### 3.79 $$\int \cos ^2(x) \cot ^3(x) \, dx$$

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

[Out]

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

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Rubi [A]  time = 0.0285206, antiderivative size = 22, normalized size of antiderivative = 1., 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 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]

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 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 \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*}

Mathematica [A]  time = 0.0225152, 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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Maple [A]  time = 0.013, size = 29, normalized size = 1.3 \begin{align*} -{\frac{ \left ( \cos \left ( x \right ) \right ) ^{6}}{2\, \left ( \sin \left ( x \right ) \right ) ^{2}}}-{\frac{ \left ( \cos \left ( x \right ) \right ) ^{4}}{2}}- \left ( \cos \left ( x \right ) \right ) ^{2}-2\,\ln \left ( \sin \left ( x \right ) \right ) \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

int(cot(x)^5*sin(x)^2,x)

[Out]

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

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Maxima [A]  time = 0.931084, size = 27, normalized size = 1.23 \begin{align*} \frac{1}{2} \, \sin \left (x\right )^{2} - \frac{1}{2 \, \sin \left (x\right )^{2}} - \log \left (\sin \left (x\right )^{2}\right ) \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate(cot(x)^5*sin(x)^2,x, algorithm="maxima")

[Out]

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

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Fricas [B]  time = 2.0689, size = 116, normalized size = 5.27 \begin{align*} -\frac{2 \, \cos \left (x\right )^{4} - 3 \, \cos \left (x\right )^{2} + 8 \,{\left (\cos \left (x\right )^{2} - 1\right )} \log \left (\frac{1}{2} \, \sin \left (x\right )\right ) - 1}{4 \,{\left (\cos \left (x\right )^{2} - 1\right )}} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate(cot(x)^5*sin(x)^2,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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Sympy [A]  time = 0.095172, size = 20, normalized size = 0.91 \begin{align*} - 2 \log{\left (\sin{\left (x \right )} \right )} + \frac{\sin ^{2}{\left (x \right )}}{2} - \frac{1}{2 \sin ^{2}{\left (x \right )}} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate(cot(x)**5*sin(x)**2,x)

[Out]

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

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Giac [B]  time = 1.07039, size = 49, normalized size = 2.23 \begin{align*} -\frac{1}{2} \, \cos \left (x\right )^{2} + \frac{2 \, \cos \left (x\right )^{2} - 1}{2 \,{\left (\cos \left (x\right )^{2} - 1\right )}} - \log \left (-\cos \left (x\right )^{2} + 1\right ) \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate(cot(x)^5*sin(x)^2,x, algorithm="giac")

[Out]

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