3.4 \(\int \cot ^4(a+b x) \, dx\)

Optimal. Leaf size=27 \[ -\frac{\cot ^3(a+b x)}{3 b}+\frac{\cot (a+b x)}{b}+x \]

[Out]

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

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Rubi [A]  time = 0.0173004, antiderivative size = 27, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 8, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.25, Rules used = {3473, 8} \[ -\frac{\cot ^3(a+b x)}{3 b}+\frac{\cot (a+b x)}{b}+x \]

Antiderivative was successfully verified.

[In]

Int[Cot[a + b*x]^4,x]

[Out]

x + Cot[a + b*x]/b - Cot[a + b*x]^3/(3*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 ^4(a+b x) \, dx &=-\frac{\cot ^3(a+b x)}{3 b}-\int \cot ^2(a+b x) \, dx\\ &=\frac{\cot (a+b x)}{b}-\frac{\cot ^3(a+b x)}{3 b}+\int 1 \, dx\\ &=x+\frac{\cot (a+b x)}{b}-\frac{\cot ^3(a+b x)}{3 b}\\ \end{align*}

Mathematica [C]  time = 0.0127684, size = 33, normalized size = 1.22 \[ -\frac{\cot ^3(a+b x) \text{Hypergeometric2F1}\left (-\frac{3}{2},1,-\frac{1}{2},-\tan ^2(a+b x)\right )}{3 b} \]

Antiderivative was successfully verified.

[In]

Integrate[Cot[a + b*x]^4,x]

[Out]

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

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Maple [A]  time = 0.017, size = 32, normalized size = 1.2 \begin{align*}{\frac{1}{b} \left ( -{\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)^4,x)

[Out]

1/b*(-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 = 1.45773, size = 46, normalized size = 1.7 \begin{align*} \frac{3 \, b x + 3 \, a + \frac{3 \, \tan \left (b x + a\right )^{2} - 1}{\tan \left (b x + a\right )^{3}}}{3 \, b} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cot(b*x+a)^4,x, algorithm="maxima")

[Out]

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

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Fricas [B]  time = 1.23237, size = 198, normalized size = 7.33 \begin{align*} \frac{4 \, \cos \left (2 \, b x + 2 \, a\right )^{2} + 3 \,{\left (b x \cos \left (2 \, b x + 2 \, a\right ) - b x\right )} \sin \left (2 \, b x + 2 \, a\right ) + 2 \, \cos \left (2 \, b x + 2 \, a\right ) - 2}{3 \,{\left (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)^4,x, algorithm="fricas")

[Out]

1/3*(4*cos(2*b*x + 2*a)^2 + 3*(b*x*cos(2*b*x + 2*a) - b*x)*sin(2*b*x + 2*a) + 2*cos(2*b*x + 2*a) - 2)/((b*cos(
2*b*x + 2*a) - b)*sin(2*b*x + 2*a))

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Sympy [A]  time = 0.227265, size = 27, normalized size = 1. \begin{align*} \begin{cases} x - \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 ^{4}{\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)**4,x)

[Out]

Piecewise((x - cot(a + b*x)**3/(3*b) + cot(a + b*x)/b, Ne(b, 0)), (x*cot(a)**4, True))

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Giac [B]  time = 1.12685, size = 84, normalized size = 3.11 \begin{align*} \frac{\tan \left (\frac{1}{2} \, b x + \frac{1}{2} \, a\right )^{3} + 24 \, b x + 24 \, a + \frac{15 \, \tan \left (\frac{1}{2} \, b x + \frac{1}{2} \, a\right )^{2} - 1}{\tan \left (\frac{1}{2} \, b x + \frac{1}{2} \, a\right )^{3}} - 15 \, \tan \left (\frac{1}{2} \, b x + \frac{1}{2} \, a\right )}{24 \, b} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cot(b*x+a)^4,x, algorithm="giac")

[Out]

1/24*(tan(1/2*b*x + 1/2*a)^3 + 24*b*x + 24*a + (15*tan(1/2*b*x + 1/2*a)^2 - 1)/tan(1/2*b*x + 1/2*a)^3 - 15*tan
(1/2*b*x + 1/2*a))/b