Optimal. Leaf size=34 \[ \frac {\tan (c+d x)}{a d}-\frac {i \tan ^2(c+d x)}{2 a d} \]
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Rubi [A] time = 0.04, antiderivative size = 34, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 1, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.042, Rules used = {3487} \[ \frac {\tan (c+d x)}{a d}-\frac {i \tan ^2(c+d x)}{2 a d} \]
Antiderivative was successfully verified.
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Rule 3487
Rubi steps
\begin {align*} \int \frac {\sec ^4(c+d x)}{a+i a \tan (c+d x)} \, dx &=-\frac {i \operatorname {Subst}(\int (a-x) \, dx,x,i a \tan (c+d x))}{a^3 d}\\ &=\frac {\tan (c+d x)}{a d}-\frac {i \tan ^2(c+d x)}{2 a d}\\ \end {align*}
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Mathematica [A] time = 0.21, size = 35, normalized size = 1.03 \[ \frac {\sec (c+d x) (2 \sec (c) \sin (d x)-i \sec (c+d x))}{2 a d} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.54, size = 33, normalized size = 0.97 \[ \frac {2 i}{a d e^{\left (4 i \, d x + 4 i \, c\right )} + 2 \, a d e^{\left (2 i \, d x + 2 i \, c\right )} + a d} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 1.42, size = 27, normalized size = 0.79 \[ -\frac {i \, \tan \left (d x + c\right )^{2} - 2 \, \tan \left (d x + c\right )}{2 \, a d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.35, size = 26, normalized size = 0.76 \[ \frac {\tan \left (d x +c \right )-\frac {i \left (\tan ^{2}\left (d x +c \right )\right )}{2}}{d a} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.31, size = 27, normalized size = 0.79 \[ -\frac {i \, \tan \left (d x + c\right )^{2} - 2 \, \tan \left (d x + c\right )}{2 \, a d} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 3.33, size = 25, normalized size = 0.74 \[ -\frac {\mathrm {tan}\left (c+d\,x\right )\,\left (-2+\mathrm {tan}\left (c+d\,x\right )\,1{}\mathrm {i}\right )}{2\,a\,d} \]
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 \[ - \frac {i \int \frac {\sec ^{4}{\left (c + d x \right )}}{\tan {\left (c + d x \right )} - i}\, dx}{a} \]
Verification of antiderivative is not currently implemented for this CAS.
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