Integrand size = 12, antiderivative size = 73 \[ \int x^2 \cosh ^2(x) \coth ^2(x) \, dx=\frac {x}{4}-x^2+\frac {x^3}{2}-\frac {1}{2} x \cosh ^2(x)-x^2 \coth (x)+2 x \log \left (1-e^{2 x}\right )+\operatorname {PolyLog}\left (2,e^{2 x}\right )+\frac {1}{4} \cosh (x) \sinh (x)+\frac {1}{2} x^2 \cosh (x) \sinh (x) \]
1/4*x-x^2+1/2*x^3-1/2*x*cosh(x)^2-x^2*coth(x)+2*x*ln(1-exp(2*x))+polylog(2 ,exp(2*x))+1/4*cosh(x)*sinh(x)+1/2*x^2*cosh(x)*sinh(x)
Time = 0.09 (sec) , antiderivative size = 64, normalized size of antiderivative = 0.88 \[ \int x^2 \cosh ^2(x) \coth ^2(x) \, dx=\frac {1}{8} \left (8 x^2+4 x^3-2 x \cosh (2 x)-8 x^2 \coth (x)+16 x \log \left (1-e^{-2 x}\right )-8 \operatorname {PolyLog}\left (2,e^{-2 x}\right )+\sinh (2 x)+2 x^2 \sinh (2 x)\right ) \]
(8*x^2 + 4*x^3 - 2*x*Cosh[2*x] - 8*x^2*Coth[x] + 16*x*Log[1 - E^(-2*x)] - 8*PolyLog[2, E^(-2*x)] + Sinh[2*x] + 2*x^2*Sinh[2*x])/8
Result contains complex when optimal does not.
Time = 0.66 (sec) , antiderivative size = 98, normalized size of antiderivative = 1.34, number of steps used = 19, number of rules used = 18, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 1.500, Rules used = {5973, 3042, 25, 3792, 15, 3042, 3115, 24, 4203, 15, 26, 3042, 26, 4199, 25, 2620, 2715, 2838}
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.
| \(\displaystyle \int x^2 \cosh ^2(x) \coth ^2(x) \, dx\) |
| \(\Big \downarrow \) 5973 |
| \(\displaystyle \int x^2 \cosh ^2(x)dx+\int x^2 \coth ^2(x)dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int x^2 \sin \left (i x+\frac {\pi }{2}\right )^2dx+\int -x^2 \tan \left (i x+\frac {\pi }{2}\right )^2dx\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \int x^2 \sin \left (i x+\frac {\pi }{2}\right )^2dx-\int x^2 \tan \left (i x+\frac {\pi }{2}\right )^2dx\) |
| \(\Big \downarrow \) 3792 |
| \(\displaystyle \frac {\int x^2dx}{2}-\int x^2 \tan \left (i x+\frac {\pi }{2}\right )^2dx+\frac {1}{2} \int \cosh ^2(x)dx+\frac {1}{2} x^2 \sinh (x) \cosh (x)-\frac {1}{2} x \cosh ^2(x)\) |
| \(\Big \downarrow \) 15 |
| \(\displaystyle -\int x^2 \tan \left (i x+\frac {\pi }{2}\right )^2dx+\frac {1}{2} \int \cosh ^2(x)dx+\frac {x^3}{6}+\frac {1}{2} x^2 \sinh (x) \cosh (x)-\frac {1}{2} x \cosh ^2(x)\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle -\int x^2 \tan \left (i x+\frac {\pi }{2}\right )^2dx+\frac {1}{2} \int \sin \left (i x+\frac {\pi }{2}\right )^2dx+\frac {x^3}{6}+\frac {1}{2} x^2 \sinh (x) \cosh (x)-\frac {1}{2} x \cosh ^2(x)\) |
| \(\Big \downarrow \) 3115 |
| \(\displaystyle -\int x^2 \tan \left (i x+\frac {\pi }{2}\right )^2dx+\frac {1}{2} \left (\frac {\int 1dx}{2}+\frac {1}{2} \sinh (x) \cosh (x)\right )+\frac {x^3}{6}+\frac {1}{2} x^2 \sinh (x) \cosh (x)-\frac {1}{2} x \cosh ^2(x)\) |
| \(\Big \downarrow \) 24 |
| \(\displaystyle -\int x^2 \tan \left (i x+\frac {\pi }{2}\right )^2dx+\frac {x^3}{6}+\frac {1}{2} x^2 \sinh (x) \cosh (x)-\frac {1}{2} x \cosh ^2(x)+\frac {1}{2} \left (\frac {x}{2}+\frac {1}{2} \sinh (x) \cosh (x)\right )\) |
| \(\Big \downarrow \) 4203 |
| \(\displaystyle \int x^2dx-2 i \int i x \coth (x)dx+\frac {x^3}{6}-x^2 \coth (x)+\frac {1}{2} x^2 \sinh (x) \cosh (x)-\frac {1}{2} x \cosh ^2(x)+\frac {1}{2} \left (\frac {x}{2}+\frac {1}{2} \sinh (x) \cosh (x)\right )\) |
| \(\Big \downarrow \) 15 |
| \(\displaystyle -2 i \int i x \coth (x)dx+\frac {x^3}{2}-x^2 \coth (x)+\frac {1}{2} x^2 \sinh (x) \cosh (x)-\frac {1}{2} x \cosh ^2(x)+\frac {1}{2} \left (\frac {x}{2}+\frac {1}{2} \sinh (x) \cosh (x)\right )\) |
| \(\Big \downarrow \) 26 |
| \(\displaystyle 2 \int x \coth (x)dx+\frac {x^3}{2}-x^2 \coth (x)+\frac {1}{2} x^2 \sinh (x) \cosh (x)-\frac {1}{2} x \cosh ^2(x)+\frac {1}{2} \left (\frac {x}{2}+\frac {1}{2} \sinh (x) \cosh (x)\right )\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle 2 \int -i x \tan \left (i x+\frac {\pi }{2}\right )dx+\frac {x^3}{2}-x^2 \coth (x)+\frac {1}{2} x^2 \sinh (x) \cosh (x)-\frac {1}{2} x \cosh ^2(x)+\frac {1}{2} \left (\frac {x}{2}+\frac {1}{2} \sinh (x) \cosh (x)\right )\) |
| \(\Big \downarrow \) 26 |
| \(\displaystyle -2 i \int x \tan \left (i x+\frac {\pi }{2}\right )dx+\frac {x^3}{2}-x^2 \coth (x)+\frac {1}{2} x^2 \sinh (x) \cosh (x)-\frac {1}{2} x \cosh ^2(x)+\frac {1}{2} \left (\frac {x}{2}+\frac {1}{2} \sinh (x) \cosh (x)\right )\) |
| \(\Big \downarrow \) 4199 |
| \(\displaystyle -2 i \left (2 i \int -\frac {e^{2 x} x}{1-e^{2 x}}dx-\frac {i x^2}{2}\right )+\frac {x^3}{2}-x^2 \coth (x)+\frac {1}{2} x^2 \sinh (x) \cosh (x)-\frac {1}{2} x \cosh ^2(x)+\frac {1}{2} \left (\frac {x}{2}+\frac {1}{2} \sinh (x) \cosh (x)\right )\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle -2 i \left (-2 i \int \frac {e^{2 x} x}{1-e^{2 x}}dx-\frac {i x^2}{2}\right )+\frac {x^3}{2}-x^2 \coth (x)+\frac {1}{2} x^2 \sinh (x) \cosh (x)-\frac {1}{2} x \cosh ^2(x)+\frac {1}{2} \left (\frac {x}{2}+\frac {1}{2} \sinh (x) \cosh (x)\right )\) |
| \(\Big \downarrow \) 2620 |
| \(\displaystyle -2 i \left (-2 i \left (\frac {1}{2} \int \log \left (1-e^{2 x}\right )dx-\frac {1}{2} x \log \left (1-e^{2 x}\right )\right )-\frac {i x^2}{2}\right )+\frac {x^3}{2}-x^2 \coth (x)+\frac {1}{2} x^2 \sinh (x) \cosh (x)-\frac {1}{2} x \cosh ^2(x)+\frac {1}{2} \left (\frac {x}{2}+\frac {1}{2} \sinh (x) \cosh (x)\right )\) |
| \(\Big \downarrow \) 2715 |
| \(\displaystyle -2 i \left (-2 i \left (\frac {1}{4} \int e^{-2 x} \log \left (1-e^{2 x}\right )de^{2 x}-\frac {1}{2} x \log \left (1-e^{2 x}\right )\right )-\frac {i x^2}{2}\right )+\frac {x^3}{2}-x^2 \coth (x)+\frac {1}{2} x^2 \sinh (x) \cosh (x)-\frac {1}{2} x \cosh ^2(x)+\frac {1}{2} \left (\frac {x}{2}+\frac {1}{2} \sinh (x) \cosh (x)\right )\) |
| \(\Big \downarrow \) 2838 |
| \(\displaystyle -2 i \left (-2 i \left (-\frac {\operatorname {PolyLog}\left (2,e^{2 x}\right )}{4}-\frac {1}{2} x \log \left (1-e^{2 x}\right )\right )-\frac {i x^2}{2}\right )+\frac {x^3}{2}-x^2 \coth (x)+\frac {1}{2} x^2 \sinh (x) \cosh (x)-\frac {1}{2} x \cosh ^2(x)+\frac {1}{2} \left (\frac {x}{2}+\frac {1}{2} \sinh (x) \cosh (x)\right )\) |
x^3/2 - (x*Cosh[x]^2)/2 - x^2*Coth[x] - (2*I)*((-1/2*I)*x^2 - (2*I)*(-1/2* (x*Log[1 - E^(2*x)]) - PolyLog[2, E^(2*x)]/4)) + (x^2*Cosh[x]*Sinh[x])/2 + (x/2 + (Cosh[x]*Sinh[x])/2)/2
3.5.19.3.1 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[Log[(a_) + (b_.)*((F_)^((e_.)*((c_.) + (d_.)*(x_))))^(n_.)], x_Symbol] :> Simp[1/(d*e*n*Log[F]) Subst[Int[Log[a + b*x]/x, x], x, (F^(e*(c + d*x) ))^n], x] /; FreeQ[{F, a, b, c, d, e, n}, x] && GtQ[a, 0]
Int[Log[(c_.)*((d_) + (e_.)*(x_)^(n_.))]/(x_), x_Symbol] :> Simp[-PolyLog[2 , (-c)*e*x^n]/n, x] /; FreeQ[{c, d, e, n}, x] && EqQ[c*d, 1]
Int[((b_.)*sin[(c_.) + (d_.)*(x_)])^(n_), x_Symbol] :> Simp[(-b)*Cos[c + d* x]*((b*Sin[c + d*x])^(n - 1)/(d*n)), x] + Simp[b^2*((n - 1)/n) Int[(b*Sin [c + d*x])^(n - 2), x], x] /; FreeQ[{b, c, d}, x] && GtQ[n, 1] && IntegerQ[ 2*n]
Int[((c_.) + (d_.)*(x_))^(m_)*((b_.)*sin[(e_.) + (f_.)*(x_)])^(n_), x_Symbo l] :> Simp[d*m*(c + d*x)^(m - 1)*((b*Sin[e + f*x])^n/(f^2*n^2)), x] + (-Sim p[b*(c + d*x)^m*Cos[e + f*x]*((b*Sin[e + f*x])^(n - 1)/(f*n)), x] + Simp[b^ 2*((n - 1)/n) Int[(c + d*x)^m*(b*Sin[e + f*x])^(n - 2), x], x] - Simp[d^2 *m*((m - 1)/(f^2*n^2)) Int[(c + d*x)^(m - 2)*(b*Sin[e + f*x])^n, x], x]) /; FreeQ[{b, c, d, e, f}, x] && GtQ[n, 1] && GtQ[m, 1]
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_.)*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 = 0.33 (sec) , antiderivative size = 87, normalized size of antiderivative = 1.19
| method | result | size |
| risch | \(\frac {x^{3}}{2}+\left (\frac {1}{16}-\frac {1}{8} x +\frac {1}{8} x^{2}\right ) {\mathrm e}^{2 x}+\left (-\frac {1}{16}-\frac {1}{8} x -\frac {1}{8} x^{2}\right ) {\mathrm e}^{-2 x}-\frac {2 x^{2}}{{\mathrm e}^{2 x}-1}-2 x^{2}+2 x \ln \left (1-{\mathrm e}^{x}\right )+2 \operatorname {polylog}\left (2, {\mathrm e}^{x}\right )+2 x \ln \left ({\mathrm e}^{x}+1\right )+2 \operatorname {polylog}\left (2, -{\mathrm e}^{x}\right )\) | \(87\) |
1/2*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^2 /(exp(x)^2-1)-2*x^2+2*x*ln(1-exp(x))+2*polylog(2,exp(x))+2*x*ln(exp(x)+1)+ 2*polylog(2,-exp(x))
Leaf count of result is larger than twice the leaf count of optimal. 617 vs. \(2 (60) = 120\).
Time = 0.26 (sec) , antiderivative size = 617, normalized size of antiderivative = 8.45 \[ \int x^2 \cosh ^2(x) \coth ^2(x) \, dx=\text {Too large to display} \]
1/16*((2*x^2 - 2*x + 1)*cosh(x)^6 + 6*(2*x^2 - 2*x + 1)*cosh(x)*sinh(x)^5 + (2*x^2 - 2*x + 1)*sinh(x)^6 + (8*x^3 - 34*x^2 + 2*x - 1)*cosh(x)^4 + (8* x^3 + 15*(2*x^2 - 2*x + 1)*cosh(x)^2 - 34*x^2 + 2*x - 1)*sinh(x)^4 + 4*(5* (2*x^2 - 2*x + 1)*cosh(x)^3 + (8*x^3 - 34*x^2 + 2*x - 1)*cosh(x))*sinh(x)^ 3 - (8*x^3 + 2*x^2 + 2*x + 1)*cosh(x)^2 + (15*(2*x^2 - 2*x + 1)*cosh(x)^4 - 8*x^3 + 6*(8*x^3 - 34*x^2 + 2*x - 1)*cosh(x)^2 - 2*x^2 - 2*x - 1)*sinh(x )^2 + 2*x^2 + 32*(cosh(x)^4 + 4*cosh(x)*sinh(x)^3 + sinh(x)^4 + (6*cosh(x) ^2 - 1)*sinh(x)^2 - cosh(x)^2 + 2*(2*cosh(x)^3 - cosh(x))*sinh(x))*dilog(c osh(x) + sinh(x)) + 32*(cosh(x)^4 + 4*cosh(x)*sinh(x)^3 + sinh(x)^4 + (6*c osh(x)^2 - 1)*sinh(x)^2 - cosh(x)^2 + 2*(2*cosh(x)^3 - cosh(x))*sinh(x))*d ilog(-cosh(x) - sinh(x)) + 32*(x*cosh(x)^4 + 4*x*cosh(x)*sinh(x)^3 + x*sin h(x)^4 - x*cosh(x)^2 + (6*x*cosh(x)^2 - x)*sinh(x)^2 + 2*(2*x*cosh(x)^3 - x*cosh(x))*sinh(x))*log(cosh(x) + sinh(x) + 1) + 32*(x*cosh(x)^4 + 4*x*cos h(x)*sinh(x)^3 + x*sinh(x)^4 - x*cosh(x)^2 + (6*x*cosh(x)^2 - x)*sinh(x)^2 + 2*(2*x*cosh(x)^3 - x*cosh(x))*sinh(x))*log(-cosh(x) - sinh(x) + 1) + 2* (3*(2*x^2 - 2*x + 1)*cosh(x)^5 + 2*(8*x^3 - 34*x^2 + 2*x - 1)*cosh(x)^3 - (8*x^3 + 2*x^2 + 2*x + 1)*cosh(x))*sinh(x) + 2*x + 1)/(cosh(x)^4 + 4*cosh( x)*sinh(x)^3 + sinh(x)^4 + (6*cosh(x)^2 - 1)*sinh(x)^2 - cosh(x)^2 + 2*(2* cosh(x)^3 - cosh(x))*sinh(x))
\[ \int x^2 \cosh ^2(x) \coth ^2(x) \, dx=\int x^{2} \cosh ^{2}{\left (x \right )} \coth ^{2}{\left (x \right )}\, dx \]
Exception generated. \[ \int x^2 \cosh ^2(x) \coth ^2(x) \, dx=\text {Exception raised: RuntimeError} \]
\[ \int x^2 \cosh ^2(x) \coth ^2(x) \, dx=\int { x^{2} \cosh \left (x\right )^{2} \coth \left (x\right )^{2} \,d x } \]
Timed out. \[ \int x^2 \cosh ^2(x) \coth ^2(x) \, dx=\int x^2\,{\mathrm {cosh}\left (x\right )}^2\,{\mathrm {coth}\left (x\right )}^2 \,d x \]