Optimal. Leaf size=50 \[ \frac{2 i}{d \left (a^3+i a^3 \tan (c+d x)\right )}-\frac{i \log (\cos (c+d x))}{a^3 d}-\frac{x}{a^3} \]
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Rubi [A] time = 0.0486496, antiderivative size = 50, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.083, Rules used = {3487, 43} \[ \frac{2 i}{d \left (a^3+i a^3 \tan (c+d x)\right )}-\frac{i \log (\cos (c+d x))}{a^3 d}-\frac{x}{a^3} \]
Antiderivative was successfully verified.
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Rule 3487
Rule 43
Rubi steps
\begin{align*} \int \frac{\sec ^4(c+d x)}{(a+i a \tan (c+d x))^3} \, dx &=-\frac{i \operatorname{Subst}\left (\int \frac{a-x}{(a+x)^2} \, dx,x,i a \tan (c+d x)\right )}{a^3 d}\\ &=-\frac{i \operatorname{Subst}\left (\int \left (\frac{1}{-a-x}+\frac{2 a}{(a+x)^2}\right ) \, dx,x,i a \tan (c+d x)\right )}{a^3 d}\\ &=-\frac{x}{a^3}-\frac{i \log (\cos (c+d x))}{a^3 d}+\frac{2 i}{d \left (a^3+i a^3 \tan (c+d x)\right )}\\ \end{align*}
Mathematica [A] time = 0.214589, size = 88, normalized size = 1.76 \[ \frac{\sec ^2(c+d x) (\cos (2 (c+d x))+i \sin (2 (c+d x))) (\log (\cos (c+d x))+\tan (c+d x) (i \log (\cos (c+d x))+d x+i)-i d x-1)}{a^3 d (\tan (c+d x)-i)^3} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.08, size = 40, normalized size = 0.8 \begin{align*}{\frac{i\ln \left ( \tan \left ( dx+c \right ) -i \right ) }{d{a}^{3}}}+2\,{\frac{1}{d{a}^{3} \left ( \tan \left ( dx+c \right ) -i \right ) }} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 0.98214, size = 89, normalized size = 1.78 \begin{align*} -\frac{\frac{4 \,{\left (-i \, \tan \left (d x + c\right ) - 1\right )}}{2 i \, a^{3} \tan \left (d x + c\right )^{2} + 4 \, a^{3} \tan \left (d x + c\right ) - 2 i \, a^{3}} - \frac{i \, \log \left (i \, \tan \left (d x + c\right ) + 1\right )}{a^{3}}}{d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 2.26739, size = 157, normalized size = 3.14 \begin{align*} -\frac{{\left (2 \, d x e^{\left (2 i \, d x + 2 i \, c\right )} + i \, e^{\left (2 i \, d x + 2 i \, c\right )} \log \left (e^{\left (2 i \, d x + 2 i \, c\right )} + 1\right ) - i\right )} e^{\left (-2 i \, d x - 2 i \, c\right )}}{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 = 1.20099, size = 138, normalized size = 2.76 \begin{align*} -\frac{-\frac{2 i \, \log \left (\tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right ) - i\right )}{a^{3}} + \frac{i \, \log \left ({\left | \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right ) + 1 \right |}\right )}{a^{3}} + \frac{i \, \log \left ({\left | \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right ) - 1 \right |}\right )}{a^{3}} + \frac{3 i \, \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{2} + 10 \, \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right ) - 3 i}{a^{3}{\left (\tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right ) - i\right )}^{2}}}{d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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