(2 (127) = 254\).

Time = 0.09 (sec) , antiderivative size = 272, normalized size of antiderivative = 1.81 \[ \int \frac {x^3 \text {csch}(x) \text {sech}(x)}{\sqrt {a \text {sech}^2(x)}} \, dx=\frac {{\left (6 \, \sqrt {\frac {a}{e^{\left (4 \, x\right )} + 2 \, e^{\left (2 \, x\right )} + 1}} {\left (e^{\left (2 \, x\right )} + 1\right )} e^{x} {\rm polylog}\left (4, \cosh \left (x\right ) + \sinh \left (x\right )\right ) - 6 \, \sqrt {\frac {a}{e^{\left (4 \, x\right )} + 2 \, e^{\left (2 \, x\right )} + 1}} {\left (e^{\left (2 \, x\right )} + 1\right )} e^{x} {\rm polylog}\left (4, -\cosh \left (x\right ) - \sinh \left (x\right )\right ) - 6 \, {\left (x e^{\left (2 \, x\right )} + x\right )} \sqrt {\frac {a}{e^{\left (4 \, x\right )} + 2 \, e^{\left (2 \, x\right )} + 1}} e^{x} {\rm polylog}\left (3, \cosh \left (x\right ) + \sinh \left (x\right )\right ) + 6 \, {\left (x e^{\left (2 \, x\right )} + x\right )} \sqrt {\frac {a}{e^{\left (4 \, x\right )} + 2 \, e^{\left (2 \, x\right )} + 1}} e^{x} {\rm polylog}\left (3, -\cosh \left (x\right ) - \sinh \left (x\right )\right ) + {\left (3 \, {\left (x^{2} e^{\left (2 \, x\right )} + x^{2}\right )} {\rm Li}_2\left (\cosh \left (x\right ) + \sinh \left (x\right )\right ) - 3 \, {\left (x^{2} e^{\left (2 \, x\right )} + x^{2}\right )} {\rm Li}_2\left (-\cosh \left (x\right ) - \sinh \left (x\right )\right ) - {\left (x^{3} e^{\left (2 \, x\right )} + x^{3}\right )} \log \left (\cosh \left (x\right ) + \sinh \left (x\right ) + 1\right ) + {\left (x^{3} e^{\left (2 \, x\right )} + x^{3}\right )} \log \left (-\cosh \left (x\right ) - \sinh \left (x\right ) + 1\right )\right )} \sqrt {\frac {a}{e^{\left (4 \, x\right )} + 2 \, e^{\left (2 \, x\right )} + 1}} e^{x}\right )} e^{\left (-x\right )}}{a} \] Input:

integrate(x^3*csch(x)*sech(x)/(a*sech(x)^2)^(1/2),x, algorithm="fricas")
 

Output:

(6*sqrt(a/(e^(4*x) + 2*e^(2*x) + 1))*(e^(2*x) + 1)*e^x*polylog(4, cosh(x) 
+ sinh(x)) - 6*sqrt(a/(e^(4*x) + 2*e^(2*x) + 1))*(e^(2*x) + 1)*e^x*polylog 
(4, -cosh(x) - sinh(x)) - 6*(x*e^(2*x) + x)*sqrt(a/(e^(4*x) + 2*e^(2*x) + 
1))*e^x*polylog(3, cosh(x) + sinh(x)) + 6*(x*e^(2*x) + x)*sqrt(a/(e^(4*x) 
+ 2*e^(2*x) + 1))*e^x*polylog(3, -cosh(x) - sinh(x)) + (3*(x^2*e^(2*x) + x 
^2)*dilog(cosh(x) + sinh(x)) - 3*(x^2*e^(2*x) + x^2)*dilog(-cosh(x) - sinh 
(x)) - (x^3*e^(2*x) + x^3)*log(cosh(x) + sinh(x) + 1) + (x^3*e^(2*x) + x^3 
)*log(-cosh(x) - sinh(x) + 1))*sqrt(a/(e^(4*x) + 2*e^(2*x) + 1))*e^x)*e^(- 
x)/a
 

Sympy [F]

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

integrate(x**3*csch(x)*sech(x)/(a*sech(x)**2)**(1/2),x)
 

Output:

Integral(x**3*csch(x)*sech(x)/sqrt(a*sech(x)**2), x)
 

Maxima [A] (verification not implemented)

Time = 0.14 (sec) , antiderivative size = 80, normalized size of antiderivative = 0.53 \[ \int \frac {x^3 \text {csch}(x) \text {sech}(x)}{\sqrt {a \text {sech}^2(x)}} \, dx=-\frac {x^{3} \log \left (e^{x} + 1\right ) + 3 \, x^{2} {\rm Li}_2\left (-e^{x}\right ) - 6 \, x {\rm Li}_{3}(-e^{x}) + 6 \, {\rm Li}_{4}(-e^{x})}{\sqrt {a}} + \frac {x^{3} \log \left (-e^{x} + 1\right ) + 3 \, x^{2} {\rm Li}_2\left (e^{x}\right ) - 6 \, x {\rm Li}_{3}(e^{x}) + 6 \, {\rm Li}_{4}(e^{x})}{\sqrt {a}} \] Input:

integrate(x^3*csch(x)*sech(x)/(a*sech(x)^2)^(1/2),x, algorithm="maxima")
 

Output:

-(x^3*log(e^x + 1) + 3*x^2*dilog(-e^x) - 6*x*polylog(3, -e^x) + 6*polylog( 
4, -e^x))/sqrt(a) + (x^3*log(-e^x + 1) + 3*x^2*dilog(e^x) - 6*x*polylog(3, 
 e^x) + 6*polylog(4, e^x))/sqrt(a)
 

Giac [F]

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

integrate(x^3*csch(x)*sech(x)/(a*sech(x)^2)^(1/2),x, algorithm="giac")
 

Output:

integrate(x^3*csch(x)*sech(x)/sqrt(a*sech(x)^2), x)
 

Mupad [F(-1)]

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

int(x^3/(cosh(x)*sinh(x)*(a/cosh(x)^2)^(1/2)),x)
 

Output:

int(x^3/(cosh(x)*sinh(x)*(a/cosh(x)^2)^(1/2)), x)
 

Reduce [F]

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

int(x^3*csch(x)*sech(x)/(a*sech(x)^2)^(1/2),x)
 

Output:

(sqrt(a)*int(csch(x)*x**3,x))/a