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

Optimal. Leaf size=3 \[ \tan ^{-1}(\sin (x)) \]

[Out]

ArcTan[Sin[x]]

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Rubi [A]  time = 0.0321606, antiderivative size = 3, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 13, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.154, Rules used = {4338, 203} \[ \tan ^{-1}(\sin (x)) \]

Antiderivative was successfully verified.

[In]

Int[(Cot[x]*Csc[x])/(1 + Csc[x]^2),x]

[Out]

ArcTan[Sin[x]]

Rule 4338

Int[(u_)*(F_)[(c_.)*((a_.) + (b_.)*(x_))], x_Symbol] :> With[{d = FreeFactors[Sin[c*(a + b*x)], x]}, Dist[1/(b
*c), Subst[Int[SubstFor[1/x, Sin[c*(a + b*x)]/d, u, x], x], x, Sin[c*(a + b*x)]/d], x] /; FunctionOfQ[Sin[c*(a
+ b*x)]/d, u, x, True]] /; FreeQ[{a, b, c}, x] && (EqQ[F, Cot] || EqQ[F, cot])

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

Mathematica [A]  time = 0.0153911, size = 3, normalized size = 1. \[ \tan ^{-1}(\sin (x)) \]

Antiderivative was successfully verified.

[In]

Integrate[(Cot[x]*Csc[x])/(1 + Csc[x]^2),x]

[Out]

ArcTan[Sin[x]]

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Maple [A]  time = 0.015, size = 6, normalized size = 2. \begin{align*} -\arctan \left ( \csc \left ( x \right ) \right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-arctan(csc(x))

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Maxima [A]  time = 1.45583, size = 4, normalized size = 1.33 \begin{align*} \arctan \left (\sin \left (x\right )\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

arctan(sin(x))

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Fricas [A]  time = 2.37124, size = 22, normalized size = 7.33 \begin{align*} \arctan \left (\sin \left (x\right )\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

arctan(sin(x))

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Sympy [A]  time = 0.231967, size = 5, normalized size = 1.67 \begin{align*} - \operatorname{atan}{\left (\csc{\left (x \right )} \right )} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-atan(csc(x))

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Giac [A]  time = 1.10593, size = 4, normalized size = 1.33 \begin{align*} \arctan \left (\sin \left (x\right )\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

arctan(sin(x))