Optimal. Leaf size=69 \[ \frac{\sin (e+f x) \left (a+b \sin ^n(e+f x)\right )^p \left (\frac{b \sin ^n(e+f x)}{a}+1\right )^{-p} \, _2F_1\left (\frac{1}{n},-p;1+\frac{1}{n};-\frac{b \sin ^n(e+f x)}{a}\right )}{f} \]
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Rubi [A] time = 0.0493514, antiderivative size = 69, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.143, Rules used = {3223, 246, 245} \[ \frac{\sin (e+f x) \left (a+b \sin ^n(e+f x)\right )^p \left (\frac{b \sin ^n(e+f x)}{a}+1\right )^{-p} \, _2F_1\left (\frac{1}{n},-p;1+\frac{1}{n};-\frac{b \sin ^n(e+f x)}{a}\right )}{f} \]
Antiderivative was successfully verified.
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Rule 3223
Rule 246
Rule 245
Rubi steps
\begin{align*} \int \cos (e+f x) \left (a+b \sin ^n(e+f x)\right )^p \, dx &=\frac{\operatorname{Subst}\left (\int \left (a+b x^n\right )^p \, dx,x,\sin (e+f x)\right )}{f}\\ &=\frac{\left (\left (a+b \sin ^n(e+f x)\right )^p \left (1+\frac{b \sin ^n(e+f x)}{a}\right )^{-p}\right ) \operatorname{Subst}\left (\int \left (1+\frac{b x^n}{a}\right )^p \, dx,x,\sin (e+f x)\right )}{f}\\ &=\frac{\, _2F_1\left (\frac{1}{n},-p;1+\frac{1}{n};-\frac{b \sin ^n(e+f x)}{a}\right ) \sin (e+f x) \left (a+b \sin ^n(e+f x)\right )^p \left (1+\frac{b \sin ^n(e+f x)}{a}\right )^{-p}}{f}\\ \end{align*}
Mathematica [A] time = 0.0218818, size = 69, normalized size = 1. \[ \frac{\sin (e+f x) \left (a+b \sin ^n(e+f x)\right )^p \left (\frac{b \sin ^n(e+f x)}{a}+1\right )^{-p} \, _2F_1\left (\frac{1}{n},-p;1+\frac{1}{n};-\frac{b \sin ^n(e+f x)}{a}\right )}{f} \]
Antiderivative was successfully verified.
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Maple [F] time = 0.7, size = 0, normalized size = 0. \begin{align*} \int \cos \left ( fx+e \right ) \left ( a+b \left ( \sin \left ( fx+e \right ) \right ) ^{n} \right ) ^{p}\, 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 (b \sin \left (f x + e\right )^{n} + a\right )}^{p} \cos \left (f x + e\right )\,{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 (b \sin \left (f x + e\right )^{n} + a\right )}^{p} \cos \left (f x + e\right ), 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 (b \sin \left (f x + e\right )^{n} + a\right )}^{p} \cos \left (f x + e\right )\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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