3.38 \(\int (c \csc (a+b x))^n \, dx\)

Optimal. Leaf size=72 \[ \frac{c \cos (a+b x) (c \csc (a+b x))^{n-1} \text{Hypergeometric2F1}\left (\frac{1}{2},\frac{1-n}{2},\frac{3-n}{2},\sin ^2(a+b x)\right )}{b (1-n) \sqrt{\cos ^2(a+b x)}} \]

[Out]

(c*Cos[a + b*x]*(c*Csc[a + b*x])^(-1 + n)*Hypergeometric2F1[1/2, (1 - n)/2, (3 - n)/2, Sin[a + b*x]^2])/(b*(1
- n)*Sqrt[Cos[a + b*x]^2])

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Rubi [A]  time = 0.0297544, antiderivative size = 72, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 10, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.2, Rules used = {3772, 2643} \[ \frac{c \cos (a+b x) (c \csc (a+b x))^{n-1} \, _2F_1\left (\frac{1}{2},\frac{1-n}{2};\frac{3-n}{2};\sin ^2(a+b x)\right )}{b (1-n) \sqrt{\cos ^2(a+b x)}} \]

Antiderivative was successfully verified.

[In]

Int[(c*Csc[a + b*x])^n,x]

[Out]

(c*Cos[a + b*x]*(c*Csc[a + b*x])^(-1 + n)*Hypergeometric2F1[1/2, (1 - n)/2, (3 - n)/2, Sin[a + b*x]^2])/(b*(1
- n)*Sqrt[Cos[a + b*x]^2])

Rule 3772

Int[(csc[(c_.) + (d_.)*(x_)]*(b_.))^(n_), x_Symbol] :> Simp[(b*Csc[c + d*x])^(n - 1)*((Sin[c + d*x]/b)^(n - 1)
*Int[1/(Sin[c + d*x]/b)^n, x]), x] /; FreeQ[{b, c, d, n}, x] &&  !IntegerQ[n]

Rule 2643

Int[((b_.)*sin[(c_.) + (d_.)*(x_)])^(n_), x_Symbol] :> Simp[(Cos[c + d*x]*(b*Sin[c + d*x])^(n + 1)*Hypergeomet
ric2F1[1/2, (n + 1)/2, (n + 3)/2, Sin[c + d*x]^2])/(b*d*(n + 1)*Sqrt[Cos[c + d*x]^2]), x] /; FreeQ[{b, c, d, n
}, x] &&  !IntegerQ[2*n]

Rubi steps

\begin{align*} \int (c \csc (a+b x))^n \, dx &=(c \csc (a+b x))^n \left (\frac{\sin (a+b x)}{c}\right )^n \int \left (\frac{\sin (a+b x)}{c}\right )^{-n} \, dx\\ &=\frac{\cos (a+b x) (c \csc (a+b x))^n \, _2F_1\left (\frac{1}{2},\frac{1-n}{2};\frac{3-n}{2};\sin ^2(a+b x)\right ) \sin (a+b x)}{b (1-n) \sqrt{\cos ^2(a+b x)}}\\ \end{align*}

Mathematica [A]  time = 0.0761917, size = 65, normalized size = 0.9 \[ -\frac{\sin (a+b x) \cos (a+b x) \sin ^2(a+b x)^{\frac{n-1}{2}} (c \csc (a+b x))^n \text{Hypergeometric2F1}\left (\frac{1}{2},\frac{n+1}{2},\frac{3}{2},\cos ^2(a+b x)\right )}{b} \]

Antiderivative was successfully verified.

[In]

Integrate[(c*Csc[a + b*x])^n,x]

[Out]

-((Cos[a + b*x]*(c*Csc[a + b*x])^n*Hypergeometric2F1[1/2, (1 + n)/2, 3/2, Cos[a + b*x]^2]*Sin[a + b*x]*(Sin[a
+ b*x]^2)^((-1 + n)/2))/b)

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Maple [F]  time = 0.619, size = 0, normalized size = 0. \begin{align*} \int \left ( c\csc \left ( bx+a \right ) \right ) ^{n}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((c*csc(b*x+a))^n,x)

[Out]

int((c*csc(b*x+a))^n,x)

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \left (c \csc \left (b x + a\right )\right )^{n}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((c*csc(b*x+a))^n,x, algorithm="maxima")

[Out]

integrate((c*csc(b*x + a))^n, x)

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Fricas [F]  time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (\left (c \csc \left (b x + a\right )\right )^{n}, x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((c*csc(b*x+a))^n,x, algorithm="fricas")

[Out]

integral((c*csc(b*x + a))^n, x)

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \left (c \csc{\left (a + b x \right )}\right )^{n}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((c*csc(b*x+a))**n,x)

[Out]

Integral((c*csc(a + b*x))**n, x)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((c*csc(b*x+a))^n,x, algorithm="giac")

[Out]

integrate((c*csc(b*x + a))^n, x)