Optimal. Leaf size=114 \[ -\frac {16 i a^{11}}{3 d (a-i a \tan (c+d x))^3}+\frac {16 i a^{10}}{d (a-i a \tan (c+d x))^2}-\frac {24 i a^9}{d (a-i a \tan (c+d x))}+\frac {a^8 \tan (c+d x)}{d}+\frac {8 i a^8 \log (\cos (c+d x))}{d}-8 a^8 x \]
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Rubi [A] time = 0.07, antiderivative size = 114, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.083, Rules used = {3487, 43} \[ -\frac {16 i a^{11}}{3 d (a-i a \tan (c+d x))^3}+\frac {16 i a^{10}}{d (a-i a \tan (c+d x))^2}-\frac {24 i a^9}{d (a-i a \tan (c+d x))}+\frac {a^8 \tan (c+d x)}{d}+\frac {8 i a^8 \log (\cos (c+d x))}{d}-8 a^8 x \]
Antiderivative was successfully verified.
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Rule 43
Rule 3487
Rubi steps
\begin {align*} \int \cos ^6(c+d x) (a+i a \tan (c+d x))^8 \, dx &=-\frac {\left (i a^7\right ) \operatorname {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 ) \operatorname {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*}
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Mathematica [B] time = 2.71, size = 414, normalized size = 3.63 \[ -\frac {a^8 \sec (c) \sec (c+d x) (\cos (3 c+11 d x)+i \sin (3 c+11 d x)) \left (-12 i d x \sin (c+2 d x)+11 \sin (c+2 d x)-12 i d x \sin (3 c+2 d x)+14 \sin (3 c+2 d x)-12 i d x \sin (3 c+4 d x)-4 \sin (3 c+4 d x)-12 i d x \sin (5 c+4 d x)-\sin (5 c+4 d x)+12 d x \cos (3 c+2 d x)+10 i \cos (3 c+2 d x)+12 d x \cos (3 c+4 d x)-2 i \cos (3 c+4 d x)+12 d x \cos (5 c+4 d x)+i \cos (5 c+4 d x)+\cos (c+2 d x) \left (-6 i \log \left (\cos ^2(c+d x)\right )+12 d x+7 i\right )-6 i \cos (3 c+2 d x) \log \left (\cos ^2(c+d x)\right )-6 i \cos (3 c+4 d x) \log \left (\cos ^2(c+d x)\right )-6 i \cos (5 c+4 d x) \log \left (\cos ^2(c+d x)\right )-6 \sin (c+2 d x) \log \left (\cos ^2(c+d x)\right )-6 \sin (3 c+2 d x) \log \left (\cos ^2(c+d x)\right )-6 \sin (3 c+4 d x) \log \left (\cos ^2(c+d x)\right )-6 \sin (5 c+4 d x) \log \left (\cos ^2(c+d x)\right )+12 i \cos (c)\right )}{6 d (\cos (d x)+i \sin (d x))^8} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.62, size = 112, normalized size = 0.98 \[ \frac {-2 i \, a^{8} e^{\left (8 i \, d x + 8 i \, c\right )} + 4 i \, a^{8} e^{\left (6 i \, d x + 6 i \, c\right )} - 12 i \, a^{8} e^{\left (4 i \, d x + 4 i \, c\right )} - 18 i \, a^{8} e^{\left (2 i \, d x + 2 i \, c\right )} + 6 i \, a^{8} + {\left (24 i \, a^{8} e^{\left (2 i \, d x + 2 i \, c\right )} + 24 i \, a^{8}\right )} \log \left (e^{\left (2 i \, d x + 2 i \, c\right )} + 1\right )}{3 \, {\left (d e^{\left (2 i \, d x + 2 i \, c\right )} + d\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 7.48, size = 799, normalized size = 7.01 \[ \text {result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.62, size = 319, normalized size = 2.80 \[ \frac {a^{8} \cos \left (d x +c \right ) \left (\sin ^{7}\left (d x +c \right )\right )}{d}-\frac {8 a^{8} c}{d}+\frac {4 i a^{8} \left (\sin ^{2}\left (d x +c \right )\right )}{d}+\frac {29 a^{8} \sin \left (d x +c \right ) \left (\cos ^{5}\left (d x +c \right )\right )}{6 d}-\frac {233 a^{8} \left (\cos ^{3}\left (d x +c \right )\right ) \sin \left (d x +c \right )}{24 d}+\frac {111 a^{8} \sin \left (d x +c \right ) \cos \left (d x +c \right )}{8 d}+\frac {32 i a^{8} \left (\sin ^{6}\left (d x +c \right )\right )}{3 d}+\frac {2 i a^{8} \left (\sin ^{4}\left (d x +c \right )\right )}{d}+\frac {28 i a^{8} \left (\sin ^{2}\left (d x +c \right )\right ) \left (\cos ^{4}\left (d x +c \right )\right )}{3 d}-\frac {35 a^{8} \left (\sin ^{3}\left (d x +c \right )\right ) \left (\cos ^{3}\left (d x +c \right )\right )}{3 d}+\frac {a^{8} \left (\sin ^{9}\left (d x +c \right )\right )}{d \cos \left (d x +c \right )}+\frac {35 a^{8} \cos \left (d x +c \right ) \left (\sin ^{5}\left (d x +c \right )\right )}{6 d}+\frac {175 a^{8} \cos \left (d x +c \right ) \left (\sin ^{3}\left (d x +c \right )\right )}{24 d}+\frac {14 i a^{8} \left (\cos ^{4}\left (d x +c \right )\right )}{3 d}+\frac {8 i a^{8} \ln \left (\cos \left (d x +c \right )\right )}{d}-\frac {4 i a^{8} \left (\cos ^{6}\left (d x +c \right )\right )}{3 d}-8 a^{8} x \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.70, size = 146, normalized size = 1.28 \[ -\frac {384 \, {\left (d x + c\right )} a^{8} + 192 i \, a^{8} \log \left (\tan \left (d x + c\right )^{2} + 1\right ) - 48 \, a^{8} \tan \left (d x + c\right ) - \frac {1152 \, a^{8} \tan \left (d x + c\right )^{5} - 1920 i \, a^{8} \tan \left (d x + c\right )^{4} + 512 \, a^{8} \tan \left (d x + c\right )^{3} - 1536 i \, a^{8} \tan \left (d x + c\right )^{2} + 384 \, a^{8} \tan \left (d x + c\right ) - 640 i \, a^{8}}{\tan \left (d x + c\right )^{6} + 3 \, \tan \left (d x + c\right )^{4} + 3 \, \tan \left (d x + c\right )^{2} + 1}}{48 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 3.41, size = 103, normalized size = 0.90 \[ \frac {a^8\,\mathrm {tan}\left (c+d\,x\right )}{d}-\frac {24\,a^8\,{\mathrm {tan}\left (c+d\,x\right )}^2+a^8\,\mathrm {tan}\left (c+d\,x\right )\,32{}\mathrm {i}-\frac {40\,a^8}{3}}{d\,\left (-{\mathrm {tan}\left (c+d\,x\right )}^3-{\mathrm {tan}\left (c+d\,x\right )}^2\,3{}\mathrm {i}+3\,\mathrm {tan}\left (c+d\,x\right )+1{}\mathrm {i}\right )}-\frac {a^8\,\ln \left (\mathrm {tan}\left (c+d\,x\right )+1{}\mathrm {i}\right )\,8{}\mathrm {i}}{d} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.84, size = 177, normalized size = 1.55 \[ - \frac {2 i a^{8}}{- d e^{2 i c} e^{2 i d x} - d} + \frac {8 i a^{8} \log {\left (e^{2 i d x} + e^{- 2 i c} \right )}}{d} + \begin {cases} - \frac {2 i a^{8} d^{2} e^{6 i c} e^{6 i d x} - 6 i a^{8} d^{2} e^{4 i c} e^{4 i d x} + 18 i a^{8} d^{2} e^{2 i c} e^{2 i d x}}{3 d^{3}} & \text {for}\: 3 d^{3} \neq 0 \\x \left (4 a^{8} e^{6 i c} - 8 a^{8} e^{4 i c} + 12 a^{8} e^{2 i c}\right ) & \text {otherwise} \end {cases} \]
Verification of antiderivative is not currently implemented for this CAS.
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