Optimal. Leaf size=12 \[ \frac{1}{4} \tan (4 x)-x \]
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Rubi [A] time = 0.0053044, antiderivative size = 12, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 6, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.333, Rules used = {3473, 8} \[ \frac{1}{4} \tan (4 x)-x \]
Antiderivative was successfully verified.
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Rule 3473
Rule 8
Rubi steps
\begin{align*} \int \tan ^2(4 x) \, dx &=\frac{1}{4} \tan (4 x)-\int 1 \, dx\\ &=-x+\frac{1}{4} \tan (4 x)\\ \end{align*}
Mathematica [A] time = 0.0081634, size = 18, normalized size = 1.5 \[ \frac{1}{4} \tan (4 x)-\frac{1}{4} \tan ^{-1}(\tan (4 x)) \]
Antiderivative was successfully verified.
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Maple [A] time = 0.003, size = 11, normalized size = 0.9 \begin{align*} -x+{\frac{\tan \left ( 4\,x \right ) }{4}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.46556, size = 14, normalized size = 1.17 \begin{align*} -x + \frac{1}{4} \, \tan \left (4 \, x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 2.01629, size = 26, normalized size = 2.17 \begin{align*} -x + \frac{1}{4} \, \tan \left (4 \, x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0.06253, size = 12, normalized size = 1. \begin{align*} - x + \frac{\sin{\left (4 x \right )}}{4 \cos{\left (4 x \right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.06407, size = 14, normalized size = 1.17 \begin{align*} -x + \frac{1}{4} \, \tan \left (4 \, x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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