Optimal. Leaf size=83 \[ -\frac{\cos (a+b x) \cot (a+b x) \sin ^2(a+b x)^{\frac{1-m}{2}} \sin ^m(2 a+2 b x) \text{Hypergeometric2F1}\left (\frac{1-m}{2},\frac{m+2}{2},\frac{m+4}{2},\cos ^2(a+b x)\right )}{b (m+2)} \]
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Rubi [A] time = 0.0608929, antiderivative size = 83, 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 = {4309, 2576} \[ -\frac{\cos (a+b x) \cot (a+b x) \sin ^2(a+b x)^{\frac{1-m}{2}} \sin ^m(2 a+2 b x) \, _2F_1\left (\frac{1-m}{2},\frac{m+2}{2};\frac{m+4}{2};\cos ^2(a+b x)\right )}{b (m+2)} \]
Antiderivative was successfully verified.
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Rule 4309
Rule 2576
Rubi steps
\begin{align*} \int \cos (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 ^{1+m}(a+b x) \sin ^m(a+b x) \, dx\\ &=-\frac{\cos (a+b x) \cot (a+b x) \, _2F_1\left (\frac{1-m}{2},\frac{2+m}{2};\frac{4+m}{2};\cos ^2(a+b x)\right ) \sin ^2(a+b x)^{\frac{1-m}{2}} \sin ^m(2 a+2 b x)}{b (2+m)}\\ \end{align*}
Mathematica [C] time = 0.244453, size = 149, normalized size = 1.8 \[ \frac{2^{-m-1} e^{i (a+b x)} \left (-i e^{-2 i (a+b x)} \left (-1+e^{4 i (a+b x)}\right )\right )^{m+1} \left ((2 m-1) \text{Hypergeometric2F1}\left (1,\frac{1}{4} (2 m+3),\frac{1}{4} (3-2 m),e^{4 i (a+b x)}\right )+(2 m+1) e^{2 i (a+b x)} \text{Hypergeometric2F1}\left (1,\frac{1}{4} (2 m+5),\frac{1}{4} (5-2 m),e^{4 i (a+b x)}\right )\right )}{b \left (4 m^2-1\right )} \]
Warning: Unable to verify antiderivative.
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Maple [F] time = 0.914, size = 0, normalized size = 0. \begin{align*} \int \cos \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} \cos \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} \cos \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(-1)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \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} \cos \left (b x + a\right )\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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