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

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Rubi [A]  time = 0.02, antiderivative size = 27, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, integrand size = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, 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 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 ^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*}

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Mathematica [C]  time = 0.01, size = 33, normalized size = 1.22 \[ -\frac {\cot ^3(a+b x) \, _2F_1\left (-\frac {3}{2},1;-\frac {1}{2};-\tan ^2(a+b x)\right )}{3 b} \]

Antiderivative was successfully verified.

[In]

Integrate[Cot[a + b*x]^4,x]

[Out]

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

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fricas [B]  time = 0.44, size = 84, normalized size = 3.11 \[ \frac {4 \, \cos \left (2 \, b x + 2 \, a\right )^{2} + 3 \, {\left (b x \cos \left (2 \, b x + 2 \, a\right ) - b x\right )} \sin \left (2 \, b x + 2 \, a\right ) + 2 \, \cos \left (2 \, b x + 2 \, a\right ) - 2}{3 \, {\left (b \cos \left (2 \, b x + 2 \, a\right ) - b\right )} \sin \left (2 \, b x + 2 \, a\right )} \]

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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giac [B]  time = 0.51, size = 62, normalized size = 2.30 \[ \frac {\tan \left (\frac {1}{2} \, b x + \frac {1}{2} \, a\right )^{3} + 24 \, b x + 24 \, a + \frac {15 \, \tan \left (\frac {1}{2} \, b x + \frac {1}{2} \, a\right )^{2} - 1}{\tan \left (\frac {1}{2} \, b x + \frac {1}{2} \, a\right )^{3}} - 15 \, \tan \left (\frac {1}{2} \, b x + \frac {1}{2} \, a\right )}{24 \, b} \]

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

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maple [A]  time = 0.03, size = 32, normalized size = 1.19 \[ \frac {-\frac {\left (\cot ^{3}\left (b x +a \right )\right )}{3}+\cot \left (b x +a \right )-\frac {\pi }{2}+\mathrm {arccot}\left (\cot \left (b x +a \right )\right )}{b} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(cot(b*x+a)^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 = 0.42, size = 34, normalized size = 1.26 \[ \frac {3 \, b x + 3 \, a + \frac {3 \, \tan \left (b x + a\right )^{2} - 1}{\tan \left (b x + a\right )^{3}}}{3 \, b} \]

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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mupad [B]  time = 0.12, size = 23, normalized size = 0.85 \[ x+\frac {\mathrm {cot}\left (a+b\,x\right )-\frac {{\mathrm {cot}\left (a+b\,x\right )}^3}{3}}{b} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(cot(a + b*x)^4,x)

[Out]

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

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sympy [A]  time = 0.19, size = 27, normalized size = 1.00 \[ \begin {cases} x - \frac {\cot ^{3}{\left (a + b x \right )}}{3 b} + \frac {\cot {\left (a + b x \right )}}{b} & \text {for}\: b \neq 0 \\x \cot ^{4}{\relax (a )} & \text {otherwise} \end {cases} \]

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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