Optimal. Leaf size=48 \[ -\frac {2 a \cot (c+d x) \, _2F_1\left (\frac {1}{2},1-n;\frac {3}{2};1-\csc (c+d x)\right )}{d \sqrt {a \csc (c+d x)+a}} \]
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Rubi [A] time = 0.06, antiderivative size = 48, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 23, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.087, Rules used = {3806, 65} \[ -\frac {2 a \cot (c+d x) \, _2F_1\left (\frac {1}{2},1-n;\frac {3}{2};1-\csc (c+d x)\right )}{d \sqrt {a \csc (c+d x)+a}} \]
Antiderivative was successfully verified.
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Rule 65
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 {2 a \cot (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*}
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Mathematica [A] time = 0.17, size = 48, normalized size = 1.00 \[ -\frac {2 a \cot (c+d x) \, _2F_1\left (\frac {1}{2},1-n;\frac {3}{2};1-\csc (c+d x)\right )}{d \sqrt {a (\csc (c+d x)+1)}} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.94, 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.
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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.
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maple [F] time = 1.95, 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.
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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.
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mupad [F] time = 0.00, size = -1, normalized size = -0.02 \[ \int \sqrt {a+\frac {a}{\sin \left (c+d\,x\right )}}\,{\left (\frac {1}{\sin \left (c+d\,x\right )}\right )}^n \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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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.
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