3.293 \(\int (\cot (x)+\csc (x))^5 \, dx\)

Optimal. Leaf size=28 \[ \frac{4}{1-\cos (x)}-\frac{2}{(1-\cos (x))^2}+\log (1-\cos (x)) \]

[Out]

-2/(1 - Cos[x])^2 + 4/(1 - Cos[x]) + Log[1 - Cos[x]]

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Rubi [A]  time = 0.0510641, antiderivative size = 28, 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} \[ \frac{4}{1-\cos (x)}-\frac{2}{(1-\cos (x))^2}+\log (1-\cos (x)) \]

Antiderivative was successfully verified.

[In]

Int[(Cot[x] + Csc[x])^5,x]

[Out]

-2/(1 - Cos[x])^2 + 4/(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))^5 \, dx &=\int (1+\cos (x))^5 \csc ^5(x) \, dx\\ &=-\operatorname{Subst}\left (\int \frac{(1+x)^2}{(1-x)^3} \, dx,x,\cos (x)\right )\\ &=-\operatorname{Subst}\left (\int \left (\frac{1}{1-x}-\frac{4}{(-1+x)^3}-\frac{4}{(-1+x)^2}\right ) \, dx,x,\cos (x)\right )\\ &=-\frac{2}{(1-\cos (x))^2}+\frac{4}{1-\cos (x)}+\log (1-\cos (x))\\ \end{align*}

Mathematica [A]  time = 0.0734023, size = 32, normalized size = 1.14 \[ -\frac{1}{2} \csc ^4\left (\frac{x}{2}\right )+2 \csc ^2\left (\frac{x}{2}\right )+2 \log \left (\sin \left (\frac{x}{2}\right )\right ) \]

Antiderivative was successfully verified.

[In]

Integrate[(Cot[x] + Csc[x])^5,x]

[Out]

2*Csc[x/2]^2 - Csc[x/2]^4/2 + 2*Log[Sin[x/2]]

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Maple [B]  time = 0.048, size = 105, normalized size = 3.8 \begin{align*} -{\frac{ \left ( \cot \left ( x \right ) \right ) ^{4}}{4}}+{\frac{ \left ( \cot \left ( x \right ) \right ) ^{2}}{2}}+\ln \left ( \sin \left ( x \right ) \right ) -{\frac{5\, \left ( \cos \left ( x \right ) \right ) ^{5}}{4\, \left ( \sin \left ( x \right ) \right ) ^{4}}}+{\frac{5\, \left ( \cos \left ( x \right ) \right ) ^{5}}{8\, \left ( \sin \left ( x \right ) \right ) ^{2}}}+{\frac{5\, \left ( \cos \left ( x \right ) \right ) ^{3}}{8}}+{\frac{5\,\cos \left ( x \right ) }{8}}+\ln \left ( \csc \left ( x \right ) -\cot \left ( x \right ) \right ) -{\frac{5\, \left ( \cos \left ( x \right ) \right ) ^{4}}{2\, \left ( \sin \left ( x \right ) \right ) ^{4}}}-{\frac{5\, \left ( \cos \left ( x \right ) \right ) ^{3}}{2\, \left ( \sin \left ( x \right ) \right ) ^{4}}}-{\frac{5\, \left ( \cos \left ( x \right ) \right ) ^{3}}{4\, \left ( \sin \left ( x \right ) \right ) ^{2}}}-{\frac{5}{4\, \left ( \sin \left ( x \right ) \right ) ^{4}}}+ \left ( -{\frac{ \left ( \csc \left ( x \right ) \right ) ^{3}}{4}}-{\frac{3\,\csc \left ( x \right ) }{8}} \right ) \cot \left ( x \right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((cot(x)+csc(x))^5,x)

[Out]

-1/4*cot(x)^4+1/2*cot(x)^2+ln(sin(x))-5/4/sin(x)^4*cos(x)^5+5/8/sin(x)^2*cos(x)^5+5/8*cos(x)^3+5/8*cos(x)+ln(c
sc(x)-cot(x))-5/2/sin(x)^4*cos(x)^4-5/2/sin(x)^4*cos(x)^3-5/4/sin(x)^2*cos(x)^3-5/4/sin(x)^4+(-1/4*csc(x)^3-3/
8*csc(x))*cot(x)

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Maxima [B]  time = 1.01307, size = 169, normalized size = 6.04 \begin{align*} -\frac{5}{2} \, \cot \left (x\right )^{4} - \frac{5 \,{\left (5 \, \cos \left (x\right )^{3} - 3 \, \cos \left (x\right )\right )}}{8 \,{\left (\cos \left (x\right )^{4} - 2 \, \cos \left (x\right )^{2} + 1\right )}} + \frac{3 \, \cos \left (x\right )^{3} - 5 \, \cos \left (x\right )}{8 \,{\left (\cos \left (x\right )^{4} - 2 \, \cos \left (x\right )^{2} + 1\right )}} - \frac{5 \,{\left (\cos \left (x\right )^{3} + \cos \left (x\right )\right )}}{4 \,{\left (\cos \left (x\right )^{4} - 2 \, \cos \left (x\right )^{2} + 1\right )}} + \frac{4 \, \sin \left (x\right )^{2} - 1}{4 \, \sin \left (x\right )^{4}} - \frac{5}{4 \, \sin \left (x\right )^{4}} + \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))^5,x, algorithm="maxima")

[Out]

-5/2*cot(x)^4 - 5/8*(5*cos(x)^3 - 3*cos(x))/(cos(x)^4 - 2*cos(x)^2 + 1) + 1/8*(3*cos(x)^3 - 5*cos(x))/(cos(x)^
4 - 2*cos(x)^2 + 1) - 5/4*(cos(x)^3 + cos(x))/(cos(x)^4 - 2*cos(x)^2 + 1) + 1/4*(4*sin(x)^2 - 1)/sin(x)^4 - 5/
4/sin(x)^4 + 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 = 2.03381, size = 126, normalized size = 4.5 \begin{align*} \frac{{\left (\cos \left (x\right )^{2} - 2 \, \cos \left (x\right ) + 1\right )} \log \left (-\frac{1}{2} \, \cos \left (x\right ) + \frac{1}{2}\right ) - 4 \, \cos \left (x\right ) + 2}{\cos \left (x\right )^{2} - 2 \, \cos \left (x\right ) + 1} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((cot(x)+csc(x))^5,x, algorithm="fricas")

[Out]

((cos(x)^2 - 2*cos(x) + 1)*log(-1/2*cos(x) + 1/2) - 4*cos(x) + 2)/(cos(x)^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((cot(x)+csc(x))**5,x)

[Out]

Timed out

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Giac [A]  time = 1.14025, size = 30, normalized size = 1.07 \begin{align*} -\frac{2 \,{\left (2 \, \cos \left (x\right ) - 1\right )}}{{\left (\cos \left (x\right ) - 1\right )}^{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))^5,x, algorithm="giac")

[Out]

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