Integrand size = 10, antiderivative size = 245 \[ \int \frac {1}{a+b \sinh ^8(x)} \, dx=-\frac {\text {arctanh}\left (\frac {\sqrt {\sqrt [4]{-a}-\sqrt [4]{b}} \tanh (x)}{\sqrt [8]{-a}}\right )}{4 (-a)^{7/8} \sqrt {\sqrt [4]{-a}-\sqrt [4]{b}}}-\frac {\text {arctanh}\left (\frac {\sqrt {\sqrt [4]{-a}-i \sqrt [4]{b}} \tanh (x)}{\sqrt [8]{-a}}\right )}{4 (-a)^{7/8} \sqrt {\sqrt [4]{-a}-i \sqrt [4]{b}}}-\frac {\text {arctanh}\left (\frac {\sqrt {\sqrt [4]{-a}+i \sqrt [4]{b}} \tanh (x)}{\sqrt [8]{-a}}\right )}{4 (-a)^{7/8} \sqrt {\sqrt [4]{-a}+i \sqrt [4]{b}}}-\frac {\text {arctanh}\left (\frac {\sqrt {\sqrt [4]{-a}+\sqrt [4]{b}} \tanh (x)}{\sqrt [8]{-a}}\right )}{4 (-a)^{7/8} \sqrt {\sqrt [4]{-a}+\sqrt [4]{b}}} \]
-1/4*arctanh(((-a)^(1/4)-b^(1/4))^(1/2)*tanh(x)/(-a)^(1/8))/(-a)^(7/8)/((- a)^(1/4)-b^(1/4))^(1/2)-1/4*arctanh(((-a)^(1/4)-I*b^(1/4))^(1/2)*tanh(x)/( -a)^(1/8))/(-a)^(7/8)/((-a)^(1/4)-I*b^(1/4))^(1/2)-1/4*arctanh(((-a)^(1/4) +I*b^(1/4))^(1/2)*tanh(x)/(-a)^(1/8))/(-a)^(7/8)/((-a)^(1/4)+I*b^(1/4))^(1 /2)-1/4*arctanh(((-a)^(1/4)+b^(1/4))^(1/2)*tanh(x)/(-a)^(1/8))/(-a)^(7/8)/ ((-a)^(1/4)+b^(1/4))^(1/2)
Result contains higher order function than in optimal. Order 9 vs. order 3 in optimal.
Time = 5.08 (sec) , antiderivative size = 160, normalized size of antiderivative = 0.65 \[ \int \frac {1}{a+b \sinh ^8(x)} \, dx=16 \text {RootSum}\left [b-8 b \text {$\#$1}+28 b \text {$\#$1}^2-56 b \text {$\#$1}^3+256 a \text {$\#$1}^4+70 b \text {$\#$1}^4-56 b \text {$\#$1}^5+28 b \text {$\#$1}^6-8 b \text {$\#$1}^7+b \text {$\#$1}^8\&,\frac {x \text {$\#$1}^3+\log (-\cosh (x)-\sinh (x)+\cosh (x) \text {$\#$1}-\sinh (x) \text {$\#$1}) \text {$\#$1}^3}{-b+7 b \text {$\#$1}-21 b \text {$\#$1}^2+128 a \text {$\#$1}^3+35 b \text {$\#$1}^3-35 b \text {$\#$1}^4+21 b \text {$\#$1}^5-7 b \text {$\#$1}^6+b \text {$\#$1}^7}\&\right ] \]
16*RootSum[b - 8*b*#1 + 28*b*#1^2 - 56*b*#1^3 + 256*a*#1^4 + 70*b*#1^4 - 5 6*b*#1^5 + 28*b*#1^6 - 8*b*#1^7 + b*#1^8 & , (x*#1^3 + Log[-Cosh[x] - Sinh [x] + Cosh[x]*#1 - Sinh[x]*#1]*#1^3)/(-b + 7*b*#1 - 21*b*#1^2 + 128*a*#1^3 + 35*b*#1^3 - 35*b*#1^4 + 21*b*#1^5 - 7*b*#1^6 + b*#1^7) & ]
Time = 0.71 (sec) , antiderivative size = 257, normalized size of antiderivative = 1.05, 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.
\(\displaystyle \int \frac {1}{a+b \sinh ^8(x)} \, dx\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \int \frac {1}{a+b \sin (i x)^8}dx\) |
\(\Big \downarrow \) 3690 |
\(\displaystyle \frac {\int \frac {1}{1-\frac {\sqrt [4]{b} \sinh ^2(x)}{\sqrt [4]{-a}}}dx}{4 a}+\frac {\int \frac {1}{1-\frac {i \sqrt [4]{b} \sinh ^2(x)}{\sqrt [4]{-a}}}dx}{4 a}+\frac {\int \frac {1}{\frac {i \sqrt [4]{b} \sinh ^2(x)}{\sqrt [4]{-a}}+1}dx}{4 a}+\frac {\int \frac {1}{\frac {\sqrt [4]{b} \sinh ^2(x)}{\sqrt [4]{-a}}+1}dx}{4 a}\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \frac {\int \frac {1}{1-\frac {\sqrt [4]{b} \sin (i x)^2}{\sqrt [4]{-a}}}dx}{4 a}+\frac {\int \frac {1}{1-\frac {i \sqrt [4]{b} \sin (i x)^2}{\sqrt [4]{-a}}}dx}{4 a}+\frac {\int \frac {1}{\frac {i \sqrt [4]{b} \sin (i x)^2}{\sqrt [4]{-a}}+1}dx}{4 a}+\frac {\int \frac {1}{\frac {\sqrt [4]{b} \sin (i x)^2}{\sqrt [4]{-a}}+1}dx}{4 a}\) |
\(\Big \downarrow \) 3660 |
\(\displaystyle \frac {\int \frac {1}{1-\left (1-\frac {\sqrt [4]{b}}{\sqrt [4]{-a}}\right ) \tanh ^2(x)}d\tanh (x)}{4 a}+\frac {\int \frac {1}{1-\left (1-\frac {i \sqrt [4]{b}}{\sqrt [4]{-a}}\right ) \tanh ^2(x)}d\tanh (x)}{4 a}+\frac {\int \frac {1}{1-\left (\frac {i \sqrt [4]{b}}{\sqrt [4]{-a}}+1\right ) \tanh ^2(x)}d\tanh (x)}{4 a}+\frac {\int \frac {1}{1-\left (1-\frac {a \sqrt [4]{b}}{(-a)^{5/4}}\right ) \tanh ^2(x)}d\tanh (x)}{4 a}\) |
\(\Big \downarrow \) 219 |
\(\displaystyle \frac {\sqrt [8]{-a} \text {arctanh}\left (\frac {\sqrt {\sqrt [4]{-a}-i \sqrt [4]{b}} \tanh (x)}{\sqrt [8]{-a}}\right )}{4 a \sqrt {\sqrt [4]{-a}-i \sqrt [4]{b}}}+\frac {\sqrt [8]{-a} \text {arctanh}\left (\frac {\sqrt {\sqrt [4]{-a}+i \sqrt [4]{b}} \tanh (x)}{\sqrt [8]{-a}}\right )}{4 a \sqrt {\sqrt [4]{-a}+i \sqrt [4]{b}}}+\frac {\sqrt [8]{-a} \text {arctanh}\left (\frac {\sqrt {\sqrt [4]{-a}+\sqrt [4]{b}} \tanh (x)}{\sqrt [8]{-a}}\right )}{4 a \sqrt {\sqrt [4]{-a}+\sqrt [4]{b}}}+\frac {(-a)^{5/8} \text {arctanh}\left (\frac {\sqrt {a \sqrt [4]{b}+(-a)^{5/4}} \tanh (x)}{(-a)^{5/8}}\right )}{4 a \sqrt {a \sqrt [4]{b}+(-a)^{5/4}}}\) |
((-a)^(1/8)*ArcTanh[(Sqrt[(-a)^(1/4) - I*b^(1/4)]*Tanh[x])/(-a)^(1/8)])/(4 *a*Sqrt[(-a)^(1/4) - I*b^(1/4)]) + ((-a)^(1/8)*ArcTanh[(Sqrt[(-a)^(1/4) + I*b^(1/4)]*Tanh[x])/(-a)^(1/8)])/(4*a*Sqrt[(-a)^(1/4) + I*b^(1/4)]) + ((-a )^(1/8)*ArcTanh[(Sqrt[(-a)^(1/4) + b^(1/4)]*Tanh[x])/(-a)^(1/8)])/(4*a*Sqr t[(-a)^(1/4) + b^(1/4)]) + ((-a)^(5/8)*ArcTanh[(Sqrt[(-a)^(5/4) + a*b^(1/4 )]*Tanh[x])/(-a)^(5/8)])/(4*a*Sqrt[(-a)^(5/4) + a*b^(1/4)])
3.3.69.3.1 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[((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 = 1.48 (sec) , antiderivative size = 162, normalized size of antiderivative = 0.66
method | result | size |
default | \(\frac {\left (\munderset {\textit {\_R} =\operatorname {RootOf}\left (a \,\textit {\_Z}^{16}-8 a \,\textit {\_Z}^{14}+28 a \,\textit {\_Z}^{12}-56 a \,\textit {\_Z}^{10}+\left (70 a +256 b \right ) \textit {\_Z}^{8}-56 a \,\textit {\_Z}^{6}+28 a \,\textit {\_Z}^{4}-8 a \,\textit {\_Z}^{2}+a \right )}{\sum }\frac {\left (-\textit {\_R}^{14}+7 \textit {\_R}^{12}-21 \textit {\_R}^{10}+35 \textit {\_R}^{8}-35 \textit {\_R}^{6}+21 \textit {\_R}^{4}-7 \textit {\_R}^{2}+1\right ) \ln \left (\tanh \left (\frac {x}{2}\right )-\textit {\_R} \right )}{\textit {\_R}^{15} a -7 \textit {\_R}^{13} a +21 \textit {\_R}^{11} a -35 \textit {\_R}^{9} a +35 \textit {\_R}^{7} a +128 \textit {\_R}^{7} b -21 \textit {\_R}^{5} a +7 \textit {\_R}^{3} a -\textit {\_R} a}\right )}{8}\) | \(162\) |
risch | \(\munderset {\textit {\_R} =\operatorname {RootOf}\left (1+\left (16777216 a^{8}+16777216 a^{7} b \right ) \textit {\_Z}^{8}-1048576 a^{6} \textit {\_Z}^{6}+24576 a^{4} \textit {\_Z}^{4}-256 a^{2} \textit {\_Z}^{2}\right )}{\sum }\textit {\_R} \ln \left ({\mathrm e}^{2 x}+\left (\frac {4194304 a^{8}}{b}+4194304 a^{7}\right ) \textit {\_R}^{7}+\left (-\frac {524288 a^{7}}{b}-524288 a^{6}\right ) \textit {\_R}^{6}+\left (-\frac {196608 a^{6}}{b}+65536 a^{5}\right ) \textit {\_R}^{5}+\left (\frac {24576 a^{5}}{b}-8192 a^{4}\right ) \textit {\_R}^{4}+\left (\frac {3072 a^{4}}{b}+1024 a^{3}\right ) \textit {\_R}^{3}+\left (-\frac {384 a^{3}}{b}-128 a^{2}\right ) \textit {\_R}^{2}+\left (-\frac {16 a^{2}}{b}+16 a \right ) \textit {\_R} +\frac {2 a}{b}-1\right )\) | \(184\) |
1/8*sum((-_R^14+7*_R^12-21*_R^10+35*_R^8-35*_R^6+21*_R^4-7*_R^2+1)/(_R^15* a-7*_R^13*a+21*_R^11*a-35*_R^9*a+35*_R^7*a+128*_R^7*b-21*_R^5*a+7*_R^3*a-_ R*a)*ln(tanh(1/2*x)-_R),_R=RootOf(a*_Z^16-8*a*_Z^14+28*a*_Z^12-56*a*_Z^10+ (70*a+256*b)*_Z^8-56*a*_Z^6+28*a*_Z^4-8*a*_Z^2+a))
Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 661332 vs. \(2 (165) = 330\).
Time = 3.12 (sec) , antiderivative size = 661332, normalized size of antiderivative = 2699.31 \[ \int \frac {1}{a+b \sinh ^8(x)} \, dx=\text {Too large to display} \]
\[ \int \frac {1}{a+b \sinh ^8(x)} \, dx=\int \frac {1}{a + b \sinh ^{8}{\left (x \right )}}\, dx \]
\[ \int \frac {1}{a+b \sinh ^8(x)} \, dx=\int { \frac {1}{b \sinh \left (x\right )^{8} + a} \,d x } \]
\[ \int \frac {1}{a+b \sinh ^8(x)} \, dx=\int { \frac {1}{b \sinh \left (x\right )^{8} + a} \,d x } \]
Timed out. \[ \int \frac {1}{a+b \sinh ^8(x)} \, dx=\text {Hanged} \]