Optimal. Leaf size=62 \[ \frac {i a^2 \tan (c+d x)}{d}-\frac {2 a^2 \log (\cos (c+d x))}{d}-2 i a^2 x+\frac {(a+i a \tan (c+d x))^2}{2 d} \]
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Rubi [A] time = 0.04, antiderivative size = 62, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, integrand size = 22, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.136, Rules used = {3527, 3477, 3475} \[ \frac {i a^2 \tan (c+d x)}{d}-\frac {2 a^2 \log (\cos (c+d x))}{d}-2 i a^2 x+\frac {(a+i a \tan (c+d x))^2}{2 d} \]
Antiderivative was successfully verified.
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Rule 3475
Rule 3477
Rule 3527
Rubi steps
\begin {align*} \int \tan (c+d x) (a+i a \tan (c+d x))^2 \, dx &=\frac {(a+i a \tan (c+d x))^2}{2 d}-i \int (a+i a \tan (c+d x))^2 \, dx\\ &=-2 i a^2 x+\frac {i a^2 \tan (c+d x)}{d}+\frac {(a+i a \tan (c+d x))^2}{2 d}+\left (2 a^2\right ) \int \tan (c+d x) \, dx\\ &=-2 i a^2 x-\frac {2 a^2 \log (\cos (c+d x))}{d}+\frac {i a^2 \tan (c+d x)}{d}+\frac {(a+i a \tan (c+d x))^2}{2 d}\\ \end {align*}
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Mathematica [A] time = 0.15, size = 51, normalized size = 0.82 \[ \frac {a^2 \left (-4 i \tan ^{-1}(\tan (c+d x))-\tan ^2(c+d x)+4 i \tan (c+d x)-4 \log (\cos (c+d x))\right )}{2 d} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.42, size = 93, normalized size = 1.50 \[ -\frac {2 \, {\left (3 \, a^{2} e^{\left (2 i \, d x + 2 i \, c\right )} + 2 \, a^{2} + {\left (a^{2} e^{\left (4 i \, d x + 4 i \, c\right )} + 2 \, a^{2} e^{\left (2 i \, d x + 2 i \, c\right )} + a^{2}\right )} \log \left (e^{\left (2 i \, d x + 2 i \, c\right )} + 1\right )\right )}}{d e^{\left (4 i \, d x + 4 i \, c\right )} + 2 \, d e^{\left (2 i \, d x + 2 i \, c\right )} + d} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 0.61, size = 116, normalized size = 1.87 \[ -\frac {2 \, {\left (a^{2} e^{\left (4 i \, d x + 4 i \, c\right )} \log \left (e^{\left (2 i \, d x + 2 i \, c\right )} + 1\right ) + 2 \, a^{2} e^{\left (2 i \, d x + 2 i \, c\right )} \log \left (e^{\left (2 i \, d x + 2 i \, c\right )} + 1\right ) + 3 \, a^{2} e^{\left (2 i \, d x + 2 i \, c\right )} + a^{2} \log \left (e^{\left (2 i \, d x + 2 i \, c\right )} + 1\right ) + 2 \, a^{2}\right )}}{d e^{\left (4 i \, d x + 4 i \, c\right )} + 2 \, d e^{\left (2 i \, d x + 2 i \, c\right )} + d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.02, size = 67, normalized size = 1.08 \[ \frac {2 i a^{2} \tan \left (d x +c \right )}{d}-\frac {a^{2} \left (\tan ^{2}\left (d x +c \right )\right )}{2 d}+\frac {a^{2} \ln \left (1+\tan ^{2}\left (d x +c \right )\right )}{d}-\frac {2 i a^{2} \arctan \left (\tan \left (d x +c \right )\right )}{d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.79, size = 55, normalized size = 0.89 \[ -\frac {a^{2} \tan \left (d x + c\right )^{2} + 4 i \, {\left (d x + c\right )} a^{2} - 2 \, a^{2} \log \left (\tan \left (d x + c\right )^{2} + 1\right ) - 4 i \, a^{2} \tan \left (d x + c\right )}{2 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 3.74, size = 40, normalized size = 0.65 \[ \frac {a^2\,\left (4\,\ln \left (\mathrm {tan}\left (c+d\,x\right )+1{}\mathrm {i}\right )-{\mathrm {tan}\left (c+d\,x\right )}^2+\mathrm {tan}\left (c+d\,x\right )\,4{}\mathrm {i}\right )}{2\,d} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.36, size = 97, normalized size = 1.56 \[ - \frac {2 a^{2} \log {\left (e^{2 i d x} + e^{- 2 i c} \right )}}{d} + \frac {6 i a^{2} e^{2 i c} e^{2 i d x} + 4 i a^{2}}{- i d e^{4 i c} e^{4 i d x} - 2 i d e^{2 i c} e^{2 i d x} - i d} \]
Verification of antiderivative is not currently implemented for this CAS.
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