Optimal. Leaf size=127 \[ \frac{\cos (e+f x) (c+d \sin (e+f x))^{n+1} F_1\left (n+1;-\frac{1}{2},-\frac{1}{2};n+2;\frac{c+d \sin (e+f x)}{c-d},\frac{c+d \sin (e+f x)}{c+d}\right )}{d f (n+1) \sqrt{1-\frac{c+d \sin (e+f x)}{c-d}} \sqrt{1-\frac{c+d \sin (e+f x)}{c+d}}} \]
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Rubi [A] time = 0.0930213, antiderivative size = 127, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.095, Rules used = {2704, 138} \[ \frac{\cos (e+f x) (c+d \sin (e+f x))^{n+1} F_1\left (n+1;-\frac{1}{2},-\frac{1}{2};n+2;\frac{c+d \sin (e+f x)}{c-d},\frac{c+d \sin (e+f x)}{c+d}\right )}{d f (n+1) \sqrt{1-\frac{c+d \sin (e+f x)}{c-d}} \sqrt{1-\frac{c+d \sin (e+f x)}{c+d}}} \]
Antiderivative was successfully verified.
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Rule 2704
Rule 138
Rubi steps
\begin{align*} \int \cos ^2(e+f x) (c+d \sin (e+f x))^n \, dx &=\frac{\cos (e+f x) \operatorname{Subst}\left (\int (c+d x)^n \sqrt{-\frac{d}{c-d}-\frac{d x}{c-d}} \sqrt{\frac{d}{c+d}-\frac{d x}{c+d}} \, dx,x,\sin (e+f x)\right )}{f \sqrt{1-\frac{c+d \sin (e+f x)}{c-d}} \sqrt{1-\frac{c+d \sin (e+f x)}{c+d}}}\\ &=\frac{F_1\left (1+n;-\frac{1}{2},-\frac{1}{2};2+n;\frac{c+d \sin (e+f x)}{c-d},\frac{c+d \sin (e+f x)}{c+d}\right ) \cos (e+f x) (c+d \sin (e+f x))^{1+n}}{d f (1+n) \sqrt{1-\frac{c+d \sin (e+f x)}{c-d}} \sqrt{1-\frac{c+d \sin (e+f x)}{c+d}}}\\ \end{align*}
Mathematica [F] time = 0.310351, size = 0, normalized size = 0. \[ \int \cos ^2(e+f x) (c+d \sin (e+f x))^n \, dx \]
Verification is Not applicable to the result.
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Maple [F] time = 0.234, size = 0, normalized size = 0. \begin{align*} \int \left ( \cos \left ( fx+e \right ) \right ) ^{2} \left ( c+d\sin \left ( fx+e \right ) \right ) ^{n}\, 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 (d \sin \left (f x + e\right ) + c\right )}^{n} \cos \left (f x + e\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 (d \sin \left (f x + e\right ) + c\right )}^{n} \cos \left (f x + e\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 (d \sin \left (f x + e\right ) + c\right )}^{n} \cos \left (f x + e\right )^{2}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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