3.376 \(\int \cot (e+f x) (b \csc (e+f x))^m \, dx\)

Optimal. Leaf size=18 \[ -\frac{(b \csc (e+f x))^m}{f m} \]

[Out]

-((b*Csc[e + f*x])^m/(f*m))

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Rubi [A]  time = 0.0214582, antiderivative size = 18, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 17, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.118, Rules used = {2606, 32} \[ -\frac{(b \csc (e+f x))^m}{f m} \]

Antiderivative was successfully verified.

[In]

Int[Cot[e + f*x]*(b*Csc[e + f*x])^m,x]

[Out]

-((b*Csc[e + f*x])^m/(f*m))

Rule 2606

Int[((a_.)*sec[(e_.) + (f_.)*(x_)])^(m_.)*((b_.)*tan[(e_.) + (f_.)*(x_)])^(n_.), x_Symbol] :> Dist[a/f, Subst[
Int[(a*x)^(m - 1)*(-1 + x^2)^((n - 1)/2), x], x, Sec[e + f*x]], x] /; FreeQ[{a, e, f, m}, x] && IntegerQ[(n -
1)/2] &&  !(IntegerQ[m/2] && LtQ[0, m, n + 1])

Rule 32

Int[((a_.) + (b_.)*(x_))^(m_), x_Symbol] :> Simp[(a + b*x)^(m + 1)/(b*(m + 1)), x] /; FreeQ[{a, b, m}, x] && N
eQ[m, -1]

Rubi steps

\begin{align*} \int \cot (e+f x) (b \csc (e+f x))^m \, dx &=-\frac{b \operatorname{Subst}\left (\int (b x)^{-1+m} \, dx,x,\csc (e+f x)\right )}{f}\\ &=-\frac{(b \csc (e+f x))^m}{f m}\\ \end{align*}

Mathematica [A]  time = 0.0174604, size = 18, normalized size = 1. \[ -\frac{(b \csc (e+f x))^m}{f m} \]

Antiderivative was successfully verified.

[In]

Integrate[Cot[e + f*x]*(b*Csc[e + f*x])^m,x]

[Out]

-((b*Csc[e + f*x])^m/(f*m))

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Maple [A]  time = 0.01, size = 19, normalized size = 1.1 \begin{align*} -{\frac{ \left ( b\csc \left ( fx+e \right ) \right ) ^{m}}{fm}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(cot(f*x+e)*(b*csc(f*x+e))^m,x)

[Out]

-(b*csc(f*x+e))^m/f/m

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Maxima [A]  time = 0.969372, size = 28, normalized size = 1.56 \begin{align*} -\frac{b^{m} \sin \left (f x + e\right )^{-m}}{f m} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cot(f*x+e)*(b*csc(f*x+e))^m,x, algorithm="maxima")

[Out]

-b^m*sin(f*x + e)^(-m)/(f*m)

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Fricas [A]  time = 1.60248, size = 36, normalized size = 2. \begin{align*} -\frac{\left (\frac{b}{\sin \left (f x + e\right )}\right )^{m}}{f m} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cot(f*x+e)*(b*csc(f*x+e))^m,x, algorithm="fricas")

[Out]

-(b/sin(f*x + e))^m/(f*m)

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Sympy [A]  time = 0.523902, size = 56, normalized size = 3.11 \begin{align*} \begin{cases} x \cot{\left (e \right )} & \text{for}\: f = 0 \wedge m = 0 \\x \left (b \csc{\left (e \right )}\right )^{m} \cot{\left (e \right )} & \text{for}\: f = 0 \\- \frac{\log{\left (\tan ^{2}{\left (e + f x \right )} + 1 \right )}}{2 f} + \frac{\log{\left (\tan{\left (e + f x \right )} \right )}}{f} & \text{for}\: m = 0 \\- \frac{b^{m} \csc ^{m}{\left (e + f x \right )}}{f m} & \text{otherwise} \end{cases} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cot(f*x+e)*(b*csc(f*x+e))**m,x)

[Out]

Piecewise((x*cot(e), Eq(f, 0) & Eq(m, 0)), (x*(b*csc(e))**m*cot(e), Eq(f, 0)), (-log(tan(e + f*x)**2 + 1)/(2*f
) + log(tan(e + f*x))/f, Eq(m, 0)), (-b**m*csc(e + f*x)**m/(f*m), True))

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \left (b \csc \left (f x + e\right )\right )^{m} \cot \left (f x + e\right )\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cot(f*x+e)*(b*csc(f*x+e))^m,x, algorithm="giac")

[Out]

integrate((b*csc(f*x + e))^m*cot(f*x + e), x)