Optimal. Leaf size=35 \[ -\frac{\log (a+b \sec (c+d x))}{a d}-\frac{\log (\cos (c+d x))}{a d} \]
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Rubi [A] time = 0.0315418, antiderivative size = 35, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.21, Rules used = {3885, 36, 29, 31} \[ -\frac{\log (a+b \sec (c+d x))}{a d}-\frac{\log (\cos (c+d x))}{a d} \]
Antiderivative was successfully verified.
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Rule 3885
Rule 36
Rule 29
Rule 31
Rubi steps
\begin{align*} \int \frac{\tan (c+d x)}{a+b \sec (c+d x)} \, dx &=\frac{\operatorname{Subst}\left (\int \frac{1}{x (a+x)} \, dx,x,b \sec (c+d x)\right )}{d}\\ &=\frac{\operatorname{Subst}\left (\int \frac{1}{x} \, dx,x,b \sec (c+d x)\right )}{a d}-\frac{\operatorname{Subst}\left (\int \frac{1}{a+x} \, dx,x,b \sec (c+d x)\right )}{a d}\\ &=-\frac{\log (\cos (c+d x))}{a d}-\frac{\log (a+b \sec (c+d x))}{a d}\\ \end{align*}
Mathematica [A] time = 0.0368242, size = 19, normalized size = 0.54 \[ -\frac{\log (a \cos (c+d x)+b)}{a d} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.02, size = 35, normalized size = 1. \begin{align*} -{\frac{\ln \left ( a+b\sec \left ( dx+c \right ) \right ) }{ad}}+{\frac{\ln \left ( \sec \left ( dx+c \right ) \right ) }{ad}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 0.963872, size = 26, normalized size = 0.74 \begin{align*} -\frac{\log \left (a \cos \left (d x + c\right ) + b\right )}{a d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 0.927896, size = 43, normalized size = 1.23 \begin{align*} -\frac{\log \left (a \cos \left (d x + c\right ) + b\right )}{a d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 6.82994, size = 82, normalized size = 2.34 \begin{align*} \begin{cases} \frac{\tilde{\infty } x \tan{\left (c \right )}}{\sec{\left (c \right )}} & \text{for}\: a = 0 \wedge b = 0 \wedge d = 0 \\\frac{x \tan{\left (c \right )}}{a + b \sec{\left (c \right )}} & \text{for}\: d = 0 \\- \frac{1}{b d \sec{\left (c + d x \right )}} & \text{for}\: a = 0 \\\frac{\log{\left (\tan ^{2}{\left (c + d x \right )} + 1 \right )}}{2 a d} & \text{for}\: b = 0 \\- \frac{\log{\left (\frac{a}{b} + \sec{\left (c + d x \right )} \right )}}{a d} + \frac{\log{\left (\tan ^{2}{\left (c + d x \right )} + 1 \right )}}{2 a d} & \text{otherwise} \end{cases} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] time = 1.35047, size = 131, normalized size = 3.74 \begin{align*} -\frac{\frac{{\left (a - b\right )} \log \left ({\left | a + b + \frac{a{\left (\cos \left (d x + c\right ) - 1\right )}}{\cos \left (d x + c\right ) + 1} - \frac{b{\left (\cos \left (d x + c\right ) - 1\right )}}{\cos \left (d x + c\right ) + 1} \right |}\right )}{a^{2} - a b} - \frac{\log \left ({\left | -\frac{\cos \left (d x + c\right ) - 1}{\cos \left (d x + c\right ) + 1} + 1 \right |}\right )}{a}}{d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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