Optimal. Leaf size=69 \[ -\frac {2 a \cos (c+d x) (-\csc (c+d x))^{-n} \csc ^{n+1}(c+d x) \, _2F_1\left (\frac {1}{2},1-n;\frac {3}{2};\csc (c+d x)+1\right )}{d \sqrt {a-a \csc (c+d x)}} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.07, antiderivative size = 69, normalized size of antiderivative = 1.00, 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]
[Out]
Rule 65
Rule 67
Rule 3806
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 = 0.99, size = 73, normalized size = 1.06 \[ -\frac {2 a \cos (c+d x) \csc ^{2 n+1}(c+d x) \left (-\csc ^2(c+d x)\right )^{-n} \, _2F_1\left (\frac {1}{2},1-n;\frac {3}{2};\csc (c+d x)+1\right )}{d \sqrt {a-a \csc (c+d x)}} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
fricas [F] time = 0.65, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\sqrt {-a \csc \left (d x + c\right ) + a} \csc \left (d x + c\right )^{n}, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \sqrt {-a \csc \left (d x + c\right ) + a} \csc \left (d x + c\right )^{n}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [F] time = 2.45, size = 0, normalized size = 0.00 \[ \int \left (\csc ^{n}\left (d x +c \right )\right ) \sqrt {a -a \csc \left (d x +c \right )}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \sqrt {-a \csc \left (d x + c\right ) + a} \csc \left (d x + c\right )^{n}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [F] time = 0.00, size = -1, normalized size = -0.01 \[ \int \sqrt {a-\frac {a}{\sin \left (c+d\,x\right )}}\,{\left (\frac {1}{\sin \left (c+d\,x\right )}\right )}^n \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \sqrt {- a \left (\csc {\left (c + d x \right )} - 1\right )} \csc ^{n}{\left (c + d x \right )}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________