Optimal. Leaf size=27 \[ \frac{i}{2 a d (a+i a \tan (c+d x))^2} \]
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Rubi [A] time = 0.0531279, antiderivative size = 27, normalized size of antiderivative = 1., 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} \[ \frac{i}{2 a d (a+i a \tan (c+d x))^2} \]
Antiderivative was successfully verified.
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Rule 3487
Rule 32
Rubi steps
\begin{align*} \int \frac{\sec ^2(c+d x)}{(a+i a \tan (c+d x))^3} \, dx &=-\frac{i \operatorname{Subst}\left (\int \frac{1}{(a+x)^3} \, dx,x,i a \tan (c+d x)\right )}{a d}\\ &=\frac{i}{2 a d (a+i a \tan (c+d x))^2}\\ \end{align*}
Mathematica [A] time = 0.0702105, size = 42, normalized size = 1.56 \[ -\frac{i (\tan (c+d x)-3 i) \sec ^2(c+d x)}{8 a^3 d (\tan (c+d x)-i)^3} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.038, size = 24, normalized size = 0.9 \begin{align*}{\frac{{\frac{i}{2}}}{ad \left ( a+ia\tan \left ( dx+c \right ) \right ) ^{2}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 0.946087, size = 28, normalized size = 1.04 \begin{align*} \frac{i}{2 \,{\left (i \, a \tan \left (d x + c\right ) + a\right )}^{2} a d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 2.22497, size = 86, normalized size = 3.19 \begin{align*} \frac{{\left (2 i \, e^{\left (2 i \, d x + 2 i \, c\right )} + i\right )} e^{\left (-4 i \, d x - 4 i \, c\right )}}{8 \, a^{3} 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 [B] time = 1.16061, size = 77, normalized size = 2.85 \begin{align*} -\frac{2 \,{\left (\tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{3} - i \, \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{2} - \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )\right )}}{a^{3} d{\left (\tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right ) - i\right )}^{4}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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