Optimal. Leaf size=27 \[ \frac{i}{3 a d (a+i a \tan (c+d x))^3} \]
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Rubi [A] time = 0.039359, 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} \[ \frac{i}{3 a d (a+i a \tan (c+d x))^3} \]
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))^4} \, dx &=-\frac{i \operatorname{Subst}\left (\int \frac{1}{(a+x)^4} \, dx,x,i a \tan (c+d x)\right )}{a d}\\ &=\frac{i}{3 a d (a+i a \tan (c+d x))^3}\\ \end{align*}
Mathematica [B] time = 0.143224, size = 56, normalized size = 2.07 \[ \frac{i \sec ^4(c+d x) (2 i \sin (2 (c+d x))+4 \cos (2 (c+d x))+3)}{24 a^4 d (\tan (c+d x)-i)^4} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.039, size = 24, normalized size = 0.9 \begin{align*}{\frac{{\frac{i}{3}}}{ad \left ( a+ia\tan \left ( dx+c \right ) \right ) ^{3}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 0.963541, size = 28, normalized size = 1.04 \begin{align*} \frac{i}{3 \,{\left (i \, a \tan \left (d x + c\right ) + a\right )}^{3} a d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 2.38951, size = 123, normalized size = 4.56 \begin{align*} \frac{{\left (3 i \, e^{\left (4 i \, d x + 4 i \, c\right )} + 3 i \, e^{\left (2 i \, d x + 2 i \, c\right )} + i\right )} e^{\left (-6 i \, d x - 6 i \, c\right )}}{24 \, a^{4} 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 = 1.19635, size = 115, normalized size = 4.26 \begin{align*} -\frac{2 \,{\left (3 \, \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{5} - 6 i \, \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{4} - 10 \, \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{3} + 6 i \, \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{2} + 3 \, \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )\right )}}{3 \, a^{4} d{\left (\tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right ) - i\right )}^{6}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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