3.1.81 \(\int \frac {1}{1+\cos ^6(x)} \, dx\) [81]

Optimal. Leaf size=83 \[ \frac {\text {ArcTan}\left (\frac {\tan (x)}{\sqrt {2}}\right )}{3 \sqrt {2}}+\frac {\text {ArcTan}\left (\frac {\tan (x)}{\sqrt {1-\sqrt [3]{-1}}}\right )}{3 \sqrt {1-\sqrt [3]{-1}}}+\frac {\text {ArcTan}\left (\frac {\tan (x)}{\sqrt {1+(-1)^{2/3}}}\right )}{3 \sqrt {1+(-1)^{2/3}}} \]

[Out]

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)

________________________________________________________________________________________

Rubi [A]
time = 0.08, antiderivative size = 103, normalized size of antiderivative = 1.24, number of steps used = 7, number of rules used = 3, integrand size = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.375, Rules used = {3290, 3260, 209} \begin {gather*} -\frac {\text {ArcTan}\left (\sqrt {1-\sqrt [3]{-1}} \cot (x)\right )}{3 \sqrt {1-\sqrt [3]{-1}}}-\frac {\text {ArcTan}\left (\sqrt {1+(-1)^{2/3}} \cot (x)\right )}{3 \sqrt {1+(-1)^{2/3}}}-\frac {\text {ArcTan}\left (\frac {\sin (x) \cos (x)}{\cos ^2(x)+\sqrt {2}+1}\right )}{3 \sqrt {2}}+\frac {x}{3 \sqrt {2}} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(1 + Cos[x]^6)^(-1),x]

[Out]

x/(3*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)]) - ArcTan[(Cos[x]*Sin[x])/(1 + Sqrt[2] + Cos[x]^2)]/(3*Sqrt[2])

Rule 209

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1/(Rt[a, 2]*Rt[b, 2]))*ArcTan[Rt[b, 2]*(x/Rt[a, 2])], x] /;
FreeQ[{a, b}, x] && PosQ[a/b] && (GtQ[a, 0] || GtQ[b, 0])

Rule 3260

Int[((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)]^2)^(-1), x_Symbol] :> With[{ff = FreeFactors[Tan[e + f*x], x]}, Dist
[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]

Rule 3290

Int[((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)]^(n_))^(-1), x_Symbol] :> Module[{k}, Dist[2/(a*n), Sum[Int[1/(1 - Si
n[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]

Rubi steps

\begin {align*} \int \frac {1}{1+\cos ^6(x)} \, dx &=\frac {1}{3} \int \frac {1}{1+\cos ^2(x)} \, dx+\frac {1}{3} \int \frac {1}{1-\sqrt [3]{-1} \cos ^2(x)} \, dx+\frac {1}{3} \int \frac {1}{1+(-1)^{2/3} \cos ^2(x)} \, dx\\ &=-\left (\frac {1}{3} \text {Subst}\left (\int \frac {1}{1+2 x^2} \, dx,x,\cot (x)\right )\right )-\frac {1}{3} \text {Subst}\left (\int \frac {1}{1+\left (1-\sqrt [3]{-1}\right ) x^2} \, dx,x,\cot (x)\right )-\frac {1}{3} \text {Subst}\left (\int \frac {1}{1+\left (1+(-1)^{2/3}\right ) x^2} \, dx,x,\cot (x)\right )\\ &=\frac {x}{3 \sqrt {2}}-\frac {\tan ^{-1}\left (\sqrt {1-\sqrt [3]{-1}} \cot (x)\right )}{3 \sqrt {1-\sqrt [3]{-1}}}-\frac {\tan ^{-1}\left (\sqrt {1+(-1)^{2/3}} \cot (x)\right )}{3 \sqrt {1+(-1)^{2/3}}}-\frac {\tan ^{-1}\left (\frac {\cos (x) \sin (x)}{1+\sqrt {2}+\cos ^2(x)}\right )}{3 \sqrt {2}}\\ \end {align*}

________________________________________________________________________________________

Mathematica [A]
time = 0.18, size = 79, normalized size = 0.95 \begin {gather*} \frac {1}{12} \left (-2 \sqrt {3} \text {ArcTan}\left (\frac {1-2 \tan (x)}{\sqrt {3}}\right )+2 \sqrt {2} \text {ArcTan}\left (\frac {\tan (x)}{\sqrt {2}}\right )+2 \sqrt {3} \text {ArcTan}\left (\frac {1+2 \tan (x)}{\sqrt {3}}\right )+\log (2-\sin (2 x))-\log (2+\sin (2 x))\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(1 + Cos[x]^6)^(-1),x]

[Out]

(-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

________________________________________________________________________________________

Maple [A]
time = 0.12, size = 73, normalized size = 0.88

method	result	size
default	\(\frac {\ln \left (\tan ^{2}\left (x \right )-\tan \left (x \right )+1\right )}{12}+\frac {\sqrt {3}\, \arctan \left (\frac {\left (2 \tan \left (x \right )-1\right ) \sqrt {3}}{3}\right )}{6}+\frac {\arctan \left (\frac {\tan \left (x \right ) \sqrt {2}}{2}\right ) \sqrt {2}}{6}-\frac {\ln \left (\tan ^{2}\left (x \right )+\tan \left (x \right )+1\right )}{12}+\frac {\sqrt {3}\, \arctan \left (\frac {\left (2 \tan \left (x \right )+1\right ) \sqrt {3}}{3}\right )}{6}\)	\(73\)
risch	\(\frac {i \sqrt {2}\, \ln \left ({\mathrm e}^{2 i x}+2 \sqrt {2}+3\right )}{12}-\frac {i \sqrt {2}\, \ln \left ({\mathrm e}^{2 i x}-2 \sqrt {2}+3\right )}{12}+\frac {\ln \left ({\mathrm e}^{2 i x}-2 i-i \sqrt {3}\right )}{12}+\frac {i \ln \left ({\mathrm e}^{2 i x}-2 i-i \sqrt {3}\right ) \sqrt {3}}{12}+\frac {\ln \left ({\mathrm e}^{2 i x}-2 i+i \sqrt {3}\right )}{12}-\frac {i \ln \left ({\mathrm e}^{2 i x}-2 i+i \sqrt {3}\right ) \sqrt {3}}{12}-\frac {\ln \left ({\mathrm e}^{2 i x}+2 i+i \sqrt {3}\right )}{12}+\frac {i \ln \left ({\mathrm e}^{2 i x}+2 i+i \sqrt {3}\right ) \sqrt {3}}{12}-\frac {\ln \left ({\mathrm e}^{2 i x}+2 i-i \sqrt {3}\right )}{12}-\frac {i \ln \left ({\mathrm e}^{2 i x}+2 i-i \sqrt {3}\right ) \sqrt {3}}{12}\)	\(192\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(1+cos(x)^6),x,method=_RETURNVERBOSE)

[Out]

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*tan(x)*2^(1/2))*2^(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))

________________________________________________________________________________________

Maxima [A]
time = 0.48, size = 72, normalized size = 0.87 \begin {gather*} \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 ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(1+cos(x)^6),x, algorithm="maxima")

[Out]

1/6*sqrt(3)*arctan(1/3*sqrt(3)*(2*tan(x) + 1)) + 1/6*sqrt(3)*arctan(1/3*sqrt(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)

________________________________________________________________________________________

Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 138 vs. \(2 (58) = 116\).
time = 0.48, size = 138, normalized size = 1.66 \begin {gather*} \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 ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(1+cos(x)^6),x, algorithm="fricas")

[Out]

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*sqr
t(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*log(-cos(x)^4 + cos(x)^2 - 2*cos(x)*
sin(x) + 1)

________________________________________________________________________________________

Sympy [F(-1)] Timed out
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(1+cos(x)**6),x)

[Out]

Timed out

________________________________________________________________________________________

Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 185 vs. \(2 (58) = 116\).
time = 0.46, size = 185, normalized size = 2.23 \begin {gather*} \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 ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(1+cos(x)^6),x, algorithm="giac")

[Out]

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(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)

________________________________________________________________________________________

Mupad [B]
time = 2.39, size = 99, normalized size = 1.19 \begin {gather*} \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 } \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(cos(x)^6 + 1),x)

[Out]

atan((tan(x)*1i)/2 + (3^(1/2)*tan(x))/2)*(3^(1/2)/6 + 1i/6) - atan((tan(x)*1i)/2 - (3^(1/2)*tan(x))/2)*(3^(1/2
)/6 - 1i/6) + (2^(1/2)*atan((2^(1/2)*tan(x))/2))/6 + ((x - atan(tan(x)))*((2^(1/2)*pi)/6 + pi*(3^(1/2)/6 - 1i/
6) + pi*(3^(1/2)/6 + 1i/6)))/pi

________________________________________________________________________________________