Integrand size = 10, antiderivative size = 33 \[ \int x \cosh ^2(x) \coth ^2(x) \, dx=\frac {3 x^2}{4}-\frac {\cosh ^2(x)}{4}-x \coth (x)+\log (\sinh (x))+\frac {1}{2} x \cosh (x) \sinh (x) \]
Time = 0.04 (sec) , antiderivative size = 33, normalized size of antiderivative = 1.00 \[ \int x \cosh ^2(x) \coth ^2(x) \, dx=\frac {3 x^2}{4}-\frac {1}{8} \cosh (2 x)-x \coth (x)+\log (\sinh (x))+\frac {1}{4} x \sinh (2 x) \]
Time = 0.35 (sec) , antiderivative size = 33, normalized size of antiderivative = 1.00, number of steps used = 11, number of rules used = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 1.100, Rules used = {5973, 3042, 25, 3791, 15, 4203, 15, 26, 3042, 26, 3956}
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.
| \(\displaystyle \int x \cosh ^2(x) \coth ^2(x) \, dx\) |
| \(\Big \downarrow \) 5973 |
| \(\displaystyle \int x \cosh ^2(x)dx+\int x \coth ^2(x)dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int x \sin \left (i x+\frac {\pi }{2}\right )^2dx+\int -x \tan \left (i x+\frac {\pi }{2}\right )^2dx\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \int x \sin \left (i x+\frac {\pi }{2}\right )^2dx-\int x \tan \left (i x+\frac {\pi }{2}\right )^2dx\) |
| \(\Big \downarrow \) 3791 |
| \(\displaystyle \frac {\int xdx}{2}-\int x \tan \left (i x+\frac {\pi }{2}\right )^2dx-\frac {1}{4} \cosh ^2(x)+\frac {1}{2} x \sinh (x) \cosh (x)\) |
| \(\Big \downarrow \) 15 |
| \(\displaystyle -\int x \tan \left (i x+\frac {\pi }{2}\right )^2dx+\frac {x^2}{4}-\frac {\cosh ^2(x)}{4}+\frac {1}{2} x \sinh (x) \cosh (x)\) |
| \(\Big \downarrow \) 4203 |
| \(\displaystyle \int xdx-i \int i \coth (x)dx+\frac {x^2}{4}-\frac {\cosh ^2(x)}{4}-x \coth (x)+\frac {1}{2} x \sinh (x) \cosh (x)\) |
| \(\Big \downarrow \) 15 |
| \(\displaystyle -i \int i \coth (x)dx+\frac {3 x^2}{4}-\frac {\cosh ^2(x)}{4}-x \coth (x)+\frac {1}{2} x \sinh (x) \cosh (x)\) |
| \(\Big \downarrow \) 26 |
| \(\displaystyle \int \coth (x)dx+\frac {3 x^2}{4}-\frac {\cosh ^2(x)}{4}-x \coth (x)+\frac {1}{2} x \sinh (x) \cosh (x)\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int -i \tan \left (i x+\frac {\pi }{2}\right )dx+\frac {3 x^2}{4}-\frac {\cosh ^2(x)}{4}-x \coth (x)+\frac {1}{2} x \sinh (x) \cosh (x)\) |
| \(\Big \downarrow \) 26 |
| \(\displaystyle -i \int \tan \left (i x+\frac {\pi }{2}\right )dx+\frac {3 x^2}{4}-\frac {\cosh ^2(x)}{4}-x \coth (x)+\frac {1}{2} x \sinh (x) \cosh (x)\) |
| \(\Big \downarrow \) 3956 |
| \(\displaystyle \frac {3 x^2}{4}-\frac {\cosh ^2(x)}{4}-x \coth (x)+\log (\sinh (x))+\frac {1}{2} x \sinh (x) \cosh (x)\) |
3.5.18.3.1 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[((c_.) + (d_.)*(x_))*((b_.)*sin[(e_.) + (f_.)*(x_)])^(n_), x_Symbol] :> Simp[d*((b*Sin[e + f*x])^n/(f^2*n^2)), x] + (-Simp[b*(c + d*x)*Cos[e + f*x ]*((b*Sin[e + f*x])^(n - 1)/(f*n)), x] + Simp[b^2*((n - 1)/n) Int[(c + d* x)*(b*Sin[e + f*x])^(n - 2), x], x]) /; FreeQ[{b, c, d, e, f}, x] && GtQ[n, 1]
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_.)*((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 = 48, normalized size of antiderivative = 1.45
| method | result | size |
| risch | \(\frac {3 x^{2}}{4}+\left (-\frac {1}{16}+\frac {x}{8}\right ) {\mathrm e}^{2 x}+\left (-\frac {1}{16}-\frac {x}{8}\right ) {\mathrm e}^{-2 x}-2 x -\frac {2 x}{{\mathrm e}^{2 x}-1}+\ln \left ({\mathrm e}^{2 x}-1\right )\) | \(48\) |
Leaf count of result is larger than twice the leaf count of optimal. 336 vs. \(2 (27) = 54\).
Time = 0.27 (sec) , antiderivative size = 336, normalized size of antiderivative = 10.18 \[ \int x \cosh ^2(x) \coth ^2(x) \, dx=\frac {{\left (2 \, x - 1\right )} \cosh \left (x\right )^{6} + 6 \, {\left (2 \, x - 1\right )} \cosh \left (x\right ) \sinh \left (x\right )^{5} + {\left (2 \, x - 1\right )} \sinh \left (x\right )^{6} + {\left (12 \, x^{2} - 34 \, x + 1\right )} \cosh \left (x\right )^{4} + {\left (15 \, {\left (2 \, x - 1\right )} \cosh \left (x\right )^{2} + 12 \, x^{2} - 34 \, x + 1\right )} \sinh \left (x\right )^{4} + 4 \, {\left (5 \, {\left (2 \, x - 1\right )} \cosh \left (x\right )^{3} + {\left (12 \, x^{2} - 34 \, x + 1\right )} \cosh \left (x\right )\right )} \sinh \left (x\right )^{3} - {\left (12 \, x^{2} + 2 \, x + 1\right )} \cosh \left (x\right )^{2} + {\left (15 \, {\left (2 \, x - 1\right )} \cosh \left (x\right )^{4} + 6 \, {\left (12 \, x^{2} - 34 \, x + 1\right )} \cosh \left (x\right )^{2} - 12 \, x^{2} - 2 \, x - 1\right )} \sinh \left (x\right )^{2} + 16 \, {\left (\cosh \left (x\right )^{4} + 4 \, \cosh \left (x\right ) \sinh \left (x\right )^{3} + \sinh \left (x\right )^{4} + {\left (6 \, \cosh \left (x\right )^{2} - 1\right )} \sinh \left (x\right )^{2} - \cosh \left (x\right )^{2} + 2 \, {\left (2 \, \cosh \left (x\right )^{3} - \cosh \left (x\right )\right )} \sinh \left (x\right )\right )} \log \left (\frac {2 \, \sinh \left (x\right )}{\cosh \left (x\right ) - \sinh \left (x\right )}\right ) + 2 \, {\left (3 \, {\left (2 \, x - 1\right )} \cosh \left (x\right )^{5} + 2 \, {\left (12 \, x^{2} - 34 \, x + 1\right )} \cosh \left (x\right )^{3} - {\left (12 \, x^{2} + 2 \, x + 1\right )} \cosh \left (x\right )\right )} \sinh \left (x\right ) + 2 \, x + 1}{16 \, {\left (\cosh \left (x\right )^{4} + 4 \, \cosh \left (x\right ) \sinh \left (x\right )^{3} + \sinh \left (x\right )^{4} + {\left (6 \, \cosh \left (x\right )^{2} - 1\right )} \sinh \left (x\right )^{2} - \cosh \left (x\right )^{2} + 2 \, {\left (2 \, \cosh \left (x\right )^{3} - \cosh \left (x\right )\right )} \sinh \left (x\right )\right )}} \]
1/16*((2*x - 1)*cosh(x)^6 + 6*(2*x - 1)*cosh(x)*sinh(x)^5 + (2*x - 1)*sinh (x)^6 + (12*x^2 - 34*x + 1)*cosh(x)^4 + (15*(2*x - 1)*cosh(x)^2 + 12*x^2 - 34*x + 1)*sinh(x)^4 + 4*(5*(2*x - 1)*cosh(x)^3 + (12*x^2 - 34*x + 1)*cosh (x))*sinh(x)^3 - (12*x^2 + 2*x + 1)*cosh(x)^2 + (15*(2*x - 1)*cosh(x)^4 + 6*(12*x^2 - 34*x + 1)*cosh(x)^2 - 12*x^2 - 2*x - 1)*sinh(x)^2 + 16*(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))*log(2*sinh(x)/(cosh(x) - sinh(x ))) + 2*(3*(2*x - 1)*cosh(x)^5 + 2*(12*x^2 - 34*x + 1)*cosh(x)^3 - (12*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 - co sh(x))*sinh(x))
\[ \int x \cosh ^2(x) \coth ^2(x) \, dx=\int x \cosh ^{2}{\left (x \right )} \coth ^{2}{\left (x \right )}\, dx \]
Exception generated. \[ \int x \cosh ^2(x) \coth ^2(x) \, dx=\text {Exception raised: RuntimeError} \]
Leaf count of result is larger than twice the leaf count of optimal. 101 vs. \(2 (27) = 54\).
Time = 0.27 (sec) , antiderivative size = 101, normalized size of antiderivative = 3.06 \[ \int x \cosh ^2(x) \coth ^2(x) \, dx=\frac {12 \, x^{2} e^{\left (4 \, x\right )} - 12 \, x^{2} e^{\left (2 \, x\right )} + 2 \, x e^{\left (6 \, x\right )} - 34 \, x e^{\left (4 \, x\right )} - 2 \, x e^{\left (2 \, x\right )} + 16 \, e^{\left (4 \, x\right )} \log \left (e^{\left (2 \, x\right )} - 1\right ) - 16 \, e^{\left (2 \, x\right )} \log \left (e^{\left (2 \, x\right )} - 1\right ) + 2 \, x - e^{\left (6 \, x\right )} + e^{\left (4 \, x\right )} - e^{\left (2 \, x\right )} + 1}{16 \, {\left (e^{\left (4 \, x\right )} - e^{\left (2 \, x\right )}\right )}} \]
1/16*(12*x^2*e^(4*x) - 12*x^2*e^(2*x) + 2*x*e^(6*x) - 34*x*e^(4*x) - 2*x*e ^(2*x) + 16*e^(4*x)*log(e^(2*x) - 1) - 16*e^(2*x)*log(e^(2*x) - 1) + 2*x - e^(6*x) + e^(4*x) - e^(2*x) + 1)/(e^(4*x) - e^(2*x))
Time = 0.13 (sec) , antiderivative size = 48, normalized size of antiderivative = 1.45 \[ \int x \cosh ^2(x) \coth ^2(x) \, dx=\ln \left ({\mathrm {e}}^{2\,x}-1\right )-2\,x-{\mathrm {e}}^{-2\,x}\,\left (\frac {x}{8}+\frac {1}{16}\right )+{\mathrm {e}}^{2\,x}\,\left (\frac {x}{8}-\frac {1}{16}\right )-\frac {2\,x}{{\mathrm {e}}^{2\,x}-1}+\frac {3\,x^2}{4} \]