3.29 \(\int \csc ^n(c+d x) \sqrt{a-a \csc (c+d x)} \, dx\)

Optimal. Leaf size=69 \[ -\frac{2 a \cos (c+d x) (-\csc (c+d x))^{-n} \csc ^{n+1}(c+d x) \text{Hypergeometric2F1}\left (\frac{1}{2},1-n,\frac{3}{2},\csc (c+d x)+1\right )}{d \sqrt{a-a \csc (c+d x)}} \]

[Out]

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

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Rubi [A]  time = 0.0701908, antiderivative size = 69, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.125, Rules used = {3806, 67, 65} \[ -\frac{2 a \cos (c+d x) (-\csc (c+d x))^{-n} \csc ^{n+1}(c+d x) \, _2F_1\left (\frac{1}{2},1-n;\frac{3}{2};\csc (c+d x)+1\right )}{d \sqrt{a-a \csc (c+d x)}} \]

Antiderivative was successfully verified.

[In]

Int[Csc[c + d*x]^n*Sqrt[a - a*Csc[c + d*x]],x]

[Out]

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

Rule 3806

Int[(csc[(e_.) + (f_.)*(x_)]*(d_.))^(n_)*Sqrt[csc[(e_.) + (f_.)*(x_)]*(b_.) + (a_)], x_Symbol] :> Dist[(a^2*d*
Cot[e + f*x])/(f*Sqrt[a + b*Csc[e + f*x]]*Sqrt[a - b*Csc[e + f*x]]), Subst[Int[(d*x)^(n - 1)/Sqrt[a - b*x], x]
, x, Csc[e + f*x]], x] /; FreeQ[{a, b, d, e, f, n}, x] && EqQ[a^2 - b^2, 0]

Rule 67

Int[((b_.)*(x_))^(m_)*((c_) + (d_.)*(x_))^(n_), x_Symbol] :> Dist[((-((b*c)/d))^IntPart[m]*(b*x)^FracPart[m])/
(-((d*x)/c))^FracPart[m], Int[(-((d*x)/c))^m*(c + d*x)^n, x], x] /; FreeQ[{b, c, d, m, n}, x] &&  !IntegerQ[m]
&&  !IntegerQ[n] &&  !GtQ[c, 0] &&  !GtQ[-(d/(b*c)), 0]

Rule 65

Int[((b_.)*(x_))^(m_)*((c_) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[((c + d*x)^(n + 1)*Hypergeometric2F1[-m, n +
1, n + 2, 1 + (d*x)/c])/(d*(n + 1)*(-(d/(b*c)))^m), x] /; FreeQ[{b, c, d, m, n}, x] &&  !IntegerQ[n] && (Inte
gerQ[m] || GtQ[-(d/(b*c)), 0])

Rubi steps

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

Mathematica [A]  time = 0.913166, size = 73, normalized size = 1.06 \[ -\frac{2 a \cos (c+d x) \csc ^{2 n+1}(c+d x) \left (-\csc ^2(c+d x)\right )^{-n} \text{Hypergeometric2F1}\left (\frac{1}{2},1-n,\frac{3}{2},\csc (c+d x)+1\right )}{d \sqrt{a-a \csc (c+d x)}} \]

Antiderivative was successfully verified.

[In]

Integrate[Csc[c + d*x]^n*Sqrt[a - a*Csc[c + d*x]],x]

[Out]

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

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Maple [F]  time = 0.484, size = 0, normalized size = 0. \begin{align*} \int \left ( \csc \left ( dx+c \right ) \right ) ^{n}\sqrt{a-a\csc \left ( dx+c \right ) }\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(csc(d*x+c)^n*(a-a*csc(d*x+c))^(1/2),x)

[Out]

int(csc(d*x+c)^n*(a-a*csc(d*x+c))^(1/2),x)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(csc(d*x+c)^n*(a-a*csc(d*x+c))^(1/2),x, algorithm="maxima")

[Out]

integrate(sqrt(-a*csc(d*x + c) + a)*csc(d*x + c)^n, x)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(csc(d*x+c)^n*(a-a*csc(d*x+c))^(1/2),x, algorithm="fricas")

[Out]

integral(sqrt(-a*csc(d*x + c) + a)*csc(d*x + c)^n, x)

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \sqrt{- a \left (\csc{\left (c + d x \right )} - 1\right )} \csc ^{n}{\left (c + d x \right )}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(csc(d*x+c)**n*(a-a*csc(d*x+c))**(1/2),x)

[Out]

Integral(sqrt(-a*(csc(c + d*x) - 1))*csc(c + d*x)**n, x)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(csc(d*x+c)^n*(a-a*csc(d*x+c))^(1/2),x, algorithm="giac")

[Out]

integrate(sqrt(-a*csc(d*x + c) + a)*csc(d*x + c)^n, x)