3.300 \(\int \frac{1}{(\cot (x)+\csc (x))^3} \, dx\)

Optimal. Leaf size=14 \[ \frac{2}{\cos (x)+1}+\log (\cos (x)+1) \]

[Out]

2/(1 + Cos[x]) + Log[1 + Cos[x]]

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Rubi [A]  time = 0.0465222, antiderivative size = 14, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 7, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.429, Rules used = {4392, 2667, 43} \[ \frac{2}{\cos (x)+1}+\log (\cos (x)+1) \]

Antiderivative was successfully verified.

[In]

Int[(Cot[x] + Csc[x])^(-3),x]

[Out]

2/(1 + Cos[x]) + Log[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 2667

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

Rule 43

Int[((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_.), x_Symbol] :> Int[ExpandIntegrand[(a + b*x)^m*(c + d
*x)^n, x], x] /; FreeQ[{a, b, c, d, n}, x] && NeQ[b*c - a*d, 0] && IGtQ[m, 0] && ( !IntegerQ[n] || (EqQ[c, 0]
&& LeQ[7*m + 4*n + 4, 0]) || LtQ[9*m + 5*(n + 1), 0] || GtQ[m + n + 2, 0])

Rubi steps

\begin{align*} \int \frac{1}{(\cot (x)+\csc (x))^3} \, dx &=\int \frac{\sin ^3(x)}{(1+\cos (x))^3} \, dx\\ &=-\operatorname{Subst}\left (\int \frac{1-x}{(1+x)^2} \, dx,x,\cos (x)\right )\\ &=-\operatorname{Subst}\left (\int \left (\frac{1}{-1-x}+\frac{2}{(1+x)^2}\right ) \, dx,x,\cos (x)\right )\\ &=\frac{2}{1+\cos (x)}+\log (1+\cos (x))\\ \end{align*}

Mathematica [A]  time = 0.0139392, size = 18, normalized size = 1.29 \[ \sec ^2\left (\frac{x}{2}\right )+2 \log \left (\cos \left (\frac{x}{2}\right )\right ) \]

Antiderivative was successfully verified.

[In]

Integrate[(Cot[x] + Csc[x])^(-3),x]

[Out]

2*Log[Cos[x/2]] + Sec[x/2]^2

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Maple [A]  time = 0.073, size = 15, normalized size = 1.1 \begin{align*} 2\, \left ( 1+\cos \left ( x \right ) \right ) ^{-1}+\ln \left ( 1+\cos \left ( x \right ) \right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(cot(x)+csc(x))^3,x)

[Out]

2/(1+cos(x))+ln(1+cos(x))

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Maxima [A]  time = 1.51203, size = 38, normalized size = 2.71 \begin{align*} \frac{\sin \left (x\right )^{2}}{{\left (\cos \left (x\right ) + 1\right )}^{2}} - \log \left (\frac{\sin \left (x\right )^{2}}{{\left (\cos \left (x\right ) + 1\right )}^{2}} + 1\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(cot(x)+csc(x))^3,x, algorithm="maxima")

[Out]

sin(x)^2/(cos(x) + 1)^2 - log(sin(x)^2/(cos(x) + 1)^2 + 1)

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Fricas [A]  time = 1.93883, size = 74, normalized size = 5.29 \begin{align*} \frac{{\left (\cos \left (x\right ) + 1\right )} \log \left (\frac{1}{2} \, \cos \left (x\right ) + \frac{1}{2}\right ) + 2}{\cos \left (x\right ) + 1} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(cot(x)+csc(x))^3,x, algorithm="fricas")

[Out]

((cos(x) + 1)*log(1/2*cos(x) + 1/2) + 2)/(cos(x) + 1)

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Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(cot(x)+csc(x))**3,x)

[Out]

Timed out

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Giac [A]  time = 1.13826, size = 19, normalized size = 1.36 \begin{align*} \frac{2}{\cos \left (x\right ) + 1} + \log \left (\cos \left (x\right ) + 1\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(cot(x)+csc(x))^3,x, algorithm="giac")

[Out]

2/(cos(x) + 1) + log(cos(x) + 1)