Optimal. Leaf size=20 \[ -\frac{1}{b d (a+b \tan (c+d x))} \]
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Rubi [A] time = 0.0391909, antiderivative size = 20, 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, 32} \[ -\frac{1}{b d (a+b \tan (c+d x))} \]
Antiderivative was successfully verified.
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Rule 3506
Rule 32
Rubi steps
\begin{align*} \int \frac{\sec ^2(c+d x)}{(a+b \tan (c+d x))^2} \, dx &=\frac{\operatorname{Subst}\left (\int \frac{1}{(a+x)^2} \, dx,x,b \tan (c+d x)\right )}{b d}\\ &=-\frac{1}{b d (a+b \tan (c+d x))}\\ \end{align*}
Mathematica [A] time = 0.0443234, size = 32, normalized size = 1.6 \[ \frac{\sin (c+d x)}{a d (a \cos (c+d x)+b \sin (c+d x))} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.043, size = 21, normalized size = 1.1 \begin{align*} -{\frac{1}{bd \left ( a+b\tan \left ( dx+c \right ) \right ) }} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.12928, size = 27, normalized size = 1.35 \begin{align*} -\frac{1}{{\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 = 1.76802, size = 132, normalized size = 6.6 \begin{align*} -\frac{b \cos \left (d x + c\right ) - a \sin \left (d x + c\right )}{{\left (a^{3} + a b^{2}\right )} d \cos \left (d x + c\right ) +{\left (a^{2} b + b^{3}\right )} d \sin \left (d x + c\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 \frac{\sec ^{2}{\left (c + d x \right )}}{\left (a + b \tan{\left (c + d x \right )}\right )^{2}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.33803, size = 27, normalized size = 1.35 \begin{align*} -\frac{1}{{\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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