Integrand size = 22, antiderivative size = 31 \[ \int \frac {1}{(a \cos (c+d x)+i a \sin (c+d x))^3} \, dx=\frac {i}{3 d (a \cos (c+d x)+i a \sin (c+d x))^3} \]
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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} \[ \int \frac {1}{(a \cos (c+d x)+i a \sin (c+d x))^3} \, dx=\frac {i}{3 d (a \cos (c+d x)+i a \sin (c+d x))^3} \]
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Rule 3150
Rubi steps \begin{align*} \text {integral}& = \frac {i}{3 d (a \cos (c+d x)+i a \sin (c+d x))^3} \\ \end{align*}
Time = 0.22 (sec) , antiderivative size = 31, normalized size of antiderivative = 1.00 \[ \int \frac {1}{(a \cos (c+d x)+i a \sin (c+d x))^3} \, dx=\frac {i}{3 d (a \cos (c+d x)+i a \sin (c+d x))^3} \]
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Time = 0.56 (sec) , antiderivative size = 19, normalized size of antiderivative = 0.61
method | result | size |
risch | \(\frac {i {\mathrm e}^{-3 i \left (d x +c \right )}}{3 d \,a^{3}}\) | \(19\) |
derivativedivides | \(\frac {\frac {4 i}{\left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-i\right )^{2}}+\frac {2}{\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-i}-\frac {8}{3 \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-i\right )^{3}}}{d \,a^{3}}\) | \(57\) |
default | \(\frac {\frac {4 i}{\left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-i\right )^{2}}+\frac {2}{\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-i}-\frac {8}{3 \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-i\right )^{3}}}{d \,a^{3}}\) | \(57\) |
norman | \(\frac {-\frac {4 i \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}}{a d}+\frac {2 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{a d}-\frac {20 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{3}}{3 a d}+\frac {2 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{5}}{a d}+\frac {2 i}{3 a d}+\frac {6 i \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{4}}{a d}}{\left (1+\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}\right )^{3} a^{2}}\) | \(125\) |
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none
Time = 0.24 (sec) , antiderivative size = 17, normalized size of antiderivative = 0.55 \[ \int \frac {1}{(a \cos (c+d x)+i a \sin (c+d x))^3} \, dx=\frac {i \, e^{\left (-3 i \, d x - 3 i \, c\right )}}{3 \, a^{3} d} \]
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Time = 0.08 (sec) , antiderivative size = 44, normalized size of antiderivative = 1.42 \[ \int \frac {1}{(a \cos (c+d x)+i a \sin (c+d x))^3} \, dx=\begin {cases} \frac {i e^{- 3 i c} e^{- 3 i d x}}{3 a^{3} d} & \text {for}\: a^{3} d e^{3 i c} \neq 0 \\\frac {x e^{- 3 i c}}{a^{3}} & \text {otherwise} \end {cases} \]
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none
Time = 0.23 (sec) , antiderivative size = 29, normalized size of antiderivative = 0.94 \[ \int \frac {1}{(a \cos (c+d x)+i a \sin (c+d x))^3} \, dx=\frac {i \, \cos \left (3 \, d x + 3 \, c\right ) + \sin \left (3 \, d x + 3 \, c\right )}{3 \, a^{3} d} \]
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none
Time = 0.29 (sec) , antiderivative size = 36, normalized size of antiderivative = 1.16 \[ \int \frac {1}{(a \cos (c+d x)+i a \sin (c+d x))^3} \, dx=\frac {2 \, {\left (3 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - 1\right )}}{3 \, a^{3} d {\left (\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - i\right )}^{3}} \]
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Time = 24.44 (sec) , antiderivative size = 68, normalized size of antiderivative = 2.19 \[ \int \frac {1}{(a \cos (c+d x)+i a \sin (c+d x))^3} \, dx=-\frac {2\,\left ({\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2\,3{}\mathrm {i}-\mathrm {i}\right )}{3\,a^3\,d\,\left (-{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3\,1{}\mathrm {i}-3\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2+\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )\,3{}\mathrm {i}+1\right )} \]
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