Optimal. Leaf size=26 \[ \frac {(a \sin (c+d x)+a)^{m+1}}{a d (m+1)} \]
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Rubi [A] time = 0.03, antiderivative size = 26, normalized size of antiderivative = 1.00, 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 32
Rule 2667
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*}
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Mathematica [A] time = 0.04, size = 26, normalized size = 1.00 \[ \frac {(a (\sin (c+d x)+1))^{m+1}}{a d (m+1)} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.46, size = 28, normalized size = 1.08 \[ \frac {{\left (a \sin \left (d x + c\right ) + a\right )}^{m} {\left (\sin \left (d x + c\right ) + 1\right )}}{d m + d} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.92, size = 26, normalized size = 1.00 \[ \frac {{\left (a \sin \left (d x + c\right ) + a\right )}^{m + 1}}{a d {\left (m + 1\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.02, size = 27, normalized size = 1.04 \[ \frac {\left (a +a \sin \left (d x +c \right )\right )^{1+m}}{a d \left (1+m \right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.54, size = 26, normalized size = 1.00 \[ \frac {{\left (a \sin \left (d x + c\right ) + a\right )}^{m + 1}}{a d {\left (m + 1\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.22, size = 29, normalized size = 1.12 \[ \frac {{\left (a\,\left (\sin \left (c+d\,x\right )+1\right )\right )}^m\,\left (\sin \left (c+d\,x\right )+1\right )}{d\,\left (m+1\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 2.47, size = 80, normalized size = 3.08 \[ \begin {cases} \frac {x \cos {\relax (c )}}{a \sin {\relax (c )} + a} & \text {for}\: d = 0 \wedge m = -1 \\x \left (a \sin {\relax (c )} + a\right )^{m} \cos {\relax (c )} & \text {for}\: d = 0 \\\frac {\log {\left (\sin {\left (c + d x \right )} + 1 \right )}}{a d} & \text {for}\: m = -1 \\\frac {\left (a \sin {\left (c + d x \right )} + a\right )^{m} \sin {\left (c + d x \right )}}{d m + d} + \frac {\left (a \sin {\left (c + d x \right )} + a\right )^{m}}{d m + d} & \text {otherwise} \end {cases} \]
Verification of antiderivative is not currently implemented for this CAS.
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