Optimal. Leaf size=26 \[ \frac{(a \sin (c+d x)+a)^{m+1}}{a d (m+1)} \]
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Rubi [A] time = 0.0275411, antiderivative size = 26, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.105, Rules used = {2667, 32} \[ \frac{(a \sin (c+d x)+a)^{m+1}}{a d (m+1)} \]
Antiderivative was successfully verified.
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Rule 2667
Rule 32
Rubi steps
\begin{align*} \int \cos (c+d x) (a+a \sin (c+d x))^m \, dx &=\frac{\operatorname{Subst}\left (\int (a+x)^m \, dx,x,a \sin (c+d x)\right )}{a d}\\ &=\frac{(a+a \sin (c+d x))^{1+m}}{a d (1+m)}\\ \end{align*}
Mathematica [A] time = 0.0321921, size = 26, normalized size = 1. \[ \frac{(a (\sin (c+d x)+1))^{m+1}}{a d (m+1)} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.001, size = 27, normalized size = 1. \begin{align*}{\frac{ \left ( a+a\sin \left ( dx+c \right ) \right ) ^{1+m}}{da \left ( 1+m \right ) }} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F(-2)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: ValueError} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 2.29077, size = 72, normalized size = 2.77 \begin{align*} \frac{{\left (a \sin \left (d x + c\right ) + a\right )}^{m}{\left (\sin \left (d x + c\right ) + 1\right )}}{d m + d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 2.64415, size = 80, normalized size = 3.08 \begin{align*} \begin{cases} \frac{x \cos{\left (c \right )}}{a \sin{\left (c \right )} + a} & \text{for}\: d = 0 \wedge m = -1 \\x \left (a \sin{\left (c \right )} + a\right )^{m} \cos{\left (c \right )} & \text{for}\: d = 0 \\\frac{\log{\left (\sin{\left (c + d x \right )} + 1 \right )}}{a d} & \text{for}\: m = -1 \\\frac{\left (a \sin{\left (c + d x \right )} + a\right )^{m} \sin{\left (c + d x \right )}}{d m + d} + \frac{\left (a \sin{\left (c + d x \right )} + a\right )^{m}}{d m + d} & \text{otherwise} \end{cases} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.07156, size = 35, normalized size = 1.35 \begin{align*} \frac{{\left (a \sin \left (d x + c\right ) + a\right )}^{m + 1}}{a d{\left (m + 1\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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