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 = 0.12, antiderivative size = 37, normalized size of antiderivative = 1.00, 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*}
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Mathematica [A] time = 0.47, size = 37, normalized size = 1.00 \[ -\frac {B \cos (e+f x) (a-a \sin (e+f x))^m (c (\sin (e+f x)+1))^n}{f} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.50, size = 37, normalized size = 1.00 \[ -\frac {{\left (-a \sin \left (f x + e\right ) + a\right )}^{m} {\left (c \sin \left (f x + e\right ) + c\right )}^{n} B \cos \left (f x + e\right )}{f} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 8.40, size = 0, normalized size = 0.00 \[ \int \left (a -a \sin \left (f x +e \right )\right )^{m} \left (c +c \sin \left (f x +e \right )\right )^{n} \left (B \left (m -n \right )+B \left (n +m +1\right ) \sin \left (f x +e \right )\right )\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int {\left (B {\left (m + n + 1\right )} \sin \left (f x + e\right ) + B {\left (m - n\right )}\right )} {\left (-a \sin \left (f x + e\right ) + a\right )}^{m} {\left (c \sin \left (f x + e\right ) + c\right )}^{n}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 13.50, size = 37, normalized size = 1.00 \[ -\frac {B\,\cos \left (e+f\,x\right )\,{\left (-a\,\left (\sin \left (e+f\,x\right )-1\right )\right )}^m\,{\left (c\,\left (\sin \left (e+f\,x\right )+1\right )\right )}^n}{f} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
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