3.296 \(\int (\cot (x)+\csc (x))^2 \, dx\)

Optimal. Leaf size=16 \[ -x-\frac{2 \sin (x)}{1-\cos (x)} \]

[Out]

-x - (2*Sin[x])/(1 - Cos[x])

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Rubi [A]  time = 0.0688719, antiderivative size = 16, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 7, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.571, Rules used = {4392, 2670, 2680, 8} \[ -x-\frac{2 \sin (x)}{1-\cos (x)} \]

Antiderivative was successfully verified.

[In]

Int[(Cot[x] + Csc[x])^2,x]

[Out]

-x - (2*Sin[x])/(1 - Cos[x])

Rule 4392

Int[(cot[(c_.) + (d_.)*(x_)]^(n_.)*(a_.) + csc[(c_.) + (d_.)*(x_)]^(n_.)*(b_.))^(p_)*(u_.), x_Symbol] :> Int[A
ctivateTrig[u]*Csc[c + d*x]^(n*p)*(b + a*Cos[c + d*x]^n)^p, x] /; FreeQ[{a, b, c, d}, x] && IntegersQ[n, p]

Rule 2670

Int[(cos[(e_.) + (f_.)*(x_)]*(g_.))^(p_)*((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_), x_Symbol] :> Dist[(a/g)^
(2*m), Int[(g*Cos[e + f*x])^(2*m + p)/(a - b*Sin[e + f*x])^m, x], x] /; FreeQ[{a, b, e, f, g}, x] && EqQ[a^2 -
b^2, 0] && IntegerQ[m] && LtQ[p, -1] && GeQ[2*m + p, 0]

Rule 2680

Int[(cos[(e_.) + (f_.)*(x_)]*(g_.))^(p_)*((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_), x_Symbol] :> Simp[(2*g*(
g*Cos[e + f*x])^(p - 1)*(a + b*Sin[e + f*x])^(m + 1))/(b*f*(2*m + p + 1)), x] + Dist[(g^2*(p - 1))/(b^2*(2*m +
p + 1)), Int[(g*Cos[e + f*x])^(p - 2)*(a + b*Sin[e + f*x])^(m + 2), x], x] /; FreeQ[{a, b, e, f, g}, x] && Eq
Q[a^2 - b^2, 0] && LeQ[m, -2] && GtQ[p, 1] && NeQ[2*m + p + 1, 0] &&  !ILtQ[m + p + 1, 0] && IntegersQ[2*m, 2*
p]

Rule 8

Int[a_, x_Symbol] :> Simp[a*x, x] /; FreeQ[a, x]

Rubi steps

\begin{align*} \int (\cot (x)+\csc (x))^2 \, dx &=\int (1+\cos (x))^2 \csc ^2(x) \, dx\\ &=\int \frac{\sin ^2(x)}{(1-\cos (x))^2} \, dx\\ &=-\frac{2 \sin (x)}{1-\cos (x)}-\int 1 \, dx\\ &=-x-\frac{2 \sin (x)}{1-\cos (x)}\\ \end{align*}

Mathematica [A]  time = 0.0236012, size = 12, normalized size = 0.75 \[ -x-2 \cot \left (\frac{x}{2}\right ) \]

Antiderivative was successfully verified.

[In]

Integrate[(Cot[x] + Csc[x])^2,x]

[Out]

-x - 2*Cot[x/2]

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Maple [A]  time = 0.012, size = 15, normalized size = 0.9 \begin{align*} -2\,\cot \left ( x \right ) -x-2\, \left ( \sin \left ( x \right ) \right ) ^{-1} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-2*cot(x)-x-2/sin(x)

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Maxima [A]  time = 1.49362, size = 22, normalized size = 1.38 \begin{align*} -x - \frac{2}{\sin \left (x\right )} - \frac{2}{\tan \left (x\right )} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-x - 2/sin(x) - 2/tan(x)

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Fricas [A]  time = 1.95338, size = 47, normalized size = 2.94 \begin{align*} -\frac{x \sin \left (x\right ) + 2 \, \cos \left (x\right ) + 2}{\sin \left (x\right )} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-(x*sin(x) + 2*cos(x) + 2)/sin(x)

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Sympy [A]  time = 8.48301, size = 17, normalized size = 1.06 \begin{align*} - x - \cot{\left (x \right )} - 2 \csc{\left (x \right )} - \frac{\cos{\left (x \right )}}{\sin{\left (x \right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-x - cot(x) - 2*csc(x) - cos(x)/sin(x)

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Giac [A]  time = 1.15252, size = 16, normalized size = 1. \begin{align*} -x - \frac{2}{\tan \left (\frac{1}{2} \, x\right )} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-x - 2/tan(1/2*x)