Optimal. Leaf size=152 \[ \frac{e (e \cos (c+d x))^{-m-1} (a+b \sin (c+d x))^{m+1} \left (1-\frac{a+b \sin (c+d x)}{a-b}\right )^{\frac{m+1}{2}} \left (1-\frac{a+b \sin (c+d x)}{a+b}\right )^{\frac{m+1}{2}} F_1\left (m+1;\frac{m+1}{2},\frac{m+1}{2};m+2;\frac{a+b \sin (c+d x)}{a-b},\frac{a+b \sin (c+d x)}{a+b}\right )}{b d (m+1)} \]
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Rubi [A] time = 0.103209, antiderivative size = 152, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 25, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.08, Rules used = {2704, 138} \[ \frac{e (e \cos (c+d x))^{-m-1} (a+b \sin (c+d x))^{m+1} \left (1-\frac{a+b \sin (c+d x)}{a-b}\right )^{\frac{m+1}{2}} \left (1-\frac{a+b \sin (c+d x)}{a+b}\right )^{\frac{m+1}{2}} F_1\left (m+1;\frac{m+1}{2},\frac{m+1}{2};m+2;\frac{a+b \sin (c+d x)}{a-b},\frac{a+b \sin (c+d x)}{a+b}\right )}{b d (m+1)} \]
Antiderivative was successfully verified.
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Rule 2704
Rule 138
Rubi steps
\begin{align*} \int (e \cos (c+d x))^{-m} (a+b \sin (c+d x))^m \, dx &=\frac{\left (e (e \cos (c+d x))^{-1-m} \left (1-\frac{a+b \sin (c+d x)}{a-b}\right )^{\frac{1+m}{2}} \left (1-\frac{a+b \sin (c+d x)}{a+b}\right )^{\frac{1+m}{2}}\right ) \operatorname{Subst}\left (\int (a+b x)^m \left (-\frac{b}{a-b}-\frac{b x}{a-b}\right )^{\frac{1}{2} (-1-m)} \left (\frac{b}{a+b}-\frac{b x}{a+b}\right )^{\frac{1}{2} (-1-m)} \, dx,x,\sin (c+d x)\right )}{d}\\ &=\frac{e F_1\left (1+m;\frac{1+m}{2},\frac{1+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))^{-1-m} (a+b \sin (c+d x))^{1+m} \left (1-\frac{a+b \sin (c+d x)}{a-b}\right )^{\frac{1+m}{2}} \left (1-\frac{a+b \sin (c+d x)}{a+b}\right )^{\frac{1+m}{2}}}{b d (1+m)}\\ \end{align*}
Mathematica [F] time = 1.58231, size = 0, normalized size = 0. \[ \int (e \cos (c+d x))^{-m} (a+b \sin (c+d x))^m \, dx \]
Verification is Not applicable to the result.
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Maple [F] time = 0.389, size = 0, normalized size = 0. \begin{align*} \int{\frac{ \left ( a+b\sin \left ( dx+c \right ) \right ) ^{m}}{ \left ( e\cos \left ( dx+c \right ) \right ) ^{m}}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{{\left (b \sin \left (d x + c\right ) + a\right )}^{m}}{\left (e \cos \left (d x + c\right )\right )^{m}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [F] time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (\frac{{\left (b \sin \left (d x + c\right ) + a\right )}^{m}}{\left (e \cos \left (d x + c\right )\right )^{m}}, x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \left (e \cos{\left (c + d x \right )}\right )^{- m} \left (a + b \sin{\left (c + d x \right )}\right )^{m}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{{\left (b \sin \left (d x + c\right ) + a\right )}^{m}}{\left (e \cos \left (d x + c\right )\right )^{m}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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