Optimal. Leaf size=134 \[ \frac {\tan ^5(c+d x)}{5 a^8 d}+\frac {2 i \tan ^4(c+d x)}{a^8 d}-\frac {10 \tan ^3(c+d x)}{a^8 d}-\frac {36 i \tan ^2(c+d x)}{a^8 d}+\frac {129 \tan (c+d x)}{a^8 d}+\frac {64 i}{d \left (a^8+i a^8 \tan (c+d x)\right )}-\frac {192 i \log (\cos (c+d x))}{a^8 d}-\frac {192 x}{a^8} \]
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Rubi [A] time = 0.08, antiderivative size = 134, 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} \[ \frac {\tan ^5(c+d x)}{5 a^8 d}+\frac {2 i \tan ^4(c+d x)}{a^8 d}-\frac {10 \tan ^3(c+d x)}{a^8 d}-\frac {36 i \tan ^2(c+d x)}{a^8 d}+\frac {129 \tan (c+d x)}{a^8 d}+\frac {64 i}{d \left (a^8+i a^8 \tan (c+d x)\right )}-\frac {192 i \log (\cos (c+d x))}{a^8 d}-\frac {192 x}{a^8} \]
Antiderivative was successfully verified.
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Rule 43
Rule 3487
Rubi steps
\begin {align*} \int \frac {\sec ^{14}(c+d x)}{(a+i a \tan (c+d x))^8} \, dx &=-\frac {i \operatorname {Subst}\left (\int \frac {(a-x)^6}{(a+x)^2} \, dx,x,i a \tan (c+d x)\right )}{a^{13} d}\\ &=-\frac {i \operatorname {Subst}\left (\int \left (129 a^4-72 a^3 x+30 a^2 x^2-8 a x^3+x^4+\frac {64 a^6}{(a+x)^2}-\frac {192 a^5}{a+x}\right ) \, dx,x,i a \tan (c+d x)\right )}{a^{13} d}\\ &=-\frac {192 x}{a^8}-\frac {192 i \log (\cos (c+d x))}{a^8 d}+\frac {129 \tan (c+d x)}{a^8 d}-\frac {36 i \tan ^2(c+d x)}{a^8 d}-\frac {10 \tan ^3(c+d x)}{a^8 d}+\frac {2 i \tan ^4(c+d x)}{a^8 d}+\frac {\tan ^5(c+d x)}{5 a^8 d}+\frac {64 i}{d \left (a^8+i a^8 \tan (c+d x)\right )}\\ \end {align*}
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Mathematica [B] time = 2.87, size = 599, normalized size = 4.47 \[ \frac {\sec (c) \sec ^{13}(c+d x) (-\cos (7 (c+d x))-i \sin (7 (c+d x))) (300 i d x \sin (c+2 d x)-985 \sin (c+2 d x)+300 i d x \sin (3 c+2 d x)+320 \sin (3 c+2 d x)+240 i d x \sin (3 c+4 d x)-512 \sin (3 c+4 d x)+240 i d x \sin (5 c+4 d x)+10 \sin (5 c+4 d x)+60 i d x \sin (5 c+6 d x)-97 \sin (5 c+6 d x)+60 i d x \sin (7 c+6 d x)-10 \sin (7 c+6 d x)+900 d x \cos (3 c+2 d x)-220 i \cos (3 c+2 d x)+360 d x \cos (3 c+4 d x)+238 i \cos (3 c+4 d x)+360 d x \cos (5 c+4 d x)-110 i \cos (5 c+4 d x)+60 d x \cos (5 c+6 d x)+77 i \cos (5 c+6 d x)+60 d x \cos (7 c+6 d x)-10 i \cos (7 c+6 d x)+900 i \cos (3 c+2 d x) \log (\cos (c+d x))+10 \cos (c) (120 i \log (\cos (c+d x))+120 d x-7 i)+5 \cos (c+2 d x) (180 i \log (\cos (c+d x))+180 d x+43 i)+360 i \cos (3 c+4 d x) \log (\cos (c+d x))+360 i \cos (5 c+4 d x) \log (\cos (c+d x))+60 i \cos (5 c+6 d x) \log (\cos (c+d x))+60 i \cos (7 c+6 d x) \log (\cos (c+d x))-300 \sin (c+2 d x) \log (\cos (c+d x))-300 \sin (3 c+2 d x) \log (\cos (c+d x))-240 \sin (3 c+4 d x) \log (\cos (c+d x))-240 \sin (5 c+4 d x) \log (\cos (c+d x))-60 \sin (5 c+6 d x) \log (\cos (c+d x))-60 \sin (7 c+6 d x) \log (\cos (c+d x))+870 \sin (c))}{20 a^8 d (\tan (c+d x)-i)^8} \]
Antiderivative was successfully verified.
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fricas [B] time = 0.64, size = 269, normalized size = 2.01 \[ -\frac {1920 \, d x e^{\left (12 i \, d x + 12 i \, c\right )} + {\left (9600 \, d x - 960 i\right )} e^{\left (10 i \, d x + 10 i \, c\right )} + {\left (19200 \, d x - 4320 i\right )} e^{\left (8 i \, d x + 8 i \, c\right )} + {\left (19200 \, d x - 7520 i\right )} e^{\left (6 i \, d x + 6 i \, c\right )} + {\left (9600 \, d x - 6160 i\right )} e^{\left (4 i \, d x + 4 i \, c\right )} + {\left (1920 \, d x - 2192 i\right )} e^{\left (2 i \, d x + 2 i \, c\right )} - {\left (-960 i \, e^{\left (12 i \, d x + 12 i \, c\right )} - 4800 i \, e^{\left (10 i \, d x + 10 i \, c\right )} - 9600 i \, e^{\left (8 i \, d x + 8 i \, c\right )} - 9600 i \, e^{\left (6 i \, d x + 6 i \, c\right )} - 4800 i \, e^{\left (4 i \, d x + 4 i \, c\right )} - 960 i \, e^{\left (2 i \, d x + 2 i \, c\right )}\right )} \log \left (e^{\left (2 i \, d x + 2 i \, c\right )} + 1\right ) - 160 i}{5 \, {\left (a^{8} d e^{\left (12 i \, d x + 12 i \, c\right )} + 5 \, a^{8} d e^{\left (10 i \, d x + 10 i \, c\right )} + 10 \, a^{8} d e^{\left (8 i \, d x + 8 i \, c\right )} + 10 \, a^{8} d e^{\left (6 i \, d x + 6 i \, c\right )} + 5 \, a^{8} d e^{\left (4 i \, d x + 4 i \, c\right )} + a^{8} d e^{\left (2 i \, d x + 2 i \, c\right )}\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 7.38, size = 250, normalized size = 1.87 \[ -\frac {2 \, {\left (\frac {480 i \, \log \left (\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 1\right )}{a^{8}} - \frac {960 i \, \log \left (\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - i\right )}{a^{8}} + \frac {480 i \, \log \left (\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 1\right )}{a^{8}} - \frac {5 \, {\left (-288 i \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - 640 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 288 i\right )}}{a^{8} {\left (\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - i\right )}^{2}} + \frac {-1096 i \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{10} + 645 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{9} + 5840 i \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{8} - 2780 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{7} - 12120 i \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{6} + 4286 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} + 12120 i \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{4} - 2780 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} - 5840 i \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + 645 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 1096 i}{{\left (\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - 1\right )}^{5} a^{8}}\right )}}{5 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.50, size = 120, normalized size = 0.90 \[ \frac {129 \tan \left (d x +c \right )}{a^{8} d}+\frac {\tan ^{5}\left (d x +c \right )}{5 a^{8} d}+\frac {2 i \left (\tan ^{4}\left (d x +c \right )\right )}{a^{8} d}-\frac {10 \left (\tan ^{3}\left (d x +c \right )\right )}{a^{8} d}-\frac {36 i \left (\tan ^{2}\left (d x +c \right )\right )}{a^{8} d}+\frac {64}{a^{8} d \left (\tan \left (d x +c \right )-i\right )}+\frac {192 i \ln \left (\tan \left (d x +c \right )-i\right )}{a^{8} d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.44, size = 232, normalized size = 1.73 \[ \frac {\frac {5 \, {\left (2240 \, \tan \left (d x + c\right )^{6} - 13440 i \, \tan \left (d x + c\right )^{5} - 33600 \, \tan \left (d x + c\right )^{4} + 44800 i \, \tan \left (d x + c\right )^{3} + 33600 \, \tan \left (d x + c\right )^{2} - 13440 i \, \tan \left (d x + c\right ) - 2240\right )}}{35 \, a^{8} \tan \left (d x + c\right )^{7} - 245 i \, a^{8} \tan \left (d x + c\right )^{6} - 735 \, a^{8} \tan \left (d x + c\right )^{5} + 1225 i \, a^{8} \tan \left (d x + c\right )^{4} + 1225 \, a^{8} \tan \left (d x + c\right )^{3} - 735 i \, a^{8} \tan \left (d x + c\right )^{2} - 245 \, a^{8} \tan \left (d x + c\right ) + 35 i \, a^{8}} + \frac {\tan \left (d x + c\right )^{5} + 10 i \, \tan \left (d x + c\right )^{4} - 50 \, \tan \left (d x + c\right )^{3} - 180 i \, \tan \left (d x + c\right )^{2} + 645 \, \tan \left (d x + c\right )}{a^{8}} + \frac {960 i \, \log \left (i \, \tan \left (d x + c\right ) + 1\right )}{a^{8}}}{5 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 3.45, size = 105, normalized size = 0.78 \[ \frac {\frac {129\,\mathrm {tan}\left (c+d\,x\right )}{a^8}-\frac {10\,{\mathrm {tan}\left (c+d\,x\right )}^3}{a^8}+\frac {{\mathrm {tan}\left (c+d\,x\right )}^5}{5\,a^8}+\frac {\ln \left (\mathrm {tan}\left (c+d\,x\right )-\mathrm {i}\right )\,192{}\mathrm {i}}{a^8}+\frac {64{}\mathrm {i}}{a^8\,\left (1+\mathrm {tan}\left (c+d\,x\right )\,1{}\mathrm {i}\right )}-\frac {{\mathrm {tan}\left (c+d\,x\right )}^2\,36{}\mathrm {i}}{a^8}+\frac {{\mathrm {tan}\left (c+d\,x\right )}^4\,2{}\mathrm {i}}{a^8}}{d} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
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