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

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Rubi [A]  time = 0.02, antiderivative size = 45, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 2, integrand size = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, 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 8

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

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*}

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Mathematica [C]  time = 0.03, size = 33, normalized size = 0.73 \[ -\frac {\cot ^5(a+b x) \, _2F_1\left (-\frac {5}{2},1;-\frac {3}{2};-\tan ^2(a+b x)\right )}{5 b} \]

Antiderivative was successfully verified.

[In]

Integrate[Cot[a + b*x]^6,x]

[Out]

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

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fricas [B]  time = 0.61, size = 123, normalized size = 2.73 \[ -\frac {23 \, \cos \left (2 \, b x + 2 \, a\right )^{3} - \cos \left (2 \, b x + 2 \, a\right )^{2} + 15 \, {\left (b x \cos \left (2 \, b x + 2 \, a\right )^{2} - 2 \, b x \cos \left (2 \, b x + 2 \, a\right ) + b x\right )} \sin \left (2 \, b x + 2 \, a\right ) - 11 \, \cos \left (2 \, b x + 2 \, a\right ) + 13}{15 \, {\left (b \cos \left (2 \, b x + 2 \, a\right )^{2} - 2 \, b \cos \left (2 \, b x + 2 \, a\right ) + b\right )} \sin \left (2 \, b x + 2 \, a\right )} \]

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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giac [B]  time = 1.14, size = 91, normalized size = 2.02 \[ \frac {3 \, \tan \left (\frac {1}{2} \, b x + \frac {1}{2} \, a\right )^{5} - 35 \, \tan \left (\frac {1}{2} \, b x + \frac {1}{2} \, a\right )^{3} - 480 \, b x - 480 \, a - \frac {330 \, \tan \left (\frac {1}{2} \, b x + \frac {1}{2} \, a\right )^{4} - 35 \, \tan \left (\frac {1}{2} \, b x + \frac {1}{2} \, a\right )^{2} + 3}{\tan \left (\frac {1}{2} \, b x + \frac {1}{2} \, a\right )^{5}} + 330 \, \tan \left (\frac {1}{2} \, b x + \frac {1}{2} \, a\right )}{480 \, b} \]

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

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maple [A]  time = 0.02, size = 46, normalized size = 1.02 \[ \frac {-\frac {\left (\cot ^{5}\left (b x +a \right )\right )}{5}+\frac {\left (\cot ^{3}\left (b x +a \right )\right )}{3}-\cot \left (b x +a \right )+\frac {\pi }{2}-\mathrm {arccot}\left (\cot \left (b x +a \right )\right )}{b} \]

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 = 0.63, size = 44, normalized size = 0.98 \[ -\frac {15 \, b x + 15 \, a + \frac {15 \, \tan \left (b x + a\right )^{4} - 5 \, \tan \left (b x + a\right )^{2} + 3}{\tan \left (b x + a\right )^{5}}}{15 \, b} \]

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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mupad [B]  time = 0.09, size = 36, normalized size = 0.80 \[ -x-\frac {\frac {{\mathrm {cot}\left (a+b\,x\right )}^5}{5}-\frac {{\mathrm {cot}\left (a+b\,x\right )}^3}{3}+\mathrm {cot}\left (a+b\,x\right )}{b} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(cot(a + b*x)^6,x)

[Out]

- x - (cot(a + b*x) - cot(a + b*x)^3/3 + cot(a + b*x)^5/5)/b

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sympy [A]  time = 0.41, size = 39, normalized size = 0.87 \[ \begin {cases} - x - \frac {\cot ^{5}{\left (a + b x \right )}}{5 b} + \frac {\cot ^{3}{\left (a + b x \right )}}{3 b} - \frac {\cot {\left (a + b x \right )}}{b} & \text {for}\: b \neq 0 \\x \cot ^{6}{\relax (a )} & \text {otherwise} \end {cases} \]

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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