\(\int \csc ^n(c+d x) \sqrt {a-a \csc (c+d x)} \, dx\) [29]

Optimal result

Integrand size = 24, antiderivative size = 69 \[ \int \csc ^n(c+d x) \sqrt {a-a \csc (c+d x)} \, dx=-\frac {2 a \cos (c+d x) (-\csc (c+d x))^{-n} \csc ^{1+n}(c+d x) \operatorname {Hypergeometric2F1}\left (\frac {1}{2},1-n,\frac {3}{2},1+\csc (c+d x)\right )}{d \sqrt {a-a \csc (c+d x)}} \]

[Out]

-2*a*cos(d*x+c)*csc(d*x+c)^(1+n)*hypergeom([1/2, 1-n],[3/2],1+csc(d*x+c))/d/((-csc(d*x+c))^n)/(a-a*csc(d*x+c))
^(1/2)

Rubi [A] (verified)

Time = 0.09 (sec) , antiderivative size = 69, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.125, Rules used = {3891, 69, 67} \[ \int \csc ^n(c+d x) \sqrt {a-a \csc (c+d x)} \, dx=-\frac {2 a \cos (c+d x) (-\csc (c+d x))^{-n} \csc ^{n+1}(c+d x) \operatorname {Hypergeometric2F1}\left (\frac {1}{2},1-n,\frac {3}{2},\csc (c+d x)+1\right )}{d \sqrt {a-a \csc (c+d x)}} \]

[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 67

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

Rule 69

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 3891

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]

Rubi steps \begin{align*} \text {integral}& = \frac {\left (a^2 \cot (c+d x)\right ) \text {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 ) \text {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) \operatorname {Hypergeometric2F1}\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] (verified)

Time = 2.90 (sec) , antiderivative size = 73, normalized size of antiderivative = 1.06 \[ \int \csc ^n(c+d x) \sqrt {a-a \csc (c+d x)} \, dx=-\frac {2 a \cos (c+d x) \csc ^{1+2 n}(c+d x) \left (-\csc ^2(c+d x)\right )^{-n} \operatorname {Hypergeometric2F1}\left (\frac {1}{2},1-n,\frac {3}{2},1+\csc (c+d x)\right )}{d \sqrt {a-a \csc (c+d x)}} \]

[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]])

Maple [F]

\[\int \csc \left (d x +c \right )^{n} \sqrt {a -a \csc \left (d x +c \right )}d x\]

[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)

Fricas [F]

\[ \int \csc ^n(c+d x) \sqrt {a-a \csc (c+d x)} \, dx=\int { \sqrt {-a \csc \left (d x + c\right ) + a} \csc \left (d x + c\right )^{n} \,d x } \]

[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)

Sympy [F]

\[ \int \csc ^n(c+d x) \sqrt {a-a \csc (c+d x)} \, dx=\int \sqrt {- a \left (\csc {\left (c + d x \right )} - 1\right )} \csc ^{n}{\left (c + d x \right )}\, dx \]

[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)

Maxima [F]

\[ \int \csc ^n(c+d x) \sqrt {a-a \csc (c+d x)} \, dx=\int { \sqrt {-a \csc \left (d x + c\right ) + a} \csc \left (d x + c\right )^{n} \,d x } \]

[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)

Giac [F]

\[ \int \csc ^n(c+d x) \sqrt {a-a \csc (c+d x)} \, dx=\int { \sqrt {-a \csc \left (d x + c\right ) + a} \csc \left (d x + c\right )^{n} \,d x } \]

[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)

Mupad [F(-1)]

Timed out. \[ \int \csc ^n(c+d x) \sqrt {a-a \csc (c+d x)} \, dx=\int \sqrt {a-\frac {a}{\sin \left (c+d\,x\right )}}\,{\left (\frac {1}{\sin \left (c+d\,x\right )}\right )}^n \,d x \]

[In]

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

[Out]

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