Optimal. Leaf size=142 \[ \frac {e (e \cos (c+d x))^{-m} (a+b \sin (c+d x))^{m+1} \left (1-\frac {a+b \sin (c+d x)}{a-b}\right )^{m/2} \left (1-\frac {a+b \sin (c+d x)}{a+b}\right )^{m/2} F_1\left (m+1;\frac {m}{2},\frac {m}{2};m+2;\frac {a+b \sin (c+d x)}{a-b},\frac {a+b \sin (c+d x)}{a+b}\right )}{b d (m+1)} \]
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Rubi [A] time = 0.10, antiderivative size = 142, 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 = {2704, 138} \[ \frac {e (e \cos (c+d x))^{-m} (a+b \sin (c+d x))^{m+1} \left (1-\frac {a+b \sin (c+d x)}{a-b}\right )^{m/2} \left (1-\frac {a+b \sin (c+d x)}{a+b}\right )^{m/2} F_1\left (m+1;\frac {m}{2},\frac {m}{2};m+2;\frac {a+b \sin (c+d x)}{a-b},\frac {a+b \sin (c+d x)}{a+b}\right )}{b d (m+1)} \]
Antiderivative was successfully verified.
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Rule 138
Rule 2704
Rubi steps
\begin {align*} \int (e \cos (c+d x))^{1-m} (a+b \sin (c+d x))^m \, dx &=\frac {\left (e (e \cos (c+d x))^{-m} \left (1-\frac {a+b \sin (c+d x)}{a-b}\right )^{m/2} \left (1-\frac {a+b \sin (c+d x)}{a+b}\right )^{m/2}\right ) \operatorname {Subst}\left (\int (a+b x)^m \left (-\frac {b}{a-b}-\frac {b x}{a-b}\right )^{-m/2} \left (\frac {b}{a+b}-\frac {b x}{a+b}\right )^{-m/2} \, dx,x,\sin (c+d x)\right )}{d}\\ &=\frac {e F_1\left (1+m;\frac {m}{2},\frac {m}{2};2+m;\frac {a+b \sin (c+d x)}{a-b},\frac {a+b \sin (c+d x)}{a+b}\right ) (e \cos (c+d x))^{-m} (a+b \sin (c+d x))^{1+m} \left (1-\frac {a+b \sin (c+d x)}{a-b}\right )^{m/2} \left (1-\frac {a+b \sin (c+d x)}{a+b}\right )^{m/2}}{b d (1+m)}\\ \end {align*}
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Mathematica [F] time = 5.63, size = 0, normalized size = 0.00 \[ \int (e \cos (c+d x))^{1-m} (a+b \sin (c+d x))^m \, dx \]
Verification is Not applicable to the result.
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fricas [F] time = 0.77, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\left (e \cos \left (d x + c\right )\right )^{-m + 1} {\left (b \sin \left (d x + c\right ) + a\right )}^{m}, x\right ) \]
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 \[ \int \left (e \cos \left (d x + c\right )\right )^{-m + 1} {\left (b \sin \left (d x + c\right ) + a\right )}^{m}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.33, size = 0, normalized size = 0.00 \[ \int \left (e \cos \left (d x +c \right )\right )^{1-m} \left (a +b \sin \left (d x +c \right )\right )^{m}\, dx \]
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 \[ \int \left (e \cos \left (d x + c\right )\right )^{-m + 1} {\left (b \sin \left (d x + c\right ) + a\right )}^{m}\,{d x} \]
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 \[ \int {\left (e\,\cos \left (c+d\,x\right )\right )}^{1-m}\,{\left (a+b\,\sin \left (c+d\,x\right )\right )}^m \,d x \]
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 \[ \int \left (e \cos {\left (c + d x \right )}\right )^{1 - m} \left (a + b \sin {\left (c + d x \right )}\right )^{m}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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