Optimal. Leaf size=35 \[ -\frac{\cos (c+d x) \sin ^{-m-1}(c+d x) (a \sin (c+d x)+a)^m}{d} \]
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Rubi [A] time = 0.0944697, antiderivative size = 35, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 37, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.027, Rules used = {2974} \[ -\frac{\cos (c+d x) \sin ^{-m-1}(c+d x) (a \sin (c+d x)+a)^m}{d} \]
Antiderivative was successfully verified.
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Rule 2974
Rubi steps
\begin{align*} \int \sin ^{-2-m}(c+d x) (a+a \sin (c+d x))^m (1+m-m \sin (c+d x)) \, dx &=-\frac{\cos (c+d x) \sin ^{-1-m}(c+d x) (a+a \sin (c+d x))^m}{d}\\ \end{align*}
Mathematica [A] time = 0.362636, size = 35, normalized size = 1. \[ -\frac{\cos (c+d x) \sin ^{-m-1}(c+d x) (a (\sin (c+d x)+1))^m}{d} \]
Antiderivative was successfully verified.
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Maple [F] time = 0.454, size = 0, normalized size = 0. \begin{align*} \int \left ( \sin \left ( dx+c \right ) \right ) ^{-2-m} \left ( a+a\sin \left ( dx+c \right ) \right ) ^{m} \left ( 1+m-m\sin \left ( dx+c \right ) \right ) \, 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 (m \sin \left (d x + c\right ) - m - 1\right )}{\left (a \sin \left (d x + c\right ) + a\right )}^{m} \sin \left (d x + c\right )^{-m - 2}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.46566, size = 101, normalized size = 2.89 \begin{align*} -\frac{{\left (a \sin \left (d x + c\right ) + a\right )}^{m} \sin \left (d x + c\right )^{-m - 2} \cos \left (d x + c\right ) \sin \left (d x + c\right )}{d} \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 -{\left (m \sin \left (d x + c\right ) - m - 1\right )}{\left (a \sin \left (d x + c\right ) + a\right )}^{m} \sin \left (d x + c\right )^{-m - 2}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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