Integrand size = 8, antiderivative size = 103 \[ \int \frac {1}{1+\sin ^6(x)} \, dx=\frac {x}{3 \sqrt {2}}+\frac {\arctan \left (\frac {\cos (x) \sin (x)}{1+\sqrt {2}+\sin ^2(x)}\right )}{3 \sqrt {2}}+\frac {\arctan \left (\sqrt {1-\sqrt [3]{-1}} \tan (x)\right )}{3 \sqrt {1-\sqrt [3]{-1}}}+\frac {\arctan \left (\sqrt {1+(-1)^{2/3}} \tan (x)\right )}{3 \sqrt {1+(-1)^{2/3}}} \]
1/6*x*2^(1/2)+1/6*arctan(cos(x)*sin(x)/(1+sin(x)^2+2^(1/2)))*2^(1/2)+1/3*a rctan((1-(-1)^(1/3))^(1/2)*tan(x))/(1-(-1)^(1/3))^(1/2)+1/3*arctan((1+(-1) ^(2/3))^(1/2)*tan(x))/(1+(-1)^(2/3))^(1/2)
Time = 5.09 (sec) , antiderivative size = 79, normalized size of antiderivative = 0.77 \[ \int \frac {1}{1+\sin ^6(x)} \, dx=\frac {1}{12} \left (-2 \sqrt {3} \arctan \left (\frac {1-2 \tan (x)}{\sqrt {3}}\right )+2 \sqrt {2} \arctan \left (\sqrt {2} \tan (x)\right )+2 \sqrt {3} \arctan \left (\frac {1+2 \tan (x)}{\sqrt {3}}\right )-\log (2-\sin (2 x))+\log (2+\sin (2 x))\right ) \]
(-2*Sqrt[3]*ArcTan[(1 - 2*Tan[x])/Sqrt[3]] + 2*Sqrt[2]*ArcTan[Sqrt[2]*Tan[ x]] + 2*Sqrt[3]*ArcTan[(1 + 2*Tan[x])/Sqrt[3]] - Log[2 - Sin[2*x]] + Log[2 + Sin[2*x]])/12
Time = 0.33 (sec) , antiderivative size = 83, normalized size of antiderivative = 0.81, 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.
\(\displaystyle \int \frac {1}{\sin ^6(x)+1} \, dx\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \int \frac {1}{\sin (x)^6+1}dx\) |
\(\Big \downarrow \) 3690 |
\(\displaystyle \frac {1}{3} \int \frac {1}{\sin ^2(x)+1}dx+\frac {1}{3} \int \frac {1}{1-\sqrt [3]{-1} \sin ^2(x)}dx+\frac {1}{3} \int \frac {1}{(-1)^{2/3} \sin ^2(x)+1}dx\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \frac {1}{3} \int \frac {1}{\sin (x)^2+1}dx+\frac {1}{3} \int \frac {1}{1-\sqrt [3]{-1} \sin (x)^2}dx+\frac {1}{3} \int \frac {1}{(-1)^{2/3} \sin (x)^2+1}dx\) |
\(\Big \downarrow \) 3660 |
\(\displaystyle \frac {1}{3} \int \frac {1}{2 \tan ^2(x)+1}d\tan (x)+\frac {1}{3} \int \frac {1}{\left (1-\sqrt [3]{-1}\right ) \tan ^2(x)+1}d\tan (x)+\frac {1}{3} \int \frac {1}{\left (1+(-1)^{2/3}\right ) \tan ^2(x)+1}d\tan (x)\) |
\(\Big \downarrow \) 216 |
\(\displaystyle \frac {\arctan \left (\sqrt {2} \tan (x)\right )}{3 \sqrt {2}}+\frac {\arctan \left (\sqrt {1-\sqrt [3]{-1}} \tan (x)\right )}{3 \sqrt {1-\sqrt [3]{-1}}}+\frac {\arctan \left (\sqrt {1+(-1)^{2/3}} \tan (x)\right )}{3 \sqrt {1+(-1)^{2/3}}}\) |
ArcTan[Sqrt[2]*Tan[x]]/(3*Sqrt[2]) + ArcTan[Sqrt[1 - (-1)^(1/3)]*Tan[x]]/( 3*Sqrt[1 - (-1)^(1/3)]) + ArcTan[Sqrt[1 + (-1)^(2/3)]*Tan[x]]/(3*Sqrt[1 + (-1)^(2/3)])
3.3.56.3.1 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 = 2.17 (sec) , antiderivative size = 72, normalized size of antiderivative = 0.70
method | result | size |
default | \(-\frac {\ln \left (\tan ^{2}\left (x \right )-\tan \left (x \right )+1\right )}{12}+\frac {\sqrt {3}\, \arctan \left (\frac {\left (2 \tan \left (x \right )-1\right ) \sqrt {3}}{3}\right )}{6}+\frac {\arctan \left (\sqrt {2}\, \tan \left (x \right )\right ) \sqrt {2}}{6}+\frac {\ln \left (\tan ^{2}\left (x \right )+\tan \left (x \right )+1\right )}{12}+\frac {\sqrt {3}\, \arctan \left (\frac {\left (2 \tan \left (x \right )+1\right ) \sqrt {3}}{3}\right )}{6}\) | \(72\) |
risch | \(-\frac {\ln \left ({\mathrm e}^{2 i x}-2 i-i \sqrt {3}\right )}{12}+\frac {i \ln \left ({\mathrm e}^{2 i x}-2 i-i \sqrt {3}\right ) \sqrt {3}}{12}-\frac {\ln \left ({\mathrm e}^{2 i x}-2 i+i \sqrt {3}\right )}{12}-\frac {i \ln \left ({\mathrm e}^{2 i x}-2 i+i \sqrt {3}\right ) \sqrt {3}}{12}+\frac {i \sqrt {2}\, \ln \left ({\mathrm e}^{2 i x}-2 \sqrt {2}-3\right )}{12}-\frac {i \sqrt {2}\, \ln \left ({\mathrm e}^{2 i x}+2 \sqrt {2}-3\right )}{12}+\frac {\ln \left ({\mathrm e}^{2 i x}+2 i+i \sqrt {3}\right )}{12}+\frac {i \ln \left ({\mathrm e}^{2 i x}+2 i+i \sqrt {3}\right ) \sqrt {3}}{12}+\frac {\ln \left ({\mathrm e}^{2 i x}+2 i-i \sqrt {3}\right )}{12}-\frac {i \ln \left ({\mathrm e}^{2 i x}+2 i-i \sqrt {3}\right ) \sqrt {3}}{12}\) | \(192\) |
-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 /6*arctan(2^(1/2)*tan(x))*2^(1/2)+1/12*ln(tan(x)^2+tan(x)+1)+1/6*3^(1/2)*a rctan(1/3*(2*tan(x)+1)*3^(1/2))
Time = 0.29 (sec) , antiderivative size = 138, normalized size of antiderivative = 1.34 \[ \int \frac {1}{1+\sin ^6(x)} \, dx=\frac {1}{12} \, \sqrt {3} \arctan \left (\frac {4 \, \sqrt {3} \cos \left (x\right ) \sin \left (x\right ) + \sqrt {3}}{3 \, {\left (2 \, \cos \left (x\right )^{2} - 1\right )}}\right ) + \frac {1}{12} \, \sqrt {3} \arctan \left (\frac {4 \, \sqrt {3} \cos \left (x\right ) \sin \left (x\right ) - \sqrt {3}}{3 \, {\left (2 \, \cos \left (x\right )^{2} - 1\right )}}\right ) - \frac {1}{12} \, \sqrt {2} \arctan \left (\frac {3 \, \sqrt {2} \cos \left (x\right )^{2} - 2 \, \sqrt {2}}{4 \, \cos \left (x\right ) \sin \left (x\right )}\right ) + \frac {1}{24} \, \log \left (-\cos \left (x\right )^{4} + \cos \left (x\right )^{2} + 2 \, \cos \left (x\right ) \sin \left (x\right ) + 1\right ) - \frac {1}{24} \, \log \left (-\cos \left (x\right )^{4} + \cos \left (x\right )^{2} - 2 \, \cos \left (x\right ) \sin \left (x\right ) + 1\right ) \]
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 - 2*sqrt(2))/(cos( x)*sin(x))) + 1/24*log(-cos(x)^4 + cos(x)^2 + 2*cos(x)*sin(x) + 1) - 1/24* log(-cos(x)^4 + cos(x)^2 - 2*cos(x)*sin(x) + 1)
Timed out. \[ \int \frac {1}{1+\sin ^6(x)} \, dx=\text {Timed out} \]
Time = 0.33 (sec) , antiderivative size = 71, normalized size of antiderivative = 0.69 \[ \int \frac {1}{1+\sin ^6(x)} \, dx=\frac {1}{6} \, \sqrt {3} \arctan \left (\frac {1}{3} \, \sqrt {3} {\left (2 \, \tan \left (x\right ) + 1\right )}\right ) + \frac {1}{6} \, \sqrt {3} \arctan \left (\frac {1}{3} \, \sqrt {3} {\left (2 \, \tan \left (x\right ) - 1\right )}\right ) + \frac {1}{6} \, \sqrt {2} \arctan \left (\sqrt {2} \tan \left (x\right )\right ) + \frac {1}{12} \, \log \left (\tan \left (x\right )^{2} + \tan \left (x\right ) + 1\right ) - \frac {1}{12} \, \log \left (\tan \left (x\right )^{2} - \tan \left (x\right ) + 1\right ) \]
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(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. \(2 (73) = 146\).
Time = 0.30 (sec) , antiderivative size = 185, normalized size of antiderivative = 1.80 \[ \int \frac {1}{1+\sin ^6(x)} \, dx=\frac {1}{6} \, \sqrt {3} {\left (x + \arctan \left (-\frac {\sqrt {3} \sin \left (2 \, x\right ) + \cos \left (2 \, x\right ) - 2 \, \sin \left (2 \, x\right ) + 1}{\sqrt {3} \cos \left (2 \, x\right ) + \sqrt {3} - 2 \, \cos \left (2 \, x\right ) - \sin \left (2 \, x\right ) + 2}\right )\right )} + \frac {1}{6} \, \sqrt {3} {\left (x + \arctan \left (-\frac {\sqrt {3} \sin \left (2 \, x\right ) - \cos \left (2 \, x\right ) - 2 \, \sin \left (2 \, x\right ) - 1}{\sqrt {3} \cos \left (2 \, x\right ) + \sqrt {3} - 2 \, \cos \left (2 \, x\right ) + \sin \left (2 \, x\right ) + 2}\right )\right )} + \frac {1}{6} \, \sqrt {2} {\left (x + \arctan \left (-\frac {\sqrt {2} \sin \left (2 \, x\right ) - 2 \, \sin \left (2 \, x\right )}{\sqrt {2} \cos \left (2 \, x\right ) + \sqrt {2} - 2 \, \cos \left (2 \, x\right ) + 2}\right )\right )} + \frac {1}{12} \, \log \left (\tan \left (x\right )^{2} + \tan \left (x\right ) + 1\right ) - \frac {1}{12} \, \log \left (\tan \left (x\right )^{2} - \tan \left (x\right ) + 1\right ) \]
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) - 2*sin(2*x))/(sqrt(2)*cos(2*x) + sqrt(2) - 2*cos(2*x) + 2))) + 1/12*log(tan(x)^2 + tan(x) + 1) - 1/12*log(tan(x)^2 - tan(x) + 1)
Time = 13.96 (sec) , antiderivative size = 98, normalized size of antiderivative = 0.95 \[ \int \frac {1}{1+\sin ^6(x)} \, dx=\frac {\sqrt {2}\,\mathrm {atan}\left (\sqrt {2}\,\mathrm {tan}\left (x\right )\right )}{6}+\mathrm {atan}\left (\frac {\sqrt {3}\,\mathrm {tan}\left (x\right )}{2}+\frac {\mathrm {tan}\left (x\right )\,1{}\mathrm {i}}{2}\right )\,\left (\frac {\sqrt {3}}{6}-\frac {1}{6}{}\mathrm {i}\right )-\mathrm {atan}\left (-\frac {\sqrt {3}\,\mathrm {tan}\left (x\right )}{2}+\frac {\mathrm {tan}\left (x\right )\,1{}\mathrm {i}}{2}\right )\,\left (\frac {\sqrt {3}}{6}+\frac {1}{6}{}\mathrm {i}\right )+\frac {\left (x-\mathrm {atan}\left (\mathrm {tan}\left (x\right )\right )\right )\,\left (\frac {\pi \,\sqrt {2}}{6}+\pi \,\left (\frac {\sqrt {3}}{6}-\frac {1}{6}{}\mathrm {i}\right )+\pi \,\left (\frac {\sqrt {3}}{6}+\frac {1}{6}{}\mathrm {i}\right )\right )}{\pi } \]