Integrand size = 24, antiderivative size = 114 \[ \int \cos ^6(c+d x) (a+i a \tan (c+d x))^8 \, dx=-8 a^8 x+\frac {8 i a^8 \log (\cos (c+d x))}{d}+\frac {a^8 \tan (c+d x)}{d}-\frac {16 i a^{11}}{3 d (a-i a \tan (c+d x))^3}+\frac {16 i a^{10}}{d (a-i a \tan (c+d x))^2}-\frac {24 i a^9}{d (a-i a \tan (c+d x))} \]
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Time = 0.15 (sec) , antiderivative size = 114, 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} \[ \int \cos ^6(c+d x) (a+i a \tan (c+d x))^8 \, dx=-\frac {16 i a^{11}}{3 d (a-i a \tan (c+d x))^3}+\frac {16 i a^{10}}{d (a-i a \tan (c+d x))^2}-\frac {24 i a^9}{d (a-i a \tan (c+d x))}+\frac {a^8 \tan (c+d x)}{d}+\frac {8 i a^8 \log (\cos (c+d x))}{d}-8 a^8 x \]
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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)^4}{(a-x)^4} \, dx,x,i a \tan (c+d x)\right )}{d} \\ & = -\frac {\left (i a^7\right ) \text {Subst}\left (\int \left (1+\frac {16 a^4}{(a-x)^4}-\frac {32 a^3}{(a-x)^3}+\frac {24 a^2}{(a-x)^2}-\frac {8 a}{a-x}\right ) \, dx,x,i a \tan (c+d x)\right )}{d} \\ & = -8 a^8 x+\frac {8 i a^8 \log (\cos (c+d x))}{d}+\frac {a^8 \tan (c+d x)}{d}-\frac {16 i a^{11}}{3 d (a-i a \tan (c+d x))^3}+\frac {16 i a^{10}}{d (a-i a \tan (c+d x))^2}-\frac {24 i a^9}{d (a-i a \tan (c+d x))} \\ \end{align*}
Time = 0.92 (sec) , antiderivative size = 77, normalized size of antiderivative = 0.68 \[ \int \cos ^6(c+d x) (a+i a \tan (c+d x))^8 \, dx=-\frac {i a^7 \left (8 a \log (i+\tan (c+d x))+i a \tan (c+d x)+\frac {8 i a \left (-5+12 i \tan (c+d x)+9 \tan ^2(c+d x)\right )}{3 (i+\tan (c+d x))^3}\right )}{d} \]
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Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 318 vs. \(2 (105 ) = 210\).
Time = 0.84 (sec) , antiderivative size = 319, normalized size of antiderivative = 2.80
\[\frac {32 i a^{8} \left (\sin ^{6}\left (d x +c \right )\right )}{3 d}+\frac {14 i a^{8} \left (\cos ^{4}\left (d x +c \right )\right )}{3 d}+\frac {28 i a^{8} \left (\sin ^{2}\left (d x +c \right )\right ) \left (\cos ^{4}\left (d x +c \right )\right )}{3 d}+\frac {8 i a^{8} \ln \left (\cos \left (d x +c \right )\right )}{d}+\frac {a^{8} \left (\sin ^{7}\left (d x +c \right )\right ) \cos \left (d x +c \right )}{d}+\frac {2 i a^{8} \left (\sin ^{4}\left (d x +c \right )\right )}{d}-\frac {4 i a^{8} \left (\cos ^{6}\left (d x +c \right )\right )}{3 d}-\frac {35 a^{8} \left (\sin ^{3}\left (d x +c \right )\right ) \left (\cos ^{3}\left (d x +c \right )\right )}{3 d}-\frac {233 a^{8} \sin \left (d x +c \right ) \left (\cos ^{3}\left (d x +c \right )\right )}{24 d}+\frac {29 a^{8} \left (\cos ^{5}\left (d x +c \right )\right ) \sin \left (d x +c \right )}{6 d}+\frac {4 i a^{8} \left (\sin ^{2}\left (d x +c \right )\right )}{d}-8 a^{8} x +\frac {a^{8} \left (\sin ^{9}\left (d x +c \right )\right )}{d \cos \left (d x +c \right )}+\frac {35 a^{8} \cos \left (d x +c \right ) \left (\sin ^{5}\left (d x +c \right )\right )}{6 d}+\frac {175 a^{8} \cos \left (d x +c \right ) \left (\sin ^{3}\left (d x +c \right )\right )}{24 d}+\frac {111 a^{8} \sin \left (d x +c \right ) \cos \left (d x +c \right )}{8 d}-\frac {8 a^{8} c}{d}\]
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Time = 0.24 (sec) , antiderivative size = 113, normalized size of antiderivative = 0.99 \[ \int \cos ^6(c+d x) (a+i a \tan (c+d x))^8 \, dx=-\frac {2 \, {\left (i \, a^{8} e^{\left (8 i \, d x + 8 i \, c\right )} - 2 i \, a^{8} e^{\left (6 i \, d x + 6 i \, c\right )} + 6 i \, a^{8} e^{\left (4 i \, d x + 4 i \, c\right )} + 9 i \, a^{8} e^{\left (2 i \, d x + 2 i \, c\right )} - 3 i \, a^{8} + 12 \, {\left (-i \, a^{8} e^{\left (2 i \, d x + 2 i \, c\right )} - i \, a^{8}\right )} \log \left (e^{\left (2 i \, d x + 2 i \, c\right )} + 1\right )\right )}}{3 \, {\left (d e^{\left (2 i \, d x + 2 i \, c\right )} + d\right )}} \]
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Time = 0.41 (sec) , antiderivative size = 172, normalized size of antiderivative = 1.51 \[ \int \cos ^6(c+d x) (a+i a \tan (c+d x))^8 \, dx=\frac {2 i a^{8}}{d e^{2 i c} e^{2 i d x} + d} + \frac {8 i a^{8} \log {\left (e^{2 i d x} + e^{- 2 i c} \right )}}{d} + \begin {cases} \frac {- 2 i a^{8} d^{2} e^{6 i c} e^{6 i d x} + 6 i a^{8} d^{2} e^{4 i c} e^{4 i d x} - 18 i a^{8} d^{2} e^{2 i c} e^{2 i d x}}{3 d^{3}} & \text {for}\: d^{3} \neq 0 \\x \left (4 a^{8} e^{6 i c} - 8 a^{8} e^{4 i c} + 12 a^{8} e^{2 i c}\right ) & \text {otherwise} \end {cases} \]
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Time = 0.34 (sec) , antiderivative size = 146, normalized size of antiderivative = 1.28 \[ \int \cos ^6(c+d x) (a+i a \tan (c+d x))^8 \, dx=-\frac {24 \, {\left (d x + c\right )} a^{8} + 12 i \, a^{8} \log \left (\tan \left (d x + c\right )^{2} + 1\right ) - 3 \, a^{8} \tan \left (d x + c\right ) - \frac {8 \, {\left (9 \, a^{8} \tan \left (d x + c\right )^{5} - 15 i \, a^{8} \tan \left (d x + c\right )^{4} + 4 \, a^{8} \tan \left (d x + c\right )^{3} - 12 i \, a^{8} \tan \left (d x + c\right )^{2} + 3 \, a^{8} \tan \left (d x + c\right ) - 5 i \, a^{8}\right )}}{\tan \left (d x + c\right )^{6} + 3 \, \tan \left (d x + c\right )^{4} + 3 \, \tan \left (d x + c\right )^{2} + 1}}{3 \, d} \]
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Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 799 vs. \(2 (98) = 196\).
Time = 1.09 (sec) , antiderivative size = 799, normalized size of antiderivative = 7.01 \[ \int \cos ^6(c+d x) (a+i a \tan (c+d x))^8 \, dx=\text {Too large to display} \]
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Time = 4.22 (sec) , antiderivative size = 103, normalized size of antiderivative = 0.90 \[ \int \cos ^6(c+d x) (a+i a \tan (c+d x))^8 \, dx=\frac {a^8\,\mathrm {tan}\left (c+d\,x\right )}{d}-\frac {24\,a^8\,{\mathrm {tan}\left (c+d\,x\right )}^2+a^8\,\mathrm {tan}\left (c+d\,x\right )\,32{}\mathrm {i}-\frac {40\,a^8}{3}}{d\,\left (-{\mathrm {tan}\left (c+d\,x\right )}^3-{\mathrm {tan}\left (c+d\,x\right )}^2\,3{}\mathrm {i}+3\,\mathrm {tan}\left (c+d\,x\right )+1{}\mathrm {i}\right )}-\frac {a^8\,\ln \left (\mathrm {tan}\left (c+d\,x\right )+1{}\mathrm {i}\right )\,8{}\mathrm {i}}{d} \]
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