Integrand size = 10, antiderivative size = 69 \[ \int \frac {1}{1-\sinh ^8(x)} \, dx=\frac {\text {arctanh}\left (\sqrt {1-i} \tanh (x)\right )}{4 \sqrt {1-i}}+\frac {\text {arctanh}\left (\sqrt {1+i} \tanh (x)\right )}{4 \sqrt {1+i}}+\frac {\text {arctanh}\left (\sqrt {2} \tanh (x)\right )}{4 \sqrt {2}}+\frac {\tanh (x)}{4} \]
1/4*arctanh((1-I)^(1/2)*tanh(x))/(1-I)^(1/2)+1/4*arctanh((1+I)^(1/2)*tanh( x))/(1+I)^(1/2)+1/8*arctanh(2^(1/2)*tanh(x))*2^(1/2)+1/4*tanh(x)
Time = 2.41 (sec) , antiderivative size = 64, normalized size of antiderivative = 0.93 \[ \int \frac {1}{1-\sinh ^8(x)} \, dx=\frac {1}{8} \left (\frac {2 \text {arctanh}\left (\sqrt {1-i} \tanh (x)\right )}{\sqrt {1-i}}+\frac {2 \text {arctanh}\left (\sqrt {1+i} \tanh (x)\right )}{\sqrt {1+i}}+\sqrt {2} \text {arctanh}\left (\sqrt {2} \tanh (x)\right )+2 \tanh (x)\right ) \]
((2*ArcTanh[Sqrt[1 - I]*Tanh[x]])/Sqrt[1 - I] + (2*ArcTanh[Sqrt[1 + I]*Tan h[x]])/Sqrt[1 + I] + Sqrt[2]*ArcTanh[Sqrt[2]*Tanh[x]] + 2*Tanh[x])/8
Time = 0.43 (sec) , antiderivative size = 69, normalized size of antiderivative = 1.00, number of steps used = 10, number of rules used = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.900, Rules used = {3042, 3690, 3042, 3654, 3042, 3660, 219, 4254, 24}
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.
| \(\displaystyle \int \frac {1}{1-\sinh ^8(x)} \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {1}{1-\sin (i x)^8}dx\) |
| \(\Big \downarrow \) 3690 |
| \(\displaystyle \frac {1}{4} \int \frac {1}{1-\sinh ^2(x)}dx+\frac {1}{4} \int \frac {1}{1-i \sinh ^2(x)}dx+\frac {1}{4} \int \frac {1}{i \sinh ^2(x)+1}dx+\frac {1}{4} \int \frac {1}{\sinh ^2(x)+1}dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {1}{4} \int \frac {1}{1-\sin (i x)^2}dx+\frac {1}{4} \int \frac {1}{1-i \sin (i x)^2}dx+\frac {1}{4} \int \frac {1}{i \sin (i x)^2+1}dx+\frac {1}{4} \int \frac {1}{\sin (i x)^2+1}dx\) |
| \(\Big \downarrow \) 3654 |
| \(\displaystyle \frac {1}{4} \int \frac {1}{1-i \sin (i x)^2}dx+\frac {1}{4} \int \frac {1}{i \sin (i x)^2+1}dx+\frac {1}{4} \int \frac {1}{\sin (i x)^2+1}dx+\frac {1}{4} \int \text {sech}^2(x)dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {1}{4} \int \frac {1}{1-i \sin (i x)^2}dx+\frac {1}{4} \int \frac {1}{i \sin (i x)^2+1}dx+\frac {1}{4} \int \frac {1}{\sin (i x)^2+1}dx+\frac {1}{4} \int \csc \left (i x+\frac {\pi }{2}\right )^2dx\) |
| \(\Big \downarrow \) 3660 |
| \(\displaystyle \frac {1}{4} \int \csc \left (i x+\frac {\pi }{2}\right )^2dx+\frac {1}{4} \int \frac {1}{1-2 \tanh ^2(x)}d\tanh (x)+\frac {1}{4} \int \frac {1}{1-(1+i) \tanh ^2(x)}d\tanh (x)+\frac {1}{4} \int \frac {1}{1-(1-i) \tanh ^2(x)}d\tanh (x)\) |
| \(\Big \downarrow \) 219 |
| \(\displaystyle \frac {1}{4} \int \csc \left (i x+\frac {\pi }{2}\right )^2dx+\frac {\text {arctanh}\left (\sqrt {1-i} \tanh (x)\right )}{4 \sqrt {1-i}}+\frac {\text {arctanh}\left (\sqrt {1+i} \tanh (x)\right )}{4 \sqrt {1+i}}+\frac {\text {arctanh}\left (\sqrt {2} \tanh (x)\right )}{4 \sqrt {2}}\) |
| \(\Big \downarrow \) 4254 |
| \(\displaystyle \frac {1}{4} i \int 1d(-i \tanh (x))+\frac {\text {arctanh}\left (\sqrt {1-i} \tanh (x)\right )}{4 \sqrt {1-i}}+\frac {\text {arctanh}\left (\sqrt {1+i} \tanh (x)\right )}{4 \sqrt {1+i}}+\frac {\text {arctanh}\left (\sqrt {2} \tanh (x)\right )}{4 \sqrt {2}}\) |
| \(\Big \downarrow \) 24 |
| \(\displaystyle \frac {\text {arctanh}\left (\sqrt {1-i} \tanh (x)\right )}{4 \sqrt {1-i}}+\frac {\text {arctanh}\left (\sqrt {1+i} \tanh (x)\right )}{4 \sqrt {1+i}}+\frac {\text {arctanh}\left (\sqrt {2} \tanh (x)\right )}{4 \sqrt {2}}+\frac {\tanh (x)}{4}\) |
ArcTanh[Sqrt[1 - I]*Tanh[x]]/(4*Sqrt[1 - I]) + ArcTanh[Sqrt[1 + I]*Tanh[x] ]/(4*Sqrt[1 + I]) + ArcTanh[Sqrt[2]*Tanh[x]]/(4*Sqrt[2]) + Tanh[x]/4
3.3.75.3.1 Defintions of rubi rules used
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[(u_.)*((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)]^2)^(p_), x_Symbol] :> Simp[ a^p Int[ActivateTrig[u*cos[e + f*x]^(2*p)], x], x] /; FreeQ[{a, b, e, f, p}, x] && EqQ[a + b, 0] && IntegerQ[p]
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]
Int[csc[(c_.) + (d_.)*(x_)]^(n_), x_Symbol] :> Simp[-d^(-1) Subst[Int[Exp andIntegrand[(1 + x^2)^(n/2 - 1), x], x], x, Cot[c + d*x]], x] /; FreeQ[{c, d}, x] && IGtQ[n/2, 0]
Result contains higher order function than in optimal. Order 9 vs. order 3.
Time = 3.56 (sec) , antiderivative size = 81, normalized size of antiderivative = 1.17
| method | result | size |
| risch | \(-\frac {1}{2 \left ({\mathrm e}^{2 x}+1\right )}+\frac {\sqrt {2}\, \ln \left ({\mathrm e}^{2 x}-3+2 \sqrt {2}\right )}{16}-\frac {\sqrt {2}\, \ln \left ({\mathrm e}^{2 x}-3-2 \sqrt {2}\right )}{16}+\left (\munderset {\textit {\_R} =\operatorname {RootOf}\left (8192 \textit {\_Z}^{4}-128 \textit {\_Z}^{2}+1\right )}{\sum }\textit {\_R} \ln \left (2048 \textit {\_R}^{3}-256 \textit {\_R}^{2}+{\mathrm e}^{2 x}+1\right )\right )\) | \(81\) |
| default | \(\frac {\sqrt {2}\, \operatorname {arctanh}\left (\frac {\left (2 \tanh \left (\frac {x}{2}\right )-2\right ) \sqrt {2}}{4}\right )}{8}+\frac {\left (\munderset {\textit {\_R} =\operatorname {RootOf}\left (2 \textit {\_Z}^{4}-2 \textit {\_Z}^{2}+1\right )}{\sum }\textit {\_R} \ln \left (\tanh \left (\frac {x}{2}\right )^{2}+\left (-4 \textit {\_R}^{3}+4 \textit {\_R} \right ) \tanh \left (\frac {x}{2}\right )+1\right )\right )}{8}+\frac {\sqrt {2}\, \operatorname {arctanh}\left (\frac {\left (2 \tanh \left (\frac {x}{2}\right )+2\right ) \sqrt {2}}{4}\right )}{8}+\frac {\tanh \left (\frac {x}{2}\right )}{2 \tanh \left (\frac {x}{2}\right )^{2}+2}\) | \(99\) |
-1/2/(exp(2*x)+1)+1/16*2^(1/2)*ln(exp(2*x)-3+2*2^(1/2))-1/16*2^(1/2)*ln(ex p(2*x)-3-2*2^(1/2))+sum(_R*ln(2048*_R^3-256*_R^2+exp(2*x)+1),_R=RootOf(819 2*_Z^4-128*_Z^2+1))
Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 347 vs. \(2 (41) = 82\).
Time = 0.28 (sec) , antiderivative size = 347, normalized size of antiderivative = 5.03 \[ \int \frac {1}{1-\sinh ^8(x)} \, dx=\frac {\sqrt {i + 1} {\left (\sqrt {2} \cosh \left (x\right )^{2} + 2 \, \sqrt {2} \cosh \left (x\right ) \sinh \left (x\right ) + \sqrt {2} \sinh \left (x\right )^{2} + \sqrt {2}\right )} \log \left (\cosh \left (x\right )^{2} + 2 \, \cosh \left (x\right ) \sinh \left (x\right ) + \sinh \left (x\right )^{2} + \left (i + 1\right ) \, \sqrt {2} \sqrt {i + 1} - 2 i - 1\right ) - \sqrt {i + 1} {\left (\sqrt {2} \cosh \left (x\right )^{2} + 2 \, \sqrt {2} \cosh \left (x\right ) \sinh \left (x\right ) + \sqrt {2} \sinh \left (x\right )^{2} + \sqrt {2}\right )} \log \left (\cosh \left (x\right )^{2} + 2 \, \cosh \left (x\right ) \sinh \left (x\right ) + \sinh \left (x\right )^{2} - \left (i + 1\right ) \, \sqrt {2} \sqrt {i + 1} - 2 i - 1\right ) + \sqrt {-i + 1} {\left (\sqrt {2} \cosh \left (x\right )^{2} + 2 \, \sqrt {2} \cosh \left (x\right ) \sinh \left (x\right ) + \sqrt {2} \sinh \left (x\right )^{2} + \sqrt {2}\right )} \log \left (\cosh \left (x\right )^{2} + 2 \, \cosh \left (x\right ) \sinh \left (x\right ) + \sinh \left (x\right )^{2} - \left (i - 1\right ) \, \sqrt {2} \sqrt {-i + 1} + 2 i - 1\right ) - \sqrt {-i + 1} {\left (\sqrt {2} \cosh \left (x\right )^{2} + 2 \, \sqrt {2} \cosh \left (x\right ) \sinh \left (x\right ) + \sqrt {2} \sinh \left (x\right )^{2} + \sqrt {2}\right )} \log \left (\cosh \left (x\right )^{2} + 2 \, \cosh \left (x\right ) \sinh \left (x\right ) + \sinh \left (x\right )^{2} + \left (i - 1\right ) \, \sqrt {2} \sqrt {-i + 1} + 2 i - 1\right ) + {\left (\sqrt {2} \cosh \left (x\right )^{2} + 2 \, \sqrt {2} \cosh \left (x\right ) \sinh \left (x\right ) + \sqrt {2} \sinh \left (x\right )^{2} + \sqrt {2}\right )} \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 ) - 8}{16 \, {\left (\cosh \left (x\right )^{2} + 2 \, \cosh \left (x\right ) \sinh \left (x\right ) + \sinh \left (x\right )^{2} + 1\right )}} \]
1/16*(sqrt(I + 1)*(sqrt(2)*cosh(x)^2 + 2*sqrt(2)*cosh(x)*sinh(x) + sqrt(2) *sinh(x)^2 + sqrt(2))*log(cosh(x)^2 + 2*cosh(x)*sinh(x) + sinh(x)^2 + (I + 1)*sqrt(2)*sqrt(I + 1) - 2*I - 1) - sqrt(I + 1)*(sqrt(2)*cosh(x)^2 + 2*sq rt(2)*cosh(x)*sinh(x) + sqrt(2)*sinh(x)^2 + sqrt(2))*log(cosh(x)^2 + 2*cos h(x)*sinh(x) + sinh(x)^2 - (I + 1)*sqrt(2)*sqrt(I + 1) - 2*I - 1) + sqrt(- I + 1)*(sqrt(2)*cosh(x)^2 + 2*sqrt(2)*cosh(x)*sinh(x) + sqrt(2)*sinh(x)^2 + sqrt(2))*log(cosh(x)^2 + 2*cosh(x)*sinh(x) + sinh(x)^2 - (I - 1)*sqrt(2) *sqrt(-I + 1) + 2*I - 1) - sqrt(-I + 1)*(sqrt(2)*cosh(x)^2 + 2*sqrt(2)*cos h(x)*sinh(x) + sqrt(2)*sinh(x)^2 + sqrt(2))*log(cosh(x)^2 + 2*cosh(x)*sinh (x) + sinh(x)^2 + (I - 1)*sqrt(2)*sqrt(-I + 1) + 2*I - 1) + (sqrt(2)*cosh( x)^2 + 2*sqrt(2)*cosh(x)*sinh(x) + sqrt(2)*sinh(x)^2 + 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)) - 8)/(cosh(x )^2 + 2*cosh(x)*sinh(x) + sinh(x)^2 + 1)
Timed out. \[ \int \frac {1}{1-\sinh ^8(x)} \, dx=\text {Timed out} \]
\[ \int \frac {1}{1-\sinh ^8(x)} \, dx=\int { -\frac {1}{\sinh \left (x\right )^{8} - 1} \,d x } \]
-1/16*sqrt(2)*log(-(sqrt(2) - e^x + 1)/(sqrt(2) + e^x - 1)) + 1/16*sqrt(2) *log(-(sqrt(2) - e^x - 1)/(sqrt(2) + e^x + 1)) - 1/2/(e^(2*x) + 1) + 8*int egrate(e^(4*x)/(e^(8*x) - 4*e^(6*x) + 22*e^(4*x) - 4*e^(2*x) + 1), x)
Time = 0.41 (sec) , antiderivative size = 48, normalized size of antiderivative = 0.70 \[ \int \frac {1}{1-\sinh ^8(x)} \, dx=-\frac {1}{16} \, \sqrt {2} \log \left (\frac {{\left | -4 \, \sqrt {2} + 2 \, e^{\left (2 \, x\right )} - 6 \right |}}{{\left | 4 \, \sqrt {2} + 2 \, e^{\left (2 \, x\right )} - 6 \right |}}\right ) - \frac {1}{2 \, {\left (e^{\left (2 \, x\right )} + 1\right )}} \]
-1/16*sqrt(2)*log(abs(-4*sqrt(2) + 2*e^(2*x) - 6)/abs(4*sqrt(2) + 2*e^(2*x ) - 6)) - 1/2/(e^(2*x) + 1)
Time = 6.06 (sec) , antiderivative size = 273, normalized size of antiderivative = 3.96 \[ \int \frac {1}{1-\sinh ^8(x)} \, dx=\frac {\sqrt {2}\,\ln \left (582732658686033920\,{\mathrm {e}}^{2\,x}-70697326355677184\,\sqrt {2}+412054214575915008\,\sqrt {2}\,{\mathrm {e}}^{2\,x}-99981117754441728\right )}{16}-\frac {\sqrt {2}\,\ln \left (582732658686033920\,{\mathrm {e}}^{2\,x}+70697326355677184\,\sqrt {2}-412054214575915008\,\sqrt {2}\,{\mathrm {e}}^{2\,x}-99981117754441728\right )}{16}-\frac {1}{2\,\left ({\mathrm {e}}^{2\,x}+1\right )}-\frac {\sqrt {2}\,\sqrt {1-\mathrm {i}}\,\ln \left ({\mathrm {e}}^{2\,x}\,\left (155613434002538496+429723297714798592{}\mathrm {i}\right )+\sqrt {2}\,\sqrt {1-\mathrm {i}}\,\left (-54684829282729984+21956972328779776{}\mathrm {i}\right )+\sqrt {2}\,\sqrt {1-\mathrm {i}}\,{\mathrm {e}}^{2\,x}\,\left (12296353929494528-271474128182050816{}\mathrm {i}\right )+70836483296067584-69311013991743488{}\mathrm {i}\right )}{16}+\frac {\sqrt {2}\,\sqrt {1-\mathrm {i}}\,\ln \left ({\mathrm {e}}^{2\,x}\,\left (155613434002538496+429723297714798592{}\mathrm {i}\right )+\sqrt {2}\,\sqrt {1-\mathrm {i}}\,\left (54684829282729984-21956972328779776{}\mathrm {i}\right )+\sqrt {2}\,\sqrt {1-\mathrm {i}}\,{\mathrm {e}}^{2\,x}\,\left (-12296353929494528+271474128182050816{}\mathrm {i}\right )+70836483296067584-69311013991743488{}\mathrm {i}\right )}{16}-\frac {\sqrt {2}\,\sqrt {1+1{}\mathrm {i}}\,\ln \left ({\mathrm {e}}^{2\,x}\,\left (155613434002538496-429723297714798592{}\mathrm {i}\right )+\sqrt {2}\,\sqrt {1+1{}\mathrm {i}}\,\left (-54684829282729984-21956972328779776{}\mathrm {i}\right )+\sqrt {2}\,\sqrt {1+1{}\mathrm {i}}\,{\mathrm {e}}^{2\,x}\,\left (12296353929494528+271474128182050816{}\mathrm {i}\right )+70836483296067584+69311013991743488{}\mathrm {i}\right )}{16}+\frac {\sqrt {2}\,\sqrt {1+1{}\mathrm {i}}\,\ln \left ({\mathrm {e}}^{2\,x}\,\left (155613434002538496-429723297714798592{}\mathrm {i}\right )+\sqrt {2}\,\sqrt {1+1{}\mathrm {i}}\,\left (54684829282729984+21956972328779776{}\mathrm {i}\right )+\sqrt {2}\,\sqrt {1+1{}\mathrm {i}}\,{\mathrm {e}}^{2\,x}\,\left (-12296353929494528-271474128182050816{}\mathrm {i}\right )+70836483296067584+69311013991743488{}\mathrm {i}\right )}{16} \]
(2^(1/2)*log(582732658686033920*exp(2*x) - 70697326355677184*2^(1/2) + 412 054214575915008*2^(1/2)*exp(2*x) - 99981117754441728))/16 - (2^(1/2)*log(5 82732658686033920*exp(2*x) + 70697326355677184*2^(1/2) - 41205421457591500 8*2^(1/2)*exp(2*x) - 99981117754441728))/16 - 1/(2*(exp(2*x) + 1)) - (2^(1 /2)*(1 - 1i)^(1/2)*log(exp(2*x)*(155613434002538496 + 429723297714798592i) - 2^(1/2)*(1 - 1i)^(1/2)*(54684829282729984 - 21956972328779776i) + 2^(1/ 2)*(1 - 1i)^(1/2)*exp(2*x)*(12296353929494528 - 271474128182050816i) + (70 836483296067584 - 69311013991743488i)))/16 + (2^(1/2)*(1 - 1i)^(1/2)*log(e xp(2*x)*(155613434002538496 + 429723297714798592i) + 2^(1/2)*(1 - 1i)^(1/2 )*(54684829282729984 - 21956972328779776i) - 2^(1/2)*(1 - 1i)^(1/2)*exp(2* x)*(12296353929494528 - 271474128182050816i) + (70836483296067584 - 693110 13991743488i)))/16 - (2^(1/2)*(1 + 1i)^(1/2)*log(exp(2*x)*(155613434002538 496 - 429723297714798592i) - 2^(1/2)*(1 + 1i)^(1/2)*(54684829282729984 + 2 1956972328779776i) + 2^(1/2)*(1 + 1i)^(1/2)*exp(2*x)*(12296353929494528 + 271474128182050816i) + (70836483296067584 + 69311013991743488i)))/16 + (2^ (1/2)*(1 + 1i)^(1/2)*log(exp(2*x)*(155613434002538496 - 429723297714798592 i) + 2^(1/2)*(1 + 1i)^(1/2)*(54684829282729984 + 21956972328779776i) - 2^( 1/2)*(1 + 1i)^(1/2)*exp(2*x)*(12296353929494528 + 271474128182050816i) + ( 70836483296067584 + 69311013991743488i)))/16