\(\int x^3 \text {csch}^2(a+b x) \text {sech}(a+b x) \, dx\) [197]

Optimal result

Integrand size = 18, antiderivative size = 237 \[ \int x^3 \text {csch}^2(a+b x) \text {sech}(a+b x) \, dx=-\frac {2 x^3 \arctan \left (e^{a+b x}\right )}{b}-\frac {6 x^2 \text {arctanh}\left (e^{a+b x}\right )}{b^2}-\frac {x^3 \text {csch}(a+b x)}{b}-\frac {6 x \operatorname {PolyLog}\left (2,-e^{a+b x}\right )}{b^3}+\frac {3 i x^2 \operatorname {PolyLog}\left (2,-i e^{a+b x}\right )}{b^2}-\frac {3 i x^2 \operatorname {PolyLog}\left (2,i e^{a+b x}\right )}{b^2}+\frac {6 x \operatorname {PolyLog}\left (2,e^{a+b x}\right )}{b^3}+\frac {6 \operatorname {PolyLog}\left (3,-e^{a+b x}\right )}{b^4}-\frac {6 i x \operatorname {PolyLog}\left (3,-i e^{a+b x}\right )}{b^3}+\frac {6 i x \operatorname {PolyLog}\left (3,i e^{a+b x}\right )}{b^3}-\frac {6 \operatorname {PolyLog}\left (3,e^{a+b x}\right )}{b^4}+\frac {6 i \operatorname {PolyLog}\left (4,-i e^{a+b x}\right )}{b^4}-\frac {6 i \operatorname {PolyLog}\left (4,i e^{a+b x}\right )}{b^4} \] Output:

-2*x^3*arctan(exp(b*x+a))/b-6*x^2*arctanh(exp(b*x+a))/b^2-x^3*csch(b*x+a)/ 
b-6*x*polylog(2,-exp(b*x+a))/b^3+3*I*x^2*polylog(2,-I*exp(b*x+a))/b^2-3*I* 
x^2*polylog(2,I*exp(b*x+a))/b^2+6*x*polylog(2,exp(b*x+a))/b^3+6*polylog(3, 
-exp(b*x+a))/b^4-6*I*x*polylog(3,-I*exp(b*x+a))/b^3+6*I*x*polylog(3,I*exp( 
b*x+a))/b^3-6*polylog(3,exp(b*x+a))/b^4+6*I*polylog(4,-I*exp(b*x+a))/b^4-6 
*I*polylog(4,I*exp(b*x+a))/b^4
 

Mathematica [A] (verified)

Time = 1.77 (sec) , antiderivative size = 334, normalized size of antiderivative = 1.41 \[ \int x^3 \text {csch}^2(a+b x) \text {sech}(a+b x) \, dx=\frac {-2 b^3 x^3 \text {csch}(a)+6 b^2 x^2 \log \left (1-e^{a+b x}\right )-2 i b^3 x^3 \log \left (1-i e^{a+b x}\right )+2 i b^3 x^3 \log \left (1+i e^{a+b x}\right )-6 b^2 x^2 \log \left (1+e^{a+b x}\right )-12 b x \operatorname {PolyLog}\left (2,-e^{a+b x}\right )+6 i b^2 x^2 \operatorname {PolyLog}\left (2,-i e^{a+b x}\right )-6 i b^2 x^2 \operatorname {PolyLog}\left (2,i e^{a+b x}\right )+12 b x \operatorname {PolyLog}\left (2,e^{a+b x}\right )+12 \operatorname {PolyLog}\left (3,-e^{a+b x}\right )-12 i b x \operatorname {PolyLog}\left (3,-i e^{a+b x}\right )+12 i b x \operatorname {PolyLog}\left (3,i e^{a+b x}\right )-12 \operatorname {PolyLog}\left (3,e^{a+b x}\right )+12 i \operatorname {PolyLog}\left (4,-i e^{a+b x}\right )-12 i \operatorname {PolyLog}\left (4,i e^{a+b x}\right )+b^3 x^3 \text {csch}\left (\frac {a}{2}\right ) \text {csch}\left (\frac {1}{2} (a+b x)\right ) \sinh \left (\frac {b x}{2}\right )+b^3 x^3 \text {sech}\left (\frac {a}{2}\right ) \text {sech}\left (\frac {1}{2} (a+b x)\right ) \sinh \left (\frac {b x}{2}\right )}{2 b^4} \] Input:

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

Output:

(-2*b^3*x^3*Csch[a] + 6*b^2*x^2*Log[1 - E^(a + b*x)] - (2*I)*b^3*x^3*Log[1 
 - I*E^(a + b*x)] + (2*I)*b^3*x^3*Log[1 + I*E^(a + b*x)] - 6*b^2*x^2*Log[1 
 + E^(a + b*x)] - 12*b*x*PolyLog[2, -E^(a + b*x)] + (6*I)*b^2*x^2*PolyLog[ 
2, (-I)*E^(a + b*x)] - (6*I)*b^2*x^2*PolyLog[2, I*E^(a + b*x)] + 12*b*x*Po 
lyLog[2, E^(a + b*x)] + 12*PolyLog[3, -E^(a + b*x)] - (12*I)*b*x*PolyLog[3 
, (-I)*E^(a + b*x)] + (12*I)*b*x*PolyLog[3, I*E^(a + b*x)] - 12*PolyLog[3, 
 E^(a + b*x)] + (12*I)*PolyLog[4, (-I)*E^(a + b*x)] - (12*I)*PolyLog[4, I* 
E^(a + b*x)] + b^3*x^3*Csch[a/2]*Csch[(a + b*x)/2]*Sinh[(b*x)/2] + b^3*x^3 
*Sech[a/2]*Sech[(a + b*x)/2]*Sinh[(b*x)/2])/(2*b^4)
 

Rubi [A] (verified)

Time = 0.82 (sec) , antiderivative size = 274, normalized size of antiderivative = 1.16, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.222, Rules used = {5985, 25, 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.

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

\(\Big \downarrow \) 5985

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

\(\Big \downarrow \) 25

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

\(\Big \downarrow \) 2010

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

\(\Big \downarrow \) 2009

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

Input:

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

Output:

-((x^3*ArcTan[Sinh[a + b*x]])/b) - (x^3*Csch[a + b*x])/b + 3*((-2*x^3*ArcT 
an[E^(a + b*x)])/(3*b) + (x^3*ArcTan[Sinh[a + b*x]])/(3*b) - (2*x^2*ArcTan 
h[E^(a + b*x)])/b^2 - (2*x*PolyLog[2, -E^(a + b*x)])/b^3 + (I*x^2*PolyLog[ 
2, (-I)*E^(a + b*x)])/b^2 - (I*x^2*PolyLog[2, I*E^(a + b*x)])/b^2 + (2*x*P 
olyLog[2, E^(a + b*x)])/b^3 + (2*PolyLog[3, -E^(a + b*x)])/b^4 - ((2*I)*x* 
PolyLog[3, (-I)*E^(a + b*x)])/b^3 + ((2*I)*x*PolyLog[3, I*E^(a + b*x)])/b^ 
3 - (2*PolyLog[3, E^(a + b*x)])/b^4 + ((2*I)*PolyLog[4, (-I)*E^(a + b*x)]) 
/b^4 - ((2*I)*PolyLog[4, I*E^(a + b*x)])/b^4)
 

Defintions of rubi rules used

rule 25

Int[-(Fx_), x_Symbol] :> Simp[Identity[-1]   Int[Fx, x], 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]

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

Input:

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

Output:

int(x^3*csch(b*x+a)^2*sech(b*x+a),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. 1309 vs. \(2 (197) = 394\).

Time = 0.12 (sec) , antiderivative size = 1309, normalized size of antiderivative = 5.52 \[ \int x^3 \text {csch}^2(a+b x) \text {sech}(a+b x) \, dx=\text {Too large to display} \] Input:

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

Output:

-(2*b^3*x^3*cosh(b*x + a) + 2*b^3*x^3*sinh(b*x + a) - 6*(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)*dilog( 
cosh(b*x + a) + sinh(b*x + a)) + 3*(I*b^2*x^2*cosh(b*x + a)^2 + 2*I*b^2*x^ 
2*cosh(b*x + a)*sinh(b*x + a) + I*b^2*x^2*sinh(b*x + a)^2 - I*b^2*x^2)*dil 
og(I*cosh(b*x + a) + I*sinh(b*x + a)) + 3*(-I*b^2*x^2*cosh(b*x + a)^2 - 2* 
I*b^2*x^2*cosh(b*x + a)*sinh(b*x + a) - I*b^2*x^2*sinh(b*x + a)^2 + I*b^2* 
x^2)*dilog(-I*cosh(b*x + a) - I*sinh(b*x + a)) + 6*(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)*dilog(-cosh 
(b*x + a) - sinh(b*x + a)) + 3*(b^2*x^2*cosh(b*x + a)^2 + 2*b^2*x^2*cosh(b 
*x + a)*sinh(b*x + a) + b^2*x^2*sinh(b*x + a)^2 - b^2*x^2)*log(cosh(b*x + 
a) + sinh(b*x + a) + 1) - (I*a^3*cosh(b*x + a)^2 + 2*I*a^3*cosh(b*x + a)*s 
inh(b*x + a) + I*a^3*sinh(b*x + a)^2 - I*a^3)*log(cosh(b*x + a) + sinh(b*x 
 + a) + I) - (-I*a^3*cosh(b*x + a)^2 - 2*I*a^3*cosh(b*x + a)*sinh(b*x + a) 
 - I*a^3*sinh(b*x + a)^2 + I*a^3)*log(cosh(b*x + a) + sinh(b*x + a) - I) - 
 3*(a^2*cosh(b*x + a)^2 + 2*a^2*cosh(b*x + a)*sinh(b*x + a) + a^2*sinh(b*x 
 + a)^2 - a^2)*log(cosh(b*x + a) + sinh(b*x + a) - 1) - (-I*b^3*x^3 - I*a^ 
3 + (I*b^3*x^3 + I*a^3)*cosh(b*x + a)^2 - 2*(-I*b^3*x^3 - I*a^3)*cosh(b*x 
+ a)*sinh(b*x + a) + (I*b^3*x^3 + I*a^3)*sinh(b*x + a)^2)*log(I*cosh(b*x + 
 a) + I*sinh(b*x + a) + 1) - (I*b^3*x^3 + I*a^3 + (-I*b^3*x^3 - I*a^3)*cos 
h(b*x + a)^2 - 2*(I*b^3*x^3 + I*a^3)*cosh(b*x + a)*sinh(b*x + a) + (-I*...
 

Sympy [F]

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

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

Output:

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

Maxima [F]

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

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

Output:

-2*x^3*e^(b*x + a)/(b*e^(2*b*x + 2*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^4 + 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^4 - 8*integrate(1/4*x^3*e^(b*x + a)/(e^(2*b*x + 2*a) + 1), x)
 

Giac [F]

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

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

Output:

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

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

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

Output:

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