Integrand size = 24, antiderivative size = 32 \[ \int \cos ^3(c+d x) (a+i a \tan (c+d x))^3 \, dx=-\frac {i \cos ^3(c+d x) (a+i a \tan (c+d x))^3}{3 d} \]
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Time = 0.05 (sec) , antiderivative size = 32, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.042, Rules used = {3569} \[ \int \cos ^3(c+d x) (a+i a \tan (c+d x))^3 \, dx=-\frac {i \cos ^3(c+d x) (a+i a \tan (c+d x))^3}{3 d} \]
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Rule 3569
Rubi steps \begin{align*} \text {integral}& = -\frac {i \cos ^3(c+d x) (a+i a \tan (c+d x))^3}{3 d} \\ \end{align*}
Time = 0.19 (sec) , antiderivative size = 31, normalized size of antiderivative = 0.97 \[ \int \cos ^3(c+d x) (a+i a \tan (c+d x))^3 \, dx=-\frac {i a^3 (\cos (c+d x)+i \sin (c+d x))^3}{3 d} \]
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Time = 6.81 (sec) , antiderivative size = 19, normalized size of antiderivative = 0.59
method | result | size |
risch | \(-\frac {i a^{3} {\mathrm e}^{3 i \left (d x +c \right )}}{3 d}\) | \(19\) |
derivativedivides | \(\frac {\frac {i a^{3} \left (2+\sin ^{2}\left (d x +c \right )\right ) \cos \left (d x +c \right )}{3}-a^{3} \left (\sin ^{3}\left (d x +c \right )\right )-i a^{3} \left (\cos ^{3}\left (d x +c \right )\right )+\frac {a^{3} \left (2+\cos ^{2}\left (d x +c \right )\right ) \sin \left (d x +c \right )}{3}}{d}\) | \(76\) |
default | \(\frac {\frac {i a^{3} \left (2+\sin ^{2}\left (d x +c \right )\right ) \cos \left (d x +c \right )}{3}-a^{3} \left (\sin ^{3}\left (d x +c \right )\right )-i a^{3} \left (\cos ^{3}\left (d x +c \right )\right )+\frac {a^{3} \left (2+\cos ^{2}\left (d x +c \right )\right ) \sin \left (d x +c \right )}{3}}{d}\) | \(76\) |
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none
Time = 0.23 (sec) , antiderivative size = 17, normalized size of antiderivative = 0.53 \[ \int \cos ^3(c+d x) (a+i a \tan (c+d x))^3 \, dx=-\frac {i \, a^{3} e^{\left (3 i \, d x + 3 i \, c\right )}}{3 \, d} \]
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Time = 0.13 (sec) , antiderivative size = 36, normalized size of antiderivative = 1.12 \[ \int \cos ^3(c+d x) (a+i a \tan (c+d x))^3 \, dx=\begin {cases} - \frac {i a^{3} e^{3 i c} e^{3 i d x}}{3 d} & \text {for}\: d \neq 0 \\a^{3} x e^{3 i c} & \text {otherwise} \end {cases} \]
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Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 75 vs. \(2 (26) = 52\).
Time = 0.31 (sec) , antiderivative size = 75, normalized size of antiderivative = 2.34 \[ \int \cos ^3(c+d x) (a+i a \tan (c+d x))^3 \, dx=-\frac {3 i \, a^{3} \cos \left (d x + c\right )^{3} + 3 \, a^{3} \sin \left (d x + c\right )^{3} + i \, {\left (\cos \left (d x + c\right )^{3} - 3 \, \cos \left (d x + c\right )\right )} a^{3} + {\left (\sin \left (d x + c\right )^{3} - 3 \, \sin \left (d x + c\right )\right )} a^{3}}{3 \, d} \]
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Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 901 vs. \(2 (26) = 52\).
Time = 0.71 (sec) , antiderivative size = 901, normalized size of antiderivative = 28.16 \[ \int \cos ^3(c+d x) (a+i a \tan (c+d x))^3 \, dx=\text {Too large to display} \]
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Time = 3.98 (sec) , antiderivative size = 66, normalized size of antiderivative = 2.06 \[ \int \cos ^3(c+d x) (a+i a \tan (c+d x))^3 \, dx=-\frac {2\,a^3\,\left (3\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2-1\right )}{3\,d\,\left (-{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3-{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2\,3{}\mathrm {i}+3\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )+1{}\mathrm {i}\right )} \]
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