Optimal. Leaf size=95 \[ -\frac{a^3 2^{\frac{p}{2}+\frac{7}{2}} (\sin (c+d x)+1)^{\frac{1}{2} (-p-1)} (e \cos (c+d x))^{p+1} \, _2F_1\left (\frac{1}{2} (-p-5),\frac{p+1}{2};\frac{p+3}{2};\frac{1}{2} (1-\sin (c+d x))\right )}{d e (p+1)} \]
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Rubi [A] time = 0.0801343, antiderivative size = 95, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 23, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.087, Rules used = {2688, 69} \[ -\frac{a^3 2^{\frac{p}{2}+\frac{7}{2}} (\sin (c+d x)+1)^{\frac{1}{2} (-p-1)} (e \cos (c+d x))^{p+1} \, _2F_1\left (\frac{1}{2} (-p-5),\frac{p+1}{2};\frac{p+3}{2};\frac{1}{2} (1-\sin (c+d x))\right )}{d e (p+1)} \]
Antiderivative was successfully verified.
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Rule 2688
Rule 69
Rubi steps
\begin{align*} \int (e \cos (c+d x))^p (a+a \sin (c+d x))^3 \, dx &=\frac{\left (a^3 (e \cos (c+d x))^{1+p} (1-\sin (c+d x))^{\frac{1}{2} (-1-p)} (1+\sin (c+d x))^{\frac{1}{2} (-1-p)}\right ) \operatorname{Subst}\left (\int (1-x)^{\frac{1}{2} (-1+p)} (1+x)^{3+\frac{1}{2} (-1+p)} \, dx,x,\sin (c+d x)\right )}{d e}\\ &=-\frac{2^{\frac{7}{2}+\frac{p}{2}} a^3 (e \cos (c+d x))^{1+p} \, _2F_1\left (\frac{1}{2} (-5-p),\frac{1+p}{2};\frac{3+p}{2};\frac{1}{2} (1-\sin (c+d x))\right ) (1+\sin (c+d x))^{\frac{1}{2} (-1-p)}}{d e (1+p)}\\ \end{align*}
Mathematica [A] time = 0.105313, size = 94, normalized size = 0.99 \[ -\frac{a^3 2^{\frac{p+7}{2}} \cos (c+d x) (\sin (c+d x)+1)^{\frac{1}{2} (-p-1)} (e \cos (c+d x))^p \, _2F_1\left (\frac{1}{2} (-p-5),\frac{p+1}{2};\frac{p+3}{2};\frac{1}{2} (1-\sin (c+d x))\right )}{d (p+1)} \]
Antiderivative was successfully verified.
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Maple [F] time = 3.378, size = 0, normalized size = 0. \begin{align*} \int \left ( e\cos \left ( dx+c \right ) \right ) ^{p} \left ( a+a\sin \left ( dx+c \right ) \right ) ^{3}\, 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 )}^{3} \left (e \cos \left (d x + c\right )\right )^{p}\,{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 (3 \, a^{3} \cos \left (d x + c\right )^{2} - 4 \, a^{3} +{\left (a^{3} \cos \left (d x + c\right )^{2} - 4 \, a^{3}\right )} \sin \left (d x + c\right )\right )} \left (e \cos \left (d x + c\right )\right )^{p}, 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 )}^{3} \left (e \cos \left (d x + c\right )\right )^{p}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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