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

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 = 0.0129672, 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 = 0.0948977, 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 = 0.011, size = 27, normalized size = 1. \begin{align*} -{\frac{ \left ( \cot \left ( bx+a \right ) \right ) ^{2}}{2\,b}}-{\frac{\ln \left ( \sin \left ( bx+a \right ) \right ) }{b}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(cos(b*x+a)^3/sin(b*x+a)^3,x)

[Out]

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

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Maxima [A]  time = 0.977599, 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(cos(b*x+a)^3/sin(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 = 2.39515, size = 108, normalized size = 3.86 \begin{align*} -\frac{2 \,{\left (\cos \left (b x + a\right )^{2} - 1\right )} \log \left (\frac{1}{2} \, \sin \left (b x + a\right )\right ) - 1}{2 \,{\left (b \cos \left (b x + a\right )^{2} - b\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cos(b*x+a)^3/sin(b*x+a)^3,x, algorithm="fricas")

[Out]

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

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Sympy [A]  time = 1.0051, size = 42, normalized size = 1.5 \begin{align*} \begin{cases} - \frac{\log{\left (\sin{\left (a + b x \right )} \right )}}{b} - \frac{\cos ^{2}{\left (a + b x \right )}}{2 b \sin ^{2}{\left (a + b x \right )}} & \text{for}\: b \neq 0 \\\frac{x \cos ^{3}{\left (a \right )}}{\sin ^{3}{\left (a \right )}} & \text{otherwise} \end{cases} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cos(b*x+a)**3/sin(b*x+a)**3,x)

[Out]

Piecewise((-log(sin(a + b*x))/b - cos(a + b*x)**2/(2*b*sin(a + b*x)**2), Ne(b, 0)), (x*cos(a)**3/sin(a)**3, Tr
ue))

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Giac [A]  time = 1.13655, size = 49, normalized size = 1.75 \begin{align*} \frac{\frac{\sin \left (b x + a\right )^{2} - 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(cos(b*x+a)^3/sin(b*x+a)^3,x, algorithm="giac")

[Out]

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