3.45 \(\int (d \cot (e+f x))^n \csc ^2(e+f x) \, dx\)

Optimal. Leaf size=25 \[ -\frac{(d \cot (e+f x))^{n+1}}{d f (n+1)} \]

[Out]

-((d*Cot[e + f*x])^(1 + n)/(d*f*(1 + n)))

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Rubi [A]  time = 0.042452, antiderivative size = 25, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.105, Rules used = {2607, 32} \[ -\frac{(d \cot (e+f x))^{n+1}}{d f (n+1)} \]

Antiderivative was successfully verified.

[In]

Int[(d*Cot[e + f*x])^n*Csc[e + f*x]^2,x]

[Out]

-((d*Cot[e + f*x])^(1 + n)/(d*f*(1 + n)))

Rule 2607

Int[sec[(e_.) + (f_.)*(x_)]^(m_)*((b_.)*tan[(e_.) + (f_.)*(x_)])^(n_.), x_Symbol] :> Dist[1/f, Subst[Int[(b*x)
^n*(1 + x^2)^(m/2 - 1), x], x, Tan[e + f*x]], x] /; FreeQ[{b, e, f, n}, x] && IntegerQ[m/2] &&  !(IntegerQ[(n
- 1)/2] && LtQ[0, n, m - 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 (d \cot (e+f x))^n \csc ^2(e+f x) \, dx &=\frac{\operatorname{Subst}\left (\int (-d x)^n \, dx,x,-\cot (e+f x)\right )}{f}\\ &=-\frac{(d \cot (e+f x))^{1+n}}{d f (1+n)}\\ \end{align*}

Mathematica [A]  time = 0.0193327, size = 26, normalized size = 1.04 \[ -\frac{\cot (e+f x) (d \cot (e+f x))^n}{f (n+1)} \]

Antiderivative was successfully verified.

[In]

Integrate[(d*Cot[e + f*x])^n*Csc[e + f*x]^2,x]

[Out]

-((Cot[e + f*x]*(d*Cot[e + f*x])^n)/(f*(1 + n)))

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Maple [A]  time = 0.03, size = 26, normalized size = 1. \begin{align*} -{\frac{ \left ( d\cot \left ( fx+e \right ) \right ) ^{1+n}}{fd \left ( 1+n \right ) }} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((d*cot(f*x+e))^n*csc(f*x+e)^2,x)

[Out]

-(d*cot(f*x+e))^(1+n)/d/f/(1+n)

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Maxima [F(-2)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: ValueError} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((d*cot(f*x+e))^n*csc(f*x+e)^2,x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 1.72804, size = 97, normalized size = 3.88 \begin{align*} -\frac{\left (\frac{d \cos \left (f x + e\right )}{\sin \left (f x + e\right )}\right )^{n} \cos \left (f x + e\right )}{{\left (f n + f\right )} \sin \left (f x + e\right )} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((d*cot(f*x+e))^n*csc(f*x+e)^2,x, algorithm="fricas")

[Out]

-(d*cos(f*x + e)/sin(f*x + e))^n*cos(f*x + e)/((f*n + f)*sin(f*x + e))

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \left (d \cot{\left (e + f x \right )}\right )^{n} \csc ^{2}{\left (e + f x \right )}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((d*cot(f*x+e))**n*csc(f*x+e)**2,x)

[Out]

Integral((d*cot(e + f*x))**n*csc(e + f*x)**2, x)

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \left (d \cot \left (f x + e\right )\right )^{n} \csc \left (f x + e\right )^{2}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((d*cot(f*x+e))^n*csc(f*x+e)^2,x, algorithm="giac")

[Out]

integrate((d*cot(f*x + e))^n*csc(f*x + e)^2, x)