Integrand size = 18, antiderivative size = 148 \[ \int x^2 \text {csch}^3(a+b x) \text {sech}(a+b x) \, dx=\frac {x^2}{2 b}+\frac {2 x^2 \text {arctanh}\left (e^{2 a+2 b x}\right )}{b}-\frac {x \coth (a+b x)}{b^2}-\frac {x^2 \coth ^2(a+b x)}{2 b}+\frac {\log (\sinh (a+b x))}{b^3}+\frac {x \operatorname {PolyLog}\left (2,-e^{2 a+2 b x}\right )}{b^2}-\frac {x \operatorname {PolyLog}\left (2,e^{2 a+2 b x}\right )}{b^2}-\frac {\operatorname {PolyLog}\left (3,-e^{2 a+2 b x}\right )}{2 b^3}+\frac {\operatorname {PolyLog}\left (3,e^{2 a+2 b x}\right )}{2 b^3} \]
1/2*x^2/b+2*x^2*arctanh(exp(2*b*x+2*a))/b-x*coth(b*x+a)/b^2-1/2*x^2*coth(b *x+a)^2/b+ln(sinh(b*x+a))/b^3+x*polylog(2,-exp(2*b*x+2*a))/b^2-x*polylog(2 ,exp(2*b*x+2*a))/b^2-1/2*polylog(3,-exp(2*b*x+2*a))/b^3+1/2*polylog(3,exp( 2*b*x+2*a))/b^3
Leaf count is larger than twice the leaf count of optimal. \(388\) vs. \(2(148)=296\).
Time = 2.95 (sec) , antiderivative size = 388, normalized size of antiderivative = 2.62 \[ \int x^2 \text {csch}^3(a+b x) \text {sech}(a+b x) \, dx=\frac {1}{6} \left (-\frac {3 x^2 \text {csch}^2(a+b x)}{b}+\frac {2 e^{2 a} \left (-6 b e^{-2 a} x-6 b \left (1-e^{-2 a}\right ) x+2 b^3 e^{-2 a} x^3-3 b^2 e^{-2 a} \left (-1+e^{2 a}\right ) x^2 \log \left (1-e^{-a-b x}\right )-3 b^2 e^{-2 a} \left (-1+e^{2 a}\right ) x^2 \log \left (1+e^{-a-b x}\right )+3 \left (1-e^{-2 a}\right ) \log \left (1-e^{a+b x}\right )+3 \left (1-e^{-2 a}\right ) \log \left (1+e^{a+b x}\right )+6 b \left (1-e^{-2 a}\right ) x \operatorname {PolyLog}\left (2,-e^{-a-b x}\right )+6 b \left (1-e^{-2 a}\right ) x \operatorname {PolyLog}\left (2,e^{-a-b x}\right )+6 \left (1-e^{-2 a}\right ) \operatorname {PolyLog}\left (3,-e^{-a-b x}\right )+6 \left (1-e^{-2 a}\right ) \operatorname {PolyLog}\left (3,e^{-a-b x}\right )\right )}{b^3 \left (-1+e^{2 a}\right )}+\frac {2 b^2 x^2 \left (\frac {2 b x}{1+e^{2 a}}+3 \log \left (1+e^{-2 (a+b x)}\right )\right )-6 b x \operatorname {PolyLog}\left (2,-e^{-2 (a+b x)}\right )-3 \operatorname {PolyLog}\left (3,-e^{-2 (a+b x)}\right )}{b^3}-2 x^3 \text {csch}(a) \text {sech}(a)+\frac {6 x \text {csch}(a) \text {csch}(a+b x) \sinh (b x)}{b^2}\right ) \]
((-3*x^2*Csch[a + b*x]^2)/b + (2*E^(2*a)*((-6*b*x)/E^(2*a) - 6*b*(1 - E^(- 2*a))*x + (2*b^3*x^3)/E^(2*a) - (3*b^2*(-1 + E^(2*a))*x^2*Log[1 - E^(-a - b*x)])/E^(2*a) - (3*b^2*(-1 + E^(2*a))*x^2*Log[1 + E^(-a - b*x)])/E^(2*a) + 3*(1 - E^(-2*a))*Log[1 - E^(a + b*x)] + 3*(1 - E^(-2*a))*Log[1 + E^(a + b*x)] + 6*b*(1 - E^(-2*a))*x*PolyLog[2, -E^(-a - b*x)] + 6*b*(1 - E^(-2*a) )*x*PolyLog[2, E^(-a - b*x)] + 6*(1 - E^(-2*a))*PolyLog[3, -E^(-a - b*x)] + 6*(1 - E^(-2*a))*PolyLog[3, E^(-a - b*x)]))/(b^3*(-1 + E^(2*a))) + (2*b^ 2*x^2*((2*b*x)/(1 + E^(2*a)) + 3*Log[1 + E^(-2*(a + b*x))]) - 6*b*x*PolyLo g[2, -E^(-2*(a + b*x))] - 3*PolyLog[3, -E^(-2*(a + b*x))])/b^3 - 2*x^3*Csc h[a]*Sech[a] + (6*x*Csch[a]*Csch[a + b*x]*Sinh[b*x])/b^2)/6
Time = 0.49 (sec) , antiderivative size = 148, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.222, 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.
| \(\displaystyle \int x^2 \text {csch}^3(a+b x) \text {sech}(a+b x) \, dx\) |
| \(\Big \downarrow \) 5985 |
| \(\displaystyle -2 \int -\frac {1}{2} x \left (\frac {\coth ^2(a+b x)}{b}+\frac {2 \log (\tanh (a+b x))}{b}\right )dx-\frac {x^2 \coth ^2(a+b x)}{2 b}-\frac {x^2 \log (\tanh (a+b x))}{b}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \int x \left (\frac {\coth ^2(a+b x)}{b}+\frac {2 \log (\tanh (a+b x))}{b}\right )dx-\frac {x^2 \coth ^2(a+b x)}{2 b}-\frac {x^2 \log (\tanh (a+b x))}{b}\) |
| \(\Big \downarrow \) 2010 |
| \(\displaystyle \int \left (\frac {x \coth ^2(a+b x)}{b}+\frac {2 x \log (\tanh (a+b x))}{b}\right )dx-\frac {x^2 \coth ^2(a+b x)}{2 b}-\frac {x^2 \log (\tanh (a+b x))}{b}\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {2 x^2 \text {arctanh}\left (e^{2 a+2 b x}\right )}{b}-\frac {\operatorname {PolyLog}\left (3,-e^{2 a+2 b x}\right )}{2 b^3}+\frac {\operatorname {PolyLog}\left (3,e^{2 a+2 b x}\right )}{2 b^3}+\frac {\log (\sinh (a+b x))}{b^3}+\frac {x \operatorname {PolyLog}\left (2,-e^{2 a+2 b x}\right )}{b^2}-\frac {x \operatorname {PolyLog}\left (2,e^{2 a+2 b x}\right )}{b^2}-\frac {x \coth (a+b x)}{b^2}-\frac {x^2 \coth ^2(a+b x)}{2 b}+\frac {x^2}{2 b}\) |
x^2/(2*b) + (2*x^2*ArcTanh[E^(2*a + 2*b*x)])/b - (x*Coth[a + b*x])/b^2 - ( x^2*Coth[a + b*x]^2)/(2*b) + Log[Sinh[a + b*x]]/b^3 + (x*PolyLog[2, -E^(2* a + 2*b*x)])/b^2 - (x*PolyLog[2, E^(2*a + 2*b*x)])/b^2 - PolyLog[3, -E^(2* a + 2*b*x)]/(2*b^3) + PolyLog[3, E^(2*a + 2*b*x)]/(2*b^3)
3.6.9.3.1 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]
Time = 2.62 (sec) , antiderivative size = 266, normalized size of antiderivative = 1.80
| method | result | size |
| risch | \(-\frac {2 x \left ({\mathrm e}^{2 b x +2 a} b x +{\mathrm e}^{2 b x +2 a}-1\right )}{b^{2} \left ({\mathrm e}^{2 b x +2 a}-1\right )^{2}}-\frac {a^{2} \ln \left ({\mathrm e}^{b x +a}-1\right )}{b^{3}}-\frac {\ln \left (1-{\mathrm e}^{b x +a}\right ) x^{2}}{b}-\frac {2 x \operatorname {polylog}\left (2, {\mathrm e}^{b x +a}\right )}{b^{2}}+\frac {x^{2} \ln \left (1+{\mathrm e}^{2 b x +2 a}\right )}{b}+\frac {x \operatorname {polylog}\left (2, -{\mathrm e}^{2 b x +2 a}\right )}{b^{2}}-\frac {\ln \left ({\mathrm e}^{b x +a}+1\right ) x^{2}}{b}-\frac {2 x \operatorname {polylog}\left (2, -{\mathrm e}^{b x +a}\right )}{b^{2}}+\frac {\ln \left (1-{\mathrm e}^{b x +a}\right ) a^{2}}{b^{3}}+\frac {\ln \left ({\mathrm e}^{b x +a}-1\right )}{b^{3}}+\frac {\ln \left ({\mathrm e}^{b x +a}+1\right )}{b^{3}}-\frac {2 \ln \left ({\mathrm e}^{b x +a}\right )}{b^{3}}+\frac {2 \operatorname {polylog}\left (3, {\mathrm e}^{b x +a}\right )}{b^{3}}-\frac {\operatorname {polylog}\left (3, -{\mathrm e}^{2 b x +2 a}\right )}{2 b^{3}}+\frac {2 \operatorname {polylog}\left (3, -{\mathrm e}^{b x +a}\right )}{b^{3}}\) | \(266\) |
-2*x*(exp(2*b*x+2*a)*b*x+exp(2*b*x+2*a)-1)/b^2/(exp(2*b*x+2*a)-1)^2-1/b^3* a^2*ln(exp(b*x+a)-1)-1/b*ln(1-exp(b*x+a))*x^2-2*x*polylog(2,exp(b*x+a))/b^ 2+x^2*ln(1+exp(2*b*x+2*a))/b+x*polylog(2,-exp(2*b*x+2*a))/b^2-1/b*ln(exp(b *x+a)+1)*x^2-2*x*polylog(2,-exp(b*x+a))/b^2+1/b^3*ln(1-exp(b*x+a))*a^2+1/b ^3*ln(exp(b*x+a)-1)+1/b^3*ln(exp(b*x+a)+1)-2/b^3*ln(exp(b*x+a))+2*polylog( 3,exp(b*x+a))/b^3-1/2*polylog(3,-exp(2*b*x+2*a))/b^3+2*polylog(3,-exp(b*x+ a))/b^3
Result contains complex when optimal does not.
Time = 0.27 (sec) , antiderivative size = 2562, normalized size of antiderivative = 17.31 \[ \int x^2 \text {csch}^3(a+b x) \text {sech}(a+b x) \, dx=\text {Too large to display} \]
-(2*(b*x + a)*cosh(b*x + a)^4 + 8*(b*x + a)*cosh(b*x + a)*sinh(b*x + a)^3 + 2*(b*x + a)*sinh(b*x + a)^4 + 2*(b^2*x^2 - b*x - 2*a)*cosh(b*x + a)^2 + 2*(b^2*x^2 + 6*(b*x + a)*cosh(b*x + a)^2 - b*x - 2*a)*sinh(b*x + a)^2 + 2* (b*x*cosh(b*x + a)^4 + 4*b*x*cosh(b*x + a)*sinh(b*x + a)^3 + b*x*sinh(b*x + a)^4 - 2*b*x*cosh(b*x + a)^2 + 2*(3*b*x*cosh(b*x + a)^2 - b*x)*sinh(b*x + a)^2 + b*x + 4*(b*x*cosh(b*x + a)^3 - b*x*cosh(b*x + a))*sinh(b*x + a))* dilog(cosh(b*x + a) + sinh(b*x + a)) - 2*(b*x*cosh(b*x + a)^4 + 4*b*x*cosh (b*x + a)*sinh(b*x + a)^3 + b*x*sinh(b*x + a)^4 - 2*b*x*cosh(b*x + a)^2 + 2*(3*b*x*cosh(b*x + a)^2 - b*x)*sinh(b*x + a)^2 + b*x + 4*(b*x*cosh(b*x + a)^3 - b*x*cosh(b*x + a))*sinh(b*x + a))*dilog(I*cosh(b*x + a) + I*sinh(b* x + a)) - 2*(b*x*cosh(b*x + a)^4 + 4*b*x*cosh(b*x + a)*sinh(b*x + a)^3 + b *x*sinh(b*x + a)^4 - 2*b*x*cosh(b*x + a)^2 + 2*(3*b*x*cosh(b*x + a)^2 - b* x)*sinh(b*x + a)^2 + b*x + 4*(b*x*cosh(b*x + a)^3 - b*x*cosh(b*x + a))*sin h(b*x + a))*dilog(-I*cosh(b*x + a) - I*sinh(b*x + a)) + 2*(b*x*cosh(b*x + a)^4 + 4*b*x*cosh(b*x + a)*sinh(b*x + a)^3 + b*x*sinh(b*x + a)^4 - 2*b*x*c osh(b*x + a)^2 + 2*(3*b*x*cosh(b*x + a)^2 - b*x)*sinh(b*x + a)^2 + b*x + 4 *(b*x*cosh(b*x + a)^3 - b*x*cosh(b*x + a))*sinh(b*x + a))*dilog(-cosh(b*x + a) - sinh(b*x + a)) + ((b^2*x^2 - 1)*cosh(b*x + a)^4 + 4*(b^2*x^2 - 1)*c osh(b*x + a)*sinh(b*x + a)^3 + (b^2*x^2 - 1)*sinh(b*x + a)^4 + b^2*x^2 - 2 *(b^2*x^2 - 1)*cosh(b*x + a)^2 - 2*(b^2*x^2 - 3*(b^2*x^2 - 1)*cosh(b*x ...
\[ \int x^2 \text {csch}^3(a+b x) \text {sech}(a+b x) \, dx=\int x^{2} \operatorname {csch}^{3}{\left (a + b x \right )} \operatorname {sech}{\left (a + b x \right )}\, dx \]
Time = 0.23 (sec) , antiderivative size = 243, normalized size of antiderivative = 1.64 \[ \int x^2 \text {csch}^3(a+b x) \text {sech}(a+b x) \, dx=-\frac {2 \, {\left ({\left (b x^{2} e^{\left (2 \, a\right )} + x e^{\left (2 \, a\right )}\right )} e^{\left (2 \, b x\right )} - x\right )}}{b^{2} e^{\left (4 \, b x + 4 \, a\right )} - 2 \, b^{2} e^{\left (2 \, b x + 2 \, a\right )} + b^{2}} - \frac {2 \, x}{b^{2}} + \frac {2 \, b^{2} x^{2} \log \left (e^{\left (2 \, b x + 2 \, a\right )} + 1\right ) + 2 \, b x {\rm Li}_2\left (-e^{\left (2 \, b x + 2 \, a\right )}\right ) - {\rm Li}_{3}(-e^{\left (2 \, b x + 2 \, a\right )})}{2 \, b^{3}} - \frac {b^{2} x^{2} \log \left (e^{\left (b x + a\right )} + 1\right ) + 2 \, b x {\rm Li}_2\left (-e^{\left (b x + a\right )}\right ) - 2 \, {\rm Li}_{3}(-e^{\left (b x + a\right )})}{b^{3}} - \frac {b^{2} x^{2} \log \left (-e^{\left (b x + a\right )} + 1\right ) + 2 \, b x {\rm Li}_2\left (e^{\left (b x + a\right )}\right ) - 2 \, {\rm Li}_{3}(e^{\left (b x + a\right )})}{b^{3}} + \frac {\log \left (e^{\left (b x + a\right )} + 1\right )}{b^{3}} + \frac {\log \left (e^{\left (b x + a\right )} - 1\right )}{b^{3}} \]
-2*((b*x^2*e^(2*a) + x*e^(2*a))*e^(2*b*x) - x)/(b^2*e^(4*b*x + 4*a) - 2*b^ 2*e^(2*b*x + 2*a) + b^2) - 2*x/b^2 + 1/2*(2*b^2*x^2*log(e^(2*b*x + 2*a) + 1) + 2*b*x*dilog(-e^(2*b*x + 2*a)) - polylog(3, -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^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 + log(e^(b*x + a) + 1)/b^3 + log(e^(b* x + a) - 1)/b^3
\[ \int x^2 \text {csch}^3(a+b x) \text {sech}(a+b x) \, dx=\int { x^{2} \operatorname {csch}\left (b x + a\right )^{3} \operatorname {sech}\left (b x + a\right ) \,d x } \]
Timed out. \[ \int x^2 \text {csch}^3(a+b x) \text {sech}(a+b x) \, dx=\int \frac {x^2}{\mathrm {cosh}\left (a+b\,x\right )\,{\mathrm {sinh}\left (a+b\,x\right )}^3} \,d x \]