Optimal. Leaf size=55 \[ \frac{2 (a \sin (c+d x)+a)^{m+2}}{a^2 d (m+2)}-\frac{(a \sin (c+d x)+a)^{m+3}}{a^3 d (m+3)} \]
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Rubi [A] time = 0.0576335, antiderivative size = 55, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.095, Rules used = {2667, 43} \[ \frac{2 (a \sin (c+d x)+a)^{m+2}}{a^2 d (m+2)}-\frac{(a \sin (c+d x)+a)^{m+3}}{a^3 d (m+3)} \]
Antiderivative was successfully verified.
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Rule 2667
Rule 43
Rubi steps
\begin{align*} \int \cos ^3(c+d x) (a+a \sin (c+d x))^m \, dx &=\frac{\operatorname{Subst}\left (\int (a-x) (a+x)^{1+m} \, dx,x,a \sin (c+d x)\right )}{a^3 d}\\ &=\frac{\operatorname{Subst}\left (\int \left (2 a (a+x)^{1+m}-(a+x)^{2+m}\right ) \, dx,x,a \sin (c+d x)\right )}{a^3 d}\\ &=\frac{2 (a+a \sin (c+d x))^{2+m}}{a^2 d (2+m)}-\frac{(a+a \sin (c+d x))^{3+m}}{a^3 d (3+m)}\\ \end{align*}
Mathematica [A] time = 0.111585, size = 52, normalized size = 0.95 \[ -\frac{(\sin (c+d x)+1)^2 ((m+2) \sin (c+d x)-m-4) (a (\sin (c+d x)+1))^m}{d (m+2) (m+3)} \]
Antiderivative was successfully verified.
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Maple [F] time = 0.837, size = 0, normalized size = 0. \begin{align*} \int \left ( \cos \left ( dx+c \right ) \right ) ^{3} \left ( a+a\sin \left ( dx+c \right ) \right ) ^{m}\, dx \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 = 2.67177, size = 153, normalized size = 2.78 \begin{align*} \frac{{\left (m \cos \left (d x + c\right )^{2} +{\left ({\left (m + 2\right )} \cos \left (d x + c\right )^{2} + 4\right )} \sin \left (d x + c\right ) + 4\right )}{\left (a \sin \left (d x + c\right ) + a\right )}^{m}}{d m^{2} + 5 \, d m + 6 \, 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 [B] time = 1.09057, size = 185, normalized size = 3.36 \begin{align*} -\frac{{\left (a \sin \left (d x + c\right ) + a\right )}^{3}{\left (a \sin \left (d x + c\right ) + a\right )}^{m} m - 2 \,{\left (a \sin \left (d x + c\right ) + a\right )}^{2}{\left (a \sin \left (d x + c\right ) + a\right )}^{m} a m + 2 \,{\left (a \sin \left (d x + c\right ) + a\right )}^{3}{\left (a \sin \left (d x + c\right ) + a\right )}^{m} - 6 \,{\left (a \sin \left (d x + c\right ) + a\right )}^{2}{\left (a \sin \left (d x + c\right ) + a\right )}^{m} a}{{\left (a^{2} m^{2} + 5 \, a^{2} m + 6 \, a^{2}\right )} a d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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