Integrand size = 11, antiderivative size = 44 \[ \int \frac {\text {csch}^7(x)}{1+\tanh (x)} \, dx=-\frac {1}{16} \text {arctanh}(\cosh (x))+\frac {1}{16} \coth (x) \text {csch}(x)-\frac {1}{24} \coth (x) \text {csch}^3(x)+\frac {\text {csch}^5(x)}{5}-\frac {1}{6} \coth (x) \text {csch}^5(x) \]
-1/16*arctanh(cosh(x))+1/16*coth(x)*csch(x)-1/24*coth(x)*csch(x)^3+1/5*csc h(x)^5-1/6*coth(x)*csch(x)^5
Leaf count is larger than twice the leaf count of optimal. \(124\) vs. \(2(44)=88\).
Time = 0.38 (sec) , antiderivative size = 124, normalized size of antiderivative = 2.82 \[ \int \frac {\text {csch}^7(x)}{1+\tanh (x)} \, dx=\frac {72 \coth \left (\frac {x}{2}\right )+30 \text {csch}^2\left (\frac {x}{2}\right )-120 \log \left (\cosh \left (\frac {x}{2}\right )\right )+120 \log \left (\sinh \left (\frac {x}{2}\right )\right )+30 \text {sech}^2\left (\frac {x}{2}\right )-5 \text {sech}^6\left (\frac {x}{2}\right )-288 \text {csch}^3(x) \sinh ^4\left (\frac {x}{2}\right )-384 \text {csch}^5(x) \sinh ^6\left (\frac {x}{2}\right )-18 \text {csch}^4\left (\frac {x}{2}\right ) \sinh (x)+\text {csch}^6\left (\frac {x}{2}\right ) (-5+6 \sinh (x))-72 \tanh \left (\frac {x}{2}\right )}{1920} \]
(72*Coth[x/2] + 30*Csch[x/2]^2 - 120*Log[Cosh[x/2]] + 120*Log[Sinh[x/2]] + 30*Sech[x/2]^2 - 5*Sech[x/2]^6 - 288*Csch[x]^3*Sinh[x/2]^4 - 384*Csch[x]^ 5*Sinh[x/2]^6 - 18*Csch[x/2]^4*Sinh[x] + Csch[x/2]^6*(-5 + 6*Sinh[x]) - 72 *Tanh[x/2])/1920
Result contains complex when optimal does not.
Time = 0.47 (sec) , antiderivative size = 58, normalized size of antiderivative = 1.32, number of steps used = 11, number of rules used = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 1.000, Rules used = {3042, 26, 4001, 26, 3042, 26, 3587, 3042, 26, 3586, 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.
\(\displaystyle \int \frac {\text {csch}^7(x)}{\tanh (x)+1} \, dx\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \int -\frac {i}{\sin (i x)^7 (1-i \tan (i x))}dx\) |
\(\Big \downarrow \) 26 |
\(\displaystyle -i \int \frac {1}{\sin (i x)^7 (1-i \tan (i x))}dx\) |
\(\Big \downarrow \) 4001 |
\(\displaystyle -i \int \frac {i \coth (x) \text {csch}^6(x)}{\cosh (x)+\sinh (x)}dx\) |
\(\Big \downarrow \) 26 |
\(\displaystyle \int \frac {\coth (x) \text {csch}^6(x)}{\sinh (x)+\cosh (x)}dx\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \int -\frac {i \cos (i x)}{\sin (i x)^7 (\cos (i x)-i \sin (i x))}dx\) |
\(\Big \downarrow \) 26 |
\(\displaystyle -i \int \frac {\cos (i x)}{(\cos (i x)-i \sin (i x)) \sin (i x)^7}dx\) |
\(\Big \downarrow \) 3587 |
\(\displaystyle \int \coth (x) \text {csch}^6(x) (\cosh (x)-\sinh (x))dx\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \int -\frac {i \cos (i x) (i \sin (i x)+\cos (i x))}{\sin (i x)^7}dx\) |
\(\Big \downarrow \) 26 |
\(\displaystyle -i \int \frac {\cos (i x) (\cos (i x)+i \sin (i x))}{\sin (i x)^7}dx\) |
\(\Big \downarrow \) 3586 |
\(\displaystyle -i \int \left (i \coth ^2(x) \text {csch}^5(x)-i \coth (x) \text {csch}^5(x)\right )dx\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle -i \left (-\frac {1}{16} i \text {arctanh}(\cosh (x))+\frac {1}{5} i \text {csch}^5(x)-\frac {1}{6} i \coth (x) \text {csch}^5(x)-\frac {1}{24} i \coth (x) \text {csch}^3(x)+\frac {1}{16} i \coth (x) \text {csch}(x)\right )\) |
(-I)*((-1/16*I)*ArcTanh[Cosh[x]] + (I/16)*Coth[x]*Csch[x] - (I/24)*Coth[x] *Csch[x]^3 + (I/5)*Csch[x]^5 - (I/6)*Coth[x]*Csch[x]^5)
3.1.79.3.1 Defintions of rubi rules used
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[cos[(c_.) + (d_.)*(x_)]^(m_.)*sin[(c_.) + (d_.)*(x_)]^(n_.)*(cos[(c_.) + (d_.)*(x_)]*(a_.) + (b_.)*sin[(c_.) + (d_.)*(x_)])^(p_.), x_Symbol] :> In t[ExpandTrig[cos[c + d*x]^m*sin[c + d*x]^n*(a*cos[c + d*x] + b*sin[c + d*x] )^p, x], x] /; FreeQ[{a, b, c, d, m, n}, x] && IGtQ[p, 0]
Int[cos[(c_.) + (d_.)*(x_)]^(m_.)*sin[(c_.) + (d_.)*(x_)]^(n_.)*(cos[(c_.) + (d_.)*(x_)]*(a_.) + (b_.)*sin[(c_.) + (d_.)*(x_)])^(p_), x_Symbol] :> Sim p[a^p*b^p Int[(Cos[c + d*x]^m*Sin[c + d*x]^n)/(b*Cos[c + d*x] + a*Sin[c + d*x])^p, x], x] /; FreeQ[{a, b, c, d, m, n}, x] && EqQ[a^2 + b^2, 0] && IL tQ[p, 0]
Int[sin[(e_.) + (f_.)*(x_)]^(m_.)*((a_) + (b_.)*tan[(e_.) + (f_.)*(x_)])^(n _.), x_Symbol] :> Int[Sin[e + f*x]^m*((a*Cos[e + f*x] + b*Sin[e + f*x])^n/C os[e + f*x]^n), x] /; FreeQ[{a, b, e, f}, x] && IntegerQ[(m - 1)/2] && ILtQ [n, 0] && ((LtQ[m, 5] && GtQ[n, -4]) || (EqQ[m, 5] && EqQ[n, -1]))
Time = 4.12 (sec) , antiderivative size = 60, normalized size of antiderivative = 1.36
method | result | size |
risch | \(\frac {{\mathrm e}^{x} \left (15 \,{\mathrm e}^{10 x}-85 \,{\mathrm e}^{8 x}+198 \,{\mathrm e}^{6 x}-1338 \,{\mathrm e}^{4 x}-85 \,{\mathrm e}^{2 x}+15\right )}{120 \left ({\mathrm e}^{2 x}-1\right )^{6}}+\frac {\ln \left ({\mathrm e}^{x}-1\right )}{16}-\frac {\ln \left ({\mathrm e}^{x}+1\right )}{16}\) | \(60\) |
default | \(\frac {\tanh \left (\frac {x}{2}\right )^{6}}{384}-\frac {\tanh \left (\frac {x}{2}\right )^{5}}{160}-\frac {\tanh \left (\frac {x}{2}\right )^{4}}{128}+\frac {\tanh \left (\frac {x}{2}\right )^{3}}{32}-\frac {\tanh \left (\frac {x}{2}\right )^{2}}{128}-\frac {\tanh \left (\frac {x}{2}\right )}{16}+\frac {\ln \left (\tanh \left (\frac {x}{2}\right )\right )}{16}+\frac {1}{128 \tanh \left (\frac {x}{2}\right )^{4}}+\frac {1}{160 \tanh \left (\frac {x}{2}\right )^{5}}+\frac {1}{128 \tanh \left (\frac {x}{2}\right )^{2}}-\frac {1}{32 \tanh \left (\frac {x}{2}\right )^{3}}-\frac {1}{384 \tanh \left (\frac {x}{2}\right )^{6}}+\frac {1}{16 \tanh \left (\frac {x}{2}\right )}\) | \(103\) |
1/120*exp(x)*(15*exp(10*x)-85*exp(8*x)+198*exp(6*x)-1338*exp(4*x)-85*exp(2 *x)+15)/(exp(2*x)-1)^6+1/16*ln(exp(x)-1)-1/16*ln(exp(x)+1)
Leaf count of result is larger than twice the leaf count of optimal. 1260 vs. \(2 (34) = 68\).
Time = 0.26 (sec) , antiderivative size = 1260, normalized size of antiderivative = 28.64 \[ \int \frac {\text {csch}^7(x)}{1+\tanh (x)} \, dx=\text {Too large to display} \]
1/240*(30*cosh(x)^11 + 330*cosh(x)*sinh(x)^10 + 30*sinh(x)^11 + 10*(165*co sh(x)^2 - 17)*sinh(x)^9 - 170*cosh(x)^9 + 90*(55*cosh(x)^3 - 17*cosh(x))*s inh(x)^8 + 36*(275*cosh(x)^4 - 170*cosh(x)^2 + 11)*sinh(x)^7 + 396*cosh(x) ^7 + 84*(165*cosh(x)^5 - 170*cosh(x)^3 + 33*cosh(x))*sinh(x)^6 + 12*(1155* cosh(x)^6 - 1785*cosh(x)^4 + 693*cosh(x)^2 - 223)*sinh(x)^5 - 2676*cosh(x) ^5 + 60*(165*cosh(x)^7 - 357*cosh(x)^5 + 231*cosh(x)^3 - 223*cosh(x))*sinh (x)^4 + 10*(495*cosh(x)^8 - 1428*cosh(x)^6 + 1386*cosh(x)^4 - 2676*cosh(x) ^2 - 17)*sinh(x)^3 - 170*cosh(x)^3 + 6*(275*cosh(x)^9 - 1020*cosh(x)^7 + 1 386*cosh(x)^5 - 4460*cosh(x)^3 - 85*cosh(x))*sinh(x)^2 - 15*(cosh(x)^12 + 12*cosh(x)*sinh(x)^11 + sinh(x)^12 + 6*(11*cosh(x)^2 - 1)*sinh(x)^10 - 6*c osh(x)^10 + 20*(11*cosh(x)^3 - 3*cosh(x))*sinh(x)^9 + 15*(33*cosh(x)^4 - 1 8*cosh(x)^2 + 1)*sinh(x)^8 + 15*cosh(x)^8 + 24*(33*cosh(x)^5 - 30*cosh(x)^ 3 + 5*cosh(x))*sinh(x)^7 + 4*(231*cosh(x)^6 - 315*cosh(x)^4 + 105*cosh(x)^ 2 - 5)*sinh(x)^6 - 20*cosh(x)^6 + 24*(33*cosh(x)^7 - 63*cosh(x)^5 + 35*cos h(x)^3 - 5*cosh(x))*sinh(x)^5 + 15*(33*cosh(x)^8 - 84*cosh(x)^6 + 70*cosh( x)^4 - 20*cosh(x)^2 + 1)*sinh(x)^4 + 15*cosh(x)^4 + 20*(11*cosh(x)^9 - 36* cosh(x)^7 + 42*cosh(x)^5 - 20*cosh(x)^3 + 3*cosh(x))*sinh(x)^3 + 6*(11*cos h(x)^10 - 45*cosh(x)^8 + 70*cosh(x)^6 - 50*cosh(x)^4 + 15*cosh(x)^2 - 1)*s inh(x)^2 - 6*cosh(x)^2 + 12*(cosh(x)^11 - 5*cosh(x)^9 + 10*cosh(x)^7 - 10* cosh(x)^5 + 5*cosh(x)^3 - cosh(x))*sinh(x) + 1)*log(cosh(x) + sinh(x) +...
\[ \int \frac {\text {csch}^7(x)}{1+\tanh (x)} \, dx=\int \frac {\operatorname {csch}^{7}{\left (x \right )}}{\tanh {\left (x \right )} + 1}\, dx \]
Leaf count of result is larger than twice the leaf count of optimal. 98 vs. \(2 (34) = 68\).
Time = 0.19 (sec) , antiderivative size = 98, normalized size of antiderivative = 2.23 \[ \int \frac {\text {csch}^7(x)}{1+\tanh (x)} \, dx=-\frac {15 \, e^{\left (-x\right )} - 85 \, e^{\left (-3 \, x\right )} + 198 \, e^{\left (-5 \, x\right )} - 1338 \, e^{\left (-7 \, x\right )} - 85 \, e^{\left (-9 \, x\right )} + 15 \, e^{\left (-11 \, x\right )}}{120 \, {\left (6 \, e^{\left (-2 \, x\right )} - 15 \, e^{\left (-4 \, x\right )} + 20 \, e^{\left (-6 \, x\right )} - 15 \, e^{\left (-8 \, x\right )} + 6 \, e^{\left (-10 \, x\right )} - e^{\left (-12 \, x\right )} - 1\right )}} - \frac {1}{16} \, \log \left (e^{\left (-x\right )} + 1\right ) + \frac {1}{16} \, \log \left (e^{\left (-x\right )} - 1\right ) \]
-1/120*(15*e^(-x) - 85*e^(-3*x) + 198*e^(-5*x) - 1338*e^(-7*x) - 85*e^(-9* x) + 15*e^(-11*x))/(6*e^(-2*x) - 15*e^(-4*x) + 20*e^(-6*x) - 15*e^(-8*x) + 6*e^(-10*x) - e^(-12*x) - 1) - 1/16*log(e^(-x) + 1) + 1/16*log(e^(-x) - 1 )
Time = 0.26 (sec) , antiderivative size = 61, normalized size of antiderivative = 1.39 \[ \int \frac {\text {csch}^7(x)}{1+\tanh (x)} \, dx=\frac {15 \, e^{\left (11 \, x\right )} - 85 \, e^{\left (9 \, x\right )} + 198 \, e^{\left (7 \, x\right )} - 1338 \, e^{\left (5 \, x\right )} - 85 \, e^{\left (3 \, x\right )} + 15 \, e^{x}}{120 \, {\left (e^{\left (2 \, x\right )} - 1\right )}^{6}} - \frac {1}{16} \, \log \left (e^{x} + 1\right ) + \frac {1}{16} \, \log \left ({\left | e^{x} - 1 \right |}\right ) \]
1/120*(15*e^(11*x) - 85*e^(9*x) + 198*e^(7*x) - 1338*e^(5*x) - 85*e^(3*x) + 15*e^x)/(e^(2*x) - 1)^6 - 1/16*log(e^x + 1) + 1/16*log(abs(e^x - 1))
Time = 0.09 (sec) , antiderivative size = 207, normalized size of antiderivative = 4.70 \[ \int \frac {\text {csch}^7(x)}{1+\tanh (x)} \, dx=\frac {\ln \left (\frac {1}{8}-\frac {{\mathrm {e}}^x}{8}\right )}{16}-\frac {\ln \left (-\frac {{\mathrm {e}}^x}{8}-\frac {1}{8}\right )}{16}-\frac {\frac {16\,{\mathrm {e}}^{3\,x}}{3}+\frac {16\,{\mathrm {e}}^{5\,x}}{3}}{15\,{\mathrm {e}}^{4\,x}-6\,{\mathrm {e}}^{2\,x}-20\,{\mathrm {e}}^{6\,x}+15\,{\mathrm {e}}^{8\,x}-6\,{\mathrm {e}}^{10\,x}+{\mathrm {e}}^{12\,x}+1}-\frac {\frac {8\,{\mathrm {e}}^{3\,x}}{3}+\frac {8\,{\mathrm {e}}^x}{5}}{5\,{\mathrm {e}}^{2\,x}-10\,{\mathrm {e}}^{4\,x}+10\,{\mathrm {e}}^{6\,x}-5\,{\mathrm {e}}^{8\,x}+{\mathrm {e}}^{10\,x}-1}-\frac {6\,{\mathrm {e}}^x}{5\,\left (6\,{\mathrm {e}}^{4\,x}-4\,{\mathrm {e}}^{2\,x}-4\,{\mathrm {e}}^{6\,x}+{\mathrm {e}}^{8\,x}+1\right )}+\frac {{\mathrm {e}}^x}{8\,\left ({\mathrm {e}}^{2\,x}-1\right )}+\frac {{\mathrm {e}}^x}{15\,\left (3\,{\mathrm {e}}^{2\,x}-3\,{\mathrm {e}}^{4\,x}+{\mathrm {e}}^{6\,x}-1\right )}-\frac {{\mathrm {e}}^x}{12\,\left ({\mathrm {e}}^{4\,x}-2\,{\mathrm {e}}^{2\,x}+1\right )} \]
log(1/8 - exp(x)/8)/16 - log(- exp(x)/8 - 1/8)/16 - ((16*exp(3*x))/3 + (16 *exp(5*x))/3)/(15*exp(4*x) - 6*exp(2*x) - 20*exp(6*x) + 15*exp(8*x) - 6*ex p(10*x) + exp(12*x) + 1) - ((8*exp(3*x))/3 + (8*exp(x))/5)/(5*exp(2*x) - 1 0*exp(4*x) + 10*exp(6*x) - 5*exp(8*x) + exp(10*x) - 1) - (6*exp(x))/(5*(6* exp(4*x) - 4*exp(2*x) - 4*exp(6*x) + exp(8*x) + 1)) + exp(x)/(8*(exp(2*x) - 1)) + exp(x)/(15*(3*exp(2*x) - 3*exp(4*x) + exp(6*x) - 1)) - exp(x)/(12* (exp(4*x) - 2*exp(2*x) + 1))