Optimal. Leaf size=34 \[ \frac {(e \cos (c+d x))^{-m} (a+a \sin (c+d x))^m}{d e m} \]
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Rubi [A]
time = 0.03, 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 = {2750}
\begin {gather*} \frac {(a \sin (c+d x)+a)^m (e \cos (c+d x))^{-m}}{d e m} \end {gather*}
Antiderivative was successfully verified.
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Rule 2750
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 = 0.06, size = 34, normalized size = 1.00 \begin {gather*} \frac {(e \cos (c+d x))^{-m} (a (1+\sin (c+d x)))^m}{d e m} \end {gather*}
Antiderivative was successfully verified.
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Maple [F]
time = 0.20, 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 = 0.49, size = 62, normalized size = 1.82 \begin {gather*} \frac {a^{m} 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 ) - m - 1\right )}}{d m} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.37, size = 40, normalized size = 1.18 \begin {gather*} \frac {\left (\cos \left (d x + c\right ) e\right )^{-m - 1} {\left (a \sin \left (d x + c\right ) + a\right )}^{m} \cos \left (d x + c\right )}{d m} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \left (a \left (\sin {\left (c + d x \right )} + 1\right )\right )^{m} \left (e \cos {\left (c + d x \right )}\right )^{- m - 1}\, dx \end {gather*}
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 \begin {gather*} \text {could not integrate} \end {gather*}
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 \begin {gather*} \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} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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