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 = 0.03, antiderivative size = 3, normalized size of antiderivative = 1.00, 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 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 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])

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*}

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Mathematica [A]  time = 0.01, size = 3, normalized size = 1.00 \[ \tan ^{-1}(\sin (x)) \]

Antiderivative was successfully verified.

[In]

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

[Out]

ArcTan[Sin[x]]

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fricas [A]  time = 0.71, size = 3, normalized size = 1.00 \[ \arctan \left (\sin \relax (x)\right ) \]

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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giac [A]  time = 0.15, size = 3, normalized size = 1.00 \[ \arctan \left (\sin \relax (x)\right ) \]

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))

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maple [A]  time = 0.05, size = 6, normalized size = 2.00 \[ -\arctan \left (\csc \relax (x )\right ) \]

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 = 0.40, size = 3, normalized size = 1.00 \[ \arctan \left (\sin \relax (x)\right ) \]

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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mupad [B]  time = 3.22, size = 26, normalized size = 8.67 \[ \mathrm {atan}\left (\frac {{\mathrm {tan}\left (\frac {x}{2}\right )}^3}{2}+\frac {5\,\mathrm {tan}\left (\frac {x}{2}\right )}{2}\right )-\mathrm {atan}\left (\frac {\mathrm {tan}\left (\frac {x}{2}\right )}{2}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

atan((5*tan(x/2))/2 + tan(x/2)^3/2) - atan(tan(x/2)/2)

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sympy [A]  time = 0.19, size = 5, normalized size = 1.67 \[ - \operatorname {atan}{\left (\csc {\relax (x )} \right )} \]

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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