Optimal. Leaf size=29 \[ \frac{\tan (c+d x)}{d \left (a^2+i a^2 \tan (c+d x)\right )^2} \]
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Rubi [A] time = 0.0412489, antiderivative size = 29, 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, 34} \[ \frac{\tan (c+d x)}{d \left (a^2+i a^2 \tan (c+d x)\right )^2} \]
Antiderivative was successfully verified.
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Rule 3487
Rule 34
Rubi steps
\begin{align*} \int \frac{\sec ^4(c+d x)}{(a+i a \tan (c+d x))^4} \, dx &=-\frac{i \operatorname{Subst}\left (\int \frac{a-x}{(a+x)^3} \, dx,x,i a \tan (c+d x)\right )}{a^3 d}\\ &=\frac{\tan (c+d x)}{d \left (a^2+i a^2 \tan (c+d x)\right )^2}\\ \end{align*}
Mathematica [A] time = 0.0511495, size = 32, normalized size = 1.1 \[ \frac{i \sec ^4(c+d x)}{4 d (a+i a \tan (c+d x))^4} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.084, size = 36, normalized size = 1.2 \begin{align*}{\frac{1}{{a}^{4}d} \left ({\frac{-i}{ \left ( \tan \left ( dx+c \right ) -i \right ) ^{2}}}- \left ( \tan \left ( dx+c \right ) -i \right ) ^{-1} \right ) } \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [B] time = 1.00374, size = 90, normalized size = 3.1 \begin{align*} -\frac{3 \,{\left (\tan \left (d x + c\right )^{2} - i \, \tan \left (d x + c\right )\right )}}{{\left (3 \, a^{4} \tan \left (d x + c\right )^{3} - 9 i \, a^{4} \tan \left (d x + c\right )^{2} - 9 \, a^{4} \tan \left (d x + c\right ) + 3 i \, a^{4}\right )} d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 2.37237, size = 49, normalized size = 1.69 \begin{align*} \frac{i \, e^{\left (-4 i \, d x - 4 i \, c\right )}}{4 \, 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 [A] time = 1.17843, size = 59, normalized size = 2.03 \begin{align*} -\frac{2 \,{\left (\tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{3} - \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )\right )}}{a^{4} 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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