Integrand size = 10, antiderivative size = 171 \[ \int \frac {1}{a+b \cosh ^6(x)} \, dx=\frac {\text {arctanh}\left (\frac {\sqrt [6]{a} \tanh (x)}{\sqrt {\sqrt [3]{a}+\sqrt [3]{b}}}\right )}{3 a^{5/6} \sqrt {\sqrt [3]{a}+\sqrt [3]{b}}}+\frac {\text {arctanh}\left (\frac {\sqrt [6]{a} \tanh (x)}{\sqrt {\sqrt [3]{a}-\sqrt [3]{-1} \sqrt [3]{b}}}\right )}{3 a^{5/6} \sqrt {\sqrt [3]{a}-\sqrt [3]{-1} \sqrt [3]{b}}}+\frac {\text {arctanh}\left (\frac {\sqrt [6]{a} \tanh (x)}{\sqrt {\sqrt [3]{a}+(-1)^{2/3} \sqrt [3]{b}}}\right )}{3 a^{5/6} \sqrt {\sqrt [3]{a}+(-1)^{2/3} \sqrt [3]{b}}} \]
1/3*arctanh(a^(1/6)*tanh(x)/(a^(1/3)+b^(1/3))^(1/2))/a^(5/6)/(a^(1/3)+b^(1 /3))^(1/2)+1/3*arctanh(a^(1/6)*tanh(x)/(a^(1/3)-(-1)^(1/3)*b^(1/3))^(1/2)) /a^(5/6)/(a^(1/3)-(-1)^(1/3)*b^(1/3))^(1/2)+1/3*arctanh(a^(1/6)*tanh(x)/(a ^(1/3)+(-1)^(2/3)*b^(1/3))^(1/2))/a^(5/6)/(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 = 5.04 (sec) , antiderivative size = 132, normalized size of antiderivative = 0.77 \[ \int \frac {1}{a+b \cosh ^6(x)} \, dx=\frac {16}{3} \text {RootSum}\left [b+6 b \text {$\#$1}+15 b \text {$\#$1}^2+64 a \text {$\#$1}^3+20 b \text {$\#$1}^3+15 b \text {$\#$1}^4+6 b \text {$\#$1}^5+b \text {$\#$1}^6\&,\frac {x \text {$\#$1}^2+\log (-\cosh (x)-\sinh (x)+\cosh (x) \text {$\#$1}-\sinh (x) \text {$\#$1}) \text {$\#$1}^2}{b+5 b \text {$\#$1}+32 a \text {$\#$1}^2+10 b \text {$\#$1}^2+10 b \text {$\#$1}^3+5 b \text {$\#$1}^4+b \text {$\#$1}^5}\&\right ] \]
(16*RootSum[b + 6*b*#1 + 15*b*#1^2 + 64*a*#1^3 + 20*b*#1^3 + 15*b*#1^4 + 6 *b*#1^5 + b*#1^6 & , (x*#1^2 + Log[-Cosh[x] - Sinh[x] + Cosh[x]*#1 - Sinh[ x]*#1]*#1^2)/(b + 5*b*#1 + 32*a*#1^2 + 10*b*#1^2 + 10*b*#1^3 + 5*b*#1^4 + b*#1^5) & ])/3
Time = 0.54 (sec) , antiderivative size = 171, normalized size of antiderivative = 1.00, 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.
| \(\displaystyle \int \frac {1}{a+b \cosh ^6(x)} \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {1}{a+b \sin \left (\frac {\pi }{2}+i x\right )^6}dx\) |
| \(\Big \downarrow \) 3690 |
| \(\displaystyle \frac {\int \frac {1}{\frac {\sqrt [3]{b} \cosh ^2(x)}{\sqrt [3]{a}}+1}dx}{3 a}+\frac {\int \frac {1}{1-\frac {\sqrt [3]{-1} \sqrt [3]{b} \cosh ^2(x)}{\sqrt [3]{a}}}dx}{3 a}+\frac {\int \frac {1}{\frac {(-1)^{2/3} \sqrt [3]{b} \cosh ^2(x)}{\sqrt [3]{a}}+1}dx}{3 a}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\int \frac {1}{\frac {\sqrt [3]{b} \sin \left (i x+\frac {\pi }{2}\right )^2}{\sqrt [3]{a}}+1}dx}{3 a}+\frac {\int \frac {1}{1-\frac {\sqrt [3]{-1} \sqrt [3]{b} \sin \left (i x+\frac {\pi }{2}\right )^2}{\sqrt [3]{a}}}dx}{3 a}+\frac {\int \frac {1}{\frac {(-1)^{2/3} \sqrt [3]{b} \sin \left (i x+\frac {\pi }{2}\right )^2}{\sqrt [3]{a}}+1}dx}{3 a}\) |
| \(\Big \downarrow \) 3660 |
| \(\displaystyle \frac {\int \frac {1}{1-\left (\frac {\sqrt [3]{b}}{\sqrt [3]{a}}+1\right ) \coth ^2(x)}d\coth (x)}{3 a}+\frac {\int \frac {1}{1-\left (1-\frac {\sqrt [3]{-1} \sqrt [3]{b}}{\sqrt [3]{a}}\right ) \coth ^2(x)}d\coth (x)}{3 a}+\frac {\int \frac {1}{1-\left (\frac {(-1)^{2/3} \sqrt [3]{b}}{\sqrt [3]{a}}+1\right ) \coth ^2(x)}d\coth (x)}{3 a}\) |
| \(\Big \downarrow \) 219 |
| \(\displaystyle \frac {\text {arctanh}\left (\frac {\sqrt {\sqrt [3]{a}+\sqrt [3]{b}} \coth (x)}{\sqrt [6]{a}}\right )}{3 a^{5/6} \sqrt {\sqrt [3]{a}+\sqrt [3]{b}}}+\frac {\text {arctanh}\left (\frac {\sqrt {\sqrt [3]{a}-\sqrt [3]{-1} \sqrt [3]{b}} \coth (x)}{\sqrt [6]{a}}\right )}{3 a^{5/6} \sqrt {\sqrt [3]{a}-\sqrt [3]{-1} \sqrt [3]{b}}}+\frac {\text {arctanh}\left (\frac {\sqrt {\sqrt [3]{a}+(-1)^{2/3} \sqrt [3]{b}} \coth (x)}{\sqrt [6]{a}}\right )}{3 a^{5/6} \sqrt {\sqrt [3]{a}+(-1)^{2/3} \sqrt [3]{b}}}\) |
ArcTanh[(Sqrt[a^(1/3) + b^(1/3)]*Coth[x])/a^(1/6)]/(3*a^(5/6)*Sqrt[a^(1/3) + b^(1/3)]) + ArcTanh[(Sqrt[a^(1/3) - (-1)^(1/3)*b^(1/3)]*Coth[x])/a^(1/6 )]/(3*a^(5/6)*Sqrt[a^(1/3) - (-1)^(1/3)*b^(1/3)]) + ArcTanh[(Sqrt[a^(1/3) + (-1)^(2/3)*b^(1/3)]*Coth[x])/a^(1/6)]/(3*a^(5/6)*Sqrt[a^(1/3) + (-1)^(2/ 3)*b^(1/3)])
3.1.65.3.1 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 = 0.94 (sec) , antiderivative size = 140, normalized size of antiderivative = 0.82
| method | result | size |
| risch | \(\munderset {\textit {\_R} =\operatorname {RootOf}\left (-1+\left (46656 a^{6}+46656 a^{5} b \right ) \textit {\_Z}^{6}-3888 a^{4} \textit {\_Z}^{4}+108 a^{2} \textit {\_Z}^{2}\right )}{\sum }\textit {\_R} \ln \left ({\mathrm e}^{2 x}+\left (-\frac {15552 a^{6}}{b}-15552 a^{5}\right ) \textit {\_R}^{5}+\left (\frac {2592 a^{5}}{b}+2592 a^{4}\right ) \textit {\_R}^{4}+\left (\frac {864 a^{4}}{b}-432 a^{3}\right ) \textit {\_R}^{3}+\left (-\frac {144 a^{3}}{b}+72 a^{2}\right ) \textit {\_R}^{2}+\left (-\frac {12 a^{2}}{b}-12 a \right ) \textit {\_R} +\frac {2 a}{b}+1\right )\) | \(140\) |
| default | \(\frac {\left (\munderset {\textit {\_R} =\operatorname {RootOf}\left (\left (a +b \right ) \textit {\_Z}^{12}+\left (-6 a +6 b \right ) \textit {\_Z}^{10}+\left (15 a +15 b \right ) \textit {\_Z}^{8}+\left (-20 a +20 b \right ) \textit {\_Z}^{6}+\left (15 a +15 b \right ) \textit {\_Z}^{4}+\left (-6 a +6 b \right ) \textit {\_Z}^{2}+a +b \right )}{\sum }\frac {\left (-\textit {\_R}^{10}+5 \textit {\_R}^{8}-10 \textit {\_R}^{6}+10 \textit {\_R}^{4}-5 \textit {\_R}^{2}+1\right ) \ln \left (\tanh \left (\frac {x}{2}\right )-\textit {\_R} \right )}{\textit {\_R}^{11} a +\textit {\_R}^{11} b -5 \textit {\_R}^{9} a +5 \textit {\_R}^{9} b +10 \textit {\_R}^{7} a +10 \textit {\_R}^{7} b -10 \textit {\_R}^{5} a +10 \textit {\_R}^{5} b +5 \textit {\_R}^{3} a +5 \textit {\_R}^{3} b -\textit {\_R} a +\textit {\_R} b}\right )}{6}\) | \(177\) |
sum(_R*ln(exp(2*x)+(-15552*a^6/b-15552*a^5)*_R^5+(2592*a^5/b+2592*a^4)*_R^ 4+(864*a^4/b-432*a^3)*_R^3+(-144/b*a^3+72*a^2)*_R^2+(-12*a^2/b-12*a)*_R+2* a/b+1),_R=RootOf(-1+(46656*a^6+46656*a^5*b)*_Z^6-3888*a^4*_Z^4+108*a^2*_Z^ 2))
Result contains complex when optimal does not.
Time = 1.21 (sec) , antiderivative size = 15201, normalized size of antiderivative = 88.89 \[ \int \frac {1}{a+b \cosh ^6(x)} \, dx=\text {Too large to display} \]
\[ \int \frac {1}{a+b \cosh ^6(x)} \, dx=\int \frac {1}{a + b \cosh ^{6}{\left (x \right )}}\, dx \]
\[ \int \frac {1}{a+b \cosh ^6(x)} \, dx=\int { \frac {1}{b \cosh \left (x\right )^{6} + a} \,d x } \]
\[ \int \frac {1}{a+b \cosh ^6(x)} \, dx=\int { \frac {1}{b \cosh \left (x\right )^{6} + a} \,d x } \]
Time = 61.87 (sec) , antiderivative size = 844, normalized size of antiderivative = 4.94 \[ \int \frac {1}{a+b \cosh ^6(x)} \, dx=\sum _{k=1}^6\ln \left (\mathrm {root}\left (46656\,a^5\,b\,d^6+46656\,a^6\,d^6-3888\,a^4\,d^4+108\,a^2\,d^2-1,d,k\right )\,\left (\mathrm {root}\left (46656\,a^5\,b\,d^6+46656\,a^6\,d^6-3888\,a^4\,d^4+108\,a^2\,d^2-1,d,k\right )\,\left (\mathrm {root}\left (46656\,a^5\,b\,d^6+46656\,a^6\,d^6-3888\,a^4\,d^4+108\,a^2\,d^2-1,d,k\right )\,\left (\mathrm {root}\left (46656\,a^5\,b\,d^6+46656\,a^6\,d^6-3888\,a^4\,d^4+108\,a^2\,d^2-1,d,k\right )\,\left (\frac {1459166279268040704\,\left (327680\,a^7\,{\mathrm {e}}^{2\,x}+298496\,a^6\,b+65536\,a^7+158\,a^2\,b^5+91315\,a^3\,b^4+348176\,a^4\,b^3+489952\,a^5\,b^2+196\,a^2\,b^5\,{\mathrm {e}}^{2\,x}+274019\,a^3\,b^4\,{\mathrm {e}}^{2\,x}+1132876\,a^4\,b^3\,{\mathrm {e}}^{2\,x}+1770440\,a^5\,b^2\,{\mathrm {e}}^{2\,x}+1239040\,a^6\,b\,{\mathrm {e}}^{2\,x}\right )}{b^{10}\,{\left (a+b\right )}^3}+\frac {\mathrm {root}\left (46656\,a^5\,b\,d^6+46656\,a^6\,d^6-3888\,a^4\,d^4+108\,a^2\,d^2-1,d,k\right )\,\left (262144\,a^7\,{\mathrm {e}}^{2\,x}+203520\,a^6\,b+65536\,a^7+453\,a^3\,b^4+72022\,a^4\,b^3+209472\,a^5\,b^2+630\,a^3\,b^4\,{\mathrm {e}}^{2\,x}+254512\,a^4\,b^3\,{\mathrm {e}}^{2\,x}+767508\,a^5\,b^2\,{\mathrm {e}}^{2\,x}+775680\,a^6\,b\,{\mathrm {e}}^{2\,x}\right )\,17509995351216488448}{b^{10}\,{\left (a+b\right )}^2}\right )-\frac {486388759756013568\,\left (655360\,a^5\,{\mathrm {e}}^{2\,x}-9\,a\,b^4+370176\,a^4\,b+196608\,a^5-24408\,a^2\,b^3+149088\,a^3\,b^2-63676\,a^2\,b^3\,{\mathrm {e}}^{2\,x}+526248\,a^3\,b^2\,{\mathrm {e}}^{2\,x}-10\,a\,b^4\,{\mathrm {e}}^{2\,x}+1245184\,a^4\,b\,{\mathrm {e}}^{2\,x}\right )}{b^{10}\,{\left (a+b\right )}^2}\right )-\frac {40532396646334464\,\left (655360\,a^5\,{\mathrm {e}}^{2\,x}-b^5\,{\mathrm {e}}^{2\,x}-24677\,a\,b^4+773120\,a^4\,b+262144\,a^5-b^5+198071\,a^2\,b^3+733696\,a^3\,b^2+477713\,a^2\,b^3\,{\mathrm {e}}^{2\,x}+1770640\,a^3\,b^2\,{\mathrm {e}}^{2\,x}-53861\,a\,b^4\,{\mathrm {e}}^{2\,x}+1894400\,a^4\,b\,{\mathrm {e}}^{2\,x}\right )}{b^{10}\,{\left (a+b\right )}^3}\right )+\frac {13510798882111488\,\left (655360\,a^3\,{\mathrm {e}}^{2\,x}+11382\,b^3\,{\mathrm {e}}^{2\,x}+144416\,a\,b^2+269056\,a^2\,b+131072\,a^3+6459\,b^3+677524\,a\,b^2\,{\mathrm {e}}^{2\,x}+1321472\,a^2\,b\,{\mathrm {e}}^{2\,x}\right )}{b^{10}\,{\left (a+b\right )}^2}\right )+\frac {1125899906842624\,\left (851968\,a^4\,{\mathrm {e}}^{2\,x}+6006\,b^4\,{\mathrm {e}}^{2\,x}+211497\,a\,b^3+597504\,a^3\,b+196608\,a^4+3840\,b^4+608544\,a^2\,b^2+2562504\,a^2\,b^2\,{\mathrm {e}}^{2\,x}+864565\,a\,b^3\,{\mathrm {e}}^{2\,x}+2555904\,a^3\,b\,{\mathrm {e}}^{2\,x}\right )}{b^{10}\,{\left (a+b\right )}^2\,\left (a^2+b\,a\right )}\right )\,\mathrm {root}\left (46656\,a^5\,b\,d^6+46656\,a^6\,d^6-3888\,a^4\,d^4+108\,a^2\,d^2-1,d,k\right ) \]
symsum(log(root(46656*a^5*b*d^6 + 46656*a^6*d^6 - 3888*a^4*d^4 + 108*a^2*d ^2 - 1, d, k)*(root(46656*a^5*b*d^6 + 46656*a^6*d^6 - 3888*a^4*d^4 + 108*a ^2*d^2 - 1, d, k)*(root(46656*a^5*b*d^6 + 46656*a^6*d^6 - 3888*a^4*d^4 + 1 08*a^2*d^2 - 1, d, k)*(root(46656*a^5*b*d^6 + 46656*a^6*d^6 - 3888*a^4*d^4 + 108*a^2*d^2 - 1, d, k)*((1459166279268040704*(327680*a^7*exp(2*x) + 298 496*a^6*b + 65536*a^7 + 158*a^2*b^5 + 91315*a^3*b^4 + 348176*a^4*b^3 + 489 952*a^5*b^2 + 196*a^2*b^5*exp(2*x) + 274019*a^3*b^4*exp(2*x) + 1132876*a^4 *b^3*exp(2*x) + 1770440*a^5*b^2*exp(2*x) + 1239040*a^6*b*exp(2*x)))/(b^10* (a + b)^3) + (17509995351216488448*root(46656*a^5*b*d^6 + 46656*a^6*d^6 - 3888*a^4*d^4 + 108*a^2*d^2 - 1, d, k)*(262144*a^7*exp(2*x) + 203520*a^6*b + 65536*a^7 + 453*a^3*b^4 + 72022*a^4*b^3 + 209472*a^5*b^2 + 630*a^3*b^4*e xp(2*x) + 254512*a^4*b^3*exp(2*x) + 767508*a^5*b^2*exp(2*x) + 775680*a^6*b *exp(2*x)))/(b^10*(a + b)^2)) - (486388759756013568*(655360*a^5*exp(2*x) - 9*a*b^4 + 370176*a^4*b + 196608*a^5 - 24408*a^2*b^3 + 149088*a^3*b^2 - 63 676*a^2*b^3*exp(2*x) + 526248*a^3*b^2*exp(2*x) - 10*a*b^4*exp(2*x) + 12451 84*a^4*b*exp(2*x)))/(b^10*(a + b)^2)) - (40532396646334464*(655360*a^5*exp (2*x) - b^5*exp(2*x) - 24677*a*b^4 + 773120*a^4*b + 262144*a^5 - b^5 + 198 071*a^2*b^3 + 733696*a^3*b^2 + 477713*a^2*b^3*exp(2*x) + 1770640*a^3*b^2*e xp(2*x) - 53861*a*b^4*exp(2*x) + 1894400*a^4*b*exp(2*x)))/(b^10*(a + b)^3) ) + (13510798882111488*(655360*a^3*exp(2*x) + 11382*b^3*exp(2*x) + 1444...