Optimal. Leaf size=126 \[ \frac {\tan ^3(c+d x)}{3 a^8 d}+\frac {4 i \tan ^2(c+d x)}{a^8 d}-\frac {31 \tan (c+d x)}{a^8 d}-\frac {80 i}{d \left (a^8+i a^8 \tan (c+d x)\right )}+\frac {80 i \log (\cos (c+d x))}{a^8 d}+\frac {80 x}{a^8}+\frac {16 i}{d \left (a^4+i a^4 \tan (c+d x)\right )^2} \]
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Rubi [A] time = 0.08, antiderivative size = 126, 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} \[ \frac {\tan ^3(c+d x)}{3 a^8 d}+\frac {4 i \tan ^2(c+d x)}{a^8 d}-\frac {31 \tan (c+d x)}{a^8 d}-\frac {80 i}{d \left (a^8+i a^8 \tan (c+d x)\right )}+\frac {16 i}{d \left (a^4+i a^4 \tan (c+d x)\right )^2}+\frac {80 i \log (\cos (c+d x))}{a^8 d}+\frac {80 x}{a^8} \]
Antiderivative was successfully verified.
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Rule 43
Rule 3487
Rubi steps
\begin {align*} \int \frac {\sec ^{12}(c+d x)}{(a+i a \tan (c+d x))^8} \, dx &=-\frac {i \operatorname {Subst}\left (\int \frac {(a-x)^5}{(a+x)^3} \, dx,x,i a \tan (c+d x)\right )}{a^{11} d}\\ &=-\frac {i \operatorname {Subst}\left (\int \left (-31 a^2+8 a x-x^2+\frac {32 a^5}{(a+x)^3}-\frac {80 a^4}{(a+x)^2}+\frac {80 a^3}{a+x}\right ) \, dx,x,i a \tan (c+d x)\right )}{a^{11} d}\\ &=\frac {80 x}{a^8}+\frac {80 i \log (\cos (c+d x))}{a^8 d}-\frac {31 \tan (c+d x)}{a^8 d}+\frac {4 i \tan ^2(c+d x)}{a^8 d}+\frac {\tan ^3(c+d x)}{3 a^8 d}+\frac {16 i}{d \left (a^4+i a^4 \tan (c+d x)\right )^2}-\frac {80 i}{d \left (a^8+i a^8 \tan (c+d x)\right )}\\ \end {align*}
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Mathematica [B] time = 1.55, size = 537, normalized size = 4.26 \[ \frac {\sec (c) \sec ^{11}(c+d x) (\cos (6 (c+d x))+i \sin (6 (c+d x))) (120 i d x \sin (2 c+d x)+87 \sin (2 c+d x)+180 i d x \sin (2 c+3 d x)-96 \sin (2 c+3 d x)+180 i d x \sin (4 c+3 d x)+45 \sin (4 c+3 d x)+60 i d x \sin (4 c+5 d x)-44 \sin (4 c+5 d x)+60 i d x \sin (6 c+5 d x)+3 \sin (6 c+5 d x)+180 d x \cos (2 c+3 d x)+66 i \cos (2 c+3 d x)+180 d x \cos (4 c+3 d x)-75 i \cos (4 c+3 d x)+60 d x \cos (4 c+5 d x)+50 i \cos (4 c+5 d x)+60 d x \cos (6 c+5 d x)+3 i \cos (6 c+5 d x)+180 i \cos (2 c+3 d x) \log (\cos (c+d x))+3 \cos (2 c+d x) (80 i \log (\cos (c+d x))+80 d x-71 i)+\cos (d x) (240 i \log (\cos (c+d x))+240 d x-119 i)+180 i \cos (4 c+3 d x) \log (\cos (c+d x))+60 i \cos (4 c+5 d x) \log (\cos (c+d x))+60 i \cos (6 c+5 d x) \log (\cos (c+d x))-120 \sin (d x) \log (\cos (c+d x))-120 \sin (2 c+d x) \log (\cos (c+d x))-180 \sin (2 c+3 d x) \log (\cos (c+d x))-180 \sin (4 c+3 d x) \log (\cos (c+d x))-60 \sin (4 c+5 d x) \log (\cos (c+d x))-60 \sin (6 c+5 d x) \log (\cos (c+d x))+120 i d x \sin (d x)-101 \sin (d x))}{12 a^8 d (\tan (c+d x)-i)^8} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.66, size = 195, normalized size = 1.55 \[ \frac {480 \, d x e^{\left (10 i \, d x + 10 i \, c\right )} + {\left (1440 \, d x - 240 i\right )} e^{\left (8 i \, d x + 8 i \, c\right )} + {\left (1440 \, d x - 600 i\right )} e^{\left (6 i \, d x + 6 i \, c\right )} + {\left (480 \, d x - 440 i\right )} e^{\left (4 i \, d x + 4 i \, c\right )} + {\left (240 i \, e^{\left (10 i \, d x + 10 i \, c\right )} + 720 i \, e^{\left (8 i \, d x + 8 i \, c\right )} + 720 i \, e^{\left (6 i \, d x + 6 i \, c\right )} + 240 i \, e^{\left (4 i \, d x + 4 i \, c\right )}\right )} \log \left (e^{\left (2 i \, d x + 2 i \, c\right )} + 1\right ) - 60 i \, e^{\left (2 i \, d x + 2 i \, c\right )} + 12 i}{3 \, {\left (a^{8} d e^{\left (10 i \, d x + 10 i \, c\right )} + 3 \, a^{8} d e^{\left (8 i \, d x + 8 i \, c\right )} + 3 \, a^{8} d e^{\left (6 i \, d x + 6 i \, c\right )} + a^{8} d e^{\left (4 i \, d x + 4 i \, c\right )}\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 4.97, size = 223, normalized size = 1.77 \[ -\frac {2 \, {\left (-\frac {120 i \, \log \left (\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 1\right )}{a^{8}} + \frac {240 i \, \log \left (\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - i\right )}{a^{8}} - \frac {120 i \, \log \left (\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 1\right )}{a^{8}} + \frac {220 i \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{6} - 93 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} - 684 i \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{4} + 190 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 684 i \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - 93 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 220 i}{{\left (\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - 1\right )}^{3} a^{8}} + \frac {-500 i \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{4} - 2144 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 3384 i \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + 2144 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 500 i}{a^{8} {\left (\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - i\right )}^{4}}\right )}}{3 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.45, size = 107, normalized size = 0.85 \[ -\frac {31 \tan \left (d x +c \right )}{a^{8} d}+\frac {\tan ^{3}\left (d x +c \right )}{3 a^{8} d}+\frac {4 i \left (\tan ^{2}\left (d x +c \right )\right )}{a^{8} d}-\frac {16 i}{a^{8} d \left (\tan \left (d x +c \right )-i\right )^{2}}-\frac {80}{a^{8} d \left (\tan \left (d x +c \right )-i\right )}-\frac {80 i \ln \left (\tan \left (d x +c \right )-i\right )}{a^{8} d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.36, size = 213, normalized size = 1.69 \[ -\frac {\frac {3 \, {\left (1680 \, \tan \left (d x + c\right )^{6} - 9744 i \, \tan \left (d x + c\right )^{5} - 23520 \, \tan \left (d x + c\right )^{4} + 30240 i \, \tan \left (d x + c\right )^{3} + 21840 \, \tan \left (d x + c\right )^{2} - 8400 i \, \tan \left (d x + c\right ) - 1344\right )}}{21 \, a^{8} \tan \left (d x + c\right )^{7} - 147 i \, a^{8} \tan \left (d x + c\right )^{6} - 441 \, a^{8} \tan \left (d x + c\right )^{5} + 735 i \, a^{8} \tan \left (d x + c\right )^{4} + 735 \, a^{8} \tan \left (d x + c\right )^{3} - 441 i \, a^{8} \tan \left (d x + c\right )^{2} - 147 \, a^{8} \tan \left (d x + c\right ) + 21 i \, a^{8}} - \frac {\tan \left (d x + c\right )^{3} + 12 i \, \tan \left (d x + c\right )^{2} - 93 \, \tan \left (d x + c\right )}{a^{8}} + \frac {240 i \, \log \left (i \, \tan \left (d x + c\right ) + 1\right )}{a^{8}}}{3 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 3.44, size = 114, normalized size = 0.90 \[ \frac {{\mathrm {tan}\left (c+d\,x\right )}^3}{3\,a^8\,d}-\frac {31\,\mathrm {tan}\left (c+d\,x\right )}{a^8\,d}+\frac {{\mathrm {tan}\left (c+d\,x\right )}^2\,4{}\mathrm {i}}{a^8\,d}-\frac {\ln \left (\mathrm {tan}\left (c+d\,x\right )-\mathrm {i}\right )\,80{}\mathrm {i}}{a^8\,d}-\frac {\frac {64}{a^8}+\frac {\mathrm {tan}\left (c+d\,x\right )\,80{}\mathrm {i}}{a^8}}{d\,\left ({\mathrm {tan}\left (c+d\,x\right )}^2\,1{}\mathrm {i}+2\,\mathrm {tan}\left (c+d\,x\right )-\mathrm {i}\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \frac {\int \frac {\sec ^{12}{\left (c + d x \right )}}{\tan ^{8}{\left (c + d x \right )} - 8 i \tan ^{7}{\left (c + d x \right )} - 28 \tan ^{6}{\left (c + d x \right )} + 56 i \tan ^{5}{\left (c + d x \right )} + 70 \tan ^{4}{\left (c + d x \right )} - 56 i \tan ^{3}{\left (c + d x \right )} - 28 \tan ^{2}{\left (c + d x \right )} + 8 i \tan {\left (c + d x \right )} + 1}\, dx}{a^{8}} \]
Verification of antiderivative is not currently implemented for this CAS.
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