Optimal. Leaf size=34 \[ \frac {(a \sin (c+d x)+a)^m (e \cos (c+d x))^{-m}}{d e m} \]
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Rubi [A] time = 0.05, antiderivative size = 34, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, integrand size = 27, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.037, Rules used = {2671} \[ \frac {(a \sin (c+d x)+a)^m (e \cos (c+d x))^{-m}}{d e m} \]
Antiderivative was successfully verified.
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Rule 2671
Rubi steps
\begin {align*} \int (e \cos (c+d x))^{-1-m} (a+a \sin (c+d x))^m \, dx &=\frac {(e \cos (c+d x))^{-m} (a+a \sin (c+d x))^m}{d e m}\\ \end {align*}
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Mathematica [A] time = 0.05, size = 34, normalized size = 1.00 \[ \frac {(a (\sin (c+d x)+1))^m (e \cos (c+d x))^{-m}}{d e m} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.46, size = 39, normalized size = 1.15 \[ \frac {\left (e \cos \left (d x + c\right )\right )^{-m - 1} {\left (a \sin \left (d x + c\right ) + a\right )}^{m} \cos \left (d x + c\right )}{d m} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \left (e \cos \left (d x + c\right )\right )^{-m - 1} {\left (a \sin \left (d x + c\right ) + a\right )}^{m}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.30, size = 0, normalized size = 0.00 \[ \int \left (e \cos \left (d x +c \right )\right )^{-1-m} \left (a +a \sin \left (d x +c \right )\right )^{m}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.46, size = 65, normalized size = 1.91 \[ \frac {a^{m} e^{-m - 1} e^{\left (m \log \left (\frac {\sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} + 1\right ) - m \log \left (-\frac {\sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} + 1\right )\right )}}{d m} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.29, size = 34, normalized size = 1.00 \[ \frac {{\left (a\,\left (\sin \left (c+d\,x\right )+1\right )\right )}^m}{d\,e\,m\,{\left (e\,\cos \left (c+d\,x\right )\right )}^m} \]
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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