Optimal. Leaf size=148 \[ \frac{\sin (e+f x) \left (a+b \sin ^n(e+f x)\right )^p \left (\frac{b \sin ^n(e+f x)}{a}+1\right )^{-p} \, _2F_1\left (\frac{1}{n},-p;1+\frac{1}{n};-\frac{b \sin ^n(e+f x)}{a}\right )}{f}-\frac{\sin ^3(e+f x) \left (a+b \sin ^n(e+f x)\right )^p \left (\frac{b \sin ^n(e+f x)}{a}+1\right )^{-p} \, _2F_1\left (\frac{3}{n},-p;\frac{n+3}{n};-\frac{b \sin ^n(e+f x)}{a}\right )}{3 f} \]
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Rubi [A] time = 0.11801, antiderivative size = 148, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 6, integrand size = 23, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.261, Rules used = {3223, 1893, 246, 245, 365, 364} \[ \frac{\sin (e+f x) \left (a+b \sin ^n(e+f x)\right )^p \left (\frac{b \sin ^n(e+f x)}{a}+1\right )^{-p} \, _2F_1\left (\frac{1}{n},-p;1+\frac{1}{n};-\frac{b \sin ^n(e+f x)}{a}\right )}{f}-\frac{\sin ^3(e+f x) \left (a+b \sin ^n(e+f x)\right )^p \left (\frac{b \sin ^n(e+f x)}{a}+1\right )^{-p} \, _2F_1\left (\frac{3}{n},-p;\frac{n+3}{n};-\frac{b \sin ^n(e+f x)}{a}\right )}{3 f} \]
Antiderivative was successfully verified.
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Rule 3223
Rule 1893
Rule 246
Rule 245
Rule 365
Rule 364
Rubi steps
\begin{align*} \int \cos ^3(e+f x) \left (a+b \sin ^n(e+f x)\right )^p \, dx &=\frac{\operatorname{Subst}\left (\int \left (1-x^2\right ) \left (a+b x^n\right )^p \, dx,x,\sin (e+f x)\right )}{f}\\ &=\frac{\operatorname{Subst}\left (\int \left (\left (a+b x^n\right )^p-x^2 \left (a+b x^n\right )^p\right ) \, dx,x,\sin (e+f x)\right )}{f}\\ &=\frac{\operatorname{Subst}\left (\int \left (a+b x^n\right )^p \, dx,x,\sin (e+f x)\right )}{f}-\frac{\operatorname{Subst}\left (\int x^2 \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{\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 x^2 \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}-\frac{\, _2F_1\left (\frac{3}{n},-p;\frac{3+n}{n};-\frac{b \sin ^n(e+f x)}{a}\right ) \sin ^3(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}}{3 f}\\ \end{align*}
Mathematica [A] time = 0.0856171, size = 114, normalized size = 0.77 \[ -\frac{\sin (e+f x) \left (a+b \sin ^n(e+f x)\right )^p \left (\frac{b \sin ^n(e+f x)}{a}+1\right )^{-p} \left (\sin ^2(e+f x) \, _2F_1\left (\frac{3}{n},-p;\frac{n+3}{n};-\frac{b \sin ^n(e+f x)}{a}\right )-3 \, _2F_1\left (\frac{1}{n},-p;1+\frac{1}{n};-\frac{b \sin ^n(e+f x)}{a}\right )\right )}{3 f} \]
Antiderivative was successfully verified.
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Maple [F] time = 0.757, size = 0, normalized size = 0. \begin{align*} \int \left ( \cos \left ( fx+e \right ) \right ) ^{3} \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 )^{3}\,{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 )^{3}, 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 )^{3}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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