Optimal. Leaf size=14 \[ \frac{\tan (a+b x)}{b}-x \]
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Rubi [A] time = 0.0081899, 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} \[ \frac{\tan (a+b x)}{b}-x \]
Antiderivative was successfully verified.
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Rule 3473
Rule 8
Rubi steps
\begin{align*} \int \tan ^2(a+b x) \, dx &=\frac{\tan (a+b x)}{b}-\int 1 \, dx\\ &=-x+\frac{\tan (a+b x)}{b}\\ \end{align*}
Mathematica [A] time = 0.0071568, size = 23, normalized size = 1.64 \[ \frac{\tan (a+b x)}{b}-\frac{\tan ^{-1}(\tan (a+b x))}{b} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.016, size = 19, normalized size = 1.4 \begin{align*}{\frac{\tan \left ( bx+a \right ) -bx-a}{b}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.48968, size = 24, normalized size = 1.71 \begin{align*} -\frac{b x + a - \tan \left (b x + a\right )}{b} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] time = 1.82419, size = 72, normalized size = 5.14 \begin{align*} -\frac{b x \cos \left (b x + a\right ) - \sin \left (b x + a\right )}{b \cos \left (b x + a\right )} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \sin ^{2}{\left (a + b x \right )} \sec ^{2}{\left (a + b x \right )}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.1317, size = 24, normalized size = 1.71 \begin{align*} -\frac{b x + a - \tan \left (b x + a\right )}{b} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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