Optimal. Leaf size=40 \[ -\frac {\cos (e+f x) (a-a \sin (e+f x))^m (c+d \sin (e+f x))^{-1-m}}{f} \]
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Rubi [A]
time = 0.12, antiderivative size = 40, normalized size of antiderivative = 1.00, number of steps
used = 1, number of rules used = 1, integrand size = 51, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.020, Rules used = {3053}
\begin {gather*} -\frac {\cos (e+f x) (a-a \sin (e+f x))^m (c+d \sin (e+f x))^{-m-1}}{f} \end {gather*}
Antiderivative was successfully verified.
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Rule 3053
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 = 0.53, size = 40, normalized size = 1.00 \begin {gather*} -\frac {\cos (e+f x) (a-a \sin (e+f x))^m (c+d \sin (e+f x))^{-1-m}}{f} \end {gather*}
Antiderivative was successfully verified.
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Maple [F]
time = 1.52, 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]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.46, size = 62, normalized size = 1.55 \begin {gather*} -\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} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: SystemError} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F(-1)] Timed out
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 14.62, size = 99, normalized size = 2.48 \begin {gather*} -\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 )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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