Optimal. Leaf size=27 \[ \frac {i (a-i a \tan (c+d x))^4}{4 a^7 d} \]
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Rubi [A] time = 0.04, antiderivative size = 27, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.083, Rules used = {3487, 32} \[ \frac {i (a-i a \tan (c+d x))^4}{4 a^7 d} \]
Antiderivative was successfully verified.
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Rule 32
Rule 3487
Rubi steps
\begin {align*} \int \frac {\sec ^8(c+d x)}{(a+i a \tan (c+d x))^3} \, dx &=-\frac {i \operatorname {Subst}\left (\int (a-x)^3 \, dx,x,i a \tan (c+d x)\right )}{a^7 d}\\ &=\frac {i (a-i a \tan (c+d x))^4}{4 a^7 d}\\ \end {align*}
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Mathematica [B] time = 0.47, size = 84, normalized size = 3.11 \[ \frac {\sec (c) \sec ^4(c+d x) (2 \sin (c+2 d x)-2 \sin (3 c+2 d x)+\sin (3 c+4 d x)-2 i \cos (c+2 d x)-2 i \cos (3 c+2 d x)-3 \sin (c)-3 i \cos (c))}{4 a^3 d} \]
Antiderivative was successfully verified.
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fricas [B] time = 0.58, size = 69, normalized size = 2.56 \[ \frac {4 i}{a^{3} d e^{\left (8 i \, d x + 8 i \, c\right )} + 4 \, a^{3} d e^{\left (6 i \, d x + 6 i \, c\right )} + 6 \, a^{3} d e^{\left (4 i \, d x + 4 i \, c\right )} + 4 \, a^{3} d e^{\left (2 i \, d x + 2 i \, c\right )} + a^{3} d} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 1.43, size = 47, normalized size = 1.74 \[ -\frac {-i \, \tan \left (d x + c\right )^{4} + 4 \, \tan \left (d x + c\right )^{3} + 6 i \, \tan \left (d x + c\right )^{2} - 4 \, \tan \left (d x + c\right )}{4 \, a^{3} d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.39, size = 47, normalized size = 1.74 \[ \frac {\tan \left (d x +c \right )+\frac {i \left (\tan ^{4}\left (d x +c \right )\right )}{4}-\left (\tan ^{3}\left (d x +c \right )\right )-\frac {3 i \left (\tan ^{2}\left (d x +c \right )\right )}{2}}{d \,a^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [B] time = 0.36, size = 47, normalized size = 1.74 \[ -\frac {-i \, \tan \left (d x + c\right )^{4} + 4 \, \tan \left (d x + c\right )^{3} + 6 i \, \tan \left (d x + c\right )^{2} - 4 \, \tan \left (d x + c\right )}{4 \, a^{3} d} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 3.34, size = 77, normalized size = 2.85 \[ -\frac {\sin \left (c+d\,x\right )\,\left (-4\,{\cos \left (c+d\,x\right )}^3+{\cos \left (c+d\,x\right )}^2\,\sin \left (c+d\,x\right )\,6{}\mathrm {i}+4\,\cos \left (c+d\,x\right )\,{\sin \left (c+d\,x\right )}^2-{\sin \left (c+d\,x\right )}^3\,1{}\mathrm {i}\right )}{4\,a^3\,d\,{\cos \left (c+d\,x\right )}^4} \]
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 \[ \frac {i \int \frac {\sec ^{8}{\left (c + d x \right )}}{\tan ^{3}{\left (c + d x \right )} - 3 i \tan ^{2}{\left (c + d x \right )} - 3 \tan {\left (c + d x \right )} + i}\, dx}{a^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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