Integrand size = 28, antiderivative size = 449 \[ \int \frac {(e+f x)^m \cos ^4(c+d x)}{a+a \sin (c+d x)} \, dx=\frac {(e+f x)^{1+m}}{2 a f (1+m)}+\frac {e^{i \left (c-\frac {d e}{f}\right )} (e+f x)^m \left (-\frac {i d (e+f x)}{f}\right )^{-m} \Gamma \left (1+m,-\frac {i d (e+f x)}{f}\right )}{8 a d}+\frac {e^{-i \left (c-\frac {d e}{f}\right )} (e+f x)^m \left (\frac {i d (e+f x)}{f}\right )^{-m} \Gamma \left (1+m,\frac {i d (e+f x)}{f}\right )}{8 a d}-\frac {i 2^{-3-m} e^{2 i \left (c-\frac {d e}{f}\right )} (e+f x)^m \left (-\frac {i d (e+f x)}{f}\right )^{-m} \Gamma \left (1+m,-\frac {2 i d (e+f x)}{f}\right )}{a d}+\frac {i 2^{-3-m} e^{-2 i \left (c-\frac {d e}{f}\right )} (e+f x)^m \left (\frac {i d (e+f x)}{f}\right )^{-m} \Gamma \left (1+m,\frac {2 i d (e+f x)}{f}\right )}{a d}+\frac {3^{-1-m} e^{3 i \left (c-\frac {d e}{f}\right )} (e+f x)^m \left (-\frac {i d (e+f x)}{f}\right )^{-m} \Gamma \left (1+m,-\frac {3 i d (e+f x)}{f}\right )}{8 a d}+\frac {3^{-1-m} e^{-3 i \left (c-\frac {d e}{f}\right )} (e+f x)^m \left (\frac {i d (e+f x)}{f}\right )^{-m} \Gamma \left (1+m,\frac {3 i d (e+f x)}{f}\right )}{8 a d} \]
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Time = 0.42 (sec) , antiderivative size = 449, normalized size of antiderivative = 1.00, number of steps used = 14, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.214, Rules used = {4619, 3393, 3388, 2212, 4491, 3389} \[ \int \frac {(e+f x)^m \cos ^4(c+d x)}{a+a \sin (c+d x)} \, dx=\frac {e^{i \left (c-\frac {d e}{f}\right )} (e+f x)^m \left (-\frac {i d (e+f x)}{f}\right )^{-m} \Gamma \left (m+1,-\frac {i d (e+f x)}{f}\right )}{8 a d}-\frac {i 2^{-m-3} e^{2 i \left (c-\frac {d e}{f}\right )} (e+f x)^m \left (-\frac {i d (e+f x)}{f}\right )^{-m} \Gamma \left (m+1,-\frac {2 i d (e+f x)}{f}\right )}{a d}+\frac {3^{-m-1} e^{3 i \left (c-\frac {d e}{f}\right )} (e+f x)^m \left (-\frac {i d (e+f x)}{f}\right )^{-m} \Gamma \left (m+1,-\frac {3 i d (e+f x)}{f}\right )}{8 a d}+\frac {e^{-i \left (c-\frac {d e}{f}\right )} (e+f x)^m \left (\frac {i d (e+f x)}{f}\right )^{-m} \Gamma \left (m+1,\frac {i d (e+f x)}{f}\right )}{8 a d}+\frac {i 2^{-m-3} e^{-2 i \left (c-\frac {d e}{f}\right )} (e+f x)^m \left (\frac {i d (e+f x)}{f}\right )^{-m} \Gamma \left (m+1,\frac {2 i d (e+f x)}{f}\right )}{a d}+\frac {3^{-m-1} e^{-3 i \left (c-\frac {d e}{f}\right )} (e+f x)^m \left (\frac {i d (e+f x)}{f}\right )^{-m} \Gamma \left (m+1,\frac {3 i d (e+f x)}{f}\right )}{8 a d}+\frac {(e+f x)^{m+1}}{2 a f (m+1)} \]
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Rule 2212
Rule 3388
Rule 3389
Rule 3393
Rule 4491
Rule 4619
Rubi steps \begin{align*} \text {integral}& = \frac {\int (e+f x)^m \cos ^2(c+d x) \, dx}{a}-\frac {\int (e+f x)^m \cos ^2(c+d x) \sin (c+d x) \, dx}{a} \\ & = \frac {\int \left (\frac {1}{2} (e+f x)^m+\frac {1}{2} (e+f x)^m \cos (2 c+2 d x)\right ) \, dx}{a}-\frac {\int \left (\frac {1}{4} (e+f x)^m \sin (c+d x)+\frac {1}{4} (e+f x)^m \sin (3 c+3 d x)\right ) \, dx}{a} \\ & = \frac {(e+f x)^{1+m}}{2 a f (1+m)}-\frac {\int (e+f x)^m \sin (c+d x) \, dx}{4 a}-\frac {\int (e+f x)^m \sin (3 c+3 d x) \, dx}{4 a}+\frac {\int (e+f x)^m \cos (2 c+2 d x) \, dx}{2 a} \\ & = \frac {(e+f x)^{1+m}}{2 a f (1+m)}-\frac {i \int e^{-i (c+d x)} (e+f x)^m \, dx}{8 a}+\frac {i \int e^{i (c+d x)} (e+f x)^m \, dx}{8 a}-\frac {i \int e^{-i (3 c+3 d x)} (e+f x)^m \, dx}{8 a}+\frac {i \int e^{i (3 c+3 d x)} (e+f x)^m \, dx}{8 a}+\frac {\int e^{-i (2 c+2 d x)} (e+f x)^m \, dx}{4 a}+\frac {\int e^{i (2 c+2 d x)} (e+f x)^m \, dx}{4 a} \\ & = \frac {(e+f x)^{1+m}}{2 a f (1+m)}+\frac {e^{i \left (c-\frac {d e}{f}\right )} (e+f x)^m \left (-\frac {i d (e+f x)}{f}\right )^{-m} \Gamma \left (1+m,-\frac {i d (e+f x)}{f}\right )}{8 a d}+\frac {e^{-i \left (c-\frac {d e}{f}\right )} (e+f x)^m \left (\frac {i d (e+f x)}{f}\right )^{-m} \Gamma \left (1+m,\frac {i d (e+f x)}{f}\right )}{8 a d}-\frac {i 2^{-3-m} e^{2 i \left (c-\frac {d e}{f}\right )} (e+f x)^m \left (-\frac {i d (e+f x)}{f}\right )^{-m} \Gamma \left (1+m,-\frac {2 i d (e+f x)}{f}\right )}{a d}+\frac {i 2^{-3-m} e^{-2 i \left (c-\frac {d e}{f}\right )} (e+f x)^m \left (\frac {i d (e+f x)}{f}\right )^{-m} \Gamma \left (1+m,\frac {2 i d (e+f x)}{f}\right )}{a d}+\frac {3^{-1-m} e^{3 i \left (c-\frac {d e}{f}\right )} (e+f x)^m \left (-\frac {i d (e+f x)}{f}\right )^{-m} \Gamma \left (1+m,-\frac {3 i d (e+f x)}{f}\right )}{8 a d}+\frac {3^{-1-m} e^{-3 i \left (c-\frac {d e}{f}\right )} (e+f x)^m \left (\frac {i d (e+f x)}{f}\right )^{-m} \Gamma \left (1+m,\frac {3 i d (e+f x)}{f}\right )}{8 a d} \\ \end{align*}
Time = 9.37 (sec) , antiderivative size = 405, normalized size of antiderivative = 0.90 \[ \int \frac {(e+f x)^m \cos ^4(c+d x)}{a+a \sin (c+d x)} \, dx=\frac {i (e+f x)^m \left (-\frac {12 i d (e+f x)}{f (1+m)}-3 i e^{i \left (c-\frac {d e}{f}\right )} \left (-\frac {i d (e+f x)}{f}\right )^{-m} \Gamma \left (1+m,-\frac {i d (e+f x)}{f}\right )-3 i e^{-i \left (c-\frac {d e}{f}\right )} \left (\frac {i d (e+f x)}{f}\right )^{-m} \Gamma \left (1+m,\frac {i d (e+f x)}{f}\right )-3\ 2^{-m} e^{2 i \left (c-\frac {d e}{f}\right )} \left (-\frac {i d (e+f x)}{f}\right )^{-m} \Gamma \left (1+m,-\frac {2 i d (e+f x)}{f}\right )+3\ 2^{-m} e^{-2 i \left (c-\frac {d e}{f}\right )} \left (\frac {i d (e+f x)}{f}\right )^{-m} \Gamma \left (1+m,\frac {2 i d (e+f x)}{f}\right )-i 3^{-m} e^{3 i \left (c-\frac {d e}{f}\right )} \left (-\frac {i d (e+f x)}{f}\right )^{-m} \Gamma \left (1+m,-\frac {3 i d (e+f x)}{f}\right )-i 3^{-m} e^{-3 i \left (c-\frac {d e}{f}\right )} \left (\frac {i d (e+f x)}{f}\right )^{-m} \Gamma \left (1+m,\frac {3 i d (e+f x)}{f}\right )\right ) \left (\cos \left (\frac {1}{2} (c+d x)\right )+\sin \left (\frac {1}{2} (c+d x)\right )\right )^2}{24 a d (1+\sin (c+d x))} \]
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\[\int \frac {\left (f x +e \right )^{m} \left (\cos ^{4}\left (d x +c \right )\right )}{a +a \sin \left (d x +c \right )}d x\]
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Time = 0.11 (sec) , antiderivative size = 340, normalized size of antiderivative = 0.76 \[ \int \frac {(e+f x)^m \cos ^4(c+d x)}{a+a \sin (c+d x)} \, dx=\frac {3 \, {\left (f m + f\right )} e^{\left (-\frac {f m \log \left (\frac {i \, d}{f}\right ) - i \, d e + i \, c f}{f}\right )} \Gamma \left (m + 1, \frac {i \, d f x + i \, d e}{f}\right ) - 3 \, {\left (i \, f m + i \, f\right )} e^{\left (-\frac {f m \log \left (-\frac {2 i \, d}{f}\right ) + 2 i \, d e - 2 i \, c f}{f}\right )} \Gamma \left (m + 1, -\frac {2 \, {\left (i \, d f x + i \, d e\right )}}{f}\right ) + {\left (f m + f\right )} e^{\left (-\frac {f m \log \left (-\frac {3 i \, d}{f}\right ) + 3 i \, d e - 3 i \, c f}{f}\right )} \Gamma \left (m + 1, -\frac {3 \, {\left (i \, d f x + i \, d e\right )}}{f}\right ) + 3 \, {\left (f m + f\right )} e^{\left (-\frac {f m \log \left (-\frac {i \, d}{f}\right ) + i \, d e - i \, c f}{f}\right )} \Gamma \left (m + 1, \frac {-i \, d f x - i \, d e}{f}\right ) - 3 \, {\left (-i \, f m - i \, f\right )} e^{\left (-\frac {f m \log \left (\frac {2 i \, d}{f}\right ) - 2 i \, d e + 2 i \, c f}{f}\right )} \Gamma \left (m + 1, -\frac {2 \, {\left (-i \, d f x - i \, d e\right )}}{f}\right ) + {\left (f m + f\right )} e^{\left (-\frac {f m \log \left (\frac {3 i \, d}{f}\right ) - 3 i \, d e + 3 i \, c f}{f}\right )} \Gamma \left (m + 1, -\frac {3 \, {\left (-i \, d f x - i \, d e\right )}}{f}\right ) + 12 \, {\left (d f x + d e\right )} {\left (f x + e\right )}^{m}}{24 \, {\left (a d f m + a d f\right )}} \]
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\[ \int \frac {(e+f x)^m \cos ^4(c+d x)}{a+a \sin (c+d x)} \, dx=\frac {\int \frac {\left (e + f x\right )^{m} \cos ^{4}{\left (c + d x \right )}}{\sin {\left (c + d x \right )} + 1}\, dx}{a} \]
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\[ \int \frac {(e+f x)^m \cos ^4(c+d x)}{a+a \sin (c+d x)} \, dx=\int { \frac {{\left (f x + e\right )}^{m} \cos \left (d x + c\right )^{4}}{a \sin \left (d x + c\right ) + a} \,d x } \]
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\[ \int \frac {(e+f x)^m \cos ^4(c+d x)}{a+a \sin (c+d x)} \, dx=\int { \frac {{\left (f x + e\right )}^{m} \cos \left (d x + c\right )^{4}}{a \sin \left (d x + c\right ) + a} \,d x } \]
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Timed out. \[ \int \frac {(e+f x)^m \cos ^4(c+d x)}{a+a \sin (c+d x)} \, dx=\int \frac {{\cos \left (c+d\,x\right )}^4\,{\left (e+f\,x\right )}^m}{a+a\,\sin \left (c+d\,x\right )} \,d x \]
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