3.44 \(\int (d \cot (e+f x))^n \csc ^4(e+f x) \, dx\)

Optimal. Leaf size=51 \[ -\frac{(d \cot (e+f x))^{n+3}}{d^3 f (n+3)}-\frac{(d \cot (e+f x))^{n+1}}{d f (n+1)} \]

[Out]

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

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Rubi [A]  time = 0.0527326, antiderivative size = 51, 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, 14} \[ -\frac{(d \cot (e+f x))^{n+3}}{d^3 f (n+3)}-\frac{(d \cot (e+f x))^{n+1}}{d f (n+1)} \]

Antiderivative was successfully verified.

[In]

Int[(d*Cot[e + f*x])^n*Csc[e + f*x]^4,x]

[Out]

-((d*Cot[e + f*x])^(1 + n)/(d*f*(1 + n))) - (d*Cot[e + f*x])^(3 + n)/(d^3*f*(3 + 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 14

Int[(u_)*((c_.)*(x_))^(m_.), x_Symbol] :> Int[ExpandIntegrand[(c*x)^m*u, x], x] /; FreeQ[{c, m}, x] && SumQ[u]
&&  !LinearQ[u, x] &&  !MatchQ[u, (a_) + (b_.)*(v_) /; FreeQ[{a, b}, x] && InverseFunctionQ[v]]

Rubi steps

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

Mathematica [A]  time = 0.127219, size = 45, normalized size = 0.88 \[ -\frac{\cot (e+f x) \left ((n+1) \csc ^2(e+f x)+2\right ) (d \cot (e+f x))^n}{f (n+1) (n+3)} \]

Antiderivative was successfully verified.

[In]

Integrate[(d*Cot[e + f*x])^n*Csc[e + f*x]^4,x]

[Out]

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

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Maple [C]  time = 0.757, size = 10907, normalized size = 213.9 \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)^4,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)^4,x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 1.73169, size = 204, normalized size = 4. \begin{align*} \frac{{\left (2 \, \cos \left (f x + e\right )^{3} -{\left (n + 3\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 (f n^{2} -{\left (f n^{2} + 4 \, f n + 3 \, f\right )} \cos \left (f x + e\right )^{2} + 4 \, f n + 3 \, 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)^4,x, algorithm="fricas")

[Out]

(2*cos(f*x + e)^3 - (n + 3)*cos(f*x + e))*(d*cos(f*x + e)/sin(f*x + e))^n/((f*n^2 - (f*n^2 + 4*f*n + 3*f)*cos(
f*x + e)^2 + 4*f*n + 3*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)**4,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 )^{4}\,{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)^4,x, algorithm="giac")

[Out]

integrate((d*cot(f*x + e))^n*csc(f*x + e)^4, x)