Optimal. Leaf size=72 \[ \frac {c \cos (a+b x) (c \csc (a+b x))^{n-1} \, _2F_1\left (\frac {1}{2},\frac {1-n}{2};\frac {3-n}{2};\sin ^2(a+b x)\right )}{b (1-n) \sqrt {\cos ^2(a+b x)}} \]
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Rubi [A] time = 0.03, antiderivative size = 72, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.200, 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.
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Rule 2643
Rule 3772
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*}
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Mathematica [A] time = 0.07, size = 65, normalized size = 0.90 \[ -\frac {\sin (a+b x) \cos (a+b x) \sin ^2(a+b x)^{\frac {n-1}{2}} (c \csc (a+b x))^n \, _2F_1\left (\frac {1}{2},\frac {n+1}{2};\frac {3}{2};\cos ^2(a+b x)\right )}{b} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.62, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\left (c \csc \left (b x + a\right )\right )^{n}, x\right ) \]
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 \[ \int \left (c \csc \left (b x + a\right )\right )^{n}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 2.62, size = 0, normalized size = 0.00 \[ \int \left (c \csc \left (b x +a \right )\right )^{n}\, dx \]
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 \[ \int \left (c \csc \left (b x + a\right )\right )^{n}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.01 \[ \int {\left (\frac {c}{\sin \left (a+b\,x\right )}\right )}^n \,d x \]
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 \[ \int \left (c \csc {\left (a + b x \right )}\right )^{n}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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