Integrand size = 12, antiderivative size = 96 \[ \int x^2 \cosh ^2(x) \coth ^3(x) \, dx=\frac {3 x^2}{4}-\frac {2 x^3}{3}-x \coth (x)-\frac {1}{2} x^2 \coth ^2(x)+2 x^2 \log \left (1-e^{2 x}\right )+\log (\sinh (x))+2 x \operatorname {PolyLog}\left (2,e^{2 x}\right )-\operatorname {PolyLog}\left (3,e^{2 x}\right )-\frac {1}{2} x \cosh (x) \sinh (x)+\frac {\sinh ^2(x)}{4}+\frac {1}{2} x^2 \sinh ^2(x) \]
3/4*x^2-2/3*x^3-x*coth(x)-1/2*x^2*coth(x)^2+2*x^2*ln(1-exp(2*x))+ln(sinh(x ))+2*x*polylog(2,exp(2*x))-polylog(3,exp(2*x))-1/2*x*cosh(x)*sinh(x)+1/4*s inh(x)^2+1/2*x^2*sinh(x)^2
Time = 0.19 (sec) , antiderivative size = 135, normalized size of antiderivative = 1.41 \[ \int x^2 \cosh ^2(x) \coth ^3(x) \, dx=-x+\frac {2 x^3}{3}+\frac {1}{8} \left (1+2 x^2\right ) \cosh (2 x)-x \coth (x)-\frac {1}{2} x^2 \text {csch}^2(x)+2 x^2 \log \left (1-e^{-x}\right )+2 x^2 \log \left (1+e^{-x}\right )+\log \left (1-e^x\right )+\log \left (1+e^x\right )-4 x \operatorname {PolyLog}\left (2,-e^{-x}\right )-4 x \operatorname {PolyLog}\left (2,e^{-x}\right )-4 \operatorname {PolyLog}\left (3,-e^{-x}\right )-4 \operatorname {PolyLog}\left (3,e^{-x}\right )-\frac {1}{4} x \sinh (2 x) \]
-x + (2*x^3)/3 + ((1 + 2*x^2)*Cosh[2*x])/8 - x*Coth[x] - (x^2*Csch[x]^2)/2 + 2*x^2*Log[1 - E^(-x)] + 2*x^2*Log[1 + E^(-x)] + Log[1 - E^x] + Log[1 + E^x] - 4*x*PolyLog[2, -E^(-x)] - 4*x*PolyLog[2, E^(-x)] - 4*PolyLog[3, -E^ (-x)] - 4*PolyLog[3, E^(-x)] - (x*Sinh[2*x])/4
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.
| \(\displaystyle \int x^2 \cosh ^2(x) \coth ^3(x) \, dx\) |
| \(\Big \downarrow \) 5973 |
| \(\displaystyle \int x^2 \coth ^3(x)dx+\int x^2 \cosh ^2(x) \coth (x)dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int x^2 \cosh ^2(x) \coth (x)dx+\int i x^2 \tan \left (i x+\frac {\pi }{2}\right )^3dx\) |
| \(\Big \downarrow \) 26 |
| \(\displaystyle \int x^2 \cosh ^2(x) \coth (x)dx+i \int x^2 \tan \left (i x+\frac {\pi }{2}\right )^3dx\) |
| \(\Big \downarrow \) 4203 |
| \(\displaystyle \int x^2 \cosh ^2(x) \coth (x)dx+i \left (-\int i x^2 \coth (x)dx+i \int -x \coth ^2(x)dx+\frac {1}{2} i x^2 \coth ^2(x)\right )\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \int x^2 \cosh ^2(x) \coth (x)dx+i \left (-\int i x^2 \coth (x)dx-i \int x \coth ^2(x)dx+\frac {1}{2} i x^2 \coth ^2(x)\right )\) |
| \(\Big \downarrow \) 26 |
| \(\displaystyle \int x^2 \cosh ^2(x) \coth (x)dx+i \left (-i \int x^2 \coth (x)dx-i \int x \coth ^2(x)dx+\frac {1}{2} i x^2 \coth ^2(x)\right )\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int x^2 \cosh ^2(x) \coth (x)dx+i \left (-i \int -i x^2 \tan \left (i x+\frac {\pi }{2}\right )dx-i \int -x \tan \left (i x+\frac {\pi }{2}\right )^2dx+\frac {1}{2} i x^2 \coth ^2(x)\right )\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \int x^2 \cosh ^2(x) \coth (x)dx+i \left (-i \int -i x^2 \tan \left (i x+\frac {\pi }{2}\right )dx+i \int x \tan \left (i x+\frac {\pi }{2}\right )^2dx+\frac {1}{2} i x^2 \coth ^2(x)\right )\) |
| \(\Big \downarrow \) 26 |
| \(\displaystyle \int x^2 \cosh ^2(x) \coth (x)dx+i \left (-\int x^2 \tan \left (i x+\frac {\pi }{2}\right )dx+i \int x \tan \left (i x+\frac {\pi }{2}\right )^2dx+\frac {1}{2} i x^2 \coth ^2(x)\right )\) |
| \(\Big \downarrow \) 4199 |
| \(\displaystyle \int x^2 \cosh ^2(x) \coth (x)dx+i \left (-2 i \int -\frac {e^{2 x} x^2}{1-e^{2 x}}dx+i \int x \tan \left (i x+\frac {\pi }{2}\right )^2dx+\frac {i x^3}{3}+\frac {1}{2} i x^2 \coth ^2(x)\right )\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \int x^2 \cosh ^2(x) \coth (x)dx+i \left (2 i \int \frac {e^{2 x} x^2}{1-e^{2 x}}dx+i \int x \tan \left (i x+\frac {\pi }{2}\right )^2dx+\frac {i x^3}{3}+\frac {1}{2} i x^2 \coth ^2(x)\right )\) |
| \(\Big \downarrow \) 2620 |
| \(\displaystyle \int x^2 \cosh ^2(x) \coth (x)dx+i \left (2 i \left (\int x \log \left (1-e^{2 x}\right )dx-\frac {1}{2} x^2 \log \left (1-e^{2 x}\right )\right )+i \int x \tan \left (i x+\frac {\pi }{2}\right )^2dx+\frac {i x^3}{3}+\frac {1}{2} i x^2 \coth ^2(x)\right )\) |
| \(\Big \downarrow \) 3011 |
| \(\displaystyle \int x^2 \cosh ^2(x) \coth (x)dx+i \left (2 i \left (\frac {1}{2} \int \operatorname {PolyLog}\left (2,e^{2 x}\right )dx-\frac {1}{2} x \operatorname {PolyLog}\left (2,e^{2 x}\right )-\frac {1}{2} x^2 \log \left (1-e^{2 x}\right )\right )+i \int x \tan \left (i x+\frac {\pi }{2}\right )^2dx+\frac {i x^3}{3}+\frac {1}{2} i x^2 \coth ^2(x)\right )\) |
| \(\Big \downarrow \) 2720 |
| \(\displaystyle \int x^2 \cosh ^2(x) \coth (x)dx+i \left (2 i \left (\frac {1}{4} \int e^{-2 x} \operatorname {PolyLog}\left (2,e^{2 x}\right )de^{2 x}-\frac {1}{2} x \operatorname {PolyLog}\left (2,e^{2 x}\right )-\frac {1}{2} x^2 \log \left (1-e^{2 x}\right )\right )+i \int x \tan \left (i x+\frac {\pi }{2}\right )^2dx+\frac {i x^3}{3}+\frac {1}{2} i x^2 \coth ^2(x)\right )\) |
| \(\Big \downarrow \) 4203 |
| \(\displaystyle \int x^2 \cosh ^2(x) \coth (x)dx+i \left (2 i \left (\frac {1}{4} \int e^{-2 x} \operatorname {PolyLog}\left (2,e^{2 x}\right )de^{2 x}-\frac {1}{2} x \operatorname {PolyLog}\left (2,e^{2 x}\right )-\frac {1}{2} x^2 \log \left (1-e^{2 x}\right )\right )+i (-\int xdx+i \int i \coth (x)dx+x \coth (x))+\frac {i x^3}{3}+\frac {1}{2} i x^2 \coth ^2(x)\right )\) |
| \(\Big \downarrow \) 15 |
| \(\displaystyle \int x^2 \cosh ^2(x) \coth (x)dx+i \left (2 i \left (\frac {1}{4} \int e^{-2 x} \operatorname {PolyLog}\left (2,e^{2 x}\right )de^{2 x}-\frac {1}{2} x \operatorname {PolyLog}\left (2,e^{2 x}\right )-\frac {1}{2} x^2 \log \left (1-e^{2 x}\right )\right )+i \left (i \int i \coth (x)dx-\frac {x^2}{2}+x \coth (x)\right )+\frac {i x^3}{3}+\frac {1}{2} i x^2 \coth ^2(x)\right )\) |
| \(\Big \downarrow \) 26 |
| \(\displaystyle \int x^2 \cosh ^2(x) \coth (x)dx+i \left (2 i \left (\frac {1}{4} \int e^{-2 x} \operatorname {PolyLog}\left (2,e^{2 x}\right )de^{2 x}-\frac {1}{2} x \operatorname {PolyLog}\left (2,e^{2 x}\right )-\frac {1}{2} x^2 \log \left (1-e^{2 x}\right )\right )+i \left (-\int \coth (x)dx-\frac {x^2}{2}+x \coth (x)\right )+\frac {i x^3}{3}+\frac {1}{2} i x^2 \coth ^2(x)\right )\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int x^2 \cosh ^2(x) \coth (x)dx+i \left (2 i \left (\frac {1}{4} \int e^{-2 x} \operatorname {PolyLog}\left (2,e^{2 x}\right )de^{2 x}-\frac {1}{2} x \operatorname {PolyLog}\left (2,e^{2 x}\right )-\frac {1}{2} x^2 \log \left (1-e^{2 x}\right )\right )+i \left (-\int -i \tan \left (i x+\frac {\pi }{2}\right )dx-\frac {x^2}{2}+x \coth (x)\right )+\frac {i x^3}{3}+\frac {1}{2} i x^2 \coth ^2(x)\right )\) |
| \(\Big \downarrow \) 26 |
| \(\displaystyle \int x^2 \cosh ^2(x) \coth (x)dx+i \left (2 i \left (\frac {1}{4} \int e^{-2 x} \operatorname {PolyLog}\left (2,e^{2 x}\right )de^{2 x}-\frac {1}{2} x \operatorname {PolyLog}\left (2,e^{2 x}\right )-\frac {1}{2} x^2 \log \left (1-e^{2 x}\right )\right )+i \left (i \int \tan \left (i x+\frac {\pi }{2}\right )dx-\frac {x^2}{2}+x \coth (x)\right )+\frac {i x^3}{3}+\frac {1}{2} i x^2 \coth ^2(x)\right )\) |
| \(\Big \downarrow \) 3956 |
| \(\displaystyle \int x^2 \cosh ^2(x) \coth (x)dx+i \left (2 i \left (\frac {1}{4} \int e^{-2 x} \operatorname {PolyLog}\left (2,e^{2 x}\right )de^{2 x}-\frac {1}{2} x \operatorname {PolyLog}\left (2,e^{2 x}\right )-\frac {1}{2} x^2 \log \left (1-e^{2 x}\right )\right )+\frac {i x^3}{3}+\frac {1}{2} i x^2 \coth ^2(x)+i \left (-\frac {x^2}{2}+x \coth (x)-\log (\sinh (x))\right )\right )\) |
| \(\Big \downarrow \) 5973 |
| \(\displaystyle i \left (2 i \left (\frac {1}{4} \int e^{-2 x} \operatorname {PolyLog}\left (2,e^{2 x}\right )de^{2 x}-\frac {1}{2} x \operatorname {PolyLog}\left (2,e^{2 x}\right )-\frac {1}{2} x^2 \log \left (1-e^{2 x}\right )\right )+\frac {i x^3}{3}+\frac {1}{2} i x^2 \coth ^2(x)+i \left (-\frac {x^2}{2}+x \coth (x)-\log (\sinh (x))\right )\right )+\int x^2 \coth (x)dx+\int x^2 \cosh (x) \sinh (x)dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle i \left (2 i \left (\frac {1}{4} \int e^{-2 x} \operatorname {PolyLog}\left (2,e^{2 x}\right )de^{2 x}-\frac {1}{2} x \operatorname {PolyLog}\left (2,e^{2 x}\right )-\frac {1}{2} x^2 \log \left (1-e^{2 x}\right )\right )+\frac {i x^3}{3}+\frac {1}{2} i x^2 \coth ^2(x)+i \left (-\frac {x^2}{2}+x \coth (x)-\log (\sinh (x))\right )\right )+\int -i x^2 \tan \left (i x+\frac {\pi }{2}\right )dx+\int x^2 \cosh (x) \sinh (x)dx\) |
| \(\Big \downarrow \) 26 |
| \(\displaystyle i \left (2 i \left (\frac {1}{4} \int e^{-2 x} \operatorname {PolyLog}\left (2,e^{2 x}\right )de^{2 x}-\frac {1}{2} x \operatorname {PolyLog}\left (2,e^{2 x}\right )-\frac {1}{2} x^2 \log \left (1-e^{2 x}\right )\right )+\frac {i x^3}{3}+\frac {1}{2} i x^2 \coth ^2(x)+i \left (-\frac {x^2}{2}+x \coth (x)-\log (\sinh (x))\right )\right )-i \int x^2 \tan \left (i x+\frac {\pi }{2}\right )dx+\int x^2 \cosh (x) \sinh (x)dx\) |
| \(\Big \downarrow \) 4199 |
| \(\displaystyle i \left (2 i \left (\frac {1}{4} \int e^{-2 x} \operatorname {PolyLog}\left (2,e^{2 x}\right )de^{2 x}-\frac {1}{2} x \operatorname {PolyLog}\left (2,e^{2 x}\right )-\frac {1}{2} x^2 \log \left (1-e^{2 x}\right )\right )+\frac {i x^3}{3}+\frac {1}{2} i x^2 \coth ^2(x)+i \left (-\frac {x^2}{2}+x \coth (x)-\log (\sinh (x))\right )\right )+\int x^2 \cosh (x) \sinh (x)dx-i \left (2 i \int -\frac {e^{2 x} x^2}{1-e^{2 x}}dx-\frac {i x^3}{3}\right )\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle i \left (2 i \left (\frac {1}{4} \int e^{-2 x} \operatorname {PolyLog}\left (2,e^{2 x}\right )de^{2 x}-\frac {1}{2} x \operatorname {PolyLog}\left (2,e^{2 x}\right )-\frac {1}{2} x^2 \log \left (1-e^{2 x}\right )\right )+\frac {i x^3}{3}+\frac {1}{2} i x^2 \coth ^2(x)+i \left (-\frac {x^2}{2}+x \coth (x)-\log (\sinh (x))\right )\right )+\int x^2 \cosh (x) \sinh (x)dx-i \left (-2 i \int \frac {e^{2 x} x^2}{1-e^{2 x}}dx-\frac {i x^3}{3}\right )\) |
| \(\Big \downarrow \) 2620 |
| \(\displaystyle i \left (2 i \left (\frac {1}{4} \int e^{-2 x} \operatorname {PolyLog}\left (2,e^{2 x}\right )de^{2 x}-\frac {1}{2} x \operatorname {PolyLog}\left (2,e^{2 x}\right )-\frac {1}{2} x^2 \log \left (1-e^{2 x}\right )\right )+\frac {i x^3}{3}+\frac {1}{2} i x^2 \coth ^2(x)+i \left (-\frac {x^2}{2}+x \coth (x)-\log (\sinh (x))\right )\right )+\int x^2 \cosh (x) \sinh (x)dx-i \left (-2 i \left (\int x \log \left (1-e^{2 x}\right )dx-\frac {1}{2} x^2 \log \left (1-e^{2 x}\right )\right )-\frac {i x^3}{3}\right )\) |
| \(\Big \downarrow \) 3011 |
| \(\displaystyle -i \left (-2 i \left (\frac {1}{2} \int \operatorname {PolyLog}\left (2,e^{2 x}\right )dx-\frac {1}{2} x \operatorname {PolyLog}\left (2,e^{2 x}\right )-\frac {1}{2} x^2 \log \left (1-e^{2 x}\right )\right )-\frac {i x^3}{3}\right )+i \left (2 i \left (\frac {1}{4} \int e^{-2 x} \operatorname {PolyLog}\left (2,e^{2 x}\right )de^{2 x}-\frac {1}{2} x \operatorname {PolyLog}\left (2,e^{2 x}\right )-\frac {1}{2} x^2 \log \left (1-e^{2 x}\right )\right )+\frac {i x^3}{3}+\frac {1}{2} i x^2 \coth ^2(x)+i \left (-\frac {x^2}{2}+x \coth (x)-\log (\sinh (x))\right )\right )+\int x^2 \cosh (x) \sinh (x)dx\) |
| \(\Big \downarrow \) 2720 |
| \(\displaystyle -i \left (-2 i \left (\frac {1}{4} \int e^{-2 x} \operatorname {PolyLog}\left (2,e^{2 x}\right )de^{2 x}-\frac {1}{2} x \operatorname {PolyLog}\left (2,e^{2 x}\right )-\frac {1}{2} x^2 \log \left (1-e^{2 x}\right )\right )-\frac {i x^3}{3}\right )+i \left (2 i \left (\frac {1}{4} \int e^{-2 x} \operatorname {PolyLog}\left (2,e^{2 x}\right )de^{2 x}-\frac {1}{2} x \operatorname {PolyLog}\left (2,e^{2 x}\right )-\frac {1}{2} x^2 \log \left (1-e^{2 x}\right )\right )+\frac {i x^3}{3}+\frac {1}{2} i x^2 \coth ^2(x)+i \left (-\frac {x^2}{2}+x \coth (x)-\log (\sinh (x))\right )\right )+\int x^2 \cosh (x) \sinh (x)dx\) |
| \(\Big \downarrow \) 5895 |
| \(\displaystyle -i \left (-2 i \left (\frac {1}{4} \int e^{-2 x} \operatorname {PolyLog}\left (2,e^{2 x}\right )de^{2 x}-\frac {1}{2} x \operatorname {PolyLog}\left (2,e^{2 x}\right )-\frac {1}{2} x^2 \log \left (1-e^{2 x}\right )\right )-\frac {i x^3}{3}\right )+i \left (2 i \left (\frac {1}{4} \int e^{-2 x} \operatorname {PolyLog}\left (2,e^{2 x}\right )de^{2 x}-\frac {1}{2} x \operatorname {PolyLog}\left (2,e^{2 x}\right )-\frac {1}{2} x^2 \log \left (1-e^{2 x}\right )\right )+\frac {i x^3}{3}+\frac {1}{2} i x^2 \coth ^2(x)+i \left (-\frac {x^2}{2}+x \coth (x)-\log (\sinh (x))\right )\right )-\int x \sinh ^2(x)dx+\frac {1}{2} x^2 \sinh ^2(x)\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle -i \left (-2 i \left (\frac {1}{4} \int e^{-2 x} \operatorname {PolyLog}\left (2,e^{2 x}\right )de^{2 x}-\frac {1}{2} x \operatorname {PolyLog}\left (2,e^{2 x}\right )-\frac {1}{2} x^2 \log \left (1-e^{2 x}\right )\right )-\frac {i x^3}{3}\right )+i \left (2 i \left (\frac {1}{4} \int e^{-2 x} \operatorname {PolyLog}\left (2,e^{2 x}\right )de^{2 x}-\frac {1}{2} x \operatorname {PolyLog}\left (2,e^{2 x}\right )-\frac {1}{2} x^2 \log \left (1-e^{2 x}\right )\right )+\frac {i x^3}{3}+\frac {1}{2} i x^2 \coth ^2(x)+i \left (-\frac {x^2}{2}+x \coth (x)-\log (\sinh (x))\right )\right )-\int -x \sin (i x)^2dx+\frac {1}{2} x^2 \sinh ^2(x)\) |
3.5.22.3.1 Defintions of rubi rules used
Int[(a_.)*(x_)^(m_.), x_Symbol] :> Simp[a*(x^(m + 1)/(m + 1)), x] /; FreeQ[ {a, m}, x] && NeQ[m, -1]
Int[(Complex[0, a_])*(Fx_), x_Symbol] :> Simp[(Complex[Identity[0], a]) I nt[Fx, x], x] /; FreeQ[a, x] && EqQ[a^2, 1]
Int[(((F_)^((g_.)*((e_.) + (f_.)*(x_))))^(n_.)*((c_.) + (d_.)*(x_))^(m_.))/ ((a_) + (b_.)*((F_)^((g_.)*((e_.) + (f_.)*(x_))))^(n_.)), x_Symbol] :> Simp [((c + d*x)^m/(b*f*g*n*Log[F]))*Log[1 + b*((F^(g*(e + f*x)))^n/a)], x] - Si mp[d*(m/(b*f*g*n*Log[F])) Int[(c + d*x)^(m - 1)*Log[1 + b*((F^(g*(e + f*x )))^n/a)], x], x] /; FreeQ[{F, a, b, c, d, e, f, g, n}, x] && IGtQ[m, 0]
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]]]
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]
Int[tan[(c_.) + (d_.)*(x_)], x_Symbol] :> Simp[-Log[RemoveContent[Cos[c + d *x], x]]/d, x] /; FreeQ[{c, d}, x]
Int[((c_.) + (d_.)*(x_))^(m_.)*tan[(e_.) + Pi*(k_.) + (Complex[0, fz_])*(f_ .)*(x_)], x_Symbol] :> Simp[(-I)*((c + d*x)^(m + 1)/(d*(m + 1))), x] + Simp [2*I Int[((c + d*x)^m*(E^(2*((-I)*e + f*fz*x))/(1 + E^(2*((-I)*e + f*fz*x ))/E^(2*I*k*Pi))))/E^(2*I*k*Pi), x], x] /; FreeQ[{c, d, e, f, fz}, x] && In tegerQ[4*k] && IGtQ[m, 0]
Int[((c_.) + (d_.)*(x_))^(m_.)*((b_.)*tan[(e_.) + (f_.)*(x_)])^(n_), x_Symb ol] :> Simp[b*(c + d*x)^m*((b*Tan[e + f*x])^(n - 1)/(f*(n - 1))), x] + (-Si mp[b*d*(m/(f*(n - 1))) Int[(c + d*x)^(m - 1)*(b*Tan[e + f*x])^(n - 1), x] , x] - Simp[b^2 Int[(c + d*x)^m*(b*Tan[e + f*x])^(n - 2), x], x]) /; Free Q[{b, c, d, e, f}, x] && GtQ[n, 1] && GtQ[m, 0]
Int[Cosh[(a_.) + (b_.)*(x_)^(n_.)]*(x_)^(m_.)*Sinh[(a_.) + (b_.)*(x_)^(n_.) ]^(p_.), x_Symbol] :> Simp[x^(m - n + 1)*(Sinh[a + b*x^n]^(p + 1)/(b*n*(p + 1))), x] - Simp[(m - n + 1)/(b*n*(p + 1)) Int[x^(m - n)*Sinh[a + b*x^n]^ (p + 1), x], x] /; FreeQ[{a, b, p}, x] && LtQ[0, n, m + 1] && NeQ[p, -1]
Int[Cosh[(a_.) + (b_.)*(x_)]^(n_.)*Coth[(a_.) + (b_.)*(x_)]^(p_.)*((c_.) + (d_.)*(x_))^(m_.), x_Symbol] :> Int[(c + d*x)^m*Cosh[a + b*x]^n*Coth[a + b* x]^(p - 2), x] + Int[(c + d*x)^m*Cosh[a + b*x]^(n - 2)*Coth[a + b*x]^p, x] /; FreeQ[{a, b, c, d, m}, x] && IGtQ[n, 0] && IGtQ[p, 0]
Time = 0.45 (sec) , antiderivative size = 127, normalized size of antiderivative = 1.32
| method | result | size |
| risch | \(-\frac {2 x^{3}}{3}+\left (\frac {1}{16}-\frac {1}{8} x +\frac {1}{8} x^{2}\right ) {\mathrm e}^{2 x}+\left (\frac {1}{16}+\frac {1}{8} x +\frac {1}{8} x^{2}\right ) {\mathrm e}^{-2 x}-\frac {2 x \left ({\mathrm e}^{2 x} x +{\mathrm e}^{2 x}-1\right )}{\left ({\mathrm e}^{2 x}-1\right )^{2}}+\ln \left ({\mathrm e}^{x}-1\right )-2 \ln \left ({\mathrm e}^{x}\right )+\ln \left ({\mathrm e}^{x}+1\right )+2 x^{2} \ln \left (1-{\mathrm e}^{x}\right )+4 x \operatorname {polylog}\left (2, {\mathrm e}^{x}\right )-4 \operatorname {polylog}\left (3, {\mathrm e}^{x}\right )+2 x^{2} \ln \left ({\mathrm e}^{x}+1\right )+4 x \operatorname {polylog}\left (2, -{\mathrm e}^{x}\right )-4 \operatorname {polylog}\left (3, -{\mathrm e}^{x}\right )\) | \(127\) |
-2/3*x^3+(1/16-1/8*x+1/8*x^2)*exp(x)^2+(1/16+1/8*x+1/8*x^2)/exp(x)^2-2*x*( x*exp(x)^2+exp(x)^2-1)/(exp(x)^2-1)^2+ln(exp(x)-1)-2*ln(exp(x))+ln(exp(x)+ 1)+2*x^2*ln(1-exp(x))+4*x*polylog(2,exp(x))-4*polylog(3,exp(x))+2*x^2*ln(e xp(x)+1)+4*x*polylog(2,-exp(x))-4*polylog(3,-exp(x))
Leaf count of result is larger than twice the leaf count of optimal. 1512 vs. \(2 (80) = 160\).
Time = 0.29 (sec) , antiderivative size = 1512, normalized size of antiderivative = 15.75 \[ \int x^2 \cosh ^2(x) \coth ^3(x) \, dx=\text {Too large to display} \]
1/48*(3*(2*x^2 - 2*x + 1)*cosh(x)^8 + 24*(2*x^2 - 2*x + 1)*cosh(x)*sinh(x) ^7 + 3*(2*x^2 - 2*x + 1)*sinh(x)^8 - 2*(16*x^3 + 6*x^2 + 42*x + 3)*cosh(x) ^6 - 2*(16*x^3 - 42*(2*x^2 - 2*x + 1)*cosh(x)^2 + 6*x^2 + 42*x + 3)*sinh(x )^6 + 12*(14*(2*x^2 - 2*x + 1)*cosh(x)^3 - (16*x^3 + 6*x^2 + 42*x + 3)*cos h(x))*sinh(x)^5 + 2*(32*x^3 - 42*x^2 + 48*x + 3)*cosh(x)^4 + 2*(105*(2*x^2 - 2*x + 1)*cosh(x)^4 + 32*x^3 - 15*(16*x^3 + 6*x^2 + 42*x + 3)*cosh(x)^2 - 42*x^2 + 48*x + 3)*sinh(x)^4 + 8*(21*(2*x^2 - 2*x + 1)*cosh(x)^5 - 5*(16 *x^3 + 6*x^2 + 42*x + 3)*cosh(x)^3 + (32*x^3 - 42*x^2 + 48*x + 3)*cosh(x)) *sinh(x)^3 - 2*(16*x^3 + 6*x^2 + 6*x + 3)*cosh(x)^2 + 2*(42*(2*x^2 - 2*x + 1)*cosh(x)^6 - 15*(16*x^3 + 6*x^2 + 42*x + 3)*cosh(x)^4 - 16*x^3 + 6*(32* x^3 - 42*x^2 + 48*x + 3)*cosh(x)^2 - 6*x^2 - 6*x - 3)*sinh(x)^2 + 6*x^2 + 192*(x*cosh(x)^6 + 6*x*cosh(x)*sinh(x)^5 + x*sinh(x)^6 - 2*x*cosh(x)^4 + ( 15*x*cosh(x)^2 - 2*x)*sinh(x)^4 + 4*(5*x*cosh(x)^3 - 2*x*cosh(x))*sinh(x)^ 3 + x*cosh(x)^2 + (15*x*cosh(x)^4 - 12*x*cosh(x)^2 + x)*sinh(x)^2 + 2*(3*x *cosh(x)^5 - 4*x*cosh(x)^3 + x*cosh(x))*sinh(x))*dilog(cosh(x) + sinh(x)) + 192*(x*cosh(x)^6 + 6*x*cosh(x)*sinh(x)^5 + x*sinh(x)^6 - 2*x*cosh(x)^4 + (15*x*cosh(x)^2 - 2*x)*sinh(x)^4 + 4*(5*x*cosh(x)^3 - 2*x*cosh(x))*sinh(x )^3 + x*cosh(x)^2 + (15*x*cosh(x)^4 - 12*x*cosh(x)^2 + x)*sinh(x)^2 + 2*(3 *x*cosh(x)^5 - 4*x*cosh(x)^3 + x*cosh(x))*sinh(x))*dilog(-cosh(x) - sinh(x )) + 48*((2*x^2 + 1)*cosh(x)^6 + 6*(2*x^2 + 1)*cosh(x)*sinh(x)^5 + (2*x...
\[ \int x^2 \cosh ^2(x) \coth ^3(x) \, dx=\int x^{2} \cosh ^{2}{\left (x \right )} \coth ^{3}{\left (x \right )}\, dx \]
Leaf count of result is larger than twice the leaf count of optimal. 174 vs. \(2 (80) = 160\).
Time = 0.24 (sec) , antiderivative size = 174, normalized size of antiderivative = 1.81 \[ \int x^2 \cosh ^2(x) \coth ^3(x) \, dx=-\frac {4}{3} \, x^{3} + 2 \, x^{2} \log \left (e^{x} + 1\right ) + 2 \, x^{2} \log \left (-e^{x} + 1\right ) + 4 \, x {\rm Li}_2\left (-e^{x}\right ) + 4 \, x {\rm Li}_2\left (e^{x}\right ) - 2 \, x + \frac {32 \, x^{3} - 12 \, x^{2} + 3 \, {\left (2 \, x^{2} - 2 \, x + 1\right )} e^{\left (6 \, x\right )} + 2 \, {\left (16 \, x^{3} - 6 \, x^{2} + 6 \, x - 3\right )} e^{\left (4 \, x\right )} - 2 \, {\left (32 \, x^{3} + 42 \, x^{2} + 48 \, x - 3\right )} e^{\left (2 \, x\right )} + 3 \, {\left (2 \, x^{2} + 2 \, x + 1\right )} e^{\left (-2 \, x\right )} + 84 \, x - 6}{48 \, {\left (e^{\left (4 \, x\right )} - 2 \, e^{\left (2 \, x\right )} + 1\right )}} + \log \left (e^{x} + 1\right ) + \log \left (e^{x} - 1\right ) - 4 \, {\rm Li}_{3}(-e^{x}) - 4 \, {\rm Li}_{3}(e^{x}) \]
-4/3*x^3 + 2*x^2*log(e^x + 1) + 2*x^2*log(-e^x + 1) + 4*x*dilog(-e^x) + 4* x*dilog(e^x) - 2*x + 1/48*(32*x^3 - 12*x^2 + 3*(2*x^2 - 2*x + 1)*e^(6*x) + 2*(16*x^3 - 6*x^2 + 6*x - 3)*e^(4*x) - 2*(32*x^3 + 42*x^2 + 48*x - 3)*e^( 2*x) + 3*(2*x^2 + 2*x + 1)*e^(-2*x) + 84*x - 6)/(e^(4*x) - 2*e^(2*x) + 1) + log(e^x + 1) + log(e^x - 1) - 4*polylog(3, -e^x) - 4*polylog(3, e^x)
\[ \int x^2 \cosh ^2(x) \coth ^3(x) \, dx=\int { x^{2} \cosh \left (x\right )^{2} \coth \left (x\right )^{3} \,d x } \]
Timed out. \[ \int x^2 \cosh ^2(x) \coth ^3(x) \, dx=\int x^2\,{\mathrm {cosh}\left (x\right )}^2\,{\mathrm {coth}\left (x\right )}^3 \,d x \]