Optimal. Leaf size=38 \[ -\frac {a^2 \tan (c+d x)}{d}-\frac {2 i a^2 \log (\cos (c+d x))}{d}+2 a^2 x \]
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Rubi [A] time = 0.02, antiderivative size = 38, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.133, Rules used = {3477, 3475} \[ -\frac {a^2 \tan (c+d x)}{d}-\frac {2 i a^2 \log (\cos (c+d x))}{d}+2 a^2 x \]
Antiderivative was successfully verified.
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Rule 3475
Rule 3477
Rubi steps
\begin {align*} \int (a+i a \tan (c+d x))^2 \, dx &=2 a^2 x-\frac {a^2 \tan (c+d x)}{d}+\left (2 i a^2\right ) \int \tan (c+d x) \, dx\\ &=2 a^2 x-\frac {2 i a^2 \log (\cos (c+d x))}{d}-\frac {a^2 \tan (c+d x)}{d}\\ \end {align*}
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Mathematica [B] time = 0.85, size = 100, normalized size = 2.63 \[ -\frac {a^2 \sec (c) \sec (c+d x) \left (-4 d x \cos (2 c+d x)+\cos (d x) \left (-4 d x+i \log \left (\cos ^2(c+d x)\right )\right )+i \cos (2 c+d x) \log \left (\cos ^2(c+d x)\right )+4 \cos (c) \cos (c+d x) \tan ^{-1}(\tan (3 c+d x))+2 \sin (d x)\right )}{2 d} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.44, size = 55, normalized size = 1.45 \[ \frac {-2 i \, a^{2} + {\left (-2 i \, a^{2} e^{\left (2 i \, d x + 2 i \, c\right )} - 2 i \, a^{2}\right )} \log \left (e^{\left (2 i \, d x + 2 i \, c\right )} + 1\right )}{d e^{\left (2 i \, d x + 2 i \, c\right )} + d} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.44, size = 65, normalized size = 1.71 \[ \frac {-2 i \, a^{2} e^{\left (2 i \, d x + 2 i \, c\right )} \log \left (e^{\left (2 i \, d x + 2 i \, c\right )} + 1\right ) - 2 i \, a^{2} \log \left (e^{\left (2 i \, d x + 2 i \, c\right )} + 1\right ) - 2 i \, a^{2}}{d e^{\left (2 i \, d x + 2 i \, c\right )} + d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.02, size = 51, normalized size = 1.34 \[ \frac {i a^{2} \ln \left (1+\tan ^{2}\left (d x +c \right )\right )}{d}+\frac {2 a^{2} \arctan \left (\tan \left (d x +c \right )\right )}{d}-\frac {a^{2} \tan \left (d x +c \right )}{d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.94, size = 41, normalized size = 1.08 \[ a^{2} x + \frac {{\left (d x + c - \tan \left (d x + c\right )\right )} a^{2}}{d} + \frac {2 i \, a^{2} \log \left (\sec \left (d x + c\right )\right )}{d} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 3.68, size = 29, normalized size = 0.76 \[ \frac {a^2\,\left (-\mathrm {tan}\left (c+d\,x\right )+\ln \left (\mathrm {tan}\left (c+d\,x\right )+1{}\mathrm {i}\right )\,2{}\mathrm {i}\right )}{d} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.25, size = 53, normalized size = 1.39 \[ \frac {2 i a^{2}}{- d e^{2 i c} e^{2 i d x} - d} - \frac {2 i a^{2} \log {\left (e^{2 i d x} + e^{- 2 i c} \right )}}{d} \]
Verification of antiderivative is not currently implemented for this CAS.
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