3.3 \(\int \cot ^3(a+b x) \, dx\)

Optimal. Leaf size=28 \[ -\frac{\cot ^2(a+b x)}{2 b}-\frac{\log (\sin (a+b x))}{b} \]

[Out]

-Cot[a + b*x]^2/(2*b) - Log[Sin[a + b*x]]/b

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Rubi [A]  time = 0.0126411, antiderivative size = 28, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 8, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.25, Rules used = {3473, 3475} \[ -\frac{\cot ^2(a+b x)}{2 b}-\frac{\log (\sin (a+b x))}{b} \]

Antiderivative was successfully verified.

[In]

Int[Cot[a + b*x]^3,x]

[Out]

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

Mathematica [A]  time = 0.0917749, size = 34, normalized size = 1.21 \[ -\frac{\cot ^2(a+b x)+2 \log (\tan (a+b x))+2 \log (\cos (a+b x))}{2 b} \]

Antiderivative was successfully verified.

[In]

Integrate[Cot[a + b*x]^3,x]

[Out]

-(Cot[a + b*x]^2 + 2*Log[Cos[a + b*x]] + 2*Log[Tan[a + b*x]])/(2*b)

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Maple [A]  time = 0.016, size = 31, normalized size = 1.1 \begin{align*} -{\frac{ \left ( \cot \left ( bx+a \right ) \right ) ^{2}}{2\,b}}+{\frac{\ln \left ( \left ( \cot \left ( bx+a \right ) \right ) ^{2}+1 \right ) }{2\,b}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(cot(b*x+a)^3,x)

[Out]

-1/2*cot(b*x+a)^2/b+1/2/b*ln(cot(b*x+a)^2+1)

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Maxima [A]  time = 0.983791, size = 31, normalized size = 1.11 \begin{align*} -\frac{\frac{1}{\sin \left (b x + a\right )^{2}} + \log \left (\sin \left (b x + a\right )^{2}\right )}{2 \, b} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cot(b*x+a)^3,x, algorithm="maxima")

[Out]

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

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Fricas [A]  time = 1.81369, size = 126, normalized size = 4.5 \begin{align*} -\frac{{\left (\cos \left (2 \, b x + 2 \, a\right ) - 1\right )} \log \left (-\frac{1}{2} \, \cos \left (2 \, b x + 2 \, a\right ) + \frac{1}{2}\right ) - 2}{2 \,{\left (b \cos \left (2 \, b x + 2 \, a\right ) - b\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cot(b*x+a)^3,x, algorithm="fricas")

[Out]

-1/2*((cos(2*b*x + 2*a) - 1)*log(-1/2*cos(2*b*x + 2*a) + 1/2) - 2)/(b*cos(2*b*x + 2*a) - b)

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Sympy [A]  time = 0.522513, size = 53, normalized size = 1.89 \begin{align*} \begin{cases} \tilde{\infty } x & \text{for}\: a = 0 \wedge b = 0 \\x \cot ^{3}{\left (a \right )} & \text{for}\: b = 0 \\\tilde{\infty } x & \text{for}\: a = - b x \\\frac{\log{\left (\tan ^{2}{\left (a + b x \right )} + 1 \right )}}{2 b} - \frac{\log{\left (\tan{\left (a + b x \right )} \right )}}{b} - \frac{1}{2 b \tan ^{2}{\left (a + b x \right )}} & \text{otherwise} \end{cases} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cot(b*x+a)**3,x)

[Out]

Piecewise((zoo*x, Eq(a, 0) & Eq(b, 0)), (x*cot(a)**3, Eq(b, 0)), (zoo*x, Eq(a, -b*x)), (log(tan(a + b*x)**2 +
1)/(2*b) - log(tan(a + b*x))/b - 1/(2*b*tan(a + b*x)**2), True))

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Giac [B]  time = 1.12876, size = 159, normalized size = 5.68 \begin{align*} \frac{\frac{{\left (\frac{4 \,{\left (\cos \left (b x + a\right ) - 1\right )}}{\cos \left (b x + a\right ) + 1} + 1\right )}{\left (\cos \left (b x + a\right ) + 1\right )}}{\cos \left (b x + a\right ) - 1} + \frac{\cos \left (b x + a\right ) - 1}{\cos \left (b x + a\right ) + 1} - 4 \, \log \left (\frac{{\left | -\cos \left (b x + a\right ) + 1 \right |}}{{\left | \cos \left (b x + a\right ) + 1 \right |}}\right ) + 8 \, \log \left ({\left | -\frac{\cos \left (b x + a\right ) - 1}{\cos \left (b x + a\right ) + 1} + 1 \right |}\right )}{8 \, b} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cot(b*x+a)^3,x, algorithm="giac")

[Out]

1/8*((4*(cos(b*x + a) - 1)/(cos(b*x + a) + 1) + 1)*(cos(b*x + a) + 1)/(cos(b*x + a) - 1) + (cos(b*x + a) - 1)/
(cos(b*x + a) + 1) - 4*log(abs(-cos(b*x + a) + 1)/abs(cos(b*x + a) + 1)) + 8*log(abs(-(cos(b*x + a) - 1)/(cos(
b*x + a) + 1) + 1)))/b