Optimal. Leaf size=97 \[ \frac {2^{1-\frac {m}{2}} (1-\sin (c+d x))^{\frac {m}{2}-1} (a \sin (c+d x)+a)^m (e \cos (c+d x))^{2-m} \, _2F_1\left (\frac {m}{2},\frac {m+2}{2};\frac {m+4}{2};\frac {1}{2} (\sin (c+d x)+1)\right )}{d e (m+2)} \]
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Rubi [A] time = 0.11, antiderivative size = 97, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, integrand size = 27, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.111, Rules used = {2689, 70, 69} \[ \frac {2^{1-\frac {m}{2}} (1-\sin (c+d x))^{\frac {m}{2}-1} (a \sin (c+d x)+a)^m (e \cos (c+d x))^{2-m} \, _2F_1\left (\frac {m}{2},\frac {m+2}{2};\frac {m+4}{2};\frac {1}{2} (\sin (c+d x)+1)\right )}{d e (m+2)} \]
Antiderivative was successfully verified.
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Rule 69
Rule 70
Rule 2689
Rubi steps
\begin {align*} \int (e \cos (c+d x))^{1-m} (a+a \sin (c+d x))^m \, dx &=\frac {\left (a^2 (e \cos (c+d x))^{2-m} (a-a \sin (c+d x))^{\frac {1}{2} (-2+m)} (a+a \sin (c+d x))^{\frac {1}{2} (-2+m)}\right ) \operatorname {Subst}\left (\int (a-a x)^{-m/2} (a+a x)^{m/2} \, dx,x,\sin (c+d x)\right )}{d e}\\ &=\frac {\left (2^{-m/2} a^2 (e \cos (c+d x))^{2-m} (a-a \sin (c+d x))^{\frac {1}{2} (-2+m)-\frac {m}{2}} \left (\frac {a-a \sin (c+d x)}{a}\right )^{m/2} (a+a \sin (c+d x))^{\frac {1}{2} (-2+m)}\right ) \operatorname {Subst}\left (\int \left (\frac {1}{2}-\frac {x}{2}\right )^{-m/2} (a+a x)^{m/2} \, dx,x,\sin (c+d x)\right )}{d e}\\ &=\frac {2^{1-\frac {m}{2}} (e \cos (c+d x))^{2-m} \, _2F_1\left (\frac {m}{2},\frac {2+m}{2};\frac {4+m}{2};\frac {1}{2} (1+\sin (c+d x))\right ) (1-\sin (c+d x))^{-1+\frac {m}{2}} (a+a \sin (c+d x))^m}{d e (2+m)}\\ \end {align*}
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Mathematica [A] time = 0.26, size = 97, normalized size = 1.00 \[ \frac {2^{\frac {m}{2}+1} (\sin (c+d x)+1)^{-\frac {m}{2}-1} (a (\sin (c+d x)+1))^m (e \cos (c+d x))^{2-m} \, _2F_1\left (1-\frac {m}{2},-\frac {m}{2};2-\frac {m}{2};\frac {1}{2} (1-\sin (c+d x))\right )}{d e (m-2)} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.48, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\left (e \cos \left (d x + c\right )\right )^{-m + 1} {\left (a \sin \left (d x + c\right ) + a\right )}^{m}, x\right ) \]
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 = 0.31, 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 [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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mupad [F] time = 0.00, size = -1, normalized size = -0.01 \[ \int {\left (e\,\cos \left (c+d\,x\right )\right )}^{1-m}\,{\left (a+a\,\sin \left (c+d\,x\right )\right )}^m \,d x \]
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 \[ \int \left (a \left (\sin {\left (c + d x \right )} + 1\right )\right )^{m} \left (e \cos {\left (c + d x \right )}\right )^{1 - m}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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