Optimal. Leaf size=109 \[ -\frac {i (a-i a \tan (c+d x))^{10}}{10 a^{13} d}+\frac {2 i (a-i a \tan (c+d x))^9}{3 a^{12} d}-\frac {3 i (a-i a \tan (c+d x))^8}{2 a^{11} d}+\frac {8 i (a-i a \tan (c+d x))^7}{7 a^{10} d} \]
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Rubi [A] time = 0.07, antiderivative size = 109, 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} \[ -\frac {i (a-i a \tan (c+d x))^{10}}{10 a^{13} d}+\frac {2 i (a-i a \tan (c+d x))^9}{3 a^{12} d}-\frac {3 i (a-i a \tan (c+d x))^8}{2 a^{11} d}+\frac {8 i (a-i a \tan (c+d x))^7}{7 a^{10} d} \]
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))^3} \, dx &=-\frac {i \operatorname {Subst}\left (\int (a-x)^6 (a+x)^3 \, dx,x,i a \tan (c+d x)\right )}{a^{13} d}\\ &=-\frac {i \operatorname {Subst}\left (\int \left (8 a^3 (a-x)^6-12 a^2 (a-x)^7+6 a (a-x)^8-(a-x)^9\right ) \, dx,x,i a \tan (c+d x)\right )}{a^{13} d}\\ &=\frac {8 i (a-i a \tan (c+d x))^7}{7 a^{10} d}-\frac {3 i (a-i a \tan (c+d x))^8}{2 a^{11} d}+\frac {2 i (a-i a \tan (c+d x))^9}{3 a^{12} d}-\frac {i (a-i a \tan (c+d x))^{10}}{10 a^{13} d}\\ \end {align*}
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Mathematica [A] time = 1.02, size = 117, normalized size = 1.07 \[ \frac {\sec (c) \sec ^{10}(c+d x) (105 \sin (c+2 d x)-105 \sin (3 c+2 d x)+120 \sin (3 c+4 d x)+45 \sin (5 c+6 d x)+10 \sin (7 c+8 d x)+\sin (9 c+10 d x)-105 i \cos (c+2 d x)-105 i \cos (3 c+2 d x)-126 \sin (c)-126 i \cos (c))}{840 a^3 d} \]
Antiderivative was successfully verified.
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fricas [B] time = 0.56, size = 194, normalized size = 1.78 \[ \frac {15360 i \, e^{\left (6 i \, d x + 6 i \, c\right )} + 5760 i \, e^{\left (4 i \, d x + 4 i \, c\right )} + 1280 i \, e^{\left (2 i \, d x + 2 i \, c\right )} + 128 i}{105 \, {\left (a^{3} d e^{\left (20 i \, d x + 20 i \, c\right )} + 10 \, a^{3} d e^{\left (18 i \, d x + 18 i \, c\right )} + 45 \, a^{3} d e^{\left (16 i \, d x + 16 i \, c\right )} + 120 \, a^{3} d e^{\left (14 i \, d x + 14 i \, c\right )} + 210 \, a^{3} d e^{\left (12 i \, d x + 12 i \, c\right )} + 252 \, a^{3} d e^{\left (10 i \, d x + 10 i \, c\right )} + 210 \, a^{3} d e^{\left (8 i \, d x + 8 i \, c\right )} + 120 \, a^{3} d e^{\left (6 i \, d x + 6 i \, c\right )} + 45 \, a^{3} d e^{\left (4 i \, d x + 4 i \, c\right )} + 10 \, a^{3} d e^{\left (2 i \, d x + 2 i \, c\right )} + a^{3} d\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 2.36, size = 87, normalized size = 0.80 \[ -\frac {-21 i \, \tan \left (d x + c\right )^{10} + 70 \, \tan \left (d x + c\right )^{9} + 240 \, \tan \left (d x + c\right )^{7} + 210 i \, \tan \left (d x + c\right )^{6} + 252 \, \tan \left (d x + c\right )^{5} + 420 i \, \tan \left (d x + c\right )^{4} + 315 i \, \tan \left (d x + c\right )^{2} - 210 \, \tan \left (d x + c\right )}{210 \, a^{3} d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.45, size = 89, normalized size = 0.82 \[ \frac {\tan \left (d x +c \right )+\frac {i \left (\tan ^{10}\left (d x +c \right )\right )}{10}-\frac {\left (\tan ^{9}\left (d x +c \right )\right )}{3}-\frac {8 \left (\tan ^{7}\left (d x +c \right )\right )}{7}-i \left (\tan ^{6}\left (d x +c \right )\right )-\frac {6 \left (\tan ^{5}\left (d x +c \right )\right )}{5}-2 i \left (\tan ^{4}\left (d x +c \right )\right )-\frac {3 i \left (\tan ^{2}\left (d x +c \right )\right )}{2}}{d \,a^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.45, size = 87, normalized size = 0.80 \[ \frac {42 i \, \tan \left (d x + c\right )^{10} - 140 \, \tan \left (d x + c\right )^{9} - 480 \, \tan \left (d x + c\right )^{7} - 420 i \, \tan \left (d x + c\right )^{6} - 504 \, \tan \left (d x + c\right )^{5} - 840 i \, \tan \left (d x + c\right )^{4} - 630 i \, \tan \left (d x + c\right )^{2} + 420 \, \tan \left (d x + c\right )}{420 \, a^{3} d} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 3.60, size = 119, normalized size = 1.09 \[ \frac {{\cos \left (c+d\,x\right )}^{10}\,84{}\mathrm {i}+128\,\sin \left (c+d\,x\right )\,{\cos \left (c+d\,x\right )}^9+64\,\sin \left (c+d\,x\right )\,{\cos \left (c+d\,x\right )}^7+48\,\sin \left (c+d\,x\right )\,{\cos \left (c+d\,x\right )}^5+40\,\sin \left (c+d\,x\right )\,{\cos \left (c+d\,x\right )}^3-{\cos \left (c+d\,x\right )}^2\,105{}\mathrm {i}-70\,\sin \left (c+d\,x\right )\,\cos \left (c+d\,x\right )+21{}\mathrm {i}}{210\,a^3\,d\,{\cos \left (c+d\,x\right )}^{10}} \]
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 \[ \frac {i \int \frac {\sec ^{14}{\left (c + d x \right )}}{\tan ^{3}{\left (c + d x \right )} - 3 i \tan ^{2}{\left (c + d x \right )} - 3 \tan {\left (c + d x \right )} + i}\, dx}{a^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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