Integrand size = 22, antiderivative size = 31 \[ \int (a \cos (c+d x)+i a \sin (c+d x))^4 \, dx=-\frac {i (a \cos (c+d x)+i a \sin (c+d x))^4}{4 d} \]
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Time = 0.02 (sec) , antiderivative size = 31, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.045, Rules used = {3150} \[ \int (a \cos (c+d x)+i a \sin (c+d x))^4 \, dx=-\frac {i (a \cos (c+d x)+i a \sin (c+d x))^4}{4 d} \]
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Rule 3150
Rubi steps \begin{align*} \text {integral}& = -\frac {i (a \cos (c+d x)+i a \sin (c+d x))^4}{4 d} \\ \end{align*}
Time = 0.52 (sec) , antiderivative size = 31, normalized size of antiderivative = 1.00 \[ \int (a \cos (c+d x)+i a \sin (c+d x))^4 \, dx=-\frac {i (a \cos (c+d x)+i a \sin (c+d x))^4}{4 d} \]
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Time = 1.95 (sec) , antiderivative size = 19, normalized size of antiderivative = 0.61
method | result | size |
risch | \(-\frac {i a^{4} {\mathrm e}^{4 i \left (d x +c \right )}}{4 d}\) | \(19\) |
parallelrisch | \(-\frac {a^{4} \left (i \cos \left (4 d x +4 c \right )-i-\sin \left (4 d x +4 c \right )\right )}{4 d}\) | \(35\) |
derivativedivides | \(\frac {a^{4} \left (-\frac {\left (\sin \left (d x +c \right )^{3}+\frac {3 \sin \left (d x +c \right )}{2}\right ) \cos \left (d x +c \right )}{4}+\frac {3 d x}{8}+\frac {3 c}{8}\right )-i a^{4} \sin \left (d x +c \right )^{4}-6 a^{4} \left (-\frac {\sin \left (d x +c \right ) \cos \left (d x +c \right )^{3}}{4}+\frac {\sin \left (d x +c \right ) \cos \left (d x +c \right )}{8}+\frac {d x}{8}+\frac {c}{8}\right )-i a^{4} \cos \left (d x +c \right )^{4}+a^{4} \left (\frac {\left (\cos \left (d x +c \right )^{3}+\frac {3 \cos \left (d x +c \right )}{2}\right ) \sin \left (d x +c \right )}{4}+\frac {3 d x}{8}+\frac {3 c}{8}\right )}{d}\) | \(151\) |
default | \(\frac {a^{4} \left (-\frac {\left (\sin \left (d x +c \right )^{3}+\frac {3 \sin \left (d x +c \right )}{2}\right ) \cos \left (d x +c \right )}{4}+\frac {3 d x}{8}+\frac {3 c}{8}\right )-i a^{4} \sin \left (d x +c \right )^{4}-6 a^{4} \left (-\frac {\sin \left (d x +c \right ) \cos \left (d x +c \right )^{3}}{4}+\frac {\sin \left (d x +c \right ) \cos \left (d x +c \right )}{8}+\frac {d x}{8}+\frac {c}{8}\right )-i a^{4} \cos \left (d x +c \right )^{4}+a^{4} \left (\frac {\left (\cos \left (d x +c \right )^{3}+\frac {3 \cos \left (d x +c \right )}{2}\right ) \sin \left (d x +c \right )}{4}+\frac {3 d x}{8}+\frac {3 c}{8}\right )}{d}\) | \(151\) |
norman | \(\frac {\frac {2 a^{4} \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{d}-\frac {14 a^{4} \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{3}}{d}+\frac {14 a^{4} \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{5}}{d}-\frac {2 a^{4} \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{7}}{d}-\frac {16 i a^{4} \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{4}}{d}+\frac {8 i a^{4} \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}}{d}+\frac {8 i a^{4} \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{6}}{d}}{\left (1+\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}\right )^{4}}\) | \(152\) |
parts | \(\frac {a^{4} \left (-\frac {\left (\sin \left (d x +c \right )^{3}+\frac {3 \sin \left (d x +c \right )}{2}\right ) \cos \left (d x +c \right )}{4}+\frac {3 d x}{8}+\frac {3 c}{8}\right )}{d}+\frac {a^{4} \left (\frac {\left (\cos \left (d x +c \right )^{3}+\frac {3 \cos \left (d x +c \right )}{2}\right ) \sin \left (d x +c \right )}{4}+\frac {3 d x}{8}+\frac {3 c}{8}\right )}{d}-\frac {i a^{4} \sin \left (d x +c \right )^{4}}{d}-\frac {i a^{4} \cos \left (d x +c \right )^{4}}{d}-\frac {6 a^{4} \left (-\frac {\sin \left (d x +c \right ) \cos \left (d x +c \right )^{3}}{4}+\frac {\sin \left (d x +c \right ) \cos \left (d x +c \right )}{8}+\frac {d x}{8}+\frac {c}{8}\right )}{d}\) | \(162\) |
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Time = 0.26 (sec) , antiderivative size = 17, normalized size of antiderivative = 0.55 \[ \int (a \cos (c+d x)+i a \sin (c+d x))^4 \, dx=-\frac {i \, a^{4} e^{\left (4 i \, d x + 4 i \, c\right )}}{4 \, d} \]
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Time = 0.09 (sec) , antiderivative size = 36, normalized size of antiderivative = 1.16 \[ \int (a \cos (c+d x)+i a \sin (c+d x))^4 \, dx=\begin {cases} - \frac {i a^{4} e^{4 i c} e^{4 i d x}}{4 d} & \text {for}\: d \neq 0 \\a^{4} x e^{4 i c} & \text {otherwise} \end {cases} \]
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Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 132 vs. \(2 (25) = 50\).
Time = 0.23 (sec) , antiderivative size = 132, normalized size of antiderivative = 4.26 \[ \int (a \cos (c+d x)+i a \sin (c+d x))^4 \, dx=-\frac {i \, a^{4} \cos \left (d x + c\right )^{4}}{d} - \frac {i \, a^{4} \sin \left (d x + c\right )^{4}}{d} + \frac {{\left (12 \, d x + 12 \, c + \sin \left (4 \, d x + 4 \, c\right ) + 8 \, \sin \left (2 \, d x + 2 \, c\right )\right )} a^{4}}{32 \, d} + \frac {{\left (12 \, d x + 12 \, c + \sin \left (4 \, d x + 4 \, c\right ) - 8 \, \sin \left (2 \, d x + 2 \, c\right )\right )} a^{4}}{32 \, d} - \frac {3 \, {\left (4 \, d x + 4 \, c - \sin \left (4 \, d x + 4 \, c\right )\right )} a^{4}}{16 \, d} \]
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Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 52 vs. \(2 (25) = 50\).
Time = 0.31 (sec) , antiderivative size = 52, normalized size of antiderivative = 1.68 \[ \int (a \cos (c+d x)+i a \sin (c+d x))^4 \, dx=-\frac {i \, a^{4} e^{\left (4 i \, d x + 4 i \, c\right )}}{8 \, d} - \frac {i \, a^{4} e^{\left (-4 i \, d x - 4 i \, c\right )}}{8 \, d} + \frac {a^{4} \sin \left (4 \, d x + 4 \, c\right )}{4 \, d} \]
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Time = 27.28 (sec) , antiderivative size = 84, normalized size of antiderivative = 2.71 \[ \int (a \cos (c+d x)+i a \sin (c+d x))^4 \, dx=-\frac {2\,a^4\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )\,\left ({\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2-1\right )}{d\,\left ({\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4+{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3\,4{}\mathrm {i}-6\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2-\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )\,4{}\mathrm {i}+1\right )} \]
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