Integrand size = 10, antiderivative size = 133 \[ \int \frac {1}{1-\sinh ^3(x)} \, dx=\frac {2 (-1)^{5/6} \arctan \left (\frac {i-(-1)^{5/6} \tanh \left (\frac {x}{2}\right )}{\sqrt {1+(-1)^{2/3}}}\right )}{3 \sqrt {1+(-1)^{2/3}}}+\frac {1}{3} \sqrt {2} \text {arctanh}\left (\frac {1+\tanh \left (\frac {x}{2}\right )}{\sqrt {2}}\right )-\frac {1}{3} (-1)^{5/6} \log \left (1+(-1)^{5/6}+(-1)^{2/3} \tanh \left (\frac {x}{2}\right )\right )+\frac {1}{3} (-1)^{5/6} \log \left (1+\sqrt [6]{-1}+(-1)^{5/6} \tanh \left (\frac {x}{2}\right )\right ) \] Output:
2/3*(-1)^(5/6)*arctan((I-(-1)^(5/6)*tanh(1/2*x))/(1+(-1)^(2/3))^(1/2))/(1+ (-1)^(2/3))^(1/2)+1/3*2^(1/2)*arctanh(1/2*(1+tanh(1/2*x))*2^(1/2))-1/3*(-1 )^(5/6)*ln(1+(-1)^(5/6)+(-1)^(2/3)*tanh(1/2*x))+1/3*(-1)^(5/6)*ln(1+(-1)^( 1/6)+(-1)^(5/6)*tanh(1/2*x))
Time = 5.25 (sec) , antiderivative size = 156, normalized size of antiderivative = 1.17 \[ \int \frac {1}{1-\sinh ^3(x)} \, dx=\frac {\sqrt {-1+i \sqrt {3}} \left (1+i \sqrt {3}\right ) \arctan \left (\frac {2+\left (-1-i \sqrt {3}\right ) \tanh \left (\frac {x}{2}\right )}{\sqrt {-2-2 i \sqrt {3}}}\right )+\sqrt {-1-i \sqrt {3}} \left (1-i \sqrt {3}\right ) \arctan \left (\frac {2+i \left (i+\sqrt {3}\right ) \tanh \left (\frac {x}{2}\right )}{\sqrt {-2+2 i \sqrt {3}}}\right )+2 \text {arctanh}\left (\frac {1+\tanh \left (\frac {x}{2}\right )}{\sqrt {2}}\right )}{3 \sqrt {2}} \] Input:
Integrate[(1 - Sinh[x]^3)^(-1),x]
Output:
(Sqrt[-1 + I*Sqrt[3]]*(1 + I*Sqrt[3])*ArcTan[(2 + (-1 - I*Sqrt[3])*Tanh[x/ 2])/Sqrt[-2 - (2*I)*Sqrt[3]]] + Sqrt[-1 - I*Sqrt[3]]*(1 - I*Sqrt[3])*ArcTa n[(2 + I*(I + Sqrt[3])*Tanh[x/2])/Sqrt[-2 + (2*I)*Sqrt[3]]] + 2*ArcTanh[(1 + Tanh[x/2])/Sqrt[2]])/(3*Sqrt[2])
Time = 0.43 (sec) , antiderivative size = 133, 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.
| \(\displaystyle \int \frac {1}{1-\sinh ^3(x)} \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {1}{1-i \sin (i x)^3}dx\) |
| \(\Big \downarrow \) 3692 |
| \(\displaystyle \int \left (-\frac {(-1)^{5/6}}{3 \left (\sqrt [6]{-1} \sinh (x)-(-1)^{5/6}\right )}-\frac {(-1)^{5/6}}{3 \left ((-1)^{5/6} \sinh (x)-(-1)^{5/6}\right )}-\frac {(-1)^{5/6}}{3 \left (-(-1)^{5/6}-i \sinh (x)\right )}\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {2 (-1)^{5/6} \arctan \left (\frac {-(-1)^{5/6} \tanh \left (\frac {x}{2}\right )+i}{\sqrt {1+(-1)^{2/3}}}\right )}{3 \sqrt {1+(-1)^{2/3}}}+\frac {1}{3} \sqrt {2} \text {arctanh}\left (\frac {\tanh \left (\frac {x}{2}\right )+1}{\sqrt {2}}\right )-\frac {1}{3} (-1)^{5/6} \log \left ((-1)^{2/3} \tanh \left (\frac {x}{2}\right )+(-1)^{5/6}+1\right )+\frac {1}{3} (-1)^{5/6} \log \left ((-1)^{5/6} \tanh \left (\frac {x}{2}\right )+\sqrt [6]{-1}+1\right )\) |
Input:
Int[(1 - Sinh[x]^3)^(-1),x]
Output:
(2*(-1)^(5/6)*ArcTan[(I - (-1)^(5/6)*Tanh[x/2])/Sqrt[1 + (-1)^(2/3)]])/(3* Sqrt[1 + (-1)^(2/3)]) + (Sqrt[2]*ArcTanh[(1 + Tanh[x/2])/Sqrt[2]])/3 - ((- 1)^(5/6)*Log[1 + (-1)^(5/6) + (-1)^(2/3)*Tanh[x/2]])/3 + ((-1)^(5/6)*Log[1 + (-1)^(1/6) + (-1)^(5/6)*Tanh[x/2]])/3
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 = 0.22 (sec) , antiderivative size = 60, normalized size of antiderivative = 0.45
| method | result | size |
| risch | \(\frac {\sqrt {2}\, \ln \left ({\mathrm e}^{x}+\sqrt {2}-1\right )}{6}-\frac {\sqrt {2}\, \ln \left ({\mathrm e}^{x}-1-\sqrt {2}\right )}{6}+\left (\munderset {\textit {\_R} =\operatorname {RootOf}\left (81 \textit {\_Z}^{4}-9 \textit {\_Z}^{2}+1\right )}{\sum }\textit {\_R} \ln \left (9 \textit {\_R}^{2}-3 \textit {\_R} +{\mathrm e}^{x}\right )\right )\) | \(60\) |
| default | \(\frac {2 \left (\munderset {\textit {\_R} =\operatorname {RootOf}\left (\textit {\_Z}^{4}-2 \textit {\_Z}^{3}+2 \textit {\_Z}^{2}+2 \textit {\_Z} +1\right )}{\sum }\frac {\left (-\textit {\_R}^{2}+\textit {\_R} +1\right ) \ln \left (\tanh \left (\frac {x}{2}\right )-\textit {\_R} \right )}{2 \textit {\_R}^{3}-3 \textit {\_R}^{2}+2 \textit {\_R} +1}\right )}{3}+\frac {\sqrt {2}\, \operatorname {arctanh}\left (\frac {\left (2 \tanh \left (\frac {x}{2}\right )+2\right ) \sqrt {2}}{4}\right )}{3}\) | \(80\) |
Input:
int(1/(1-sinh(x)^3),x,method=_RETURNVERBOSE)
Output:
1/6*2^(1/2)*ln(exp(x)+2^(1/2)-1)-1/6*2^(1/2)*ln(exp(x)-1-2^(1/2))+sum(_R*l n(9*_R^2-3*_R+exp(x)),_R=RootOf(81*_Z^4-9*_Z^2+1))
Time = 0.11 (sec) , antiderivative size = 158, normalized size of antiderivative = 1.19 \[ \int \frac {1}{1-\sinh ^3(x)} \, dx=-\frac {1}{6} \, \sqrt {3} \log \left (\frac {{\left (\sqrt {3} + 3\right )} \cosh \left (x\right ) - {\left (\sqrt {3} + 1\right )} \sinh \left (x\right ) + \sqrt {3} + 1}{\cosh \left (x\right ) - \sinh \left (x\right )}\right ) + \frac {1}{6} \, \sqrt {3} \log \left (-\frac {{\left (\sqrt {3} - 3\right )} \cosh \left (x\right ) - {\left (\sqrt {3} - 1\right )} \sinh \left (x\right ) + \sqrt {3} - 1}{\cosh \left (x\right ) - \sinh \left (x\right )}\right ) + \frac {1}{6} \, \sqrt {2} \log \left (-\frac {{\left (\sqrt {2} - 2\right )} \cosh \left (x\right ) - {\left (\sqrt {2} - 1\right )} \sinh \left (x\right ) - \sqrt {2} + 1}{\sinh \left (x\right ) - 1}\right ) + \frac {1}{3} \, \arctan \left ({\left (\sqrt {3} + 1\right )} \cosh \left (x\right ) + {\left (\sqrt {3} + 1\right )} \sinh \left (x\right ) - 1\right ) - \frac {1}{3} \, \arctan \left ({\left (\sqrt {3} - 1\right )} \cosh \left (x\right ) + {\left (\sqrt {3} - 1\right )} \sinh \left (x\right ) + 1\right ) \] Input:
integrate(1/(1-sinh(x)^3),x, algorithm="fricas")
Output:
-1/6*sqrt(3)*log(((sqrt(3) + 3)*cosh(x) - (sqrt(3) + 1)*sinh(x) + sqrt(3) + 1)/(cosh(x) - sinh(x))) + 1/6*sqrt(3)*log(-((sqrt(3) - 3)*cosh(x) - (sqr t(3) - 1)*sinh(x) + sqrt(3) - 1)/(cosh(x) - sinh(x))) + 1/6*sqrt(2)*log(-( (sqrt(2) - 2)*cosh(x) - (sqrt(2) - 1)*sinh(x) - sqrt(2) + 1)/(sinh(x) - 1) ) + 1/3*arctan((sqrt(3) + 1)*cosh(x) + (sqrt(3) + 1)*sinh(x) - 1) - 1/3*ar ctan((sqrt(3) - 1)*cosh(x) + (sqrt(3) - 1)*sinh(x) + 1)
Result contains complex when optimal does not.
Time = 41.81 (sec) , antiderivative size = 5697, normalized size of antiderivative = 42.83 \[ \int \frac {1}{1-\sinh ^3(x)} \, dx=\text {Too large to display} \] Input:
integrate(1/(1-sinh(x)**3),x)
Output:
1993064208509905040030222651967612223670550485947450269720751264236*sqrt(1 + sqrt(3)*I)*log(tanh(x/2) + 1 + sqrt(2))/(-13808353887544825059365004867 017916698238206831962200180747387258840 + 97639806709065714749513852448015 99061639534598550631929830715751530*sqrt(2) - 8455855303065317721456178609 505793922179731446324180990310407413822*sqrt(1 + sqrt(3)*I) - 199306420850 9905040030222651967612223670550485947450269720751264236*sqrt(6)*I*sqrt(1 + sqrt(3)*I) + 281861843435510590715205953650193130739324381544139366343680 2471274*sqrt(3)*I*sqrt(1 + sqrt(3)*I) + 5979192625529715120090667955902836 671011651457842350809162253792708*sqrt(2)*sqrt(1 + sqrt(3)*I)) + 469769739 059184317858676589416988551232207302573565610572800411879*sqrt(6)*I*sqrt(1 + sqrt(3)*I)*log(tanh(x/2) + 1 + sqrt(2))/(-13808353887544825059365004867 017916698238206831962200180747387258840 + 97639806709065714749513852448015 99061639534598550631929830715751530*sqrt(2) - 8455855303065317721456178609 505793922179731446324180990310407413822*sqrt(1 + sqrt(3)*I) - 199306420850 9905040030222651967612223670550485947450269720751264236*sqrt(6)*I*sqrt(1 + sqrt(3)*I) + 281861843435510590715205953650193130739324381544139366343680 2471274*sqrt(3)*I*sqrt(1 + sqrt(3)*I) + 5979192625529715120090667955902836 671011651457842350809162253792708*sqrt(2)*sqrt(1 + sqrt(3)*I)) + 325466022 3635523824983795081600533020546511532850210643276905250510*log(tanh(x/2) + 1 + sqrt(2))/(-1380835388754482505936500486701791669823820683196220018...
\[ \int \frac {1}{1-\sinh ^3(x)} \, dx=\int { -\frac {1}{\sinh \left (x\right )^{3} - 1} \,d x } \] Input:
integrate(1/(1-sinh(x)^3),x, algorithm="maxima")
Output:
-1/6*sqrt(2)*log(-(sqrt(2) - e^x + 1)/(sqrt(2) + e^x - 1)) + integrate(2/3 *(e^(3*x) + 4*e^(2*x) - e^x)/(e^(4*x) + 2*e^(3*x) + 2*e^(2*x) - 2*e^x + 1) , x)
Time = 0.13 (sec) , antiderivative size = 106, normalized size of antiderivative = 0.80 \[ \int \frac {1}{1-\sinh ^3(x)} \, dx=-\frac {1}{6} \, \pi - \frac {1}{6} \, \sqrt {3} \log \left ({\left (\sqrt {3} + e^{x} + 1\right )}^{2} + e^{\left (2 \, x\right )}\right ) + \frac {1}{6} \, \sqrt {3} \log \left ({\left (\sqrt {3} - e^{x} - 1\right )}^{2} + e^{\left (2 \, x\right )}\right ) - \frac {1}{6} \, \sqrt {2} \log \left (\frac {{\left | -2 \, \sqrt {2} + 2 \, e^{x} - 2 \right |}}{{\left | 2 \, \sqrt {2} + 2 \, e^{x} - 2 \right |}}\right ) - \frac {1}{3} \, \arctan \left (-{\left (\sqrt {3} + 1\right )} e^{x} + 1\right ) - \frac {1}{3} \, \arctan \left ({\left (\sqrt {3} - 1\right )} e^{x} + 1\right ) \] Input:
integrate(1/(1-sinh(x)^3),x, algorithm="giac")
Output:
-1/6*pi - 1/6*sqrt(3)*log((sqrt(3) + e^x + 1)^2 + e^(2*x)) + 1/6*sqrt(3)*l og((sqrt(3) - e^x - 1)^2 + e^(2*x)) - 1/6*sqrt(2)*log(abs(-2*sqrt(2) + 2*e ^x - 2)/abs(2*sqrt(2) + 2*e^x - 2)) - 1/3*arctan(-(sqrt(3) + 1)*e^x + 1) - 1/3*arctan((sqrt(3) - 1)*e^x + 1)
Time = 3.22 (sec) , antiderivative size = 225, normalized size of antiderivative = 1.69 \[ \int \frac {1}{1-\sinh ^3(x)} \, dx=\frac {\mathrm {atan}\left (\frac {77824\,{\mathrm {e}}^x+32768\,\sqrt {3}-45056\,\sqrt {3}\,{\mathrm {e}}^x-57344}{77824\,{\mathrm {e}}^x-45056\,\sqrt {3}\,{\mathrm {e}}^x}\right )}{3}+\frac {\mathrm {atan}\left (\frac {77824\,{\mathrm {e}}^x-32768\,\sqrt {3}+45056\,\sqrt {3}\,{\mathrm {e}}^x-57344}{77824\,{\mathrm {e}}^x+45056\,\sqrt {3}\,{\mathrm {e}}^x}\right )}{3}+\frac {\pi \,\mathrm {sign}\left (77824\,{\mathrm {e}}^x+32768\,\sqrt {3}-45056\,\sqrt {3}\,{\mathrm {e}}^x-57344\right )}{3}-\frac {\sqrt {2}\,\ln \left (59392\,{\mathrm {e}}^x-17408\,\sqrt {2}-41984\,\sqrt {2}\,{\mathrm {e}}^x+24576\right )}{6}+\frac {\sqrt {2}\,\ln \left (59392\,{\mathrm {e}}^x+17408\,\sqrt {2}+41984\,\sqrt {2}\,{\mathrm {e}}^x+24576\right )}{6}+\frac {\sqrt {3}\,\ln \left ({\left (77824\,{\mathrm {e}}^x-32768\,\sqrt {3}+45056\,\sqrt {3}\,{\mathrm {e}}^x-57344\right )}^2+{\left (77824\,{\mathrm {e}}^x+45056\,\sqrt {3}\,{\mathrm {e}}^x\right )}^2\right )}{6}-\frac {\sqrt {3}\,\ln \left ({\left (77824\,{\mathrm {e}}^x+32768\,\sqrt {3}-45056\,\sqrt {3}\,{\mathrm {e}}^x-57344\right )}^2+{\left (77824\,{\mathrm {e}}^x-45056\,\sqrt {3}\,{\mathrm {e}}^x\right )}^2\right )}{6} \] Input:
int(-1/(sinh(x)^3 - 1),x)
Output:
atan((77824*exp(x) + 32768*3^(1/2) - 45056*3^(1/2)*exp(x) - 57344)/(77824* exp(x) - 45056*3^(1/2)*exp(x)))/3 + atan((77824*exp(x) - 32768*3^(1/2) + 4 5056*3^(1/2)*exp(x) - 57344)/(77824*exp(x) + 45056*3^(1/2)*exp(x)))/3 + (p i*sign(77824*exp(x) + 32768*3^(1/2) - 45056*3^(1/2)*exp(x) - 57344))/3 - ( 2^(1/2)*log(59392*exp(x) - 17408*2^(1/2) - 41984*2^(1/2)*exp(x) + 24576))/ 6 + (2^(1/2)*log(59392*exp(x) + 17408*2^(1/2) + 41984*2^(1/2)*exp(x) + 245 76))/6 + (3^(1/2)*log((77824*exp(x) - 32768*3^(1/2) + 45056*3^(1/2)*exp(x) - 57344)^2 + (77824*exp(x) + 45056*3^(1/2)*exp(x))^2))/6 - (3^(1/2)*log(( 77824*exp(x) + 32768*3^(1/2) - 45056*3^(1/2)*exp(x) - 57344)^2 + (77824*ex p(x) - 45056*3^(1/2)*exp(x))^2))/6
\[ \int \frac {1}{1-\sinh ^3(x)} \, dx=-\sqrt {2}\, \mathrm {log}\left (e^{x}-\sqrt {2}-1\right )+\sqrt {2}\, \mathrm {log}\left (e^{x}+\sqrt {2}-1\right )-16 \left (\int \frac {e^{2 x}}{e^{6 x}-3 e^{4 x}-8 e^{3 x}+3 e^{2 x}-1}d x \right )+8 \left (\int \frac {e^{x}}{e^{6 x}-3 e^{4 x}-8 e^{3 x}+3 e^{2 x}-1}d x \right )-\frac {2 \,\mathrm {log}\left (e^{4 x}+2 e^{3 x}+2 e^{2 x}-2 e^{x}+1\right )}{3}+\frac {4 \,\mathrm {log}\left (e^{x}-\sqrt {2}-1\right )}{3}+\frac {4 \,\mathrm {log}\left (e^{x}+\sqrt {2}-1\right )}{3} \] Input:
int(1/(1-sinh(x)^3),x)
Output:
( - 3*sqrt(2)*log(e**x - sqrt(2) - 1) + 3*sqrt(2)*log(e**x + sqrt(2) - 1) - 48*int(e**(2*x)/(e**(6*x) - 3*e**(4*x) - 8*e**(3*x) + 3*e**(2*x) - 1),x) + 24*int(e**x/(e**(6*x) - 3*e**(4*x) - 8*e**(3*x) + 3*e**(2*x) - 1),x) - 2*log(e**(4*x) + 2*e**(3*x) + 2*e**(2*x) - 2*e**x + 1) + 4*log(e**x - sqrt (2) - 1) + 4*log(e**x + sqrt(2) - 1))/3