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))+ln(1-cos(x))

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Rubi [A]  time = 0.05, antiderivative size = 28, 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} \[ \frac {4}{1-\cos (x)}-\frac {2}{(1-\cos (x))^2}+\log (1-\cos (x)) \]

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 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 (\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*}

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Mathematica [A]  time = 0.07, 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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fricas [A]  time = 1.58, size = 37, normalized size = 1.32 \[ \frac {{\left (\cos \relax (x)^{2} - 2 \, \cos \relax (x) + 1\right )} \log \left (-\frac {1}{2} \, \cos \relax (x) + \frac {1}{2}\right ) - 4 \, \cos \relax (x) + 2}{\cos \relax (x)^{2} - 2 \, \cos \relax (x) + 1} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((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 = 0.79 \[ -\frac {2 \, {\left (2 \, \cos \relax (x) - 1\right )}}{{\left (\cos \relax (x) - 1\right )}^{2}} + \log \left (-\cos \relax (x) + 1\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((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 [B]  time = 0.11, size = 105, normalized size = 3.75 \[ -\frac {\left (\cot ^{4}\relax (x )\right )}{4}+\frac {\left (\cot ^{2}\relax (x )\right )}{2}+\ln \left (\sin \relax (x )\right )-\frac {5 \left (\cos ^{5}\relax (x )\right )}{4 \sin \relax (x )^{4}}+\frac {5 \left (\cos ^{5}\relax (x )\right )}{8 \sin \relax (x )^{2}}+\frac {5 \left (\cos ^{3}\relax (x )\right )}{8}+\frac {5 \cos \relax (x )}{8}+\ln \left (\csc \relax (x )-\cot \relax (x )\right )-\frac {5 \left (\cos ^{4}\relax (x )\right )}{2 \sin \relax (x )^{4}}-\frac {5 \left (\cos ^{3}\relax (x )\right )}{2 \sin \relax (x )^{4}}-\frac {5 \left (\cos ^{3}\relax (x )\right )}{4 \sin \relax (x )^{2}}-\frac {5}{4 \sin \relax (x )^{4}}+\left (-\frac {\left (\csc ^{3}\relax (x )\right )}{4}-\frac {3 \csc \relax (x )}{8}\right ) \cot \relax (x ) \]

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 = 0.31, size = 125, normalized size = 4.46 \[ -\frac {5}{2} \, \cot \relax (x)^{4} - \frac {5 \, {\left (5 \, \cos \relax (x)^{3} - 3 \, \cos \relax (x)\right )}}{8 \, {\left (\cos \relax (x)^{4} - 2 \, \cos \relax (x)^{2} + 1\right )}} + \frac {3 \, \cos \relax (x)^{3} - 5 \, \cos \relax (x)}{8 \, {\left (\cos \relax (x)^{4} - 2 \, \cos \relax (x)^{2} + 1\right )}} - \frac {5 \, {\left (\cos \relax (x)^{3} + \cos \relax (x)\right )}}{4 \, {\left (\cos \relax (x)^{4} - 2 \, \cos \relax (x)^{2} + 1\right )}} + \frac {4 \, \sin \relax (x)^{2} - 1}{4 \, \sin \relax (x)^{4}} - \frac {5}{4 \, \sin \relax (x)^{4}} + \frac {1}{2} \, \log \left (\sin \relax (x)^{2}\right ) - \frac {1}{2} \, \log \left (\cos \relax (x) + 1\right ) + \frac {1}{2} \, \log \left (\cos \relax (x) - 1\right ) \]

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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mupad [B]  time = 2.42, size = 34, normalized size = 1.21 \[ 2\,\ln \left (\mathrm {tan}\left (\frac {x}{2}\right )\right )-\ln \left ({\mathrm {tan}\left (\frac {x}{2}\right )}^2+1\right )+\frac {{\mathrm {tan}\left (\frac {x}{2}\right )}^2-\frac {1}{2}}{{\mathrm {tan}\left (\frac {x}{2}\right )}^4} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((cot(x) + 1/sin(x))^5,x)

[Out]

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

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sympy [B]  time = 107.21, size = 68, normalized size = 2.43 \[ \frac {\log {\left (\cos {\relax (x )} - 1 \right )}}{2} - \frac {\log {\left (\cos {\relax (x )} + 1 \right )}}{2} - \frac {\log {\left (\csc ^{2}{\relax (x )} \right )}}{2} - \frac {5 \cot ^{4}{\relax (x )}}{2} - \frac {3 \csc ^{4}{\relax (x )}}{2} + \csc ^{2}{\relax (x )} - \frac {32 \cos ^{3}{\relax (x )}}{8 \cos ^{4}{\relax (x )} - 16 \cos ^{2}{\relax (x )} + 8} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

log(cos(x) - 1)/2 - log(cos(x) + 1)/2 - log(csc(x)**2)/2 - 5*cot(x)**4/2 - 3*csc(x)**4/2 + csc(x)**2 - 32*cos(
x)**3/(8*cos(x)**4 - 16*cos(x)**2 + 8)

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