Integrand size = 23, antiderivative size = 95 \[ \int (e \cos (c+d x))^p (a+a \sin (c+d x))^3 \, dx=-\frac {2^{\frac {7}{2}+\frac {p}{2}} a^3 (e \cos (c+d x))^{1+p} \operatorname {Hypergeometric2F1}\left (\frac {1}{2} (-5-p),\frac {1+p}{2},\frac {3+p}{2},\frac {1}{2} (1-\sin (c+d x))\right ) (1+\sin (c+d x))^{\frac {1}{2} (-1-p)}}{d e (1+p)} \]
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Time = 0.06 (sec) , antiderivative size = 95, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.087, Rules used = {2767, 71} \[ \int (e \cos (c+d x))^p (a+a \sin (c+d x))^3 \, dx=-\frac {a^3 2^{\frac {p}{2}+\frac {7}{2}} (\sin (c+d x)+1)^{\frac {1}{2} (-p-1)} (e \cos (c+d x))^{p+1} \operatorname {Hypergeometric2F1}\left (\frac {1}{2} (-p-5),\frac {p+1}{2},\frac {p+3}{2},\frac {1}{2} (1-\sin (c+d x))\right )}{d e (p+1)} \]
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Rule 71
Rule 2767
Rubi steps \begin{align*} \text {integral}& = \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 ) \text {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} \operatorname {Hypergeometric2F1}\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*}
Time = 0.07 (sec) , antiderivative size = 94, normalized size of antiderivative = 0.99 \[ \int (e \cos (c+d x))^p (a+a \sin (c+d x))^3 \, dx=-\frac {2^{\frac {7+p}{2}} a^3 \cos (c+d x) (e \cos (c+d x))^p \operatorname {Hypergeometric2F1}\left (\frac {1}{2} (-5-p),\frac {1+p}{2},\frac {3+p}{2},\frac {1}{2} (1-\sin (c+d x))\right ) (1+\sin (c+d x))^{\frac {1}{2} (-1-p)}}{d (1+p)} \]
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\[\int \left (e \cos \left (d x +c \right )\right )^{p} \left (a +a \sin \left (d x +c \right )\right )^{3}d x\]
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\[ \int (e \cos (c+d x))^p (a+a \sin (c+d x))^3 \, dx=\int { {\left (a \sin \left (d x + c\right ) + a\right )}^{3} \left (e \cos \left (d x + c\right )\right )^{p} \,d x } \]
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\[ \int (e \cos (c+d x))^p (a+a \sin (c+d x))^3 \, dx=a^{3} \left (\int \left (e \cos {\left (c + d x \right )}\right )^{p}\, dx + \int 3 \left (e \cos {\left (c + d x \right )}\right )^{p} \sin {\left (c + d x \right )}\, dx + \int 3 \left (e \cos {\left (c + d x \right )}\right )^{p} \sin ^{2}{\left (c + d x \right )}\, dx + \int \left (e \cos {\left (c + d x \right )}\right )^{p} \sin ^{3}{\left (c + d x \right )}\, dx\right ) \]
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\[ \int (e \cos (c+d x))^p (a+a \sin (c+d x))^3 \, dx=\int { {\left (a \sin \left (d x + c\right ) + a\right )}^{3} \left (e \cos \left (d x + c\right )\right )^{p} \,d x } \]
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\[ \int (e \cos (c+d x))^p (a+a \sin (c+d x))^3 \, dx=\int { {\left (a \sin \left (d x + c\right ) + a\right )}^{3} \left (e \cos \left (d x + c\right )\right )^{p} \,d x } \]
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Timed out. \[ \int (e \cos (c+d x))^p (a+a \sin (c+d x))^3 \, dx=\int {\left (e\,\cos \left (c+d\,x\right )\right )}^p\,{\left (a+a\,\sin \left (c+d\,x\right )\right )}^3 \,d x \]
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