Optimal. Leaf size=37 \[ -\frac{B \cos (e+f x) (a-a \sin (e+f x))^m (c \sin (e+f x)+c)^n}{f} \]
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Rubi [A] time = 0.123455, antiderivative size = 37, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 46, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.022, Rules used = {2970} \[ -\frac{B \cos (e+f x) (a-a \sin (e+f x))^m (c \sin (e+f x)+c)^n}{f} \]
Antiderivative was successfully verified.
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Rule 2970
Rubi steps
\begin{align*} \int (a-a \sin (e+f x))^m (c+c \sin (e+f x))^n (B (m-n)+B (1+m+n) \sin (e+f x)) \, dx &=-\frac{B \cos (e+f x) (a-a \sin (e+f x))^m (c+c \sin (e+f x))^n}{f}\\ \end{align*}
Mathematica [A] time = 0.463715, size = 37, normalized size = 1. \[ -\frac{B \cos (e+f x) (a-a \sin (e+f x))^m (c (\sin (e+f x)+1))^n}{f} \]
Antiderivative was successfully verified.
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Maple [F] time = 2.543, size = 0, normalized size = 0. \begin{align*} \int \left ( a-a\sin \left ( fx+e \right ) \right ) ^{m} \left ( c+c\sin \left ( fx+e \right ) \right ) ^{n} \left ( B \left ( m-n \right ) +B \left ( m+n+1 \right ) \sin \left ( fx+e \right ) \right ) \, 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 (B{\left (m + n + 1\right )} \sin \left (f x + e\right ) + B{\left (m - n\right )}\right )}{\left (-a \sin \left (f x + e\right ) + a\right )}^{m}{\left (c \sin \left (f x + e\right ) + c\right )}^{n}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 2.07714, size = 89, normalized size = 2.41 \begin{align*} -\frac{{\left (-a \sin \left (f x + e\right ) + a\right )}^{m}{\left (c \sin \left (f x + e\right ) + c\right )}^{n} B \cos \left (f x + e\right )}{f} \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(-2)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: NotImplementedError} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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