Optimal. Leaf size=142 \[ -\frac{2 (a \sin (c+d x)+a)^{m+2} (e \cos (c+d x))^{-m-2}}{a^2 d e m \left (4-m^2\right )}-\frac{(a \sin (c+d x)+a)^m (e \cos (c+d x))^{-m-2}}{d e (2-m)}+\frac{2 (a \sin (c+d x)+a)^{m+1} (e \cos (c+d x))^{-m-2}}{a d e (2-m) m} \]
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Rubi [A] time = 0.222483, antiderivative size = 142, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 27, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.074, Rules used = {2672, 2671} \[ -\frac{2 (a \sin (c+d x)+a)^{m+2} (e \cos (c+d x))^{-m-2}}{a^2 d e m \left (4-m^2\right )}-\frac{(a \sin (c+d x)+a)^m (e \cos (c+d x))^{-m-2}}{d e (2-m)}+\frac{2 (a \sin (c+d x)+a)^{m+1} (e \cos (c+d x))^{-m-2}}{a d e (2-m) m} \]
Antiderivative was successfully verified.
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Rule 2672
Rule 2671
Rubi steps
\begin{align*} \int (e \cos (c+d x))^{-3-m} (a+a \sin (c+d x))^m \, dx &=-\frac{(e \cos (c+d x))^{-2-m} (a+a \sin (c+d x))^m}{d e (2-m)}+\frac{2 \int (e \cos (c+d x))^{-3-m} (a+a \sin (c+d x))^{1+m} \, dx}{a (2-m)}\\ &=-\frac{(e \cos (c+d x))^{-2-m} (a+a \sin (c+d x))^m}{d e (2-m)}+\frac{2 (e \cos (c+d x))^{-2-m} (a+a \sin (c+d x))^{1+m}}{a d e (2-m) m}-\frac{2 \int (e \cos (c+d x))^{-3-m} (a+a \sin (c+d x))^{2+m} \, dx}{a^2 (2-m) m}\\ &=-\frac{(e \cos (c+d x))^{-2-m} (a+a \sin (c+d x))^m}{d e (2-m)}+\frac{2 (e \cos (c+d x))^{-2-m} (a+a \sin (c+d x))^{1+m}}{a d e (2-m) m}-\frac{2 (e \cos (c+d x))^{-2-m} (a+a \sin (c+d x))^{2+m}}{a^2 d e m \left (4-m^2\right )}\\ \end{align*}
Mathematica [A] time = 0.136861, size = 76, normalized size = 0.54 \[ \frac{\sec ^2(c+d x) \left (-2 m \sin (c+d x)+2 \sin ^2(c+d x)+m^2-2\right ) (a (\sin (c+d x)+1))^m (e \cos (c+d x))^{-m}}{d e^3 (m-2) m (m+2)} \]
Antiderivative was successfully verified.
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Maple [F] time = 0.157, size = 0, normalized size = 0. \begin{align*} \int \left ( e\cos \left ( dx+c \right ) \right ) ^{-3-m} \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] time = 0., size = 0, normalized size = 0. \begin{align*} \int \left (e \cos \left (d x + c\right )\right )^{-m - 3}{\left (a \sin \left (d x + c\right ) + a\right )}^{m}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 2.33089, size = 184, normalized size = 1.3 \begin{align*} \frac{{\left (m^{2} \cos \left (d x + c\right ) - 2 \, \cos \left (d x + c\right )^{3} - 2 \, m \cos \left (d x + c\right ) \sin \left (d x + c\right )\right )} \left (e \cos \left (d x + c\right )\right )^{-m - 3}{\left (a \sin \left (d x + c\right ) + a\right )}^{m}}{d m^{3} - 4 \, d m} \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 (e \cos \left (d x + c\right )\right )^{-m - 3}{\left (a \sin \left (d x + c\right ) + a\right )}^{m}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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