Optimal. Leaf size=201 \[ \frac{6 (a \sin (c+d x)+a)^{m+2} (e \cos (c+d x))^{-m-3}}{a^2 d e (3-m) \left (1-m^2\right )}-\frac{6 (a \sin (c+d x)+a)^{m+3} (e \cos (c+d x))^{-m-3}}{a^3 d e \left (m^4-10 m^2+9\right )}-\frac{(a \sin (c+d x)+a)^m (e \cos (c+d x))^{-m-3}}{d e (3-m)}-\frac{3 (a \sin (c+d x)+a)^{m+1} (e \cos (c+d x))^{-m-3}}{a d e (1-m) (3-m)} \]
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Rubi [A] time = 0.321228, antiderivative size = 201, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 2, integrand size = 27, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.074, Rules used = {2672, 2671} \[ \frac{6 (a \sin (c+d x)+a)^{m+2} (e \cos (c+d x))^{-m-3}}{a^2 d e (3-m) \left (1-m^2\right )}-\frac{6 (a \sin (c+d x)+a)^{m+3} (e \cos (c+d x))^{-m-3}}{a^3 d e \left (m^4-10 m^2+9\right )}-\frac{(a \sin (c+d x)+a)^m (e \cos (c+d x))^{-m-3}}{d e (3-m)}-\frac{3 (a \sin (c+d x)+a)^{m+1} (e \cos (c+d x))^{-m-3}}{a d e (1-m) (3-m)} \]
Antiderivative was successfully verified.
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Rule 2672
Rule 2671
Rubi steps
\begin{align*} \int (e \cos (c+d x))^{-4-m} (a+a \sin (c+d x))^m \, dx &=-\frac{(e \cos (c+d x))^{-3-m} (a+a \sin (c+d x))^m}{d e (3-m)}+\frac{3 \int (e \cos (c+d x))^{-4-m} (a+a \sin (c+d x))^{1+m} \, dx}{a (3-m)}\\ &=-\frac{(e \cos (c+d x))^{-3-m} (a+a \sin (c+d x))^m}{d e (3-m)}-\frac{3 (e \cos (c+d x))^{-3-m} (a+a \sin (c+d x))^{1+m}}{a d e (1-m) (3-m)}+\frac{6 \int (e \cos (c+d x))^{-4-m} (a+a \sin (c+d x))^{2+m} \, dx}{a^2 (1-m) (3-m)}\\ &=-\frac{(e \cos (c+d x))^{-3-m} (a+a \sin (c+d x))^m}{d e (3-m)}-\frac{3 (e \cos (c+d x))^{-3-m} (a+a \sin (c+d x))^{1+m}}{a d e (1-m) (3-m)}+\frac{6 (e \cos (c+d x))^{-3-m} (a+a \sin (c+d x))^{2+m}}{a^2 d e (1-m) (3-m) (1+m)}-\frac{6 \int (e \cos (c+d x))^{-4-m} (a+a \sin (c+d x))^{3+m} \, dx}{a^3 (1-m) (3-m) (1+m)}\\ &=-\frac{(e \cos (c+d x))^{-3-m} (a+a \sin (c+d x))^m}{d e (3-m)}-\frac{3 (e \cos (c+d x))^{-3-m} (a+a \sin (c+d x))^{1+m}}{a d e (1-m) (3-m)}+\frac{6 (e \cos (c+d x))^{-3-m} (a+a \sin (c+d x))^{2+m}}{a^2 d e (1-m) (3-m) (1+m)}-\frac{6 (e \cos (c+d x))^{-3-m} (a+a \sin (c+d x))^{3+m}}{a^3 d e \left (9-10 m^2+m^4\right )}\\ \end{align*}
Mathematica [A] time = 0.196414, size = 101, normalized size = 0.5 \[ \frac{\sec ^3(c+d x) \left (-3 \left (m^2-3\right ) \sin (c+d x)+6 m \sin ^2(c+d x)-6 \sin ^3(c+d x)+m \left (m^2-7\right )\right ) (a (\sin (c+d x)+1))^m (e \cos (c+d x))^{-m}}{d e^4 (m-3) (m-1) (m+1) (m+3)} \]
Antiderivative was successfully verified.
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Maple [F] time = 0.171, size = 0, normalized size = 0. \begin{align*} \int \left ( e\cos \left ( dx+c \right ) \right ) ^{-4-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 - 4}{\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 = 2.37926, size = 247, normalized size = 1.23 \begin{align*} -\frac{{\left (6 \, m \cos \left (d x + c\right )^{3} -{\left (m^{3} - m\right )} \cos \left (d x + c\right ) - 3 \,{\left (2 \, \cos \left (d x + c\right )^{3} -{\left (m^{2} - 1\right )} \cos \left (d x + c\right )\right )} \sin \left (d x + c\right )\right )} \left (e \cos \left (d x + c\right )\right )^{-m - 4}{\left (a \sin \left (d x + c\right ) + a\right )}^{m}}{d m^{4} - 10 \, d m^{2} + 9 \, 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 (e \cos \left (d x + c\right )\right )^{-m - 4}{\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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