3.302 \(\int \frac {1}{(\cot (x)+\csc (x))^5} \, dx\)

Optimal. Leaf size=24 \[ -\frac {4}{\cos (x)+1}+\frac {2}{(\cos (x)+1)^2}-\log (\cos (x)+1) \]

[Out]

2/(1+cos(x))^2-4/(1+cos(x))-ln(1+cos(x))

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Rubi [A]  time = 0.05, antiderivative size = 24, normalized size of antiderivative = 1.00, 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} \[ -\frac {4}{\cos (x)+1}+\frac {2}{(\cos (x)+1)^2}-\log (\cos (x)+1) \]

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 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])

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 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]

Rubi steps

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

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Mathematica [A]  time = 0.01, size = 32, normalized size = 1.33 \[ \frac {1}{2} \sec ^4\left (\frac {x}{2}\right )-2 \sec ^2\left (\frac {x}{2}\right )-2 \log \left (\cos \left (\frac {x}{2}\right )\right ) \]

Antiderivative was successfully verified.

[In]

Integrate[(Cot[x] + Csc[x])^(-5),x]

[Out]

-2*Log[Cos[x/2]] - 2*Sec[x/2]^2 + Sec[x/2]^4/2

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fricas [A]  time = 1.99, size = 38, normalized size = 1.58 \[ -\frac {{\left (\cos \relax (x)^{2} + 2 \, \cos \relax (x) + 1\right )} \log \left (\frac {1}{2} \, \cos \relax (x) + \frac {1}{2}\right ) + 4 \, \cos \relax (x) + 2}{\cos \relax (x)^{2} + 2 \, \cos \relax (x) + 1} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(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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giac [A]  time = 0.12, size = 22, normalized size = 0.92 \[ -\frac {2 \, {\left (2 \, \cos \relax (x) + 1\right )}}{{\left (\cos \relax (x) + 1\right )}^{2}} - \log \left (\cos \relax (x) + 1\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(cot(x)+csc(x))^5,x, algorithm="giac")

[Out]

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

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maple [A]  time = 0.16, size = 25, normalized size = 1.04 \[ \frac {2}{\left (1+\cos \relax (x )\right )^{2}}-\frac {4}{1+\cos \relax (x )}-\ln \left (1+\cos \relax (x )\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(cot(x)+csc(x))^5,x)

[Out]

2/(1+cos(x))^2-4/(1+cos(x))-ln(1+cos(x))

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maxima [A]  time = 0.41, size = 39, normalized size = 1.62 \[ -\frac {\sin \relax (x)^{2}}{{\left (\cos \relax (x) + 1\right )}^{2}} + \frac {\sin \relax (x)^{4}}{2 \, {\left (\cos \relax (x) + 1\right )}^{4}} + \log \left (\frac {\sin \relax (x)^{2}}{{\left (\cos \relax (x) + 1\right )}^{2}} + 1\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(cot(x)+csc(x))^5,x, algorithm="maxima")

[Out]

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

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mupad [B]  time = 2.41, size = 26, normalized size = 1.08 \[ \ln \left ({\mathrm {tan}\left (\frac {x}{2}\right )}^2+1\right )-{\mathrm {tan}\left (\frac {x}{2}\right )}^2+\frac {{\mathrm {tan}\left (\frac {x}{2}\right )}^4}{2} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(cot(x) + 1/sin(x))^5,x)

[Out]

log(tan(x/2)^2 + 1) - tan(x/2)^2 + tan(x/2)^4/2

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sympy [F(-1)]  time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(cot(x)+csc(x))**5,x)

[Out]

Timed out

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