3.256 \(\int \frac {1}{1+\sin ^6(x)} \, dx\)

Optimal. Leaf size=103 \[ \frac {x}{3 \sqrt {2}}+\frac {\tan ^{-1}\left (\sqrt {1-\sqrt [3]{-1}} \tan (x)\right )}{3 \sqrt {1-\sqrt [3]{-1}}}+\frac {\tan ^{-1}\left (\sqrt {1+(-1)^{2/3}} \tan (x)\right )}{3 \sqrt {1+(-1)^{2/3}}}+\frac {\tan ^{-1}\left (\frac {\sin (x) \cos (x)}{\sin ^2(x)+\sqrt {2}+1}\right )}{3 \sqrt {2}} \]

[Out]

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*arctan((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)

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Rubi [A]  time = 0.10, antiderivative size = 103, normalized size of antiderivative = 1.00, 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 = {3211, 3181, 203} \[ \frac {x}{3 \sqrt {2}}+\frac {\tan ^{-1}\left (\sqrt {1-\sqrt [3]{-1}} \tan (x)\right )}{3 \sqrt {1-\sqrt [3]{-1}}}+\frac {\tan ^{-1}\left (\sqrt {1+(-1)^{2/3}} \tan (x)\right )}{3 \sqrt {1+(-1)^{2/3}}}+\frac {\tan ^{-1}\left (\frac {\sin (x) \cos (x)}{\sin ^2(x)+\sqrt {2}+1}\right )}{3 \sqrt {2}} \]

Antiderivative was successfully verified.

[In]

Int[(1 + Sin[x]^6)^(-1),x]

[Out]

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

Rule 203

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

Rule 3181

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 3211

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

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Mathematica [A]  time = 0.16, size = 79, normalized size = 0.77 \[ \frac {1}{12} \left (-2 \sqrt {3} \tan ^{-1}\left (\frac {1-2 \tan (x)}{\sqrt {3}}\right )+2 \sqrt {2} \tan ^{-1}\left (\sqrt {2} \tan (x)\right )+2 \sqrt {3} \tan ^{-1}\left (\frac {2 \tan (x)+1}{\sqrt {3}}\right )-\log (2-\sin (2 x))+\log (\sin (2 x)+2)\right ) \]

Antiderivative was successfully verified.

[In]

Integrate[(1 + Sin[x]^6)^(-1),x]

[Out]

(-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

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fricas [A]  time = 0.51, size = 138, normalized size = 1.34 \[ \frac {1}{12} \, \sqrt {3} \arctan \left (\frac {4 \, \sqrt {3} \cos \relax (x) \sin \relax (x) + \sqrt {3}}{3 \, {\left (2 \, \cos \relax (x)^{2} - 1\right )}}\right ) + \frac {1}{12} \, \sqrt {3} \arctan \left (\frac {4 \, \sqrt {3} \cos \relax (x) \sin \relax (x) - \sqrt {3}}{3 \, {\left (2 \, \cos \relax (x)^{2} - 1\right )}}\right ) - \frac {1}{12} \, \sqrt {2} \arctan \left (\frac {3 \, \sqrt {2} \cos \relax (x)^{2} - 2 \, \sqrt {2}}{4 \, \cos \relax (x) \sin \relax (x)}\right ) + \frac {1}{24} \, \log \left (-\cos \relax (x)^{4} + \cos \relax (x)^{2} + 2 \, \cos \relax (x) \sin \relax (x) + 1\right ) - \frac {1}{24} \, \log \left (-\cos \relax (x)^{4} + \cos \relax (x)^{2} - 2 \, \cos \relax (x) \sin \relax (x) + 1\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(1+sin(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 - 2*sqrt(2))/(co
s(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)

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giac [B]  time = 0.16, size = 185, normalized size = 1.80 \[ \frac {1}{6} \, \sqrt {3} {\left (x + \arctan \left (-\frac {\sqrt {3} \sin \left (2 \, x\right ) + \cos \left (2 \, x\right ) - 2 \, \sin \left (2 \, x\right ) + 1}{\sqrt {3} \cos \left (2 \, x\right ) + \sqrt {3} - 2 \, \cos \left (2 \, x\right ) - \sin \left (2 \, x\right ) + 2}\right )\right )} + \frac {1}{6} \, \sqrt {3} {\left (x + \arctan \left (-\frac {\sqrt {3} \sin \left (2 \, x\right ) - \cos \left (2 \, x\right ) - 2 \, \sin \left (2 \, x\right ) - 1}{\sqrt {3} \cos \left (2 \, x\right ) + \sqrt {3} - 2 \, \cos \left (2 \, x\right ) + \sin \left (2 \, x\right ) + 2}\right )\right )} + \frac {1}{6} \, \sqrt {2} {\left (x + \arctan \left (-\frac {\sqrt {2} \sin \left (2 \, x\right ) - 2 \, \sin \left (2 \, x\right )}{\sqrt {2} \cos \left (2 \, x\right ) + \sqrt {2} - 2 \, \cos \left (2 \, x\right ) + 2}\right )\right )} + \frac {1}{12} \, \log \left (\tan \relax (x)^{2} + \tan \relax (x) + 1\right ) - \frac {1}{12} \, \log \left (\tan \relax (x)^{2} - \tan \relax (x) + 1\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(1+sin(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) - 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)

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maple [A]  time = 0.18, size = 72, normalized size = 0.70 \[ \frac {\arctan \left (\sqrt {2}\, \tan \relax (x )\right ) \sqrt {2}}{6}+\frac {\ln \left (\tan ^{2}\relax (x )+\tan \relax (x )+1\right )}{12}+\frac {\sqrt {3}\, \arctan \left (\frac {\left (1+2 \tan \relax (x )\right ) \sqrt {3}}{3}\right )}{6}-\frac {\ln \left (\tan ^{2}\relax (x )-\tan \relax (x )+1\right )}{12}+\frac {\sqrt {3}\, \arctan \left (\frac {\left (2 \tan \relax (x )-1\right ) \sqrt {3}}{3}\right )}{6} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(1+sin(x)^6),x)

[Out]

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)*arctan(1/3*(1+2*tan(x))*3^(1/2))-1/1
2*ln(tan(x)^2-tan(x)+1)+1/6*3^(1/2)*arctan(1/3*(2*tan(x)-1)*3^(1/2))

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maxima [A]  time = 0.44, size = 71, normalized size = 0.69 \[ \frac {1}{6} \, \sqrt {3} \arctan \left (\frac {1}{3} \, \sqrt {3} {\left (2 \, \tan \relax (x) + 1\right )}\right ) + \frac {1}{6} \, \sqrt {3} \arctan \left (\frac {1}{3} \, \sqrt {3} {\left (2 \, \tan \relax (x) - 1\right )}\right ) + \frac {1}{6} \, \sqrt {2} \arctan \left (\sqrt {2} \tan \relax (x)\right ) + \frac {1}{12} \, \log \left (\tan \relax (x)^{2} + \tan \relax (x) + 1\right ) - \frac {1}{12} \, \log \left (\tan \relax (x)^{2} - \tan \relax (x) + 1\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(1+sin(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(sqrt(2)*tan(x)) + 1/12*log(tan(x)^2 + tan(x) + 1) - 1/12*log(tan(x)^2 - tan(x) + 1)

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mupad [B]  time = 14.23, size = 98, normalized size = 0.95 \[ \frac {\sqrt {2}\,\mathrm {atan}\left (\sqrt {2}\,\mathrm {tan}\relax (x)\right )}{6}+\mathrm {atan}\left (\frac {\sqrt {3}\,\mathrm {tan}\relax (x)}{2}+\frac {\mathrm {tan}\relax (x)\,1{}\mathrm {i}}{2}\right )\,\left (\frac {\sqrt {3}}{6}-\frac {1}{6}{}\mathrm {i}\right )-\mathrm {atan}\left (-\frac {\sqrt {3}\,\mathrm {tan}\relax (x)}{2}+\frac {\mathrm {tan}\relax (x)\,1{}\mathrm {i}}{2}\right )\,\left (\frac {\sqrt {3}}{6}+\frac {1}{6}{}\mathrm {i}\right )+\frac {\left (x-\mathrm {atan}\left (\mathrm {tan}\relax (x)\right )\right )\,\left (\frac {\pi \,\sqrt {2}}{6}+\pi \,\left (\frac {\sqrt {3}}{6}-\frac {1}{6}{}\mathrm {i}\right )+\pi \,\left (\frac {\sqrt {3}}{6}+\frac {1}{6}{}\mathrm {i}\right )\right )}{\pi } \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(sin(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)))/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

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sympy [F(-1)]  time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Timed out

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