Optimal. Leaf size=42 \[ \frac{(a \sin (c+d x)+a)^{m+1} \, _2F_1(2,m+1;m+2;\sin (c+d x)+1)}{a d (m+1)} \]
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Rubi [A] time = 0.0579094, antiderivative size = 42, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 25, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.12, Rules used = {2833, 12, 65} \[ \frac{(a \sin (c+d x)+a)^{m+1} \, _2F_1(2,m+1;m+2;\sin (c+d x)+1)}{a d (m+1)} \]
Antiderivative was successfully verified.
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Rule 2833
Rule 12
Rule 65
Rubi steps
\begin{align*} \int \cot (c+d x) \csc (c+d x) (a+a \sin (c+d x))^m \, dx &=\frac{\operatorname{Subst}\left (\int \frac{a^2 (a+x)^m}{x^2} \, dx,x,a \sin (c+d x)\right )}{a d}\\ &=\frac{a \operatorname{Subst}\left (\int \frac{(a+x)^m}{x^2} \, dx,x,a \sin (c+d x)\right )}{d}\\ &=\frac{\, _2F_1(2,1+m;2+m;1+\sin (c+d x)) (a+a \sin (c+d x))^{1+m}}{a d (1+m)}\\ \end{align*}
Mathematica [A] time = 0.0553152, size = 42, normalized size = 1. \[ \frac{(a \sin (c+d x)+a)^{m+1} \, _2F_1(2,m+1;m+2;\sin (c+d x)+1)}{a d (m+1)} \]
Antiderivative was successfully verified.
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Maple [F] time = 0.668, size = 0, normalized size = 0. \begin{align*} \int \cos \left ( dx+c \right ) \left ( \csc \left ( dx+c \right ) \right ) ^{2} \left ( a+a\sin \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{\left (a \sin \left (d x + c\right ) + a\right )}^{m} \cos \left (d x + c\right ) \csc \left (d x + c\right )^{2}\,{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 ({\left (a \sin \left (d x + c\right ) + a\right )}^{m} \cos \left (d x + c\right ) \csc \left (d x + c\right )^{2}, x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F(-1)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \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{\left (a \sin \left (d x + c\right ) + a\right )}^{m} \cos \left (d x + c\right ) \csc \left (d x + c\right )^{2}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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