Integrand size = 31, antiderivative size = 68 \[ \int \frac {\cos ^3(c+d x)}{(a \cos (c+d x)+i a \sin (c+d x))^2} \, dx=\frac {2 i \cos ^5(c+d x)}{5 a^2 d}+\frac {\sin (c+d x)}{a^2 d}-\frac {\sin ^3(c+d x)}{a^2 d}+\frac {2 \sin ^5(c+d x)}{5 a^2 d} \]
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Time = 0.20 (sec) , antiderivative size = 68, normalized size of antiderivative = 1.00, number of steps used = 10, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.226, Rules used = {3171, 3169, 2713, 2645, 30, 2644, 14} \[ \int \frac {\cos ^3(c+d x)}{(a \cos (c+d x)+i a \sin (c+d x))^2} \, dx=\frac {2 \sin ^5(c+d x)}{5 a^2 d}-\frac {\sin ^3(c+d x)}{a^2 d}+\frac {\sin (c+d x)}{a^2 d}+\frac {2 i \cos ^5(c+d x)}{5 a^2 d} \]
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Rule 14
Rule 30
Rule 2644
Rule 2645
Rule 2713
Rule 3169
Rule 3171
Rubi steps \begin{align*} \text {integral}& = -\frac {\int \cos ^3(c+d x) (i a \cos (c+d x)+a \sin (c+d x))^2 \, dx}{a^4} \\ & = -\frac {\int \left (-a^2 \cos ^5(c+d x)+2 i a^2 \cos ^4(c+d x) \sin (c+d x)+a^2 \cos ^3(c+d x) \sin ^2(c+d x)\right ) \, dx}{a^4} \\ & = -\frac {(2 i) \int \cos ^4(c+d x) \sin (c+d x) \, dx}{a^2}+\frac {\int \cos ^5(c+d x) \, dx}{a^2}-\frac {\int \cos ^3(c+d x) \sin ^2(c+d x) \, dx}{a^2} \\ & = \frac {(2 i) \text {Subst}\left (\int x^4 \, dx,x,\cos (c+d x)\right )}{a^2 d}-\frac {\text {Subst}\left (\int x^2 \left (1-x^2\right ) \, dx,x,\sin (c+d x)\right )}{a^2 d}-\frac {\text {Subst}\left (\int \left (1-2 x^2+x^4\right ) \, dx,x,-\sin (c+d x)\right )}{a^2 d} \\ & = \frac {2 i \cos ^5(c+d x)}{5 a^2 d}+\frac {\sin (c+d x)}{a^2 d}-\frac {2 \sin ^3(c+d x)}{3 a^2 d}+\frac {\sin ^5(c+d x)}{5 a^2 d}-\frac {\text {Subst}\left (\int \left (x^2-x^4\right ) \, dx,x,\sin (c+d x)\right )}{a^2 d} \\ & = \frac {2 i \cos ^5(c+d x)}{5 a^2 d}+\frac {\sin (c+d x)}{a^2 d}-\frac {\sin ^3(c+d x)}{a^2 d}+\frac {2 \sin ^5(c+d x)}{5 a^2 d} \\ \end{align*}
Time = 0.77 (sec) , antiderivative size = 111, normalized size of antiderivative = 1.63 \[ \int \frac {\cos ^3(c+d x)}{(a \cos (c+d x)+i a \sin (c+d x))^2} \, dx=\frac {i \cos (c+d x)}{4 a^2 d}+\frac {i \cos (3 (c+d x))}{8 a^2 d}+\frac {i \cos (5 (c+d x))}{40 a^2 d}+\frac {\sin (c+d x)}{2 a^2 d}+\frac {\sin (3 (c+d x))}{8 a^2 d}+\frac {\sin (5 (c+d x))}{40 a^2 d} \]
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Time = 0.85 (sec) , antiderivative size = 67, normalized size of antiderivative = 0.99
method | result | size |
risch | \(\frac {i {\mathrm e}^{-3 i \left (d x +c \right )}}{8 a^{2} d}+\frac {i {\mathrm e}^{-5 i \left (d x +c \right )}}{40 a^{2} d}+\frac {i \cos \left (d x +c \right )}{4 a^{2} d}+\frac {\sin \left (d x +c \right )}{2 a^{2} d}\) | \(67\) |
derivativedivides | \(\frac {\frac {2}{8 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )+8 i}-\frac {2 i}{\left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-i\right )^{4}}+\frac {5 i}{2 \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-i\right )^{2}}+\frac {4}{5 \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-i\right )^{5}}-\frac {3}{\left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-i\right )^{3}}+\frac {7}{4 \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-i\right )}}{a^{2} d}\) | \(108\) |
default | \(\frac {\frac {2}{8 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )+8 i}-\frac {2 i}{\left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-i\right )^{4}}+\frac {5 i}{2 \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-i\right )^{2}}+\frac {4}{5 \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-i\right )^{5}}-\frac {3}{\left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-i\right )^{3}}+\frac {7}{4 \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-i\right )}}{a^{2} d}\) | \(108\) |
parallelrisch | \(\frac {4 i \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{4}-2 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{5}+4 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{3}-\frac {4 i}{5}+\frac {6 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{5}}{a^{2} d \left (4 i \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{5}-\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{6}+5 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{4}-4 i \tan \left (\frac {d x}{2}+\frac {c}{2}\right )+5 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}-1\right )}\) | \(132\) |
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Time = 0.25 (sec) , antiderivative size = 52, normalized size of antiderivative = 0.76 \[ \int \frac {\cos ^3(c+d x)}{(a \cos (c+d x)+i a \sin (c+d x))^2} \, dx=\frac {{\left (-5 i \, e^{\left (6 i \, d x + 6 i \, c\right )} + 15 i \, e^{\left (4 i \, d x + 4 i \, c\right )} + 5 i \, e^{\left (2 i \, d x + 2 i \, c\right )} + i\right )} e^{\left (-5 i \, d x - 5 i \, c\right )}}{40 \, a^{2} d} \]
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Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 163 vs. \(2 (60) = 120\).
Time = 0.20 (sec) , antiderivative size = 163, normalized size of antiderivative = 2.40 \[ \int \frac {\cos ^3(c+d x)}{(a \cos (c+d x)+i a \sin (c+d x))^2} \, dx=\begin {cases} \frac {\left (- 2560 i a^{6} d^{3} e^{10 i c} e^{i d x} + 7680 i a^{6} d^{3} e^{8 i c} e^{- i d x} + 2560 i a^{6} d^{3} e^{6 i c} e^{- 3 i d x} + 512 i a^{6} d^{3} e^{4 i c} e^{- 5 i d x}\right ) e^{- 9 i c}}{20480 a^{8} d^{4}} & \text {for}\: a^{8} d^{4} e^{9 i c} \neq 0 \\\frac {x \left (e^{6 i c} + 3 e^{4 i c} + 3 e^{2 i c} + 1\right ) e^{- 5 i c}}{8 a^{2}} & \text {otherwise} \end {cases} \]
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Exception generated. \[ \int \frac {\cos ^3(c+d x)}{(a \cos (c+d x)+i a \sin (c+d x))^2} \, dx=\text {Exception raised: RuntimeError} \]
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Time = 0.31 (sec) , antiderivative size = 93, normalized size of antiderivative = 1.37 \[ \int \frac {\cos ^3(c+d x)}{(a \cos (c+d x)+i a \sin (c+d x))^2} \, dx=\frac {\frac {5}{a^{2} {\left (\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + i\right )}} + \frac {35 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{4} - 90 i \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} - 120 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + 70 i \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 21}{a^{2} {\left (\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - i\right )}^{5}}}{20 \, d} \]
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Time = 24.29 (sec) , antiderivative size = 90, normalized size of antiderivative = 1.32 \[ \int \frac {\cos ^3(c+d x)}{(a \cos (c+d x)+i a \sin (c+d x))^2} \, dx=-\frac {2\,\left (-5\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^5+{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4\,10{}\mathrm {i}+10\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3+3\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )-2{}\mathrm {i}\right )}{5\,a^2\,d\,{\left (\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )-\mathrm {i}\right )}^5\,\left (\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )+1{}\mathrm {i}\right )} \]
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