Optimal. Leaf size=68 \[ \frac{\cos (a+b x) (c \sin (a+b x))^{m+1} \, _2F_1\left (\frac{1}{2},\frac{m+1}{2};\frac{m+3}{2};\sin ^2(a+b x)\right )}{b c (m+1) \sqrt{\cos ^2(a+b x)}} \]
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Rubi [A] time = 0.0146984, antiderivative size = 68, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 10, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.1, Rules used = {2643} \[ \frac{\cos (a+b x) (c \sin (a+b x))^{m+1} \, _2F_1\left (\frac{1}{2},\frac{m+1}{2};\frac{m+3}{2};\sin ^2(a+b x)\right )}{b c (m+1) \sqrt{\cos ^2(a+b x)}} \]
Antiderivative was successfully verified.
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Rule 2643
Rubi steps
\begin{align*} \int (c \sin (a+b x))^m \, dx &=\frac{\cos (a+b x) \, _2F_1\left (\frac{1}{2},\frac{1+m}{2};\frac{3+m}{2};\sin ^2(a+b x)\right ) (c \sin (a+b x))^{1+m}}{b c (1+m) \sqrt{\cos ^2(a+b x)}}\\ \end{align*}
Mathematica [A] time = 0.0389735, size = 63, normalized size = 0.93 \[ \frac{\sqrt{\cos ^2(a+b x)} \tan (a+b x) (c \sin (a+b x))^m \, _2F_1\left (\frac{1}{2},\frac{m+1}{2};\frac{m+3}{2};\sin ^2(a+b x)\right )}{b (m+1)} \]
Antiderivative was successfully verified.
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Maple [F] time = 0.457, size = 0, normalized size = 0. \begin{align*} \int \left ( c\sin \left ( bx+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 \left (c \sin \left (b x + a\right )\right )^{m}\,{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 (\left (c \sin \left (b x + a\right )\right )^{m}, 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 \left (c \sin{\left (a + b x \right )}\right )^{m}\, 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 \left (c \sin \left (b x + a\right )\right )^{m}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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