Integrand size = 24, antiderivative size = 90 \[ \int \cos ^6(c+d x) (a+i a \tan (c+d x))^3 \, dx=\frac {a^3 x}{8}-\frac {i a^6}{6 d (a-i a \tan (c+d x))^3}-\frac {i a^5}{8 d (a-i a \tan (c+d x))^2}-\frac {i a^4}{8 d (a-i a \tan (c+d x))} \]
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Time = 0.09 (sec) , antiderivative size = 90, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.125, Rules used = {3568, 46, 212} \[ \int \cos ^6(c+d x) (a+i a \tan (c+d x))^3 \, dx=-\frac {i a^6}{6 d (a-i a \tan (c+d x))^3}-\frac {i a^5}{8 d (a-i a \tan (c+d x))^2}-\frac {i a^4}{8 d (a-i a \tan (c+d x))}+\frac {a^3 x}{8} \]
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Rule 46
Rule 212
Rule 3568
Rubi steps \begin{align*} \text {integral}& = -\frac {\left (i a^7\right ) \text {Subst}\left (\int \frac {1}{(a-x)^4 (a+x)} \, dx,x,i a \tan (c+d x)\right )}{d} \\ & = -\frac {\left (i a^7\right ) \text {Subst}\left (\int \left (\frac {1}{2 a (a-x)^4}+\frac {1}{4 a^2 (a-x)^3}+\frac {1}{8 a^3 (a-x)^2}+\frac {1}{8 a^3 \left (a^2-x^2\right )}\right ) \, dx,x,i a \tan (c+d x)\right )}{d} \\ & = -\frac {i a^6}{6 d (a-i a \tan (c+d x))^3}-\frac {i a^5}{8 d (a-i a \tan (c+d x))^2}-\frac {i a^4}{8 d (a-i a \tan (c+d x))}-\frac {\left (i a^4\right ) \text {Subst}\left (\int \frac {1}{a^2-x^2} \, dx,x,i a \tan (c+d x)\right )}{8 d} \\ & = \frac {a^3 x}{8}-\frac {i a^6}{6 d (a-i a \tan (c+d x))^3}-\frac {i a^5}{8 d (a-i a \tan (c+d x))^2}-\frac {i a^4}{8 d (a-i a \tan (c+d x))} \\ \end{align*}
Time = 0.24 (sec) , antiderivative size = 65, normalized size of antiderivative = 0.72 \[ \int \cos ^6(c+d x) (a+i a \tan (c+d x))^3 \, dx=\frac {a^3 \left (-10+9 i \tan (c+d x)+3 \tan ^2(c+d x)+3 \arctan (\tan (c+d x)) (i+\tan (c+d x))^3\right )}{24 d (i+\tan (c+d x))^3} \]
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Time = 47.77 (sec) , antiderivative size = 62, normalized size of antiderivative = 0.69
method | result | size |
risch | \(\frac {a^{3} x}{8}-\frac {i a^{3} {\mathrm e}^{6 i \left (d x +c \right )}}{48 d}-\frac {3 i a^{3} {\mathrm e}^{4 i \left (d x +c \right )}}{32 d}-\frac {3 i a^{3} {\mathrm e}^{2 i \left (d x +c \right )}}{16 d}\) | \(62\) |
derivativedivides | \(\frac {-i a^{3} \left (-\frac {\left (\cos ^{4}\left (d x +c \right )\right ) \left (\sin ^{2}\left (d x +c \right )\right )}{6}-\frac {\left (\cos ^{4}\left (d x +c \right )\right )}{12}\right )-3 a^{3} \left (-\frac {\left (\cos ^{5}\left (d x +c \right )\right ) \sin \left (d x +c \right )}{6}+\frac {\left (\cos ^{3}\left (d x +c \right )+\frac {3 \cos \left (d x +c \right )}{2}\right ) \sin \left (d x +c \right )}{24}+\frac {d x}{16}+\frac {c}{16}\right )-\frac {i a^{3} \left (\cos ^{6}\left (d x +c \right )\right )}{2}+a^{3} \left (\frac {\left (\cos ^{5}\left (d x +c \right )+\frac {5 \left (\cos ^{3}\left (d x +c \right )\right )}{4}+\frac {15 \cos \left (d x +c \right )}{8}\right ) \sin \left (d x +c \right )}{6}+\frac {5 d x}{16}+\frac {5 c}{16}\right )}{d}\) | \(156\) |
default | \(\frac {-i a^{3} \left (-\frac {\left (\cos ^{4}\left (d x +c \right )\right ) \left (\sin ^{2}\left (d x +c \right )\right )}{6}-\frac {\left (\cos ^{4}\left (d x +c \right )\right )}{12}\right )-3 a^{3} \left (-\frac {\left (\cos ^{5}\left (d x +c \right )\right ) \sin \left (d x +c \right )}{6}+\frac {\left (\cos ^{3}\left (d x +c \right )+\frac {3 \cos \left (d x +c \right )}{2}\right ) \sin \left (d x +c \right )}{24}+\frac {d x}{16}+\frac {c}{16}\right )-\frac {i a^{3} \left (\cos ^{6}\left (d x +c \right )\right )}{2}+a^{3} \left (\frac {\left (\cos ^{5}\left (d x +c \right )+\frac {5 \left (\cos ^{3}\left (d x +c \right )\right )}{4}+\frac {15 \cos \left (d x +c \right )}{8}\right ) \sin \left (d x +c \right )}{6}+\frac {5 d x}{16}+\frac {5 c}{16}\right )}{d}\) | \(156\) |
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Time = 0.23 (sec) , antiderivative size = 55, normalized size of antiderivative = 0.61 \[ \int \cos ^6(c+d x) (a+i a \tan (c+d x))^3 \, dx=\frac {12 \, a^{3} d x - 2 i \, a^{3} e^{\left (6 i \, d x + 6 i \, c\right )} - 9 i \, a^{3} e^{\left (4 i \, d x + 4 i \, c\right )} - 18 i \, a^{3} e^{\left (2 i \, d x + 2 i \, c\right )}}{96 \, d} \]
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Time = 0.20 (sec) , antiderivative size = 131, normalized size of antiderivative = 1.46 \[ \int \cos ^6(c+d x) (a+i a \tan (c+d x))^3 \, dx=\frac {a^{3} x}{8} + \begin {cases} \frac {- 512 i a^{3} d^{2} e^{6 i c} e^{6 i d x} - 2304 i a^{3} d^{2} e^{4 i c} e^{4 i d x} - 4608 i a^{3} d^{2} e^{2 i c} e^{2 i d x}}{24576 d^{3}} & \text {for}\: d^{3} \neq 0 \\x \left (\frac {a^{3} e^{6 i c}}{8} + \frac {3 a^{3} e^{4 i c}}{8} + \frac {3 a^{3} e^{2 i c}}{8}\right ) & \text {otherwise} \end {cases} \]
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Time = 0.46 (sec) , antiderivative size = 105, normalized size of antiderivative = 1.17 \[ \int \cos ^6(c+d x) (a+i a \tan (c+d x))^3 \, dx=\frac {3 \, {\left (d x + c\right )} a^{3} + \frac {3 \, a^{3} \tan \left (d x + c\right )^{5} + 8 \, a^{3} \tan \left (d x + c\right )^{3} + 6 i \, a^{3} \tan \left (d x + c\right )^{2} + 21 \, a^{3} \tan \left (d x + c\right ) - 10 i \, a^{3}}{\tan \left (d x + c\right )^{6} + 3 \, \tan \left (d x + c\right )^{4} + 3 \, \tan \left (d x + c\right )^{2} + 1}}{24 \, d} \]
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Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 457 vs. \(2 (70) = 140\).
Time = 0.81 (sec) , antiderivative size = 457, normalized size of antiderivative = 5.08 \[ \int \cos ^6(c+d x) (a+i a \tan (c+d x))^3 \, dx=\frac {12 \, a^{3} d x e^{\left (8 i \, d x + 4 i \, c\right )} + 48 \, a^{3} d x e^{\left (6 i \, d x + 2 i \, c\right )} + 48 \, a^{3} d x e^{\left (2 i \, d x - 2 i \, c\right )} + 72 \, a^{3} d x e^{\left (4 i \, d x\right )} + 12 \, a^{3} d x e^{\left (-4 i \, c\right )} - 3 i \, a^{3} e^{\left (8 i \, d x + 4 i \, c\right )} \log \left (e^{\left (2 i \, d x + 2 i \, c\right )} + 1\right ) - 12 i \, a^{3} e^{\left (6 i \, d x + 2 i \, c\right )} \log \left (e^{\left (2 i \, d x + 2 i \, c\right )} + 1\right ) - 12 i \, a^{3} e^{\left (2 i \, d x - 2 i \, c\right )} \log \left (e^{\left (2 i \, d x + 2 i \, c\right )} + 1\right ) - 18 i \, a^{3} e^{\left (4 i \, d x\right )} \log \left (e^{\left (2 i \, d x + 2 i \, c\right )} + 1\right ) - 3 i \, a^{3} e^{\left (-4 i \, c\right )} \log \left (e^{\left (2 i \, d x + 2 i \, c\right )} + 1\right ) + 3 i \, a^{3} e^{\left (8 i \, d x + 4 i \, c\right )} \log \left (e^{\left (2 i \, d x\right )} + e^{\left (-2 i \, c\right )}\right ) + 12 i \, a^{3} e^{\left (6 i \, d x + 2 i \, c\right )} \log \left (e^{\left (2 i \, d x\right )} + e^{\left (-2 i \, c\right )}\right ) + 12 i \, a^{3} e^{\left (2 i \, d x - 2 i \, c\right )} \log \left (e^{\left (2 i \, d x\right )} + e^{\left (-2 i \, c\right )}\right ) + 18 i \, a^{3} e^{\left (4 i \, d x\right )} \log \left (e^{\left (2 i \, d x\right )} + e^{\left (-2 i \, c\right )}\right ) + 3 i \, a^{3} e^{\left (-4 i \, c\right )} \log \left (e^{\left (2 i \, d x\right )} + e^{\left (-2 i \, c\right )}\right ) - 2 i \, a^{3} e^{\left (14 i \, d x + 10 i \, c\right )} - 17 i \, a^{3} e^{\left (12 i \, d x + 8 i \, c\right )} - 66 i \, a^{3} e^{\left (10 i \, d x + 6 i \, c\right )} - 134 i \, a^{3} e^{\left (8 i \, d x + 4 i \, c\right )} - 146 i \, a^{3} e^{\left (6 i \, d x + 2 i \, c\right )} - 18 i \, a^{3} e^{\left (2 i \, d x - 2 i \, c\right )} - 81 i \, a^{3} e^{\left (4 i \, d x\right )}}{96 \, {\left (d e^{\left (8 i \, d x + 4 i \, c\right )} + 4 \, d e^{\left (6 i \, d x + 2 i \, c\right )} + 4 \, d e^{\left (2 i \, d x - 2 i \, c\right )} + 6 \, d e^{\left (4 i \, d x\right )} + d e^{\left (-4 i \, c\right )}\right )}} \]
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Time = 4.15 (sec) , antiderivative size = 77, normalized size of antiderivative = 0.86 \[ \int \cos ^6(c+d x) (a+i a \tan (c+d x))^3 \, dx=\frac {a^3\,x}{8}-\frac {\frac {a^3\,{\mathrm {tan}\left (c+d\,x\right )}^2}{8}+\frac {a^3\,\mathrm {tan}\left (c+d\,x\right )\,3{}\mathrm {i}}{8}-\frac {5\,a^3}{12}}{d\,\left (-{\mathrm {tan}\left (c+d\,x\right )}^3-{\mathrm {tan}\left (c+d\,x\right )}^2\,3{}\mathrm {i}+3\,\mathrm {tan}\left (c+d\,x\right )+1{}\mathrm {i}\right )} \]
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