Integrand size = 20, antiderivative size = 197 \[ \int x^2 \text {csch}^3(a+b x) \text {sech}^2(a+b x) \, dx=\frac {4 x \arctan \left (e^{a+b x}\right )}{b^2}+\frac {3 x^2 \text {arctanh}\left (e^{a+b x}\right )}{b}-\frac {\text {arctanh}(\cosh (a+b x))}{b^3}-\frac {x \text {csch}(a+b x)}{b^2}+\frac {3 x \operatorname {PolyLog}\left (2,-e^{a+b x}\right )}{b^2}-\frac {2 i \operatorname {PolyLog}\left (2,-i e^{a+b x}\right )}{b^3}+\frac {2 i \operatorname {PolyLog}\left (2,i e^{a+b x}\right )}{b^3}-\frac {3 x \operatorname {PolyLog}\left (2,e^{a+b x}\right )}{b^2}-\frac {3 \operatorname {PolyLog}\left (3,-e^{a+b x}\right )}{b^3}+\frac {3 \operatorname {PolyLog}\left (3,e^{a+b x}\right )}{b^3}-\frac {3 x^2 \text {sech}(a+b x)}{2 b}-\frac {x^2 \text {csch}^2(a+b x) \text {sech}(a+b x)}{2 b} \]
4*x*arctan(exp(b*x+a))/b^2+3*x^2*arctanh(exp(b*x+a))/b-arctanh(cosh(b*x+a) )/b^3-x*csch(b*x+a)/b^2+3*x*polylog(2,-exp(b*x+a))/b^2-2*I*polylog(2,-I*ex p(b*x+a))/b^3+2*I*polylog(2,I*exp(b*x+a))/b^3-3*x*polylog(2,exp(b*x+a))/b^ 2-3*polylog(3,-exp(b*x+a))/b^3+3*polylog(3,exp(b*x+a))/b^3-3/2*x^2*sech(b* x+a)/b-1/2*x^2*csch(b*x+a)^2*sech(b*x+a)/b
Time = 6.34 (sec) , antiderivative size = 341, normalized size of antiderivative = 1.73 \[ \int x^2 \text {csch}^3(a+b x) \text {sech}^2(a+b x) \, dx=-\frac {x \text {csch}(a)}{b^2}-\frac {x^2 \text {csch}^2\left (\frac {a}{2}+\frac {b x}{2}\right )}{8 b}+\frac {2 i \left (b x \left (\log \left (1-i e^{a+b x}\right )-\log \left (1+i e^{a+b x}\right )\right )-\operatorname {PolyLog}\left (2,-i e^{a+b x}\right )+\operatorname {PolyLog}\left (2,i e^{a+b x}\right )\right )}{b^3}+\frac {\frac {\log \left (1-e^{a+b x}\right )}{b}-\frac {3}{2} b x^2 \log \left (1-e^{a+b x}\right )-\frac {\log \left (1+e^{a+b x}\right )}{b}+\frac {3}{2} b x^2 \log \left (1+e^{a+b x}\right )+3 x \operatorname {PolyLog}\left (2,-e^{a+b x}\right )-3 x \operatorname {PolyLog}\left (2,e^{a+b x}\right )-\frac {3 \operatorname {PolyLog}\left (3,-e^{a+b x}\right )}{b}+\frac {3 \operatorname {PolyLog}\left (3,e^{a+b x}\right )}{b}}{b^2}-\frac {x^2 \text {sech}^2\left (\frac {a}{2}+\frac {b x}{2}\right )}{8 b}-\frac {x^2 \text {sech}(a+b x)}{b}+\frac {x \text {csch}\left (\frac {a}{2}\right ) \text {csch}\left (\frac {a}{2}+\frac {b x}{2}\right ) \sinh \left (\frac {b x}{2}\right )}{2 b^2}+\frac {x \text {sech}\left (\frac {a}{2}\right ) \text {sech}\left (\frac {a}{2}+\frac {b x}{2}\right ) \sinh \left (\frac {b x}{2}\right )}{2 b^2} \]
-((x*Csch[a])/b^2) - (x^2*Csch[a/2 + (b*x)/2]^2)/(8*b) + ((2*I)*(b*x*(Log[ 1 - I*E^(a + b*x)] - Log[1 + I*E^(a + b*x)]) - PolyLog[2, (-I)*E^(a + b*x) ] + PolyLog[2, I*E^(a + b*x)]))/b^3 + (Log[1 - E^(a + b*x)]/b - (3*b*x^2*L og[1 - E^(a + b*x)])/2 - Log[1 + E^(a + b*x)]/b + (3*b*x^2*Log[1 + E^(a + b*x)])/2 + 3*x*PolyLog[2, -E^(a + b*x)] - 3*x*PolyLog[2, E^(a + b*x)] - (3 *PolyLog[3, -E^(a + b*x)])/b + (3*PolyLog[3, E^(a + b*x)])/b)/b^2 - (x^2*S ech[a/2 + (b*x)/2]^2)/(8*b) - (x^2*Sech[a + b*x])/b + (x*Csch[a/2]*Csch[a/ 2 + (b*x)/2]*Sinh[(b*x)/2])/(2*b^2) + (x*Sech[a/2]*Sech[a/2 + (b*x)/2]*Sin h[(b*x)/2])/(2*b^2)
Time = 0.75 (sec) , antiderivative size = 197, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.200, Rules used = {5985, 27, 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.
| \(\displaystyle \int x^2 \text {csch}^3(a+b x) \text {sech}^2(a+b x) \, dx\) |
| \(\Big \downarrow \) 5985 |
| \(\displaystyle -2 \int \frac {1}{2} x \left (-\frac {\text {sech}(a+b x) \text {csch}^2(a+b x)}{b}+\frac {3 \text {arctanh}(\cosh (a+b x))}{b}-\frac {3 \text {sech}(a+b x)}{b}\right )dx+\frac {3 x^2 \text {arctanh}(\cosh (a+b x))}{2 b}-\frac {3 x^2 \text {sech}(a+b x)}{2 b}-\frac {x^2 \text {csch}^2(a+b x) \text {sech}(a+b x)}{2 b}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle -\int x \left (-\frac {\text {sech}(a+b x) \text {csch}^2(a+b x)}{b}+\frac {3 \text {arctanh}(\cosh (a+b x))}{b}-\frac {3 \text {sech}(a+b x)}{b}\right )dx+\frac {3 x^2 \text {arctanh}(\cosh (a+b x))}{2 b}-\frac {3 x^2 \text {sech}(a+b x)}{2 b}-\frac {x^2 \text {csch}^2(a+b x) \text {sech}(a+b x)}{2 b}\) |
| \(\Big \downarrow \) 2010 |
| \(\displaystyle -\int \left (\frac {3 x \text {arctanh}(\cosh (a+b x))}{b}-\frac {x \left (\text {csch}^2(a+b x)+3\right ) \text {sech}(a+b x)}{b}\right )dx+\frac {3 x^2 \text {arctanh}(\cosh (a+b x))}{2 b}-\frac {3 x^2 \text {sech}(a+b x)}{2 b}-\frac {x^2 \text {csch}^2(a+b x) \text {sech}(a+b x)}{2 b}\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {4 x \arctan \left (e^{a+b x}\right )}{b^2}-\frac {\text {arctanh}(\cosh (a+b x))}{b^3}+\frac {3 x^2 \text {arctanh}\left (e^{a+b x}\right )}{b}-\frac {2 i \operatorname {PolyLog}\left (2,-i e^{a+b x}\right )}{b^3}+\frac {2 i \operatorname {PolyLog}\left (2,i e^{a+b x}\right )}{b^3}-\frac {3 \operatorname {PolyLog}\left (3,-e^{a+b x}\right )}{b^3}+\frac {3 \operatorname {PolyLog}\left (3,e^{a+b x}\right )}{b^3}+\frac {3 x \operatorname {PolyLog}\left (2,-e^{a+b x}\right )}{b^2}-\frac {3 x \operatorname {PolyLog}\left (2,e^{a+b x}\right )}{b^2}-\frac {x \text {csch}(a+b x)}{b^2}-\frac {3 x^2 \text {sech}(a+b x)}{2 b}-\frac {x^2 \text {csch}^2(a+b x) \text {sech}(a+b x)}{2 b}\) |
(4*x*ArcTan[E^(a + b*x)])/b^2 + (3*x^2*ArcTanh[E^(a + b*x)])/b - ArcTanh[C osh[a + b*x]]/b^3 - (x*Csch[a + b*x])/b^2 + (3*x*PolyLog[2, -E^(a + b*x)]) /b^2 - ((2*I)*PolyLog[2, (-I)*E^(a + b*x)])/b^3 + ((2*I)*PolyLog[2, I*E^(a + b*x)])/b^3 - (3*x*PolyLog[2, E^(a + b*x)])/b^2 - (3*PolyLog[3, -E^(a + b*x)])/b^3 + (3*PolyLog[3, E^(a + b*x)])/b^3 - (3*x^2*Sech[a + b*x])/(2*b) - (x^2*Csch[a + b*x]^2*Sech[a + b*x])/(2*b)
3.6.16.3.1 Defintions of rubi rules used
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
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]]
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]
\[\int x^{2} \operatorname {csch}\left (b x +a \right )^{3} \operatorname {sech}\left (b x +a \right )^{2}d x\]
Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 3804 vs. \(2 (173) = 346\).
Time = 0.32 (sec) , antiderivative size = 3804, normalized size of antiderivative = 19.31 \[ \int x^2 \text {csch}^3(a+b x) \text {sech}^2(a+b x) \, dx=\text {Too large to display} \]
1/2*(4*b^2*x^2*cosh(b*x + a)^3 - 2*(3*b^2*x^2 + 2*b*x)*cosh(b*x + a)^5 - 1 0*(3*b^2*x^2 + 2*b*x)*cosh(b*x + a)*sinh(b*x + a)^4 - 2*(3*b^2*x^2 + 2*b*x )*sinh(b*x + a)^5 + 4*(b^2*x^2 - 5*(3*b^2*x^2 + 2*b*x)*cosh(b*x + a)^2)*si nh(b*x + a)^3 + 4*(3*b^2*x^2*cosh(b*x + a) - 5*(3*b^2*x^2 + 2*b*x)*cosh(b* x + a)^3)*sinh(b*x + a)^2 - 2*(3*b^2*x^2 - 2*b*x)*cosh(b*x + a) - 6*(b*x*c osh(b*x + a)^6 + 6*b*x*cosh(b*x + a)*sinh(b*x + a)^5 + b*x*sinh(b*x + a)^6 - b*x*cosh(b*x + a)^4 + (15*b*x*cosh(b*x + a)^2 - b*x)*sinh(b*x + a)^4 - b*x*cosh(b*x + a)^2 + 4*(5*b*x*cosh(b*x + a)^3 - b*x*cosh(b*x + a))*sinh(b *x + a)^3 + (15*b*x*cosh(b*x + a)^4 - 6*b*x*cosh(b*x + a)^2 - b*x)*sinh(b* x + a)^2 + b*x + 2*(3*b*x*cosh(b*x + a)^5 - 2*b*x*cosh(b*x + a)^3 - b*x*co sh(b*x + a))*sinh(b*x + a))*dilog(cosh(b*x + a) + sinh(b*x + a)) - 4*(-I*c osh(b*x + a)^6 - 6*I*cosh(b*x + a)*sinh(b*x + a)^5 - I*sinh(b*x + a)^6 + ( -15*I*cosh(b*x + a)^2 + I)*sinh(b*x + a)^4 + I*cosh(b*x + a)^4 + 4*(-5*I*c osh(b*x + a)^3 + I*cosh(b*x + a))*sinh(b*x + a)^3 + (-15*I*cosh(b*x + a)^4 + 6*I*cosh(b*x + a)^2 + I)*sinh(b*x + a)^2 + I*cosh(b*x + a)^2 + 2*(-3*I* cosh(b*x + a)^5 + 2*I*cosh(b*x + a)^3 + I*cosh(b*x + a))*sinh(b*x + a) - I )*dilog(I*cosh(b*x + a) + I*sinh(b*x + a)) - 4*(I*cosh(b*x + a)^6 + 6*I*co sh(b*x + a)*sinh(b*x + a)^5 + I*sinh(b*x + a)^6 + (15*I*cosh(b*x + a)^2 - I)*sinh(b*x + a)^4 - I*cosh(b*x + a)^4 + 4*(5*I*cosh(b*x + a)^3 - I*cosh(b *x + a))*sinh(b*x + a)^3 + (15*I*cosh(b*x + a)^4 - 6*I*cosh(b*x + a)^2 ...
\[ \int x^2 \text {csch}^3(a+b x) \text {sech}^2(a+b x) \, dx=\int x^{2} \operatorname {csch}^{3}{\left (a + b x \right )} \operatorname {sech}^{2}{\left (a + b x \right )}\, dx \]
\[ \int x^2 \text {csch}^3(a+b x) \text {sech}^2(a+b x) \, dx=\int { x^{2} \operatorname {csch}\left (b x + a\right )^{3} \operatorname {sech}\left (b x + a\right )^{2} \,d x } \]
(2*b*x^2*e^(3*b*x + 3*a) - (3*b*x^2*e^(5*a) + 2*x*e^(5*a))*e^(5*b*x) - (3* b*x^2*e^a - 2*x*e^a)*e^(b*x))/(b^2*e^(6*b*x + 6*a) - b^2*e^(4*b*x + 4*a) - b^2*e^(2*b*x + 2*a) + b^2) + 3/2*(b^2*x^2*log(e^(b*x + a) + 1) + 2*b*x*di log(-e^(b*x + a)) - 2*polylog(3, -e^(b*x + a)))/b^3 - 3/2*(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 - log(e^(b*x + a) + 1)/b^3 + log(e^(b*x + a) - 1)/b^3 + 32*integrate(1/8* x*e^(b*x + a)/(b*e^(2*b*x + 2*a) + b), x)
Exception generated. \[ \int x^2 \text {csch}^3(a+b x) \text {sech}^2(a+b x) \, dx=\text {Exception raised: AttributeError} \]
Timed out. \[ \int x^2 \text {csch}^3(a+b x) \text {sech}^2(a+b x) \, dx=\int \frac {x^2}{{\mathrm {cosh}\left (a+b\,x\right )}^2\,{\mathrm {sinh}\left (a+b\,x\right )}^3} \,d x \]