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

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Rubi [A]  time = 0.05, antiderivative size = 20, 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} \[ -\frac {2}{1-\cos (x)}-\log (1-\cos (x)) \]

Antiderivative was successfully verified.

[In]

Int[(Cot[x] + Csc[x])^3,x]

[Out]

-2/(1 - Cos[x]) - Log[1 - Cos[x]]

Rule 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))^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*}

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Mathematica [A]  time = 0.04, size = 20, normalized size = 1.00 \[ -\csc ^2\left (\frac {x}{2}\right )-2 \log \left (\sin \left (\frac {x}{2}\right )\right ) \]

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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fricas [A]  time = 1.68, size = 22, normalized size = 1.10 \[ -\frac {{\left (\cos \relax (x) - 1\right )} \log \left (-\frac {1}{2} \, \cos \relax (x) + \frac {1}{2}\right ) - 2}{\cos \relax (x) - 1} \]

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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giac [A]  time = 0.15, size = 18, normalized size = 0.90 \[ \frac {2}{\cos \relax (x) - 1} - \log \left (-\cos \relax (x) + 1\right ) \]

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)

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maple [B]  time = 0.08, size = 49, normalized size = 2.45 \[ -\frac {\left (\cot ^{2}\relax (x )\right )}{2}-\ln \left (\sin \relax (x )\right )-\frac {3 \left (\cos ^{3}\relax (x )\right )}{2 \sin \relax (x )^{2}}-\frac {3 \cos \relax (x )}{2}-\ln \left (\csc \relax (x )-\cot \relax (x )\right )-\frac {3}{2 \sin \relax (x )^{2}}-\frac {\cot \relax (x ) \csc \relax (x )}{2} \]

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 = 0.31, size = 46, normalized size = 2.30 \[ -\frac {3}{2} \, \cot \relax (x)^{2} + \frac {2 \, \cos \relax (x)}{\cos \relax (x)^{2} - 1} - \frac {1}{2 \, \sin \relax (x)^{2}} - \frac {1}{2} \, \log \left (\sin \relax (x)^{2}\right ) + \frac {1}{2} \, \log \left (\cos \relax (x) + 1\right ) - \frac {1}{2} \, \log \left (\cos \relax (x) - 1\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((cot(x)+csc(x))^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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mupad [B]  time = 2.39, size = 25, normalized size = 1.25 \[ \ln \left ({\mathrm {tan}\left (\frac {x}{2}\right )}^2+1\right )-2\,\ln \left (\mathrm {tan}\left (\frac {x}{2}\right )\right )-\frac {1}{{\mathrm {tan}\left (\frac {x}{2}\right )}^2} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((cot(x) + 1/sin(x))^3,x)

[Out]

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

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sympy [B]  time = 14.64, size = 46, normalized size = 2.30 \[ - \frac {\log {\left (\cos {\relax (x )} - 1 \right )}}{2} + \frac {\log {\left (\cos {\relax (x )} + 1 \right )}}{2} + \frac {\log {\left (- \csc ^{2}{\relax (x )} \right )}}{2} - 2 \csc ^{2}{\relax (x )} + \frac {4 \cos {\relax (x )}}{2 \cos ^{2}{\relax (x )} - 2} \]

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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