Integrand size = 10, antiderivative size = 228 \[ \int \frac {1}{1-\sinh ^5(x)} \, dx=-\frac {2 \sqrt [10]{-1} \arctan \left (\frac {i+\sqrt [10]{-1} \tanh \left (\frac {x}{2}\right )}{\sqrt {1-\sqrt [5]{-1}}}\right )}{5 \sqrt {1-\sqrt [5]{-1}}}-\frac {2 \text {arctanh}\left (\frac {(-1)^{3/5}-\tanh \left (\frac {x}{2}\right )}{\sqrt {1-\sqrt [5]{-1}}}\right )}{5 \sqrt {1-\sqrt [5]{-1}}}+\frac {1}{5} \sqrt {2} \text {arctanh}\left (\frac {1+\tanh \left (\frac {x}{2}\right )}{\sqrt {2}}\right )+\frac {2 \text {arctanh}\left (\frac {(-1)^{4/5}+\tanh \left (\frac {x}{2}\right )}{\sqrt {1-(-1)^{3/5}}}\right )}{5 \sqrt {1-(-1)^{3/5}}}-\frac {2 \sqrt [10]{-1} \text {arctanh}\left (\frac {(-1)^{3/10} \left (1+(-1)^{4/5} \tanh \left (\frac {x}{2}\right )\right )}{\sqrt {\sqrt [5]{-1}+(-1)^{3/5}}}\right )}{5 \sqrt {\sqrt [5]{-1}+(-1)^{3/5}}} \]
1/5*arctanh(1/2*(1+tanh(1/2*x))*2^(1/2))*2^(1/2)-2/5*(-1)^(1/10)*arctan((I +(-1)^(1/10)*tanh(1/2*x))/(1-(-1)^(1/5))^(1/2))/(1-(-1)^(1/5))^(1/2)-2/5*a rctanh(((-1)^(3/5)-tanh(1/2*x))/(1-(-1)^(1/5))^(1/2))/(1-(-1)^(1/5))^(1/2) +2/5*arctanh(((-1)^(4/5)+tanh(1/2*x))/(1-(-1)^(3/5))^(1/2))/(1-(-1)^(3/5)) ^(1/2)-2/5*(-1)^(1/10)*arctanh((-1)^(3/10)*(1+(-1)^(4/5)*tanh(1/2*x))/((-1 )^(1/5)+(-1)^(3/5))^(1/2))/((-1)^(1/5)+(-1)^(3/5))^(1/2)
Result contains higher order function than in optimal. Order 9 vs. order 3 in optimal.
Time = 5.05 (sec) , antiderivative size = 437, normalized size of antiderivative = 1.92 \[ \int \frac {1}{1-\sinh ^5(x)} \, dx=\frac {1}{10} \left (2 \sqrt {2} \text {arctanh}\left (\frac {1+\tanh \left (\frac {x}{2}\right )}{\sqrt {2}}\right )+\text {RootSum}\left [1-2 \text {$\#$1}-2 \text {$\#$1}^3+14 \text {$\#$1}^4+2 \text {$\#$1}^5+2 \text {$\#$1}^7+\text {$\#$1}^8\&,\frac {-x-2 \log \left (-\cosh \left (\frac {x}{2}\right )-\sinh \left (\frac {x}{2}\right )+\cosh \left (\frac {x}{2}\right ) \text {$\#$1}-\sinh \left (\frac {x}{2}\right ) \text {$\#$1}\right )+4 x \text {$\#$1}+8 \log \left (-\cosh \left (\frac {x}{2}\right )-\sinh \left (\frac {x}{2}\right )+\cosh \left (\frac {x}{2}\right ) \text {$\#$1}-\sinh \left (\frac {x}{2}\right ) \text {$\#$1}\right ) \text {$\#$1}-9 x \text {$\#$1}^2-18 \log \left (-\cosh \left (\frac {x}{2}\right )-\sinh \left (\frac {x}{2}\right )+\cosh \left (\frac {x}{2}\right ) \text {$\#$1}-\sinh \left (\frac {x}{2}\right ) \text {$\#$1}\right ) \text {$\#$1}^2+24 x \text {$\#$1}^3+48 \log \left (-\cosh \left (\frac {x}{2}\right )-\sinh \left (\frac {x}{2}\right )+\cosh \left (\frac {x}{2}\right ) \text {$\#$1}-\sinh \left (\frac {x}{2}\right ) \text {$\#$1}\right ) \text {$\#$1}^3+9 x \text {$\#$1}^4+18 \log \left (-\cosh \left (\frac {x}{2}\right )-\sinh \left (\frac {x}{2}\right )+\cosh \left (\frac {x}{2}\right ) \text {$\#$1}-\sinh \left (\frac {x}{2}\right ) \text {$\#$1}\right ) \text {$\#$1}^4+4 x \text {$\#$1}^5+8 \log \left (-\cosh \left (\frac {x}{2}\right )-\sinh \left (\frac {x}{2}\right )+\cosh \left (\frac {x}{2}\right ) \text {$\#$1}-\sinh \left (\frac {x}{2}\right ) \text {$\#$1}\right ) \text {$\#$1}^5+x \text {$\#$1}^6+2 \log \left (-\cosh \left (\frac {x}{2}\right )-\sinh \left (\frac {x}{2}\right )+\cosh \left (\frac {x}{2}\right ) \text {$\#$1}-\sinh \left (\frac {x}{2}\right ) \text {$\#$1}\right ) \text {$\#$1}^6}{-1-3 \text {$\#$1}^2+28 \text {$\#$1}^3+5 \text {$\#$1}^4+7 \text {$\#$1}^6+4 \text {$\#$1}^7}\&\right ]\right ) \]
(2*Sqrt[2]*ArcTanh[(1 + Tanh[x/2])/Sqrt[2]] + RootSum[1 - 2*#1 - 2*#1^3 + 14*#1^4 + 2*#1^5 + 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] + Cos h[x/2]*#1 - Sinh[x/2]*#1]*#1 - 9*x*#1^2 - 18*Log[-Cosh[x/2] - Sinh[x/2] + Cosh[x/2]*#1 - Sinh[x/2]*#1]*#1^2 + 24*x*#1^3 + 48*Log[-Cosh[x/2] - Sinh[x /2] + Cosh[x/2]*#1 - Sinh[x/2]*#1]*#1^3 + 9*x*#1^4 + 18*Log[-Cosh[x/2] - S inh[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 - 3*#1^2 + 28*#1^3 + 5*#1^4 + 7*#1^6 + 4*#1^7) & ])/10
Time = 0.68 (sec) , antiderivative size = 233, normalized size of antiderivative = 1.02, 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.
| \(\displaystyle \int \frac {1}{1-\sinh ^5(x)} \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {1}{1+i \sin (i x)^5}dx\) |
| \(\Big \downarrow \) 3692 |
| \(\displaystyle \int \left (\frac {\sqrt [10]{-1}}{5 \left (\sqrt [10]{-1}-\sqrt [10]{-1} \sinh (x)\right )}+\frac {\sqrt [10]{-1}}{5 \left ((-1)^{3/10} \sinh (x)+\sqrt [10]{-1}\right )}+\frac {\sqrt [10]{-1}}{5 \left ((-1)^{7/10} \sinh (x)+\sqrt [10]{-1}\right )}+\frac {\sqrt [10]{-1}}{5 \left (\sqrt [10]{-1}-(-1)^{9/10} \sinh (x)\right )}+\frac {\sqrt [10]{-1}}{5 \left (\sqrt [10]{-1}-i \sinh (x)\right )}\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle -\frac {2 \sqrt [10]{-1} \arctan \left (\frac {\sqrt [10]{-1} \tanh \left (\frac {x}{2}\right )+i}{\sqrt {1-\sqrt [5]{-1}}}\right )}{5 \sqrt {1-\sqrt [5]{-1}}}+\frac {1}{5} \sqrt {2} \text {arctanh}\left (\frac {\tanh \left (\frac {x}{2}\right )+1}{\sqrt {2}}\right )+\frac {2 \text {arctanh}\left (\frac {\tanh \left (\frac {x}{2}\right )+(-1)^{4/5}}{\sqrt {1-(-1)^{3/5}}}\right )}{5 \sqrt {1-(-1)^{3/5}}}-\frac {2 \text {arctanh}\left (\frac {(-1)^{3/5} \left ((-1)^{2/5} \tanh \left (\frac {x}{2}\right )+1\right )}{\sqrt {1-\sqrt [5]{-1}}}\right )}{5 \sqrt {1-\sqrt [5]{-1}}}-\frac {2 \sqrt [10]{-1} \text {arctanh}\left (\frac {(-1)^{3/10} \left ((-1)^{4/5} \tanh \left (\frac {x}{2}\right )+1\right )}{\sqrt {\sqrt [5]{-1}+(-1)^{3/5}}}\right )}{5 \sqrt {\sqrt [5]{-1}+(-1)^{3/5}}}\) |
(-2*(-1)^(1/10)*ArcTan[(I + (-1)^(1/10)*Tanh[x/2])/Sqrt[1 - (-1)^(1/5)]])/ (5*Sqrt[1 - (-1)^(1/5)]) + (Sqrt[2]*ArcTanh[(1 + Tanh[x/2])/Sqrt[2]])/5 + (2*ArcTanh[((-1)^(4/5) + Tanh[x/2])/Sqrt[1 - (-1)^(3/5)]])/(5*Sqrt[1 - (-1 )^(3/5)]) - (2*ArcTanh[((-1)^(3/5)*(1 + (-1)^(2/5)*Tanh[x/2]))/Sqrt[1 - (- 1)^(1/5)]])/(5*Sqrt[1 - (-1)^(1/5)]) - (2*(-1)^(1/10)*ArcTanh[((-1)^(3/10) *(1 + (-1)^(4/5)*Tanh[x/2]))/Sqrt[(-1)^(1/5) + (-1)^(3/5)]])/(5*Sqrt[(-1)^ (1/5) + (-1)^(3/5)])
3.3.73.3.1 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 = 0.21 (sec) , antiderivative size = 90, normalized size of antiderivative = 0.39
| method | result | size |
| risch | \(\left (\munderset {\textit {\_R} =\operatorname {RootOf}\left (390625 \textit {\_Z}^{8}-31250 \textit {\_Z}^{6}+2500 \textit {\_Z}^{4}-75 \textit {\_Z}^{2}+1\right )}{\sum }\textit {\_R} \ln \left (15625 \textit {\_R}^{6}-3125 \textit {\_R}^{5}-625 \textit {\_R}^{4}+125 \textit {\_R}^{3}+50 \textit {\_R}^{2}-10 \textit {\_R} +{\mathrm e}^{x}\right )\right )+\frac {\sqrt {2}\, \ln \left ({\mathrm e}^{x}+\sqrt {2}-1\right )}{10}-\frac {\sqrt {2}\, \ln \left ({\mathrm e}^{x}-1-\sqrt {2}\right )}{10}\) | \(90\) |
| default | \(\frac {\sqrt {2}\, \operatorname {arctanh}\left (\frac {\left (2 \tanh \left (\frac {x}{2}\right )+2\right ) \sqrt {2}}{4}\right )}{5}+\frac {2 \left (\munderset {\textit {\_R} =\operatorname {RootOf}\left (\textit {\_Z}^{8}-2 \textit {\_Z}^{7}-2 \textit {\_Z}^{5}+14 \textit {\_Z}^{4}+2 \textit {\_Z}^{3}+2 \textit {\_Z} +1\right )}{\sum }\frac {\left (-2 \textit {\_R}^{6}+3 \textit {\_R}^{5}+2 \textit {\_R}^{4}-2 \textit {\_R}^{3}-2 \textit {\_R}^{2}+3 \textit {\_R} +2\right ) \ln \left (\tanh \left (\frac {x}{2}\right )-\textit {\_R} \right )}{4 \textit {\_R}^{7}-7 \textit {\_R}^{6}-5 \textit {\_R}^{4}+28 \textit {\_R}^{3}+3 \textit {\_R}^{2}+1}\right )}{5}\) | \(124\) |
sum(_R*ln(15625*_R^6-3125*_R^5-625*_R^4+125*_R^3+50*_R^2-10*_R+exp(x)),_R= RootOf(390625*_Z^8-31250*_Z^6+2500*_Z^4-75*_Z^2+1))+1/10*2^(1/2)*ln(exp(x) +2^(1/2)-1)-1/10*2^(1/2)*ln(exp(x)-1-2^(1/2))
Leaf count of result is larger than twice the leaf count of optimal. 591 vs. \(2 (153) = 306\).
Time = 0.32 (sec) , antiderivative size = 591, normalized size of antiderivative = 2.59 \[ \int \frac {1}{1-\sinh ^5(x)} \, dx=-\frac {1}{10} \, \sqrt {2 \, \sqrt {2 \, \sqrt {5} - 5} + 2} \log \left (\sqrt {2 \, \sqrt {5} - 5} {\left (\sqrt {5} + 1\right )} + {\left (\sqrt {5} + 1\right )} \sqrt {2 \, \sqrt {2 \, \sqrt {5} - 5} + 2} + \sqrt {5} + 4 \, \cosh \left (x\right ) + 4 \, \sinh \left (x\right ) + 1\right ) + \frac {1}{10} \, \sqrt {2 \, \sqrt {2 \, \sqrt {5} - 5} + 2} \log \left (\sqrt {2 \, \sqrt {5} - 5} {\left (\sqrt {5} + 1\right )} - {\left (\sqrt {5} + 1\right )} \sqrt {2 \, \sqrt {2 \, \sqrt {5} - 5} + 2} + \sqrt {5} + 4 \, \cosh \left (x\right ) + 4 \, \sinh \left (x\right ) + 1\right ) - \frac {1}{10} \, \sqrt {-2 \, \sqrt {2 \, \sqrt {5} - 5} + 2} \log \left (-\sqrt {2 \, \sqrt {5} - 5} {\left (\sqrt {5} + 1\right )} + {\left (\sqrt {5} + 1\right )} \sqrt {-2 \, \sqrt {2 \, \sqrt {5} - 5} + 2} + \sqrt {5} + 4 \, \cosh \left (x\right ) + 4 \, \sinh \left (x\right ) + 1\right ) + \frac {1}{10} \, \sqrt {-2 \, \sqrt {2 \, \sqrt {5} - 5} + 2} \log \left (-\sqrt {2 \, \sqrt {5} - 5} {\left (\sqrt {5} + 1\right )} - {\left (\sqrt {5} + 1\right )} \sqrt {-2 \, \sqrt {2 \, \sqrt {5} - 5} + 2} + \sqrt {5} + 4 \, \cosh \left (x\right ) + 4 \, \sinh \left (x\right ) + 1\right ) + \frac {1}{10} \, \sqrt {-2 \, \sqrt {-2 \, \sqrt {5} - 5} + 2} \log \left ({\left (\sqrt {5} - 1\right )} \sqrt {-2 \, \sqrt {5} - 5} + {\left (\sqrt {5} - 1\right )} \sqrt {-2 \, \sqrt {-2 \, \sqrt {5} - 5} + 2} - \sqrt {5} + 4 \, \cosh \left (x\right ) + 4 \, \sinh \left (x\right ) + 1\right ) - \frac {1}{10} \, \sqrt {-2 \, \sqrt {-2 \, \sqrt {5} - 5} + 2} \log \left ({\left (\sqrt {5} - 1\right )} \sqrt {-2 \, \sqrt {5} - 5} - {\left (\sqrt {5} - 1\right )} \sqrt {-2 \, \sqrt {-2 \, \sqrt {5} - 5} + 2} - \sqrt {5} + 4 \, \cosh \left (x\right ) + 4 \, \sinh \left (x\right ) + 1\right ) + \frac {1}{10} \, \sqrt {2 \, \sqrt {-2 \, \sqrt {5} - 5} + 2} \log \left (-{\left (\sqrt {5} - 1\right )} \sqrt {-2 \, \sqrt {5} - 5} + {\left (\sqrt {5} - 1\right )} \sqrt {2 \, \sqrt {-2 \, \sqrt {5} - 5} + 2} - \sqrt {5} + 4 \, \cosh \left (x\right ) + 4 \, \sinh \left (x\right ) + 1\right ) - \frac {1}{10} \, \sqrt {2 \, \sqrt {-2 \, \sqrt {5} - 5} + 2} \log \left (-{\left (\sqrt {5} - 1\right )} \sqrt {-2 \, \sqrt {5} - 5} - {\left (\sqrt {5} - 1\right )} \sqrt {2 \, \sqrt {-2 \, \sqrt {5} - 5} + 2} - \sqrt {5} + 4 \, \cosh \left (x\right ) + 4 \, \sinh \left (x\right ) + 1\right ) + \frac {1}{10} \, \sqrt {2} \log \left (-\frac {{\left (\sqrt {2} - 2\right )} \cosh \left (x\right ) - {\left (\sqrt {2} - 1\right )} \sinh \left (x\right ) - \sqrt {2} + 1}{\sinh \left (x\right ) - 1}\right ) \]
-1/10*sqrt(2*sqrt(2*sqrt(5) - 5) + 2)*log(sqrt(2*sqrt(5) - 5)*(sqrt(5) + 1 ) + (sqrt(5) + 1)*sqrt(2*sqrt(2*sqrt(5) - 5) + 2) + sqrt(5) + 4*cosh(x) + 4*sinh(x) + 1) + 1/10*sqrt(2*sqrt(2*sqrt(5) - 5) + 2)*log(sqrt(2*sqrt(5) - 5)*(sqrt(5) + 1) - (sqrt(5) + 1)*sqrt(2*sqrt(2*sqrt(5) - 5) + 2) + sqrt(5 ) + 4*cosh(x) + 4*sinh(x) + 1) - 1/10*sqrt(-2*sqrt(2*sqrt(5) - 5) + 2)*log (-sqrt(2*sqrt(5) - 5)*(sqrt(5) + 1) + (sqrt(5) + 1)*sqrt(-2*sqrt(2*sqrt(5) - 5) + 2) + sqrt(5) + 4*cosh(x) + 4*sinh(x) + 1) + 1/10*sqrt(-2*sqrt(2*sq rt(5) - 5) + 2)*log(-sqrt(2*sqrt(5) - 5)*(sqrt(5) + 1) - (sqrt(5) + 1)*sqr t(-2*sqrt(2*sqrt(5) - 5) + 2) + sqrt(5) + 4*cosh(x) + 4*sinh(x) + 1) + 1/1 0*sqrt(-2*sqrt(-2*sqrt(5) - 5) + 2)*log((sqrt(5) - 1)*sqrt(-2*sqrt(5) - 5) + (sqrt(5) - 1)*sqrt(-2*sqrt(-2*sqrt(5) - 5) + 2) - sqrt(5) + 4*cosh(x) + 4*sinh(x) + 1) - 1/10*sqrt(-2*sqrt(-2*sqrt(5) - 5) + 2)*log((sqrt(5) - 1) *sqrt(-2*sqrt(5) - 5) - (sqrt(5) - 1)*sqrt(-2*sqrt(-2*sqrt(5) - 5) + 2) - sqrt(5) + 4*cosh(x) + 4*sinh(x) + 1) + 1/10*sqrt(2*sqrt(-2*sqrt(5) - 5) + 2)*log(-(sqrt(5) - 1)*sqrt(-2*sqrt(5) - 5) + (sqrt(5) - 1)*sqrt(2*sqrt(-2* sqrt(5) - 5) + 2) - sqrt(5) + 4*cosh(x) + 4*sinh(x) + 1) - 1/10*sqrt(2*sqr t(-2*sqrt(5) - 5) + 2)*log(-(sqrt(5) - 1)*sqrt(-2*sqrt(5) - 5) - (sqrt(5) - 1)*sqrt(2*sqrt(-2*sqrt(5) - 5) + 2) - sqrt(5) + 4*cosh(x) + 4*sinh(x) + 1) + 1/10*sqrt(2)*log(-((sqrt(2) - 2)*cosh(x) - (sqrt(2) - 1)*sinh(x) - sq rt(2) + 1)/(sinh(x) - 1))
\[ \int \frac {1}{1-\sinh ^5(x)} \, dx=- \int \frac {1}{\sinh ^{5}{\left (x \right )} - 1}\, dx \]
\[ \int \frac {1}{1-\sinh ^5(x)} \, dx=\int { -\frac {1}{\sinh \left (x\right )^{5} - 1} \,d x } \]
-1/10*sqrt(2)*log(-(sqrt(2) - e^x + 1)/(sqrt(2) + e^x - 1)) + integrate(2/ 5*(e^(7*x) + 4*e^(6*x) + 9*e^(5*x) + 24*e^(4*x) - 9*e^(3*x) + 4*e^(2*x) - e^x)/(e^(8*x) + 2*e^(7*x) + 2*e^(5*x) + 14*e^(4*x) - 2*e^(3*x) - 2*e^x + 1 ), x)
Leaf count of result is larger than twice the leaf count of optimal. 4948 vs. \(2 (153) = 306\).
Time = 2.43 (sec) , antiderivative size = 4948, normalized size of antiderivative = 21.70 \[ \int \frac {1}{1-\sinh ^5(x)} \, dx=\text {Too large to display} \]
-8/25*5^(3/4)*sqrt(-1/32*sqrt(5) + 5/64)*arctan(5*(5^(3/4) + sqrt(5) + 5^( 1/4) + 4*e^x + 1)/(5^(3/4)*sqrt(-2*sqrt(5) + 5) + 5*sqrt(5)*sqrt(-2*sqrt(5 ) + 5) + 5*5^(1/4)*sqrt(-2*sqrt(5) + 5) + 5*sqrt(-2*sqrt(5) + 5))) + 8/25* 5^(3/4)*sqrt(-1/32*sqrt(5) + 5/64)*arctan(5*(5^(3/4) - sqrt(5) + 5^(1/4) - 4*e^x - 1)/(5^(3/4)*sqrt(-2*sqrt(5) + 5) - 5*sqrt(5)*sqrt(-2*sqrt(5) + 5) + 5*5^(1/4)*sqrt(-2*sqrt(5) + 5) - 5*sqrt(-2*sqrt(5) + 5))) - 1/10*sqrt(s qrt(5) + 2)*log((302427386195713850867712*sqrt(5)*(2*sqrt(5) + 5)^3 + 1728 15649254693629067264*(2*sqrt(5) + 5)^(7/2) + 226820539646785388150784*sqrt (5)*(2*sqrt(5) + 5)^(5/2)*sqrt(sqrt(5) + 2) + 151213693097856925433856*(2* sqrt(5) + 5)^3*sqrt(sqrt(5) + 2) + 70881418639620433797120*sqrt(5)*(2*sqrt (5) + 5)^2*(sqrt(5) + 2) + 56705134911696347037696*(2*sqrt(5) + 5)^(5/2)*( sqrt(5) + 2) + 11813569773270072299520*sqrt(5)*(2*sqrt(5) + 5)^(3/2)*(sqrt (5) + 2)^(3/2) + 11813569773270072299520*(2*sqrt(5) + 5)^2*(sqrt(5) + 2)^( 3/2) + 1107522166244069278080*sqrt(5)*(2*sqrt(5) + 5)*(sqrt(5) + 2)^2 + 14 76696221658759037440*(2*sqrt(5) + 5)^(3/2)*(sqrt(5) + 2)^2 + 5537610831220 3463904*sqrt(5)*sqrt(2*sqrt(5) + 5)*(sqrt(5) + 2)^(5/2) + 1107522166244069 27808*(2*sqrt(5) + 5)*(sqrt(5) + 2)^(5/2) + 1153668923170905498*sqrt(5)*(s qrt(5) + 2)^3 + 4614675692683621992*sqrt(2*sqrt(5) + 5)*(sqrt(5) + 2)^3 + 82404923083636107*(sqrt(5) + 2)^(7/2) - 622619531678741564620800*sqrt(5)*( 2*sqrt(5) + 5)^(5/2) - 415079687785827709747200*(2*sqrt(5) + 5)^3 - 389...
Timed out. \[ \int \frac {1}{1-\sinh ^5(x)} \, dx=\text {Hanged} \]