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 = 0.0431182, antiderivative size = 34, normalized size of antiderivative = 1., 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*}
Mathematica [A] time = 0.163767, 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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Maple [A] time = 0.054, size = 26, normalized size = 0.8 \begin{align*}{\frac{\tan \left ( dx+c \right ) -{\frac{i}{2}} \left ( \tan \left ( dx+c \right ) \right ) ^{2}}{ad}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.13882, size = 36, normalized size = 1.06 \begin{align*} -\frac{i \, \tan \left (d x + c\right )^{2} - 2 \, \tan \left (d x + c\right )}{2 \, a d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.78664, size = 88, normalized size = 2.59 \begin{align*} \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} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F(-2)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: AttributeError} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.15863, size = 36, normalized size = 1.06 \begin{align*} -\frac{i \, \tan \left (d x + c\right )^{2} - 2 \, \tan \left (d x + c\right )}{2 \, a d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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