3.259 \(\int \frac {1}{1-\sin ^6(x)} \, dx\)

Optimal. Leaf size=71 \[ \frac {\tan ^{-1}\left (\sqrt {1+\sqrt [3]{-1}} \tan (x)\right )}{3 \sqrt {1+\sqrt [3]{-1}}}+\frac {\tan ^{-1}\left (\sqrt {1-(-1)^{2/3}} \tan (x)\right )}{3 \sqrt {1-(-1)^{2/3}}}+\frac {\tan (x)}{3} \]

[Out]

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)+1/3*tan(x)

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Rubi [A]  time = 0.14, antiderivative size = 71, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 6, integrand size = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.600, Rules used = {3211, 3181, 203, 3175, 3767, 8} \[ \frac {\tan ^{-1}\left (\sqrt {1+\sqrt [3]{-1}} \tan (x)\right )}{3 \sqrt {1+\sqrt [3]{-1}}}+\frac {\tan ^{-1}\left (\sqrt {1-(-1)^{2/3}} \tan (x)\right )}{3 \sqrt {1-(-1)^{2/3}}}+\frac {\tan (x)}{3} \]

Antiderivative was successfully verified.

[In]

Int[(1 - Sin[x]^6)^(-1),x]

[Out]

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)]) + Tan[x]/3

Rule 8

Int[a_, x_Symbol] :> Simp[a*x, x] /; FreeQ[a, x]

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 3175

Int[(u_.)*((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)]^2)^(p_), x_Symbol] :> Dist[a^p, Int[ActivateTrig[u*cos[e + f*x
]^(2*p)], x], x] /; FreeQ[{a, b, e, f, p}, x] && EqQ[a + b, 0] && IntegerQ[p]

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]

Rule 3767

Int[csc[(c_.) + (d_.)*(x_)]^(n_), x_Symbol] :> -Dist[d^(-1), Subst[Int[ExpandIntegrand[(1 + x^2)^(n/2 - 1), x]
, x], x, Cot[c + d*x]], x] /; FreeQ[{c, d}, x] && IGtQ[n/2, 0]

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} \int \sec ^2(x) \, dx+\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 {\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 {1}{3} \operatorname {Subst}(\int 1 \, dx,x,-\tan (x))\\ &=\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 (x)}{3}\\ \end {align*}

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Mathematica [C]  time = 0.28, size = 117, normalized size = 1.65 \[ \frac {\cos (x) (-8 \cos (2 x)+\cos (4 x)+15) \left (-6 \sin (x)+i \sqrt [4]{-3} \left (\sqrt {3}+3 i\right ) \cos (x) \tan ^{-1}\left (\frac {1}{2} \sqrt [4]{-\frac {1}{3}} \left (\sqrt {3}-3 i\right ) \tan (x)\right )+\sqrt [4]{-3} \left (\sqrt {3}-3 i\right ) \cos (x) \tan ^{-1}\left (\frac {(-1)^{3/4} \left (\sqrt {3}+3 i\right ) \tan (x)}{2 \sqrt [4]{3}}\right )\right )}{144 \left (\sin ^6(x)-1\right )} \]

Antiderivative was successfully verified.

[In]

Integrate[(1 - Sin[x]^6)^(-1),x]

[Out]

(Cos[x]*(15 - 8*Cos[2*x] + Cos[4*x])*(I*(-3)^(1/4)*(3*I + Sqrt[3])*ArcTan[((-1/3)^(1/4)*(-3*I + Sqrt[3])*Tan[x
])/2]*Cos[x] + (-3)^(1/4)*(-3*I + Sqrt[3])*ArcTan[((-1)^(3/4)*(3*I + Sqrt[3])*Tan[x])/(2*3^(1/4))]*Cos[x] - 6*
Sin[x]))/(144*(-1 + Sin[x]^6))

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fricas [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, algorithm="fricas")

[Out]

Timed out

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giac [B]  time = 0.15, size = 197, normalized size = 2.77 \[ \frac {1}{18} \, {\left (\pi \left \lfloor \frac {x}{\pi } + \frac {1}{2} \right \rfloor - \arctan \left (-\frac {3 \, \left (\frac {1}{3}\right )^{\frac {3}{4}} {\left (\left (\frac {1}{3}\right )^{\frac {1}{4}} {\left (\sqrt {6} - \sqrt {2}\right )} + 4 \, \tan \relax (x)\right )}}{\sqrt {6} + \sqrt {2}}\right )\right )} \sqrt {6 \, \sqrt {3} + 9} + \frac {1}{18} \, {\left (\pi \left \lfloor \frac {x}{\pi } + \frac {1}{2} \right \rfloor + \arctan \left (-\frac {3 \, \left (\frac {1}{3}\right )^{\frac {3}{4}} {\left (\left (\frac {1}{3}\right )^{\frac {1}{4}} {\left (\sqrt {6} - \sqrt {2}\right )} - 4 \, \tan \relax (x)\right )}}{\sqrt {6} + \sqrt {2}}\right )\right )} \sqrt {6 \, \sqrt {3} + 9} + \frac {1}{36} \, \sqrt {6 \, \sqrt {3} - 9} \log \left (\frac {1}{2} \, {\left (\sqrt {6} \left (\frac {1}{3}\right )^{\frac {1}{4}} - \sqrt {2} \left (\frac {1}{3}\right )^{\frac {1}{4}}\right )} \tan \relax (x) + \tan \relax (x)^{2} + \sqrt {\frac {1}{3}}\right ) - \frac {1}{36} \, \sqrt {6 \, \sqrt {3} - 9} \log \left (-\frac {1}{2} \, {\left (\sqrt {6} \left (\frac {1}{3}\right )^{\frac {1}{4}} - \sqrt {2} \left (\frac {1}{3}\right )^{\frac {1}{4}}\right )} \tan \relax (x) + \tan \relax (x)^{2} + \sqrt {\frac {1}{3}}\right ) + \frac {1}{3} \, \tan \relax (x) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(1-sin(x)^6),x, algorithm="giac")

[Out]

1/18*(pi*floor(x/pi + 1/2) - arctan(-3*(1/3)^(3/4)*((1/3)^(1/4)*(sqrt(6) - sqrt(2)) + 4*tan(x))/(sqrt(6) + sqr
t(2))))*sqrt(6*sqrt(3) + 9) + 1/18*(pi*floor(x/pi + 1/2) + arctan(-3*(1/3)^(3/4)*((1/3)^(1/4)*(sqrt(6) - sqrt(
2)) - 4*tan(x))/(sqrt(6) + sqrt(2))))*sqrt(6*sqrt(3) + 9) + 1/36*sqrt(6*sqrt(3) - 9)*log(1/2*(sqrt(6)*(1/3)^(1
/4) - sqrt(2)*(1/3)^(1/4))*tan(x) + tan(x)^2 + sqrt(1/3)) - 1/36*sqrt(6*sqrt(3) - 9)*log(-1/2*(sqrt(6)*(1/3)^(
1/4) - sqrt(2)*(1/3)^(1/4))*tan(x) + tan(x)^2 + sqrt(1/3)) + 1/3*tan(x)

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maple [B]  time = 0.45, size = 255, normalized size = 3.59 \[ \frac {\tan \relax (x )}{3}+\frac {\sqrt {3}\, \sqrt {2 \sqrt {3}-3}\, \ln \left (\sqrt {3}+3 \left (\tan ^{2}\relax (x )\right )+\sqrt {2 \sqrt {3}-3}\, \sqrt {3}\, \tan \relax (x )\right )}{36}+\frac {\arctan \left (\frac {6 \tan \relax (x )+\sqrt {2 \sqrt {3}-3}\, \sqrt {3}}{\sqrt {6 \sqrt {3}+9}}\right ) \sqrt {3}}{3 \sqrt {6 \sqrt {3}+9}}+\frac {\arctan \left (\frac {6 \tan \relax (x )+\sqrt {2 \sqrt {3}-3}\, \sqrt {3}}{\sqrt {6 \sqrt {3}+9}}\right )}{2 \sqrt {6 \sqrt {3}+9}}-\frac {\sqrt {3}\, \sqrt {2 \sqrt {3}-3}\, \ln \left (-\sqrt {2 \sqrt {3}-3}\, \sqrt {3}\, \tan \relax (x )+3 \left (\tan ^{2}\relax (x )\right )+\sqrt {3}\right )}{36}+\frac {\arctan \left (\frac {-\sqrt {2 \sqrt {3}-3}\, \sqrt {3}+6 \tan \relax (x )}{\sqrt {6 \sqrt {3}+9}}\right ) \sqrt {3}}{3 \sqrt {6 \sqrt {3}+9}}+\frac {\arctan \left (\frac {-\sqrt {2 \sqrt {3}-3}\, \sqrt {3}+6 \tan \relax (x )}{\sqrt {6 \sqrt {3}+9}}\right )}{2 \sqrt {6 \sqrt {3}+9}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/3*tan(x)+1/36*3^(1/2)*(2*3^(1/2)-3)^(1/2)*ln(3^(1/2)+3*tan(x)^2+(2*3^(1/2)-3)^(1/2)*3^(1/2)*tan(x))+1/3/(6*3
^(1/2)+9)^(1/2)*arctan((6*tan(x)+(2*3^(1/2)-3)^(1/2)*3^(1/2))/(6*3^(1/2)+9)^(1/2))*3^(1/2)+1/2/(6*3^(1/2)+9)^(
1/2)*arctan((6*tan(x)+(2*3^(1/2)-3)^(1/2)*3^(1/2))/(6*3^(1/2)+9)^(1/2))-1/36*3^(1/2)*(2*3^(1/2)-3)^(1/2)*ln(-(
2*3^(1/2)-3)^(1/2)*3^(1/2)*tan(x)+3*tan(x)^2+3^(1/2))+1/3/(6*3^(1/2)+9)^(1/2)*arctan((-(2*3^(1/2)-3)^(1/2)*3^(
1/2)+6*tan(x))/(6*3^(1/2)+9)^(1/2))*3^(1/2)+1/2/(6*3^(1/2)+9)^(1/2)*arctan((-(2*3^(1/2)-3)^(1/2)*3^(1/2)+6*tan
(x))/(6*3^(1/2)+9)^(1/2))

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maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \[ -\frac {-4 \, {\left (\cos \left (2 \, x\right )^{2} + \sin \left (2 \, x\right )^{2} + 2 \, \cos \left (2 \, x\right ) + 1\right )} \int \frac {{\left (\cos \left (6 \, x\right ) - 10 \, \cos \left (4 \, x\right ) + \cos \left (2 \, x\right )\right )} \cos \left (8 \, x\right ) + {\left (110 \, \cos \left (4 \, x\right ) - 16 \, \cos \left (2 \, x\right ) + 1\right )} \cos \left (6 \, x\right ) - 8 \, \cos \left (6 \, x\right )^{2} + 10 \, {\left (11 \, \cos \left (2 \, x\right ) - 1\right )} \cos \left (4 \, x\right ) - 300 \, \cos \left (4 \, x\right )^{2} - 8 \, \cos \left (2 \, x\right )^{2} + {\left (\sin \left (6 \, x\right ) - 10 \, \sin \left (4 \, x\right ) + \sin \left (2 \, x\right )\right )} \sin \left (8 \, x\right ) + 2 \, {\left (55 \, \sin \left (4 \, x\right ) - 8 \, \sin \left (2 \, x\right )\right )} \sin \left (6 \, x\right ) - 8 \, \sin \left (6 \, x\right )^{2} - 300 \, \sin \left (4 \, x\right )^{2} + 110 \, \sin \left (4 \, x\right ) \sin \left (2 \, x\right ) - 8 \, \sin \left (2 \, x\right )^{2} + \cos \left (2 \, x\right )}{2 \, {\left (8 \, \cos \left (6 \, x\right ) - 30 \, \cos \left (4 \, x\right ) + 8 \, \cos \left (2 \, x\right ) - 1\right )} \cos \left (8 \, x\right ) - \cos \left (8 \, x\right )^{2} + 16 \, {\left (30 \, \cos \left (4 \, x\right ) - 8 \, \cos \left (2 \, x\right ) + 1\right )} \cos \left (6 \, x\right ) - 64 \, \cos \left (6 \, x\right )^{2} + 60 \, {\left (8 \, \cos \left (2 \, x\right ) - 1\right )} \cos \left (4 \, x\right ) - 900 \, \cos \left (4 \, x\right )^{2} - 64 \, \cos \left (2 \, x\right )^{2} + 4 \, {\left (4 \, \sin \left (6 \, x\right ) - 15 \, \sin \left (4 \, x\right ) + 4 \, \sin \left (2 \, x\right )\right )} \sin \left (8 \, x\right ) - \sin \left (8 \, x\right )^{2} + 32 \, {\left (15 \, \sin \left (4 \, x\right ) - 4 \, \sin \left (2 \, x\right )\right )} \sin \left (6 \, x\right ) - 64 \, \sin \left (6 \, x\right )^{2} - 900 \, \sin \left (4 \, x\right )^{2} + 480 \, \sin \left (4 \, x\right ) \sin \left (2 \, x\right ) - 64 \, \sin \left (2 \, x\right )^{2} + 16 \, \cos \left (2 \, x\right ) - 1}\,{d x} - 2 \, \sin \left (2 \, x\right )}{3 \, {\left (\cos \left (2 \, x\right )^{2} + \sin \left (2 \, x\right )^{2} + 2 \, \cos \left (2 \, x\right ) + 1\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(1-sin(x)^6),x, algorithm="maxima")

[Out]

-1/3*(3*(cos(2*x)^2 + sin(2*x)^2 + 2*cos(2*x) + 1)*integrate(-4/3*((cos(6*x) - 10*cos(4*x) + cos(2*x))*cos(8*x
) + (110*cos(4*x) - 16*cos(2*x) + 1)*cos(6*x) - 8*cos(6*x)^2 + 10*(11*cos(2*x) - 1)*cos(4*x) - 300*cos(4*x)^2
- 8*cos(2*x)^2 + (sin(6*x) - 10*sin(4*x) + sin(2*x))*sin(8*x) + 2*(55*sin(4*x) - 8*sin(2*x))*sin(6*x) - 8*sin(
6*x)^2 - 300*sin(4*x)^2 + 110*sin(4*x)*sin(2*x) - 8*sin(2*x)^2 + cos(2*x))/(2*(8*cos(6*x) - 30*cos(4*x) + 8*co
s(2*x) - 1)*cos(8*x) - cos(8*x)^2 + 16*(30*cos(4*x) - 8*cos(2*x) + 1)*cos(6*x) - 64*cos(6*x)^2 + 60*(8*cos(2*x
) - 1)*cos(4*x) - 900*cos(4*x)^2 - 64*cos(2*x)^2 + 4*(4*sin(6*x) - 15*sin(4*x) + 4*sin(2*x))*sin(8*x) - sin(8*
x)^2 + 32*(15*sin(4*x) - 4*sin(2*x))*sin(6*x) - 64*sin(6*x)^2 - 900*sin(4*x)^2 + 480*sin(4*x)*sin(2*x) - 64*si
n(2*x)^2 + 16*cos(2*x) - 1), x) - 2*sin(2*x))/(cos(2*x)^2 + sin(2*x)^2 + 2*cos(2*x) + 1)

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mupad [B]  time = 14.21, size = 99, normalized size = 1.39 \[ \frac {\mathrm {tan}\relax (x)}{3}-\frac {\sqrt {6}\,\mathrm {atan}\left (3^{1/4}\,\sqrt {6}\,\mathrm {tan}\relax (x)\,\left (\frac {1}{4}-\frac {1}{4}{}\mathrm {i}\right )+3^{3/4}\,\sqrt {6}\,\mathrm {tan}\relax (x)\,\left (\frac {1}{12}+\frac {1}{12}{}\mathrm {i}\right )\right )\,\left (3^{1/4}\,\left (1+1{}\mathrm {i}\right )+3^{3/4}\,\left (-1+1{}\mathrm {i}\right )\right )\,1{}\mathrm {i}}{36}+\frac {\sqrt {6}\,\mathrm {atan}\left (3^{1/4}\,\sqrt {6}\,\mathrm {tan}\relax (x)\,\left (\frac {1}{4}+\frac {1}{4}{}\mathrm {i}\right )+3^{3/4}\,\sqrt {6}\,\mathrm {tan}\relax (x)\,\left (\frac {1}{12}-\frac {1}{12}{}\mathrm {i}\right )\right )\,\left (3^{1/4}\,\left (1-\mathrm {i}\right )+3^{3/4}\,\left (-1-\mathrm {i}\right )\right )\,1{}\mathrm {i}}{36} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(-1/(sin(x)^6 - 1),x)

[Out]

tan(x)/3 - (6^(1/2)*atan(3^(1/4)*6^(1/2)*tan(x)*(1/4 - 1i/4) + 3^(3/4)*6^(1/2)*tan(x)*(1/12 + 1i/12))*(3^(1/4)
*(1 + 1i) - 3^(3/4)*(1 - 1i))*1i)/36 + (6^(1/2)*atan(3^(1/4)*6^(1/2)*tan(x)*(1/4 + 1i/4) + 3^(3/4)*6^(1/2)*tan
(x)*(1/12 - 1i/12))*(3^(1/4)*(1 - 1i) - 3^(3/4)*(1 + 1i))*1i)/36

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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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