Integrand size = 13, antiderivative size = 41 \[ \int \frac {\coth ^4(x)}{a+a \cosh (x)} \, dx=\frac {\coth ^5(x)}{5 a}-\frac {\text {csch}(x)}{a}-\frac {2 \text {csch}^3(x)}{3 a}-\frac {\text {csch}^5(x)}{5 a} \]
Time = 0.15 (sec) , antiderivative size = 41, normalized size of antiderivative = 1.00 \[ \int \frac {\coth ^4(x)}{a+a \cosh (x)} \, dx=-\frac {(-25+8 \cosh (x)+36 \cosh (2 x)+24 \cosh (3 x)-3 \cosh (4 x)) \text {csch}^3(x)}{120 a (1+\cosh (x))} \]
-1/120*((-25 + 8*Cosh[x] + 36*Cosh[2*x] + 24*Cosh[3*x] - 3*Cosh[4*x])*Csch [x]^3)/(a*(1 + Cosh[x]))
Result contains complex when optimal does not.
Time = 0.37 (sec) , antiderivative size = 46, normalized size of antiderivative = 1.12, number of steps used = 11, number of rules used = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.769, Rules used = {3042, 3185, 25, 3042, 25, 3086, 210, 2009, 3087, 15}
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.
| \(\displaystyle \int \frac {\coth ^4(x)}{a \cosh (x)+a} \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {\tan \left (-\frac {\pi }{2}+i x\right )^4}{a-a \sin \left (-\frac {\pi }{2}+i x\right )}dx\) |
| \(\Big \downarrow \) 3185 |
| \(\displaystyle \frac {\int \coth ^5(x) \text {csch}(x)dx}{a}+\frac {\int -\coth ^4(x) \text {csch}^2(x)dx}{a}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {\int \coth ^5(x) \text {csch}(x)dx}{a}-\frac {\int \coth ^4(x) \text {csch}^2(x)dx}{a}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\int \sec \left (i x-\frac {\pi }{2}\right ) \tan \left (i x-\frac {\pi }{2}\right )^5dx}{a}-\frac {\int -\sec \left (i x-\frac {\pi }{2}\right )^2 \tan \left (i x-\frac {\pi }{2}\right )^4dx}{a}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {\int \sec \left (i x-\frac {\pi }{2}\right )^2 \tan \left (i x-\frac {\pi }{2}\right )^4dx}{a}+\frac {\int \sec \left (i x-\frac {\pi }{2}\right ) \tan \left (i x-\frac {\pi }{2}\right )^5dx}{a}\) |
| \(\Big \downarrow \) 3086 |
| \(\displaystyle \frac {\int \sec \left (i x-\frac {\pi }{2}\right )^2 \tan \left (i x-\frac {\pi }{2}\right )^4dx}{a}-\frac {i \int \left (-\text {csch}^2(x)-1\right )^2d(-i \text {csch}(x))}{a}\) |
| \(\Big \downarrow \) 210 |
| \(\displaystyle \frac {\int \sec \left (i x-\frac {\pi }{2}\right )^2 \tan \left (i x-\frac {\pi }{2}\right )^4dx}{a}-\frac {i \int \left (\text {csch}^4(x)+2 \text {csch}^2(x)+1\right )d(-i \text {csch}(x))}{a}\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {\int \sec \left (i x-\frac {\pi }{2}\right )^2 \tan \left (i x-\frac {\pi }{2}\right )^4dx}{a}-\frac {i \left (-\frac {1}{5} i \text {csch}^5(x)-\frac {2}{3} i \text {csch}^3(x)-i \text {csch}(x)\right )}{a}\) |
| \(\Big \downarrow \) 3087 |
| \(\displaystyle -\frac {i \int \coth ^4(x)d(i \coth (x))}{a}-\frac {i \left (-\frac {1}{5} i \text {csch}^5(x)-\frac {2}{3} i \text {csch}^3(x)-i \text {csch}(x)\right )}{a}\) |
| \(\Big \downarrow \) 15 |
| \(\displaystyle \frac {\coth ^5(x)}{5 a}-\frac {i \left (-\frac {1}{5} i \text {csch}^5(x)-\frac {2}{3} i \text {csch}^3(x)-i \text {csch}(x)\right )}{a}\) |
3.2.96.3.1 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[((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> Int[ExpandIntegrand[(a + b*x^2 )^p, x], x] /; FreeQ[{a, b}, x] && IGtQ[p, 0]
Int[((a_.)*sec[(e_.) + (f_.)*(x_)])^(m_.)*((b_.)*tan[(e_.) + (f_.)*(x_)])^( n_.), x_Symbol] :> Simp[a/f Subst[Int[(a*x)^(m - 1)*(-1 + x^2)^((n - 1)/2 ), x], x, Sec[e + f*x]], x] /; FreeQ[{a, e, f, m}, x] && IntegerQ[(n - 1)/2 ] && !(IntegerQ[m/2] && LtQ[0, m, n + 1])
Int[sec[(e_.) + (f_.)*(x_)]^(m_)*((b_.)*tan[(e_.) + (f_.)*(x_)])^(n_.), x_S ymbol] :> Simp[1/f Subst[Int[(b*x)^n*(1 + x^2)^(m/2 - 1), x], x, Tan[e + f*x]], x] /; FreeQ[{b, e, f, n}, x] && IntegerQ[m/2] && !(IntegerQ[(n - 1) /2] && LtQ[0, n, m - 1])
Int[((g_.)*tan[(e_.) + (f_.)*(x_)])^(p_.)/((a_) + (b_.)*sin[(e_.) + (f_.)*( x_)]), x_Symbol] :> Simp[1/a Int[Sec[e + f*x]^2*(g*Tan[e + f*x])^p, x], x ] - Simp[1/(b*g) Int[Sec[e + f*x]*(g*Tan[e + f*x])^(p + 1), x], x] /; Fre eQ[{a, b, e, f, g, p}, x] && EqQ[a^2 - b^2, 0] && NeQ[p, -1]
Time = 0.20 (sec) , antiderivative size = 45, normalized size of antiderivative = 1.10
| method | result | size |
| default | \(\frac {\frac {\tanh \left (\frac {x}{2}\right )^{5}}{5}+\frac {4 \tanh \left (\frac {x}{2}\right )^{3}}{3}+6 \tanh \left (\frac {x}{2}\right )-\frac {4}{\tanh \left (\frac {x}{2}\right )}-\frac {1}{3 \tanh \left (\frac {x}{2}\right )^{3}}}{16 a}\) | \(45\) |
| risch | \(-\frac {2 \left (15 \,{\mathrm e}^{7 x}+15 \,{\mathrm e}^{6 x}-5 \,{\mathrm e}^{5 x}-25 \,{\mathrm e}^{4 x}+13 \,{\mathrm e}^{3 x}+21 \,{\mathrm e}^{2 x}+9 \,{\mathrm e}^{x}-3\right )}{15 \left ({\mathrm e}^{x}+1\right )^{5} a \left ({\mathrm e}^{x}-1\right )^{3}}\) | \(60\) |
Leaf count of result is larger than twice the leaf count of optimal. 224 vs. \(2 (35) = 70\).
Time = 0.25 (sec) , antiderivative size = 224, normalized size of antiderivative = 5.46 \[ \int \frac {\coth ^4(x)}{a+a \cosh (x)} \, dx=-\frac {2 \, {\left (15 \, \cosh \left (x\right )^{4} + 6 \, {\left (10 \, \cosh \left (x\right ) + 3\right )} \sinh \left (x\right )^{3} + 15 \, \sinh \left (x\right )^{4} + 12 \, \cosh \left (x\right )^{3} + 2 \, {\left (45 \, \cosh \left (x\right )^{2} + 18 \, \cosh \left (x\right ) + 2\right )} \sinh \left (x\right )^{2} + 4 \, \cosh \left (x\right )^{2} + 2 \, {\left (30 \, \cosh \left (x\right )^{3} + 27 \, \cosh \left (x\right )^{2} - 14 \, \cosh \left (x\right ) - 23\right )} \sinh \left (x\right ) - 4 \, \cosh \left (x\right ) + 13\right )}}{15 \, {\left (a \cosh \left (x\right )^{5} + a \sinh \left (x\right )^{5} + 2 \, a \cosh \left (x\right )^{4} + {\left (5 \, a \cosh \left (x\right ) + 2 \, a\right )} \sinh \left (x\right )^{4} - 3 \, a \cosh \left (x\right )^{3} + {\left (10 \, a \cosh \left (x\right )^{2} + 8 \, a \cosh \left (x\right ) - a\right )} \sinh \left (x\right )^{3} - 8 \, a \cosh \left (x\right )^{2} + {\left (10 \, a \cosh \left (x\right )^{3} + 12 \, a \cosh \left (x\right )^{2} - 9 \, a \cosh \left (x\right ) - 8 \, a\right )} \sinh \left (x\right )^{2} + 2 \, a \cosh \left (x\right ) + {\left (5 \, a \cosh \left (x\right )^{4} + 8 \, a \cosh \left (x\right )^{3} - 3 \, a \cosh \left (x\right )^{2} - 8 \, a \cosh \left (x\right ) - 2 \, a\right )} \sinh \left (x\right ) + 6 \, a\right )}} \]
-2/15*(15*cosh(x)^4 + 6*(10*cosh(x) + 3)*sinh(x)^3 + 15*sinh(x)^4 + 12*cos h(x)^3 + 2*(45*cosh(x)^2 + 18*cosh(x) + 2)*sinh(x)^2 + 4*cosh(x)^2 + 2*(30 *cosh(x)^3 + 27*cosh(x)^2 - 14*cosh(x) - 23)*sinh(x) - 4*cosh(x) + 13)/(a* cosh(x)^5 + a*sinh(x)^5 + 2*a*cosh(x)^4 + (5*a*cosh(x) + 2*a)*sinh(x)^4 - 3*a*cosh(x)^3 + (10*a*cosh(x)^2 + 8*a*cosh(x) - a)*sinh(x)^3 - 8*a*cosh(x) ^2 + (10*a*cosh(x)^3 + 12*a*cosh(x)^2 - 9*a*cosh(x) - 8*a)*sinh(x)^2 + 2*a *cosh(x) + (5*a*cosh(x)^4 + 8*a*cosh(x)^3 - 3*a*cosh(x)^2 - 8*a*cosh(x) - 2*a)*sinh(x) + 6*a)
\[ \int \frac {\coth ^4(x)}{a+a \cosh (x)} \, dx=\frac {\int \frac {\coth ^{4}{\left (x \right )}}{\cosh {\left (x \right )} + 1}\, dx}{a} \]
Leaf count of result is larger than twice the leaf count of optimal. 469 vs. \(2 (35) = 70\).
Time = 0.20 (sec) , antiderivative size = 469, normalized size of antiderivative = 11.44 \[ \int \frac {\coth ^4(x)}{a+a \cosh (x)} \, dx=-\frac {6 \, e^{\left (-x\right )}}{5 \, {\left (2 \, a e^{\left (-x\right )} - 2 \, a e^{\left (-2 \, x\right )} - 6 \, a e^{\left (-3 \, x\right )} + 6 \, a e^{\left (-5 \, x\right )} + 2 \, a e^{\left (-6 \, x\right )} - 2 \, a e^{\left (-7 \, x\right )} - a e^{\left (-8 \, x\right )} + a\right )}} - \frac {14 \, e^{\left (-2 \, x\right )}}{5 \, {\left (2 \, a e^{\left (-x\right )} - 2 \, a e^{\left (-2 \, x\right )} - 6 \, a e^{\left (-3 \, x\right )} + 6 \, a e^{\left (-5 \, x\right )} + 2 \, a e^{\left (-6 \, x\right )} - 2 \, a e^{\left (-7 \, x\right )} - a e^{\left (-8 \, x\right )} + a\right )}} - \frac {26 \, e^{\left (-3 \, x\right )}}{15 \, {\left (2 \, a e^{\left (-x\right )} - 2 \, a e^{\left (-2 \, x\right )} - 6 \, a e^{\left (-3 \, x\right )} + 6 \, a e^{\left (-5 \, x\right )} + 2 \, a e^{\left (-6 \, x\right )} - 2 \, a e^{\left (-7 \, x\right )} - a e^{\left (-8 \, x\right )} + a\right )}} + \frac {10 \, e^{\left (-4 \, x\right )}}{3 \, {\left (2 \, a e^{\left (-x\right )} - 2 \, a e^{\left (-2 \, x\right )} - 6 \, a e^{\left (-3 \, x\right )} + 6 \, a e^{\left (-5 \, x\right )} + 2 \, a e^{\left (-6 \, x\right )} - 2 \, a e^{\left (-7 \, x\right )} - a e^{\left (-8 \, x\right )} + a\right )}} + \frac {2 \, e^{\left (-5 \, x\right )}}{3 \, {\left (2 \, a e^{\left (-x\right )} - 2 \, a e^{\left (-2 \, x\right )} - 6 \, a e^{\left (-3 \, x\right )} + 6 \, a e^{\left (-5 \, x\right )} + 2 \, a e^{\left (-6 \, x\right )} - 2 \, a e^{\left (-7 \, x\right )} - a e^{\left (-8 \, x\right )} + a\right )}} - \frac {2 \, e^{\left (-6 \, x\right )}}{2 \, a e^{\left (-x\right )} - 2 \, a e^{\left (-2 \, x\right )} - 6 \, a e^{\left (-3 \, x\right )} + 6 \, a e^{\left (-5 \, x\right )} + 2 \, a e^{\left (-6 \, x\right )} - 2 \, a e^{\left (-7 \, x\right )} - a e^{\left (-8 \, x\right )} + a} - \frac {2 \, e^{\left (-7 \, x\right )}}{2 \, a e^{\left (-x\right )} - 2 \, a e^{\left (-2 \, x\right )} - 6 \, a e^{\left (-3 \, x\right )} + 6 \, a e^{\left (-5 \, x\right )} + 2 \, a e^{\left (-6 \, x\right )} - 2 \, a e^{\left (-7 \, x\right )} - a e^{\left (-8 \, x\right )} + a} + \frac {2}{5 \, {\left (2 \, a e^{\left (-x\right )} - 2 \, a e^{\left (-2 \, x\right )} - 6 \, a e^{\left (-3 \, x\right )} + 6 \, a e^{\left (-5 \, x\right )} + 2 \, a e^{\left (-6 \, x\right )} - 2 \, a e^{\left (-7 \, x\right )} - a e^{\left (-8 \, x\right )} + a\right )}} \]
-6/5*e^(-x)/(2*a*e^(-x) - 2*a*e^(-2*x) - 6*a*e^(-3*x) + 6*a*e^(-5*x) + 2*a *e^(-6*x) - 2*a*e^(-7*x) - a*e^(-8*x) + a) - 14/5*e^(-2*x)/(2*a*e^(-x) - 2 *a*e^(-2*x) - 6*a*e^(-3*x) + 6*a*e^(-5*x) + 2*a*e^(-6*x) - 2*a*e^(-7*x) - a*e^(-8*x) + a) - 26/15*e^(-3*x)/(2*a*e^(-x) - 2*a*e^(-2*x) - 6*a*e^(-3*x) + 6*a*e^(-5*x) + 2*a*e^(-6*x) - 2*a*e^(-7*x) - a*e^(-8*x) + a) + 10/3*e^( -4*x)/(2*a*e^(-x) - 2*a*e^(-2*x) - 6*a*e^(-3*x) + 6*a*e^(-5*x) + 2*a*e^(-6 *x) - 2*a*e^(-7*x) - a*e^(-8*x) + a) + 2/3*e^(-5*x)/(2*a*e^(-x) - 2*a*e^(- 2*x) - 6*a*e^(-3*x) + 6*a*e^(-5*x) + 2*a*e^(-6*x) - 2*a*e^(-7*x) - a*e^(-8 *x) + a) - 2*e^(-6*x)/(2*a*e^(-x) - 2*a*e^(-2*x) - 6*a*e^(-3*x) + 6*a*e^(- 5*x) + 2*a*e^(-6*x) - 2*a*e^(-7*x) - a*e^(-8*x) + a) - 2*e^(-7*x)/(2*a*e^( -x) - 2*a*e^(-2*x) - 6*a*e^(-3*x) + 6*a*e^(-5*x) + 2*a*e^(-6*x) - 2*a*e^(- 7*x) - a*e^(-8*x) + a) + 2/5/(2*a*e^(-x) - 2*a*e^(-2*x) - 6*a*e^(-3*x) + 6 *a*e^(-5*x) + 2*a*e^(-6*x) - 2*a*e^(-7*x) - a*e^(-8*x) + a)
Time = 0.25 (sec) , antiderivative size = 59, normalized size of antiderivative = 1.44 \[ \int \frac {\coth ^4(x)}{a+a \cosh (x)} \, dx=-\frac {15 \, e^{\left (2 \, x\right )} - 24 \, e^{x} + 13}{24 \, a {\left (e^{x} - 1\right )}^{3}} - \frac {165 \, e^{\left (4 \, x\right )} + 480 \, e^{\left (3 \, x\right )} + 650 \, e^{\left (2 \, x\right )} + 400 \, e^{x} + 113}{120 \, a {\left (e^{x} + 1\right )}^{5}} \]
-1/24*(15*e^(2*x) - 24*e^x + 13)/(a*(e^x - 1)^3) - 1/120*(165*e^(4*x) + 48 0*e^(3*x) + 650*e^(2*x) + 400*e^x + 113)/(a*(e^x + 1)^5)
Time = 1.84 (sec) , antiderivative size = 263, normalized size of antiderivative = 6.41 \[ \int \frac {\coth ^4(x)}{a+a \cosh (x)} \, dx=\frac {1}{6\,a\,\left (3\,{\mathrm {e}}^{2\,x}-{\mathrm {e}}^{3\,x}-3\,{\mathrm {e}}^x+1\right )}-\frac {\frac {3\,{\mathrm {e}}^{2\,x}}{8\,a}+\frac {11\,{\mathrm {e}}^{3\,x}}{40\,a}+\frac {1}{8\,a}+\frac {17\,{\mathrm {e}}^x}{40\,a}}{6\,{\mathrm {e}}^{2\,x}+4\,{\mathrm {e}}^{3\,x}+{\mathrm {e}}^{4\,x}+4\,{\mathrm {e}}^x+1}-\frac {\frac {11\,{\mathrm {e}}^{2\,x}}{40\,a}+\frac {17}{120\,a}+\frac {{\mathrm {e}}^x}{4\,a}}{3\,{\mathrm {e}}^{2\,x}+{\mathrm {e}}^{3\,x}+3\,{\mathrm {e}}^x+1}-\frac {\frac {1}{8\,a}+\frac {11\,{\mathrm {e}}^x}{40\,a}}{{\mathrm {e}}^{2\,x}+2\,{\mathrm {e}}^x+1}-\frac {\frac {17\,{\mathrm {e}}^{2\,x}}{20\,a}+\frac {{\mathrm {e}}^{3\,x}}{2\,a}+\frac {11\,{\mathrm {e}}^{4\,x}}{40\,a}+\frac {11}{40\,a}+\frac {{\mathrm {e}}^x}{2\,a}}{10\,{\mathrm {e}}^{2\,x}+10\,{\mathrm {e}}^{3\,x}+5\,{\mathrm {e}}^{4\,x}+{\mathrm {e}}^{5\,x}+5\,{\mathrm {e}}^x+1}-\frac {1}{4\,a\,\left ({\mathrm {e}}^{2\,x}-2\,{\mathrm {e}}^x+1\right )}-\frac {5}{8\,a\,\left ({\mathrm {e}}^x-1\right )}-\frac {11}{40\,a\,\left ({\mathrm {e}}^x+1\right )} \]
1/(6*a*(3*exp(2*x) - exp(3*x) - 3*exp(x) + 1)) - ((3*exp(2*x))/(8*a) + (11 *exp(3*x))/(40*a) + 1/(8*a) + (17*exp(x))/(40*a))/(6*exp(2*x) + 4*exp(3*x) + exp(4*x) + 4*exp(x) + 1) - ((11*exp(2*x))/(40*a) + 17/(120*a) + exp(x)/ (4*a))/(3*exp(2*x) + exp(3*x) + 3*exp(x) + 1) - (1/(8*a) + (11*exp(x))/(40 *a))/(exp(2*x) + 2*exp(x) + 1) - ((17*exp(2*x))/(20*a) + exp(3*x)/(2*a) + (11*exp(4*x))/(40*a) + 11/(40*a) + exp(x)/(2*a))/(10*exp(2*x) + 10*exp(3*x ) + 5*exp(4*x) + exp(5*x) + 5*exp(x) + 1) - 1/(4*a*(exp(2*x) - 2*exp(x) + 1)) - 5/(8*a*(exp(x) - 1)) - 11/(40*a*(exp(x) + 1))