Optimal. Leaf size=72 \[ \frac{\sec (a+b x) \sin ^m(2 a+2 b x) \cos ^2(a+b x)^{\frac{1-m}{2}} \text{Hypergeometric2F1}\left (\frac{1-m}{2},\frac{m}{2},\frac{m+2}{2},\sin ^2(a+b x)\right )}{b m} \]
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Rubi [A] time = 0.0702846, antiderivative size = 72, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 18, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.111, Rules used = {4310, 2577} \[ \frac{\sec (a+b x) \sin ^m(2 a+2 b x) \cos ^2(a+b x)^{\frac{1-m}{2}} \, _2F_1\left (\frac{1-m}{2},\frac{m}{2};\frac{m+2}{2};\sin ^2(a+b x)\right )}{b m} \]
Antiderivative was successfully verified.
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Rule 4310
Rule 2577
Rubi steps
\begin{align*} \int \csc (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 ^{-1+m}(a+b x) \, dx\\ &=\frac{\cos ^2(a+b x)^{\frac{1-m}{2}} \, _2F_1\left (\frac{1-m}{2},\frac{m}{2};\frac{2+m}{2};\sin ^2(a+b x)\right ) \sec (a+b x) \sin ^m(2 a+2 b x)}{b m}\\ \end{align*}
Mathematica [C] time = 0.889277, size = 254, normalized size = 3.53 \[ \frac{2 (m+2) \cos ^2\left (\frac{1}{2} (a+b x)\right ) \sin ^m(2 (a+b x)) F_1\left (\frac{m}{2};-m,2 m;\frac{m+2}{2};\tan ^2\left (\frac{1}{2} (a+b x)\right ),-\tan ^2\left (\frac{1}{2} (a+b x)\right )\right )}{b m \left ((m+2) (\cos (a+b x)+1) F_1\left (\frac{m}{2};-m,2 m;\frac{m+2}{2};\tan ^2\left (\frac{1}{2} (a+b x)\right ),-\tan ^2\left (\frac{1}{2} (a+b x)\right )\right )-4 m \sin ^2\left (\frac{1}{2} (a+b x)\right ) \left (F_1\left (\frac{m+2}{2};1-m,2 m;\frac{m+4}{2};\tan ^2\left (\frac{1}{2} (a+b x)\right ),-\tan ^2\left (\frac{1}{2} (a+b x)\right )\right )+2 F_1\left (\frac{m+2}{2};-m,2 m+1;\frac{m+4}{2};\tan ^2\left (\frac{1}{2} (a+b x)\right ),-\tan ^2\left (\frac{1}{2} (a+b x)\right )\right )\right )\right )} \]
Warning: Unable to verify antiderivative.
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Maple [F] time = 0.517, size = 0, normalized size = 0. \begin{align*} \int \csc \left ( bx+a \right ) \left ( \sin \left ( 2\,bx+2\,a \right ) \right ) ^{m}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \sin \left (2 \, b x + 2 \, a\right )^{m} \csc \left (b x + a\right )\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [F] time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (\sin \left (2 \, b x + 2 \, a\right )^{m} \csc \left (b x + a\right ), x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \sin ^{m}{\left (2 a + 2 b x \right )} \csc{\left (a + b x \right )}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \sin \left (2 \, b x + 2 \, a\right )^{m} \csc \left (b x + a\right )\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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