3.236 \(\int \frac{1}{1-\sin ^4(x)} \, dx\)

Optimal. Leaf size=25 \[ \frac{\tan ^{-1}\left (\sqrt{2} \tan (x)\right )}{2 \sqrt{2}}+\frac{\tan (x)}{2} \]

[Out]

ArcTan[Sqrt[2]*Tan[x]]/(2*Sqrt[2]) + Tan[x]/2

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Rubi [A]  time = 0.0192821, antiderivative size = 45, normalized size of antiderivative = 1.8, 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} \[ \frac{x}{2 \sqrt{2}}+\frac{\tan (x)}{2}+\frac{\tan ^{-1}\left (\frac{\sin (x) \cos (x)}{\sin ^2(x)+\sqrt{2}+1}\right )}{2 \sqrt{2}} \]

Antiderivative was successfully verified.

[In]

Int[(1 - Sin[x]^4)^(-1),x]

[Out]

x/(2*Sqrt[2]) + ArcTan[(Cos[x]*Sin[x])/(1 + Sqrt[2] + Sin[x]^2)]/(2*Sqrt[2]) + Tan[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-\sin ^4(x)} \, dx &=\operatorname{Subst}\left (\int \frac{1+x^2}{1+2 x^2} \, dx,x,\tan (x)\right )\\ &=\frac{\tan (x)}{2}+\frac{1}{2} \operatorname{Subst}\left (\int \frac{1}{1+2 x^2} \, dx,x,\tan (x)\right )\\ &=\frac{x}{2 \sqrt{2}}+\frac{\tan ^{-1}\left (\frac{\cos (x) \sin (x)}{1+\sqrt{2}+\sin ^2(x)}\right )}{2 \sqrt{2}}+\frac{\tan (x)}{2}\\ \end{align*}

Mathematica [A]  time = 0.0525903, size = 24, normalized size = 0.96 \[ \frac{1}{4} \left (\sqrt{2} \tan ^{-1}\left (\sqrt{2} \tan (x)\right )+2 \tan (x)\right ) \]

Antiderivative was successfully verified.

[In]

Integrate[(1 - Sin[x]^4)^(-1),x]

[Out]

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

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Maple [A]  time = 0.04, size = 18, normalized size = 0.7 \begin{align*}{\frac{\arctan \left ( \sqrt{2}\tan \left ( x \right ) \right ) \sqrt{2}}{4}}+{\frac{\tan \left ( x \right ) }{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]  time = 1.41544, size = 23, normalized size = 0.92 \begin{align*} \frac{1}{4} \, \sqrt{2} \arctan \left (\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-sin(x)^4),x, algorithm="maxima")

[Out]

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

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Fricas [B]  time = 1.9956, size = 138, normalized size = 5.52 \begin{align*} -\frac{\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 ) \cos \left (x\right ) - 4 \, \sin \left (x\right )}{8 \, \cos \left (x\right )} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} - \int \frac{1}{\sin ^{4}{\left (x \right )} - 1}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-Integral(1/(sin(x)**4 - 1), x)

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Giac [B]  time = 1.10003, size = 69, normalized size = 2.76 \begin{align*} \frac{1}{4} \, \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}{2} \, \tan \left (x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/4*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))) + 1/2
*tan(x)