Integrand size = 8, antiderivative size = 83 \[ \int \frac {1}{1+\cos ^6(x)} \, dx=\frac {\arctan \left (\frac {\tan (x)}{\sqrt {2}}\right )}{3 \sqrt {2}}+\frac {\arctan \left (\frac {\tan (x)}{\sqrt {1-\sqrt [3]{-1}}}\right )}{3 \sqrt {1-\sqrt [3]{-1}}}+\frac {\arctan \left (\frac {\tan (x)}{\sqrt {1+(-1)^{2/3}}}\right )}{3 \sqrt {1+(-1)^{2/3}}} \]
1/6*arctan(1/2*2^(1/2)*tan(x))*2^(1/2)+1/3*arctan(tan(x)/(1-(-1)^(1/3))^(1 /2))/(1-(-1)^(1/3))^(1/2)+1/3*arctan(tan(x)/(1+(-1)^(2/3))^(1/2))/(1+(-1)^ (2/3))^(1/2)
Time = 5.08 (sec) , antiderivative size = 79, normalized size of antiderivative = 0.95 \[ \int \frac {1}{1+\cos ^6(x)} \, dx=\frac {1}{12} \left (-2 \sqrt {3} \arctan \left (\frac {1-2 \tan (x)}{\sqrt {3}}\right )+2 \sqrt {2} \arctan \left (\frac {\tan (x)}{\sqrt {2}}\right )+2 \sqrt {3} \arctan \left (\frac {1+2 \tan (x)}{\sqrt {3}}\right )+\log (2-\sin (2 x))-\log (2+\sin (2 x))\right ) \]
(-2*Sqrt[3]*ArcTan[(1 - 2*Tan[x])/Sqrt[3]] + 2*Sqrt[2]*ArcTan[Tan[x]/Sqrt[ 2]] + 2*Sqrt[3]*ArcTan[(1 + 2*Tan[x])/Sqrt[3]] + Log[2 - Sin[2*x]] - Log[2 + Sin[2*x]])/12
Time = 0.34 (sec) , antiderivative size = 83, 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.
\(\displaystyle \int \frac {1}{\cos ^6(x)+1} \, dx\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \int \frac {1}{\sin \left (x+\frac {\pi }{2}\right )^6+1}dx\) |
\(\Big \downarrow \) 3690 |
\(\displaystyle \frac {1}{3} \int \frac {1}{\cos ^2(x)+1}dx+\frac {1}{3} \int \frac {1}{1-\sqrt [3]{-1} \cos ^2(x)}dx+\frac {1}{3} \int \frac {1}{(-1)^{2/3} \cos ^2(x)+1}dx\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \frac {1}{3} \int \frac {1}{\sin \left (x+\frac {\pi }{2}\right )^2+1}dx+\frac {1}{3} \int \frac {1}{1-\sqrt [3]{-1} \sin \left (x+\frac {\pi }{2}\right )^2}dx+\frac {1}{3} \int \frac {1}{(-1)^{2/3} \sin \left (x+\frac {\pi }{2}\right )^2+1}dx\) |
\(\Big \downarrow \) 3660 |
\(\displaystyle -\frac {1}{3} \int \frac {1}{2 \cot ^2(x)+1}d\cot (x)-\frac {1}{3} \int \frac {1}{\left (1-\sqrt [3]{-1}\right ) \cot ^2(x)+1}d\cot (x)-\frac {1}{3} \int \frac {1}{\left (1+(-1)^{2/3}\right ) \cot ^2(x)+1}d\cot (x)\) |
\(\Big \downarrow \) 216 |
\(\displaystyle -\frac {\arctan \left (\sqrt {2} \cot (x)\right )}{3 \sqrt {2}}-\frac {\arctan \left (\sqrt {1-\sqrt [3]{-1}} \cot (x)\right )}{3 \sqrt {1-\sqrt [3]{-1}}}-\frac {\arctan \left (\sqrt {1+(-1)^{2/3}} \cot (x)\right )}{3 \sqrt {1+(-1)^{2/3}}}\) |
-1/3*ArcTan[Sqrt[2]*Cot[x]]/Sqrt[2] - ArcTan[Sqrt[1 - (-1)^(1/3)]*Cot[x]]/ (3*Sqrt[1 - (-1)^(1/3)]) - ArcTan[Sqrt[1 + (-1)^(2/3)]*Cot[x]]/(3*Sqrt[1 + (-1)^(2/3)])
3.1.81.3.1 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]
Time = 1.35 (sec) , antiderivative size = 73, normalized size of antiderivative = 0.88
method | result | size |
default | \(\frac {\ln \left (\tan ^{2}\left (x \right )-\tan \left (x \right )+1\right )}{12}+\frac {\sqrt {3}\, \arctan \left (\frac {\left (2 \tan \left (x \right )-1\right ) \sqrt {3}}{3}\right )}{6}-\frac {\ln \left (\tan ^{2}\left (x \right )+\tan \left (x \right )+1\right )}{12}+\frac {\sqrt {3}\, \arctan \left (\frac {\left (2 \tan \left (x \right )+1\right ) \sqrt {3}}{3}\right )}{6}+\frac {\arctan \left (\frac {\sqrt {2}\, \tan \left (x \right )}{2}\right ) \sqrt {2}}{6}\) | \(73\) |
risch | \(\frac {i \sqrt {2}\, \ln \left ({\mathrm e}^{2 i x}+2 \sqrt {2}+3\right )}{12}-\frac {i \sqrt {2}\, \ln \left ({\mathrm e}^{2 i x}-2 \sqrt {2}+3\right )}{12}-\frac {\ln \left ({\mathrm e}^{2 i x}+2 i+i \sqrt {3}\right )}{12}+\frac {i \ln \left ({\mathrm e}^{2 i x}+2 i+i \sqrt {3}\right ) \sqrt {3}}{12}-\frac {\ln \left ({\mathrm e}^{2 i x}+2 i-i \sqrt {3}\right )}{12}-\frac {i \ln \left ({\mathrm e}^{2 i x}+2 i-i \sqrt {3}\right ) \sqrt {3}}{12}+\frac {\ln \left ({\mathrm e}^{2 i x}-2 i-i \sqrt {3}\right )}{12}+\frac {i \ln \left ({\mathrm e}^{2 i x}-2 i-i \sqrt {3}\right ) \sqrt {3}}{12}+\frac {\ln \left ({\mathrm e}^{2 i x}-2 i+i \sqrt {3}\right )}{12}-\frac {i \ln \left ({\mathrm e}^{2 i x}-2 i+i \sqrt {3}\right ) \sqrt {3}}{12}\) | \(192\) |
1/12*ln(tan(x)^2-tan(x)+1)+1/6*3^(1/2)*arctan(1/3*(2*tan(x)-1)*3^(1/2))-1/ 12*ln(tan(x)^2+tan(x)+1)+1/6*3^(1/2)*arctan(1/3*(2*tan(x)+1)*3^(1/2))+1/6* arctan(1/2*2^(1/2)*tan(x))*2^(1/2)
Leaf count of result is larger than twice the leaf count of optimal. 138 vs. \(2 (58) = 116\).
Time = 0.26 (sec) , antiderivative size = 138, normalized size of antiderivative = 1.66 \[ \int \frac {1}{1+\cos ^6(x)} \, dx=\frac {1}{12} \, \sqrt {3} \arctan \left (\frac {4 \, \sqrt {3} \cos \left (x\right ) \sin \left (x\right ) + \sqrt {3}}{3 \, {\left (2 \, \cos \left (x\right )^{2} - 1\right )}}\right ) + \frac {1}{12} \, \sqrt {3} \arctan \left (\frac {4 \, \sqrt {3} \cos \left (x\right ) \sin \left (x\right ) - \sqrt {3}}{3 \, {\left (2 \, \cos \left (x\right )^{2} - 1\right )}}\right ) - \frac {1}{12} \, \sqrt {2} \arctan \left (\frac {3 \, \sqrt {2} \cos \left (x\right )^{2} - \sqrt {2}}{4 \, \cos \left (x\right ) \sin \left (x\right )}\right ) - \frac {1}{24} \, \log \left (-\cos \left (x\right )^{4} + \cos \left (x\right )^{2} + 2 \, \cos \left (x\right ) \sin \left (x\right ) + 1\right ) + \frac {1}{24} \, \log \left (-\cos \left (x\right )^{4} + \cos \left (x\right )^{2} - 2 \, \cos \left (x\right ) \sin \left (x\right ) + 1\right ) \]
1/12*sqrt(3)*arctan(1/3*(4*sqrt(3)*cos(x)*sin(x) + sqrt(3))/(2*cos(x)^2 - 1)) + 1/12*sqrt(3)*arctan(1/3*(4*sqrt(3)*cos(x)*sin(x) - sqrt(3))/(2*cos(x )^2 - 1)) - 1/12*sqrt(2)*arctan(1/4*(3*sqrt(2)*cos(x)^2 - sqrt(2))/(cos(x) *sin(x))) - 1/24*log(-cos(x)^4 + cos(x)^2 + 2*cos(x)*sin(x) + 1) + 1/24*lo g(-cos(x)^4 + cos(x)^2 - 2*cos(x)*sin(x) + 1)
Timed out. \[ \int \frac {1}{1+\cos ^6(x)} \, dx=\text {Timed out} \]
Time = 0.31 (sec) , antiderivative size = 72, normalized size of antiderivative = 0.87 \[ \int \frac {1}{1+\cos ^6(x)} \, dx=\frac {1}{6} \, \sqrt {3} \arctan \left (\frac {1}{3} \, \sqrt {3} {\left (2 \, \tan \left (x\right ) + 1\right )}\right ) + \frac {1}{6} \, \sqrt {3} \arctan \left (\frac {1}{3} \, \sqrt {3} {\left (2 \, \tan \left (x\right ) - 1\right )}\right ) + \frac {1}{6} \, \sqrt {2} \arctan \left (\frac {1}{2} \, \sqrt {2} \tan \left (x\right )\right ) - \frac {1}{12} \, \log \left (\tan \left (x\right )^{2} + \tan \left (x\right ) + 1\right ) + \frac {1}{12} \, \log \left (\tan \left (x\right )^{2} - \tan \left (x\right ) + 1\right ) \]
1/6*sqrt(3)*arctan(1/3*sqrt(3)*(2*tan(x) + 1)) + 1/6*sqrt(3)*arctan(1/3*sq rt(3)*(2*tan(x) - 1)) + 1/6*sqrt(2)*arctan(1/2*sqrt(2)*tan(x)) - 1/12*log( tan(x)^2 + tan(x) + 1) + 1/12*log(tan(x)^2 - tan(x) + 1)
Leaf count of result is larger than twice the leaf count of optimal. 185 vs. \(2 (58) = 116\).
Time = 0.34 (sec) , antiderivative size = 185, normalized size of antiderivative = 2.23 \[ \int \frac {1}{1+\cos ^6(x)} \, dx=\frac {1}{6} \, \sqrt {3} {\left (x + \arctan \left (-\frac {\sqrt {3} \sin \left (2 \, x\right ) + \cos \left (2 \, x\right ) - 2 \, \sin \left (2 \, x\right ) + 1}{\sqrt {3} \cos \left (2 \, x\right ) + \sqrt {3} - 2 \, \cos \left (2 \, x\right ) - \sin \left (2 \, x\right ) + 2}\right )\right )} + \frac {1}{6} \, \sqrt {3} {\left (x + \arctan \left (-\frac {\sqrt {3} \sin \left (2 \, x\right ) - \cos \left (2 \, x\right ) - 2 \, \sin \left (2 \, x\right ) - 1}{\sqrt {3} \cos \left (2 \, x\right ) + \sqrt {3} - 2 \, \cos \left (2 \, x\right ) + \sin \left (2 \, x\right ) + 2}\right )\right )} + \frac {1}{6} \, \sqrt {2} {\left (x + \arctan \left (-\frac {\sqrt {2} \sin \left (2 \, x\right ) - \sin \left (2 \, x\right )}{\sqrt {2} \cos \left (2 \, x\right ) + \sqrt {2} - \cos \left (2 \, x\right ) + 1}\right )\right )} - \frac {1}{12} \, \log \left (\tan \left (x\right )^{2} + \tan \left (x\right ) + 1\right ) + \frac {1}{12} \, \log \left (\tan \left (x\right )^{2} - \tan \left (x\right ) + 1\right ) \]
1/6*sqrt(3)*(x + arctan(-(sqrt(3)*sin(2*x) + cos(2*x) - 2*sin(2*x) + 1)/(s qrt(3)*cos(2*x) + sqrt(3) - 2*cos(2*x) - sin(2*x) + 2))) + 1/6*sqrt(3)*(x + arctan(-(sqrt(3)*sin(2*x) - cos(2*x) - 2*sin(2*x) - 1)/(sqrt(3)*cos(2*x) + sqrt(3) - 2*cos(2*x) + sin(2*x) + 2))) + 1/6*sqrt(2)*(x + arctan(-(sqrt (2)*sin(2*x) - sin(2*x))/(sqrt(2)*cos(2*x) + sqrt(2) - cos(2*x) + 1))) - 1 /12*log(tan(x)^2 + tan(x) + 1) + 1/12*log(tan(x)^2 - tan(x) + 1)
Time = 3.36 (sec) , antiderivative size = 99, normalized size of antiderivative = 1.19 \[ \int \frac {1}{1+\cos ^6(x)} \, dx=\frac {\sqrt {2}\,\mathrm {atan}\left (\frac {\sqrt {2}\,\mathrm {tan}\left (x\right )}{2}\right )}{6}+\mathrm {atan}\left (\frac {\sqrt {3}\,\mathrm {tan}\left (x\right )}{2}+\frac {\mathrm {tan}\left (x\right )\,1{}\mathrm {i}}{2}\right )\,\left (\frac {\sqrt {3}}{6}+\frac {1}{6}{}\mathrm {i}\right )-\mathrm {atan}\left (-\frac {\sqrt {3}\,\mathrm {tan}\left (x\right )}{2}+\frac {\mathrm {tan}\left (x\right )\,1{}\mathrm {i}}{2}\right )\,\left (\frac {\sqrt {3}}{6}-\frac {1}{6}{}\mathrm {i}\right )+\frac {\left (x-\mathrm {atan}\left (\mathrm {tan}\left (x\right )\right )\right )\,\left (\frac {\pi \,\sqrt {2}}{6}+\pi \,\left (\frac {\sqrt {3}}{6}-\frac {1}{6}{}\mathrm {i}\right )+\pi \,\left (\frac {\sqrt {3}}{6}+\frac {1}{6}{}\mathrm {i}\right )\right )}{\pi } \]