Optimal. Leaf size=39 \[ -\frac {\cos (e+f x) (a \sin (e+f x)+a)^m (c+d \sin (e+f x))^{-m-1}}{f} \]
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Rubi [A] time = 0.17, antiderivative size = 39, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, integrand size = 55, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.018, Rules used = {2974} \[ -\frac {\cos (e+f x) (a \sin (e+f x)+a)^m (c+d \sin (e+f x))^{-m-1}}{f} \]
Antiderivative was successfully verified.
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Rule 2974
Rubi steps
\begin {align*} \int (a+a \sin (e+f x))^m (c+d \sin (e+f x))^{-2-m} (d-(c-d) m+(c+(c-d) m) \sin (e+f x)) \, dx &=-\frac {\cos (e+f x) (a+a \sin (e+f x))^m (c+d \sin (e+f x))^{-1-m}}{f}\\ \end {align*}
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Mathematica [A] time = 0.68, size = 39, normalized size = 1.00 \[ -\frac {\cos (e+f x) (a (\sin (e+f x)+1))^m (c+d \sin (e+f x))^{-m-1}}{f} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.54, size = 56, normalized size = 1.44 \[ -\frac {{\left (d \cos \left (f x + e\right ) \sin \left (f x + e\right ) + c \cos \left (f x + e\right )\right )} {\left (a \sin \left (f x + e\right ) + a\right )}^{m} {\left (d \sin \left (f x + e\right ) + c\right )}^{-m - 2}}{f} \]
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 = 6.47, size = 0, normalized size = 0.00 \[ \int \left (a +a \sin \left (f x +e \right )\right )^{m} \left (c +d \sin \left (f x +e \right )\right )^{-2-m} \left (d -\left (c -d \right ) m +\left (c +\left (c -d \right ) m \right ) \sin \left (f x +e \right )\right )\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [F(-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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mupad [B] time = 14.77, size = 98, normalized size = 2.51 \[ -\frac {{\left (a\,\left (\sin \left (e+f\,x\right )+1\right )\right )}^m\,\left (d\,\sin \left (2\,e+2\,f\,x\right )-2\,c\,\left (2\,{\sin \left (\frac {e}{2}+\frac {f\,x}{2}\right )}^2-1\right )\right )}{f\,{\left (c+d\,\sin \left (e+f\,x\right )\right )}^m\,\left (d^2\,\left (2\,{\sin \left (e+f\,x\right )}^2-1\right )+2\,c^2+d^2+4\,c\,d\,\sin \left (e+f\,x\right )\right )} \]
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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