3.6 \(\int \cot ^6(a+b x) \, dx\)

Optimal. Leaf size=45 \[ -\frac{\cot ^5(a+b x)}{5 b}+\frac{\cot ^3(a+b x)}{3 b}-\frac{\cot (a+b x)}{b}-x \]

[Out]

-x - Cot[a + b*x]/b + Cot[a + b*x]^3/(3*b) - Cot[a + b*x]^5/(5*b)

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Rubi [A]  time = 0.0244749, antiderivative size = 45, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 2, integrand size = 8, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.25, Rules used = {3473, 8} \[ -\frac{\cot ^5(a+b x)}{5 b}+\frac{\cot ^3(a+b x)}{3 b}-\frac{\cot (a+b x)}{b}-x \]

Antiderivative was successfully verified.

[In]

Int[Cot[a + b*x]^6,x]

[Out]

-x - Cot[a + b*x]/b + Cot[a + b*x]^3/(3*b) - Cot[a + b*x]^5/(5*b)

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 8

Int[a_, x_Symbol] :> Simp[a*x, x] /; FreeQ[a, x]

Rubi steps

\begin{align*} \int \cot ^6(a+b x) \, dx &=-\frac{\cot ^5(a+b x)}{5 b}-\int \cot ^4(a+b x) \, dx\\ &=\frac{\cot ^3(a+b x)}{3 b}-\frac{\cot ^5(a+b x)}{5 b}+\int \cot ^2(a+b x) \, dx\\ &=-\frac{\cot (a+b x)}{b}+\frac{\cot ^3(a+b x)}{3 b}-\frac{\cot ^5(a+b x)}{5 b}-\int 1 \, dx\\ &=-x-\frac{\cot (a+b x)}{b}+\frac{\cot ^3(a+b x)}{3 b}-\frac{\cot ^5(a+b x)}{5 b}\\ \end{align*}

Mathematica [C]  time = 0.0202553, size = 33, normalized size = 0.73 \[ -\frac{\cot ^5(a+b x) \text{Hypergeometric2F1}\left (-\frac{5}{2},1,-\frac{3}{2},-\tan ^2(a+b x)\right )}{5 b} \]

Antiderivative was successfully verified.

[In]

Integrate[Cot[a + b*x]^6,x]

[Out]

-(Cot[a + b*x]^5*Hypergeometric2F1[-5/2, 1, -3/2, -Tan[a + b*x]^2])/(5*b)

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Maple [A]  time = 0.014, size = 46, normalized size = 1. \begin{align*}{\frac{1}{b} \left ( -{\frac{ \left ( \cot \left ( bx+a \right ) \right ) ^{5}}{5}}+{\frac{ \left ( \cot \left ( bx+a \right ) \right ) ^{3}}{3}}-\cot \left ( bx+a \right ) +{\frac{\pi }{2}}-{\rm arccot} \left (\cot \left ( bx+a \right ) \right ) \right ) } \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(cot(b*x+a)^6,x)

[Out]

1/b*(-1/5*cot(b*x+a)^5+1/3*cot(b*x+a)^3-cot(b*x+a)+1/2*Pi-arccot(cot(b*x+a)))

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Maxima [A]  time = 1.52765, size = 59, normalized size = 1.31 \begin{align*} -\frac{15 \, b x + 15 \, a + \frac{15 \, \tan \left (b x + a\right )^{4} - 5 \, \tan \left (b x + a\right )^{2} + 3}{\tan \left (b x + a\right )^{5}}}{15 \, b} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cot(b*x+a)^6,x, algorithm="maxima")

[Out]

-1/15*(15*b*x + 15*a + (15*tan(b*x + a)^4 - 5*tan(b*x + a)^2 + 3)/tan(b*x + a)^5)/b

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Fricas [B]  time = 1.35594, size = 305, normalized size = 6.78 \begin{align*} -\frac{23 \, \cos \left (2 \, b x + 2 \, a\right )^{3} - \cos \left (2 \, b x + 2 \, a\right )^{2} + 15 \,{\left (b x \cos \left (2 \, b x + 2 \, a\right )^{2} - 2 \, b x \cos \left (2 \, b x + 2 \, a\right ) + b x\right )} \sin \left (2 \, b x + 2 \, a\right ) - 11 \, \cos \left (2 \, b x + 2 \, a\right ) + 13}{15 \,{\left (b \cos \left (2 \, b x + 2 \, a\right )^{2} - 2 \, b \cos \left (2 \, b x + 2 \, a\right ) + b\right )} \sin \left (2 \, b x + 2 \, a\right )} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cot(b*x+a)^6,x, algorithm="fricas")

[Out]

-1/15*(23*cos(2*b*x + 2*a)^3 - cos(2*b*x + 2*a)^2 + 15*(b*x*cos(2*b*x + 2*a)^2 - 2*b*x*cos(2*b*x + 2*a) + b*x)
*sin(2*b*x + 2*a) - 11*cos(2*b*x + 2*a) + 13)/((b*cos(2*b*x + 2*a)^2 - 2*b*cos(2*b*x + 2*a) + b)*sin(2*b*x + 2
*a))

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Sympy [A]  time = 0.492814, size = 39, normalized size = 0.87 \begin{align*} \begin{cases} - x - \frac{\cot ^{5}{\left (a + b x \right )}}{5 b} + \frac{\cot ^{3}{\left (a + b x \right )}}{3 b} - \frac{\cot{\left (a + b x \right )}}{b} & \text{for}\: b \neq 0 \\x \cot ^{6}{\left (a \right )} & \text{otherwise} \end{cases} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cot(b*x+a)**6,x)

[Out]

Piecewise((-x - cot(a + b*x)**5/(5*b) + cot(a + b*x)**3/(3*b) - cot(a + b*x)/b, Ne(b, 0)), (x*cot(a)**6, True)
)

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Giac [B]  time = 1.13303, size = 123, normalized size = 2.73 \begin{align*} \frac{3 \, \tan \left (\frac{1}{2} \, b x + \frac{1}{2} \, a\right )^{5} - 35 \, \tan \left (\frac{1}{2} \, b x + \frac{1}{2} \, a\right )^{3} - 480 \, b x - 480 \, a - \frac{330 \, \tan \left (\frac{1}{2} \, b x + \frac{1}{2} \, a\right )^{4} - 35 \, \tan \left (\frac{1}{2} \, b x + \frac{1}{2} \, a\right )^{2} + 3}{\tan \left (\frac{1}{2} \, b x + \frac{1}{2} \, a\right )^{5}} + 330 \, \tan \left (\frac{1}{2} \, b x + \frac{1}{2} \, a\right )}{480 \, b} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cot(b*x+a)^6,x, algorithm="giac")

[Out]

1/480*(3*tan(1/2*b*x + 1/2*a)^5 - 35*tan(1/2*b*x + 1/2*a)^3 - 480*b*x - 480*a - (330*tan(1/2*b*x + 1/2*a)^4 -
35*tan(1/2*b*x + 1/2*a)^2 + 3)/tan(1/2*b*x + 1/2*a)^5 + 330*tan(1/2*b*x + 1/2*a))/b