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

Optimal. Leaf size=77 \[ -\frac{\csc (e+f x) \sin ^2(e+f x)^{\frac{n+2}{2}} (d \cot (e+f x))^{n+1} \text{Hypergeometric2F1}\left (\frac{n+1}{2},\frac{n+2}{2},\frac{n+3}{2},\cos ^2(e+f x)\right )}{d f (n+1)} \]

[Out]

-(((d*Cot[e + f*x])^(1 + n)*Csc[e + f*x]*Hypergeometric2F1[(1 + n)/2, (2 + n)/2, (3 + n)/2, Cos[e + f*x]^2]*(S
in[e + f*x]^2)^((2 + n)/2))/(d*f*(1 + n)))

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Rubi [A]  time = 0.029108, antiderivative size = 77, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 17, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.059, Rules used = {2617} \[ -\frac{\csc (e+f x) \sin ^2(e+f x)^{\frac{n+2}{2}} (d \cot (e+f x))^{n+1} \, _2F_1\left (\frac{n+1}{2},\frac{n+2}{2};\frac{n+3}{2};\cos ^2(e+f x)\right )}{d f (n+1)} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-(((d*Cot[e + f*x])^(1 + n)*Csc[e + f*x]*Hypergeometric2F1[(1 + n)/2, (2 + n)/2, (3 + n)/2, Cos[e + f*x]^2]*(S
in[e + f*x]^2)^((2 + n)/2))/(d*f*(1 + n)))

Rule 2617

Int[((a_.)*sec[(e_.) + (f_.)*(x_)])^(m_.)*((b_.)*tan[(e_.) + (f_.)*(x_)])^(n_), x_Symbol] :> Simp[((a*Sec[e +
f*x])^m*(b*Tan[e + f*x])^(n + 1)*(Cos[e + f*x]^2)^((m + n + 1)/2)*Hypergeometric2F1[(n + 1)/2, (m + n + 1)/2,
(n + 3)/2, Sin[e + f*x]^2])/(b*f*(n + 1)), x] /; FreeQ[{a, b, e, f, m, n}, x] &&  !IntegerQ[(n - 1)/2] &&  !In
tegerQ[m/2]

Rubi steps

\begin{align*} \int (d \cot (e+f x))^n \csc (e+f x) \, dx &=-\frac{(d \cot (e+f x))^{1+n} \csc (e+f x) \, _2F_1\left (\frac{1+n}{2},\frac{2+n}{2};\frac{3+n}{2};\cos ^2(e+f x)\right ) \sin ^2(e+f x)^{\frac{2+n}{2}}}{d f (1+n)}\\ \end{align*}

Mathematica [A]  time = 0.13063, size = 69, normalized size = 0.9 \[ -\frac{(d \cot (e+f x))^n \left (\cos (e+f x) \sec ^2\left (\frac{1}{2} (e+f x)\right )\right )^{-n} \text{Hypergeometric2F1}\left (-n,-\frac{n}{2},1-\frac{n}{2},\tan ^2\left (\frac{1}{2} (e+f x)\right )\right )}{f n} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-(((d*Cot[e + f*x])^n*Hypergeometric2F1[-n, -n/2, 1 - n/2, Tan[(e + f*x)/2]^2])/(f*n*(Cos[e + f*x]*Sec[(e + f*
x)/2]^2)^n))

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Maple [F]  time = 0.499, size = 0, normalized size = 0. \begin{align*} \int \left ( d\cot \left ( fx+e \right ) \right ) ^{n}\csc \left ( fx+e \right ) \, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [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 )\,{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),x, algorithm="maxima")

[Out]

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

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Fricas [F]  time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (\left (d \cot \left (f x + e\right )\right )^{n} \csc \left (f x + e\right ), x\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),x, algorithm="fricas")

[Out]

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

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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{\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),x)

[Out]

Integral((d*cot(e + f*x))**n*csc(e + f*x), 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 )\,{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),x, algorithm="giac")

[Out]

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