Integrand size = 10, antiderivative size = 53 \[ \int x \cosh (2 x) \text {sech}^3(x) \, dx=3 x \arctan \left (e^x\right )-\frac {3}{2} i \operatorname {PolyLog}\left (2,-i e^x\right )+\frac {3}{2} i \operatorname {PolyLog}\left (2,i e^x\right )-\frac {\text {sech}(x)}{2}-\frac {1}{2} x \text {sech}(x) \tanh (x) \]
3*x*arctan(exp(x))-3/2*I*polylog(2,-I*exp(x))+3/2*I*polylog(2,I*exp(x))-1/ 2*sech(x)-1/2*x*sech(x)*tanh(x)
Time = 0.09 (sec) , antiderivative size = 68, normalized size of antiderivative = 1.28 \[ \int x \cosh (2 x) \text {sech}^3(x) \, dx=\frac {3}{2} i \left (x \left (\log \left (1-i e^x\right )-\log \left (1+i e^x\right )\right )-\operatorname {PolyLog}\left (2,-i e^x\right )+\operatorname {PolyLog}\left (2,i e^x\right )\right )-\frac {\text {sech}(x)}{2}-\frac {1}{2} x \text {sech}(x) \tanh (x) \]
((3*I)/2)*(x*(Log[1 - I*E^x] - Log[1 + I*E^x]) - PolyLog[2, (-I)*E^x] + Po lyLog[2, I*E^x]) - Sech[x]/2 - (x*Sech[x]*Tanh[x])/2
Time = 0.30 (sec) , antiderivative size = 53, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.200, Rules used = {5996, 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.
\(\displaystyle \int x \cosh (2 x) \text {sech}^3(x) \, dx\) |
\(\Big \downarrow \) 5996 |
\(\displaystyle \int \left (x \text {sech}(x)+x \tanh ^2(x) \text {sech}(x)\right )dx\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle 3 x \arctan \left (e^x\right )-\frac {3}{2} i \operatorname {PolyLog}\left (2,-i e^x\right )+\frac {3}{2} i \operatorname {PolyLog}\left (2,i e^x\right )-\frac {\text {sech}(x)}{2}-\frac {1}{2} x \tanh (x) \text {sech}(x)\) |
3*x*ArcTan[E^x] - ((3*I)/2)*PolyLog[2, (-I)*E^x] + ((3*I)/2)*PolyLog[2, I* E^x] - Sech[x]/2 - (x*Sech[x]*Tanh[x])/2
3.11.41.3.1 Defintions of rubi rules used
Int[((e_.) + (f_.)*(x_))^(m_.)*(F_)[(a_.) + (b_.)*(x_)]^(p_.)*(G_)[(c_.) + (d_.)*(x_)]^(q_.), x_Symbol] :> Int[ExpandTrigExpand[(e + f*x)^m*G[c + d*x] ^q, F, c + d*x, p, b/d, x], x] /; FreeQ[{a, b, c, d, e, f, m}, x] && Member Q[{Sinh, Cosh}, F] && MemberQ[{Sech, Csch}, G] && IGtQ[p, 0] && IGtQ[q, 0] && EqQ[b*c - a*d, 0] && IGtQ[b/d, 1]
Time = 3.75 (sec) , antiderivative size = 75, normalized size of antiderivative = 1.42
method | result | size |
risch | \(-\frac {{\mathrm e}^{x} \left (x \,{\mathrm e}^{2 x}+{\mathrm e}^{2 x}-x +1\right )}{\left (1+{\mathrm e}^{2 x}\right )^{2}}-\frac {3 i x \ln \left (1+i {\mathrm e}^{x}\right )}{2}+\frac {3 i x \ln \left (1-i {\mathrm e}^{x}\right )}{2}-\frac {3 i \operatorname {dilog}\left (1+i {\mathrm e}^{x}\right )}{2}+\frac {3 i \operatorname {dilog}\left (1-i {\mathrm e}^{x}\right )}{2}\) | \(75\) |
-exp(x)*(x*exp(x)^2+exp(x)^2-x+1)/(exp(x)^2+1)^2-3/2*I*x*ln(1+I*exp(x))+3/ 2*I*x*ln(1-I*exp(x))-3/2*I*dilog(1+I*exp(x))+3/2*I*dilog(1-I*exp(x))
Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 408 vs. \(2 (32) = 64\).
Time = 0.27 (sec) , antiderivative size = 408, normalized size of antiderivative = 7.70 \[ \int x \cosh (2 x) \text {sech}^3(x) \, dx=-\frac {2 \, {\left (x + 1\right )} \cosh \left (x\right )^{3} + 6 \, {\left (x + 1\right )} \cosh \left (x\right ) \sinh \left (x\right )^{2} + 2 \, {\left (x + 1\right )} \sinh \left (x\right )^{3} - 2 \, {\left (x - 1\right )} \cosh \left (x\right ) + 3 \, {\left (-i \, \cosh \left (x\right )^{4} - 4 i \, \cosh \left (x\right ) \sinh \left (x\right )^{3} - i \, \sinh \left (x\right )^{4} + 2 \, {\left (-3 i \, \cosh \left (x\right )^{2} - i\right )} \sinh \left (x\right )^{2} - 2 i \, \cosh \left (x\right )^{2} + 4 \, {\left (-i \, \cosh \left (x\right )^{3} - i \, \cosh \left (x\right )\right )} \sinh \left (x\right ) - i\right )} {\rm Li}_2\left (i \, \cosh \left (x\right ) + i \, \sinh \left (x\right )\right ) + 3 \, {\left (i \, \cosh \left (x\right )^{4} + 4 i \, \cosh \left (x\right ) \sinh \left (x\right )^{3} + i \, \sinh \left (x\right )^{4} + 2 \, {\left (3 i \, \cosh \left (x\right )^{2} + i\right )} \sinh \left (x\right )^{2} + 2 i \, \cosh \left (x\right )^{2} + 4 \, {\left (i \, \cosh \left (x\right )^{3} + i \, \cosh \left (x\right )\right )} \sinh \left (x\right ) + i\right )} {\rm Li}_2\left (-i \, \cosh \left (x\right ) - i \, \sinh \left (x\right )\right ) + 3 \, {\left (i \, x \cosh \left (x\right )^{4} + 4 i \, x \cosh \left (x\right ) \sinh \left (x\right )^{3} + i \, x \sinh \left (x\right )^{4} + 2 i \, x \cosh \left (x\right )^{2} + 2 \, {\left (3 i \, x \cosh \left (x\right )^{2} + i \, x\right )} \sinh \left (x\right )^{2} + 4 \, {\left (i \, x \cosh \left (x\right )^{3} + i \, x \cosh \left (x\right )\right )} \sinh \left (x\right ) + i \, x\right )} \log \left (i \, \cosh \left (x\right ) + i \, \sinh \left (x\right ) + 1\right ) + 3 \, {\left (-i \, x \cosh \left (x\right )^{4} - 4 i \, x \cosh \left (x\right ) \sinh \left (x\right )^{3} - i \, x \sinh \left (x\right )^{4} - 2 i \, x \cosh \left (x\right )^{2} + 2 \, {\left (-3 i \, x \cosh \left (x\right )^{2} - i \, x\right )} \sinh \left (x\right )^{2} + 4 \, {\left (-i \, x \cosh \left (x\right )^{3} - i \, x \cosh \left (x\right )\right )} \sinh \left (x\right ) - i \, x\right )} \log \left (-i \, \cosh \left (x\right ) - i \, \sinh \left (x\right ) + 1\right ) + 2 \, {\left (3 \, {\left (x + 1\right )} \cosh \left (x\right )^{2} - x + 1\right )} \sinh \left (x\right )}{2 \, {\left (\cosh \left (x\right )^{4} + 4 \, \cosh \left (x\right ) \sinh \left (x\right )^{3} + \sinh \left (x\right )^{4} + 2 \, {\left (3 \, \cosh \left (x\right )^{2} + 1\right )} \sinh \left (x\right )^{2} + 2 \, \cosh \left (x\right )^{2} + 4 \, {\left (\cosh \left (x\right )^{3} + \cosh \left (x\right )\right )} \sinh \left (x\right ) + 1\right )}} \]
-1/2*(2*(x + 1)*cosh(x)^3 + 6*(x + 1)*cosh(x)*sinh(x)^2 + 2*(x + 1)*sinh(x )^3 - 2*(x - 1)*cosh(x) + 3*(-I*cosh(x)^4 - 4*I*cosh(x)*sinh(x)^3 - I*sinh (x)^4 + 2*(-3*I*cosh(x)^2 - I)*sinh(x)^2 - 2*I*cosh(x)^2 + 4*(-I*cosh(x)^3 - I*cosh(x))*sinh(x) - I)*dilog(I*cosh(x) + I*sinh(x)) + 3*(I*cosh(x)^4 + 4*I*cosh(x)*sinh(x)^3 + I*sinh(x)^4 + 2*(3*I*cosh(x)^2 + I)*sinh(x)^2 + 2 *I*cosh(x)^2 + 4*(I*cosh(x)^3 + I*cosh(x))*sinh(x) + I)*dilog(-I*cosh(x) - I*sinh(x)) + 3*(I*x*cosh(x)^4 + 4*I*x*cosh(x)*sinh(x)^3 + I*x*sinh(x)^4 + 2*I*x*cosh(x)^2 + 2*(3*I*x*cosh(x)^2 + I*x)*sinh(x)^2 + 4*(I*x*cosh(x)^3 + I*x*cosh(x))*sinh(x) + I*x)*log(I*cosh(x) + I*sinh(x) + 1) + 3*(-I*x*cos h(x)^4 - 4*I*x*cosh(x)*sinh(x)^3 - I*x*sinh(x)^4 - 2*I*x*cosh(x)^2 + 2*(-3 *I*x*cosh(x)^2 - I*x)*sinh(x)^2 + 4*(-I*x*cosh(x)^3 - I*x*cosh(x))*sinh(x) - I*x)*log(-I*cosh(x) - I*sinh(x) + 1) + 2*(3*(x + 1)*cosh(x)^2 - x + 1)* sinh(x))/(cosh(x)^4 + 4*cosh(x)*sinh(x)^3 + sinh(x)^4 + 2*(3*cosh(x)^2 + 1 )*sinh(x)^2 + 2*cosh(x)^2 + 4*(cosh(x)^3 + cosh(x))*sinh(x) + 1)
\[ \int x \cosh (2 x) \text {sech}^3(x) \, dx=\int x \cosh {\left (2 x \right )} \operatorname {sech}^{3}{\left (x \right )}\, dx \]
\[ \int x \cosh (2 x) \text {sech}^3(x) \, dx=\int { x \cosh \left (2 \, x\right ) \operatorname {sech}\left (x\right )^{3} \,d x } \]
-((x + 1)*e^(3*x) - (x - 1)*e^x)/(e^(4*x) + 2*e^(2*x) + 1) + 12*integrate( 1/4*x*e^x/(e^(2*x) + 1), x)
\[ \int x \cosh (2 x) \text {sech}^3(x) \, dx=\int { x \cosh \left (2 \, x\right ) \operatorname {sech}\left (x\right )^{3} \,d x } \]
Timed out. \[ \int x \cosh (2 x) \text {sech}^3(x) \, dx=\int \frac {x\,\mathrm {cosh}\left (2\,x\right )}{{\mathrm {cosh}\left (x\right )}^3} \,d x \]