Integrand size = 10, antiderivative size = 95 \[ \int \frac {1}{1-\cosh ^3(x)} \, dx=-\frac {2 \sqrt [4]{-1} \arctan \left (\frac {(-1)^{3/4} \tanh \left (\frac {x}{2}\right )}{\sqrt [4]{3}}\right )}{3^{3/4} \left (1-(-1)^{2/3}\right )}-\frac {2 \sqrt [4]{-1} \text {arctanh}\left (\frac {(-1)^{3/4} \tanh \left (\frac {x}{2}\right )}{\sqrt [4]{3}}\right )}{3^{3/4} \left (1+\sqrt [3]{-1}\right )}-\frac {\sinh (x)}{3 (1-\cosh (x))} \]
-2/3*(-1)^(1/4)*arctan(1/3*(-1)^(3/4)*tanh(1/2*x)*3^(3/4))*3^(1/4)/(1-(-1) ^(2/3))-2/3*(-1)^(1/4)*arctanh(1/3*(-1)^(3/4)*tanh(1/2*x)*3^(3/4))*3^(1/4) /(1+(-1)^(1/3))-1/3*sinh(x)/(1-cosh(x))
Timed out. \[ \int \frac {1}{1-\cosh ^3(x)} \, dx=\text {\$Aborted} \]
Time = 0.33 (sec) , antiderivative size = 95, 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.
| \(\displaystyle \int \frac {1}{1-\cosh ^3(x)} \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {1}{1-\sin \left (\frac {\pi }{2}+i x\right )^3}dx\) |
| \(\Big \downarrow \) 3692 |
| \(\displaystyle \int \left (\frac {1}{3 \left (\sqrt [3]{-1} \cosh (x)+1\right )}+\frac {1}{3 \left (1-(-1)^{2/3} \cosh (x)\right )}+\frac {1}{3 (1-\cosh (x))}\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle -\frac {2 \sqrt [4]{-1} \arctan \left (\frac {(-1)^{3/4} \tanh \left (\frac {x}{2}\right )}{\sqrt [4]{3}}\right )}{3^{3/4} \left (1-(-1)^{2/3}\right )}-\frac {2 \sqrt [4]{-1} \text {arctanh}\left (\frac {(-1)^{3/4} \tanh \left (\frac {x}{2}\right )}{\sqrt [4]{3}}\right )}{3^{3/4} \left (1+\sqrt [3]{-1}\right )}-\frac {\sinh (x)}{3 (1-\cosh (x))}\) |
(-2*(-1)^(1/4)*ArcTan[((-1)^(3/4)*Tanh[x/2])/3^(1/4)])/(3^(3/4)*(1 - (-1)^ (2/3))) - (2*(-1)^(1/4)*ArcTanh[((-1)^(3/4)*Tanh[x/2])/3^(1/4)])/(3^(3/4)* (1 + (-1)^(1/3))) - Sinh[x]/(3*(1 - Cosh[x]))
3.1.59.3.1 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 = 0.07 (sec) , antiderivative size = 46, normalized size of antiderivative = 0.48
| method | result | size |
| risch | \(\frac {2}{3 \left ({\mathrm e}^{x}-1\right )}+\left (\munderset {\textit {\_R} =\operatorname {RootOf}\left (243 \textit {\_Z}^{4}-27 \textit {\_Z}^{2}+1\right )}{\sum }\textit {\_R} \ln \left (162 \textit {\_R}^{3}-27 \textit {\_R}^{2}-9 \textit {\_R} +{\mathrm e}^{x}+2\right )\right )\) | \(46\) |
| default | \(\frac {3^{\frac {1}{4}} \sqrt {2}\, \left (\ln \left (\frac {\tanh \left (\frac {x}{2}\right )^{2}+\sqrt {2}\, 3^{\frac {1}{4}} \tanh \left (\frac {x}{2}\right )+\sqrt {3}}{\tanh \left (\frac {x}{2}\right )^{2}-\sqrt {2}\, 3^{\frac {1}{4}} \tanh \left (\frac {x}{2}\right )+\sqrt {3}}\right )+2 \arctan \left (\frac {3^{\frac {3}{4}} \tanh \left (\frac {x}{2}\right ) \sqrt {2}}{3}+1\right )+2 \arctan \left (\frac {3^{\frac {3}{4}} \tanh \left (\frac {x}{2}\right ) \sqrt {2}}{3}-1\right )\right )}{12}-\frac {3^{\frac {3}{4}} \sqrt {2}\, \left (\ln \left (\frac {\tanh \left (\frac {x}{2}\right )^{2}-\sqrt {2}\, 3^{\frac {1}{4}} \tanh \left (\frac {x}{2}\right )+\sqrt {3}}{\tanh \left (\frac {x}{2}\right )^{2}+\sqrt {2}\, 3^{\frac {1}{4}} \tanh \left (\frac {x}{2}\right )+\sqrt {3}}\right )+2 \arctan \left (\frac {3^{\frac {3}{4}} \tanh \left (\frac {x}{2}\right ) \sqrt {2}}{3}+1\right )+2 \arctan \left (\frac {3^{\frac {3}{4}} \tanh \left (\frac {x}{2}\right ) \sqrt {2}}{3}-1\right )\right )}{36}+\frac {1}{3 \tanh \left (\frac {x}{2}\right )}\) | \(190\) |
Result contains complex when optimal does not.
Time = 0.26 (sec) , antiderivative size = 234, normalized size of antiderivative = 2.46 \[ \int \frac {1}{1-\cosh ^3(x)} \, dx=-\frac {{\left (\sqrt {3} \cosh \left (x\right ) + \sqrt {3} \sinh \left (x\right ) - \sqrt {3}\right )} \sqrt {-2 i \, \sqrt {3} + 6} \log \left (i \, \sqrt {3} + i \, \sqrt {-2 i \, \sqrt {3} + 6} + 2 \, \cosh \left (x\right ) + 2 \, \sinh \left (x\right ) + 1\right ) - {\left (\sqrt {3} \cosh \left (x\right ) + \sqrt {3} \sinh \left (x\right ) - \sqrt {3}\right )} \sqrt {-2 i \, \sqrt {3} + 6} \log \left (i \, \sqrt {3} - i \, \sqrt {-2 i \, \sqrt {3} + 6} + 2 \, \cosh \left (x\right ) + 2 \, \sinh \left (x\right ) + 1\right ) - {\left (\sqrt {3} \cosh \left (x\right ) + \sqrt {3} \sinh \left (x\right ) - \sqrt {3}\right )} \sqrt {2 i \, \sqrt {3} + 6} \log \left (-i \, \sqrt {3} + i \, \sqrt {2 i \, \sqrt {3} + 6} + 2 \, \cosh \left (x\right ) + 2 \, \sinh \left (x\right ) + 1\right ) + {\left (\sqrt {3} \cosh \left (x\right ) + \sqrt {3} \sinh \left (x\right ) - \sqrt {3}\right )} \sqrt {2 i \, \sqrt {3} + 6} \log \left (-i \, \sqrt {3} - i \, \sqrt {2 i \, \sqrt {3} + 6} + 2 \, \cosh \left (x\right ) + 2 \, \sinh \left (x\right ) + 1\right ) - 12}{18 \, {\left (\cosh \left (x\right ) + \sinh \left (x\right ) - 1\right )}} \]
-1/18*((sqrt(3)*cosh(x) + sqrt(3)*sinh(x) - sqrt(3))*sqrt(-2*I*sqrt(3) + 6 )*log(I*sqrt(3) + I*sqrt(-2*I*sqrt(3) + 6) + 2*cosh(x) + 2*sinh(x) + 1) - (sqrt(3)*cosh(x) + sqrt(3)*sinh(x) - sqrt(3))*sqrt(-2*I*sqrt(3) + 6)*log(I *sqrt(3) - I*sqrt(-2*I*sqrt(3) + 6) + 2*cosh(x) + 2*sinh(x) + 1) - (sqrt(3 )*cosh(x) + sqrt(3)*sinh(x) - sqrt(3))*sqrt(2*I*sqrt(3) + 6)*log(-I*sqrt(3 ) + I*sqrt(2*I*sqrt(3) + 6) + 2*cosh(x) + 2*sinh(x) + 1) + (sqrt(3)*cosh(x ) + sqrt(3)*sinh(x) - sqrt(3))*sqrt(2*I*sqrt(3) + 6)*log(-I*sqrt(3) - I*sq rt(2*I*sqrt(3) + 6) + 2*cosh(x) + 2*sinh(x) + 1) - 12)/(cosh(x) + sinh(x) - 1)
Leaf count of result is larger than twice the leaf count of optimal. 320 vs. \(2 (78) = 156\).
Time = 1.56 (sec) , antiderivative size = 320, normalized size of antiderivative = 3.37 \[ \int \frac {1}{1-\cosh ^3(x)} \, dx=- \frac {\sqrt {2} \cdot \sqrt [4]{3} \log {\left (4 \tanh ^{2}{\left (\frac {x}{2} \right )} - 4 \sqrt {2} \cdot \sqrt [4]{3} \tanh {\left (\frac {x}{2} \right )} + 4 \sqrt {3} \right )}}{12} - \frac {\sqrt {2} \cdot 3^{\frac {3}{4}} \log {\left (4 \tanh ^{2}{\left (\frac {x}{2} \right )} - 4 \sqrt {2} \cdot \sqrt [4]{3} \tanh {\left (\frac {x}{2} \right )} + 4 \sqrt {3} \right )}}{36} + \frac {\sqrt {2} \cdot 3^{\frac {3}{4}} \log {\left (4 \tanh ^{2}{\left (\frac {x}{2} \right )} + 4 \sqrt {2} \cdot \sqrt [4]{3} \tanh {\left (\frac {x}{2} \right )} + 4 \sqrt {3} \right )}}{36} + \frac {\sqrt {2} \cdot \sqrt [4]{3} \log {\left (4 \tanh ^{2}{\left (\frac {x}{2} \right )} + 4 \sqrt {2} \cdot \sqrt [4]{3} \tanh {\left (\frac {x}{2} \right )} + 4 \sqrt {3} \right )}}{12} - \frac {\sqrt {2} \cdot 3^{\frac {3}{4}} \operatorname {atan}{\left (\frac {\sqrt {2} \cdot 3^{\frac {3}{4}} \tanh {\left (\frac {x}{2} \right )}}{3} - 1 \right )}}{18} + \frac {\sqrt {2} \cdot \sqrt [4]{3} \operatorname {atan}{\left (\frac {\sqrt {2} \cdot 3^{\frac {3}{4}} \tanh {\left (\frac {x}{2} \right )}}{3} - 1 \right )}}{6} - \frac {\sqrt {2} \cdot 3^{\frac {3}{4}} \operatorname {atan}{\left (\frac {\sqrt {2} \cdot 3^{\frac {3}{4}} \tanh {\left (\frac {x}{2} \right )}}{3} + 1 \right )}}{18} + \frac {\sqrt {2} \cdot \sqrt [4]{3} \operatorname {atan}{\left (\frac {\sqrt {2} \cdot 3^{\frac {3}{4}} \tanh {\left (\frac {x}{2} \right )}}{3} + 1 \right )}}{6} + \frac {1}{3 \tanh {\left (\frac {x}{2} \right )}} \]
-sqrt(2)*3**(1/4)*log(4*tanh(x/2)**2 - 4*sqrt(2)*3**(1/4)*tanh(x/2) + 4*sq rt(3))/12 - sqrt(2)*3**(3/4)*log(4*tanh(x/2)**2 - 4*sqrt(2)*3**(1/4)*tanh( x/2) + 4*sqrt(3))/36 + sqrt(2)*3**(3/4)*log(4*tanh(x/2)**2 + 4*sqrt(2)*3** (1/4)*tanh(x/2) + 4*sqrt(3))/36 + sqrt(2)*3**(1/4)*log(4*tanh(x/2)**2 + 4* sqrt(2)*3**(1/4)*tanh(x/2) + 4*sqrt(3))/12 - sqrt(2)*3**(3/4)*atan(sqrt(2) *3**(3/4)*tanh(x/2)/3 - 1)/18 + sqrt(2)*3**(1/4)*atan(sqrt(2)*3**(3/4)*tan h(x/2)/3 - 1)/6 - sqrt(2)*3**(3/4)*atan(sqrt(2)*3**(3/4)*tanh(x/2)/3 + 1)/ 18 + sqrt(2)*3**(1/4)*atan(sqrt(2)*3**(3/4)*tanh(x/2)/3 + 1)/6 + 1/(3*tanh (x/2))
\[ \int \frac {1}{1-\cosh ^3(x)} \, dx=\int { -\frac {1}{\cosh \left (x\right )^{3} - 1} \,d x } \]
2/3/(e^x - 1) + integrate(2/3*(e^(3*x) + 4*e^(2*x) + e^x)/(e^(4*x) + 2*e^( 3*x) + 6*e^(2*x) + 2*e^x + 1), x)
Leaf count of result is larger than twice the leaf count of optimal. 275 vs. \(2 (67) = 134\).
Time = 0.32 (sec) , antiderivative size = 275, normalized size of antiderivative = 2.89 \[ \int \frac {1}{1-\cosh ^3(x)} \, dx=-\frac {1}{18} \, \sqrt {6 \, \sqrt {3} + 9} \log \left (4 \, {\left (2 \, \sqrt {3} \sqrt {6 \, \sqrt {3} + 9} - 3 \, \sqrt {6 \, \sqrt {3} + 9} + 6 \, e^{x} + 3\right )}^{2} + 4 \, {\left (\sqrt {3} \sqrt {6 \, \sqrt {3} + 9} + 3 \, \sqrt {3}\right )}^{2}\right ) + \frac {1}{18} \, \sqrt {6 \, \sqrt {3} + 9} \log \left (4 \, {\left (2 \, \sqrt {3} \sqrt {6 \, \sqrt {3} + 9} - 3 \, \sqrt {6 \, \sqrt {3} + 9} - 6 \, e^{x} - 3\right )}^{2} + 4 \, {\left (\sqrt {3} \sqrt {6 \, \sqrt {3} + 9} - 3 \, \sqrt {3}\right )}^{2}\right ) + \frac {\sqrt {3} \sqrt {6 \, \sqrt {3} + 9} \arctan \left (\frac {3 \, {\left (\sqrt {2 \, \sqrt {3} - 3} + 2 \, e^{x} + 1\right )}}{\sqrt {3} \sqrt {6 \, \sqrt {3} + 9} + 3 \, \sqrt {3}}\right )}{9 \, {\left (2 \, \sqrt {3} + 3\right )}} + \frac {\sqrt {3} \sqrt {6 \, \sqrt {3} + 9} \arctan \left (-\frac {3 \, {\left (\sqrt {2 \, \sqrt {3} - 3} - 2 \, e^{x} - 1\right )}}{\sqrt {3} \sqrt {6 \, \sqrt {3} + 9} - 3 \, \sqrt {3}}\right )}{9 \, {\left (2 \, \sqrt {3} + 3\right )}} + \frac {2}{3 \, {\left (e^{x} - 1\right )}} \]
-1/18*sqrt(6*sqrt(3) + 9)*log(4*(2*sqrt(3)*sqrt(6*sqrt(3) + 9) - 3*sqrt(6* sqrt(3) + 9) + 6*e^x + 3)^2 + 4*(sqrt(3)*sqrt(6*sqrt(3) + 9) + 3*sqrt(3))^ 2) + 1/18*sqrt(6*sqrt(3) + 9)*log(4*(2*sqrt(3)*sqrt(6*sqrt(3) + 9) - 3*sqr t(6*sqrt(3) + 9) - 6*e^x - 3)^2 + 4*(sqrt(3)*sqrt(6*sqrt(3) + 9) - 3*sqrt( 3))^2) + 1/9*sqrt(3)*sqrt(6*sqrt(3) + 9)*arctan(3*(sqrt(2*sqrt(3) - 3) + 2 *e^x + 1)/(sqrt(3)*sqrt(6*sqrt(3) + 9) + 3*sqrt(3)))/(2*sqrt(3) + 3) + 1/9 *sqrt(3)*sqrt(6*sqrt(3) + 9)*arctan(-3*(sqrt(2*sqrt(3) - 3) - 2*e^x - 1)/( sqrt(3)*sqrt(6*sqrt(3) + 9) - 3*sqrt(3)))/(2*sqrt(3) + 3) + 2/3/(e^x - 1)
Time = 4.32 (sec) , antiderivative size = 295, normalized size of antiderivative = 3.11 \[ \int \frac {1}{1-\cosh ^3(x)} \, dx=\ln \left (\frac {32\,{\mathrm {e}}^x}{3}+\sqrt {\frac {1}{18}-\frac {\sqrt {3}\,1{}\mathrm {i}}{54}}\,\left (\frac {32\,{\mathrm {e}}^x}{3}-\sqrt {\frac {1}{18}-\frac {\sqrt {3}\,1{}\mathrm {i}}{54}}\,\left (384\,{\mathrm {e}}^x+\sqrt {\frac {1}{18}-\frac {\sqrt {3}\,1{}\mathrm {i}}{54}}\,\left (1152\,{\mathrm {e}}^x+864\right )+192\right )+\frac {160}{3}\right )+\frac {128}{9}\right )\,\sqrt {\frac {1}{18}-\frac {\sqrt {3}\,1{}\mathrm {i}}{54}}+\ln \left (\frac {32\,{\mathrm {e}}^x}{3}+\sqrt {\frac {1}{18}+\frac {\sqrt {3}\,1{}\mathrm {i}}{54}}\,\left (\frac {32\,{\mathrm {e}}^x}{3}-\sqrt {\frac {1}{18}+\frac {\sqrt {3}\,1{}\mathrm {i}}{54}}\,\left (384\,{\mathrm {e}}^x+\sqrt {\frac {1}{18}+\frac {\sqrt {3}\,1{}\mathrm {i}}{54}}\,\left (1152\,{\mathrm {e}}^x+864\right )+192\right )+\frac {160}{3}\right )+\frac {128}{9}\right )\,\sqrt {\frac {1}{18}+\frac {\sqrt {3}\,1{}\mathrm {i}}{54}}-\ln \left (\frac {32\,{\mathrm {e}}^x}{3}-\sqrt {\frac {1}{18}-\frac {\sqrt {3}\,1{}\mathrm {i}}{54}}\,\left (\frac {32\,{\mathrm {e}}^x}{3}+\sqrt {\frac {1}{18}-\frac {\sqrt {3}\,1{}\mathrm {i}}{54}}\,\left (384\,{\mathrm {e}}^x-\sqrt {\frac {1}{18}-\frac {\sqrt {3}\,1{}\mathrm {i}}{54}}\,\left (1152\,{\mathrm {e}}^x+864\right )+192\right )+\frac {160}{3}\right )+\frac {128}{9}\right )\,\sqrt {\frac {1}{18}-\frac {\sqrt {3}\,1{}\mathrm {i}}{54}}-\ln \left (\frac {32\,{\mathrm {e}}^x}{3}-\sqrt {\frac {1}{18}+\frac {\sqrt {3}\,1{}\mathrm {i}}{54}}\,\left (\frac {32\,{\mathrm {e}}^x}{3}+\sqrt {\frac {1}{18}+\frac {\sqrt {3}\,1{}\mathrm {i}}{54}}\,\left (384\,{\mathrm {e}}^x-\sqrt {\frac {1}{18}+\frac {\sqrt {3}\,1{}\mathrm {i}}{54}}\,\left (1152\,{\mathrm {e}}^x+864\right )+192\right )+\frac {160}{3}\right )+\frac {128}{9}\right )\,\sqrt {\frac {1}{18}+\frac {\sqrt {3}\,1{}\mathrm {i}}{54}}+\frac {2}{3\,\left ({\mathrm {e}}^x-1\right )} \]
log((32*exp(x))/3 + (1/18 - (3^(1/2)*1i)/54)^(1/2)*((32*exp(x))/3 - (1/18 - (3^(1/2)*1i)/54)^(1/2)*(384*exp(x) + (1/18 - (3^(1/2)*1i)/54)^(1/2)*(115 2*exp(x) + 864) + 192) + 160/3) + 128/9)*(1/18 - (3^(1/2)*1i)/54)^(1/2) + log((32*exp(x))/3 + ((3^(1/2)*1i)/54 + 1/18)^(1/2)*((32*exp(x))/3 - ((3^(1 /2)*1i)/54 + 1/18)^(1/2)*(384*exp(x) + ((3^(1/2)*1i)/54 + 1/18)^(1/2)*(115 2*exp(x) + 864) + 192) + 160/3) + 128/9)*((3^(1/2)*1i)/54 + 1/18)^(1/2) - log((32*exp(x))/3 - (1/18 - (3^(1/2)*1i)/54)^(1/2)*((32*exp(x))/3 + (1/18 - (3^(1/2)*1i)/54)^(1/2)*(384*exp(x) - (1/18 - (3^(1/2)*1i)/54)^(1/2)*(115 2*exp(x) + 864) + 192) + 160/3) + 128/9)*(1/18 - (3^(1/2)*1i)/54)^(1/2) - log((32*exp(x))/3 - ((3^(1/2)*1i)/54 + 1/18)^(1/2)*((32*exp(x))/3 + ((3^(1 /2)*1i)/54 + 1/18)^(1/2)*(384*exp(x) - ((3^(1/2)*1i)/54 + 1/18)^(1/2)*(115 2*exp(x) + 864) + 192) + 160/3) + 128/9)*((3^(1/2)*1i)/54 + 1/18)^(1/2) + 2/(3*(exp(x) - 1))