Integrand size = 10, antiderivative size = 205 \[ \int \frac {1}{1-\cosh ^5(x)} \, dx=-\frac {2 \arctan \left (\frac {\tanh \left (\frac {x}{2}\right )}{\sqrt {-\frac {1-(-1)^{2/5}}{1+(-1)^{2/5}}}}\right )}{5 \sqrt {-1+(-1)^{4/5}}}+\frac {2 \arctan \left (\sqrt {-\frac {1+(-1)^{4/5}}{1-(-1)^{4/5}}} \tanh \left (\frac {x}{2}\right )\right )}{5 \sqrt {-1-(-1)^{3/5}}}+\frac {2 \text {arctanh}\left (\sqrt {\frac {1-\sqrt [5]{-1}}{1+\sqrt [5]{-1}}} \tanh \left (\frac {x}{2}\right )\right )}{5 \sqrt {1-(-1)^{2/5}}}+\frac {2 \text {arctanh}\left (\sqrt {\frac {1-(-1)^{3/5}}{1+(-1)^{3/5}}} \tanh \left (\frac {x}{2}\right )\right )}{5 \sqrt {1+\sqrt [5]{-1}}}-\frac {\sinh (x)}{5 (1-\cosh (x))} \]
-1/5*sinh(x)/(1-cosh(x))+2/5*arctanh(((1-(-1)^(3/5))/(1+(-1)^(3/5)))^(1/2) *tanh(1/2*x))/(1+(-1)^(1/5))^(1/2)+2/5*arctanh(((1-(-1)^(1/5))/(1+(-1)^(1/ 5)))^(1/2)*tanh(1/2*x))/(1-(-1)^(2/5))^(1/2)+2/5*arctan(((-1-(-1)^(4/5))/( 1-(-1)^(4/5)))^(1/2)*tanh(1/2*x))/(-1-(-1)^(3/5))^(1/2)-2/5*arctan(tanh(1/ 2*x)/((-1+(-1)^(2/5))/(1+(-1)^(2/5)))^(1/2))/(-1+(-1)^(4/5))^(1/2)
Result contains higher order function than in optimal. Order 9 vs. order 3 in optimal.
Time = 5.03 (sec) , antiderivative size = 445, normalized size of antiderivative = 2.17 \[ \int \frac {1}{1-\cosh ^5(x)} \, dx=\frac {1}{5} \coth \left (\frac {x}{2}\right )+\frac {1}{10} \text {RootSum}\left [1+2 \text {$\#$1}+8 \text {$\#$1}^2+14 \text {$\#$1}^3+30 \text {$\#$1}^4+14 \text {$\#$1}^5+8 \text {$\#$1}^6+2 \text {$\#$1}^7+\text {$\#$1}^8\&,\frac {x+2 \log \left (-\cosh \left (\frac {x}{2}\right )-\sinh \left (\frac {x}{2}\right )+\cosh \left (\frac {x}{2}\right ) \text {$\#$1}-\sinh \left (\frac {x}{2}\right ) \text {$\#$1}\right )+4 x \text {$\#$1}+8 \log \left (-\cosh \left (\frac {x}{2}\right )-\sinh \left (\frac {x}{2}\right )+\cosh \left (\frac {x}{2}\right ) \text {$\#$1}-\sinh \left (\frac {x}{2}\right ) \text {$\#$1}\right ) \text {$\#$1}+15 x \text {$\#$1}^2+30 \log \left (-\cosh \left (\frac {x}{2}\right )-\sinh \left (\frac {x}{2}\right )+\cosh \left (\frac {x}{2}\right ) \text {$\#$1}-\sinh \left (\frac {x}{2}\right ) \text {$\#$1}\right ) \text {$\#$1}^2+40 x \text {$\#$1}^3+80 \log \left (-\cosh \left (\frac {x}{2}\right )-\sinh \left (\frac {x}{2}\right )+\cosh \left (\frac {x}{2}\right ) \text {$\#$1}-\sinh \left (\frac {x}{2}\right ) \text {$\#$1}\right ) \text {$\#$1}^3+15 x \text {$\#$1}^4+30 \log \left (-\cosh \left (\frac {x}{2}\right )-\sinh \left (\frac {x}{2}\right )+\cosh \left (\frac {x}{2}\right ) \text {$\#$1}-\sinh \left (\frac {x}{2}\right ) \text {$\#$1}\right ) \text {$\#$1}^4+4 x \text {$\#$1}^5+8 \log \left (-\cosh \left (\frac {x}{2}\right )-\sinh \left (\frac {x}{2}\right )+\cosh \left (\frac {x}{2}\right ) \text {$\#$1}-\sinh \left (\frac {x}{2}\right ) \text {$\#$1}\right ) \text {$\#$1}^5+x \text {$\#$1}^6+2 \log \left (-\cosh \left (\frac {x}{2}\right )-\sinh \left (\frac {x}{2}\right )+\cosh \left (\frac {x}{2}\right ) \text {$\#$1}-\sinh \left (\frac {x}{2}\right ) \text {$\#$1}\right ) \text {$\#$1}^6}{1+8 \text {$\#$1}+21 \text {$\#$1}^2+60 \text {$\#$1}^3+35 \text {$\#$1}^4+24 \text {$\#$1}^5+7 \text {$\#$1}^6+4 \text {$\#$1}^7}\&\right ] \]
Coth[x/2]/5 + RootSum[1 + 2*#1 + 8*#1^2 + 14*#1^3 + 30*#1^4 + 14*#1^5 + 8* #1^6 + 2*#1^7 + #1^8 & , (x + 2*Log[-Cosh[x/2] - Sinh[x/2] + Cosh[x/2]*#1 - Sinh[x/2]*#1] + 4*x*#1 + 8*Log[-Cosh[x/2] - Sinh[x/2] + Cosh[x/2]*#1 - S inh[x/2]*#1]*#1 + 15*x*#1^2 + 30*Log[-Cosh[x/2] - Sinh[x/2] + Cosh[x/2]*#1 - Sinh[x/2]*#1]*#1^2 + 40*x*#1^3 + 80*Log[-Cosh[x/2] - Sinh[x/2] + Cosh[x /2]*#1 - Sinh[x/2]*#1]*#1^3 + 15*x*#1^4 + 30*Log[-Cosh[x/2] - Sinh[x/2] + Cosh[x/2]*#1 - Sinh[x/2]*#1]*#1^4 + 4*x*#1^5 + 8*Log[-Cosh[x/2] - Sinh[x/2 ] + Cosh[x/2]*#1 - Sinh[x/2]*#1]*#1^5 + x*#1^6 + 2*Log[-Cosh[x/2] - Sinh[x /2] + Cosh[x/2]*#1 - Sinh[x/2]*#1]*#1^6)/(1 + 8*#1 + 21*#1^2 + 60*#1^3 + 3 5*#1^4 + 24*#1^5 + 7*#1^6 + 4*#1^7) & ]/10
Time = 0.67 (sec) , antiderivative size = 205, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.300, Rules used = {3042, 3692, 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.
\(\displaystyle \int \frac {1}{1-\cosh ^5(x)} \, dx\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \int \frac {1}{1-\sin \left (\frac {\pi }{2}+i x\right )^5}dx\) |
\(\Big \downarrow \) 3692 |
\(\displaystyle \int \left (\frac {1}{5 \left (\sqrt [5]{-1} \cosh (x)+1\right )}+\frac {1}{5 \left (1-(-1)^{2/5} \cosh (x)\right )}+\frac {1}{5 \left ((-1)^{3/5} \cosh (x)+1\right )}+\frac {1}{5 \left (1-(-1)^{4/5} \cosh (x)\right )}+\frac {1}{5 (1-\cosh (x))}\right )dx\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle -\frac {2 \arctan \left (\frac {\tanh \left (\frac {x}{2}\right )}{\sqrt {-\frac {1-(-1)^{2/5}}{1+(-1)^{2/5}}}}\right )}{5 \sqrt {(-1)^{4/5}-1}}+\frac {2 \arctan \left (\sqrt {-\frac {1+(-1)^{4/5}}{1-(-1)^{4/5}}} \tanh \left (\frac {x}{2}\right )\right )}{5 \sqrt {-1-(-1)^{3/5}}}+\frac {2 \text {arctanh}\left (\sqrt {\frac {1-\sqrt [5]{-1}}{1+\sqrt [5]{-1}}} \tanh \left (\frac {x}{2}\right )\right )}{5 \sqrt {1-(-1)^{2/5}}}+\frac {2 \text {arctanh}\left (\sqrt {\frac {1-(-1)^{3/5}}{1+(-1)^{3/5}}} \tanh \left (\frac {x}{2}\right )\right )}{5 \sqrt {1+\sqrt [5]{-1}}}-\frac {\sinh (x)}{5 (1-\cosh (x))}\) |
(-2*ArcTan[Tanh[x/2]/Sqrt[-((1 - (-1)^(2/5))/(1 + (-1)^(2/5)))]])/(5*Sqrt[ -1 + (-1)^(4/5)]) + (2*ArcTan[Sqrt[-((1 + (-1)^(4/5))/(1 - (-1)^(4/5)))]*T anh[x/2]])/(5*Sqrt[-1 - (-1)^(3/5)]) + (2*ArcTanh[Sqrt[(1 - (-1)^(1/5))/(1 + (-1)^(1/5))]*Tanh[x/2]])/(5*Sqrt[1 - (-1)^(2/5)]) + (2*ArcTanh[Sqrt[(1 - (-1)^(3/5))/(1 + (-1)^(3/5))]*Tanh[x/2]])/(5*Sqrt[1 + (-1)^(1/5)]) - Sin h[x]/(5*(1 - Cosh[x]))
3.1.73.3.1 Defintions of rubi rules used
Int[((a_) + (b_.)*((c_.)*sin[(e_.) + (f_.)*(x_)])^(n_))^(p_), x_Symbol] :> Int[ExpandTrig[(a + b*(c*sin[e + f*x])^n)^p, x], x] /; FreeQ[{a, b, c, e, f , n}, x] && (IGtQ[p, 0] || (EqQ[p, -1] && IntegerQ[n]))
Result contains higher order function than in optimal. Order 9 vs. order 3.
Time = 0.13 (sec) , antiderivative size = 64, normalized size of antiderivative = 0.31
method | result | size |
default | \(\frac {\left (\munderset {\textit {\_R} =\operatorname {RootOf}\left (\textit {\_Z}^{8}+10 \textit {\_Z}^{4}+5\right )}{\sum }\frac {\left (-\textit {\_R}^{6}+5 \textit {\_R}^{4}-5 \textit {\_R}^{2}+5\right ) \ln \left (\tanh \left (\frac {x}{2}\right )-\textit {\_R} \right )}{\textit {\_R}^{7}+5 \textit {\_R}^{3}}\right )}{10}+\frac {1}{5 \tanh \left (\frac {x}{2}\right )}\) | \(64\) |
risch | \(\frac {2}{5 \left ({\mathrm e}^{x}-1\right )}+\left (\munderset {\textit {\_R} =\operatorname {RootOf}\left (1953125 \textit {\_Z}^{8}-156250 \textit {\_Z}^{6}+6250 \textit {\_Z}^{4}-125 \textit {\_Z}^{2}+1\right )}{\sum }\textit {\_R} \ln \left (-2343750 \textit {\_R}^{7}+234375 \textit {\_R}^{6}+140625 \textit {\_R}^{5}-15625 \textit {\_R}^{4}-4375 \textit {\_R}^{3}+500 \textit {\_R}^{2}+50 \textit {\_R} +{\mathrm e}^{x}-6\right )\right )\) | \(76\) |
1/10*sum((-_R^6+5*_R^4-5*_R^2+5)/(_R^7+5*_R^3)*ln(tanh(1/2*x)-_R),_R=RootO f(_Z^8+10*_Z^4+5))+1/5/tanh(1/2*x)
Leaf count of result is larger than twice the leaf count of optimal. 852 vs. \(2 (137) = 274\).
Time = 0.30 (sec) , antiderivative size = 852, normalized size of antiderivative = 4.16 \[ \int \frac {1}{1-\cosh ^5(x)} \, dx=\text {Too large to display} \]
-1/50*((sqrt(5)*cosh(x) + sqrt(5)*sinh(x) - sqrt(5))*sqrt(2*sqrt(5)*sqrt(2 *sqrt(5) - 5) + 10)*log(sqrt(2*sqrt(5)*sqrt(2*sqrt(5) - 5) + 10)*(3*sqrt(5 ) + 5)*sqrt(2*sqrt(5) - 5) + 5*sqrt(2*sqrt(5) - 5)*(sqrt(5) + 3) - 5*sqrt( 5) + 20*cosh(x) + 20*sinh(x) + 5) - (sqrt(5)*cosh(x) + sqrt(5)*sinh(x) - s qrt(5))*sqrt(2*sqrt(5)*sqrt(2*sqrt(5) - 5) + 10)*log(-sqrt(2*sqrt(5)*sqrt( 2*sqrt(5) - 5) + 10)*(3*sqrt(5) + 5)*sqrt(2*sqrt(5) - 5) + 5*sqrt(2*sqrt(5 ) - 5)*(sqrt(5) + 3) - 5*sqrt(5) + 20*cosh(x) + 20*sinh(x) + 5) - (sqrt(5) *cosh(x) + sqrt(5)*sinh(x) - sqrt(5))*sqrt(-2*sqrt(5)*sqrt(2*sqrt(5) - 5) + 10)*log(sqrt(-2*sqrt(5)*sqrt(2*sqrt(5) - 5) + 10)*(3*sqrt(5) + 5)*sqrt(2 *sqrt(5) - 5) - 5*sqrt(2*sqrt(5) - 5)*(sqrt(5) + 3) - 5*sqrt(5) + 20*cosh( x) + 20*sinh(x) + 5) + (sqrt(5)*cosh(x) + sqrt(5)*sinh(x) - sqrt(5))*sqrt( -2*sqrt(5)*sqrt(2*sqrt(5) - 5) + 10)*log(-sqrt(-2*sqrt(5)*sqrt(2*sqrt(5) - 5) + 10)*(3*sqrt(5) + 5)*sqrt(2*sqrt(5) - 5) - 5*sqrt(2*sqrt(5) - 5)*(sqr t(5) + 3) - 5*sqrt(5) + 20*cosh(x) + 20*sinh(x) + 5) - (sqrt(5)*cosh(x) + sqrt(5)*sinh(x) - sqrt(5))*sqrt(2*sqrt(5)*sqrt(-2*sqrt(5) - 5) + 10)*log(s qrt(2*sqrt(5)*sqrt(-2*sqrt(5) - 5) + 10)*(3*sqrt(5) - 5)*sqrt(-2*sqrt(5) - 5) + 5*(sqrt(5) - 3)*sqrt(-2*sqrt(5) - 5) + 5*sqrt(5) + 20*cosh(x) + 20*s inh(x) + 5) + (sqrt(5)*cosh(x) + sqrt(5)*sinh(x) - sqrt(5))*sqrt(2*sqrt(5) *sqrt(-2*sqrt(5) - 5) + 10)*log(-sqrt(2*sqrt(5)*sqrt(-2*sqrt(5) - 5) + 10) *(3*sqrt(5) - 5)*sqrt(-2*sqrt(5) - 5) + 5*(sqrt(5) - 3)*sqrt(-2*sqrt(5)...
Timed out. \[ \int \frac {1}{1-\cosh ^5(x)} \, dx=\text {Timed out} \]
\[ \int \frac {1}{1-\cosh ^5(x)} \, dx=\int { -\frac {1}{\cosh \left (x\right )^{5} - 1} \,d x } \]
2/5/(e^x - 1) + integrate(2/5*(e^(7*x) + 4*e^(6*x) + 15*e^(5*x) + 40*e^(4* x) + 15*e^(3*x) + 4*e^(2*x) + e^x)/(e^(8*x) + 2*e^(7*x) + 8*e^(6*x) + 14*e ^(5*x) + 30*e^(4*x) + 14*e^(3*x) + 8*e^(2*x) + 2*e^x + 1), x)
\[ \int \frac {1}{1-\cosh ^5(x)} \, dx=\int { -\frac {1}{\cosh \left (x\right )^{5} - 1} \,d x } \]
Timed out. \[ \int \frac {1}{1-\cosh ^5(x)} \, dx=\text {Hanged} \]