3.43 \(\int (d \cot (e+f x))^n \csc ^6(e+f x) \, dx\)

Optimal. Leaf size=76 \[ -\frac{2 (d \cot (e+f x))^{n+3}}{d^3 f (n+3)}-\frac{(d \cot (e+f x))^{n+5}}{d^5 f (n+5)}-\frac{(d \cot (e+f x))^{n+1}}{d f (n+1)} \]

[Out]

-((d*Cot[e + f*x])^(1 + n)/(d*f*(1 + n))) - (2*(d*Cot[e + f*x])^(3 + n))/(d^3*f*(3 + n)) - (d*Cot[e + f*x])^(5
+ n)/(d^5*f*(5 + n))

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Rubi [A]  time = 0.070749, antiderivative size = 76, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.105, Rules used = {2607, 270} \[ -\frac{2 (d \cot (e+f x))^{n+3}}{d^3 f (n+3)}-\frac{(d \cot (e+f x))^{n+5}}{d^5 f (n+5)}-\frac{(d \cot (e+f x))^{n+1}}{d f (n+1)} \]

Antiderivative was successfully verified.

[In]

Int[(d*Cot[e + f*x])^n*Csc[e + f*x]^6,x]

[Out]

-((d*Cot[e + f*x])^(1 + n)/(d*f*(1 + n))) - (2*(d*Cot[e + f*x])^(3 + n))/(d^3*f*(3 + n)) - (d*Cot[e + f*x])^(5
+ n)/(d^5*f*(5 + 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 270

Int[((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_.), x_Symbol] :> Int[ExpandIntegrand[(c*x)^m*(a + b*x^n)^p,
x], x] /; FreeQ[{a, b, c, m, n}, x] && IGtQ[p, 0]

Rubi steps

\begin{align*} \int (d \cot (e+f x))^n \csc ^6(e+f x) \, dx &=\frac{\operatorname{Subst}\left (\int (-d x)^n \left (1+x^2\right )^2 \, dx,x,-\cot (e+f x)\right )}{f}\\ &=\frac{\operatorname{Subst}\left (\int \left ((-d x)^n+\frac{2 (-d x)^{2+n}}{d^2}+\frac{(-d x)^{4+n}}{d^4}\right ) \, dx,x,-\cot (e+f x)\right )}{f}\\ &=-\frac{(d \cot (e+f x))^{1+n}}{d f (1+n)}-\frac{2 (d \cot (e+f x))^{3+n}}{d^3 f (3+n)}-\frac{(d \cot (e+f x))^{5+n}}{d^5 f (5+n)}\\ \end{align*}

Mathematica [A]  time = 0.233779, size = 73, normalized size = 0.96 \[ -\frac{\cot (e+f x) \csc ^4(e+f x) \left (-2 (n+3) \cos (2 (e+f x))+\cos (4 (e+f x))+n^2+6 n+8\right ) (d \cot (e+f x))^n}{f (n+1) (n+3) (n+5)} \]

Antiderivative was successfully verified.

[In]

Integrate[(d*Cot[e + f*x])^n*Csc[e + f*x]^6,x]

[Out]

-(((8 + 6*n + n^2 - 2*(3 + n)*Cos[2*(e + f*x)] + Cos[4*(e + f*x)])*Cot[e + f*x]*(d*Cot[e + f*x])^n*Csc[e + f*x
]^4)/(f*(1 + n)*(3 + n)*(5 + n)))

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Maple [C]  time = 2.195, size = 21900, normalized size = 288.2 \begin{align*} \text{output too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((d*cot(f*x+e))^n*csc(f*x+e)^6,x)

[Out]

result too large to display

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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)^6,x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 1.75107, size = 358, normalized size = 4.71 \begin{align*} -\frac{{\left (8 \, \cos \left (f x + e\right )^{5} - 4 \,{\left (n + 5\right )} \cos \left (f x + e\right )^{3} +{\left (n^{2} + 8 \, n + 15\right )} \cos \left (f x + e\right )\right )} \left (\frac{d \cos \left (f x + e\right )}{\sin \left (f x + e\right )}\right )^{n}}{{\left ({\left (f n^{3} + 9 \, f n^{2} + 23 \, f n + 15 \, f\right )} \cos \left (f x + e\right )^{4} + f n^{3} + 9 \, f n^{2} - 2 \,{\left (f n^{3} + 9 \, f n^{2} + 23 \, f n + 15 \, f\right )} \cos \left (f x + e\right )^{2} + 23 \, f n + 15 \, 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)^6,x, algorithm="fricas")

[Out]

-(8*cos(f*x + e)^5 - 4*(n + 5)*cos(f*x + e)^3 + (n^2 + 8*n + 15)*cos(f*x + e))*(d*cos(f*x + e)/sin(f*x + e))^n
/(((f*n^3 + 9*f*n^2 + 23*f*n + 15*f)*cos(f*x + e)^4 + f*n^3 + 9*f*n^2 - 2*(f*n^3 + 9*f*n^2 + 23*f*n + 15*f)*co
s(f*x + e)^2 + 23*f*n + 15*f)*sin(f*x + e))

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Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((d*cot(f*x+e))**n*csc(f*x+e)**6,x)

[Out]

Timed out

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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 )^{6}\,{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)^6,x, algorithm="giac")

[Out]

integrate((d*cot(f*x + e))^n*csc(f*x + e)^6, x)