Optimal. Leaf size=107 \[ -\frac {i (a-i a \tan (c+d x))^8}{8 a^9 d}+\frac {6 i (a-i a \tan (c+d x))^7}{7 a^8 d}-\frac {2 i (a-i a \tan (c+d x))^6}{a^7 d}+\frac {8 i (a-i a \tan (c+d x))^5}{5 a^6 d} \]
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Rubi [A] time = 0.07, antiderivative size = 107, 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} \[ -\frac {i (a-i a \tan (c+d x))^8}{8 a^9 d}+\frac {6 i (a-i a \tan (c+d x))^7}{7 a^8 d}-\frac {2 i (a-i a \tan (c+d x))^6}{a^7 d}+\frac {8 i (a-i a \tan (c+d x))^5}{5 a^6 d} \]
Antiderivative was successfully verified.
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Rule 43
Rule 3487
Rubi steps
\begin {align*} \int \frac {\sec ^{10}(c+d x)}{a+i a \tan (c+d x)} \, dx &=-\frac {i \operatorname {Subst}\left (\int (a-x)^4 (a+x)^3 \, dx,x,i a \tan (c+d x)\right )}{a^9 d}\\ &=-\frac {i \operatorname {Subst}\left (\int \left (8 a^3 (a-x)^4-12 a^2 (a-x)^5+6 a (a-x)^6-(a-x)^7\right ) \, dx,x,i a \tan (c+d x)\right )}{a^9 d}\\ &=\frac {8 i (a-i a \tan (c+d x))^5}{5 a^6 d}-\frac {2 i (a-i a \tan (c+d x))^6}{a^7 d}+\frac {6 i (a-i a \tan (c+d x))^7}{7 a^8 d}-\frac {i (a-i a \tan (c+d x))^8}{8 a^9 d}\\ \end {align*}
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Mathematica [A] time = 0.49, size = 71, normalized size = 0.66 \[ \frac {\sec (c) \sec ^8(c+d x) (56 \sin (c+2 d x)+28 \sin (3 c+4 d x)+8 \sin (5 c+6 d x)+\sin (7 c+8 d x)-35 \sin (c)-35 i \cos (c))}{280 a d} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.63, size = 146, normalized size = 1.36 \[ \frac {1792 i \, e^{\left (6 i \, d x + 6 i \, c\right )} + 896 i \, e^{\left (4 i \, d x + 4 i \, c\right )} + 256 i \, e^{\left (2 i \, d x + 2 i \, c\right )} + 32 i}{35 \, {\left (a d e^{\left (16 i \, d x + 16 i \, c\right )} + 8 \, a d e^{\left (14 i \, d x + 14 i \, c\right )} + 28 \, a d e^{\left (12 i \, d x + 12 i \, c\right )} + 56 \, a d e^{\left (10 i \, d x + 10 i \, c\right )} + 70 \, a d e^{\left (8 i \, d x + 8 i \, c\right )} + 56 \, a d e^{\left (6 i \, d x + 6 i \, c\right )} + 28 \, a d e^{\left (4 i \, d x + 4 i \, c\right )} + 8 \, a d e^{\left (2 i \, d x + 2 i \, c\right )} + a d\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.69, size = 87, normalized size = 0.81 \[ -\frac {35 i \, \tan \left (d x + c\right )^{8} - 40 \, \tan \left (d x + c\right )^{7} + 140 i \, \tan \left (d x + c\right )^{6} - 168 \, \tan \left (d x + c\right )^{5} + 210 i \, \tan \left (d x + c\right )^{4} - 280 \, \tan \left (d x + c\right )^{3} + 140 i \, \tan \left (d x + c\right )^{2} - 280 \, \tan \left (d x + c\right )}{280 \, a d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.39, size = 87, normalized size = 0.81 \[ \frac {\tan \left (d x +c \right )-\frac {i \left (\tan ^{8}\left (d x +c \right )\right )}{8}+\frac {\left (\tan ^{7}\left (d x +c \right )\right )}{7}-\frac {i \left (\tan ^{6}\left (d x +c \right )\right )}{2}+\frac {3 \left (\tan ^{5}\left (d x +c \right )\right )}{5}-\frac {3 i \left (\tan ^{4}\left (d x +c \right )\right )}{4}+\tan ^{3}\left (d x +c \right )-\frac {i \left (\tan ^{2}\left (d x +c \right )\right )}{2}}{d a} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.42, size = 87, normalized size = 0.81 \[ \frac {-105 i \, \tan \left (d x + c\right )^{8} + 120 \, \tan \left (d x + c\right )^{7} - 420 i \, \tan \left (d x + c\right )^{6} + 504 \, \tan \left (d x + c\right )^{5} - 630 i \, \tan \left (d x + c\right )^{4} + 840 \, \tan \left (d x + c\right )^{3} - 420 i \, \tan \left (d x + c\right )^{2} + 840 \, \tan \left (d x + c\right )}{840 \, a d} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 3.53, size = 92, normalized size = 0.86 \[ \frac {{\cos \left (c+d\,x\right )}^8\,35{}\mathrm {i}+128\,\sin \left (c+d\,x\right )\,{\cos \left (c+d\,x\right )}^7+64\,\sin \left (c+d\,x\right )\,{\cos \left (c+d\,x\right )}^5+48\,\sin \left (c+d\,x\right )\,{\cos \left (c+d\,x\right )}^3+40\,\sin \left (c+d\,x\right )\,\cos \left (c+d\,x\right )-35{}\mathrm {i}}{280\,a\,d\,{\cos \left (c+d\,x\right )}^8} \]
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 \[ - \frac {i \int \frac {\sec ^{10}{\left (c + d x \right )}}{\tan {\left (c + d x \right )} - i}\, dx}{a} \]
Verification of antiderivative is not currently implemented for this CAS.
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