Integrand size = 10, antiderivative size = 75 \[ \int \frac {1}{1-\sinh ^6(x)} \, dx=-\frac {1}{6} \arctan \left (\sqrt {3}-2 \tanh (x)\right )+\frac {1}{6} \arctan \left (\sqrt {3}+2 \tanh (x)\right )+\frac {\text {arctanh}\left (\sqrt {2} \tanh (x)\right )}{3 \sqrt {2}}+\frac {\text {arctanh}\left (\frac {\sqrt {3} \tanh (x)}{1+\tanh ^2(x)}\right )}{2 \sqrt {3}} \] Output:
1/6*arctan(-3^(1/2)+2*tanh(x))+1/6*arctan(3^(1/2)+2*tanh(x))+1/6*arctanh(2 ^(1/2)*tanh(x))*2^(1/2)+1/6*arctanh(3^(1/2)*tanh(x)/(1+tanh(x)^2))*3^(1/2)
Time = 5.17 (sec) , antiderivative size = 127, normalized size of antiderivative = 1.69 \[ \int \frac {1}{1-\sinh ^6(x)} \, dx=\frac {1}{6} \left (\sqrt {7-4 \sqrt {3}} \left (2+\sqrt {3}\right ) \arctan \left (\frac {e^{2 x}}{\sqrt {7-4 \sqrt {3}}}\right )-\left (-2+\sqrt {3}\right ) \sqrt {7+4 \sqrt {3}} \arctan \left (\frac {e^{2 x}}{\sqrt {7+4 \sqrt {3}}}\right )-\sqrt {3} \text {arctanh}\left (\frac {7+e^{4 x}}{4 \sqrt {3}}\right )+\sqrt {2} \text {arctanh}\left (\sqrt {2} \tanh (x)\right )\right ) \] Input:
Integrate[(1 - Sinh[x]^6)^(-1),x]
Output:
(Sqrt[7 - 4*Sqrt[3]]*(2 + Sqrt[3])*ArcTan[E^(2*x)/Sqrt[7 - 4*Sqrt[3]]] - ( -2 + Sqrt[3])*Sqrt[7 + 4*Sqrt[3]]*ArcTan[E^(2*x)/Sqrt[7 + 4*Sqrt[3]]] - Sq rt[3]*ArcTanh[(7 + E^(4*x))/(4*Sqrt[3])] + Sqrt[2]*ArcTanh[Sqrt[2]*Tanh[x] ])/6
Time = 0.36 (sec) , antiderivative size = 83, normalized size of antiderivative = 1.11, number of steps used = 6, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.500, Rules used = {3042, 3690, 3042, 3660, 219}
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.
| \(\displaystyle \int \frac {1}{1-\sinh ^6(x)} \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {1}{1+\sin (i x)^6}dx\) |
| \(\Big \downarrow \) 3690 |
| \(\displaystyle \frac {1}{3} \int \frac {1}{1-\sinh ^2(x)}dx+\frac {1}{3} \int \frac {1}{\sqrt [3]{-1} \sinh ^2(x)+1}dx+\frac {1}{3} \int \frac {1}{1-(-1)^{2/3} \sinh ^2(x)}dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {1}{3} \int \frac {1}{\sin (i x)^2+1}dx+\frac {1}{3} \int \frac {1}{1-\sqrt [3]{-1} \sin (i x)^2}dx+\frac {1}{3} \int \frac {1}{(-1)^{2/3} \sin (i x)^2+1}dx\) |
| \(\Big \downarrow \) 3660 |
| \(\displaystyle \frac {1}{3} \int \frac {1}{1-2 \tanh ^2(x)}d\tanh (x)+\frac {1}{3} \int \frac {1}{1-\left (1-\sqrt [3]{-1}\right ) \tanh ^2(x)}d\tanh (x)+\frac {1}{3} \int \frac {1}{1-\left (1+(-1)^{2/3}\right ) \tanh ^2(x)}d\tanh (x)\) |
| \(\Big \downarrow \) 219 |
| \(\displaystyle \frac {\text {arctanh}\left (\sqrt {2} \tanh (x)\right )}{3 \sqrt {2}}+\frac {\text {arctanh}\left (\sqrt {1-\sqrt [3]{-1}} \tanh (x)\right )}{3 \sqrt {1-\sqrt [3]{-1}}}+\frac {\text {arctanh}\left (\sqrt {1+(-1)^{2/3}} \tanh (x)\right )}{3 \sqrt {1+(-1)^{2/3}}}\) |
Input:
Int[(1 - Sinh[x]^6)^(-1),x]
Output:
ArcTanh[Sqrt[2]*Tanh[x]]/(3*Sqrt[2]) + ArcTanh[Sqrt[1 - (-1)^(1/3)]*Tanh[x ]]/(3*Sqrt[1 - (-1)^(1/3)]) + ArcTanh[Sqrt[1 + (-1)^(2/3)]*Tanh[x]]/(3*Sqr t[1 + (-1)^(2/3)])
Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1/(Rt[a, 2]*Rt[-b, 2]))* ArcTanh[Rt[-b, 2]*(x/Rt[a, 2])], x] /; FreeQ[{a, b}, x] && NegQ[a/b] && (Gt Q[a, 0] || LtQ[b, 0])
Int[((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)]^2)^(-1), x_Symbol] :> With[{ff = FreeFactors[Tan[e + f*x], x]}, Simp[ff/f Subst[Int[1/(a + (a + b)*ff^2*x^ 2), x], x, Tan[e + f*x]/ff], x]] /; FreeQ[{a, b, e, f}, x]
Int[((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)]^(n_))^(-1), x_Symbol] :> Module[{ k}, Simp[2/(a*n) Sum[Int[1/(1 - Sin[e + f*x]^2/((-1)^(4*(k/n))*Rt[-a/b, n /2])), x], {k, 1, n/2}], x]] /; FreeQ[{a, b, e, f}, x] && IntegerQ[n/2]
Result contains higher order function than in optimal. Order 9 vs. order 3.
Time = 0.86 (sec) , antiderivative size = 71, normalized size of antiderivative = 0.95
| method | result | size |
| risch | \(\frac {\sqrt {2}\, \ln \left ({\mathrm e}^{2 x}-3+2 \sqrt {2}\right )}{12}-\frac {\sqrt {2}\, \ln \left ({\mathrm e}^{2 x}-3-2 \sqrt {2}\right )}{12}+\left (\munderset {\textit {\_R} =\operatorname {RootOf}\left (1296 \textit {\_Z}^{4}-36 \textit {\_Z}^{2}+1\right )}{\sum }\textit {\_R} \ln \left (432 \textit {\_R}^{3}-72 \textit {\_R}^{2}+{\mathrm e}^{2 x}+1\right )\right )\) | \(71\) |
| default | \(\frac {\left (\munderset {\textit {\_R} =\operatorname {RootOf}\left (\textit {\_Z}^{4}+2 \textit {\_Z}^{3}+2 \textit {\_Z}^{2}-2 \textit {\_Z} +1\right )}{\sum }\frac {\left (-\textit {\_R}^{2}-\textit {\_R} +1\right ) \ln \left (\tanh \left (\frac {x}{2}\right )-\textit {\_R} \right )}{2 \textit {\_R}^{3}+3 \textit {\_R}^{2}+2 \textit {\_R} -1}\right )}{3}+\frac {\sqrt {2}\, \operatorname {arctanh}\left (\frac {\left (2 \tanh \left (\frac {x}{2}\right )-2\right ) \sqrt {2}}{4}\right )}{6}+\frac {\left (\munderset {\textit {\_R} =\operatorname {RootOf}\left (\textit {\_Z}^{4}-2 \textit {\_Z}^{3}+2 \textit {\_Z}^{2}+2 \textit {\_Z} +1\right )}{\sum }\frac {\left (-\textit {\_R}^{2}+\textit {\_R} +1\right ) \ln \left (\tanh \left (\frac {x}{2}\right )-\textit {\_R} \right )}{2 \textit {\_R}^{3}-3 \textit {\_R}^{2}+2 \textit {\_R} +1}\right )}{3}+\frac {\sqrt {2}\, \operatorname {arctanh}\left (\frac {\left (2 \tanh \left (\frac {x}{2}\right )+2\right ) \sqrt {2}}{4}\right )}{6}\) | \(160\) |
Input:
int(1/(1-sinh(x)^6),x,method=_RETURNVERBOSE)
Output:
1/12*2^(1/2)*ln(exp(2*x)-3+2*2^(1/2))-1/12*2^(1/2)*ln(exp(2*x)-3-2*2^(1/2) )+sum(_R*ln(432*_R^3-72*_R^2+exp(2*x)+1),_R=RootOf(1296*_Z^4-36*_Z^2+1))
Leaf count of result is larger than twice the leaf count of optimal. 247 vs. \(2 (57) = 114\).
Time = 0.09 (sec) , antiderivative size = 247, normalized size of antiderivative = 3.29 \[ \int \frac {1}{1-\sinh ^6(x)} \, dx=-\frac {1}{12} \, \sqrt {3} \log \left (\frac {4 \, {\left ({\left (\sqrt {3} + 2\right )} \cosh \left (x\right )^{2} - {\left (2 \, \sqrt {3} + 3\right )} \cosh \left (x\right ) \sinh \left (x\right ) + {\left (\sqrt {3} + 2\right )} \sinh \left (x\right )^{2}\right )}}{\cosh \left (x\right )^{2} - 2 \, \cosh \left (x\right ) \sinh \left (x\right ) + \sinh \left (x\right )^{2}}\right ) + \frac {1}{12} \, \sqrt {3} \log \left (-\frac {4 \, {\left ({\left (\sqrt {3} - 2\right )} \cosh \left (x\right )^{2} - {\left (2 \, \sqrt {3} - 3\right )} \cosh \left (x\right ) \sinh \left (x\right ) + {\left (\sqrt {3} - 2\right )} \sinh \left (x\right )^{2}\right )}}{\cosh \left (x\right )^{2} - 2 \, \cosh \left (x\right ) \sinh \left (x\right ) + \sinh \left (x\right )^{2}}\right ) + \frac {1}{12} \, \sqrt {2} \log \left (-\frac {3 \, {\left (2 \, \sqrt {2} - 3\right )} \cosh \left (x\right )^{2} - 4 \, {\left (3 \, \sqrt {2} - 4\right )} \cosh \left (x\right ) \sinh \left (x\right ) + 3 \, {\left (2 \, \sqrt {2} - 3\right )} \sinh \left (x\right )^{2} - 2 \, \sqrt {2} + 3}{\cosh \left (x\right )^{2} + \sinh \left (x\right )^{2} - 3}\right ) - \frac {1}{6} \, \arctan \left (-\frac {{\left (\sqrt {3} + 2\right )} \cosh \left (x\right ) + {\left (\sqrt {3} + 2\right )} \sinh \left (x\right )}{\cosh \left (x\right ) - \sinh \left (x\right )}\right ) + \frac {1}{6} \, \arctan \left (-\frac {{\left (\sqrt {3} - 2\right )} \cosh \left (x\right ) + {\left (\sqrt {3} - 2\right )} \sinh \left (x\right )}{\cosh \left (x\right ) - \sinh \left (x\right )}\right ) \] Input:
integrate(1/(1-sinh(x)^6),x, algorithm="fricas")
Output:
-1/12*sqrt(3)*log(4*((sqrt(3) + 2)*cosh(x)^2 - (2*sqrt(3) + 3)*cosh(x)*sin h(x) + (sqrt(3) + 2)*sinh(x)^2)/(cosh(x)^2 - 2*cosh(x)*sinh(x) + sinh(x)^2 )) + 1/12*sqrt(3)*log(-4*((sqrt(3) - 2)*cosh(x)^2 - (2*sqrt(3) - 3)*cosh(x )*sinh(x) + (sqrt(3) - 2)*sinh(x)^2)/(cosh(x)^2 - 2*cosh(x)*sinh(x) + sinh (x)^2)) + 1/12*sqrt(2)*log(-(3*(2*sqrt(2) - 3)*cosh(x)^2 - 4*(3*sqrt(2) - 4)*cosh(x)*sinh(x) + 3*(2*sqrt(2) - 3)*sinh(x)^2 - 2*sqrt(2) + 3)/(cosh(x) ^2 + sinh(x)^2 - 3)) - 1/6*arctan(-((sqrt(3) + 2)*cosh(x) + (sqrt(3) + 2)* sinh(x))/(cosh(x) - sinh(x))) + 1/6*arctan(-((sqrt(3) - 2)*cosh(x) + (sqrt (3) - 2)*sinh(x))/(cosh(x) - sinh(x)))
Timed out. \[ \int \frac {1}{1-\sinh ^6(x)} \, dx=\text {Timed out} \] Input:
integrate(1/(1-sinh(x)**6),x)
Output:
Timed out
\[ \int \frac {1}{1-\sinh ^6(x)} \, dx=\int { -\frac {1}{\sinh \left (x\right )^{6} - 1} \,d x } \] Input:
integrate(1/(1-sinh(x)^6),x, algorithm="maxima")
Output:
-1/12*sqrt(2)*log(-(sqrt(2) - e^x + 1)/(sqrt(2) + e^x - 1)) + 1/12*sqrt(2) *log(-(sqrt(2) - e^x - 1)/(sqrt(2) + e^x + 1)) + integrate(1/3*(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) - integ rate(1/3*(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)
Leaf count of result is larger than twice the leaf count of optimal. 143 vs. \(2 (57) = 114\).
Time = 0.14 (sec) , antiderivative size = 143, normalized size of antiderivative = 1.91 \[ \int \frac {1}{1-\sinh ^6(x)} \, dx=-\frac {1}{36} \, {\left ({\left (2 \, \sqrt {3} - 3\right )} e^{\left (4 \, x\right )} + 2 \, \sqrt {3} - 3\right )} \arctan \left (\frac {e^{\left (2 \, x\right )}}{\sqrt {3} + 2}\right ) + \frac {1}{36} \, {\left ({\left (2 \, \sqrt {3} + 3\right )} e^{\left (4 \, x\right )} + 2 \, \sqrt {3} + 3\right )} \arctan \left (-\frac {e^{\left (2 \, x\right )}}{\sqrt {3} - 2}\right ) - \frac {1}{12} \, \sqrt {3} \log \left ({\left (\sqrt {3} + 2\right )}^{2} + e^{\left (4 \, x\right )}\right ) + \frac {1}{12} \, \sqrt {3} \log \left ({\left (\sqrt {3} - 2\right )}^{2} + e^{\left (4 \, x\right )}\right ) - \frac {1}{12} \, \sqrt {2} \log \left (\frac {{\left | -4 \, \sqrt {2} + 2 \, e^{\left (2 \, x\right )} - 6 \right |}}{{\left | 4 \, \sqrt {2} + 2 \, e^{\left (2 \, x\right )} - 6 \right |}}\right ) \] Input:
integrate(1/(1-sinh(x)^6),x, algorithm="giac")
Output:
-1/36*((2*sqrt(3) - 3)*e^(4*x) + 2*sqrt(3) - 3)*arctan(e^(2*x)/(sqrt(3) + 2)) + 1/36*((2*sqrt(3) + 3)*e^(4*x) + 2*sqrt(3) + 3)*arctan(-e^(2*x)/(sqrt (3) - 2)) - 1/12*sqrt(3)*log((sqrt(3) + 2)^2 + e^(4*x)) + 1/12*sqrt(3)*log ((sqrt(3) - 2)^2 + e^(4*x)) - 1/12*sqrt(2)*log(abs(-4*sqrt(2) + 2*e^(2*x) - 6)/abs(4*sqrt(2) + 2*e^(2*x) - 6))
Time = 3.35 (sec) , antiderivative size = 285, normalized size of antiderivative = 3.80 \[ \int \frac {1}{1-\sinh ^6(x)} \, dx =\text {Too large to display} \] Input:
int(-1/(sinh(x)^6 - 1),x)
Output:
(log(exp(2*x)*(14009449395540459520 + 6177144285775790080i) + 3^(1/2)*(955 607545932677120 - 2167269359741829120i) - 3^(1/2)*exp(2*x)*(80883593776411 44320 + 3566375915854233600i) - (1655160823988879360 - 3753820658157486080 i))*1i)/12 - (log(exp(2*x)*(14009449395540459520 - 6177144285775790080i) + 3^(1/2)*(955607545932677120 + 2167269359741829120i) - 3^(1/2)*exp(2*x)*(8 088359377641144320 - 3566375915854233600i) - (1655160823988879360 + 375382 0658157486080i))*1i)/12 + atan((14009449395540459520*exp(2*x) - 9556075459 32677120*3^(1/2) + 8088359377641144320*3^(1/2)*exp(2*x) - 1655160823988879 360)/(6177144285775790080*exp(2*x) + 2167269359741829120*3^(1/2) + 3566375 915854233600*3^(1/2)*exp(2*x) + 3753820658157486080))/6 - (3^(1/2)*log((61 77144285775790080*exp(2*x) - 2167269359741829120*3^(1/2) - 356637591585423 3600*3^(1/2)*exp(2*x) + 3753820658157486080)^2 + (14009449395540459520*exp (2*x) + 955607545932677120*3^(1/2) - 8088359377641144320*3^(1/2)*exp(2*x) - 1655160823988879360)^2))/12 + (3^(1/2)*log((6177144285775790080*exp(2*x) + 2167269359741829120*3^(1/2) + 3566375915854233600*3^(1/2)*exp(2*x) + 37 53820658157486080)^2 + (14009449395540459520*exp(2*x) - 955607545932677120 *3^(1/2) + 8088359377641144320*3^(1/2)*exp(2*x) - 1655160823988879360)^2)) /12 + (2^(1/2)*log(17674880313941032960*exp(2*x) - 2144322552070144000*2^( 1/2) + 12498027726650736640*2^(1/2)*exp(2*x) - 3032530035220152320))/12 - (2^(1/2)*log(17674880313941032960*exp(2*x) + 2144322552070144000*2^(1/2...
\[ \int \frac {1}{1-\sinh ^6(x)} \, dx=-2 \sqrt {2}\, \mathrm {log}\left (e^{x}-\sqrt {2}-1\right )+2 \sqrt {2}\, \mathrm {log}\left (e^{x}-\sqrt {2}+1\right )+2 \sqrt {2}\, \mathrm {log}\left (e^{x}+\sqrt {2}-1\right )-2 \sqrt {2}\, \mathrm {log}\left (e^{x}+\sqrt {2}+1\right )+\frac {960 \left (\int \frac {e^{4 x}}{e^{12 x}-6 e^{10 x}+15 e^{8 x}-84 e^{6 x}+15 e^{4 x}-6 e^{2 x}+1}d x \right )}{7}+\frac {64 \left (\int \frac {e^{2 x}}{e^{12 x}-6 e^{10 x}+15 e^{8 x}-84 e^{6 x}+15 e^{4 x}-6 e^{2 x}+1}d x \right )}{7}+\frac {64 \left (\int \frac {1}{e^{12 x}-6 e^{10 x}+15 e^{8 x}-84 e^{6 x}+15 e^{4 x}-6 e^{2 x}+1}d x \right )}{7}-\frac {4 \,\mathrm {log}\left (e^{8 x}+14 e^{4 x}+1\right )}{21}+\frac {8 \,\mathrm {log}\left (e^{x}-\sqrt {2}-1\right )}{3}+\frac {8 \,\mathrm {log}\left (e^{x}-\sqrt {2}+1\right )}{3}+\frac {8 \,\mathrm {log}\left (e^{x}+\sqrt {2}-1\right )}{3}+\frac {8 \,\mathrm {log}\left (e^{x}+\sqrt {2}+1\right )}{3}-\frac {64 x}{7} \] Input:
int(1/(1-sinh(x)^6),x)
Output:
(2*( - 21*sqrt(2)*log(e**x - sqrt(2) - 1) + 21*sqrt(2)*log(e**x - sqrt(2) + 1) + 21*sqrt(2)*log(e**x + sqrt(2) - 1) - 21*sqrt(2)*log(e**x + sqrt(2) + 1) + 1440*int(e**(4*x)/(e**(12*x) - 6*e**(10*x) + 15*e**(8*x) - 84*e**(6 *x) + 15*e**(4*x) - 6*e**(2*x) + 1),x) + 96*int(e**(2*x)/(e**(12*x) - 6*e* *(10*x) + 15*e**(8*x) - 84*e**(6*x) + 15*e**(4*x) - 6*e**(2*x) + 1),x) + 9 6*int(1/(e**(12*x) - 6*e**(10*x) + 15*e**(8*x) - 84*e**(6*x) + 15*e**(4*x) - 6*e**(2*x) + 1),x) - 2*log(e**(8*x) + 14*e**(4*x) + 1) + 28*log(e**x - sqrt(2) - 1) + 28*log(e**x - sqrt(2) + 1) + 28*log(e**x + sqrt(2) - 1) + 2 8*log(e**x + sqrt(2) + 1) - 96*x))/21