Optimal. Leaf size=82 \[ \frac {i (a-i a \tan (c+d x))^9}{9 a^{13} d}-\frac {i (a-i a \tan (c+d x))^8}{2 a^{12} d}+\frac {4 i (a-i a \tan (c+d x))^7}{7 a^{11} d} \]
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Rubi [A] time = 0.06, antiderivative size = 82, 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} \[ \frac {i (a-i a \tan (c+d x))^9}{9 a^{13} d}-\frac {i (a-i a \tan (c+d x))^8}{2 a^{12} d}+\frac {4 i (a-i a \tan (c+d x))^7}{7 a^{11} d} \]
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))^4} \, dx &=-\frac {i \operatorname {Subst}\left (\int (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 (4 a^2 (a-x)^6-4 a (a-x)^7+(a-x)^8\right ) \, dx,x,i a \tan (c+d x)\right )}{a^{13} d}\\ &=\frac {4 i (a-i a \tan (c+d x))^7}{7 a^{11} d}-\frac {i (a-i a \tan (c+d x))^8}{2 a^{12} d}+\frac {i (a-i a \tan (c+d x))^9}{9 a^{13} d}\\ \end {align*}
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Mathematica [A] time = 0.79, size = 136, normalized size = 1.66 \[ \frac {\sec (c) \sec ^9(c+d x) (-63 \sin (2 c+d x)+42 \sin (2 c+3 d x)-42 \sin (4 c+3 d x)+36 \sin (4 c+5 d x)+9 \sin (6 c+7 d x)+\sin (8 c+9 d x)-63 i \cos (2 c+d x)-42 i \cos (2 c+3 d x)-42 i \cos (4 c+3 d x)+63 \sin (d x)-63 i \cos (d x))}{252 a^4 d} \]
Antiderivative was successfully verified.
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fricas [B] time = 0.54, size = 168, normalized size = 2.05 \[ \frac {4608 i \, e^{\left (4 i \, d x + 4 i \, c\right )} + 1152 i \, e^{\left (2 i \, d x + 2 i \, c\right )} + 128 i}{63 \, {\left (a^{4} d e^{\left (18 i \, d x + 18 i \, c\right )} + 9 \, a^{4} d e^{\left (16 i \, d x + 16 i \, c\right )} + 36 \, a^{4} d e^{\left (14 i \, d x + 14 i \, c\right )} + 84 \, a^{4} d e^{\left (12 i \, d x + 12 i \, c\right )} + 126 \, a^{4} d e^{\left (10 i \, d x + 10 i \, c\right )} + 126 \, a^{4} d e^{\left (8 i \, d x + 8 i \, c\right )} + 84 \, a^{4} d e^{\left (6 i \, d x + 6 i \, c\right )} + 36 \, a^{4} d e^{\left (4 i \, d x + 4 i \, c\right )} + 9 \, a^{4} d e^{\left (2 i \, d x + 2 i \, c\right )} + a^{4} d\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 3.64, size = 97, normalized size = 1.18 \[ \frac {14 \, \tan \left (d x + c\right )^{9} + 63 i \, \tan \left (d x + c\right )^{8} - 72 \, \tan \left (d x + c\right )^{7} + 84 i \, \tan \left (d x + c\right )^{6} - 252 \, \tan \left (d x + c\right )^{5} - 126 i \, \tan \left (d x + c\right )^{4} - 168 \, \tan \left (d x + c\right )^{3} - 252 i \, \tan \left (d x + c\right )^{2} + 126 \, \tan \left (d x + c\right )}{126 \, a^{4} d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.42, size = 99, normalized size = 1.21 \[ \frac {\tan \left (d x +c \right )+\frac {\left (\tan ^{9}\left (d x +c \right )\right )}{9}+\frac {i \left (\tan ^{8}\left (d x +c \right )\right )}{2}-\frac {4 \left (\tan ^{7}\left (d x +c \right )\right )}{7}+\frac {2 i \left (\tan ^{6}\left (d x +c \right )\right )}{3}-2 \left (\tan ^{5}\left (d x +c \right )\right )-i \left (\tan ^{4}\left (d x +c \right )\right )-\frac {4 \left (\tan ^{3}\left (d x +c \right )\right )}{3}-2 i \left (\tan ^{2}\left (d x +c \right )\right )}{d \,a^{4}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.39, size = 97, normalized size = 1.18 \[ \frac {14 \, \tan \left (d x + c\right )^{9} + 63 i \, \tan \left (d x + c\right )^{8} - 72 \, \tan \left (d x + c\right )^{7} + 84 i \, \tan \left (d x + c\right )^{6} - 252 \, \tan \left (d x + c\right )^{5} - 126 i \, \tan \left (d x + c\right )^{4} - 168 \, \tan \left (d x + c\right )^{3} - 252 i \, \tan \left (d x + c\right )^{2} + 126 \, \tan \left (d x + c\right )}{126 \, a^{4} d} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 3.52, size = 120, normalized size = 1.46 \[ \frac {{\cos \left (c+d\,x\right )}^9\,105{}\mathrm {i}+128\,\sin \left (c+d\,x\right )\,{\cos \left (c+d\,x\right )}^8+64\,\sin \left (c+d\,x\right )\,{\cos \left (c+d\,x\right )}^6+48\,\sin \left (c+d\,x\right )\,{\cos \left (c+d\,x\right )}^4-{\cos \left (c+d\,x\right )}^3\,168{}\mathrm {i}-128\,\sin \left (c+d\,x\right )\,{\cos \left (c+d\,x\right )}^2+\cos \left (c+d\,x\right )\,63{}\mathrm {i}+14\,\sin \left (c+d\,x\right )}{126\,a^4\,d\,{\cos \left (c+d\,x\right )}^9} \]
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 \[ \frac {\int \frac {\sec ^{14}{\left (c + d x \right )}}{\tan ^{4}{\left (c + d x \right )} - 4 i \tan ^{3}{\left (c + d x \right )} - 6 \tan ^{2}{\left (c + d x \right )} + 4 i \tan {\left (c + d x \right )} + 1}\, dx}{a^{4}} \]
Verification of antiderivative is not currently implemented for this CAS.
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