3.295 \(\int (\cot (x)+\csc (x))^3 \, dx\)

Optimal. Leaf size=20 \[ -\frac{2}{1-\cos (x)}-\log (1-\cos (x)) \]

[Out]

-2/(1 - Cos[x]) - Log[1 - Cos[x]]

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Rubi [A]  time = 0.0480181, antiderivative size = 20, 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} \[ -\frac{2}{1-\cos (x)}-\log (1-\cos (x)) \]

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

Mathematica [A]  time = 0.0394034, size = 20, normalized size = 1. \[ -\csc ^2\left (\frac{x}{2}\right )-2 \log \left (\sin \left (\frac{x}{2}\right )\right ) \]

Antiderivative was successfully verified.

[In]

Integrate[(Cot[x] + Csc[x])^3,x]

[Out]

-Csc[x/2]^2 - 2*Log[Sin[x/2]]

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Maple [B]  time = 0.037, size = 49, normalized size = 2.5 \begin{align*} -{\frac{ \left ( \cot \left ( x \right ) \right ) ^{2}}{2}}-\ln \left ( \sin \left ( x \right ) \right ) -{\frac{3\, \left ( \cos \left ( x \right ) \right ) ^{3}}{2\, \left ( \sin \left ( x \right ) \right ) ^{2}}}-{\frac{3\,\cos \left ( x \right ) }{2}}-\ln \left ( \csc \left ( x \right ) -\cot \left ( x \right ) \right ) -{\frac{3}{2\, \left ( \sin \left ( x \right ) \right ) ^{2}}}-{\frac{\cot \left ( x \right ) \csc \left ( x \right ) }{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/2*cot(x)^2-ln(sin(x))-3/2/sin(x)^2*cos(x)^3-3/2*cos(x)-ln(csc(x)-cot(x))-3/2/sin(x)^2-1/2*cot(x)*csc(x)

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Maxima [B]  time = 0.95826, size = 62, normalized size = 3.1 \begin{align*} -\frac{3}{2} \, \cot \left (x\right )^{2} + \frac{2 \, \cos \left (x\right )}{\cos \left (x\right )^{2} - 1} - \frac{1}{2 \, \sin \left (x\right )^{2}} - \frac{1}{2} \, \log \left (\sin \left (x\right )^{2}\right ) + \frac{1}{2} \, \log \left (\cos \left (x\right ) + 1\right ) - \frac{1}{2} \, \log \left (\cos \left (x\right ) - 1\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-3/2*cot(x)^2 + 2*cos(x)/(cos(x)^2 - 1) - 1/2/sin(x)^2 - 1/2*log(sin(x)^2) + 1/2*log(cos(x) + 1) - 1/2*log(cos
(x) - 1)

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Fricas [A]  time = 2.05848, size = 77, normalized size = 3.85 \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((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 [B]  time = 82.1944, size = 44, normalized size = 2.2 \begin{align*} - \frac{\log{\left (\cos{\left (x \right )} - 1 \right )}}{2} + \frac{\log{\left (\cos{\left (x \right )} + 1 \right )}}{2} + \frac{\log{\left (\csc ^{2}{\left (x \right )} \right )}}{2} - 2 \csc ^{2}{\left (x \right )} + \frac{4 \cos{\left (x \right )}}{2 \cos ^{2}{\left (x \right )} - 2} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-log(cos(x) - 1)/2 + log(cos(x) + 1)/2 + log(csc(x)**2)/2 - 2*csc(x)**2 + 4*cos(x)/(2*cos(x)**2 - 2)

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Giac [A]  time = 1.1422, size = 24, normalized size = 1.2 \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((cot(x)+csc(x))^3,x, algorithm="giac")

[Out]

2/(cos(x) - 1) - log(-cos(x) + 1)