Integrand size = 11, antiderivative size = 288 \[ \int \frac {1}{a-b \cosh ^3(x)} \, dx=\frac {2 \text {arctanh}\left (\frac {\sqrt {\sqrt [3]{a}+\sqrt [3]{b}} \tanh \left (\frac {x}{2}\right )}{\sqrt {\sqrt [3]{a}-\sqrt [3]{b}}}\right )}{3 a^{2/3} \sqrt {\sqrt [3]{a}-\sqrt [3]{b}} \sqrt {\sqrt [3]{a}+\sqrt [3]{b}}}+\frac {2 \text {arctanh}\left (\frac {\sqrt {\sqrt [3]{a}-\sqrt [3]{-1} \sqrt [3]{b}} \tanh \left (\frac {x}{2}\right )}{\sqrt {\sqrt [3]{a}+\sqrt [3]{-1} \sqrt [3]{b}}}\right )}{3 a^{2/3} \sqrt {\sqrt [3]{a}-\sqrt [3]{-1} \sqrt [3]{b}} \sqrt {\sqrt [3]{a}+\sqrt [3]{-1} \sqrt [3]{b}}}+\frac {2 \text {arctanh}\left (\frac {\sqrt {\sqrt [3]{a}+(-1)^{2/3} \sqrt [3]{b}} \tanh \left (\frac {x}{2}\right )}{\sqrt {\sqrt [3]{a}-(-1)^{2/3} \sqrt [3]{b}}}\right )}{3 a^{2/3} \sqrt {\sqrt [3]{a}-(-1)^{2/3} \sqrt [3]{b}} \sqrt {\sqrt [3]{a}+(-1)^{2/3} \sqrt [3]{b}}} \]
2/3*arctanh((a^(1/3)+b^(1/3))^(1/2)*tanh(1/2*x)/(a^(1/3)-b^(1/3))^(1/2))/a ^(2/3)/(a^(1/3)-b^(1/3))^(1/2)/(a^(1/3)+b^(1/3))^(1/2)+2/3*arctanh((a^(1/3 )-(-1)^(1/3)*b^(1/3))^(1/2)*tanh(1/2*x)/(a^(1/3)+(-1)^(1/3)*b^(1/3))^(1/2) )/a^(2/3)/(a^(1/3)-(-1)^(1/3)*b^(1/3))^(1/2)/(a^(1/3)+(-1)^(1/3)*b^(1/3))^ (1/2)+2/3*arctanh((a^(1/3)+(-1)^(2/3)*b^(1/3))^(1/2)*tanh(1/2*x)/(a^(1/3)- (-1)^(2/3)*b^(1/3))^(1/2))/a^(2/3)/(a^(1/3)-(-1)^(2/3)*b^(1/3))^(1/2)/(a^( 1/3)+(-1)^(2/3)*b^(1/3))^(1/2)
Result contains higher order function than in optimal. Order 9 vs. order 3 in optimal.
Time = 5.06 (sec) , antiderivative size = 105, normalized size of antiderivative = 0.36 \[ \int \frac {1}{a-b \cosh ^3(x)} \, dx=-\frac {2}{3} \text {RootSum}\left [b+3 b \text {$\#$1}^2-8 a \text {$\#$1}^3+3 b \text {$\#$1}^4+b \text {$\#$1}^6\&,\frac {x \text {$\#$1}+2 \log \left (-\cosh \left (\frac {x}{2}\right )-\sinh \left (\frac {x}{2}\right )+\cosh \left (\frac {x}{2}\right ) \text {$\#$1}-\sinh \left (\frac {x}{2}\right ) \text {$\#$1}\right ) \text {$\#$1}}{b-4 a \text {$\#$1}+2 b \text {$\#$1}^2+b \text {$\#$1}^4}\&\right ] \]
(-2*RootSum[b + 3*b*#1^2 - 8*a*#1^3 + 3*b*#1^4 + b*#1^6 & , (x*#1 + 2*Log[ -Cosh[x/2] - Sinh[x/2] + Cosh[x/2]*#1 - Sinh[x/2]*#1]*#1)/(b - 4*a*#1 + 2* b*#1^2 + b*#1^4) & ])/3
Time = 0.52 (sec) , antiderivative size = 288, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.273, 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.
\(\displaystyle \int \frac {1}{a-b \cosh ^3(x)} \, dx\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \int \frac {1}{a-b \sin \left (\frac {\pi }{2}+i x\right )^3}dx\) |
\(\Big \downarrow \) 3692 |
\(\displaystyle \int \left (\frac {1}{3 a^{2/3} \left (\sqrt [3]{a}-\sqrt [3]{b} \cosh (x)\right )}+\frac {1}{3 a^{2/3} \left (\sqrt [3]{a}+\sqrt [3]{-1} \sqrt [3]{b} \cosh (x)\right )}+\frac {1}{3 a^{2/3} \left (\sqrt [3]{a}-(-1)^{2/3} \sqrt [3]{b} \cosh (x)\right )}\right )dx\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle \frac {2 \text {arctanh}\left (\frac {\sqrt {\sqrt [3]{a}+\sqrt [3]{b}} \tanh \left (\frac {x}{2}\right )}{\sqrt {\sqrt [3]{a}-\sqrt [3]{b}}}\right )}{3 a^{2/3} \sqrt {\sqrt [3]{a}-\sqrt [3]{b}} \sqrt {\sqrt [3]{a}+\sqrt [3]{b}}}+\frac {2 \text {arctanh}\left (\frac {\sqrt {\sqrt [3]{a}-\sqrt [3]{-1} \sqrt [3]{b}} \tanh \left (\frac {x}{2}\right )}{\sqrt {\sqrt [3]{a}+\sqrt [3]{-1} \sqrt [3]{b}}}\right )}{3 a^{2/3} \sqrt {\sqrt [3]{a}-\sqrt [3]{-1} \sqrt [3]{b}} \sqrt {\sqrt [3]{a}+\sqrt [3]{-1} \sqrt [3]{b}}}+\frac {2 \text {arctanh}\left (\frac {\sqrt {\sqrt [3]{a}+(-1)^{2/3} \sqrt [3]{b}} \tanh \left (\frac {x}{2}\right )}{\sqrt {\sqrt [3]{a}-(-1)^{2/3} \sqrt [3]{b}}}\right )}{3 a^{2/3} \sqrt {\sqrt [3]{a}-(-1)^{2/3} \sqrt [3]{b}} \sqrt {\sqrt [3]{a}+(-1)^{2/3} \sqrt [3]{b}}}\) |
(2*ArcTanh[(Sqrt[a^(1/3) + b^(1/3)]*Tanh[x/2])/Sqrt[a^(1/3) - b^(1/3)]])/( 3*a^(2/3)*Sqrt[a^(1/3) - b^(1/3)]*Sqrt[a^(1/3) + b^(1/3)]) + (2*ArcTanh[(S qrt[a^(1/3) - (-1)^(1/3)*b^(1/3)]*Tanh[x/2])/Sqrt[a^(1/3) + (-1)^(1/3)*b^( 1/3)]])/(3*a^(2/3)*Sqrt[a^(1/3) - (-1)^(1/3)*b^(1/3)]*Sqrt[a^(1/3) + (-1)^ (1/3)*b^(1/3)]) + (2*ArcTanh[(Sqrt[a^(1/3) + (-1)^(2/3)*b^(1/3)]*Tanh[x/2] )/Sqrt[a^(1/3) - (-1)^(2/3)*b^(1/3)]])/(3*a^(2/3)*Sqrt[a^(1/3) - (-1)^(2/3 )*b^(1/3)]*Sqrt[a^(1/3) + (-1)^(2/3)*b^(1/3)])
3.1.57.3.1 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 = 0.53 (sec) , antiderivative size = 94, normalized size of antiderivative = 0.33
method | result | size |
default | \(\frac {\left (\munderset {\textit {\_R} =\operatorname {RootOf}\left (\left (a +b \right ) \textit {\_Z}^{6}+\left (-3 a +3 b \right ) \textit {\_Z}^{4}+\left (3 a +3 b \right ) \textit {\_Z}^{2}-a +b \right )}{\sum }\frac {\left (-\textit {\_R}^{4}+2 \textit {\_R}^{2}-1\right ) \ln \left (\tanh \left (\frac {x}{2}\right )-\textit {\_R} \right )}{\textit {\_R}^{5} a +\textit {\_R}^{5} b -2 \textit {\_R}^{3} a +2 \textit {\_R}^{3} b +\textit {\_R} a +\textit {\_R} b}\right )}{3}\) | \(94\) |
risch | \(\munderset {\textit {\_R} =\operatorname {RootOf}\left (-1+\left (729 a^{6}-729 a^{4} b^{2}\right ) \textit {\_Z}^{6}-243 a^{4} \textit {\_Z}^{4}+27 a^{2} \textit {\_Z}^{2}\right )}{\sum }\textit {\_R} \ln \left ({\mathrm e}^{x}+\left (-\frac {486 a^{6}}{b}+486 a^{4} b \right ) \textit {\_R}^{5}+\left (\frac {81 a^{5}}{b}-81 a^{3} b \right ) \textit {\_R}^{4}+\left (\frac {135 a^{4}}{b}+27 a^{2} b \right ) \textit {\_R}^{3}-\frac {27 a^{3} \textit {\_R}^{2}}{b}-\frac {9 a^{2} \textit {\_R}}{b}+\frac {2 a}{b}\right )\) | \(130\) |
1/3*sum((-_R^4+2*_R^2-1)/(_R^5*a+_R^5*b-2*_R^3*a+2*_R^3*b+_R*a+_R*b)*ln(ta nh(1/2*x)-_R),_R=RootOf((a+b)*_Z^6+(-3*a+3*b)*_Z^4+(3*a+3*b)*_Z^2-a+b))
Result contains complex when optimal does not.
Time = 0.98 (sec) , antiderivative size = 18612, normalized size of antiderivative = 64.62 \[ \int \frac {1}{a-b \cosh ^3(x)} \, dx=\text {Too large to display} \]
Timed out. \[ \int \frac {1}{a-b \cosh ^3(x)} \, dx=\text {Timed out} \]
\[ \int \frac {1}{a-b \cosh ^3(x)} \, dx=\int { -\frac {1}{b \cosh \left (x\right )^{3} - a} \,d x } \]
\[ \int \frac {1}{a-b \cosh ^3(x)} \, dx=\int { -\frac {1}{b \cosh \left (x\right )^{3} - a} \,d x } \]
Time = 6.81 (sec) , antiderivative size = 633, normalized size of antiderivative = 2.20 \[ \int \frac {1}{a-b \cosh ^3(x)} \, dx=\sum _{k=1}^6\ln \left (-\frac {\left (4\,{\mathrm {e}}^x+\mathrm {root}\left (729\,a^4\,b^2\,d^6-729\,a^6\,d^6+243\,a^4\,d^4-27\,a^2\,d^2+1,d,k\right )\,b+{\mathrm {root}\left (729\,a^4\,b^2\,d^6-729\,a^6\,d^6+243\,a^4\,d^4-27\,a^2\,d^2+1,d,k\right )}^3\,a^2\,b\,54+{\mathrm {root}\left (729\,a^4\,b^2\,d^6-729\,a^6\,d^6+243\,a^4\,d^4-27\,a^2\,d^2+1,d,k\right )}^4\,a^3\,b\,108+{\mathrm {root}\left (729\,a^4\,b^2\,d^6-729\,a^6\,d^6+243\,a^4\,d^4-27\,a^2\,d^2+1,d,k\right )}^5\,a^4\,b\,81-{\mathrm {root}\left (729\,a^4\,b^2\,d^6-729\,a^6\,d^6+243\,a^4\,d^4-27\,a^2\,d^2+1,d,k\right )}^2\,a^2\,{\mathrm {e}}^x\,24-{\mathrm {root}\left (729\,a^4\,b^2\,d^6-729\,a^6\,d^6+243\,a^4\,d^4-27\,a^2\,d^2+1,d,k\right )}^3\,a^3\,{\mathrm {e}}^x\,216-{\mathrm {root}\left (729\,a^4\,b^2\,d^6-729\,a^6\,d^6+243\,a^4\,d^4-27\,a^2\,d^2+1,d,k\right )}^4\,a^4\,{\mathrm {e}}^x\,108+{\mathrm {root}\left (729\,a^4\,b^2\,d^6-729\,a^6\,d^6+243\,a^4\,d^4-27\,a^2\,d^2+1,d,k\right )}^5\,a^5\,{\mathrm {e}}^x\,324+{\mathrm {root}\left (729\,a^4\,b^2\,d^6-729\,a^6\,d^6+243\,a^4\,d^4-27\,a^2\,d^2+1,d,k\right )}^2\,a\,b\,12+\mathrm {root}\left (729\,a^4\,b^2\,d^6-729\,a^6\,d^6+243\,a^4\,d^4-27\,a^2\,d^2+1,d,k\right )\,a\,{\mathrm {e}}^x\,20-{\mathrm {root}\left (729\,a^4\,b^2\,d^6-729\,a^6\,d^6+243\,a^4\,d^4-27\,a^2\,d^2+1,d,k\right )}^4\,a^2\,b^2\,{\mathrm {e}}^x\,27-{\mathrm {root}\left (729\,a^4\,b^2\,d^6-729\,a^6\,d^6+243\,a^4\,d^4-27\,a^2\,d^2+1,d,k\right )}^5\,a^3\,b^2\,{\mathrm {e}}^x\,405\right )\,24576}{b^5}\right )\,\mathrm {root}\left (729\,a^4\,b^2\,d^6-729\,a^6\,d^6+243\,a^4\,d^4-27\,a^2\,d^2+1,d,k\right ) \]
symsum(log(-(24576*(4*exp(x) + root(729*a^4*b^2*d^6 - 729*a^6*d^6 + 243*a^ 4*d^4 - 27*a^2*d^2 + 1, d, k)*b + 54*root(729*a^4*b^2*d^6 - 729*a^6*d^6 + 243*a^4*d^4 - 27*a^2*d^2 + 1, d, k)^3*a^2*b + 108*root(729*a^4*b^2*d^6 - 7 29*a^6*d^6 + 243*a^4*d^4 - 27*a^2*d^2 + 1, d, k)^4*a^3*b + 81*root(729*a^4 *b^2*d^6 - 729*a^6*d^6 + 243*a^4*d^4 - 27*a^2*d^2 + 1, d, k)^5*a^4*b - 24* root(729*a^4*b^2*d^6 - 729*a^6*d^6 + 243*a^4*d^4 - 27*a^2*d^2 + 1, d, k)^2 *a^2*exp(x) - 216*root(729*a^4*b^2*d^6 - 729*a^6*d^6 + 243*a^4*d^4 - 27*a^ 2*d^2 + 1, d, k)^3*a^3*exp(x) - 108*root(729*a^4*b^2*d^6 - 729*a^6*d^6 + 2 43*a^4*d^4 - 27*a^2*d^2 + 1, d, k)^4*a^4*exp(x) + 324*root(729*a^4*b^2*d^6 - 729*a^6*d^6 + 243*a^4*d^4 - 27*a^2*d^2 + 1, d, k)^5*a^5*exp(x) + 12*roo t(729*a^4*b^2*d^6 - 729*a^6*d^6 + 243*a^4*d^4 - 27*a^2*d^2 + 1, d, k)^2*a* b + 20*root(729*a^4*b^2*d^6 - 729*a^6*d^6 + 243*a^4*d^4 - 27*a^2*d^2 + 1, d, k)*a*exp(x) - 27*root(729*a^4*b^2*d^6 - 729*a^6*d^6 + 243*a^4*d^4 - 27* a^2*d^2 + 1, d, k)^4*a^2*b^2*exp(x) - 405*root(729*a^4*b^2*d^6 - 729*a^6*d ^6 + 243*a^4*d^4 - 27*a^2*d^2 + 1, d, k)^5*a^3*b^2*exp(x)))/b^5)*root(729* a^4*b^2*d^6 - 729*a^6*d^6 + 243*a^4*d^4 - 27*a^2*d^2 + 1, d, k), k, 1, 6)