Optimal. Leaf size=89 \[ \frac {(a \sin (c+d x)+a)^{m+1} (e \cos (c+d x))^{-m-1}}{a d e \left (1-m^2\right )}-\frac {(a \sin (c+d x)+a)^m (e \cos (c+d x))^{-m-1}}{d e (1-m)} \]
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Rubi [A] time = 0.12, antiderivative size = 89, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 27, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.074, Rules used = {2672, 2671} \[ \frac {(a \sin (c+d x)+a)^{m+1} (e \cos (c+d x))^{-m-1}}{a d e \left (1-m^2\right )}-\frac {(a \sin (c+d x)+a)^m (e \cos (c+d x))^{-m-1}}{d e (1-m)} \]
Antiderivative was successfully verified.
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Rule 2671
Rule 2672
Rubi steps
\begin {align*} \int (e \cos (c+d x))^{-2-m} (a+a \sin (c+d x))^m \, dx &=-\frac {(e \cos (c+d x))^{-1-m} (a+a \sin (c+d x))^m}{d e (1-m)}+\frac {\int (e \cos (c+d x))^{-2-m} (a+a \sin (c+d x))^{1+m} \, dx}{a (1-m)}\\ &=-\frac {(e \cos (c+d x))^{-1-m} (a+a \sin (c+d x))^m}{d e (1-m)}+\frac {(e \cos (c+d x))^{-1-m} (a+a \sin (c+d x))^{1+m}}{a d e \left (1-m^2\right )}\\ \end {align*}
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Mathematica [A] time = 0.12, size = 53, normalized size = 0.60 \[ \frac {(m-\sin (c+d x)) (a (\sin (c+d x)+1))^m (e \cos (c+d x))^{-m-1}}{d e (m-1) (m+1)} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.47, size = 61, normalized size = 0.69 \[ \frac {{\left (m \cos \left (d x + c\right ) - \cos \left (d x + c\right ) \sin \left (d x + c\right )\right )} \left (e \cos \left (d x + c\right )\right )^{-m - 2} {\left (a \sin \left (d x + c\right ) + a\right )}^{m}}{d m^{2} - d} \]
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 \[ \int \left (e \cos \left (d x + c\right )\right )^{-m - 2} {\left (a \sin \left (d x + c\right ) + a\right )}^{m}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.32, size = 0, normalized size = 0.00 \[ \int \left (e \cos \left (d x +c \right )\right )^{-2-m} \left (a +a \sin \left (d x +c \right )\right )^{m}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \left (e \cos \left (d x + c\right )\right )^{-m - 2} {\left (a \sin \left (d x + c\right ) + a\right )}^{m}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 5.60, size = 71, normalized size = 0.80 \[ -\frac {\left (\sin \left (2\,c+2\,d\,x\right )-2\,m\,\cos \left (c+d\,x\right )\right )\,{\left (a\,\left (\sin \left (c+d\,x\right )+1\right )\right )}^m}{d\,e^2\,\left (\cos \left (2\,c+2\,d\,x\right )+1\right )\,{\left (e\,\cos \left (c+d\,x\right )\right )}^m\,\left (m^2-1\right )} \]
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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