Optimal. Leaf size=35 \[ \frac{A x}{a}-\frac{(A-B) \tan (c+d x)}{d (a \sec (c+d x)+a)} \]
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Rubi [A] time = 0.0591123, antiderivative size = 35, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 23, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.087, Rules used = {3919, 3794} \[ \frac{A x}{a}-\frac{(A-B) \tan (c+d x)}{d (a \sec (c+d x)+a)} \]
Antiderivative was successfully verified.
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Rule 3919
Rule 3794
Rubi steps
\begin{align*} \int \frac{A+B \sec (c+d x)}{a+a \sec (c+d x)} \, dx &=\frac{A x}{a}-(A-B) \int \frac{\sec (c+d x)}{a+a \sec (c+d x)} \, dx\\ &=\frac{A x}{a}-\frac{(A-B) \tan (c+d x)}{d (a+a \sec (c+d x))}\\ \end{align*}
Mathematica [B] time = 0.13754, size = 72, normalized size = 2.06 \[ \frac{\sec \left (\frac{c}{2}\right ) \cos \left (\frac{1}{2} (c+d x)\right ) \left (2 (B-A) \sin \left (\frac{d x}{2}\right )+A d x \cos \left (c+\frac{d x}{2}\right )+A d x \cos \left (\frac{d x}{2}\right )\right )}{a d (\cos (c+d x)+1)} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.05, size = 56, normalized size = 1.6 \begin{align*} 2\,{\frac{A\arctan \left ( \tan \left ( 1/2\,dx+c/2 \right ) \right ) }{ad}}-{\frac{A}{ad}\tan \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) }+{\frac{B}{ad}\tan \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) } \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [B] time = 1.46919, size = 99, normalized size = 2.83 \begin{align*} \frac{A{\left (\frac{2 \, \arctan \left (\frac{\sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1}\right )}{a} - \frac{\sin \left (d x + c\right )}{a{\left (\cos \left (d x + c\right ) + 1\right )}}\right )} + \frac{B \sin \left (d x + c\right )}{a{\left (\cos \left (d x + c\right ) + 1\right )}}}{d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 0.450558, size = 105, normalized size = 3. \begin{align*} \frac{A d x \cos \left (d x + c\right ) + A d x -{\left (A - B\right )} \sin \left (d x + c\right )}{a d \cos \left (d x + c\right ) + a 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*} \frac{\int \frac{A}{\sec{\left (c + d x \right )} + 1}\, dx + \int \frac{B \sec{\left (c + d x \right )}}{\sec{\left (c + d x \right )} + 1}\, dx}{a} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.18523, size = 59, normalized size = 1.69 \begin{align*} \frac{\frac{{\left (d x + c\right )} A}{a} - \frac{A \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right ) - B \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )}{a}}{d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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