Optimal. Leaf size=171 \[ \frac {d n \cos (e+f x) (d \csc (e+f x))^{n-1} \, _2F_1\left (\frac {1}{2},\frac {1-n}{2};\frac {3-n}{2};\sin ^2(e+f x)\right )}{a f (1-n) \sqrt {\cos ^2(e+f x)}}+\frac {\cos (e+f x) (d \csc (e+f x))^n \, _2F_1\left (\frac {1}{2},-\frac {n}{2};\frac {2-n}{2};\sin ^2(e+f x)\right )}{a f \sqrt {\cos ^2(e+f x)}}-\frac {\cot (e+f x) (d \csc (e+f x))^n}{f (a \csc (e+f x)+a)} \]
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Rubi [A] time = 0.24, antiderivative size = 171, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 5, integrand size = 23, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.217, Rules used = {3238, 3820, 3787, 3772, 2643} \[ \frac {d n \cos (e+f x) (d \csc (e+f x))^{n-1} \, _2F_1\left (\frac {1}{2},\frac {1-n}{2};\frac {3-n}{2};\sin ^2(e+f x)\right )}{a f (1-n) \sqrt {\cos ^2(e+f x)}}+\frac {\cos (e+f x) (d \csc (e+f x))^n \, _2F_1\left (\frac {1}{2},-\frac {n}{2};\frac {2-n}{2};\sin ^2(e+f x)\right )}{a f \sqrt {\cos ^2(e+f x)}}-\frac {\cot (e+f x) (d \csc (e+f x))^n}{f (a \csc (e+f x)+a)} \]
Antiderivative was successfully verified.
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Rule 2643
Rule 3238
Rule 3772
Rule 3787
Rule 3820
Rubi steps
\begin {align*} \int \frac {(d \csc (e+f x))^n}{a+a \sin (e+f x)} \, dx &=\frac {\int \frac {(d \csc (e+f x))^{1+n}}{a+a \csc (e+f x)} \, dx}{d}\\ &=-\frac {\cot (e+f x) (d \csc (e+f x))^n}{f (a+a \csc (e+f x))}+\frac {n \int (d \csc (e+f x))^n (a-a \csc (e+f x)) \, dx}{a^2}\\ &=-\frac {\cot (e+f x) (d \csc (e+f x))^n}{f (a+a \csc (e+f x))}+\frac {n \int (d \csc (e+f x))^n \, dx}{a}-\frac {n \int (d \csc (e+f x))^{1+n} \, dx}{a d}\\ &=-\frac {\cot (e+f x) (d \csc (e+f x))^n}{f (a+a \csc (e+f x))}+\frac {\left (n (d \csc (e+f x))^n \left (\frac {\sin (e+f x)}{d}\right )^n\right ) \int \left (\frac {\sin (e+f x)}{d}\right )^{-n} \, dx}{a}-\frac {\left (n (d \csc (e+f x))^n \left (\frac {\sin (e+f x)}{d}\right )^n\right ) \int \left (\frac {\sin (e+f x)}{d}\right )^{-1-n} \, dx}{a d}\\ &=-\frac {\cot (e+f x) (d \csc (e+f x))^n}{f (a+a \csc (e+f x))}+\frac {\cos (e+f x) (d \csc (e+f x))^n \, _2F_1\left (\frac {1}{2},-\frac {n}{2};\frac {2-n}{2};\sin ^2(e+f x)\right )}{a f \sqrt {\cos ^2(e+f x)}}+\frac {n \cos (e+f x) (d \csc (e+f x))^n \, _2F_1\left (\frac {1}{2},\frac {1-n}{2};\frac {3-n}{2};\sin ^2(e+f x)\right ) \sin (e+f x)}{a f (1-n) \sqrt {\cos ^2(e+f x)}}\\ \end {align*}
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Mathematica [F] time = 2.80, size = 0, normalized size = 0.00 \[ \int \frac {(d \csc (e+f x))^n}{a+a \sin (e+f x)} \, dx \]
Verification is Not applicable to the result.
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fricas [F] time = 0.51, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {\left (d \csc \left (f x + e\right )\right )^{n}}{a \sin \left (f x + e\right ) + a}, x\right ) \]
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 \[ \int \frac {\left (d \csc \left (f x + e\right )\right )^{n}}{a \sin \left (f x + e\right ) + a}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.85, size = 0, normalized size = 0.00 \[ \int \frac {\left (d \csc \left (f x +e \right )\right )^{n}}{a +a \sin \left (f x +e \right )}\, dx \]
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 \[ \int \frac {\left (d \csc \left (f x + e\right )\right )^{n}}{a \sin \left (f x + e\right ) + a}\,{d x} \]
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 \[ \int \frac {{\left (\frac {d}{\sin \left (e+f\,x\right )}\right )}^n}{a+a\,\sin \left (e+f\,x\right )} \,d x \]
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 \[ \frac {\int \frac {\left (d \csc {\left (e + f x \right )}\right )^{n}}{\sin {\left (e + f x \right )} + 1}\, dx}{a} \]
Verification of antiderivative is not currently implemented for this CAS.
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