Optimal. Leaf size=82 \[ \frac {i (a-i a \tan (c+d x))^8}{8 a^{11} d}-\frac {4 i (a-i a \tan (c+d x))^7}{7 a^{10} d}+\frac {2 i (a-i a \tan (c+d x))^6}{3 a^9 d} \]
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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} \[ \frac {i (a-i a \tan (c+d x))^8}{8 a^{11} d}-\frac {4 i (a-i a \tan (c+d x))^7}{7 a^{10} d}+\frac {2 i (a-i a \tan (c+d x))^6}{3 a^9 d} \]
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))^3} \, dx &=-\frac {i \operatorname {Subst}\left (\int (a-x)^5 (a+x)^2 \, dx,x,i a \tan (c+d x)\right )}{a^{11} d}\\ &=-\frac {i \operatorname {Subst}\left (\int \left (4 a^2 (a-x)^5-4 a (a-x)^6+(a-x)^7\right ) \, dx,x,i a \tan (c+d x)\right )}{a^{11} d}\\ &=\frac {2 i (a-i a \tan (c+d x))^6}{3 a^9 d}-\frac {4 i (a-i a \tan (c+d x))^7}{7 a^{10} d}+\frac {i (a-i a \tan (c+d x))^8}{8 a^{11} d}\\ \end {align*}
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Mathematica [A] time = 0.66, size = 106, normalized size = 1.29 \[ \frac {\sec (c) \sec ^8(c+d x) (28 \sin (c+2 d x)-28 \sin (3 c+2 d x)+28 \sin (3 c+4 d x)+8 \sin (5 c+6 d x)+\sin (7 c+8 d x)-28 i \cos (c+2 d x)-28 i \cos (3 c+2 d x)-35 \sin (c)-35 i \cos (c))}{168 a^3 d} \]
Antiderivative was successfully verified.
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fricas [B] time = 0.58, size = 153, normalized size = 1.87 \[ \frac {896 i \, e^{\left (4 i \, d x + 4 i \, c\right )} + 256 i \, e^{\left (2 i \, d x + 2 i \, c\right )} + 32 i}{21 \, {\left (a^{3} d e^{\left (16 i \, d x + 16 i \, c\right )} + 8 \, a^{3} d e^{\left (14 i \, d x + 14 i \, c\right )} + 28 \, a^{3} d e^{\left (12 i \, d x + 12 i \, c\right )} + 56 \, a^{3} d e^{\left (10 i \, d x + 10 i \, c\right )} + 70 \, a^{3} d e^{\left (8 i \, d x + 8 i \, c\right )} + 56 \, a^{3} d e^{\left (6 i \, d x + 6 i \, c\right )} + 28 \, a^{3} d e^{\left (4 i \, d x + 4 i \, c\right )} + 8 \, a^{3} d e^{\left (2 i \, d x + 2 i \, c\right )} + a^{3} d\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 1.72, size = 87, normalized size = 1.06 \[ -\frac {-21 i \, \tan \left (d x + c\right )^{8} + 72 \, \tan \left (d x + c\right )^{7} + 28 i \, \tan \left (d x + c\right )^{6} + 168 \, \tan \left (d x + c\right )^{5} + 210 i \, \tan \left (d x + c\right )^{4} + 56 \, \tan \left (d x + c\right )^{3} + 252 i \, \tan \left (d x + c\right )^{2} - 168 \, \tan \left (d x + c\right )}{168 \, a^{3} d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.39, size = 89, normalized size = 1.09 \[ \frac {\tan \left (d x +c \right )+\frac {i \left (\tan ^{8}\left (d x +c \right )\right )}{8}-\frac {3 \left (\tan ^{7}\left (d x +c \right )\right )}{7}-\frac {i \left (\tan ^{6}\left (d x +c \right )\right )}{6}-\left (\tan ^{5}\left (d x +c \right )\right )-\frac {5 i \left (\tan ^{4}\left (d x +c \right )\right )}{4}-\frac {\left (\tan ^{3}\left (d x +c \right )\right )}{3}-\frac {3 i \left (\tan ^{2}\left (d x +c \right )\right )}{2}}{d \,a^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.44, size = 87, normalized size = 1.06 \[ -\frac {-21 i \, \tan \left (d x + c\right )^{8} + 72 \, \tan \left (d x + c\right )^{7} + 28 i \, \tan \left (d x + c\right )^{6} + 168 \, \tan \left (d x + c\right )^{5} + 210 i \, \tan \left (d x + c\right )^{4} + 56 \, \tan \left (d x + c\right )^{3} + 252 i \, \tan \left (d x + c\right )^{2} - 168 \, \tan \left (d x + c\right )}{168 \, a^{3} d} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 3.49, size = 103, normalized size = 1.26 \[ \frac {{\cos \left (c+d\,x\right )}^8\,91{}\mathrm {i}+128\,\sin \left (c+d\,x\right )\,{\cos \left (c+d\,x\right )}^7+64\,\sin \left (c+d\,x\right )\,{\cos \left (c+d\,x\right )}^5+48\,\sin \left (c+d\,x\right )\,{\cos \left (c+d\,x\right )}^3-{\cos \left (c+d\,x\right )}^2\,112{}\mathrm {i}-72\,\sin \left (c+d\,x\right )\,\cos \left (c+d\,x\right )+21{}\mathrm {i}}{168\,a^3\,d\,{\cos \left (c+d\,x\right )}^8} \]
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 \[ \frac {i \int \frac {\sec ^{12}{\left (c + d x \right )}}{\tan ^{3}{\left (c + d x \right )} - 3 i \tan ^{2}{\left (c + d x \right )} - 3 \tan {\left (c + d x \right )} + i}\, dx}{a^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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