Optimal. Leaf size=14 \[ \frac{\tan (c+d x)}{d}-x \]
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Rubi [A] time = 0.0076875, antiderivative size = 14, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 8, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.25, Rules used = {3473, 8} \[ \frac{\tan (c+d x)}{d}-x \]
Antiderivative was successfully verified.
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Rule 3473
Rule 8
Rubi steps
\begin{align*} \int \tan ^2(c+d x) \, dx &=\frac{\tan (c+d x)}{d}-\int 1 \, dx\\ &=-x+\frac{\tan (c+d x)}{d}\\ \end{align*}
Mathematica [A] time = 0.0063154, size = 23, normalized size = 1.64 \[ \frac{\tan (c+d x)}{d}-\frac{\tan ^{-1}(\tan (c+d x))}{d} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.003, size = 24, normalized size = 1.7 \begin{align*}{\frac{\tan \left ( dx+c \right ) }{d}}-{\frac{\arctan \left ( \tan \left ( dx+c \right ) \right ) }{d}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.38102, size = 24, normalized size = 1.71 \begin{align*} -\frac{d x + c - \tan \left (d x + c\right )}{d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.89881, size = 34, normalized size = 2.43 \begin{align*} -\frac{d x - \tan \left (d x + c\right )}{d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0.169093, size = 15, normalized size = 1.07 \begin{align*} \begin{cases} - x + \frac{\tan{\left (c + d x \right )}}{d} & \text{for}\: d \neq 0 \\x \tan ^{2}{\left (c \right )} & \text{otherwise} \end{cases} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] time = 1.59345, size = 305, normalized size = 21.79 \begin{align*} \frac{\pi - 4 \, d x \tan \left (d x\right ) \tan \left (c\right ) - \pi \mathrm{sgn}\left (2 \, \tan \left (d x\right )^{2} \tan \left (c\right ) + 2 \, \tan \left (d x\right ) \tan \left (c\right )^{2} - 2 \, \tan \left (d x\right ) - 2 \, \tan \left (c\right )\right ) \tan \left (d x\right ) \tan \left (c\right ) - \pi \tan \left (d x\right ) \tan \left (c\right ) + 2 \, \arctan \left (\frac{\tan \left (d x\right ) \tan \left (c\right ) - 1}{\tan \left (d x\right ) + \tan \left (c\right )}\right ) \tan \left (d x\right ) \tan \left (c\right ) + 2 \, \arctan \left (\frac{\tan \left (d x\right ) + \tan \left (c\right )}{\tan \left (d x\right ) \tan \left (c\right ) - 1}\right ) \tan \left (d x\right ) \tan \left (c\right ) + 4 \, d x + \pi \mathrm{sgn}\left (2 \, \tan \left (d x\right )^{2} \tan \left (c\right ) + 2 \, \tan \left (d x\right ) \tan \left (c\right )^{2} - 2 \, \tan \left (d x\right ) - 2 \, \tan \left (c\right )\right ) - 2 \, \arctan \left (\frac{\tan \left (d x\right ) \tan \left (c\right ) - 1}{\tan \left (d x\right ) + \tan \left (c\right )}\right ) - 2 \, \arctan \left (\frac{\tan \left (d x\right ) + \tan \left (c\right )}{\tan \left (d x\right ) \tan \left (c\right ) - 1}\right ) - 4 \, \tan \left (d x\right ) - 4 \, \tan \left (c\right )}{4 \,{\left (d \tan \left (d x\right ) \tan \left (c\right ) - d\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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