Integrand size = 8, antiderivative size = 129 \[ \int \frac {1}{1+\cos ^8(x)} \, dx=-\frac {\arctan \left (\sqrt {1-\sqrt [4]{-1}} \cot (x)\right )}{4 \sqrt {1-\sqrt [4]{-1}}}-\frac {\arctan \left (\sqrt {1+\sqrt [4]{-1}} \cot (x)\right )}{4 \sqrt {1+\sqrt [4]{-1}}}-\frac {\arctan \left (\sqrt {1-(-1)^{3/4}} \cot (x)\right )}{4 \sqrt {1-(-1)^{3/4}}}-\frac {\arctan \left (\sqrt {1+(-1)^{3/4}} \cot (x)\right )}{4 \sqrt {1+(-1)^{3/4}}} \]
-1/4*arctan(cot(x)*(1-(-1)^(1/4))^(1/2))/(1-(-1)^(1/4))^(1/2)-1/4*arctan(c ot(x)*(1+(-1)^(1/4))^(1/2))/(1+(-1)^(1/4))^(1/2)-1/4*arctan(cot(x)*(1-(-1) ^(3/4))^(1/2))/(1-(-1)^(3/4))^(1/2)-1/4*arctan(cot(x)*(1+(-1)^(3/4))^(1/2) )/(1+(-1)^(3/4))^(1/2)
Result contains higher order function than in optimal. Order 9 vs. order 3 in optimal.
Time = 5.04 (sec) , antiderivative size = 141, normalized size of antiderivative = 1.09 \[ \int \frac {1}{1+\cos ^8(x)} \, dx=8 \text {RootSum}\left [1+8 \text {$\#$1}+28 \text {$\#$1}^2+56 \text {$\#$1}^3+326 \text {$\#$1}^4+56 \text {$\#$1}^5+28 \text {$\#$1}^6+8 \text {$\#$1}^7+\text {$\#$1}^8\&,\frac {2 \arctan \left (\frac {\sin (2 x)}{\cos (2 x)-\text {$\#$1}}\right ) \text {$\#$1}^3-i \log \left (1-2 \cos (2 x) \text {$\#$1}+\text {$\#$1}^2\right ) \text {$\#$1}^3}{1+7 \text {$\#$1}+21 \text {$\#$1}^2+163 \text {$\#$1}^3+35 \text {$\#$1}^4+21 \text {$\#$1}^5+7 \text {$\#$1}^6+\text {$\#$1}^7}\&\right ] \]
8*RootSum[1 + 8*#1 + 28*#1^2 + 56*#1^3 + 326*#1^4 + 56*#1^5 + 28*#1^6 + 8* #1^7 + #1^8 & , (2*ArcTan[Sin[2*x]/(Cos[2*x] - #1)]*#1^3 - I*Log[1 - 2*Cos [2*x]*#1 + #1^2]*#1^3)/(1 + 7*#1 + 21*#1^2 + 163*#1^3 + 35*#1^4 + 21*#1^5 + 7*#1^6 + #1^7) & ]
Time = 0.42 (sec) , antiderivative size = 129, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.625, Rules used = {3042, 3690, 3042, 3660, 216}
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.
\(\displaystyle \int \frac {1}{\cos ^8(x)+1} \, dx\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \int \frac {1}{\sin \left (x+\frac {\pi }{2}\right )^8+1}dx\) |
\(\Big \downarrow \) 3690 |
\(\displaystyle \frac {1}{4} \int \frac {1}{1-\sqrt [4]{-1} \cos ^2(x)}dx+\frac {1}{4} \int \frac {1}{\sqrt [4]{-1} \cos ^2(x)+1}dx+\frac {1}{4} \int \frac {1}{1-(-1)^{3/4} \cos ^2(x)}dx+\frac {1}{4} \int \frac {1}{(-1)^{3/4} \cos ^2(x)+1}dx\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \frac {1}{4} \int \frac {1}{1-\sqrt [4]{-1} \sin \left (x+\frac {\pi }{2}\right )^2}dx+\frac {1}{4} \int \frac {1}{\sqrt [4]{-1} \sin \left (x+\frac {\pi }{2}\right )^2+1}dx+\frac {1}{4} \int \frac {1}{1-(-1)^{3/4} \sin \left (x+\frac {\pi }{2}\right )^2}dx+\frac {1}{4} \int \frac {1}{(-1)^{3/4} \sin \left (x+\frac {\pi }{2}\right )^2+1}dx\) |
\(\Big \downarrow \) 3660 |
\(\displaystyle -\frac {1}{4} \int \frac {1}{\left (1-\sqrt [4]{-1}\right ) \cot ^2(x)+1}d\cot (x)-\frac {1}{4} \int \frac {1}{\left (1+\sqrt [4]{-1}\right ) \cot ^2(x)+1}d\cot (x)-\frac {1}{4} \int \frac {1}{\left (1-(-1)^{3/4}\right ) \cot ^2(x)+1}d\cot (x)-\frac {1}{4} \int \frac {1}{\left (1+(-1)^{3/4}\right ) \cot ^2(x)+1}d\cot (x)\) |
\(\Big \downarrow \) 216 |
\(\displaystyle -\frac {\arctan \left (\sqrt {1-\sqrt [4]{-1}} \cot (x)\right )}{4 \sqrt {1-\sqrt [4]{-1}}}-\frac {\arctan \left (\sqrt {1+\sqrt [4]{-1}} \cot (x)\right )}{4 \sqrt {1+\sqrt [4]{-1}}}-\frac {\arctan \left (\sqrt {1-(-1)^{3/4}} \cot (x)\right )}{4 \sqrt {1-(-1)^{3/4}}}-\frac {\arctan \left (\sqrt {1+(-1)^{3/4}} \cot (x)\right )}{4 \sqrt {1+(-1)^{3/4}}}\) |
-1/4*ArcTan[Sqrt[1 - (-1)^(1/4)]*Cot[x]]/Sqrt[1 - (-1)^(1/4)] - ArcTan[Sqr t[1 + (-1)^(1/4)]*Cot[x]]/(4*Sqrt[1 + (-1)^(1/4)]) - ArcTan[Sqrt[1 - (-1)^ (3/4)]*Cot[x]]/(4*Sqrt[1 - (-1)^(3/4)]) - ArcTan[Sqrt[1 + (-1)^(3/4)]*Cot[ x]]/(4*Sqrt[1 + (-1)^(3/4)])
3.1.82.3.1 Defintions of rubi rules used
Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1/(Rt[a, 2]*Rt[b, 2]))*A rcTan[Rt[b, 2]*(x/Rt[a, 2])], x] /; FreeQ[{a, b}, x] && PosQ[a/b] && (GtQ[a , 0] || GtQ[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 = 3.13 (sec) , antiderivative size = 67, normalized size of antiderivative = 0.52
method | result | size |
default | \(\frac {\left (\munderset {\textit {\_R} =\operatorname {RootOf}\left (\textit {\_Z}^{8}+4 \textit {\_Z}^{6}+6 \textit {\_Z}^{4}+4 \textit {\_Z}^{2}+2\right )}{\sum }\frac {\left (\textit {\_R}^{6}+3 \textit {\_R}^{4}+3 \textit {\_R}^{2}+1\right ) \ln \left (\tan \left (x \right )-\textit {\_R} \right )}{\textit {\_R}^{7}+3 \textit {\_R}^{5}+3 \textit {\_R}^{3}+\textit {\_R}}\right )}{8}\) | \(67\) |
risch | \(\left (\munderset {\textit {\_R} =\operatorname {RootOf}\left (8192 \textit {\_Z}^{4}+\left (128-128 i\right ) \textit {\_Z}^{2}+1-i\right )}{\sum }\textit {\_R} \ln \left ({\mathrm e}^{2 i x}+\left (1024+1024 i\right ) \textit {\_R}^{3}+\left (-128+128 i\right ) \textit {\_R}^{2}+\left (16-16 i\right ) \textit {\_R} +1+2 i\right )\right )+\left (\munderset {\textit {\_R} =\operatorname {RootOf}\left (8192 \textit {\_Z}^{4}+\left (128+128 i\right ) \textit {\_Z}^{2}+1+i\right )}{\sum }\textit {\_R} \ln \left ({\mathrm e}^{2 i x}+\left (-1024+1024 i\right ) \textit {\_R}^{3}+\left (-128-128 i\right ) \textit {\_R}^{2}+\left (-16-16 i\right ) \textit {\_R} +1-2 i\right )\right )\) | \(104\) |
1/8*sum((_R^6+3*_R^4+3*_R^2+1)/(_R^7+3*_R^5+3*_R^3+_R)*ln(tan(x)-_R),_R=Ro otOf(_Z^8+4*_Z^6+6*_Z^4+4*_Z^2+2))
Leaf count of result is larger than twice the leaf count of optimal. 885 vs. \(2 (89) = 178\).
Time = 0.38 (sec) , antiderivative size = 885, normalized size of antiderivative = 6.86 \[ \int \frac {1}{1+\cos ^8(x)} \, dx=\text {Too large to display} \]
-1/32*sqrt(2)*sqrt(-sqrt(2*sqrt(2) - 3) - 1)*log(2*(sqrt(2) + 1)*cos(x)^2 + (2*(sqrt(2) + 2)*cos(x)^2 - sqrt(2) - 2)*sqrt(2*sqrt(2) - 3) + 2*(sqrt(2 *sqrt(2) - 3)*(sqrt(2) + 1)*cos(x)*sin(x) + (sqrt(2) + 1)*cos(x)*sin(x))*s qrt(-sqrt(2*sqrt(2) - 3) - 1) - sqrt(2)) + 1/32*sqrt(2)*sqrt(-sqrt(2*sqrt( 2) - 3) - 1)*log(2*(sqrt(2) + 1)*cos(x)^2 + (2*(sqrt(2) + 2)*cos(x)^2 - sq rt(2) - 2)*sqrt(2*sqrt(2) - 3) - 2*(sqrt(2*sqrt(2) - 3)*(sqrt(2) + 1)*cos( x)*sin(x) + (sqrt(2) + 1)*cos(x)*sin(x))*sqrt(-sqrt(2*sqrt(2) - 3) - 1) - sqrt(2)) - 1/32*sqrt(2)*sqrt(sqrt(2*sqrt(2) - 3) - 1)*log(-2*(sqrt(2) + 1) *cos(x)^2 + (2*(sqrt(2) + 2)*cos(x)^2 - sqrt(2) - 2)*sqrt(2*sqrt(2) - 3) + 2*(sqrt(2*sqrt(2) - 3)*(sqrt(2) + 1)*cos(x)*sin(x) - (sqrt(2) + 1)*cos(x) *sin(x))*sqrt(sqrt(2*sqrt(2) - 3) - 1) + sqrt(2)) + 1/32*sqrt(2)*sqrt(sqrt (2*sqrt(2) - 3) - 1)*log(-2*(sqrt(2) + 1)*cos(x)^2 + (2*(sqrt(2) + 2)*cos( x)^2 - sqrt(2) - 2)*sqrt(2*sqrt(2) - 3) - 2*(sqrt(2*sqrt(2) - 3)*(sqrt(2) + 1)*cos(x)*sin(x) - (sqrt(2) + 1)*cos(x)*sin(x))*sqrt(sqrt(2*sqrt(2) - 3) - 1) + sqrt(2)) + 1/32*sqrt(2)*sqrt(-sqrt(-2*sqrt(2) - 3) - 1)*log(2*(sqr t(2) - 1)*cos(x)^2 + (2*(sqrt(2) - 2)*cos(x)^2 - sqrt(2) + 2)*sqrt(-2*sqrt (2) - 3) + 2*((sqrt(2) - 1)*sqrt(-2*sqrt(2) - 3)*cos(x)*sin(x) + (sqrt(2) - 1)*cos(x)*sin(x))*sqrt(-sqrt(-2*sqrt(2) - 3) - 1) - sqrt(2)) - 1/32*sqrt (2)*sqrt(-sqrt(-2*sqrt(2) - 3) - 1)*log(2*(sqrt(2) - 1)*cos(x)^2 + (2*(sqr t(2) - 2)*cos(x)^2 - sqrt(2) + 2)*sqrt(-2*sqrt(2) - 3) - 2*((sqrt(2) - ...
Timed out. \[ \int \frac {1}{1+\cos ^8(x)} \, dx=\text {Timed out} \]
\[ \int \frac {1}{1+\cos ^8(x)} \, dx=\int { \frac {1}{\cos \left (x\right )^{8} + 1} \,d x } \]
\[ \int \frac {1}{1+\cos ^8(x)} \, dx=\int { \frac {1}{\cos \left (x\right )^{8} + 1} \,d x } \]
Time = 3.38 (sec) , antiderivative size = 1025, normalized size of antiderivative = 7.95 \[ \int \frac {1}{1+\cos ^8(x)} \, dx=\text {Too large to display} \]
atan((tan(x)*((2*2^(1/2) - 3)^(1/2)/128 - 1/128)^(1/2)*8i)/((2^(1/2)*(2*2^ (1/2) - 3)^(1/2))/2 - 2^(1/2)/2 - (2*2^(1/2) - 3)^(1/2) + 1) - (2^(1/2)*ta n(x)*((2*2^(1/2) - 3)^(1/2)/128 - 1/128)^(1/2)*4i)/((2^(1/2)*(2*2^(1/2) - 3)^(1/2))/2 - 2^(1/2)/2 - (2*2^(1/2) - 3)^(1/2) + 1) - (tan(x)*(2*2^(1/2) - 3)^(1/2)*((2*2^(1/2) - 3)^(1/2)/128 - 1/128)^(1/2)*8i)/((2^(1/2)*(2*2^(1 /2) - 3)^(1/2))/2 - 2^(1/2)/2 - (2*2^(1/2) - 3)^(1/2) + 1) + (2^(1/2)*tan( x)*(2*2^(1/2) - 3)^(1/2)*((2*2^(1/2) - 3)^(1/2)/128 - 1/128)^(1/2)*4i)/((2 ^(1/2)*(2*2^(1/2) - 3)^(1/2))/2 - 2^(1/2)/2 - (2*2^(1/2) - 3)^(1/2) + 1))* ((2*2^(1/2) - 3)^(1/2)/128 - 1/128)^(1/2)*2i - atan((tan(x)*(- (2*2^(1/2) - 3)^(1/2)/128 - 1/128)^(1/2)*8i)/((2^(1/2)*(2*2^(1/2) - 3)^(1/2))/2 + 2^( 1/2)/2 - (2*2^(1/2) - 3)^(1/2) - 1) - (2^(1/2)*tan(x)*(- (2*2^(1/2) - 3)^( 1/2)/128 - 1/128)^(1/2)*4i)/((2^(1/2)*(2*2^(1/2) - 3)^(1/2))/2 + 2^(1/2)/2 - (2*2^(1/2) - 3)^(1/2) - 1) + (tan(x)*(2*2^(1/2) - 3)^(1/2)*(- (2*2^(1/2 ) - 3)^(1/2)/128 - 1/128)^(1/2)*8i)/((2^(1/2)*(2*2^(1/2) - 3)^(1/2))/2 + 2 ^(1/2)/2 - (2*2^(1/2) - 3)^(1/2) - 1) - (2^(1/2)*tan(x)*(2*2^(1/2) - 3)^(1 /2)*(- (2*2^(1/2) - 3)^(1/2)/128 - 1/128)^(1/2)*4i)/((2^(1/2)*(2*2^(1/2) - 3)^(1/2))/2 + 2^(1/2)/2 - (2*2^(1/2) - 3)^(1/2) - 1))*(- (2*2^(1/2) - 3)^ (1/2)/128 - 1/128)^(1/2)*2i + atan((tan(x)*(- (- 2*2^(1/2) - 3)^(1/2)/128 - 1/128)^(1/2)*8i)/((2^(1/2)*(- 2*2^(1/2) - 3)^(1/2))/2 + 2^(1/2)/2 + (- 2 *2^(1/2) - 3)^(1/2) + 1) + (2^(1/2)*tan(x)*(- (- 2*2^(1/2) - 3)^(1/2)/1...