3.73 \(\int \frac{1}{1-\cos ^4(x)} \, dx\)

Optimal. Leaf size=45 \[ \frac{x}{2 \sqrt{2}}-\frac{\cot (x)}{2}-\frac{\tan ^{-1}\left (\frac{\sin (x) \cos (x)}{\cos ^2(x)+\sqrt{2}+1}\right )}{2 \sqrt{2}} \]

[Out]

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

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Rubi [A]  time = 0.0183842, antiderivative size = 45, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 10, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.3, Rules used = {3209, 388, 203} \[ \frac{x}{2 \sqrt{2}}-\frac{\cot (x)}{2}-\frac{\tan ^{-1}\left (\frac{\sin (x) \cos (x)}{\cos ^2(x)+\sqrt{2}+1}\right )}{2 \sqrt{2}} \]

Antiderivative was successfully verified.

[In]

Int[(1 - Cos[x]^4)^(-1),x]

[Out]

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

Rule 3209

Int[((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)]^4)^(p_.), x_Symbol] :> With[{ff = FreeFactors[Tan[e + f*x], x]}, Dis
t[ff/f, Subst[Int[(a + 2*a*ff^2*x^2 + (a + b)*ff^4*x^4)^p/(1 + ff^2*x^2)^(2*p + 1), x], x, Tan[e + f*x]/ff], x
]] /; FreeQ[{a, b, e, f}, x] && IntegerQ[p]

Rule 388

Int[((a_) + (b_.)*(x_)^(n_))^(p_)*((c_) + (d_.)*(x_)^(n_)), x_Symbol] :> Simp[(d*x*(a + b*x^n)^(p + 1))/(b*(n*
(p + 1) + 1)), x] - Dist[(a*d - b*c*(n*(p + 1) + 1))/(b*(n*(p + 1) + 1)), Int[(a + b*x^n)^p, x], x] /; FreeQ[{
a, b, c, d, n}, x] && NeQ[b*c - a*d, 0] && NeQ[n*(p + 1) + 1, 0]

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])

Rubi steps

\begin{align*} \int \frac{1}{1-\cos ^4(x)} \, dx &=-\operatorname{Subst}\left (\int \frac{1+x^2}{1+2 x^2} \, dx,x,\cot (x)\right )\\ &=-\frac{\cot (x)}{2}-\frac{1}{2} \operatorname{Subst}\left (\int \frac{1}{1+2 x^2} \, dx,x,\cot (x)\right )\\ &=\frac{x}{2 \sqrt{2}}-\frac{\tan ^{-1}\left (\frac{\cos (x) \sin (x)}{1+\sqrt{2}+\cos ^2(x)}\right )}{2 \sqrt{2}}-\frac{\cot (x)}{2}\\ \end{align*}

Mathematica [A]  time = 0.0543841, size = 24, normalized size = 0.53 \[ \frac{1}{4} \left (\sqrt{2} \tan ^{-1}\left (\frac{\tan (x)}{\sqrt{2}}\right )-2 \cot (x)\right ) \]

Antiderivative was successfully verified.

[In]

Integrate[(1 - Cos[x]^4)^(-1),x]

[Out]

(Sqrt[2]*ArcTan[Tan[x]/Sqrt[2]] - 2*Cot[x])/4

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Maple [A]  time = 0.02, size = 21, normalized size = 0.5 \begin{align*} -{\frac{1}{2\,\tan \left ( x \right ) }}+{\frac{\sqrt{2}}{4}\arctan \left ({\frac{\tan \left ( x \right ) \sqrt{2}}{2}} \right ) } \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(1-cos(x)^4),x)

[Out]

-1/2/tan(x)+1/4*arctan(1/2*tan(x)*2^(1/2))*2^(1/2)

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Maxima [A]  time = 1.4547, size = 27, normalized size = 0.6 \begin{align*} \frac{1}{4} \, \sqrt{2} \arctan \left (\frac{1}{2} \, \sqrt{2} \tan \left (x\right )\right ) - \frac{1}{2 \, \tan \left (x\right )} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(1-cos(x)^4),x, algorithm="maxima")

[Out]

1/4*sqrt(2)*arctan(1/2*sqrt(2)*tan(x)) - 1/2/tan(x)

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Fricas [A]  time = 2.18423, size = 135, normalized size = 3. \begin{align*} -\frac{\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 ) \sin \left (x\right ) + 4 \, \cos \left (x\right )}{8 \, \sin \left (x\right )} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(1-cos(x)^4),x, algorithm="fricas")

[Out]

-1/8*(sqrt(2)*arctan(1/4*(3*sqrt(2)*cos(x)^2 - sqrt(2))/(cos(x)*sin(x)))*sin(x) + 4*cos(x))/sin(x)

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Sympy [A]  time = 4.63275, size = 78, normalized size = 1.73 \begin{align*} \frac{\sqrt{2} \left (\operatorname{atan}{\left (\sqrt{2} \tan{\left (\frac{x}{2} \right )} - 1 \right )} + \pi \left \lfloor{\frac{\frac{x}{2} - \frac{\pi }{2}}{\pi }}\right \rfloor \right )}{4} + \frac{\sqrt{2} \left (\operatorname{atan}{\left (\sqrt{2} \tan{\left (\frac{x}{2} \right )} + 1 \right )} + \pi \left \lfloor{\frac{\frac{x}{2} - \frac{\pi }{2}}{\pi }}\right \rfloor \right )}{4} + \frac{\tan{\left (\frac{x}{2} \right )}}{4} - \frac{1}{4 \tan{\left (\frac{x}{2} \right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(1-cos(x)**4),x)

[Out]

sqrt(2)*(atan(sqrt(2)*tan(x/2) - 1) + pi*floor((x/2 - pi/2)/pi))/4 + sqrt(2)*(atan(sqrt(2)*tan(x/2) + 1) + pi*
floor((x/2 - pi/2)/pi))/4 + tan(x/2)/4 - 1/(4*tan(x/2))

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Giac [A]  time = 1.23848, size = 72, normalized size = 1.6 \begin{align*} \frac{1}{4} \, \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}{2 \, \tan \left (x\right )} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(1-cos(x)^4),x, algorithm="giac")

[Out]

1/4*sqrt(2)*(x + arctan(-(sqrt(2)*sin(2*x) - sin(2*x))/(sqrt(2)*cos(2*x) + sqrt(2) - cos(2*x) + 1))) - 1/2/tan
(x)