Optimal. Leaf size=34 \[ -\frac{i \tan ^3(c+d x) (-\cot (c+d x)+i)^3}{3 a^2 d} \]
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Rubi [A] time = 0.0636908, antiderivative size = 34, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 31, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.097, Rules used = {3088, 848, 37} \[ -\frac{i \tan ^3(c+d x) (-\cot (c+d x)+i)^3}{3 a^2 d} \]
Antiderivative was successfully verified.
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Rule 3088
Rule 848
Rule 37
Rubi steps
\begin{align*} \int \frac{\sec ^4(c+d x)}{(a \cos (c+d x)+i a \sin (c+d x))^2} \, dx &=-\frac{\operatorname{Subst}\left (\int \frac{\left (1+x^2\right )^2}{x^4 (i a+a x)^2} \, dx,x,\cot (c+d x)\right )}{d}\\ &=-\frac{\operatorname{Subst}\left (\int \frac{\left (-\frac{i}{a}+\frac{x}{a}\right )^2}{x^4} \, dx,x,\cot (c+d x)\right )}{d}\\ &=-\frac{i (i-\cot (c+d x))^3 \tan ^3(c+d x)}{3 a^2 d}\\ \end{align*}
Mathematica [A] time = 0.251387, size = 68, normalized size = 2. \[ -\frac{\sec (c) \sec ^3(c+d x) (3 \sin (2 c+d x)-2 \sin (2 c+3 d x)+3 i \cos (2 c+d x)-3 \sin (d x)+3 i \cos (d x))}{6 a^2 d} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.172, size = 36, normalized size = 1.1 \begin{align*}{\frac{1}{d{a}^{2}} \left ( \tan \left ( dx+c \right ) -{\frac{ \left ( \tan \left ( dx+c \right ) \right ) ^{3}}{3}}-i \left ( \tan \left ( dx+c \right ) \right ) ^{2} \right ) } \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.00289, size = 47, normalized size = 1.38 \begin{align*} -\frac{\tan \left (d x + c\right )^{3} + 3 i \, \tan \left (d x + c\right )^{2} - 3 \, \tan \left (d x + c\right )}{3 \, a^{2} d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] time = 0.447857, size = 139, normalized size = 4.09 \begin{align*} \frac{8 i}{3 \,{\left (a^{2} d e^{\left (6 i \, d x + 6 i \, c\right )} + 3 \, a^{2} d e^{\left (4 i \, d x + 4 i \, c\right )} + 3 \, a^{2} d e^{\left (2 i \, d x + 2 i \, c\right )} + a^{2} d\right )}} \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 = 1.10297, size = 47, normalized size = 1.38 \begin{align*} -\frac{\tan \left (d x + c\right )^{3} + 3 i \, \tan \left (d x + c\right )^{2} - 3 \, \tan \left (d x + c\right )}{3 \, a^{2} d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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