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

Optimal. Leaf size=6 \[ x-\cot (x) \]

[Out]

x - Cot[x]

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Rubi [A]  time = 0.0470889, antiderivative size = 6, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 2, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.133, Rules used = {14, 203} \[ x-\cot (x) \]

Antiderivative was successfully verified.

[In]

Int[(1 + (1 + Cot[x]^2)^(-1))*Csc[x]^2,x]

[Out]

x - Cot[x]

Rule 14

Int[(u_)*((c_.)*(x_))^(m_.), x_Symbol] :> Int[ExpandIntegrand[(c*x)^m*u, x], x] /; FreeQ[{c, m}, x] && SumQ[u]
&&  !LinearQ[u, x] &&  !MatchQ[u, (a_) + (b_.)*(v_) /; FreeQ[{a, b}, x] && InverseFunctionQ[v]]

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

Mathematica [A]  time = 0.0053662, size = 6, normalized size = 1. \[ x-\cot (x) \]

Antiderivative was successfully verified.

[In]

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

[Out]

x - Cot[x]

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Maple [A]  time = 0.025, size = 7, normalized size = 1.2 \begin{align*} x-\cot \left ( x \right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

x-cot(x)

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Maxima [A]  time = 1.4416, size = 11, normalized size = 1.83 \begin{align*} x - \frac{1}{\tan \left (x\right )} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

x - 1/tan(x)

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Fricas [B]  time = 2.31265, size = 38, normalized size = 6.33 \begin{align*} \frac{x \sin \left (x\right ) - \cos \left (x\right )}{\sin \left (x\right )} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(x*sin(x) - cos(x))/sin(x)

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Sympy [B]  time = 0.938944, size = 27, normalized size = 4.5 \begin{align*} \frac{x \csc ^{2}{\left (x \right )}}{\cot ^{2}{\left (x \right )} + 1} - \frac{\cot{\left (x \right )} \csc ^{2}{\left (x \right )}}{\cot ^{2}{\left (x \right )} + 1} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

x*csc(x)**2/(cot(x)**2 + 1) - cot(x)*csc(x)**2/(cot(x)**2 + 1)

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Giac [B]  time = 1.11885, size = 22, normalized size = 3.67 \begin{align*} x - \frac{1}{2 \, \tan \left (\frac{1}{2} \, x\right )} + \frac{1}{2} \, \tan \left (\frac{1}{2} \, x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

x - 1/2/tan(1/2*x) + 1/2*tan(1/2*x)