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

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Rubi [A]  time = 0.04, antiderivative size = 47, normalized size of antiderivative = 1.00, 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 12

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

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*}

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Mathematica [A]  time = 0.11, 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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fricas [A]  time = 1.07, size = 66, normalized size = 1.40 \[ \frac {4 \, x \cos \relax (x)^{2} + {\left (\sqrt {2} \cos \relax (x)^{2} + \sqrt {2}\right )} \arctan \left (\frac {3 \, \sqrt {2} \cos \relax (x)^{2} - \sqrt {2}}{4 \, \cos \relax (x) \sin \relax (x)}\right ) - 4 \, \cos \relax (x) \sin \relax (x) + 4 \, x}{4 \, {\left (\cos \relax (x)^{2} + 1\right )}} \]

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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giac [A]  time = 0.14, size = 60, normalized size = 1.28 \[ -\frac {1}{2} \, \sqrt {2} {\left (x + \arctan \left (-\frac {\sqrt {2} \sin \left (2 \, x\right ) - \sin \left (2 \, x\right )}{\sqrt {2} \cos \left (2 \, x\right ) + \sqrt {2} - \cos \left (2 \, x\right ) + 1}\right )\right )} + x - \frac {\tan \relax (x)}{\tan \relax (x)^{2} + 2} \]

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)

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maple [A]  time = 0.18, size = 28, normalized size = 0.60 \[ -\frac {\tan \relax (x )}{2+\tan ^{2}\relax (x )}-\frac {\sqrt {2}\, \arctan \left (\frac {\sqrt {2}\, \tan \relax (x )}{2}\right )}{2}+x \]

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

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maxima [A]  time = 0.40, size = 27, normalized size = 0.57 \[ -\frac {1}{2} \, \sqrt {2} \arctan \left (\frac {1}{2} \, \sqrt {2} \tan \relax (x)\right ) + x - \frac {\tan \relax (x)}{\tan \relax (x)^{2} + 2} \]

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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mupad [B]  time = 2.69, size = 27, normalized size = 0.57 \[ x-\frac {\mathrm {tan}\relax (x)}{{\mathrm {tan}\relax (x)}^2+2}-\frac {\sqrt {2}\,\mathrm {atan}\left (\frac {\sqrt {2}\,\mathrm {tan}\relax (x)}{2}\right )}{2} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(cot(x)^2 + 1/sin(x)^2)^2,x)

[Out]

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

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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {1}{\left (\cot ^{2}{\relax (x )} + \csc ^{2}{\relax (x )}\right )^{2}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Integral((cot(x)**2 + csc(x)**2)**(-2), x)

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