Optimal. Leaf size=121 \[ -\frac{2 \sqrt{2} (1-\sin (e+f x))^2 \cos (e+f x) (c+d \sin (e+f x))^n \left (\frac{c+d \sin (e+f x)}{c+d}\right )^{-n} F_1\left (\frac{5}{2};-\frac{1}{2},-n;\frac{7}{2};\frac{1}{2} (1-\sin (e+f x)),\frac{d (1-\sin (e+f x))}{c+d}\right )}{5 a f \sqrt{\sin (e+f x)+1}} \]
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Rubi [A] time = 0.181231, antiderivative size = 121, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 33, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.091, Rules used = {2917, 139, 138} \[ -\frac{2 \sqrt{2} (1-\sin (e+f x))^2 \cos (e+f x) (c+d \sin (e+f x))^n \left (\frac{c+d \sin (e+f x)}{c+d}\right )^{-n} F_1\left (\frac{5}{2};-\frac{1}{2},-n;\frac{7}{2};\frac{1}{2} (1-\sin (e+f x)),\frac{d (1-\sin (e+f x))}{c+d}\right )}{5 a f \sqrt{\sin (e+f x)+1}} \]
Antiderivative was successfully verified.
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Rule 2917
Rule 139
Rule 138
Rubi steps
\begin{align*} \int \frac{\cos ^4(e+f x) (c+d \sin (e+f x))^n}{a+a \sin (e+f x)} \, dx &=\frac{\cos (e+f x) \operatorname{Subst}\left (\int (1-x)^{3/2} \sqrt{1+x} (c+d x)^n \, dx,x,\sin (e+f x)\right )}{a f \sqrt{1-\sin (e+f x)} \sqrt{1+\sin (e+f x)}}\\ &=\frac{\left (\cos (e+f x) (c+d \sin (e+f x))^n \left (-\frac{c+d \sin (e+f x)}{-c-d}\right )^{-n}\right ) \operatorname{Subst}\left (\int (1-x)^{3/2} \sqrt{1+x} \left (-\frac{c}{-c-d}-\frac{d x}{-c-d}\right )^n \, dx,x,\sin (e+f x)\right )}{a f \sqrt{1-\sin (e+f x)} \sqrt{1+\sin (e+f x)}}\\ &=-\frac{2 \sqrt{2} F_1\left (\frac{5}{2};-\frac{1}{2},-n;\frac{7}{2};\frac{1}{2} (1-\sin (e+f x)),\frac{d (1-\sin (e+f x))}{c+d}\right ) \cos (e+f x) (1-\sin (e+f x))^2 (c+d \sin (e+f x))^n \left (\frac{c+d \sin (e+f x)}{c+d}\right )^{-n}}{5 a f \sqrt{1+\sin (e+f x)}}\\ \end{align*}
Mathematica [F] time = 29.2282, size = 0, normalized size = 0. \[ \int \frac{\cos ^4(e+f x) (c+d \sin (e+f x))^n}{a+a \sin (e+f x)} \, dx \]
Verification is Not applicable to the result.
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Maple [F] time = 0.403, size = 0, normalized size = 0. \begin{align*} \int{\frac{ \left ( \cos \left ( fx+e \right ) \right ) ^{4} \left ( c+d\sin \left ( fx+e \right ) \right ) ^{n}}{a+a\sin \left ( fx+e \right ) }}\, 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 (d \sin \left (f x + e\right ) + c\right )}^{n} \cos \left (f x + e\right )^{4}}{a \sin \left (f x + e\right ) + a}\,{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 (d \sin \left (f x + e\right ) + c\right )}^{n} \cos \left (f x + e\right )^{4}}{a \sin \left (f x + e\right ) + a}, 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 \frac{{\left (d \sin \left (f x + e\right ) + c\right )}^{n} \cos \left (f x + e\right )^{4}}{a \sin \left (f x + e\right ) + a}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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