Optimal. Leaf size=44 \[ \frac{(B-C) \tan (c+d x)}{a d (\sec (c+d x)+1)}+\frac{C \tanh ^{-1}(\sin (c+d x))}{a d} \]
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Rubi [A] time = 0.0728113, antiderivative size = 44, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 32, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.125, Rules used = {4050, 3770, 12, 3794} \[ \frac{(B-C) \tan (c+d x)}{a d (\sec (c+d x)+1)}+\frac{C \tanh ^{-1}(\sin (c+d x))}{a d} \]
Antiderivative was successfully verified.
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Rule 4050
Rule 3770
Rule 12
Rule 3794
Rubi steps
\begin{align*} \int \frac{B \sec (c+d x)+C \sec ^2(c+d x)}{a+a \sec (c+d x)} \, dx &=\frac{\int \frac{(a B-a C) \sec (c+d x)}{a+a \sec (c+d x)} \, dx}{a}+\frac{C \int \sec (c+d x) \, dx}{a}\\ &=\frac{C \tanh ^{-1}(\sin (c+d x))}{a d}+(B-C) \int \frac{\sec (c+d x)}{a+a \sec (c+d x)} \, dx\\ &=\frac{C \tanh ^{-1}(\sin (c+d x))}{a d}+\frac{(B-C) \tan (c+d x)}{d (a+a \sec (c+d x))}\\ \end{align*}
Mathematica [B] time = 0.184871, size = 106, normalized size = 2.41 \[ \frac{2 \cos \left (\frac{1}{2} (c+d x)\right ) \left ((B-C) \sin \left (\frac{1}{2} (c+d x)\right )+C \cos \left (\frac{1}{2} (c+d x)\right ) \left (\log \left (\sin \left (\frac{1}{2} (c+d x)\right )+\cos \left (\frac{1}{2} (c+d x)\right )\right )-\log \left (\cos \left (\frac{1}{2} (c+d x)\right )-\sin \left (\frac{1}{2} (c+d x)\right )\right )\right )\right )}{a d (\cos (c+d x)+1)} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.044, size = 78, normalized size = 1.8 \begin{align*}{\frac{B}{ad}\tan \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) }+{\frac{C}{ad}\ln \left ( \tan \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) +1 \right ) }-{\frac{C}{ad}\ln \left ( \tan \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) -1 \right ) }-{\frac{C}{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 = 0.932191, size = 134, normalized size = 3.05 \begin{align*} \frac{C{\left (\frac{\log \left (\frac{\sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} + 1\right )}{a} - \frac{\log \left (\frac{\sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} - 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 = 0.495314, size = 197, normalized size = 4.48 \begin{align*} \frac{{\left (C \cos \left (d x + c\right ) + C\right )} \log \left (\sin \left (d x + c\right ) + 1\right ) -{\left (C \cos \left (d x + c\right ) + C\right )} \log \left (-\sin \left (d x + c\right ) + 1\right ) + 2 \,{\left (B - C\right )} \sin \left (d x + c\right )}{2 \,{\left (a d \cos \left (d x + c\right ) + a d\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*} \frac{\int \frac{B \sec{\left (c + d x \right )}}{\sec{\left (c + d x \right )} + 1}\, dx + \int \frac{C \sec ^{2}{\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 = 1.14828, size = 95, normalized size = 2.16 \begin{align*} \frac{\frac{C \log \left ({\left | \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right ) + 1 \right |}\right )}{a} - \frac{C \log \left ({\left | \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right ) - 1 \right |}\right )}{a} + \frac{B \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right ) - C \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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