Integrand size = 24, antiderivative size = 55 \[ \int \cos ^6(c+d x) (a+i a \tan (c+d x))^5 \, dx=-\frac {2 i a^8}{3 d (a-i a \tan (c+d x))^3}+\frac {i a^7}{2 d (a-i a \tan (c+d x))^2} \]
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Time = 0.07 (sec) , antiderivative size = 55, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.083, Rules used = {3568, 45} \[ \int \cos ^6(c+d x) (a+i a \tan (c+d x))^5 \, dx=\frac {i a^7}{2 d (a-i a \tan (c+d x))^2}-\frac {2 i a^8}{3 d (a-i a \tan (c+d x))^3} \]
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Rule 45
Rule 3568
Rubi steps \begin{align*} \text {integral}& = -\frac {\left (i a^7\right ) \text {Subst}\left (\int \frac {a+x}{(a-x)^4} \, dx,x,i a \tan (c+d x)\right )}{d} \\ & = -\frac {\left (i a^7\right ) \text {Subst}\left (\int \left (\frac {2 a}{(a-x)^4}-\frac {1}{(a-x)^3}\right ) \, dx,x,i a \tan (c+d x)\right )}{d} \\ & = -\frac {2 i a^8}{3 d (a-i a \tan (c+d x))^3}+\frac {i a^7}{2 d (a-i a \tan (c+d x))^2} \\ \end{align*}
Time = 0.12 (sec) , antiderivative size = 36, normalized size of antiderivative = 0.65 \[ \int \cos ^6(c+d x) (a+i a \tan (c+d x))^5 \, dx=-\frac {i a^5 (-i+3 \tan (c+d x))}{6 d (i+\tan (c+d x))^3} \]
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Time = 135.10 (sec) , antiderivative size = 38, normalized size of antiderivative = 0.69
method | result | size |
risch | \(-\frac {i a^{5} {\mathrm e}^{6 i \left (d x +c \right )}}{12 d}-\frac {i a^{5} {\mathrm e}^{4 i \left (d x +c \right )}}{8 d}\) | \(38\) |
derivativedivides | \(\frac {\frac {i a^{5} \left (\sin ^{6}\left (d x +c \right )\right )}{6}+5 a^{5} \left (-\frac {\left (\sin ^{3}\left (d x +c \right )\right ) \left (\cos ^{3}\left (d x +c \right )\right )}{6}-\frac {\left (\cos ^{3}\left (d x +c \right )\right ) \sin \left (d x +c \right )}{8}+\frac {\sin \left (d x +c \right ) \cos \left (d x +c \right )}{16}+\frac {d x}{16}+\frac {c}{16}\right )-10 i a^{5} \left (-\frac {\left (\cos ^{4}\left (d x +c \right )\right ) \left (\sin ^{2}\left (d x +c \right )\right )}{6}-\frac {\left (\cos ^{4}\left (d x +c \right )\right )}{12}\right )-10 a^{5} \left (-\frac {\left (\cos ^{5}\left (d x +c \right )\right ) \sin \left (d x +c \right )}{6}+\frac {\left (\cos ^{3}\left (d x +c \right )+\frac {3 \cos \left (d x +c \right )}{2}\right ) \sin \left (d x +c \right )}{24}+\frac {d x}{16}+\frac {c}{16}\right )-\frac {5 i a^{5} \left (\cos ^{6}\left (d x +c \right )\right )}{6}+a^{5} \left (\frac {\left (\cos ^{5}\left (d x +c \right )+\frac {5 \left (\cos ^{3}\left (d x +c \right )\right )}{4}+\frac {15 \cos \left (d x +c \right )}{8}\right ) \sin \left (d x +c \right )}{6}+\frac {5 d x}{16}+\frac {5 c}{16}\right )}{d}\) | \(231\) |
default | \(\frac {\frac {i a^{5} \left (\sin ^{6}\left (d x +c \right )\right )}{6}+5 a^{5} \left (-\frac {\left (\sin ^{3}\left (d x +c \right )\right ) \left (\cos ^{3}\left (d x +c \right )\right )}{6}-\frac {\left (\cos ^{3}\left (d x +c \right )\right ) \sin \left (d x +c \right )}{8}+\frac {\sin \left (d x +c \right ) \cos \left (d x +c \right )}{16}+\frac {d x}{16}+\frac {c}{16}\right )-10 i a^{5} \left (-\frac {\left (\cos ^{4}\left (d x +c \right )\right ) \left (\sin ^{2}\left (d x +c \right )\right )}{6}-\frac {\left (\cos ^{4}\left (d x +c \right )\right )}{12}\right )-10 a^{5} \left (-\frac {\left (\cos ^{5}\left (d x +c \right )\right ) \sin \left (d x +c \right )}{6}+\frac {\left (\cos ^{3}\left (d x +c \right )+\frac {3 \cos \left (d x +c \right )}{2}\right ) \sin \left (d x +c \right )}{24}+\frac {d x}{16}+\frac {c}{16}\right )-\frac {5 i a^{5} \left (\cos ^{6}\left (d x +c \right )\right )}{6}+a^{5} \left (\frac {\left (\cos ^{5}\left (d x +c \right )+\frac {5 \left (\cos ^{3}\left (d x +c \right )\right )}{4}+\frac {15 \cos \left (d x +c \right )}{8}\right ) \sin \left (d x +c \right )}{6}+\frac {5 d x}{16}+\frac {5 c}{16}\right )}{d}\) | \(231\) |
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Time = 0.23 (sec) , antiderivative size = 34, normalized size of antiderivative = 0.62 \[ \int \cos ^6(c+d x) (a+i a \tan (c+d x))^5 \, dx=\frac {-2 i \, a^{5} e^{\left (6 i \, d x + 6 i \, c\right )} - 3 i \, a^{5} e^{\left (4 i \, d x + 4 i \, c\right )}}{24 \, d} \]
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Time = 0.26 (sec) , antiderivative size = 80, normalized size of antiderivative = 1.45 \[ \int \cos ^6(c+d x) (a+i a \tan (c+d x))^5 \, dx=\begin {cases} \frac {- 8 i a^{5} d e^{6 i c} e^{6 i d x} - 12 i a^{5} d e^{4 i c} e^{4 i d x}}{96 d^{2}} & \text {for}\: d^{2} \neq 0 \\x \left (\frac {a^{5} e^{6 i c}}{2} + \frac {a^{5} e^{4 i c}}{2}\right ) & \text {otherwise} \end {cases} \]
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Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 93 vs. \(2 (43) = 86\).
Time = 0.51 (sec) , antiderivative size = 93, normalized size of antiderivative = 1.69 \[ \int \cos ^6(c+d x) (a+i a \tan (c+d x))^5 \, dx=-\frac {3 i \, a^{5} \tan \left (d x + c\right )^{4} + 10 \, a^{5} \tan \left (d x + c\right )^{3} - 12 i \, a^{5} \tan \left (d x + c\right )^{2} - 6 \, a^{5} \tan \left (d x + c\right ) + i \, a^{5}}{6 \, {\left (\tan \left (d x + c\right )^{6} + 3 \, \tan \left (d x + c\right )^{4} + 3 \, \tan \left (d x + c\right )^{2} + 1\right )} d} \]
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Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 187 vs. \(2 (43) = 86\).
Time = 0.70 (sec) , antiderivative size = 187, normalized size of antiderivative = 3.40 \[ \int \cos ^6(c+d x) (a+i a \tan (c+d x))^5 \, dx=-\frac {2 i \, a^{5} e^{\left (18 i \, d x + 12 i \, c\right )} + 15 i \, a^{5} e^{\left (16 i \, d x + 10 i \, c\right )} + 48 i \, a^{5} e^{\left (14 i \, d x + 8 i \, c\right )} + 85 i \, a^{5} e^{\left (12 i \, d x + 6 i \, c\right )} + 90 i \, a^{5} e^{\left (10 i \, d x + 4 i \, c\right )} + 57 i \, a^{5} e^{\left (8 i \, d x + 2 i \, c\right )} + 3 i \, a^{5} e^{\left (4 i \, d x - 2 i \, c\right )} + 20 i \, a^{5} e^{\left (6 i \, d x\right )}}{24 \, {\left (d e^{\left (12 i \, d x + 6 i \, c\right )} + 6 \, d e^{\left (10 i \, d x + 4 i \, c\right )} + 15 \, d e^{\left (8 i \, d x + 2 i \, c\right )} + 15 \, d e^{\left (4 i \, d x - 2 i \, c\right )} + 6 \, d e^{\left (2 i \, d x - 4 i \, c\right )} + 20 \, d e^{\left (6 i \, d x\right )} + d e^{\left (-6 i \, c\right )}\right )}} \]
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Time = 4.17 (sec) , antiderivative size = 53, normalized size of antiderivative = 0.96 \[ \int \cos ^6(c+d x) (a+i a \tan (c+d x))^5 \, dx=\frac {a^5\,\left (1+\mathrm {tan}\left (c+d\,x\right )\,3{}\mathrm {i}\right )}{6\,d\,\left (-{\mathrm {tan}\left (c+d\,x\right )}^3-{\mathrm {tan}\left (c+d\,x\right )}^2\,3{}\mathrm {i}+3\,\mathrm {tan}\left (c+d\,x\right )+1{}\mathrm {i}\right )} \]
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