Optimal. Leaf size=18 \[ \frac{\log (a+b \tan (c+d x))}{b d} \]
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Rubi [A] time = 0.0417421, antiderivative size = 18, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.095, Rules used = {3506, 31} \[ \frac{\log (a+b \tan (c+d x))}{b d} \]
Antiderivative was successfully verified.
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Rule 3506
Rule 31
Rubi steps
\begin{align*} \int \frac{\sec ^2(c+d x)}{a+b \tan (c+d x)} \, dx &=\frac{\operatorname{Subst}\left (\int \frac{1}{a+x} \, dx,x,b \tan (c+d x)\right )}{b d}\\ &=\frac{\log (a+b \tan (c+d x))}{b d}\\ \end{align*}
Mathematica [A] time = 0.0137411, size = 18, normalized size = 1. \[ \frac{\log (a+b \tan (c+d x))}{b d} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.029, size = 19, normalized size = 1.1 \begin{align*}{\frac{\ln \left ( a+b\tan \left ( dx+c \right ) \right ) }{bd}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.13644, size = 24, normalized size = 1.33 \begin{align*} \frac{\log \left (b \tan \left (d x + c\right ) + a\right )}{b d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] time = 1.95648, size = 144, normalized size = 8. \begin{align*} \frac{\log \left (2 \, a b \cos \left (d x + c\right ) \sin \left (d x + c\right ) +{\left (a^{2} - b^{2}\right )} \cos \left (d x + c\right )^{2} + b^{2}\right ) - \log \left (\cos \left (d x + c\right )^{2}\right )}{2 \, b d} \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 \frac{\sec ^{2}{\left (c + d x \right )}}{a + b \tan{\left (c + d x \right )}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.82938, size = 26, normalized size = 1.44 \begin{align*} \frac{\log \left ({\left | b \tan \left (d x + c\right ) + a \right |}\right )}{b d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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