Optimal. Leaf size=38 \[ -\frac {\tan (c+d x)}{a^2 d}+\frac {2 i \log (\cos (c+d x))}{a^2 d}+\frac {2 x}{a^2} \]
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Rubi [A] time = 0.05, antiderivative size = 38, normalized size of antiderivative = 1.00, 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} \[ -\frac {\tan (c+d x)}{a^2 d}+\frac {2 i \log (\cos (c+d x))}{a^2 d}+\frac {2 x}{a^2} \]
Antiderivative was successfully verified.
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Rule 43
Rule 3487
Rubi steps
\begin {align*} \int \frac {\sec ^4(c+d x)}{(a+i a \tan (c+d x))^2} \, dx &=-\frac {i \operatorname {Subst}\left (\int \frac {a-x}{a+x} \, dx,x,i a \tan (c+d x)\right )}{a^3 d}\\ &=-\frac {i \operatorname {Subst}\left (\int \left (-1+\frac {2 a}{a+x}\right ) \, dx,x,i a \tan (c+d x)\right )}{a^3 d}\\ &=\frac {2 x}{a^2}+\frac {2 i \log (\cos (c+d x))}{a^2 d}-\frac {\tan (c+d x)}{a^2 d}\\ \end {align*}
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Mathematica [A] time = 0.44, size = 71, normalized size = 1.87 \[ \frac {4 \tan ^{-1}(\tan (d x))+i \sec (c) \sec (c+d x) \left (\cos (d x) \log \left (\cos ^2(c+d x)\right )+\cos (2 c+d x) \log \left (\cos ^2(c+d x)\right )+2 i \sin (d x)\right )}{2 a^2 d} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.50, size = 68, normalized size = 1.79 \[ \frac {4 \, d x e^{\left (2 i \, d x + 2 i \, c\right )} + 4 \, d x + {\left (2 i \, e^{\left (2 i \, d x + 2 i \, c\right )} + 2 i\right )} \log \left (e^{\left (2 i \, d x + 2 i \, c\right )} + 1\right ) - 2 i}{a^{2} d e^{\left (2 i \, d x + 2 i \, c\right )} + a^{2} d} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 0.98, size = 100, normalized size = 2.63 \[ \frac {2 \, {\left (\frac {i \, \log \left (\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 1\right )}{a^{2}} - \frac {2 i \, \log \left (\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - i\right )}{a^{2}} + \frac {i \, \log \left (\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 1\right )}{a^{2}} + \frac {-i \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + i}{{\left (\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - 1\right )} a^{2}}\right )}}{d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.36, size = 35, normalized size = 0.92 \[ -\frac {2 i \ln \left (\tan \left (d x +c \right )-i\right )}{a^{2} d}-\frac {\tan \left (d x +c \right )}{a^{2} d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.32, size = 32, normalized size = 0.84 \[ \frac {-\frac {2 i \, \log \left (i \, \tan \left (d x + c\right ) + 1\right )}{a^{2}} - \frac {\tan \left (d x + c\right )}{a^{2}}}{d} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 3.35, size = 28, normalized size = 0.74 \[ -\frac {\mathrm {tan}\left (c+d\,x\right )+\ln \left (\mathrm {tan}\left (c+d\,x\right )-\mathrm {i}\right )\,2{}\mathrm {i}}{a^2\,d} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ - \frac {\int \frac {\sec ^{4}{\left (c + d x \right )}}{\tan ^{2}{\left (c + d x \right )} - 2 i \tan {\left (c + d x \right )} - 1}\, dx}{a^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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