3.491 \(\int \frac{1}{(\cot ^2(x)+\csc ^2(x))^2} \, dx\)

Optimal. Leaf size=47 \[ -\frac{x}{\sqrt{2}}+x-\frac{\tan (x)}{\tan ^2(x)+2}+\frac{\tan ^{-1}\left (\frac{\sin (x) \cos (x)}{\cos ^2(x)+\sqrt{2}+1}\right )}{\sqrt{2}} \]

[Out]

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

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Rubi [A]  time = 0.0400156, antiderivative size = 47, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 4, integrand size = 11, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.364, Rules used = {470, 12, 391, 203} \[ -\frac{x}{\sqrt{2}}+x-\frac{\tan (x)}{\tan ^2(x)+2}+\frac{\tan ^{-1}\left (\frac{\sin (x) \cos (x)}{\cos ^2(x)+\sqrt{2}+1}\right )}{\sqrt{2}} \]

Antiderivative was successfully verified.

[In]

Int[(Cot[x]^2 + Csc[x]^2)^(-2),x]

[Out]

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

Rule 470

Int[((e_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_)*((c_) + (d_.)*(x_)^(n_))^(q_), x_Symbol] :> -Simp[(a*e^(2
*n - 1)*(e*x)^(m - 2*n + 1)*(a + b*x^n)^(p + 1)*(c + d*x^n)^(q + 1))/(b*n*(b*c - a*d)*(p + 1)), x] + Dist[e^(2
*n)/(b*n*(b*c - a*d)*(p + 1)), Int[(e*x)^(m - 2*n)*(a + b*x^n)^(p + 1)*(c + d*x^n)^q*Simp[a*c*(m - 2*n + 1) +
(a*d*(m - n + n*q + 1) + b*c*n*(p + 1))*x^n, x], x], x] /; FreeQ[{a, b, c, d, e, q}, x] && NeQ[b*c - a*d, 0] &
& IGtQ[n, 0] && LtQ[p, -1] && GtQ[m - n + 1, n] && IntBinomialQ[a, b, c, d, e, m, n, p, q, x]

Rule 12

Int[(a_)*(u_), x_Symbol] :> Dist[a, Int[u, x], x] /; FreeQ[a, x] &&  !MatchQ[u, (b_)*(v_) /; FreeQ[b, x]]

Rule 391

Int[1/(((a_) + (b_.)*(x_)^(n_))*((c_) + (d_.)*(x_)^(n_))), x_Symbol] :> Dist[b/(b*c - a*d), Int[1/(a + b*x^n),
x], x] - Dist[d/(b*c - a*d), Int[1/(c + d*x^n), x], x] /; FreeQ[{a, b, c, d, n}, x] && NeQ[b*c - a*d, 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}{\left (\cot ^2(x)+\csc ^2(x)\right )^2} \, dx &=\operatorname{Subst}\left (\int \frac{x^4}{\left (1+x^2\right ) \left (2+x^2\right )^2} \, dx,x,\tan (x)\right )\\ &=-\frac{\tan (x)}{2+\tan ^2(x)}+\frac{1}{2} \operatorname{Subst}\left (\int \frac{2}{\left (1+x^2\right ) \left (2+x^2\right )} \, dx,x,\tan (x)\right )\\ &=-\frac{\tan (x)}{2+\tan ^2(x)}+\operatorname{Subst}\left (\int \frac{1}{\left (1+x^2\right ) \left (2+x^2\right )} \, dx,x,\tan (x)\right )\\ &=-\frac{\tan (x)}{2+\tan ^2(x)}+\operatorname{Subst}\left (\int \frac{1}{1+x^2} \, dx,x,\tan (x)\right )-\operatorname{Subst}\left (\int \frac{1}{2+x^2} \, dx,x,\tan (x)\right )\\ &=x-\frac{x}{\sqrt{2}}+\frac{\tan ^{-1}\left (\frac{\cos (x) \sin (x)}{1+\sqrt{2}+\cos ^2(x)}\right )}{\sqrt{2}}-\frac{\tan (x)}{2+\tan ^2(x)}\\ \end{align*}

Mathematica [A]  time = 0.106036, size = 64, normalized size = 1.36 \[ \frac{(\cos (2 x)+3) \csc ^4(x) \left (6 x-2 \sin (2 x)+2 x \cos (2 x)-\sqrt{2} (\cos (2 x)+3) \tan ^{-1}\left (\frac{\tan (x)}{\sqrt{2}}\right )\right )}{8 \left (\cot ^2(x)+\csc ^2(x)\right )^2} \]

Antiderivative was successfully verified.

[In]

Integrate[(Cot[x]^2 + Csc[x]^2)^(-2),x]

[Out]

((3 + Cos[2*x])*Csc[x]^4*(6*x + 2*x*Cos[2*x] - Sqrt[2]*ArcTan[Tan[x]/Sqrt[2]]*(3 + Cos[2*x]) - 2*Sin[2*x]))/(8
*(Cot[x]^2 + Csc[x]^2)^2)

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Maple [A]  time = 0.103, size = 28, normalized size = 0.6 \begin{align*} -{\frac{\tan \left ( x \right ) }{2+ \left ( \tan \left ( x \right ) \right ) ^{2}}}-{\frac{\sqrt{2}}{2}\arctan \left ({\frac{\tan \left ( x \right ) \sqrt{2}}{2}} \right ) }+x \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(cot(x)^2+csc(x)^2)^2,x)

[Out]

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

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Maxima [A]  time = 1.49496, size = 36, normalized size = 0.77 \begin{align*} -\frac{1}{2} \, \sqrt{2} \arctan \left (\frac{1}{2} \, \sqrt{2} \tan \left (x\right )\right ) + x - \frac{\tan \left (x\right )}{\tan \left (x\right )^{2} + 2} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(cot(x)^2+csc(x)^2)^2,x, algorithm="maxima")

[Out]

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

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Fricas [A]  time = 1.9151, size = 201, normalized size = 4.28 \begin{align*} \frac{4 \, x \cos \left (x\right )^{2} +{\left (\sqrt{2} \cos \left (x\right )^{2} + \sqrt{2}\right )} \arctan \left (\frac{3 \, \sqrt{2} \cos \left (x\right )^{2} - \sqrt{2}}{4 \, \cos \left (x\right ) \sin \left (x\right )}\right ) - 4 \, \cos \left (x\right ) \sin \left (x\right ) + 4 \, x}{4 \,{\left (\cos \left (x\right )^{2} + 1\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(cot(x)^2+csc(x)^2)^2,x, algorithm="fricas")

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(cot(x)**2+csc(x)**2)**2,x)

[Out]

Timed out

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Giac [A]  time = 1.15392, size = 81, normalized size = 1.72 \begin{align*} -\frac{1}{2} \, \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 )} + x - \frac{\tan \left (x\right )}{\tan \left (x\right )^{2} + 2} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(cot(x)^2+csc(x)^2)^2,x, algorithm="giac")

[Out]

-1/2*sqrt(2)*(x + arctan(-(sqrt(2)*sin(2*x) - sin(2*x))/(sqrt(2)*cos(2*x) + sqrt(2) - cos(2*x) + 1))) + x - ta
n(x)/(tan(x)^2 + 2)