Optimal. Leaf size=85 \[ -\frac {\csc (a+b x) \sec (a+b x) \sin ^m(2 a+2 b x) \cos ^2(a+b x)^{\frac {1-m}{2}} \, _2F_1\left (\frac {1-m}{2},\frac {m-1}{2};\frac {m+1}{2};\sin ^2(a+b x)\right )}{b (1-m)} \]
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Rubi [A] time = 0.08, antiderivative size = 85, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.100, Rules used = {4310, 2577} \[ -\frac {\csc (a+b x) \sec (a+b x) \sin ^m(2 a+2 b x) \cos ^2(a+b x)^{\frac {1-m}{2}} \, _2F_1\left (\frac {1-m}{2},\frac {m-1}{2};\frac {m+1}{2};\sin ^2(a+b x)\right )}{b (1-m)} \]
Antiderivative was successfully verified.
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Rule 2577
Rule 4310
Rubi steps
\begin {align*} \int \csc ^2(a+b x) \sin ^m(2 a+2 b x) \, dx &=\left (\cos ^{-m}(a+b x) \sin ^{-m}(a+b x) \sin ^m(2 a+2 b x)\right ) \int \cos ^m(a+b x) \sin ^{-2+m}(a+b x) \, dx\\ &=-\frac {\cos ^2(a+b x)^{\frac {1-m}{2}} \csc (a+b x) \, _2F_1\left (\frac {1-m}{2},\frac {1}{2} (-1+m);\frac {1+m}{2};\sin ^2(a+b x)\right ) \sec (a+b x) \sin ^m(2 a+2 b x)}{b (1-m)}\\ \end {align*}
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Mathematica [C] time = 5.42, size = 938, normalized size = 11.04 \[ \frac {2 \left ((m+1) F_1\left (\frac {m-1}{2};-m,2 m;\frac {m+1}{2};\tan ^2\left (\frac {1}{2} (a+b x)\right ),-\tan ^2\left (\frac {1}{2} (a+b x)\right )\right ) \cot ^2\left (\frac {1}{2} (a+b x)\right )+(m-1) F_1\left (\frac {m+1}{2};-m,2 m;\frac {m+3}{2};\tan ^2\left (\frac {1}{2} (a+b x)\right ),-\tan ^2\left (\frac {1}{2} (a+b x)\right )\right )\right ) \csc ^2(a+b x) \sin ^m(2 (a+b x)) \tan \left (\frac {1}{2} (a+b x)\right )}{b \left (m (m+1) F_1\left (\frac {m-1}{2};-m,2 m;\frac {m+1}{2};\tan ^2\left (\frac {1}{2} (a+b x)\right ),-\tan ^2\left (\frac {1}{2} (a+b x)\right )\right ) (3 \cos (a+b x)-2) \sec (a+b x) \cot ^2\left (\frac {1}{2} (a+b x)\right )+2 m (m+1) F_1\left (\frac {m-1}{2};-m,2 m;\frac {m+1}{2};\tan ^2\left (\frac {1}{2} (a+b x)\right ),-\tan ^2\left (\frac {1}{2} (a+b x)\right )\right ) \tan (a+b x) \cot \left (\frac {1}{2} (a+b x)\right )-(m+1) F_1\left (\frac {m-1}{2};-m,2 m;\frac {m+1}{2};\tan ^2\left (\frac {1}{2} (a+b x)\right ),-\tan ^2\left (\frac {1}{2} (a+b x)\right )\right ) \csc ^2\left (\frac {1}{2} (a+b x)\right )+(m-1) F_1\left (\frac {m+1}{2};-m,2 m;\frac {m+3}{2};\tan ^2\left (\frac {1}{2} (a+b x)\right ),-\tan ^2\left (\frac {1}{2} (a+b x)\right )\right ) \sec ^2\left (\frac {1}{2} (a+b x)\right )-2 (m-1) m \left (F_1\left (\frac {m+1}{2};1-m,2 m;\frac {m+3}{2};\tan ^2\left (\frac {1}{2} (a+b x)\right ),-\tan ^2\left (\frac {1}{2} (a+b x)\right )\right )+2 F_1\left (\frac {m+1}{2};-m,2 m+1;\frac {m+3}{2};\tan ^2\left (\frac {1}{2} (a+b x)\right ),-\tan ^2\left (\frac {1}{2} (a+b x)\right )\right )\right ) \sec ^2\left (\frac {1}{2} (a+b x)\right )-\frac {2 (m-1) m (m+1) \left (F_1\left (\frac {m+3}{2};1-m,2 m;\frac {m+5}{2};\tan ^2\left (\frac {1}{2} (a+b x)\right ),-\tan ^2\left (\frac {1}{2} (a+b x)\right )\right )+2 F_1\left (\frac {m+3}{2};-m,2 m+1;\frac {m+5}{2};\tan ^2\left (\frac {1}{2} (a+b x)\right ),-\tan ^2\left (\frac {1}{2} (a+b x)\right )\right )\right ) \sec ^2\left (\frac {1}{2} (a+b x)\right ) \tan ^2\left (\frac {1}{2} (a+b x)\right )}{m+3}+(m-1) m F_1\left (\frac {m+1}{2};-m,2 m;\frac {m+3}{2};\tan ^2\left (\frac {1}{2} (a+b x)\right ),-\tan ^2\left (\frac {1}{2} (a+b x)\right )\right ) \tan ^2\left (\frac {1}{2} (a+b x)\right )+m (m+1) F_1\left (\frac {m-1}{2};-m,2 m;\frac {m+1}{2};\tan ^2\left (\frac {1}{2} (a+b x)\right ),-\tan ^2\left (\frac {1}{2} (a+b x)\right )\right )+(m-1) m F_1\left (\frac {m+1}{2};-m,2 m;\frac {m+3}{2};\tan ^2\left (\frac {1}{2} (a+b x)\right ),-\tan ^2\left (\frac {1}{2} (a+b x)\right )\right ) (3 \cos (a+b x)-2) \sec (a+b x)+2 (m-1) m F_1\left (\frac {m+1}{2};-m,2 m;\frac {m+3}{2};\tan ^2\left (\frac {1}{2} (a+b x)\right ),-\tan ^2\left (\frac {1}{2} (a+b x)\right )\right ) \tan \left (\frac {1}{2} (a+b x)\right ) \tan (a+b x)\right )} \]
Warning: Unable to verify antiderivative.
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fricas [F] time = 0.51, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\sin \left (2 \, b x + 2 \, a\right )^{m} \csc \left (b x + a\right )^{2}, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 4.28, size = 0, normalized size = 0.00 \[ \int \left (\csc ^{2}\left (b x +a \right )\right ) \left (\sin ^{m}\left (2 b x +2 a \right )\right )\, dx \]
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 \[ \int \sin \left (2 \, b x + 2 \, a\right )^{m} \csc \left (b x + a\right )^{2}\,{d x} \]
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 \[ \int \frac {{\sin \left (2\,a+2\,b\,x\right )}^m}{{\sin \left (a+b\,x\right )}^2} \,d x \]
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 \[ \int \sin ^{m}{\left (2 a + 2 b x \right )} \csc ^{2}{\left (a + b x \right )}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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