Optimal. Leaf size=69 \[ \frac {\cos (a+b x) \csc ^{n-1}(a+b x) \, _2F_1\left (\frac {1}{2},\frac {1-n}{2};\frac {3-n}{2};\sin ^2(a+b x)\right )}{b (1-n) \sqrt {\cos ^2(a+b x)}} \]
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Rubi [A] time = 0.03, antiderivative size = 69, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {3772, 2643} \[ \frac {\cos (a+b x) \csc ^{n-1}(a+b x) \, _2F_1\left (\frac {1}{2},\frac {1-n}{2};\frac {3-n}{2};\sin ^2(a+b x)\right )}{b (1-n) \sqrt {\cos ^2(a+b x)}} \]
Antiderivative was successfully verified.
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Rule 2643
Rule 3772
Rubi steps
\begin {align*} \int \csc ^n(a+b x) \, dx &=\csc ^n(a+b x) \sin ^n(a+b x) \int \sin ^{-n}(a+b x) \, dx\\ &=\frac {\cos (a+b x) \csc ^{-1+n}(a+b x) \, _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)}}\\ \end {align*}
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Mathematica [A] time = 0.09, size = 59, normalized size = 0.86 \[ -\frac {\cos (a+b x) \sin ^2(a+b x)^{\frac {n-1}{2}} \csc ^{n-1}(a+b x) \, _2F_1\left (\frac {1}{2},\frac {n+1}{2};\frac {3}{2};\cos ^2(a+b x)\right )}{b} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.66, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\csc \left (b x + a\right )^{n}, x\right ) \]
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 \[ \int \csc \left (b x + a\right )^{n}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 2.65, size = 0, normalized size = 0.00 \[ \int \csc ^{n}\left (b x +a \right )\, dx \]
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 \[ \int \csc \left (b x + a\right )^{n}\,{d x} \]
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 \[ \int {\left (\frac {1}{\sin \left (a+b\,x\right )}\right )}^n \,d x \]
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 \[ \int \csc ^{n}{\left (a + b x \right )}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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