3.115 \(\int \cos ^2(x) \cot ^3(x) \, dx\)

Optimal. Leaf size=22 \[ \frac {\sin ^2(x)}{2}-\frac {1}{2} \csc ^2(x)-2 \log (\sin (x)) \]

[Out]

-1/2*csc(x)^2-2*ln(sin(x))+1/2*sin(x)^2

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Rubi [A]  time = 0.05, antiderivative size = 22, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, integrand size = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.333, Rules used = {2590, 266, 43} \[ \frac {\sin ^2(x)}{2}-\frac {1}{2} \csc ^2(x)-2 \log (\sin (x)) \]

Antiderivative was successfully verified.

[In]

Int[Cos[x]^2*Cot[x]^3,x]

[Out]

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

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])

Rule 266

Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Dist[1/n, Subst[Int[x^(Simplify[(m + 1)/n] - 1)*(a
+ b*x)^p, x], x, x^n], x] /; FreeQ[{a, b, m, n, p}, x] && IntegerQ[Simplify[(m + 1)/n]]

Rule 2590

Int[sin[(e_.) + (f_.)*(x_)]^(m_.)*tan[(e_.) + (f_.)*(x_)]^(n_.), x_Symbol] :> -Dist[f^(-1), Subst[Int[(1 - x^2
)^((m + n - 1)/2)/x^n, x], x, Cos[e + f*x]], x] /; FreeQ[{e, f}, x] && IntegersQ[m, n, (m + n - 1)/2]

Rubi steps

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

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Mathematica [A]  time = 0.01, size = 20, normalized size = 0.91 \[ \frac {1}{2} \left (\sin ^2(x)-\csc ^2(x)-4 \log (\sin (x))\right ) \]

Antiderivative was successfully verified.

[In]

Integrate[Cos[x]^2*Cot[x]^3,x]

[Out]

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

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fricas [B]  time = 0.45, size = 37, normalized size = 1.68 \[ -\frac {2 \, \cos \relax (x)^{4} - 3 \, \cos \relax (x)^{2} + 8 \, {\left (\cos \relax (x)^{2} - 1\right )} \log \left (\frac {1}{2} \, \sin \relax (x)\right ) - 1}{4 \, {\left (\cos \relax (x)^{2} - 1\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cos(x)^2*cot(x)^3,x, algorithm="fricas")

[Out]

-1/4*(2*cos(x)^4 - 3*cos(x)^2 + 8*(cos(x)^2 - 1)*log(1/2*sin(x)) - 1)/(cos(x)^2 - 1)

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giac [A]  time = 0.85, size = 28, normalized size = 1.27 \[ -\frac {1}{2} \, \cos \relax (x)^{2} + \frac {1}{2 \, {\left (\cos \relax (x)^{2} - 1\right )}} - \log \left (-\cos \relax (x)^{2} + 1\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cos(x)^2*cot(x)^3,x, algorithm="giac")

[Out]

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

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maple [A]  time = 0.03, size = 29, normalized size = 1.32 \[ -\frac {\cos ^{6}\relax (x )}{2 \sin \relax (x )^{2}}-\frac {\left (\cos ^{4}\relax (x )\right )}{2}-\left (\cos ^{2}\relax (x )\right )-2 \ln \left (\sin \relax (x )\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(cos(x)^2*cot(x)^3,x)

[Out]

-1/2*cos(x)^6/sin(x)^2-1/2*cos(x)^4-cos(x)^2-2*ln(sin(x))

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maxima [A]  time = 0.42, size = 20, normalized size = 0.91 \[ \frac {1}{2} \, \sin \relax (x)^{2} - \frac {1}{2 \, \sin \relax (x)^{2}} - \log \left (\sin \relax (x)^{2}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cos(x)^2*cot(x)^3,x, algorithm="maxima")

[Out]

1/2*sin(x)^2 - 1/2/sin(x)^2 - log(sin(x)^2)

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mupad [B]  time = 0.15, size = 32, normalized size = 1.45 \[ \ln \left ({\mathrm {tan}\relax (x)}^2+1\right )-2\,\ln \left (\mathrm {tan}\relax (x)\right )-\frac {{\mathrm {tan}\relax (x)}^2+\frac {1}{2}}{{\mathrm {tan}\relax (x)}^4+{\mathrm {tan}\relax (x)}^2} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(cos(x)^2*cot(x)^3,x)

[Out]

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

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sympy [A]  time = 0.10, size = 20, normalized size = 0.91 \[ - 2 \log {\left (\sin {\relax (x )} \right )} + \frac {\sin ^{2}{\relax (x )}}{2} - \frac {1}{2 \sin ^{2}{\relax (x )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cos(x)**2*cot(x)**3,x)

[Out]

-2*log(sin(x)) + sin(x)**2/2 - 1/(2*sin(x)**2)

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