3.342 \(\int \cot ^5(x) \, dx\)

Optimal. Leaf size=20 \[ -\frac{1}{4} \cot ^4(x)+\frac{\cot ^2(x)}{2}+\log (\sin (x)) \]

[Out]

Cot[x]^2/2 - Cot[x]^4/4 + Log[Sin[x]]

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Rubi [A]  time = 0.0150477, antiderivative size = 20, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 4, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.5, Rules used = {3473, 3475} \[ -\frac{1}{4} \cot ^4(x)+\frac{\cot ^2(x)}{2}+\log (\sin (x)) \]

Antiderivative was successfully verified.

[In]

Int[Cot[x]^5,x]

[Out]

Cot[x]^2/2 - Cot[x]^4/4 + Log[Sin[x]]

Rule 3473

Int[((b_.)*tan[(c_.) + (d_.)*(x_)])^(n_), x_Symbol] :> Simp[(b*(b*Tan[c + d*x])^(n - 1))/(d*(n - 1)), x] - Dis
t[b^2, Int[(b*Tan[c + d*x])^(n - 2), x], x] /; FreeQ[{b, c, d}, x] && GtQ[n, 1]

Rule 3475

Int[tan[(c_.) + (d_.)*(x_)], x_Symbol] :> -Simp[Log[RemoveContent[Cos[c + d*x], x]]/d, x] /; FreeQ[{c, d}, x]

Rubi steps

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

Mathematica [A]  time = 0.0031104, size = 16, normalized size = 0.8 \[ -\frac{1}{4} \csc ^4(x)+\csc ^2(x)+\log (\sin (x)) \]

Antiderivative was successfully verified.

[In]

Integrate[Cot[x]^5,x]

[Out]

Csc[x]^2 - Csc[x]^4/4 + Log[Sin[x]]

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Maple [A]  time = 0.007, size = 26, normalized size = 1.3 \begin{align*} -{\frac{\ln \left ( \left ( \tan \left ( x \right ) \right ) ^{2}+1 \right ) }{2}}-{\frac{1}{4\, \left ( \tan \left ( x \right ) \right ) ^{4}}}+\ln \left ( \tan \left ( x \right ) \right ) +{\frac{1}{2\, \left ( \tan \left ( x \right ) \right ) ^{2}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/tan(x)^5,x)

[Out]

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

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Maxima [A]  time = 0.932898, size = 30, normalized size = 1.5 \begin{align*} \frac{4 \, \sin \left (x\right )^{2} - 1}{4 \, \sin \left (x\right )^{4}} + \frac{1}{2} \, \log \left (\sin \left (x\right )^{2}\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/tan(x)^5,x, algorithm="maxima")

[Out]

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

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Fricas [B]  time = 1.72129, size = 116, normalized size = 5.8 \begin{align*} \frac{2 \, \log \left (\frac{\tan \left (x\right )^{2}}{\tan \left (x\right )^{2} + 1}\right ) \tan \left (x\right )^{4} + 3 \, \tan \left (x\right )^{4} + 2 \, \tan \left (x\right )^{2} - 1}{4 \, \tan \left (x\right )^{4}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/tan(x)^5,x, algorithm="fricas")

[Out]

1/4*(2*log(tan(x)^2/(tan(x)^2 + 1))*tan(x)^4 + 3*tan(x)^4 + 2*tan(x)^2 - 1)/tan(x)^4

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Sympy [A]  time = 0.1013, size = 19, normalized size = 0.95 \begin{align*} \frac{4 \sin ^{2}{\left (x \right )} - 1}{4 \sin ^{4}{\left (x \right )}} + \log{\left (\sin{\left (x \right )} \right )} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/tan(x)**5,x)

[Out]

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

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Giac [B]  time = 1.06858, size = 50, normalized size = 2.5 \begin{align*} -\frac{3 \, \tan \left (x\right )^{4} - 2 \, \tan \left (x\right )^{2} + 1}{4 \, \tan \left (x\right )^{4}} - \frac{1}{2} \, \log \left (\tan \left (x\right )^{2} + 1\right ) + \frac{1}{2} \, \log \left (\tan \left (x\right )^{2}\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/tan(x)^5,x, algorithm="giac")

[Out]

-1/4*(3*tan(x)^4 - 2*tan(x)^2 + 1)/tan(x)^4 - 1/2*log(tan(x)^2 + 1) + 1/2*log(tan(x)^2)